Subgroups — C₂×C₇×#3492 #13239984

Order 1792.

7910 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3492	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₇×#554	C₂	1	4	yes	maximal, complemented
3	C₂×C₇×#553	C₂	1	4	yes	maximal, complemented
4	C₂×C₇×#552	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×#547	C₂	1	4	yes	maximal, complemented
6	C₂×C₇×#544	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#539	C₂	1	4	yes	maximal, complemented
8	C₂×C₇×#537	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₇×#525	C₂	1	2	yes	maximal, complemented
10	C₂×C₇×#525	C₂	1	4	yes	maximal, complemented
11	C₂×C₇×#520	C₂	1	2	yes	maximal, complemented
12	C₂³×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×C₇×#41	C₂²	1	2	yes	—
14	C₂²×C₇×#41	C₂²	1	4	yes	—
15	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₇	C₂²	1	2	yes	—
17	C₇×#554	C₂²	1	64	yes	complemented
18	C₇×#553	C₂²	1	64	yes	complemented
19	C₇×#552	C₂²	1	64	yes	complemented
20	C₇×#547	C₂²	1	64	yes	complemented
21	C₇×#544	C₂²	1	16	yes	complemented
22	C₇×#539	C₂²	1	64	yes	complemented
23	C₇×#537	C₂²	1	16	yes	complemented
24	C₂×C₇×C₄⋊Q₈	C₂²	1	4	yes	complemented
25	C₂×C₇×#126	C₂²	1	8	yes	complemented
26	C₂×C₇×#125	C₂²	1	8	yes	complemented
27	C₂×C₇×#125	C₂²	1	4	yes	complemented
28	C₇×#525	C₂²	1	32	yes	complemented
29	C₇×#525	C₂²	1	64	yes	complemented
30	C₂×C₇×#123	C₂²	1	4	yes	—
31	C₂×C₇×#123	C₂²	1	8	yes	—
32	C₂×C₇×#123	C₂²	1	8	yes	complemented
33	C₂×C₇×C₂²⋊Q₈	C₂²	1	4	yes	—
34	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
35	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
36	C₂×C₇×C₂²⋊Q₈	C₂²	1	4	yes	—
37	C₂×C₇×C₂²⋊Q₈	C₂²	1	4	yes	complemented
38	C₂×C₇×C₂²⋊Q₈	C₂²	1	8	yes	complemented
39	C₂×C₇×#121	C₂²	1	4	yes	—
40	C₂×C₇×#121	C₂²	1	2	yes	—
41	C₂×C₇×#121	C₂²	1	4	yes	—
42	C₂×C₇×#121	C₂²	1	2	yes	—
43	C₂×C₇×#121	C₂²	1	4	yes	complemented
44	C₂×C₇×#120	C₂²	1	4	yes	—
45	C₇×#520	C₂²	1	32	yes	complemented
46	C₂×C₄×C₇×Q₈	C₂²	1	4	yes	complemented
47	C₂×C₄×C₇×D₄	C₂²	1	8	yes	—
48	C₂×C₄×C₇×D₄	C₂²	1	4	yes	—
49	C₂×C₄×C₇×D₄	C₂²	1	8	yes	—
50	C₂×C₄×C₇×D₄	C₂²	1	4	yes	—
51	C₂×C₄×C₇×D₄	C₂²	1	4	yes	complemented
52	C₂×C₇×#117	C₂²	1	2	yes	—
53	C₂×C₇×#117	C₂²	1	4	yes	—
54	C₂×C₇×#117	C₂²	1	4	yes	complemented
55	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
56	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
57	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
58	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	—
59	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	—
60	C₂²×C₇×C₄⋊C₄	C₂²	1	16	yes	complemented
61	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	complemented
62	C₂×#3492	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂⁵×C₇	C₂³	1	1	yes	characteristic
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	—
68	C₂²×C₇×Q₈	C₂³	1	2	yes	—
69	C₂²×C₇×Q₈	C₂³	1	4	yes	—
70	C₂²×C₇×D₄	C₂³	1	4	yes	—
71	C₂²×C₇×D₄	C₂³	1	4	yes	—
72	C₂²×C₇×D₄	C₂³	1	2	yes	—
73	C₂²×C₇×D₄	C₂³	1	4	yes	—
74	C₂²×C₇×D₄	C₂³	1	4	yes	—
75	C₂³×C₄×C₇	—	2	8	no	—
76	C₂³×C₄×C₇	—	4	8	no	—
77	C₂³×C₄×C₇	C₂³	1	4	yes	—
78	C₂³×C₄×C₇	C₂³	1	4	yes	—
79	C₂³×C₄×C₇	C₂³	1	2	yes	—
80	C₂³×C₄×C₇	C₂³	1	2	yes	—
81	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₇	C₂³	1	8	yes	—
83	C₂³×C₄×C₇	C₂³	1	8	yes	—
84	C₂³×C₄×C₇	C₂³	1	2	yes	—
85	C₂³×C₄×C₇	C₂³	1	4	yes	—
86	C₂³×C₄×C₇	C₂³	1	2	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×#126	C₂³	1	64	yes	complemented
91	C₇×#125	C₂³	1	64	yes	complemented
92	C₇×#125	C₂³	1	32	yes	complemented
93	C₇×#123	C₂³	1	32	yes	—
94	C₇×#123	C₂³	1	64	yes	—
95	C₇×#123	C₂³	1	64	yes	complemented
96	C₇×C₂²⋊Q₈	C₂³	1	32	yes	—
97	C₇×C₂²⋊Q₈	C₂³	1	16	yes	—
98	C₇×C₂²⋊Q₈	C₂³	1	16	yes	—
99	C₇×C₂²⋊Q₈	C₂³	1	32	yes	—
100	C₇×C₂²⋊Q₈	C₂³	1	32	yes	complemented
101	C₇×C₂²⋊Q₈	C₂³	1	64	yes	complemented
102	C₇×#121	C₂³	1	32	yes	—
103	C₇×#121	C₂³	1	16	yes	—
104	C₇×#121	C₂³	1	32	yes	—
105	C₇×#121	C₂³	1	16	yes	—
106	C₇×#121	C₂³	1	32	yes	complemented
107	C₇×#120	C₂³	1	32	yes	—
108	C₄×C₇×Q₈	C₂³	1	32	yes	complemented
109	C₄×C₇×D₄	C₂³	1	64	yes	—
110	C₄×C₇×D₄	C₂³	1	32	yes	—
111	C₄×C₇×D₄	C₂³	1	64	yes	—
112	C₄×C₇×D₄	C₂³	1	32	yes	—
113	C₄×C₇×D₄	C₂³	1	32	yes	complemented
114	C₇×#117	C₂³	1	16	yes	—
115	C₇×#117	C₂³	1	32	yes	—
116	C₇×#117	C₂³	1	32	yes	complemented
117	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
118	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
119	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
120	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
121	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
122	C₂×C₇×C₄⋊C₄	C₂³	1	16	yes	—
123	C₂×C₇×C₄⋊C₄	C₂³	1	32	yes	—
124	C₂×C₇×C₄⋊C₄	C₂³	1	32	yes	—
125	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
126	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
127	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
128	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
129	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
130	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
131	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
132	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
133	C₂×C₇×C₄⋊C₄	C₂³	1	32	yes	complemented
134	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
135	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
136	C₂×C₇×C₂²⋊C₄	C₂³	1	8	yes	—
137	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
138	C₂×C₇×C₂²⋊C₄	C₂³	1	8	yes	—
139	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
140	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
141	C₂×C₄²×C₇	C₂³	1	8	yes	—
142	C₂×C₄²×C₇	C₂³	1	4	yes	—
143	#3492	C₂×C₇	1	32	yes	direct product factor
144	C₂×#554	C₂×C₇	1	4	yes	complemented
145	C₂×#553	C₂×C₇	1	4	yes	complemented
146	C₂×#552	C₂×C₇	1	4	yes	complemented
147	C₂×#547	C₂×C₇	1	4	yes	complemented
148	C₂×#544	C₂×C₇	1	1	yes	characteristic, complemented
149	C₂×#539	C₂×C₇	1	4	yes	complemented
150	C₂×#537	C₂×C₇	1	1	yes	characteristic, complemented
151	C₂×#525	C₂×C₇	1	2	yes	complemented
152	C₂×#525	C₂×C₇	1	4	yes	complemented
153	C₂×#520	C₂×C₇	1	2	yes	complemented
154	C₂³×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
155	C₂⁴×C₇	—	2	4	no	—
156	C₂⁴×C₇	—	2	4	no	—
157	C₂⁴×C₇	—	2	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	1	yes	characteristic
162	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
163	C₇×#41	—	2	64	no	—
164	C₇×#41	—	4	64	no	—
165	C₂×C₇×Q₈	—	2	8	no	—
166	C₂×C₇×Q₈	—	4	8	no	—
167	C₂×C₇×Q₈	C₂⁴	1	8	yes	—
168	C₂×C₇×Q₈	C₂⁴	1	16	yes	—
169	C₂×C₇×D₄	—	2	16	no	—
170	C₂×C₇×D₄	—	2	16	no	—
171	C₂×C₇×D₄	—	2	16	no	—
172	C₂×C₇×D₄	—	2	8	no	—
173	C₂×C₇×D₄	—	4	8	no	—
174	C₂×C₇×D₄	C₂⁴	1	16	yes	—
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	16	yes	—
179	C₂²×C₄×C₇	—	2	4	no	—
180	C₂²×C₄×C₇	—	2	16	no	—
181	C₂²×C₄×C₇	—	2	8	no	—
182	C₂²×C₄×C₇	—	2	8	no	—
183	C₂²×C₄×C₇	—	2	16	no	—
184	C₂²×C₄×C₇	—	2	32	no	—
185	C₂²×C₄×C₇	—	4	4	no	—
186	C₂²×C₄×C₇	—	4	8	no	—
187	C₂²×C₄×C₇	—	4	4	no	—
188	C₂²×C₄×C₇	—	4	32	no	—
189	C₂²×C₄×C₇	—	4	16	no	—
190	C₂²×C₄×C₇	#41	1	8	yes	complemented
191	C₂²×C₄×C₇	C₂⁴	1	16	yes	—
192	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
193	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
194	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
195	C₂²×C₄×C₇	C₂⁴	1	16	yes	—
196	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
197	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
198	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
199	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
200	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
201	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
202	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
203	C₂²×C₄×C₇	C₂⁴	1	16	yes	—
204	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
205	C₂²×C₄×C₇	C₂⁴	1	32	yes	—
206	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
207	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
208	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
209	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
210	C₇×C₄⋊C₄	C₂⁴	1	32	yes	—
211	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
212	C₇×C₄⋊C₄	C₂⁴	1	32	yes	—
213	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
214	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
215	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
216	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
217	C₇×C₄⋊C₄	C₂⁴	1	32	yes	—
218	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
219	C₇×C₂²⋊C₄	C₂⁴	1	16	yes	—
220	C₇×C₂²⋊C₄	C₂⁴	1	16	yes	—
221	C₇×C₂²⋊C₄	C₂⁴	1	32	yes	—
222	C₇×C₂²⋊C₄	C₂⁴	1	16	yes	—
223	C₇×C₂²⋊C₄	C₂⁴	1	32	yes	—
224	C₇×C₂²⋊C₄	C₂⁴	1	16	yes	—
225	C₇×C₂²⋊C₄	C₂⁴	1	16	yes	—
226	C₄²×C₇	C₂⁴	1	32	yes	—
227	C₄²×C₇	C₂⁴	1	16	yes	—
228	C₂²×#41	C₂²×C₇	1	2	yes	—
229	C₂²×#41	C₂²×C₇	1	4	yes	—
230	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
231	C₂⁴×C₄	C₂²×C₇	1	2	yes	—
232	#554	C₂²×C₇	1	64	yes	complemented
233	#553	C₂²×C₇	1	64	yes	complemented
234	#552	C₂²×C₇	1	64	yes	complemented
235	#547	C₂²×C₇	1	64	yes	complemented
236	#544	C₂²×C₇	1	16	yes	complemented
237	#539	C₂²×C₇	1	64	yes	complemented
238	#537	C₂²×C₇	1	16	yes	complemented
239	C₂×C₄⋊Q₈	C₂²×C₇	1	4	yes	complemented
240	C₂×#126	C₂²×C₇	1	8	yes	complemented
241	C₂×#125	C₂²×C₇	1	8	yes	complemented
242	C₂×#125	C₂²×C₇	1	4	yes	complemented
243	#525	C₂²×C₇	1	32	yes	complemented
244	#525	C₂²×C₇	1	64	yes	complemented
245	C₂×#123	C₂²×C₇	1	4	yes	—
246	C₂×#123	C₂²×C₇	1	8	yes	—
247	C₂×#123	C₂²×C₇	1	8	yes	complemented
248	C₂×C₂²⋊Q₈	C₂²×C₇	1	4	yes	—
249	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
250	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
251	C₂×C₂²⋊Q₈	C₂²×C₇	1	4	yes	—
252	C₂×C₂²⋊Q₈	C₂²×C₇	1	4	yes	complemented
253	C₂×C₂²⋊Q₈	C₂²×C₇	1	8	yes	complemented
254	C₂×#121	C₂²×C₇	1	4	yes	—
255	C₂×#121	C₂²×C₇	1	2	yes	—
256	C₂×#121	C₂²×C₇	1	4	yes	—
257	C₂×#121	C₂²×C₇	1	2	yes	—
258	C₂×#121	C₂²×C₇	1	4	yes	complemented
259	C₂×#120	C₂²×C₇	1	4	yes	—
260	#520	C₂²×C₇	1	32	yes	complemented
261	C₂×C₄×Q₈	C₂²×C₇	1	4	yes	complemented
262	C₂×C₄×D₄	C₂²×C₇	1	8	yes	—
263	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
264	C₂×C₄×D₄	C₂²×C₇	1	8	yes	—
265	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
266	C₂×C₄×D₄	C₂²×C₇	1	4	yes	complemented
267	C₂×#117	C₂²×C₇	1	2	yes	—
268	C₂×#117	C₂²×C₇	1	4	yes	—
269	C₂×#117	C₂²×C₇	1	4	yes	complemented
270	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
271	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
272	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
273	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
274	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
275	C₂²×C₄⋊C₄	C₂²×C₇	1	16	yes	complemented
276	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	complemented
277	C₂³×C₇	—	2	32	no	—
278	C₂³×C₇	—	2	8	no	—
279	C₂³×C₇	—	2	2	no	—
280	C₂³×C₇	—	2	2	no	—
281	C₂³×C₇	—	2	8	no	—
282	C₂³×C₇	—	2	2	no	—
283	C₂³×C₇	—	2	2	no	—
284	C₂³×C₇	—	2	8	no	—
285	C₂³×C₇	—	2	2	no	—
286	C₂³×C₇	—	2	2	no	—
287	C₂³×C₇	—	2	8	no	—
288	C₂³×C₇	—	2	8	no	—
289	C₂³×C₇	—	2	8	no	—
290	C₂³×C₇	C₂×#41	1	4	yes	—
291	C₂³×C₇	H(5,𝔽₂)	1	4	yes	—
292	C₂³×C₇	2₋¹⁺⁴	1	4	yes	—
293	C₂³×C₇	C₂⁵	1	8	yes	—
294	C₂³×C₇	C₂⁵	1	2	yes	—
295	C₂³×C₇	C₂⁵	1	2	yes	—
296	C₂³×C₇	C₂⁵	1	2	yes	—
297	C₂³×C₇	C₂⁵	1	1	yes	characteristic
298	C₇×Q₈	—	2	16	no	—
299	C₇×Q₈	—	4	16	no	—
300	C₇×D₄	—	2	32	no	—
301	C₇×D₄	—	2	32	no	—
302	C₇×D₄	—	2	32	no	—
303	C₇×D₄	—	2	16	no	—
304	C₇×D₄	—	4	16	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	8	no	—
309	C₂×C₄×C₇	—	2	8	no	—
310	C₂×C₄×C₇	—	2	32	no	—
311	C₂×C₄×C₇	—	2	8	no	—
312	C₂×C₄×C₇	—	2	8	no	—
313	C₂×C₄×C₇	—	2	16	no	—
314	C₂×C₄×C₇	—	2	16	no	—
315	C₂×C₄×C₇	—	2	32	no	—
316	C₂×C₄×C₇	—	4	8	no	—
317	C₂×C₄×C₇	—	4	16	no	—
318	C₂×C₄×C₇	—	4	8	no	—
319	C₂×C₄×C₇	—	4	32	no	—
320	C₂×C₄×C₇	C₂×#41	1	8	yes	—
321	C₂×C₄×C₇	C₂×#41	1	16	yes	complemented
322	C₂×C₄×C₇	C₂⁵	1	8	yes	—
323	C₂×C₄×C₇	C₂⁵	1	8	yes	—
324	C₂×C₄×C₇	C₂⁵	1	8	yes	—
325	C₂×C₄×C₇	C₂⁵	1	8	yes	—
326	C₂×C₄×C₇	C₂⁵	1	4	yes	—
327	C₂×C₄×C₇	C₂⁵	1	8	yes	—
328	C₂×C₄×C₇	C₂⁵	1	4	yes	—
329	C₂⁵	C₂³×C₇	1	1	yes	characteristic
330	C₂×#41	—	2	16	no	—
331	C₂×#41	—	4	16	no	—
332	C₂×#41	C₂³×C₇	1	16	yes	—
333	C₂×#41	C₂³×C₇	1	32	yes	—
334	C₂²×Q₈	C₂³×C₇	1	2	yes	—
335	C₂²×Q₈	C₂³×C₇	1	4	yes	—
336	C₂²×D₄	C₂³×C₇	1	4	yes	—
337	C₂²×D₄	C₂³×C₇	1	4	yes	—
338	C₂²×D₄	C₂³×C₇	1	4	yes	—
339	C₂²×D₄	C₂³×C₇	1	2	yes	—
340	C₂²×D₄	C₂³×C₇	1	4	yes	—
341	C₂³×C₄	—	2	8	no	—
342	C₂³×C₄	—	4	8	no	—
343	C₂³×C₄	C₂³×C₇	1	4	yes	—
344	C₂³×C₄	C₂³×C₇	1	2	yes	—
345	C₂³×C₄	C₂³×C₇	1	2	yes	—
346	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
347	C₂³×C₄	C₂³×C₇	1	8	yes	—
348	C₂³×C₄	C₂³×C₇	1	4	yes	—
349	C₂³×C₄	C₂³×C₇	1	2	yes	—
350	C₂³×C₄	C₂³×C₇	1	4	yes	—
351	C₂³×C₄	C₂³×C₇	1	2	yes	—
352	C₂³×C₄	C₂³×C₇	1	16	yes	—
353	C₂³×C₄	C₂³×C₇	1	8	yes	—
354	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
355	C₄⋊Q₈	C₂³×C₇	1	32	yes	complemented
356	#126	C₂³×C₇	1	64	yes	complemented
357	#125	C₂³×C₇	1	64	yes	complemented
358	#125	C₂³×C₇	1	32	yes	complemented
359	#123	C₂³×C₇	1	32	yes	—
360	#123	C₂³×C₇	1	64	yes	—
361	#123	C₂³×C₇	1	64	yes	complemented
362	C₂²⋊Q₈	C₂³×C₇	1	32	yes	—
363	C₂²⋊Q₈	C₂³×C₇	1	16	yes	—
364	C₂²⋊Q₈	C₂³×C₇	1	16	yes	—
365	C₂²⋊Q₈	C₂³×C₇	1	32	yes	—
366	C₂²⋊Q₈	C₂³×C₇	1	32	yes	complemented
367	C₂²⋊Q₈	C₂³×C₇	1	64	yes	complemented
368	#121	C₂³×C₇	1	32	yes	—
369	#121	C₂³×C₇	1	16	yes	—
370	#121	C₂³×C₇	1	32	yes	—
371	#121	C₂³×C₇	1	16	yes	—
372	#121	C₂³×C₇	1	32	yes	complemented
373	#120	C₂³×C₇	1	32	yes	—
374	C₄×Q₈	C₂³×C₇	1	32	yes	complemented
375	C₄×D₄	C₂³×C₇	1	64	yes	—
376	C₄×D₄	C₂³×C₇	1	32	yes	—
377	C₄×D₄	C₂³×C₇	1	64	yes	—
378	C₄×D₄	C₂³×C₇	1	32	yes	—
379	C₄×D₄	C₂³×C₇	1	32	yes	complemented
380	#117	C₂³×C₇	1	16	yes	—
381	#117	C₂³×C₇	1	32	yes	—
382	#117	C₂³×C₇	1	32	yes	complemented
383	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
384	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
385	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
386	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
387	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
388	C₂×C₄⋊C₄	C₂³×C₇	1	16	yes	—
389	C₂×C₄⋊C₄	C₂³×C₇	1	32	yes	—
390	C₂×C₄⋊C₄	C₂³×C₇	1	32	yes	—
391	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
392	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
393	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
394	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
395	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
396	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
397	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
398	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
399	C₂×C₄⋊C₄	C₂³×C₇	1	32	yes	complemented
400	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₇	1	8	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₇	1	8	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
407	C₂×C₄²	C₂³×C₇	1	8	yes	—
408	C₂×C₄²	C₂³×C₇	1	4	yes	—
409	C₂²×C₇	—	2	32	no	—
410	C₂²×C₇	—	2	2	no	—
411	C₂²×C₇	—	2	2	no	—
412	C₂²×C₇	—	2	2	no	—
413	C₂²×C₇	—	2	2	no	—
414	C₂²×C₇	—	2	2	no	—
415	C₂²×C₇	—	2	2	no	—
416	C₂²×C₇	—	2	8	no	—
417	C₂²×C₇	—	2	8	no	—
418	C₂²×C₇	—	2	8	no	—
419	C₂²×C₇	—	2	8	no	—
420	C₂²×C₇	—	2	8	no	—
421	C₂²×C₇	—	2	8	no	—
422	C₂²×C₇	—	4	16	no	—
423	C₂²×C₇	C₂²×#41	1	4	yes	—
424	C₂²×C₇	C₂²×#41	1	2	yes	—
425	C₂²×C₇	C₂×H(5,𝔽₂)	1	4	yes	—
426	C₂²×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
427	C₂²×C₇	C₂×2₋¹⁺⁴	1	4	yes	—
428	C₂²×C₇	C₂×2₋¹⁺⁴	1	2	yes	—
429	C₂²×C₇	C₂⁶	1	1	yes	characteristic
430	C₄×C₇	—	2	4	no	—
431	C₄×C₇	—	2	8	no	—
432	C₄×C₇	—	2	8	no	—
433	C₄×C₇	—	2	8	no	—
434	C₄×C₇	—	2	8	no	—
435	C₄×C₇	—	2	8	no	—
436	C₄×C₇	C₂²×#41	1	8	yes	—
437	C₂⁴	—	2	4	no	—
438	C₂⁴	—	2	4	no	—
439	C₂⁴	—	2	4	no	—
440	C₂⁴	C₂⁴×C₇	1	4	yes	—
441	C₂⁴	C₂⁴×C₇	1	4	yes	—
442	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
443	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
444	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
445	#41	—	2	64	no	—
446	#41	—	4	64	no	—
447	C₂×Q₈	—	2	8	no	—
448	C₂×Q₈	—	4	8	no	—
449	C₂×Q₈	C₂⁴×C₇	1	8	yes	—
450	C₂×Q₈	C₂⁴×C₇	1	16	yes	—
451	C₂×D₄	—	2	16	no	—
452	C₂×D₄	—	2	16	no	—
453	C₂×D₄	—	2	16	no	—
454	C₂×D₄	—	2	8	no	—
455	C₂×D₄	—	4	8	no	—
456	C₂×D₄	C₂⁴×C₇	1	16	yes	—
457	C₂×D₄	C₂⁴×C₇	1	16	yes	—
458	C₂×D₄	C₂⁴×C₇	1	16	yes	—
459	C₂×D₄	C₂⁴×C₇	1	8	yes	—
460	C₂×D₄	C₂⁴×C₇	1	16	yes	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	2	16	no	—
463	C₂²×C₄	—	2	8	no	—
464	C₂²×C₄	—	2	8	no	—
465	C₂²×C₄	—	2	16	no	—
466	C₂²×C₄	—	2	32	no	—
467	C₂²×C₄	—	4	4	no	—
468	C₂²×C₄	—	4	8	no	—
469	C₂²×C₄	—	4	4	no	—
470	C₂²×C₄	—	4	32	no	—
471	C₂²×C₄	—	4	16	no	—
472	C₂²×C₄	C₇×#41	1	8	yes	complemented
473	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
474	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
475	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
476	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
477	C₂²×C₄	C₂⁴×C₇	1	2	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	8	yes	—
482	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
483	C₂²×C₄	C₂⁴×C₇	1	16	yes	—
484	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
485	C₂²×C₄	C₂⁴×C₇	1	16	yes	—
486	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
487	C₂²×C₄	C₂⁴×C₇	1	32	yes	—
488	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
489	C₂²×C₄	C₂⁴×C₇	1	4	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	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
494	C₄⋊C₄	C₂⁴×C₇	1	32	yes	—
495	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
496	C₄⋊C₄	C₂⁴×C₇	1	16	yes	—
497	C₄⋊C₄	C₂⁴×C₇	1	16	yes	—
498	C₄⋊C₄	C₂⁴×C₇	1	16	yes	—
499	C₄⋊C₄	C₂⁴×C₇	1	32	yes	—
500	C₄⋊C₄	C₂⁴×C₇	1	16	yes	—
501	C₂²⋊C₄	C₂⁴×C₇	1	16	yes	—
502	C₂²⋊C₄	C₂⁴×C₇	1	16	yes	—
503	C₂²⋊C₄	C₂⁴×C₇	1	32	yes	—
504	C₂²⋊C₄	C₂⁴×C₇	1	16	yes	—
505	C₂²⋊C₄	C₂⁴×C₇	1	32	yes	—
506	C₂²⋊C₄	C₂⁴×C₇	1	16	yes	—
507	C₂²⋊C₄	C₂⁴×C₇	1	16	yes	—
508	C₄²	C₂⁴×C₇	1	32	yes	—
509	C₄²	C₂⁴×C₇	1	16	yes	—
510	C₂×C₇	—	2	4	no	—
511	C₂×C₇	—	2	4	no	—
512	C₂×C₇	—	2	4	no	—
513	C₂×C₇	—	4	8	no	—
514	C₂×C₇	#3492	1	4	yes	direct product factor
515	C₂×C₇	C₂³×#41	1	1	yes	characteristic
516	C₂×C₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
517	C₂×C₇	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
518	C₂³	—	2	32	no	—
519	C₂³	—	2	8	no	—
520	C₂³	—	2	2	no	—
521	C₂³	—	2	2	no	—
522	C₂³	—	2	8	no	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	8	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	8	no	—
529	C₂³	—	2	8	no	—
530	C₂³	—	2	8	no	—
531	C₂³	C₂×C₇×#41	1	4	yes	—
532	C₂³	C₇×H(5,𝔽₂)	1	4	yes	—
533	C₂³	C₇×2₋¹⁺⁴	1	4	yes	—
534	C₂³	C₂⁵×C₇	1	8	yes	—
535	C₂³	C₂⁵×C₇	1	2	yes	—
536	C₂³	C₂⁵×C₇	1	2	yes	—
537	C₂³	C₂⁵×C₇	1	2	yes	—
538	C₂³	C₂⁵×C₇	1	1	yes	characteristic
539	Q₈	—	2	16	no	—
540	Q₈	—	4	16	no	—
541	D₄	—	2	32	no	—
542	D₄	—	2	32	no	—
543	D₄	—	2	32	no	—
544	D₄	—	2	16	no	—
545	D₄	—	4	16	no	—
546	C₂×C₄	—	2	8	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	8	no	—
549	C₂×C₄	—	2	8	no	—
550	C₂×C₄	—	2	8	no	—
551	C₂×C₄	—	2	32	no	—
552	C₂×C₄	—	2	8	no	—
553	C₂×C₄	—	2	8	no	—
554	C₂×C₄	—	2	16	no	—
555	C₂×C₄	—	2	16	no	—
556	C₂×C₄	—	2	32	no	—
557	C₂×C₄	—	4	8	no	—
558	C₂×C₄	—	4	16	no	—
559	C₂×C₄	—	4	8	no	—
560	C₂×C₄	—	4	32	no	—
561	C₂×C₄	C₂×C₇×#41	1	8	yes	—
562	C₂×C₄	C₂×C₇×#41	1	16	yes	complemented
563	C₂×C₄	C₂⁵×C₇	1	8	yes	—
564	C₂×C₄	C₂⁵×C₇	1	8	yes	—
565	C₂×C₄	C₂⁵×C₇	1	8	yes	—
566	C₂×C₄	C₂⁵×C₇	1	8	yes	—
567	C₂×C₄	C₂⁵×C₇	1	4	yes	—
568	C₂×C₄	C₂⁵×C₇	1	8	yes	—
569	C₂×C₄	C₂⁵×C₇	1	4	yes	—
570	C₇	C₂×#3492	1	1	yes	characteristic, direct product factor, Sylow
571	C₂²	—	2	32	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	2	2	no	—
578	C₂²	—	2	8	no	—
579	C₂²	—	2	8	no	—
580	C₂²	—	2	8	no	—
581	C₂²	—	2	8	no	—
582	C₂²	—	2	8	no	—
583	C₂²	—	2	8	no	—
584	C₂²	—	4	16	no	—
585	C₂²	C₂²×C₇×#41	1	4	yes	—
586	C₂²	C₂²×C₇×#41	1	2	yes	—
587	C₂²	C₂×C₇×H(5,𝔽₂)	1	4	yes	—
588	C₂²	C₂×C₇×H(5,𝔽₂)	1	2	yes	—
589	C₂²	C₂×C₇×2₋¹⁺⁴	1	4	yes	—
590	C₂²	C₂×C₇×2₋¹⁺⁴	1	2	yes	—
591	C₂²	C₂⁶×C₇	1	1	yes	characteristic
592	C₄	—	2	4	no	—
593	C₄	—	2	8	no	—
594	C₄	—	2	8	no	—
595	C₄	—	2	8	no	—
596	C₄	—	2	8	no	—
597	C₄	—	2	8	no	—
598	C₄	C₂²×C₇×#41	1	8	yes	—
599	C₂	—	2	4	no	—
600	C₂	—	2	4	no	—
601	C₂	—	2	4	no	—
602	C₂	—	4	8	no	—
603	C₂	C₇×#3492	1	4	yes	direct product factor
604	C₂	C₂³×C₇×#41	1	1	yes	characteristic
605	C₂	C₂²×C₇×H(5,𝔽₂)	1	1	yes	characteristic
606	C₂	C₂²×C₇×2₋¹⁺⁴	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₇×#3492	C₂	direct (N × H)
2. C₂×C₇×#554	C₂	semidirect
3. C₂×C₇×#553	C₂	semidirect
4. C₂×C₇×#552	C₂	semidirect
5. C₂×C₇×#547	C₂	semidirect
6. C₂×C₇×#544	C₂	semidirect
7. C₂×C₇×#539	C₂	semidirect
8. C₂×C₇×#537	C₂	semidirect
9. C₂×C₇×#525	C₂	semidirect
10. C₂×C₇×#525	C₂	semidirect
11. C₂×C₇×#520	C₂	semidirect
12. C₂³×C₇×C₄⋊C₄	C₂	semidirect
17. C₇×#554	C₂²	semidirect
18. C₇×#553	C₂²	semidirect
19. C₇×#552	C₂²	semidirect
20. C₇×#547	C₂²	semidirect
21. C₇×#544	C₂²	semidirect
22. C₇×#539	C₂²	semidirect
23. C₇×#537	C₂²	semidirect
24. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
25. C₂×C₇×#126	C₂²	semidirect
26. C₂×C₇×#125	C₂²	semidirect
27. C₂×C₇×#125	C₂²	semidirect
28. C₇×#525	C₂²	semidirect
29. C₇×#525	C₂²	semidirect
32. C₂×C₇×#123	C₂²	semidirect
37. C₂×C₇×C₂²⋊Q₈	C₂²	semidirect
38. C₂×C₇×C₂²⋊Q₈	C₂²	semidirect
43. C₂×C₇×#121	C₂²	semidirect
45. C₇×#520	C₂²	semidirect
46. C₂×C₄×C₇×Q₈	C₂²	semidirect
51. C₂×C₄×C₇×D₄	C₂²	semidirect
54. C₂×C₇×#117	C₂²	semidirect
60. C₂²×C₇×C₄⋊C₄	C₂²	semidirect
61. C₂²×C₇×C₄⋊C₄	C₂²	semidirect
62. C₂×#3492	C₇	direct (N × H)
89. C₇×C₄⋊Q₈	C₂³	semidirect
90. C₇×#126	C₂³	semidirect
91. C₇×#125	C₂³	semidirect
92. C₇×#125	C₂³	semidirect
95. C₇×#123	C₂³	semidirect
100. C₇×C₂²⋊Q₈	C₂³	semidirect
101. C₇×C₂²⋊Q₈	C₂³	semidirect
106. C₇×#121	C₂³	semidirect
108. C₄×C₇×Q₈	C₂³	semidirect
113. C₄×C₇×D₄	C₂³	semidirect
116. C₇×#117	C₂³	semidirect
133. C₂×C₇×C₄⋊C₄	C₂³	semidirect
143. #3492	C₂×C₇	direct (N × H)
144. C₂×#554	C₂×C₇	semidirect
145. C₂×#553	C₂×C₇	semidirect
146. C₂×#552	C₂×C₇	semidirect
147. C₂×#547	C₂×C₇	semidirect
148. C₂×#544	C₂×C₇	semidirect
149. C₂×#539	C₂×C₇	semidirect
150. C₂×#537	C₂×C₇	semidirect
151. C₂×#525	C₂×C₇	semidirect
152. C₂×#525	C₂×C₇	semidirect
153. C₂×#520	C₂×C₇	semidirect
154. C₂³×C₄⋊C₄	C₂×C₇	semidirect
190. C₂²×C₄×C₇	#41	semidirect
232. #554	C₂²×C₇	semidirect
233. #553	C₂²×C₇	semidirect
234. #552	C₂²×C₇	semidirect
235. #547	C₂²×C₇	semidirect
236. #544	C₂²×C₇	semidirect
237. #539	C₂²×C₇	semidirect
238. #537	C₂²×C₇	semidirect
239. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
240. C₂×#126	C₂²×C₇	semidirect
241. C₂×#125	C₂²×C₇	semidirect
242. C₂×#125	C₂²×C₇	semidirect
243. #525	C₂²×C₇	semidirect
244. #525	C₂²×C₇	semidirect
247. C₂×#123	C₂²×C₇	semidirect
252. C₂×C₂²⋊Q₈	C₂²×C₇	semidirect
253. C₂×C₂²⋊Q₈	C₂²×C₇	semidirect
258. C₂×#121	C₂²×C₇	semidirect
260. #520	C₂²×C₇	semidirect
261. C₂×C₄×Q₈	C₂²×C₇	semidirect
266. C₂×C₄×D₄	C₂²×C₇	semidirect
269. C₂×#117	C₂²×C₇	semidirect
275. C₂²×C₄⋊C₄	C₂²×C₇	semidirect
276. C₂²×C₄⋊C₄	C₂²×C₇	semidirect
321. C₂×C₄×C₇	C₂×#41	semidirect
355. C₄⋊Q₈	C₂³×C₇	semidirect
356. #126	C₂³×C₇	semidirect
357. #125	C₂³×C₇	semidirect
358. #125	C₂³×C₇	semidirect
361. #123	C₂³×C₇	semidirect
366. C₂²⋊Q₈	C₂³×C₇	semidirect
367. C₂²⋊Q₈	C₂³×C₇	semidirect
372. #121	C₂³×C₇	semidirect
374. C₄×Q₈	C₂³×C₇	semidirect
379. C₄×D₄	C₂³×C₇	semidirect
382. #117	C₂³×C₇	semidirect
399. C₂×C₄⋊C₄	C₂³×C₇	semidirect
472. C₂²×C₄	C₇×#41	semidirect
514. C₂×C₇	#3492	direct (N × H)
562. C₂×C₄	C₂×C₇×#41	semidirect
570. C₇	C₂×#3492	direct (N × H)
603. C₂	C₇×#3492	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×#3492	2
C₂×C₇×#554	2
C₂×C₇×#553	2
C₂×C₇×#552	2
C₂×C₇×#547	2
C₂×C₇×#544	2
C₂×C₇×#539	2
C₂×C₇×#537	2
C₂×C₇×#525	2
C₂×C₇×#520	2
C₂³×C₇×C₄⋊C₄	2
C₂×#3492	7

Subgroups — C2×C7×#3492 #13239984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3492 #13239984