Subgroups — C₇×#61373 #13238589

Order 1792.

1778 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3503	C₂	1	2	yes	maximal, complemented
2	C₇×#3386	C₂	1	4	yes	maximal, complemented
3	C₇×#3385	C₂	1	4	yes	maximal, complemented
4	C₇×#3383	C₂	1	2	yes	maximal, complemented
5	C₇×Q₈×Q₁₆	C₂	1	2	yes	maximal, complemented
6	C₇×#3380	C₂	1	2	yes	maximal, complemented
7	C₇×Q₈×D₈	C₂	1	2	yes	maximal, complemented
8	C₇×#3164	C₂	1	2	yes	maximal, complemented
9	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3083	C₂	1	4	yes	maximal, complemented
11	C₇×#3079	C₂	1	2	yes	maximal, complemented
12	C₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#2951	C₂	1	2	yes	maximal
14	C₄×C₇×#135	C₂	1	1	yes	maximal, characteristic
15	C₂×C₇×#135	C₂²	1	1	yes	characteristic
16	C₇×Q₈²	C₂²	1	4	yes	—
17	C₇×#557	C₂²	1	4	yes	—
18	C₇×#554	—	2	4	no	—
19	C₇×#554	C₂²	1	4	yes	—
20	C₇×D₄×Q₈	—	2	4	no	—
21	C₇×D₄×Q₈	C₂²	1	4	yes	—
22	C₇×#544	—	2	4	no	—
23	C₇×#544	C₂²	1	4	yes	—
24	C₇×#533	—	2	2	no	—
25	C₇×#533	C₂²	1	1	yes	characteristic
26	C₂×C₇×C₄⋊Q₈	—	2	2	no	—
27	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
28	C₇×#520	C₂²	1	4	yes	—
29	C₄×C₇×#41	C₂²	1	2	yes	—
30	C₇×#501	C₂²	1	8	yes	—
31	C₇×#500	C₂²	1	2	yes	—
32	C₇×#500	C₂²	1	2	yes	—
33	C₇×#500	C₂²	1	2	yes	—
34	C₇×#500	C₂²	1	2	yes	—
35	C₇×#500	C₂²	1	4	yes	—
36	C₇×#499	C₂²	1	4	yes	—
37	C₇×#479	C₂²	1	8	yes	—
38	C₇×#478	C₂²	1	8	yes	—
39	C₇×#477	C₂²	1	4	yes	—
40	C₇×#477	C₂²	1	8	yes	—
41	C₇×#476	C₂²	1	4	yes	—
42	C₇×#475	C₂²	1	4	yes	—
43	C₇×#474	C₂²	1	4	yes	—
44	C₇×#474	C₂²	1	8	yes	—
45	C₇×#446	C₂²	1	2	yes	—
46	C₇×#446	C₂²	1	2	yes	—
47	C₇×C₈×Q₈	C₂²	1	2	yes	—
48	C₇×C₈×Q₈	C₂²	1	2	yes	—
49	C₇×#440	C₂²	1	8	yes	—
50	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
51	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
52	C₄×C₇×Q₁₆	C₂²	1	4	yes	—
53	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
54	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
55	C₄×C₇×D₈	C₂²	1	1	yes	characteristic
56	C₄×C₇×D₈	C₂²	1	1	yes	characteristic
57	C₄×C₇×D₈	C₂²	1	4	yes	—
58	C₇×#428	C₂²	1	4	yes	—
59	C₇×#427	C₂²	1	1	yes	characteristic
60	C₂×C₇×#107	C₂²	1	2	yes	—
61	C₂×C₇×#107	C₂²	1	2	yes	—
62	C₇×#424	C₂²	1	2	yes	—
63	C₇×#423	C₂²	1	1	yes	characteristic
64	C₇×#417	C₂²	1	4	yes	—
65	C₇×#416	C₂²	1	2	yes	—
66	C₇×#405	C₂²	1	2	yes	—
67	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
68	#61373	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×C₇×#41	C₂³	1	2	yes	—
70	C₂²×C₇×Q₈	—	2	4	no	—
71	C₂²×C₇×Q₈	—	2	2	no	—
72	C₇×#135	Q₈	1	8	yes	complemented
73	C₂×C₇×Q₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
75	C₂×C₇×D₈	C₂³	1	1	yes	characteristic
76	C₂×C₇×M₁₆	C₂³	1	2	yes	—
77	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
78	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
79	C₇×C₄⋊Q₈	—	2	4	no	—
80	C₇×C₄⋊Q₈	—	2	4	no	—
81	C₇×C₄⋊Q₈	—	2	4	no	—
82	C₇×C₄⋊Q₈	—	2	8	no	—
83	C₇×C₄⋊Q₈	—	2	2	no	—
84	C₇×C₄⋊Q₈	—	2	2	no	—
85	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
86	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
87	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
88	C₇×C₄⋊Q₈	C₂³	1	8	yes	—
89	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
90	C₇×#125	—	2	4	no	—
91	C₇×#125	—	2	8	no	—
92	C₇×#125	D₄	1	2	yes	—
93	C₇×#125	C₂³	1	8	yes	—
94	C₇×C₂²⋊Q₈	—	2	4	no	—
95	C₇×C₂²⋊Q₈	—	2	4	no	—
96	C₇×C₂²⋊Q₈	—	2	8	no	—
97	C₇×C₂²⋊Q₈	—	2	8	no	—
98	C₇×C₂²⋊Q₈	—	2	8	no	—
99	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
100	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
101	C₄×C₇×Q₈	—	2	4	no	—
102	C₄×C₇×Q₈	—	2	4	no	—
103	C₄×C₇×Q₈	C₂³	1	2	yes	—
104	C₄×C₇×Q₈	C₂³	1	2	yes	—
105	C₄×C₇×Q₈	C₂³	1	8	yes	—
106	C₄×C₇×Q₈	C₂³	1	2	yes	—
107	C₄×C₇×Q₈	C₂³	1	2	yes	—
108	C₄×C₇×D₄	—	2	2	no	—
109	C₄×C₇×D₄	—	2	2	no	—
110	C₄×C₇×D₄	—	2	8	no	—
111	C₄×C₇×D₄	C₂³	1	2	yes	—
112	C₄×C₇×D₄	C₂³	1	8	yes	—
113	C₄×C₇×D₄	C₂³	1	2	yes	—
114	C₇×#117	—	2	4	no	—
115	C₇×#117	—	2	2	no	—
116	C₇×#117	C₂³	1	2	yes	—
117	C₇×#117	C₂³	1	2	yes	—
118	C₂×C₇×C₄⋊C₄	—	2	2	no	—
119	C₂×C₇×C₄⋊C₄	—	2	4	no	—
120	C₂×C₇×C₄⋊C₄	—	2	4	no	—
121	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
122	C₂×C₄²×C₇	—	2	2	no	—
123	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
124	C₇×#107	C₂³	1	8	yes	—
125	C₇×#107	C₂³	1	4	yes	—
126	C₇×#107	C₂³	1	4	yes	—
127	C₇×#107	C₂³	1	1	yes	characteristic
128	C₇×#107	C₂³	1	4	yes	—
129	C₇×#107	C₂³	1	1	yes	characteristic
130	C₇×#107	C₂³	1	4	yes	—
131	C₇×#106	C₂³	1	8	yes	—
132	C₇×#106	C₂³	1	2	yes	—
133	C₇×C₄⋊C₈	C₂³	1	2	yes	—
134	C₇×C₄⋊C₈	C₂³	1	4	yes	—
135	C₇×C₄⋊C₈	C₂³	1	2	yes	—
136	C₇×C₄⋊C₈	C₂³	1	4	yes	—
137	C₇×#103	C₂³	1	8	yes	—
138	C₇×#103	C₂³	1	2	yes	—
139	C₇×#103	C₂³	1	2	yes	—
140	C₇×#102	C₂³	1	2	yes	—
141	C₇×#102	C₂³	1	8	yes	—
142	C₇×#102	C₂³	1	2	yes	—
143	C₇×#97	C₂³	1	4	yes	—
144	C₄×C₇×C₈	C₂³	1	4	yes	—
145	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
146	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
147	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
148	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
149	#3503	C₂×C₇	1	2	yes	complemented
150	#3386	C₂×C₇	1	4	yes	complemented
151	#3385	C₂×C₇	1	4	yes	complemented
152	#3383	C₂×C₇	1	2	yes	complemented
153	Q₈×Q₁₆	C₂×C₇	1	2	yes	complemented
154	#3380	C₂×C₇	1	2	yes	complemented
155	Q₈×D₈	C₂×C₇	1	2	yes	complemented
156	#3164	C₂×C₇	1	2	yes	complemented
157	C₂×#500	C₂×C₇	1	1	yes	characteristic, complemented
158	#3083	C₂×C₇	1	4	yes	complemented
159	#3079	C₂×C₇	1	2	yes	complemented
160	#2960	C₂×C₇	1	1	yes	characteristic, complemented
161	#2951	C₂×C₇	1	2	yes	—
162	C₄×#135	C₂×C₇	1	1	yes	characteristic
163	C₇×#41	—	2	4	no	—
164	C₇×#41	C₂×Q₈	1	8	yes	complemented
165	C₂×C₇×Q₈	—	2	4	no	—
166	C₂×C₇×Q₈	—	2	8	no	—
167	C₂×C₇×Q₈	—	2	4	no	—
168	C₂×C₇×Q₈	—	8	4	no	—
169	C₂×C₇×Q₈	—	8	8	no	—
170	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
171	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
172	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
173	C₂×C₇×D₄	—	2	2	no	—
174	C₂×C₇×D₄	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₇	—	2	2	no	—
176	C₂²×C₄×C₇	—	2	4	no	—
177	C₂²×C₄×C₇	—	2	2	no	—
178	C₂²×C₄×C₇	—	2	2	no	—
179	C₂²×C₄×C₇	—	2	8	no	—
180	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
183	C₇×Q₁₆	C₂×Q₈	1	4	yes	complemented
184	C₇×SD₁₆	C₂×Q₈	1	8	yes	complemented
185	C₇×D₈	C₂×Q₈	1	4	yes	complemented
186	C₇×M₁₆	—	2	4	no	—
187	C₂×C₇×C₈	—	2	2	no	—
188	C₂×C₇×C₈	C₂×Q₈	1	4	yes	complemented
189	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
190	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
191	C₂×C₇×C₈	C₂⁴	1	4	yes	—
192	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
193	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
194	C₇×C₄⋊C₄	—	2	4	no	—
195	C₇×C₄⋊C₄	—	2	4	no	—
196	C₇×C₄⋊C₄	—	2	8	no	—
197	C₇×C₄⋊C₄	—	2	4	no	—
198	C₇×C₄⋊C₄	—	2	8	no	—
199	C₇×C₄⋊C₄	—	2	2	no	—
200	C₇×C₄⋊C₄	—	2	2	no	—
201	C₇×C₄⋊C₄	—	2	8	no	—
202	C₇×C₄⋊C₄	—	2	8	no	—
203	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
204	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
205	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
206	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
207	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
208	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
209	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
210	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
211	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
212	C₇×C₂²⋊C₄	—	2	2	no	—
213	C₇×C₂²⋊C₄	—	2	2	no	—
214	C₇×C₂²⋊C₄	—	2	2	no	—
215	C₇×C₂²⋊C₄	—	2	8	no	—
216	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
217	C₄²×C₇	—	2	2	no	—
218	C₄²×C₇	—	2	2	no	—
219	C₄²×C₇	—	2	2	no	—
220	C₄²×C₇	—	2	8	no	—
221	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
222	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
223	C₄²×C₇	C₂⁴	1	1	yes	characteristic
224	C₄²×C₇	C₂⁴	1	1	yes	characteristic
225	C₄²×C₇	C₂⁴	1	4	yes	—
226	C₂×#135	C₂²×C₇	1	1	yes	characteristic
227	Q₈²	C₂²×C₇	1	4	yes	—
228	#557	C₂²×C₇	1	4	yes	—
229	#554	—	2	4	no	—
230	#554	C₂²×C₇	1	4	yes	—
231	D₄×Q₈	—	2	4	no	—
232	D₄×Q₈	C₂²×C₇	1	4	yes	—
233	#544	—	2	4	no	—
234	#544	C₂²×C₇	1	4	yes	—
235	#533	—	2	2	no	—
236	#533	C₂²×C₇	1	1	yes	characteristic
237	C₂×C₄⋊Q₈	—	2	2	no	—
238	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
239	#520	C₂²×C₇	1	4	yes	—
240	C₄×#41	C₂²×C₇	1	2	yes	—
241	#501	C₂²×C₇	1	8	yes	—
242	#500	C₂²×C₇	1	4	yes	—
243	#500	C₂²×C₇	1	2	yes	—
244	#500	C₂²×C₇	1	2	yes	—
245	#500	C₂²×C₇	1	2	yes	—
246	#500	C₂²×C₇	1	2	yes	—
247	#499	C₂²×C₇	1	4	yes	—
248	#479	C₂²×C₇	1	8	yes	—
249	#478	C₂²×C₇	1	8	yes	—
250	#477	C₂²×C₇	1	4	yes	—
251	#477	C₂²×C₇	1	8	yes	—
252	#476	C₂²×C₇	1	4	yes	—
253	#475	C₂²×C₇	1	4	yes	—
254	#474	C₂²×C₇	1	4	yes	—
255	#474	C₂²×C₇	1	8	yes	—
256	#446	C₂²×C₇	1	2	yes	—
257	#446	C₂²×C₇	1	2	yes	—
258	C₈×Q₈	C₂²×C₇	1	2	yes	—
259	C₈×Q₈	C₂²×C₇	1	2	yes	—
260	#440	C₂²×C₇	1	8	yes	—
261	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
262	C₄×Q₁₆	C₂²×C₇	1	4	yes	—
263	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
264	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
265	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
266	C₄×D₈	C₂²×C₇	1	1	yes	characteristic
267	C₄×D₈	C₂²×C₇	1	4	yes	—
268	C₄×D₈	C₂²×C₇	1	1	yes	characteristic
269	#428	C₂²×C₇	1	4	yes	—
270	#427	C₂²×C₇	1	1	yes	characteristic
271	C₂×#107	C₂²×C₇	1	2	yes	—
272	C₂×#107	C₂²×C₇	1	2	yes	—
273	#424	C₂²×C₇	1	2	yes	—
274	#423	C₂²×C₇	1	1	yes	characteristic
275	#417	C₂²×C₇	1	4	yes	—
276	#416	C₂²×C₇	1	2	yes	—
277	#405	C₂²×C₇	1	2	yes	—
278	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
279	C₂³×C₇	—	2	2	no	—
280	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
281	C₇×Q₈	—	2	4	no	—
282	C₇×Q₈	—	2	4	no	—
283	C₇×Q₈	—	2	2	no	—
284	C₇×Q₈	—	4	8	no	—
285	C₇×Q₈	—	8	4	no	—
286	C₇×Q₈	—	8	8	no	—
287	C₇×Q₈	C₂²×Q₈	1	4	yes	—
288	C₇×D₄	—	2	8	no	—
289	C₇×D₄	—	2	2	no	—
290	C₇×D₄	C₂²×Q₈	1	4	yes	—
291	C₂×C₄×C₇	—	2	2	no	—
292	C₂×C₄×C₇	—	2	8	no	—
293	C₂×C₄×C₇	—	2	2	no	—
294	C₂×C₄×C₇	—	2	2	no	—
295	C₂×C₄×C₇	—	2	2	no	—
296	C₂×C₄×C₇	—	2	2	no	—
297	C₂×C₄×C₇	—	2	8	no	—
298	C₂×C₄×C₇	—	4	4	no	—
299	C₂×C₄×C₇	—	4	4	no	—
300	C₂×C₄×C₇	—	8	8	no	—
301	C₂×C₄×C₇	—	8	4	no	—
302	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
303	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
304	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
305	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
306	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
307	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
308	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
309	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
310	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
311	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
312	C₇×C₈	—	2	1	no	—
313	C₇×C₈	—	2	1	no	—
314	C₇×C₈	—	2	4	no	—
315	C₇×C₈	C₂²×Q₈	1	2	yes	—
316	C₇×C₈	C₂²×Q₈	1	2	yes	—
317	C₂×#41	C₂³×C₇	1	2	yes	—
318	C₂²×Q₈	—	2	4	no	—
319	C₂²×Q₈	—	2	2	no	—
320	#135	C₇×Q₈	1	8	yes	complemented
321	C₂×Q₁₆	C₂³×C₇	1	1	yes	characteristic
322	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
323	C₂×D₈	C₂³×C₇	1	1	yes	characteristic
324	C₂×M₁₆	C₂³×C₇	1	2	yes	—
325	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
326	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
327	C₄⋊Q₈	—	2	4	no	—
328	C₄⋊Q₈	—	2	4	no	—
329	C₄⋊Q₈	—	2	4	no	—
330	C₄⋊Q₈	—	2	8	no	—
331	C₄⋊Q₈	—	2	2	no	—
332	C₄⋊Q₈	—	2	2	no	—
333	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
334	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
335	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
336	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
337	C₄⋊Q₈	C₂³×C₇	1	8	yes	—
338	#125	—	2	4	no	—
339	#125	—	2	8	no	—
340	#125	C₇×D₄	1	2	yes	—
341	#125	C₂³×C₇	1	8	yes	—
342	C₂²⋊Q₈	—	2	4	no	—
343	C₂²⋊Q₈	—	2	4	no	—
344	C₂²⋊Q₈	—	2	8	no	—
345	C₂²⋊Q₈	—	2	8	no	—
346	C₂²⋊Q₈	—	2	8	no	—
347	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
348	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
349	C₄×Q₈	—	2	4	no	—
350	C₄×Q₈	—	2	4	no	—
351	C₄×Q₈	C₂³×C₇	1	2	yes	—
352	C₄×Q₈	C₂³×C₇	1	2	yes	—
353	C₄×Q₈	C₂³×C₇	1	8	yes	—
354	C₄×Q₈	C₂³×C₇	1	2	yes	—
355	C₄×Q₈	C₂³×C₇	1	2	yes	—
356	C₄×D₄	—	2	2	no	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	8	no	—
359	C₄×D₄	C₂³×C₇	1	2	yes	—
360	C₄×D₄	C₂³×C₇	1	2	yes	—
361	C₄×D₄	C₂³×C₇	1	8	yes	—
362	#117	—	2	4	no	—
363	#117	—	2	2	no	—
364	#117	C₂³×C₇	1	2	yes	—
365	#117	C₂³×C₇	1	2	yes	—
366	C₂×C₄⋊C₄	—	2	2	no	—
367	C₂×C₄⋊C₄	—	2	4	no	—
368	C₂×C₄⋊C₄	—	2	4	no	—
369	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
370	C₂×C₄²	—	2	2	no	—
371	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
372	#107	C₂³×C₇	1	4	yes	—
373	#107	C₂³×C₇	1	4	yes	—
374	#107	C₂³×C₇	1	4	yes	—
375	#107	C₂³×C₇	1	4	yes	—
376	#107	C₂³×C₇	1	1	yes	characteristic
377	#107	C₂³×C₇	1	1	yes	characteristic
378	#107	C₂³×C₇	1	8	yes	—
379	#106	C₂³×C₇	1	2	yes	—
380	#106	C₂³×C₇	1	8	yes	—
381	C₄⋊C₈	C₂³×C₇	1	2	yes	—
382	C₄⋊C₈	C₂³×C₇	1	2	yes	—
383	C₄⋊C₈	C₂³×C₇	1	4	yes	—
384	C₄⋊C₈	C₂³×C₇	1	4	yes	—
385	#103	C₂³×C₇	1	2	yes	—
386	#103	C₂³×C₇	1	2	yes	—
387	#103	C₂³×C₇	1	8	yes	—
388	#102	C₂³×C₇	1	2	yes	—
389	#102	C₂³×C₇	1	2	yes	—
390	#102	C₂³×C₇	1	8	yes	—
391	#97	C₂³×C₇	1	4	yes	—
392	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
393	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
394	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
395	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
396	C₄×C₈	C₂³×C₇	1	4	yes	—
397	C₂²×C₇	—	2	1	no	—
398	C₂²×C₇	—	2	1	no	—
399	C₂²×C₇	—	2	4	no	—
400	C₂²×C₇	—	4	2	no	—
401	C₂²×C₇	—	4	2	no	—
402	C₂²×C₇	D₄×Q₈	1	2	yes	—
403	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
404	C₄×C₇	—	2	4	no	—
405	C₄×C₇	—	2	1	no	—
406	C₄×C₇	—	2	4	no	—
407	C₄×C₇	—	2	1	no	—
408	C₄×C₇	—	2	4	no	—
409	C₄×C₇	—	4	2	no	—
410	C₄×C₇	—	4	8	no	—
411	C₄×C₇	—	4	2	no	—
412	C₄×C₇	D₄×Q₈	1	2	yes	—
413	C₄×C₇	#576	1	2	yes	—
414	C₄×C₇	#578	1	2	yes	—
415	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
416	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
417	#41	—	2	4	no	—
418	#41	C₂×C₇×Q₈	1	8	yes	complemented
419	C₂×Q₈	—	2	4	no	—
420	C₂×Q₈	—	2	8	no	—
421	C₂×Q₈	—	2	4	no	—
422	C₂×Q₈	—	8	4	no	—
423	C₂×Q₈	—	8	8	no	—
424	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
425	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
426	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	C₂⁴×C₇	1	2	yes	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	2	8	no	—
434	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
435	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
436	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
437	Q₁₆	C₂×C₇×Q₈	1	4	yes	complemented
438	SD₁₆	C₂×C₇×Q₈	1	8	yes	complemented
439	D₈	C₂×C₇×Q₈	1	4	yes	complemented
440	M₁₆	—	2	4	no	—
441	C₂×C₈	—	2	2	no	—
442	C₂×C₈	C₂×C₇×Q₈	1	4	yes	complemented
443	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
444	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
445	C₂×C₈	C₂⁴×C₇	1	4	yes	—
446	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
447	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	8	no	—
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	—	2	2	no	—
455	C₄⋊C₄	—	2	8	no	—
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
458	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
459	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
460	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
461	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
462	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
463	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
464	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
465	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
466	C₂²⋊C₄	—	2	2	no	—
467	C₂²⋊C₄	—	2	2	no	—
468	C₂²⋊C₄	—	2	2	no	—
469	C₂²⋊C₄	—	2	8	no	—
470	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
471	C₄²	—	2	2	no	—
472	C₄²	—	2	2	no	—
473	C₄²	—	2	2	no	—
474	C₄²	—	2	8	no	—
475	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
476	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
477	C₄²	C₂⁴×C₇	1	1	yes	characteristic
478	C₄²	C₂⁴×C₇	1	1	yes	characteristic
479	C₄²	C₂⁴×C₇	1	4	yes	—
480	C₂×C₇	—	2	2	no	—
481	C₂×C₇	—	4	4	no	—
482	C₂×C₇	C₂×D₄×Q₈	1	1	yes	characteristic
483	C₂×C₇	C₂×#576	1	1	yes	characteristic
484	C₂×C₇	C₂×#578	1	1	yes	characteristic
485	C₂³	—	2	2	no	—
486	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
487	Q₈	—	2	2	no	—
488	Q₈	—	2	4	no	—
489	Q₈	—	2	4	no	—
490	Q₈	—	4	8	no	—
491	Q₈	—	8	4	no	—
492	Q₈	—	8	8	no	—
493	Q₈	C₂²×C₇×Q₈	1	4	yes	—
494	D₄	—	2	8	no	—
495	D₄	—	2	2	no	—
496	D₄	C₂²×C₇×Q₈	1	4	yes	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	8	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	8	no	—
504	C₂×C₄	—	4	4	no	—
505	C₂×C₄	—	4	4	no	—
506	C₂×C₄	—	8	8	no	—
507	C₂×C₄	—	8	4	no	—
508	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
509	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
510	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
511	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
512	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
513	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
514	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
515	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
516	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
517	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
518	C₈	—	2	1	no	—
519	C₈	—	2	1	no	—
520	C₈	—	2	4	no	—
521	C₈	C₂²×C₇×Q₈	1	2	yes	—
522	C₈	C₂²×C₇×Q₈	1	2	yes	—
523	C₇	#61373	1	1	yes	characteristic, direct product factor, Sylow
524	C₂²	—	2	1	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	2	4	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	4	2	no	—
529	C₂²	C₇×D₄×Q₈	1	2	yes	—
530	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
531	C₄	—	2	1	no	—
532	C₄	—	2	4	no	—
533	C₄	—	2	1	no	—
534	C₄	—	2	4	no	—
535	C₄	—	2	4	no	—
536	C₄	—	4	2	no	—
537	C₄	—	4	8	no	—
538	C₄	—	4	2	no	—
539	C₄	C₇×D₄×Q₈	1	2	yes	—
540	C₄	C₇×#576	1	2	yes	—
541	C₄	C₇×#578	1	2	yes	—
542	C₄	C₂³×C₇×Q₈	1	1	yes	characteristic
543	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
544	C₂	—	2	2	no	—
545	C₂	—	4	4	no	—
546	C₂	C₂×C₇×D₄×Q₈	1	1	yes	characteristic
547	C₂	C₂×C₇×#576	1	1	yes	characteristic
548	C₂	C₂×C₇×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂×C₇×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₇×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3503	C₂	semidirect
2. C₇×#3386	C₂	semidirect
3. C₇×#3385	C₂	semidirect
4. C₇×#3383	C₂	semidirect
5. C₇×Q₈×Q₁₆	C₂	semidirect
6. C₇×#3380	C₂	semidirect
7. C₇×Q₈×D₈	C₂	semidirect
8. C₇×#3164	C₂	semidirect
9. C₂×C₇×#500	C₂	semidirect
10. C₇×#3083	C₂	semidirect
11. C₇×#3079	C₂	semidirect
12. C₇×#2960	C₂	semidirect
68. #61373	C₇	direct (N × H)
72. C₇×#135	Q₈	semidirect
149. #3503	C₂×C₇	semidirect
150. #3386	C₂×C₇	semidirect
151. #3385	C₂×C₇	semidirect
152. #3383	C₂×C₇	semidirect
153. Q₈×Q₁₆	C₂×C₇	semidirect
154. #3380	C₂×C₇	semidirect
155. Q₈×D₈	C₂×C₇	semidirect
156. #3164	C₂×C₇	semidirect
157. C₂×#500	C₂×C₇	semidirect
158. #3083	C₂×C₇	semidirect
159. #3079	C₂×C₇	semidirect
160. #2960	C₂×C₇	semidirect
164. C₇×#41	C₂×Q₈	semidirect
183. C₇×Q₁₆	C₂×Q₈	semidirect
184. C₇×SD₁₆	C₂×Q₈	semidirect
185. C₇×D₈	C₂×Q₈	semidirect
188. C₂×C₇×C₈	C₂×Q₈	semidirect
320. #135	C₇×Q₈	semidirect
418. #41	C₂×C₇×Q₈	semidirect
437. Q₁₆	C₂×C₇×Q₈	semidirect
438. SD₁₆	C₂×C₇×Q₈	semidirect
439. D₈	C₂×C₇×Q₈	semidirect
442. C₂×C₈	C₂×C₇×Q₈	semidirect
523. C₇	#61373	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×#3503	2
C₇×#3386	2
C₇×#3385	2
C₇×#3383	2
C₇×Q₈×Q₁₆	2
C₇×#3380	2
C₇×Q₈×D₈	2
C₇×#3164	2
C₂×C₇×#500	2
C₇×#3083	2
C₇×#3079	2
C₇×#2960	2
C₇×#2951	2
C₄×C₇×#135	2
#61373	7

Subgroups — C7×#61373 #13238589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61373 #13238589