Subgroups — C₂×C₇×#3460 #13239895

Order 1792.

7994 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3460	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₇×#544	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#542	C₂	1	4	yes	maximal, complemented
4	C₂×C₇×#540	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×#538	C₂	1	4	yes	maximal, complemented
6	C₂×C₇×#537	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#535	C₂	1	2	yes	maximal, complemented
8	C₂×C₇×#532	C₂	1	4	yes	maximal, complemented
9	C₂×C₇×#529	C₂	1	4	yes	maximal, complemented
10	C₂×C₇×#525	C₂	1	4	yes	maximal, complemented
11	C₂×C₄×C₇×#41	C₂	1	2	yes	maximal, complemented
12	C₂²×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×C₇×#41	C₂²	1	2	yes	—
14	C₂²×C₇×#41	C₂²	1	4	yes	complemented
15	C₂³×C₇×D₄	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₇	C₂²	1	2	yes	—
17	C₇×#544	C₂²	1	16	yes	complemented
18	C₇×#542	C₂²	1	64	yes	complemented
19	C₇×#540	C₂²	1	64	yes	complemented
20	C₇×#538	C₂²	1	64	yes	complemented
21	C₇×#537	C₂²	1	16	yes	complemented
22	C₇×#535	C₂²	1	32	yes	complemented
23	C₇×#532	C₂²	1	64	yes	complemented
24	C₂×C₇×C₄⋊Q₈	C₂²	1	4	yes	complemented
25	C₂×C₇×#127	C₂²	1	4	yes	—
26	C₇×#529	C₂²	1	64	yes	complemented
27	C₂×C₇×#126	C₂²	1	8	yes	complemented
28	C₂×C₇×#125	C₂²	1	4	yes	complemented
29	C₂×C₇×#124	C₂²	1	4	yes	—
30	C₂×C₇×#124	C₂²	1	8	yes	complemented
31	C₇×#525	C₂²	1	64	yes	complemented
32	C₂×C₇×#123	C₂²	1	8	yes	complemented
33	C₂×C₇×#123	C₂²	1	8	yes	complemented
34	C₂×C₇×C₂²⋊Q₈	C₂²	1	4	yes	—
35	C₂×C₇×C₂²⋊Q₈	C₂²	1	4	yes	complemented
36	C₂×C₇×C₂²⋊Q₈	C₂²	1	4	yes	complemented
37	C₂×C₇×#121	C₂²	1	4	yes	—
38	C₂×C₇×#121	C₂²	1	4	yes	—
39	C₂×C₇×#121	C₂²	1	4	yes	—
40	C₂×C₇×#121	C₂²	1	4	yes	complemented
41	C₂×C₇×#121	C₂²	1	4	yes	complemented
42	C₂×C₇×#121	C₂²	1	8	yes	complemented
43	C₂×C₇×#120	C₂²	1	8	yes	—
44	C₄×C₇×#41	C₂²	1	32	yes	complemented
45	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	complemented
46	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	complemented
47	C₂×C₄×C₇×D₄	C₂²	1	4	yes	—
48	C₂×C₄×C₇×D₄	C₂²	1	4	yes	—
49	C₂×C₄×C₇×D₄	C₂²	1	4	yes	—
50	C₂×C₄×C₇×D₄	C₂²	1	4	yes	—
51	C₂×C₄×C₇×D₄	C₂²	1	16	yes	complemented
52	C₂×C₄×C₇×D₄	C₂²	1	2	yes	complemented
53	C₂×C₄×C₇×D₄	C₂²	1	2	yes	complemented
54	C₂×C₄×C₇×D₄	C₂²	1	8	yes	complemented
55	C₂×C₇×#117	C₂²	1	2	yes	—
56	C₂×C₇×#117	C₂²	1	4	yes	complemented
57	C₂×C₇×#117	C₂²	1	4	yes	complemented
58	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
59	C₂²×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
60	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
61	C₂²×C₄²×C₇	C₂²	1	4	yes	complemented
62	C₂×#3460	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂⁵×C₇	C₂³	1	2	yes	—
64	C₂×C₇×#41	—	2	16	no	—
65	C₂×C₇×#41	—	4	16	no	—
66	C₂×C₇×#41	C₂³	1	16	yes	—
67	C₂×C₇×#41	C₂³	1	32	yes	complemented
68	C₂²×C₇×Q₈	C₂³	1	2	yes	—
69	C₂²×C₇×Q₈	C₂³	1	4	yes	—
70	C₂²×C₇×D₄	—	2	8	no	—
71	C₂²×C₇×D₄	C₂³	1	4	yes	—
72	C₂²×C₇×D₄	C₂³	1	8	yes	—
73	C₂²×C₇×D₄	C₂³	1	4	yes	—
74	C₂²×C₇×D₄	C₂³	1	2	yes	—
75	C₂²×C₇×D₄	C₂³	1	8	yes	—
76	C₂²×C₇×D₄	C₂³	1	4	yes	—
77	C₂²×C₇×D₄	C₂³	1	8	yes	—
78	C₂³×C₄×C₇	—	4	8	no	—
79	C₂³×C₄×C₇	C₂³	1	4	yes	—
80	C₂³×C₄×C₇	C₂³	1	4	yes	—
81	C₂³×C₄×C₇	C₂³	1	4	yes	—
82	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₇	C₂³	1	4	yes	—
85	C₂³×C₄×C₇	C₂³	1	2	yes	—
86	C₂³×C₄×C₇	C₂³	1	4	yes	—
87	C₂³×C₄×C₇	C₂³	1	16	yes	—
88	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
89	C₇×C₄⋊Q₈	C₂³	1	32	yes	complemented
90	C₇×#127	C₂³	1	32	yes	—
91	C₇×#126	C₂³	1	64	yes	complemented
92	C₇×#125	C₂³	1	32	yes	complemented
93	C₇×#124	C₂³	1	32	yes	—
94	C₇×#124	C₂³	1	64	yes	complemented
95	C₇×#123	C₂³	1	64	yes	complemented
96	C₇×#123	C₂³	1	64	yes	complemented
97	C₇×C₂²⋊Q₈	C₂³	1	32	yes	—
98	C₇×C₂²⋊Q₈	C₂³	1	32	yes	complemented
99	C₇×C₂²⋊Q₈	C₂³	1	32	yes	complemented
100	C₇×#121	C₂³	1	32	yes	—
101	C₇×#121	C₂³	1	32	yes	—
102	C₇×#121	C₂³	1	32	yes	—
103	C₇×#121	C₂³	1	32	yes	complemented
104	C₇×#121	C₂³	1	32	yes	complemented
105	C₇×#121	C₂³	1	64	yes	complemented
106	C₇×#120	C₂³	1	64	yes	—
107	C₄×C₇×Q₈	C₂³	1	16	yes	complemented
108	C₄×C₇×Q₈	C₂³	1	16	yes	complemented
109	C₄×C₇×D₄	C₂³	1	32	yes	—
110	C₄×C₇×D₄	C₂³	1	32	yes	—
111	C₄×C₇×D₄	C₂³	1	32	yes	—
112	C₄×C₇×D₄	C₂³	1	32	yes	—
113	C₄×C₇×D₄	C₂³	1	16	yes	complemented
114	C₄×C₇×D₄	C₂³	1	16	yes	complemented
115	C₄×C₇×D₄	C₂³	1	64	yes	complemented
116	C₇×#117	C₂³	1	16	yes	—
117	C₇×#117	C₂³	1	32	yes	complemented
118	C₇×#117	C₂³	1	32	yes	complemented
119	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
120	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
121	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
122	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
123	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
124	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
125	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
126	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
127	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
128	C₂×C₇×C₂²⋊C₄	C₂³	1	8	yes	—
129	C₂×C₇×C₂²⋊C₄	C₂³	1	16	yes	—
130	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
131	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
132	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
133	C₂×C₇×C₂²⋊C₄	C₂³	1	8	yes	—
134	C₂×C₇×C₂²⋊C₄	C₂³	1	8	yes	—
135	C₂×C₇×C₂²⋊C₄	C₂³	1	8	yes	—
136	C₂×C₄²×C₇	C₂³	1	2	yes	—
137	C₂×C₄²×C₇	C₂³	1	8	yes	—
138	C₂×C₄²×C₇	C₂³	1	4	yes	—
139	C₂×C₄²×C₇	C₂³	1	4	yes	—
140	C₂×C₄²×C₇	C₂³	1	2	yes	—
141	C₂×C₄²×C₇	C₂³	1	4	yes	—
142	C₂×C₄²×C₇	C₂³	1	32	yes	complemented
143	#3460	C₂×C₇	1	32	yes	direct product factor
144	C₂×#544	C₂×C₇	1	1	yes	characteristic, complemented
145	C₂×#542	C₂×C₇	1	4	yes	complemented
146	C₂×#540	C₂×C₇	1	4	yes	complemented
147	C₂×#538	C₂×C₇	1	4	yes	complemented
148	C₂×#537	C₂×C₇	1	1	yes	characteristic, complemented
149	C₂×#535	C₂×C₇	1	2	yes	complemented
150	C₂×#532	C₂×C₇	1	4	yes	complemented
151	C₂×#529	C₂×C₇	1	4	yes	complemented
152	C₂×#525	C₂×C₇	1	4	yes	complemented
153	C₂×C₄×#41	C₂×C₇	1	2	yes	complemented
154	C₂²×C₄×D₄	C₂×C₇	1	1	yes	characteristic, complemented
155	C₂⁴×C₇	—	2	8	no	—
156	C₂⁴×C₇	—	4	4	no	—
157	C₂⁴×C₇	—	4	4	no	—
158	C₂⁴×C₇	C₂⁴	1	4	yes	—
159	C₂⁴×C₇	C₂⁴	1	4	yes	—
160	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
161	C₂⁴×C₇	C₂⁴	1	8	yes	—
162	C₇×#41	—	2	64	no	—
163	C₇×#41	—	4	64	no	—
164	C₂×C₇×Q₈	—	2	8	no	—
165	C₂×C₇×Q₈	—	4	8	no	—
166	C₂×C₇×Q₈	C₂⁴	1	8	yes	—
167	C₂×C₇×Q₈	C₂⁴	1	16	yes	—
168	C₂×C₇×D₄	—	2	16	no	—
169	C₂×C₇×D₄	—	2	16	no	—
170	C₂×C₇×D₄	—	2	8	no	—
171	C₂×C₇×D₄	—	2	32	no	—
172	C₂×C₇×D₄	—	4	16	no	—
173	C₂×C₇×D₄	—	4	8	no	—
174	C₂×C₇×D₄	—	4	16	no	—
175	C₂×C₇×D₄	C₂⁴	1	16	yes	—
176	C₂×C₇×D₄	C₂⁴	1	16	yes	—
177	C₂×C₇×D₄	C₂⁴	1	8	yes	—
178	C₂×C₇×D₄	C₂⁴	1	32	yes	—
179	C₂×C₇×D₄	C₂⁴	1	32	yes	—
180	C₂×C₇×D₄	C₂⁴	1	16	yes	—
181	C₂²×C₄×C₇	—	2	16	no	—
182	C₂²×C₄×C₇	—	2	4	no	—
183	C₂²×C₄×C₇	—	2	16	no	—
184	C₂²×C₄×C₇	—	2	16	no	—
185	C₂²×C₄×C₇	—	4	32	no	—
186	C₂²×C₄×C₇	—	4	8	no	—
187	C₂²×C₄×C₇	—	4	4	no	—
188	C₂²×C₄×C₇	—	4	2	no	—
189	C₂²×C₄×C₇	—	4	8	no	—
190	C₂²×C₄×C₇	#41	1	8	yes	complemented
191	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
192	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
193	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
194	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
195	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
196	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
197	C₂²×C₄×C₇	C₂⁴	1	16	yes	—
198	C₂²×C₄×C₇	C₂⁴	1	16	yes	—
199	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
200	C₂²×C₄×C₇	C₂⁴	1	16	yes	—
201	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
202	C₂²×C₄×C₇	C₂⁴	1	16	yes	—
203	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
204	C₂²×C₄×C₇	C₂⁴	1	16	yes	—
205	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
206	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
207	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
208	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
209	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
210	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
211	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
212	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
213	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
214	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
215	C₇×C₄⋊C₄	C₂⁴	1	32	yes	—
216	C₇×C₂²⋊C₄	C₂⁴	1	32	yes	—
217	C₇×C₂²⋊C₄	C₂⁴	1	16	yes	—
218	C₇×C₂²⋊C₄	C₂⁴	1	16	yes	—
219	C₇×C₂²⋊C₄	C₂⁴	1	16	yes	—
220	C₇×C₂²⋊C₄	C₂⁴	1	32	yes	—
221	C₇×C₂²⋊C₄	C₂⁴	1	32	yes	—
222	C₇×C₂²⋊C₄	C₂⁴	1	32	yes	—
223	C₄²×C₇	C₂⁴	1	8	yes	—
224	C₄²×C₇	C₂⁴	1	8	yes	—
225	C₄²×C₇	C₂⁴	1	16	yes	—
226	C₄²×C₇	C₂⁴	1	16	yes	—
227	C₄²×C₇	C₂⁴	1	16	yes	—
228	C₂²×#41	C₂²×C₇	1	2	yes	—
229	C₂²×#41	C₂²×C₇	1	4	yes	complemented
230	C₂³×D₄	C₂²×C₇	1	1	yes	characteristic
231	C₂⁴×C₄	C₂²×C₇	1	2	yes	—
232	#544	C₂²×C₇	1	16	yes	complemented
233	#542	C₂²×C₇	1	64	yes	complemented
234	#540	C₂²×C₇	1	64	yes	complemented
235	#538	C₂²×C₇	1	64	yes	complemented
236	#537	C₂²×C₇	1	16	yes	complemented
237	#535	C₂²×C₇	1	32	yes	complemented
238	#532	C₂²×C₇	1	64	yes	complemented
239	C₂×C₄⋊Q₈	C₂²×C₇	1	4	yes	complemented
240	C₂×#127	C₂²×C₇	1	4	yes	—
241	#529	C₂²×C₇	1	64	yes	complemented
242	C₂×#126	C₂²×C₇	1	8	yes	complemented
243	C₂×#125	C₂²×C₇	1	4	yes	complemented
244	C₂×#124	C₂²×C₇	1	4	yes	—
245	C₂×#124	C₂²×C₇	1	8	yes	complemented
246	#525	C₂²×C₇	1	64	yes	complemented
247	C₂×#123	C₂²×C₇	1	8	yes	complemented
248	C₂×#123	C₂²×C₇	1	8	yes	complemented
249	C₂×C₂²⋊Q₈	C₂²×C₇	1	4	yes	—
250	C₂×C₂²⋊Q₈	C₂²×C₇	1	4	yes	complemented
251	C₂×C₂²⋊Q₈	C₂²×C₇	1	4	yes	complemented
252	C₂×#121	C₂²×C₇	1	4	yes	—
253	C₂×#121	C₂²×C₇	1	4	yes	—
254	C₂×#121	C₂²×C₇	1	4	yes	—
255	C₂×#121	C₂²×C₇	1	4	yes	complemented
256	C₂×#121	C₂²×C₇	1	4	yes	complemented
257	C₂×#121	C₂²×C₇	1	8	yes	complemented
258	C₂×#120	C₂²×C₇	1	8	yes	—
259	C₄×#41	C₂²×C₇	1	32	yes	complemented
260	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	complemented
261	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	complemented
262	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
263	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
264	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
265	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
266	C₂×C₄×D₄	C₂²×C₇	1	16	yes	complemented
267	C₂×C₄×D₄	C₂²×C₇	1	2	yes	complemented
268	C₂×C₄×D₄	C₂²×C₇	1	2	yes	complemented
269	C₂×C₄×D₄	C₂²×C₇	1	8	yes	complemented
270	C₂×#117	C₂²×C₇	1	2	yes	—
271	C₂×#117	C₂²×C₇	1	4	yes	complemented
272	C₂×#117	C₂²×C₇	1	4	yes	complemented
273	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
274	C₂²×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
275	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
276	C₂²×C₄²	C₂²×C₇	1	4	yes	complemented
277	C₂³×C₇	—	2	2	no	—
278	C₂³×C₇	—	2	2	no	—
279	C₂³×C₇	—	2	8	no	—
280	C₂³×C₇	—	2	16	no	—
281	C₂³×C₇	—	2	8	no	—
282	C₂³×C₇	—	2	8	no	—
283	C₂³×C₇	—	2	8	no	—
284	C₂³×C₇	—	2	8	no	—
285	C₂³×C₇	—	2	8	no	—
286	C₂³×C₇	—	4	8	no	—
287	C₂³×C₇	—	4	8	no	—
288	C₂³×C₇	—	4	32	no	—
289	C₂³×C₇	C₂×#41	1	4	yes	—
290	C₂³×C₇	C₂⁵	1	8	yes	—
291	C₂³×C₇	C₂⁵	1	8	yes	—
292	C₂³×C₇	C₂⁵	1	2	yes	—
293	C₂³×C₇	C₂⁵	1	1	yes	characteristic
294	C₇×Q₈	—	2	16	no	—
295	C₇×Q₈	—	4	16	no	—
296	C₇×D₄	—	2	32	no	—
297	C₇×D₄	—	2	32	no	—
298	C₇×D₄	—	2	16	no	—
299	C₇×D₄	—	4	32	no	—
300	C₇×D₄	—	4	16	no	—
301	C₇×D₄	—	4	32	no	—
302	C₂×C₄×C₇	—	2	8	no	—
303	C₂×C₄×C₇	—	2	2	no	—
304	C₂×C₄×C₇	—	2	8	no	—
305	C₂×C₄×C₇	—	2	8	no	—
306	C₂×C₄×C₇	—	2	4	no	—
307	C₂×C₄×C₇	—	2	8	no	—
308	C₂×C₄×C₇	—	2	32	no	—
309	C₂×C₄×C₇	—	2	8	no	—
310	C₂×C₄×C₇	—	2	32	no	—
311	C₂×C₄×C₇	—	2	32	no	—
312	C₂×C₄×C₇	—	4	16	no	—
313	C₂×C₄×C₇	—	4	8	no	—
314	C₂×C₄×C₇	—	4	4	no	—
315	C₂×C₄×C₇	—	4	16	no	—
316	C₂×C₄×C₇	C₂×#41	1	8	yes	—
317	C₂×C₄×C₇	C₂×#41	1	16	yes	complemented
318	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
319	C₂×C₄×C₇	C₂⁵	1	4	yes	—
320	C₂×C₄×C₇	C₂⁵	1	8	yes	—
321	C₂×C₄×C₇	C₂⁵	1	8	yes	—
322	C₂×C₄×C₇	C₂⁵	1	2	yes	—
323	C₂×C₄×C₇	C₂⁵	1	8	yes	—
324	C₂×C₄×C₇	C₂⁵	1	8	yes	—
325	C₂×C₄×C₇	C₂⁵	1	4	yes	—
326	C₂×C₄×C₇	C₂⁵	1	2	yes	—
327	C₂⁵	C₂³×C₇	1	2	yes	—
328	C₂×#41	—	2	16	no	—
329	C₂×#41	—	4	16	no	—
330	C₂×#41	C₂³×C₇	1	16	yes	—
331	C₂×#41	C₂³×C₇	1	32	yes	complemented
332	C₂²×Q₈	C₂³×C₇	1	2	yes	—
333	C₂²×Q₈	C₂³×C₇	1	4	yes	—
334	C₂²×D₄	—	2	8	no	—
335	C₂²×D₄	C₂³×C₇	1	4	yes	—
336	C₂²×D₄	C₂³×C₇	1	4	yes	—
337	C₂²×D₄	C₂³×C₇	1	2	yes	—
338	C₂²×D₄	C₂³×C₇	1	8	yes	—
339	C₂²×D₄	C₂³×C₇	1	8	yes	—
340	C₂²×D₄	C₂³×C₇	1	8	yes	—
341	C₂²×D₄	C₂³×C₇	1	4	yes	—
342	C₂³×C₄	—	4	8	no	—
343	C₂³×C₄	C₂³×C₇	1	4	yes	—
344	C₂³×C₄	C₂³×C₇	1	4	yes	—
345	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
346	C₂³×C₄	C₂³×C₇	1	4	yes	—
347	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
348	C₂³×C₄	C₂³×C₇	1	2	yes	—
349	C₂³×C₄	C₂³×C₇	1	4	yes	—
350	C₂³×C₄	C₂³×C₇	1	4	yes	—
351	C₂³×C₄	C₂³×C₇	1	16	yes	—
352	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
353	C₄⋊Q₈	C₂³×C₇	1	32	yes	complemented
354	#127	C₂³×C₇	1	32	yes	—
355	#126	C₂³×C₇	1	64	yes	complemented
356	#125	C₂³×C₇	1	32	yes	complemented
357	#124	C₂³×C₇	1	32	yes	—
358	#124	C₂³×C₇	1	64	yes	complemented
359	#123	C₂³×C₇	1	64	yes	complemented
360	#123	C₂³×C₇	1	64	yes	complemented
361	C₂²⋊Q₈	C₂³×C₇	1	32	yes	—
362	C₂²⋊Q₈	C₂³×C₇	1	32	yes	complemented
363	C₂²⋊Q₈	C₂³×C₇	1	32	yes	complemented
364	#121	C₂³×C₇	1	32	yes	—
365	#121	C₂³×C₇	1	32	yes	—
366	#121	C₂³×C₇	1	32	yes	—
367	#121	C₂³×C₇	1	32	yes	complemented
368	#121	C₂³×C₇	1	32	yes	complemented
369	#121	C₂³×C₇	1	64	yes	complemented
370	#120	C₂³×C₇	1	64	yes	—
371	C₄×Q₈	C₂³×C₇	1	16	yes	complemented
372	C₄×Q₈	C₂³×C₇	1	16	yes	complemented
373	C₄×D₄	C₂³×C₇	1	32	yes	—
374	C₄×D₄	C₂³×C₇	1	32	yes	—
375	C₄×D₄	C₂³×C₇	1	32	yes	—
376	C₄×D₄	C₂³×C₇	1	32	yes	—
377	C₄×D₄	C₂³×C₇	1	16	yes	complemented
378	C₄×D₄	C₂³×C₇	1	16	yes	complemented
379	C₄×D₄	C₂³×C₇	1	64	yes	complemented
380	#117	C₂³×C₇	1	16	yes	—
381	#117	C₂³×C₇	1	32	yes	complemented
382	#117	C₂³×C₇	1	32	yes	complemented
383	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
384	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
385	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
386	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
387	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
388	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
389	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
390	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
391	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
392	C₂×C₂²⋊C₄	C₂³×C₇	1	8	yes	—
393	C₂×C₂²⋊C₄	C₂³×C₇	1	16	yes	—
394	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
395	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
396	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₇	1	8	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₇	1	8	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₇	1	8	yes	—
400	C₂×C₄²	C₂³×C₇	1	4	yes	—
401	C₂×C₄²	C₂³×C₇	1	2	yes	—
402	C₂×C₄²	C₂³×C₇	1	4	yes	—
403	C₂×C₄²	C₂³×C₇	1	2	yes	—
404	C₂×C₄²	C₂³×C₇	1	4	yes	—
405	C₂×C₄²	C₂³×C₇	1	8	yes	—
406	C₂×C₄²	C₂³×C₇	1	32	yes	complemented
407	C₂²×C₇	—	2	2	no	—
408	C₂²×C₇	—	2	2	no	—
409	C₂²×C₇	—	2	8	no	—
410	C₂²×C₇	—	2	8	no	—
411	C₂²×C₇	—	2	8	no	—
412	C₂²×C₇	—	2	8	no	—
413	C₂²×C₇	—	2	8	no	—
414	C₂²×C₇	—	2	8	no	—
415	C₂²×C₇	—	2	8	no	—
416	C₂²×C₇	—	4	32	no	—
417	C₂²×C₇	—	4	16	no	—
418	C₂²×C₇	—	4	16	no	—
419	C₂²×C₇	C₂²×#41	1	4	yes	—
420	C₂²×C₇	C₂²×#41	1	2	yes	—
421	C₂²×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
422	C₂²×C₇	#585	1	2	yes	—
423	C₂²×C₇	C₂⁶	1	1	yes	characteristic
424	C₄×C₇	—	2	8	no	—
425	C₄×C₇	—	2	2	no	—
426	C₄×C₇	—	2	8	no	—
427	C₄×C₇	—	2	8	no	—
428	C₄×C₇	—	2	4	no	—
429	C₄×C₇	—	2	8	no	—
430	C₄×C₇	C₂²×#41	1	8	yes	—
431	C₄×C₇	C₂×H(5,𝔽₂)	1	4	yes	—
432	C₂⁴	—	2	8	no	—
433	C₂⁴	—	4	4	no	—
434	C₂⁴	—	4	4	no	—
435	C₂⁴	C₂⁴×C₇	1	4	yes	—
436	C₂⁴	C₂⁴×C₇	1	4	yes	—
437	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
438	C₂⁴	C₂⁴×C₇	1	8	yes	—
439	#41	—	2	64	no	—
440	#41	—	4	64	no	—
441	C₂×Q₈	—	2	8	no	—
442	C₂×Q₈	—	4	8	no	—
443	C₂×Q₈	C₂⁴×C₇	1	8	yes	—
444	C₂×Q₈	C₂⁴×C₇	1	16	yes	—
445	C₂×D₄	—	2	16	no	—
446	C₂×D₄	—	2	16	no	—
447	C₂×D₄	—	2	8	no	—
448	C₂×D₄	—	2	32	no	—
449	C₂×D₄	—	4	8	no	—
450	C₂×D₄	—	4	16	no	—
451	C₂×D₄	—	4	16	no	—
452	C₂×D₄	C₂⁴×C₇	1	16	yes	—
453	C₂×D₄	C₂⁴×C₇	1	16	yes	—
454	C₂×D₄	C₂⁴×C₇	1	8	yes	—
455	C₂×D₄	C₂⁴×C₇	1	32	yes	—
456	C₂×D₄	C₂⁴×C₇	1	32	yes	—
457	C₂×D₄	C₂⁴×C₇	1	16	yes	—
458	C₂²×C₄	—	2	16	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	16	no	—
461	C₂²×C₄	—	2	16	no	—
462	C₂²×C₄	—	4	32	no	—
463	C₂²×C₄	—	4	8	no	—
464	C₂²×C₄	—	4	4	no	—
465	C₂²×C₄	—	4	2	no	—
466	C₂²×C₄	—	4	8	no	—
467	C₂²×C₄	C₇×#41	1	8	yes	complemented
468	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
469	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
470	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
471	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
472	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
473	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
474	C₂²×C₄	C₂⁴×C₇	1	16	yes	—
475	C₂²×C₄	C₂⁴×C₇	1	16	yes	—
476	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
477	C₂²×C₄	C₂⁴×C₇	1	16	yes	—
478	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
479	C₂²×C₄	C₂⁴×C₇	1	16	yes	—
480	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
481	C₂²×C₄	C₂⁴×C₇	1	16	yes	—
482	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
483	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
484	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
485	C₄⋊C₄	C₂⁴×C₇	1	16	yes	—
486	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
487	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
488	C₄⋊C₄	C₂⁴×C₇	1	16	yes	—
489	C₄⋊C₄	C₂⁴×C₇	1	16	yes	—
490	C₄⋊C₄	C₂⁴×C₇	1	16	yes	—
491	C₄⋊C₄	C₂⁴×C₇	1	16	yes	—
492	C₄⋊C₄	C₂⁴×C₇	1	32	yes	—
493	C₂²⋊C₄	C₂⁴×C₇	1	32	yes	—
494	C₂²⋊C₄	C₂⁴×C₇	1	16	yes	—
495	C₂²⋊C₄	C₂⁴×C₇	1	16	yes	—
496	C₂²⋊C₄	C₂⁴×C₇	1	16	yes	—
497	C₂²⋊C₄	C₂⁴×C₇	1	32	yes	—
498	C₂²⋊C₄	C₂⁴×C₇	1	32	yes	—
499	C₂²⋊C₄	C₂⁴×C₇	1	32	yes	—
500	C₄²	C₂⁴×C₇	1	8	yes	—
501	C₄²	C₂⁴×C₇	1	8	yes	—
502	C₄²	C₂⁴×C₇	1	16	yes	—
503	C₄²	C₂⁴×C₇	1	16	yes	—
504	C₄²	C₂⁴×C₇	1	16	yes	—
505	C₂×C₇	—	2	4	no	—
506	C₂×C₇	—	4	8	no	—
507	C₂×C₇	—	4	8	no	—
508	C₂×C₇	#3460	1	4	yes	direct product factor
509	C₂×C₇	C₂³×#41	1	1	yes	characteristic
510	C₂×C₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
511	C₂×C₇	C₂×#585	1	1	yes	characteristic
512	C₂³	—	2	2	no	—
513	C₂³	—	2	2	no	—
514	C₂³	—	2	8	no	—
515	C₂³	—	2	16	no	—
516	C₂³	—	2	8	no	—
517	C₂³	—	2	8	no	—
518	C₂³	—	2	8	no	—
519	C₂³	—	2	8	no	—
520	C₂³	—	2	8	no	—
521	C₂³	—	4	8	no	—
522	C₂³	—	4	8	no	—
523	C₂³	—	4	32	no	—
524	C₂³	C₂×C₇×#41	1	4	yes	—
525	C₂³	C₂⁵×C₇	1	8	yes	—
526	C₂³	C₂⁵×C₇	1	8	yes	—
527	C₂³	C₂⁵×C₇	1	2	yes	—
528	C₂³	C₂⁵×C₇	1	1	yes	characteristic
529	Q₈	—	2	16	no	—
530	Q₈	—	4	16	no	—
531	D₄	—	2	32	no	—
532	D₄	—	2	32	no	—
533	D₄	—	2	16	no	—
534	D₄	—	4	32	no	—
535	D₄	—	4	16	no	—
536	D₄	—	4	32	no	—
537	C₂×C₄	—	2	8	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	8	no	—
540	C₂×C₄	—	2	8	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	8	no	—
543	C₂×C₄	—	2	32	no	—
544	C₂×C₄	—	2	8	no	—
545	C₂×C₄	—	2	32	no	—
546	C₂×C₄	—	2	32	no	—
547	C₂×C₄	—	4	16	no	—
548	C₂×C₄	—	4	8	no	—
549	C₂×C₄	—	4	4	no	—
550	C₂×C₄	—	4	16	no	—
551	C₂×C₄	C₂×C₇×#41	1	8	yes	—
552	C₂×C₄	C₂×C₇×#41	1	16	yes	complemented
553	C₂×C₄	C₇×H(5,𝔽₂)	1	4	yes	—
554	C₂×C₄	C₂⁵×C₇	1	4	yes	—
555	C₂×C₄	C₂⁵×C₇	1	8	yes	—
556	C₂×C₄	C₂⁵×C₇	1	8	yes	—
557	C₂×C₄	C₂⁵×C₇	1	2	yes	—
558	C₂×C₄	C₂⁵×C₇	1	8	yes	—
559	C₂×C₄	C₂⁵×C₇	1	8	yes	—
560	C₂×C₄	C₂⁵×C₇	1	4	yes	—
561	C₂×C₄	C₂⁵×C₇	1	2	yes	—
562	C₇	C₂×#3460	1	1	yes	characteristic, direct product factor, Sylow
563	C₂²	—	2	2	no	—
564	C₂²	—	2	2	no	—
565	C₂²	—	2	8	no	—
566	C₂²	—	2	8	no	—
567	C₂²	—	2	8	no	—
568	C₂²	—	2	8	no	—
569	C₂²	—	2	8	no	—
570	C₂²	—	2	8	no	—
571	C₂²	—	2	8	no	—
572	C₂²	—	4	32	no	—
573	C₂²	—	4	16	no	—
574	C₂²	—	4	16	no	—
575	C₂²	C₂²×C₇×#41	1	4	yes	—
576	C₂²	C₂²×C₇×#41	1	2	yes	—
577	C₂²	C₂×C₇×H(5,𝔽₂)	1	2	yes	—
578	C₂²	C₇×#585	1	2	yes	—
579	C₂²	C₂⁶×C₇	1	1	yes	characteristic
580	C₄	—	2	8	no	—
581	C₄	—	2	2	no	—
582	C₄	—	2	8	no	—
583	C₄	—	2	8	no	—
584	C₄	—	2	4	no	—
585	C₄	—	2	8	no	—
586	C₄	C₂²×C₇×#41	1	8	yes	—
587	C₄	C₂×C₇×H(5,𝔽₂)	1	4	yes	—
588	C₂	—	2	4	no	—
589	C₂	—	4	8	no	—
590	C₂	—	4	8	no	—
591	C₂	C₇×#3460	1	4	yes	direct product factor
592	C₂	C₂³×C₇×#41	1	1	yes	characteristic
593	C₂	C₂²×C₇×H(5,𝔽₂)	1	1	yes	characteristic
594	C₂	C₂×C₇×#585	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₇
socle	C₂³×C₇	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₇×#3460	C₂	direct (N × H)
2. C₂×C₇×#544	C₂	semidirect
3. C₂×C₇×#542	C₂	semidirect
4. C₂×C₇×#540	C₂	semidirect
5. C₂×C₇×#538	C₂	semidirect
6. C₂×C₇×#537	C₂	semidirect
7. C₂×C₇×#535	C₂	semidirect
8. C₂×C₇×#532	C₂	semidirect
9. C₂×C₇×#529	C₂	semidirect
10. C₂×C₇×#525	C₂	semidirect
11. C₂×C₄×C₇×#41	C₂	semidirect
12. C₂²×C₄×C₇×D₄	C₂	semidirect
14. C₂²×C₇×#41	C₂²	semidirect
17. C₇×#544	C₂²	semidirect
18. C₇×#542	C₂²	semidirect
19. C₇×#540	C₂²	semidirect
20. C₇×#538	C₂²	semidirect
21. C₇×#537	C₂²	semidirect
22. C₇×#535	C₂²	semidirect
23. C₇×#532	C₂²	semidirect
24. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
26. C₇×#529	C₂²	semidirect
27. C₂×C₇×#126	C₂²	semidirect
28. C₂×C₇×#125	C₂²	semidirect
30. C₂×C₇×#124	C₂²	semidirect
31. C₇×#525	C₂²	semidirect
32. C₂×C₇×#123	C₂²	semidirect
33. C₂×C₇×#123	C₂²	semidirect
35. C₂×C₇×C₂²⋊Q₈	C₂²	semidirect
36. C₂×C₇×C₂²⋊Q₈	C₂²	semidirect
40. C₂×C₇×#121	C₂²	semidirect
41. C₂×C₇×#121	C₂²	semidirect
42. C₂×C₇×#121	C₂²	semidirect
44. C₄×C₇×#41	C₂²	semidirect
45. C₂×C₄×C₇×Q₈	C₂²	semidirect
46. C₂×C₄×C₇×Q₈	C₂²	semidirect
51. C₂×C₄×C₇×D₄	C₂²	semidirect
52. C₂×C₄×C₇×D₄	C₂²	semidirect
53. C₂×C₄×C₇×D₄	C₂²	semidirect
54. C₂×C₄×C₇×D₄	C₂²	semidirect
56. C₂×C₇×#117	C₂²	semidirect
57. C₂×C₇×#117	C₂²	semidirect
61. C₂²×C₄²×C₇	C₂²	semidirect
62. C₂×#3460	C₇	direct (N × H)
67. C₂×C₇×#41	C₂³	semidirect
89. C₇×C₄⋊Q₈	C₂³	semidirect
91. C₇×#126	C₂³	semidirect
92. C₇×#125	C₂³	semidirect
94. C₇×#124	C₂³	semidirect
95. C₇×#123	C₂³	semidirect
96. C₇×#123	C₂³	semidirect
98. C₇×C₂²⋊Q₈	C₂³	semidirect
99. C₇×C₂²⋊Q₈	C₂³	semidirect
103. C₇×#121	C₂³	semidirect
104. C₇×#121	C₂³	semidirect
105. C₇×#121	C₂³	semidirect
107. C₄×C₇×Q₈	C₂³	semidirect
108. C₄×C₇×Q₈	C₂³	semidirect
113. C₄×C₇×D₄	C₂³	semidirect
114. C₄×C₇×D₄	C₂³	semidirect
115. C₄×C₇×D₄	C₂³	semidirect
117. C₇×#117	C₂³	semidirect
118. C₇×#117	C₂³	semidirect
142. C₂×C₄²×C₇	C₂³	semidirect
143. #3460	C₂×C₇	direct (N × H)
144. C₂×#544	C₂×C₇	semidirect
145. C₂×#542	C₂×C₇	semidirect
146. C₂×#540	C₂×C₇	semidirect
147. C₂×#538	C₂×C₇	semidirect
148. C₂×#537	C₂×C₇	semidirect
149. C₂×#535	C₂×C₇	semidirect
150. C₂×#532	C₂×C₇	semidirect
151. C₂×#529	C₂×C₇	semidirect
152. C₂×#525	C₂×C₇	semidirect
153. C₂×C₄×#41	C₂×C₇	semidirect
154. C₂²×C₄×D₄	C₂×C₇	semidirect
190. C₂²×C₄×C₇	#41	semidirect
229. C₂²×#41	C₂²×C₇	semidirect
232. #544	C₂²×C₇	semidirect
233. #542	C₂²×C₇	semidirect
234. #540	C₂²×C₇	semidirect
235. #538	C₂²×C₇	semidirect
236. #537	C₂²×C₇	semidirect
237. #535	C₂²×C₇	semidirect
238. #532	C₂²×C₇	semidirect
239. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
241. #529	C₂²×C₇	semidirect
242. C₂×#126	C₂²×C₇	semidirect
243. C₂×#125	C₂²×C₇	semidirect
245. C₂×#124	C₂²×C₇	semidirect
246. #525	C₂²×C₇	semidirect
247. C₂×#123	C₂²×C₇	semidirect
248. C₂×#123	C₂²×C₇	semidirect
250. C₂×C₂²⋊Q₈	C₂²×C₇	semidirect
251. C₂×C₂²⋊Q₈	C₂²×C₇	semidirect
255. C₂×#121	C₂²×C₇	semidirect
256. C₂×#121	C₂²×C₇	semidirect
257. C₂×#121	C₂²×C₇	semidirect
259. C₄×#41	C₂²×C₇	semidirect
260. C₂×C₄×Q₈	C₂²×C₇	semidirect
261. C₂×C₄×Q₈	C₂²×C₇	semidirect
266. C₂×C₄×D₄	C₂²×C₇	semidirect
267. C₂×C₄×D₄	C₂²×C₇	semidirect
268. C₂×C₄×D₄	C₂²×C₇	semidirect
269. C₂×C₄×D₄	C₂²×C₇	semidirect
271. C₂×#117	C₂²×C₇	semidirect
272. C₂×#117	C₂²×C₇	semidirect
276. C₂²×C₄²	C₂²×C₇	semidirect
317. C₂×C₄×C₇	C₂×#41	semidirect
331. C₂×#41	C₂³×C₇	semidirect
353. C₄⋊Q₈	C₂³×C₇	semidirect
355. #126	C₂³×C₇	semidirect
356. #125	C₂³×C₇	semidirect
358. #124	C₂³×C₇	semidirect
359. #123	C₂³×C₇	semidirect
360. #123	C₂³×C₇	semidirect
362. C₂²⋊Q₈	C₂³×C₇	semidirect
363. C₂²⋊Q₈	C₂³×C₇	semidirect
367. #121	C₂³×C₇	semidirect
368. #121	C₂³×C₇	semidirect
369. #121	C₂³×C₇	semidirect
371. C₄×Q₈	C₂³×C₇	semidirect
372. C₄×Q₈	C₂³×C₇	semidirect
377. C₄×D₄	C₂³×C₇	semidirect
378. C₄×D₄	C₂³×C₇	semidirect
379. C₄×D₄	C₂³×C₇	semidirect
381. #117	C₂³×C₇	semidirect
382. #117	C₂³×C₇	semidirect
406. C₂×C₄²	C₂³×C₇	semidirect
467. C₂²×C₄	C₇×#41	semidirect
508. C₂×C₇	#3460	direct (N × H)
552. C₂×C₄	C₂×C₇×#41	semidirect
562. C₇	C₂×#3460	direct (N × H)
591. C₂	C₇×#3460	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#3460	2
C₂×C₇×#544	2
C₂×C₇×#542	2
C₂×C₇×#540	2
C₂×C₇×#538	2
C₂×C₇×#537	2
C₂×C₇×#535	2
C₂×C₇×#532	2
C₂×C₇×#529	2
C₂×C₇×#525	2
C₂×C₄×C₇×#41	2
C₂²×C₄×C₇×D₄	2
C₂×#3460	7

Subgroups — C2×C7×#3460 #13239895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3460 #13239895