Subgroups — C₇×#34929 #13212145

Order 1792.

2308 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#533	C₂	1	2	yes	maximal, complemented
2	C₇×#2556	C₂	1	4	yes	maximal, complemented
3	C₇×#2551	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2546	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#2542	C₂	1	1	yes	maximal, characteristic
6	C₇×#2538	C₂	1	1	yes	maximal, characteristic
7	C₇×#2506	C₂	1	2	yes	maximal, complemented
8	C₇×#2492	C₂	1	2	yes	maximal, complemented
9	C₇×#2452	C₂	1	2	yes	maximal
10	C₇×#2429	C₂	1	2	yes	maximal, complemented
11	C₇×#2396	C₂	1	4	yes	maximal, complemented
12	C₄×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₇×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
14	C₄×C₇×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
15	C₇×#2293	C₂	1	2	yes	maximal
16	C₄²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
17	C₄²×C₇×D₄	C₂	1	1	yes	maximal, characteristic
18	C₇×#533	—	2	16	no	—
19	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
20	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
21	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
22	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₇×#125	C₂²	1	4	yes	—
24	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
26	C₂×C₇×C₂²⋊Q₈	C₂²	1	4	yes	—
27	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
28	C₂×C₄×C₇×Q₈	C₂²	1	4	yes	—
29	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
31	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
36	C₂×C₇×#117	C₂²	1	4	yes	—
37	C₂²×C₄²×C₇	C₂²	1	2	yes	—
38	C₇×#400	C₂²	1	4	yes	—
39	C₇×#399	C₂²	1	4	yes	—
40	C₇×#398	C₂²	1	4	yes	—
41	C₇×#398	C₂²	1	4	yes	—
42	C₇×#396	C₂²	1	4	yes	—
43	C₇×#396	C₂²	1	4	yes	—
44	C₇×#395	C₂²	1	4	yes	—
45	C₇×#393	C₂²	1	4	yes	—
46	C₇×#391	C₂²	1	2	yes	—
47	C₇×#391	C₂²	1	2	yes	—
48	C₇×#391	C₂²	1	4	yes	—
49	C₇×#390	C₂²	1	2	yes	—
50	C₇×#390	C₂²	1	2	yes	—
51	C₇×#389	C₂²	1	2	yes	—
52	C₇×#389	C₂²	1	4	yes	—
53	C₇×#389	C₂²	1	2	yes	—
54	C₇×#389	C₂²	1	2	yes	—
55	C₇×#389	C₂²	1	2	yes	—
56	C₇×#389	C₂²	1	2	yes	—
57	C₇×#389	C₂²	1	2	yes	—
58	C₇×#389	C₂²	1	2	yes	—
59	C₇×#388	C₂²	1	4	yes	—
60	C₇×#388	C₂²	1	4	yes	—
61	C₇×#387	C₂²	1	4	yes	—
62	C₇×#385	C₂²	1	4	yes	—
63	C₇×#384	C₂²	1	2	yes	—
64	C₇×#384	C₂²	1	1	yes	characteristic
65	C₇×#380	C₂²	1	2	yes	—
66	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
67	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
68	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
69	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
70	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
71	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
72	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
73	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
74	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
75	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
76	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
77	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
78	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
79	C₄×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
80	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
81	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
82	C₄³×C₇	C₂²	1	1	yes	characteristic
83	C₄³×C₇	C₂²	1	2	yes	—
84	#34929	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₇×Q₈	C₂³	1	2	yes	—
86	C₂²×C₇×Q₈	C₂³	1	4	yes	—
87	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
88	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
89	C₂³×C₄×C₇	C₂³	1	2	yes	—
90	C₂³×C₄×C₇	C₂³	1	2	yes	—
91	C₂³×C₄×C₇	C₂³	1	2	yes	—
92	C₇×C₄⋊Q₈	—	2	8	no	—
93	C₇×C₄⋊Q₈	—	2	8	no	—
94	C₇×C₄⋊Q₈	—	2	8	no	—
95	C₇×C₄⋊Q₈	—	2	4	no	—
96	C₇×#125	—	2	16	no	—
97	C₇×C₂²⋊Q₈	—	2	8	no	—
98	C₇×C₂²⋊Q₈	—	2	8	no	—
99	C₇×C₂²⋊Q₈	—	2	16	no	—
100	C₇×C₂²⋊Q₈	—	2	8	no	—
101	C₄×C₇×Q₈	—	2	4	no	—
102	C₄×C₇×Q₈	—	2	16	no	—
103	C₄×C₇×Q₈	—	2	8	no	—
104	C₄×C₇×Q₈	—	2	4	no	—
105	C₄×C₇×Q₈	D₄	1	8	yes	complemented
106	C₄×C₇×D₄	—	2	4	no	—
107	C₄×C₇×D₄	—	2	4	no	—
108	C₄×C₇×D₄	Q₈	1	8	yes	complemented
109	C₇×#117	—	2	16	no	—
110	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
111	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
112	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
113	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
114	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
115	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
116	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
117	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
118	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
119	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
120	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
121	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
122	C₂×C₇×C₄⋊C₄	C₂³	1	4	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	2	yes	—
127	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
128	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
129	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
130	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
131	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
132	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
133	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
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	4	yes	—
137	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
138	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
139	C₂×C₄²×C₇	—	2	8	no	—
140	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
141	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
142	C₂×C₄²×C₇	C₂³	1	2	yes	—
143	C₂×C₄²×C₇	C₂³	1	2	yes	—
144	C₂×C₄²×C₇	C₂³	1	2	yes	—
145	C₂×C₄²×C₇	C₂³	1	2	yes	—
146	C₂×C₄²×C₇	C₂³	1	2	yes	—
147	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
148	C₂×C₄²×C₇	C₂³	1	2	yes	—
149	C₂×C₄²×C₇	C₂³	1	2	yes	—
150	C₂×C₄²×C₇	C₂³	1	4	yes	—
151	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
152	C₂×C₄²×C₇	C₂³	1	2	yes	—
153	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
154	C₂×C₄²×C₇	C₂³	1	2	yes	—
155	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
156	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
157	C₇×#95	C₂³	1	2	yes	—
158	C₇×#95	C₂³	1	2	yes	—
159	C₇×#95	C₂³	1	2	yes	—
160	C₇×#95	C₂³	1	4	yes	—
161	C₇×#95	C₂³	1	4	yes	—
162	C₇×#95	C₂³	1	2	yes	—
163	C₇×#95	C₂³	1	2	yes	—
164	C₇×#95	C₂³	1	2	yes	—
165	C₇×#95	C₂³	1	4	yes	—
166	C₇×#95	C₂³	1	4	yes	—
167	C₇×#95	C₂³	1	4	yes	—
168	C₇×#95	C₂³	1	4	yes	—
169	C₂×#533	C₂×C₇	1	2	yes	complemented
170	#2556	C₂×C₇	1	4	yes	complemented
171	#2551	C₂×C₇	1	1	yes	characteristic, complemented
172	#2546	C₂×C₇	1	1	yes	characteristic, complemented
173	#2542	C₂×C₇	1	1	yes	characteristic
174	#2538	C₂×C₇	1	1	yes	characteristic
175	#2506	C₂×C₇	1	2	yes	complemented
176	#2492	C₂×C₇	1	2	yes	complemented
177	#2452	C₂×C₇	1	2	yes	—
178	#2429	C₂×C₇	1	2	yes	complemented
179	#2396	C₂×C₇	1	4	yes	complemented
180	C₄×C₄⋊Q₈	C₂×C₇	1	2	yes	complemented
181	C₄×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
182	C₄×C₂²⋊Q₈	C₂×C₇	1	2	yes	complemented
183	#2293	C₂×C₇	1	2	yes	—
184	C₄²×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
185	C₄²×D₄	C₂×C₇	1	1	yes	characteristic
186	C₂⁴×C₇	C₂⁴	1	2	yes	—
187	C₂×C₇×Q₈	—	2	8	no	—
188	C₂×C₇×Q₈	—	2	4	no	—
189	C₂×C₇×Q₈	—	2	2	no	—
190	C₂×C₇×Q₈	—	2	2	no	—
191	C₂×C₇×Q₈	—	2	4	no	—
192	C₂×C₇×Q₈	—	2	8	no	—
193	C₂×C₇×Q₈	—	2	4	no	—
194	C₂×C₇×Q₈	—	4	4	no	—
195	C₂×C₇×Q₈	C₂×D₄	1	4	yes	—
196	C₂×C₇×D₄	—	2	2	no	—
197	C₂×C₇×D₄	—	2	2	no	—
198	C₂×C₇×D₄	C₂×Q₈	1	4	yes	—
199	C₂²×C₄×C₇	—	2	4	no	—
200	C₂²×C₄×C₇	—	2	4	no	—
201	C₂²×C₄×C₇	—	2	4	no	—
202	C₂²×C₄×C₇	—	2	4	no	—
203	C₂²×C₄×C₇	—	2	4	no	—
204	C₂²×C₄×C₇	—	2	4	no	—
205	C₂²×C₄×C₇	—	2	4	no	—
206	C₂²×C₄×C₇	—	4	2	no	—
207	C₂²×C₄×C₇	C₂×Q₈	1	8	yes	—
208	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
209	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
210	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
211	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
212	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
213	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
214	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
215	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
216	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
217	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
218	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
219	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
220	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
221	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
222	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
223	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
224	C₇×C₄⋊C₄	—	2	8	no	—
225	C₇×C₄⋊C₄	—	2	8	no	—
226	C₇×C₄⋊C₄	—	2	8	no	—
227	C₇×C₄⋊C₄	—	2	8	no	—
228	C₇×C₄⋊C₄	—	2	8	no	—
229	C₇×C₄⋊C₄	—	2	8	no	—
230	C₇×C₄⋊C₄	—	2	4	no	—
231	C₇×C₄⋊C₄	—	2	2	no	—
232	C₇×C₄⋊C₄	—	2	4	no	—
233	C₇×C₄⋊C₄	—	2	4	no	—
234	C₇×C₄⋊C₄	—	2	2	no	—
235	C₇×C₄⋊C₄	—	2	8	no	—
236	C₇×C₄⋊C₄	—	2	8	no	—
237	C₇×C₄⋊C₄	—	2	4	no	—
238	C₇×C₄⋊C₄	—	2	4	no	—
239	C₇×C₄⋊C₄	—	2	4	no	—
240	C₇×C₄⋊C₄	—	4	2	no	—
241	C₇×C₄⋊C₄	C₂×D₄	1	8	yes	—
242	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
243	C₇×C₄⋊C₄	C₂×Q₈	1	4	yes	—
244	C₇×C₄⋊C₄	#41	1	4	yes	—
245	C₇×C₄⋊C₄	#41	1	4	yes	—
246	C₇×C₂²⋊C₄	—	2	8	no	—
247	C₇×C₂²⋊C₄	—	2	8	no	—
248	C₇×C₂²⋊C₄	—	2	4	no	—
249	C₇×C₂²⋊C₄	—	2	4	no	—
250	C₇×C₂²⋊C₄	—	2	8	no	—
251	C₇×C₂²⋊C₄	—	4	4	no	—
252	C₇×C₂²⋊C₄	C₂×Q₈	1	8	yes	—
253	C₄²×C₇	—	2	4	no	—
254	C₄²×C₇	—	2	4	no	—
255	C₄²×C₇	—	2	4	no	—
256	C₄²×C₇	—	2	8	no	—
257	C₄²×C₇	—	2	2	no	—
258	C₄²×C₇	—	2	4	no	—
259	C₄²×C₇	—	2	2	no	—
260	C₄²×C₇	—	2	4	no	—
261	C₄²×C₇	—	2	4	no	—
262	C₄²×C₇	—	2	2	no	—
263	C₄²×C₇	—	2	4	no	—
264	C₄²×C₇	—	4	1	no	—
265	C₄²×C₇	—	4	2	no	—
266	C₄²×C₇	C₂×D₄	1	4	yes	—
267	C₄²×C₇	C₂×D₄	1	8	yes	—
268	C₄²×C₇	C₂×Q₈	1	4	yes	—
269	C₄²×C₇	#41	1	4	yes	—
270	#533	—	2	16	no	—
271	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
272	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
273	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
274	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
275	C₂×#125	C₂²×C₇	1	4	yes	—
276	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
277	C₂×C₂²⋊Q₈	C₂²×C₇	1	4	yes	—
278	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
279	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
280	C₂×C₄×Q₈	C₂²×C₇	1	4	yes	—
281	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
282	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
283	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
284	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
285	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
286	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
287	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
288	C₂×#117	C₂²×C₇	1	4	yes	—
289	C₂²×C₄²	C₂²×C₇	1	2	yes	—
290	#400	C₂²×C₇	1	4	yes	—
291	#399	C₂²×C₇	1	4	yes	—
292	#398	C₂²×C₇	1	4	yes	—
293	#398	C₂²×C₇	1	4	yes	—
294	#396	C₂²×C₇	1	4	yes	—
295	#396	C₂²×C₇	1	4	yes	—
296	#395	C₂²×C₇	1	4	yes	—
297	#393	C₂²×C₇	1	4	yes	—
298	#391	C₂²×C₇	1	2	yes	—
299	#391	C₂²×C₇	1	4	yes	—
300	#391	C₂²×C₇	1	2	yes	—
301	#390	C₂²×C₇	1	2	yes	—
302	#390	C₂²×C₇	1	2	yes	—
303	#389	C₂²×C₇	1	4	yes	—
304	#389	C₂²×C₇	1	2	yes	—
305	#389	C₂²×C₇	1	2	yes	—
306	#389	C₂²×C₇	1	2	yes	—
307	#389	C₂²×C₇	1	2	yes	—
308	#389	C₂²×C₇	1	2	yes	—
309	#389	C₂²×C₇	1	2	yes	—
310	#389	C₂²×C₇	1	2	yes	—
311	#388	C₂²×C₇	1	4	yes	—
312	#388	C₂²×C₇	1	4	yes	—
313	#387	C₂²×C₇	1	4	yes	—
314	#385	C₂²×C₇	1	4	yes	—
315	#384	C₂²×C₇	1	2	yes	—
316	#384	C₂²×C₇	1	1	yes	characteristic
317	#380	C₂²×C₇	1	2	yes	—
318	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
319	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
320	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
321	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
322	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
323	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
324	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
325	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
326	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
327	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
328	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
329	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
330	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
331	C₄×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
332	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
333	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
334	C₄³	C₂²×C₇	1	1	yes	characteristic
335	C₄³	C₂²×C₇	1	2	yes	—
336	C₂³×C₇	—	2	2	no	—
337	C₂³×C₇	—	2	2	no	—
338	C₂³×C₇	—	2	2	no	—
339	C₂³×C₇	—	2	2	no	—
340	C₂³×C₇	—	2	2	no	—
341	C₂³×C₇	—	2	2	no	—
342	C₂³×C₇	C₂²×Q₈	1	4	yes	—
343	C₂³×C₇	C₂⁵	1	1	yes	characteristic
344	C₇×Q₈	—	2	8	no	—
345	C₇×Q₈	—	4	16	no	—
346	C₇×Q₈	—	4	8	no	—
347	C₇×D₄	—	2	8	no	—
348	C₂×C₄×C₇	—	2	2	no	—
349	C₂×C₄×C₇	—	2	2	no	—
350	C₂×C₄×C₇	—	2	1	no	—
351	C₂×C₄×C₇	—	2	2	no	—
352	C₂×C₄×C₇	—	2	2	no	—
353	C₂×C₄×C₇	—	2	2	no	—
354	C₂×C₄×C₇	—	2	1	no	—
355	C₂×C₄×C₇	—	2	2	no	—
356	C₂×C₄×C₇	—	2	1	no	—
357	C₂×C₄×C₇	—	2	2	no	—
358	C₂×C₄×C₇	—	2	2	no	—
359	C₂×C₄×C₇	—	2	1	no	—
360	C₂×C₄×C₇	—	2	1	no	—
361	C₂×C₄×C₇	—	2	1	no	—
362	C₂×C₄×C₇	—	2	2	no	—
363	C₂×C₄×C₇	—	2	2	no	—
364	C₂×C₄×C₇	—	2	16	no	—
365	C₂×C₄×C₇	—	2	2	no	—
366	C₂×C₄×C₇	—	2	4	no	—
367	C₂×C₄×C₇	—	2	4	no	—
368	C₂×C₄×C₇	—	2	1	no	—
369	C₂×C₄×C₇	—	2	2	no	—
370	C₂×C₄×C₇	—	2	1	no	—
371	C₂×C₄×C₇	—	2	2	no	—
372	C₂×C₄×C₇	—	2	4	no	—
373	C₂×C₄×C₇	—	2	2	no	—
374	C₂×C₄×C₇	—	2	4	no	—
375	C₂×C₄×C₇	—	2	4	no	—
376	C₂×C₄×C₇	—	2	2	no	—
377	C₂×C₄×C₇	—	2	1	no	—
378	C₂×C₄×C₇	—	2	2	no	—
379	C₂×C₄×C₇	—	2	4	no	—
380	C₂×C₄×C₇	—	4	8	no	—
381	C₂×C₄×C₇	—	4	8	no	—
382	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
383	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
384	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
385	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
386	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
387	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
388	C₂×C₄×C₇	C₂²×Q₈	1	4	yes	—
389	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
390	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
391	C₂×C₄×C₇	C₂×#41	1	4	yes	—
392	C₂×C₄×C₇	C₂×#41	1	2	yes	—
393	C₂×C₄×C₇	C₂×#41	1	4	yes	—
394	C₂×C₄×C₇	C₂×#41	1	2	yes	—
395	C₂×C₄×C₇	C₂×#41	1	2	yes	—
396	C₂×C₄×C₇	C₂×#41	1	2	yes	—
397	C₂×C₄×C₇	C₂×#41	1	2	yes	—
398	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
399	C₂²×Q₈	C₂³×C₇	1	2	yes	—
400	C₂²×Q₈	C₂³×C₇	1	4	yes	—
401	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
402	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
403	C₂³×C₄	C₂³×C₇	1	2	yes	—
404	C₂³×C₄	C₂³×C₇	1	2	yes	—
405	C₂³×C₄	C₂³×C₇	1	2	yes	—
406	C₄⋊Q₈	—	2	8	no	—
407	C₄⋊Q₈	—	2	8	no	—
408	C₄⋊Q₈	—	2	8	no	—
409	C₄⋊Q₈	—	2	4	no	—
410	#125	—	2	16	no	—
411	C₂²⋊Q₈	—	2	8	no	—
412	C₂²⋊Q₈	—	2	8	no	—
413	C₂²⋊Q₈	—	2	16	no	—
414	C₂²⋊Q₈	—	2	8	no	—
415	C₄×Q₈	—	2	4	no	—
416	C₄×Q₈	—	2	16	no	—
417	C₄×Q₈	—	2	8	no	—
418	C₄×Q₈	—	2	4	no	—
419	C₄×Q₈	C₇×D₄	1	8	yes	complemented
420	C₄×D₄	—	2	4	no	—
421	C₄×D₄	—	2	4	no	—
422	C₄×D₄	C₇×Q₈	1	8	yes	complemented
423	#117	—	2	16	no	—
424	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
425	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
426	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
427	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
428	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
429	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
430	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
431	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
432	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
433	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
434	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
435	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
436	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
437	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
438	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
439	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
440	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
441	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
442	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
443	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
444	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
445	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
446	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
447	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
448	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
449	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
450	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
451	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
452	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
453	C₂×C₄²	—	2	8	no	—
454	C₂×C₄²	C₂³×C₇	1	4	yes	—
455	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
456	C₂×C₄²	C₂³×C₇	1	2	yes	—
457	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
458	C₂×C₄²	C₂³×C₇	1	2	yes	—
459	C₂×C₄²	C₂³×C₇	1	2	yes	—
460	C₂×C₄²	C₂³×C₇	1	2	yes	—
461	C₂×C₄²	C₂³×C₇	1	2	yes	—
462	C₂×C₄²	C₂³×C₇	1	2	yes	—
463	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
464	C₂×C₄²	C₂³×C₇	1	2	yes	—
465	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
466	C₂×C₄²	C₂³×C₇	1	2	yes	—
467	C₂×C₄²	C₂³×C₇	1	2	yes	—
468	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
469	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
470	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
471	#95	C₂³×C₇	1	2	yes	—
472	#95	C₂³×C₇	1	2	yes	—
473	#95	C₂³×C₇	1	2	yes	—
474	#95	C₂³×C₇	1	4	yes	—
475	#95	C₂³×C₇	1	4	yes	—
476	#95	C₂³×C₇	1	2	yes	—
477	#95	C₂³×C₇	1	2	yes	—
478	#95	C₂³×C₇	1	2	yes	—
479	#95	C₂³×C₇	1	4	yes	—
480	#95	C₂³×C₇	1	4	yes	—
481	#95	C₂³×C₇	1	4	yes	—
482	#95	C₂³×C₇	1	4	yes	—
483	C₂²×C₇	—	2	4	no	—
484	C₂²×C₇	—	2	4	no	—
485	C₂²×C₇	—	2	4	no	—
486	C₂²×C₇	—	4	2	no	—
487	C₂²×C₇	—	4	2	no	—
488	C₂²×C₇	—	4	2	no	—
489	C₂²×C₇	—	4	2	no	—
490	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
491	C₂²×C₇	C₂³×Q₈	1	1	yes	characteristic
492	C₂²×C₇	C₂²×#41	1	1	yes	characteristic
493	C₂²×C₇	C₂²×#41	1	1	yes	characteristic
494	C₂²×C₇	C₂²×#41	1	1	yes	characteristic
495	C₂²×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
496	C₂²×C₇	#585	1	1	yes	characteristic
497	C₄×C₇	—	2	4	no	—
498	C₄×C₇	—	2	2	no	—
499	C₄×C₇	—	2	4	no	—
500	C₄×C₇	—	2	2	no	—
501	C₄×C₇	—	2	4	no	—
502	C₄×C₇	—	2	4	no	—
503	C₄×C₇	—	2	2	no	—
504	C₄×C₇	—	2	4	no	—
505	C₄×C₇	—	4	2	no	—
506	C₄×C₇	—	4	4	no	—
507	C₄×C₇	—	4	2	no	—
508	C₄×C₇	—	4	4	no	—
509	C₄×C₇	—	4	1	no	—
510	C₄×C₇	#532	1	4	yes	—
511	C₄×C₇	#533	1	4	yes	complemented
512	C₄×C₇	D₄×Q₈	1	4	yes	—
513	C₂⁴	C₂⁴×C₇	1	2	yes	—
514	C₂×Q₈	—	2	8	no	—
515	C₂×Q₈	—	2	4	no	—
516	C₂×Q₈	—	2	2	no	—
517	C₂×Q₈	—	2	2	no	—
518	C₂×Q₈	—	2	4	no	—
519	C₂×Q₈	—	2	8	no	—
520	C₂×Q₈	—	2	4	no	—
521	C₂×Q₈	—	4	4	no	—
522	C₂×Q₈	C₂×C₇×D₄	1	4	yes	—
523	C₂×D₄	—	2	2	no	—
524	C₂×D₄	—	2	2	no	—
525	C₂×D₄	C₂×C₇×Q₈	1	4	yes	—
526	C₂²×C₄	—	2	4	no	—
527	C₂²×C₄	—	2	4	no	—
528	C₂²×C₄	—	2	4	no	—
529	C₂²×C₄	—	2	4	no	—
530	C₂²×C₄	—	2	4	no	—
531	C₂²×C₄	—	2	4	no	—
532	C₂²×C₄	—	2	4	no	—
533	C₂²×C₄	—	4	2	no	—
534	C₂²×C₄	C₂×C₇×Q₈	1	8	yes	—
535	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
536	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
537	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
538	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
539	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
540	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
541	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
542	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
543	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
544	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
545	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
546	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
547	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
548	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
549	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
550	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
551	C₄⋊C₄	—	2	8	no	—
552	C₄⋊C₄	—	2	8	no	—
553	C₄⋊C₄	—	2	8	no	—
554	C₄⋊C₄	—	2	8	no	—
555	C₄⋊C₄	—	2	8	no	—
556	C₄⋊C₄	—	2	8	no	—
557	C₄⋊C₄	—	2	4	no	—
558	C₄⋊C₄	—	2	2	no	—
559	C₄⋊C₄	—	2	4	no	—
560	C₄⋊C₄	—	2	4	no	—
561	C₄⋊C₄	—	2	2	no	—
562	C₄⋊C₄	—	2	8	no	—
563	C₄⋊C₄	—	2	8	no	—
564	C₄⋊C₄	—	2	4	no	—
565	C₄⋊C₄	—	2	4	no	—
566	C₄⋊C₄	—	2	4	no	—
567	C₄⋊C₄	—	4	2	no	—
568	C₄⋊C₄	C₂×C₇×D₄	1	8	yes	—
569	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
570	C₄⋊C₄	C₂×C₇×Q₈	1	4	yes	—
571	C₄⋊C₄	C₇×#41	1	4	yes	—
572	C₄⋊C₄	C₇×#41	1	4	yes	—
573	C₂²⋊C₄	—	2	8	no	—
574	C₂²⋊C₄	—	2	8	no	—
575	C₂²⋊C₄	—	2	4	no	—
576	C₂²⋊C₄	—	2	4	no	—
577	C₂²⋊C₄	—	2	8	no	—
578	C₂²⋊C₄	—	4	4	no	—
579	C₂²⋊C₄	C₂×C₇×Q₈	1	8	yes	—
580	C₄²	—	2	4	no	—
581	C₄²	—	2	4	no	—
582	C₄²	—	2	4	no	—
583	C₄²	—	2	8	no	—
584	C₄²	—	2	2	no	—
585	C₄²	—	2	4	no	—
586	C₄²	—	2	2	no	—
587	C₄²	—	2	4	no	—
588	C₄²	—	2	4	no	—
589	C₄²	—	2	2	no	—
590	C₄²	—	2	4	no	—
591	C₄²	—	4	1	no	—
592	C₄²	—	4	2	no	—
593	C₄²	C₂×C₇×D₄	1	4	yes	—
594	C₄²	C₂×C₇×D₄	1	8	yes	—
595	C₄²	C₂×C₇×Q₈	1	4	yes	—
596	C₄²	C₇×#41	1	4	yes	—
597	C₂×C₇	—	4	4	no	—
598	C₂×C₇	C₂×#532	1	1	yes	characteristic
599	C₂×C₇	C₂×#533	1	1	yes	characteristic
600	C₂×C₇	#3461	1	1	yes	characteristic
601	C₂×C₇	C₂×D₄×Q₈	1	1	yes	characteristic
602	C₂×C₇	D₄×#41	1	1	yes	characteristic
603	C₂×C₇	Q₈×#41	1	1	yes	characteristic
604	C₂×C₇	#3482	1	1	yes	characteristic
605	C₂³	—	2	2	no	—
606	C₂³	—	2	2	no	—
607	C₂³	—	2	2	no	—
608	C₂³	—	2	2	no	—
609	C₂³	—	2	2	no	—
610	C₂³	—	2	2	no	—
611	C₂³	C₂²×C₇×Q₈	1	4	yes	—
612	C₂³	C₂⁵×C₇	1	1	yes	characteristic
613	Q₈	—	2	8	no	—
614	Q₈	—	4	16	no	—
615	Q₈	—	4	8	no	—
616	D₄	—	2	8	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	1	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	1	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	2	1	no	—
631	C₂×C₄	—	2	2	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	16	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	4	no	—
636	C₂×C₄	—	2	4	no	—
637	C₂×C₄	—	2	1	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	1	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	4	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	4	no	—
644	C₂×C₄	—	2	4	no	—
645	C₂×C₄	—	2	2	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	2	no	—
648	C₂×C₄	—	2	4	no	—
649	C₂×C₄	—	4	8	no	—
650	C₂×C₄	—	4	8	no	—
651	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
652	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
653	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
654	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
655	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
656	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
657	C₂×C₄	C₂²×C₇×Q₈	1	4	yes	—
658	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
659	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
660	C₂×C₄	C₂×C₇×#41	1	4	yes	—
661	C₂×C₄	C₂×C₇×#41	1	2	yes	—
662	C₂×C₄	C₂×C₇×#41	1	4	yes	—
663	C₂×C₄	C₂×C₇×#41	1	2	yes	—
664	C₂×C₄	C₂×C₇×#41	1	2	yes	—
665	C₂×C₄	C₂×C₇×#41	1	2	yes	—
666	C₂×C₄	C₂×C₇×#41	1	2	yes	—
667	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
668	C₇	#34929	1	1	yes	characteristic, direct product factor, Sylow
669	C₂²	—	2	4	no	—
670	C₂²	—	2	4	no	—
671	C₂²	—	2	4	no	—
672	C₂²	—	4	2	no	—
673	C₂²	—	4	2	no	—
674	C₂²	—	4	2	no	—
675	C₂²	—	4	2	no	—
676	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
677	C₂²	C₂³×C₇×Q₈	1	1	yes	characteristic
678	C₂²	C₂²×C₇×#41	1	1	yes	characteristic
679	C₂²	C₂²×C₇×#41	1	1	yes	characteristic
680	C₂²	C₂²×C₇×#41	1	1	yes	characteristic
681	C₂²	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
682	C₂²	C₇×#585	1	1	yes	characteristic
683	C₄	—	2	4	no	—
684	C₄	—	2	2	no	—
685	C₄	—	2	4	no	—
686	C₄	—	2	2	no	—
687	C₄	—	2	4	no	—
688	C₄	—	2	4	no	—
689	C₄	—	2	2	no	—
690	C₄	—	2	4	no	—
691	C₄	—	4	2	no	—
692	C₄	—	4	4	no	—
693	C₄	—	4	2	no	—
694	C₄	—	4	4	no	—
695	C₄	—	4	1	no	—
696	C₄	C₇×#532	1	4	yes	—
697	C₄	C₇×#533	1	4	yes	complemented
698	C₄	C₇×D₄×Q₈	1	4	yes	—
699	C₂	—	4	4	no	—
700	C₂	C₂×C₇×#532	1	1	yes	characteristic
701	C₂	C₂×C₇×#533	1	1	yes	characteristic
702	C₂	C₇×#3461	1	1	yes	characteristic
703	C₂	C₂×C₇×D₄×Q₈	1	1	yes	characteristic
704	C₂	C₇×D₄×#41	1	1	yes	characteristic
705	C₂	C₇×Q₈×#41	1	1	yes	characteristic
706	C₂	C₇×#3482	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
socle	C₂³×C₇	C₂⁵
prime radical	C₂²×C₇×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#533	C₂	semidirect
2. C₇×#2556	C₂	semidirect
3. C₇×#2551	C₂	semidirect
4. C₇×#2546	C₂	semidirect
7. C₇×#2506	C₂	semidirect
8. C₇×#2492	C₂	semidirect
10. C₇×#2429	C₂	semidirect
11. C₇×#2396	C₂	semidirect
12. C₄×C₇×C₄⋊Q₈	C₂	semidirect
13. C₄×C₇×C₄⋊Q₈	C₂	semidirect
14. C₄×C₇×C₂²⋊Q₈	C₂	semidirect
16. C₄²×C₇×Q₈	C₂	semidirect
84. #34929	C₇	direct (N × H)
105. C₄×C₇×Q₈	D₄	semidirect
108. C₄×C₇×D₄	Q₈	semidirect
169. C₂×#533	C₂×C₇	semidirect
170. #2556	C₂×C₇	semidirect
171. #2551	C₂×C₇	semidirect
172. #2546	C₂×C₇	semidirect
175. #2506	C₂×C₇	semidirect
176. #2492	C₂×C₇	semidirect
178. #2429	C₂×C₇	semidirect
179. #2396	C₂×C₇	semidirect
180. C₄×C₄⋊Q₈	C₂×C₇	semidirect
181. C₄×C₄⋊Q₈	C₂×C₇	semidirect
182. C₄×C₂²⋊Q₈	C₂×C₇	semidirect
184. C₄²×Q₈	C₂×C₇	semidirect
419. C₄×Q₈	C₇×D₄	semidirect
422. C₄×D₄	C₇×Q₈	semidirect
511. C₄×C₇	#533	semidirect
668. C₇	#34929	direct (N × H)
697. C₄	C₇×#533	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#533	2
C₇×#2556	2
C₇×#2551	2
C₇×#2546	2
C₇×#2542	2
C₇×#2538	2
C₇×#2506	2
C₇×#2492	2
C₇×#2452	2
C₇×#2429	2
C₇×#2396	2
C₄×C₇×C₄⋊Q₈	2
C₄×C₇×C₂²⋊Q₈	2
C₇×#2293	2
C₄²×C₇×Q₈	2
C₄²×C₇×D₄	2
#34929	7

Subgroups — C7×#34929 #13212145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#34929 #13212145