Subgroups — C₇×#60493 #13237709

Order 1792.

2030 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3428	C₂	1	1	yes	maximal, characteristic
3	C₇×#3084	C₂	1	4	yes	maximal, complemented
4	C₇×#3083	C₂	1	4	yes	maximal, complemented
5	C₇×#3082	C₂	1	4	yes	maximal, complemented
6	C₇×#3081	C₂	1	4	yes	maximal, complemented
7	C₇×#3080	C₂	1	4	yes	maximal, complemented
8	C₇×#3079	C₂	1	4	yes	maximal, complemented
9	C₇×#2911	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×#428	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#2905	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#2896	C₂	1	1	yes	maximal, characteristic
13	C₂×C₇×#417	C₂	1	1	yes	maximal, characteristic
14	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
15	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
17	C₇×#544	—	2	2	no	—
18	C₇×#544	—	2	2	no	—
19	C₇×#544	C₂²	1	4	yes	—
20	C₇×#544	C₂²	1	4	yes	complemented
21	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
22	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	complemented
23	C₂×C₇×#125	—	2	1	no	—
24	C₂×C₇×#125	C₂²	1	2	yes	complemented
25	C₂×C₇×C₂²⋊Q₈	—	2	2	no	—
26	C₇×#520	—	2	1	no	—
27	C₇×#520	—	2	1	no	—
28	C₇×#520	C₂²	1	2	yes	—
29	C₇×#520	C₂²	1	2	yes	—
30	C₇×#519	C₂²	1	2	yes	—
31	C₇×#518	—	2	2	no	—
32	C₇×#518	C₂²	1	2	yes	—
33	C₄×C₇×#41	—	2	2	no	—
34	C₄×C₇×#41	C₂²	1	4	yes	—
35	C₂×C₇×#117	—	2	2	no	—
36	C₂×C₇×#117	C₂²	1	1	yes	characteristic
37	C₂×C₇×#117	C₂²	1	1	yes	characteristic
38	C₂×C₇×#117	C₂²	1	1	yes	characteristic
39	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
40	C₇×#479	C₂²	1	8	yes	complemented
41	C₇×#479	C₂²	1	8	yes	complemented
42	C₇×#478	C₂²	1	8	yes	complemented
43	C₇×#478	C₂²	1	8	yes	complemented
44	C₇×#477	C₂²	1	8	yes	complemented
45	C₇×#476	C₂²	1	8	yes	complemented
46	C₇×#475	C₂²	1	8	yes	complemented
47	C₇×#474	C₂²	1	8	yes	complemented
48	C₇×#428	C₂²	1	4	yes	—
49	C₇×#428	C₂²	1	4	yes	complemented
50	C₇×#428	C₂²	1	2	yes	complemented
51	C₇×#428	C₂²	1	2	yes	complemented
52	C₇×#428	C₂²	1	2	yes	complemented
53	C₇×#428	C₂²	1	2	yes	complemented
54	C₇×#427	C₂²	1	4	yes	—
55	C₂×C₇×#107	C₂²	1	2	yes	complemented
56	C₂×C₇×#106	C₂²	1	2	yes	complemented
57	C₇×#424	C₂²	1	4	yes	—
58	C₇×#424	C₂²	1	4	yes	complemented
59	C₇×#423	C₂²	1	4	yes	complemented
60	C₇×#419	C₂²	1	2	yes	—
61	C₇×#418	C₂²	1	2	yes	—
62	C₇×#417	C₂²	1	2	yes	—
63	C₇×#417	C₂²	1	4	yes	—
64	C₇×#417	C₂²	1	2	yes	—
65	C₇×#417	C₂²	1	4	yes	—
66	C₇×#416	C₂²	1	4	yes	—
67	C₂×C₇×#103	C₂²	1	2	yes	—
68	C₂×C₇×#102	C₂²	1	2	yes	—
69	#60493	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×C₇×#41	—	2	1	no	—
71	C₂×C₇×#41	—	2	1	no	—
72	C₂×C₇×#41	—	2	4	no	—
73	C₂×C₇×#41	Q₈	1	8	yes	complemented
74	C₂×C₇×#41	C₂³	1	2	yes	—
75	C₂×C₇×#41	C₂³	1	2	yes	—
76	C₂²×C₇×Q₈	—	2	2	no	—
77	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
78	C₂²×C₇×D₄	—	2	1	no	—
79	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₇	—	2	1	no	—
81	C₂³×C₄×C₇	—	2	1	no	—
82	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
83	C₂×C₇×M₁₆	—	2	4	no	—
84	C₂×C₇×M₁₆	—	2	4	no	—
85	C₂×C₇×M₁₆	C₂³	1	2	yes	—
86	C₂×C₇×M₁₆	C₂³	1	2	yes	—
87	C₂×C₇×M₁₆	C₂³	1	2	yes	—
88	C₂×C₇×M₁₆	C₂³	1	2	yes	—
89	C₂²×C₇×C₈	C₂³	1	2	yes	—
90	C₂²×C₇×C₈	C₂³	1	2	yes	—
91	C₇×C₄⋊Q₈	—	2	4	no	—
92	C₇×C₄⋊Q₈	—	2	4	no	—
93	C₇×C₄⋊Q₈	—	2	4	no	—
94	C₇×C₄⋊Q₈	C₂³	1	8	yes	—
95	C₇×#125	—	2	4	no	—
96	C₇×#125	—	2	4	no	—
97	C₇×#125	—	2	4	no	—
98	C₇×#125	C₂³	1	8	yes	—
99	C₇×C₂²⋊Q₈	—	2	8	no	—
100	C₇×C₂²⋊Q₈	—	2	8	no	—
101	C₄×C₇×Q₈	—	2	2	no	—
102	C₄×C₇×Q₈	—	2	2	no	—
103	C₄×C₇×Q₈	C₂³	1	4	yes	—
104	C₄×C₇×Q₈	C₂³	1	4	yes	—
105	C₄×C₇×D₄	—	2	4	no	—
106	C₄×C₇×D₄	—	2	4	no	—
107	C₄×C₇×D₄	—	2	2	no	—
108	C₄×C₇×D₄	—	2	2	no	—
109	C₄×C₇×D₄	—	2	4	no	—
110	C₄×C₇×D₄	—	2	4	no	—
111	C₄×C₇×D₄	C₂³	1	4	yes	—
112	C₄×C₇×D₄	C₂³	1	4	yes	—
113	C₇×#117	—	2	1	no	—
114	C₇×#117	—	2	1	no	—
115	C₇×#117	—	2	1	no	—
116	C₇×#117	—	2	1	no	—
117	C₇×#117	—	2	2	no	—
118	C₇×#117	—	2	2	no	—
119	C₇×#117	—	2	4	no	—
120	C₇×#117	—	2	4	no	—
121	C₇×#117	D₄	1	2	yes	—
122	C₇×#117	D₄	1	2	yes	—
123	C₇×#117	C₂³	1	2	yes	—
124	C₇×#117	C₂³	1	2	yes	—
125	C₇×#117	C₂³	1	2	yes	—
126	C₇×#117	C₂³	1	2	yes	—
127	C₇×#117	C₂³	1	2	yes	—
128	C₇×#117	C₂³	1	2	yes	—
129	C₂×C₇×C₄⋊C₄	—	2	1	no	—
130	C₂×C₇×C₄⋊C₄	—	2	1	no	—
131	C₂×C₇×C₄⋊C₄	—	2	1	no	—
132	C₂×C₇×C₄⋊C₄	—	2	2	no	—
133	C₂×C₇×C₄⋊C₄	—	2	2	no	—
134	C₂×C₇×C₄⋊C₄	—	2	2	no	—
135	C₂×C₇×C₄⋊C₄	—	2	2	no	—
136	C₂×C₇×C₄⋊C₄	—	2	2	no	—
137	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
138	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
139	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
140	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
141	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
142	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
143	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
144	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
145	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
146	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
147	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
148	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
149	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
150	C₂×C₄²×C₇	—	2	1	no	—
151	C₂×C₄²×C₇	—	2	1	no	—
152	C₂×C₄²×C₇	—	2	4	no	—
153	C₂×C₄²×C₇	C₂³	1	2	yes	—
154	C₇×#107	C₂³	1	4	yes	—
155	C₇×#107	C₂³	1	8	yes	—
156	C₇×#107	C₂³	1	4	yes	—
157	C₇×#106	C₂³	1	4	yes	—
158	C₇×#106	C₂³	1	8	yes	—
159	C₇×#106	C₂³	1	4	yes	—
160	C₇×C₄⋊C₈	C₂³	1	8	yes	—
161	C₇×C₄⋊C₈	C₂³	1	8	yes	—
162	C₇×#103	C₂³	1	4	yes	—
163	C₇×#103	C₂³	1	8	yes	—
164	C₇×#103	C₂³	1	4	yes	—
165	C₇×#102	C₂³	1	4	yes	—
166	C₇×#102	C₂³	1	8	yes	—
167	C₇×#102	C₂³	1	4	yes	—
168	C₂×#544	C₂×C₇	1	1	yes	characteristic, complemented
169	#3428	C₂×C₇	1	1	yes	characteristic
170	#3084	C₂×C₇	1	4	yes	complemented
171	#3083	C₂×C₇	1	4	yes	complemented
172	#3082	C₂×C₇	1	4	yes	complemented
173	#3081	C₂×C₇	1	4	yes	complemented
174	#3080	C₂×C₇	1	4	yes	complemented
175	#3079	C₂×C₇	1	4	yes	complemented
176	#2911	C₂×C₇	1	1	yes	characteristic, complemented
177	C₂×#428	C₂×C₇	1	1	yes	characteristic, complemented
178	#2905	C₂×C₇	1	1	yes	characteristic, complemented
179	#2896	C₂×C₇	1	1	yes	characteristic
180	C₂×#417	C₂×C₇	1	1	yes	characteristic
181	C₂⁴×C₇	—	2	1	no	—
182	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
183	C₇×#41	—	2	8	no	—
184	C₇×#41	—	2	8	no	—
185	C₇×#41	—	2	4	no	—
186	C₇×#41	—	2	4	no	—
187	C₇×#41	C₂×Q₈	1	8	yes	—
188	C₇×#41	C₂×Q₈	1	8	yes	complemented
189	C₂×C₇×Q₈	—	2	1	no	—
190	C₂×C₇×Q₈	—	2	8	no	—
191	C₂×C₇×Q₈	—	2	2	no	—
192	C₂×C₇×Q₈	—	8	4	no	—
193	C₂×C₇×Q₈	C₂×Q₈	1	4	yes	complemented
194	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
195	C₂×C₇×D₄	—	2	8	no	—
196	C₂×C₇×D₄	—	2	2	no	—
197	C₂×C₇×D₄	—	2	2	no	—
198	C₂×C₇×D₄	—	2	1	no	—
199	C₂×C₇×D₄	—	2	2	no	—
200	C₂×C₇×D₄	C₂×Q₈	1	4	yes	complemented
201	C₂×C₇×D₄	C₂⁴	1	2	yes	—
202	C₂²×C₄×C₇	—	2	1	no	—
203	C₂²×C₄×C₇	—	2	8	no	—
204	C₂²×C₄×C₇	—	2	2	no	—
205	C₂²×C₄×C₇	—	2	2	no	—
206	C₂²×C₄×C₇	—	2	1	no	—
207	C₂²×C₄×C₇	—	2	1	no	—
208	C₂²×C₄×C₇	—	2	2	no	—
209	C₂²×C₄×C₇	—	2	1	no	—
210	C₂²×C₄×C₇	—	2	1	no	—
211	C₂²×C₄×C₇	—	2	2	no	—
212	C₂²×C₄×C₇	—	2	2	no	—
213	C₂²×C₄×C₇	—	2	4	no	—
214	C₂²×C₄×C₇	—	2	4	no	—
215	C₂²×C₄×C₇	—	8	2	no	—
216	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
217	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
218	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
219	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
220	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
221	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
222	C₂²×C₄×C₇	#41	1	4	yes	—
223	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
224	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
225	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
226	C₇×M₁₆	—	2	4	no	—
227	C₇×M₁₆	—	2	4	no	—
228	C₇×M₁₆	—	2	4	no	—
229	C₇×M₁₆	—	2	4	no	—
230	C₂×C₇×C₈	—	2	4	no	—
231	C₂×C₇×C₈	—	2	4	no	—
232	C₂×C₇×C₈	C₂⁴	1	4	yes	—
233	C₂×C₇×C₈	C₂⁴	1	4	yes	—
234	C₇×C₄⋊C₄	—	2	1	no	—
235	C₇×C₄⋊C₄	—	2	1	no	—
236	C₇×C₄⋊C₄	—	2	2	no	—
237	C₇×C₄⋊C₄	—	2	2	no	—
238	C₇×C₄⋊C₄	—	2	2	no	—
239	C₇×C₄⋊C₄	—	2	4	no	—
240	C₇×C₄⋊C₄	—	2	4	no	—
241	C₇×C₄⋊C₄	—	2	8	no	—
242	C₇×C₄⋊C₄	—	2	8	no	—
243	C₇×C₄⋊C₄	—	2	8	no	—
244	C₇×C₄⋊C₄	—	2	1	no	—
245	C₇×C₄⋊C₄	—	2	1	no	—
246	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
247	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
248	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
249	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
250	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
251	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
252	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
253	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
254	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
255	C₇×C₂²⋊C₄	—	2	2	no	—
256	C₇×C₂²⋊C₄	—	2	2	no	—
257	C₇×C₂²⋊C₄	—	2	2	no	—
258	C₇×C₂²⋊C₄	—	2	2	no	—
259	C₇×C₂²⋊C₄	—	2	4	no	—
260	C₇×C₂²⋊C₄	—	2	4	no	—
261	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
262	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
263	C₄²×C₇	—	2	2	no	—
264	C₄²×C₇	—	2	4	no	—
265	C₄²×C₇	—	2	4	no	—
266	C₄²×C₇	—	2	2	no	—
267	C₄²×C₇	—	2	2	no	—
268	C₄²×C₇	—	2	2	no	—
269	C₄²×C₇	C₂×D₄	1	4	yes	—
270	C₄²×C₇	C₂⁴	1	4	yes	—
271	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
272	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
273	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
274	#544	—	2	2	no	—
275	#544	—	2	2	no	—
276	#544	C₂²×C₇	1	4	yes	—
277	#544	C₂²×C₇	1	4	yes	complemented
278	C₂×C₄⋊Q₈	—	2	1	no	—
279	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	complemented
280	C₂×#125	—	2	1	no	—
281	C₂×#125	C₂²×C₇	1	2	yes	complemented
282	C₂×C₂²⋊Q₈	—	2	2	no	—
283	#520	—	2	1	no	—
284	#520	—	2	1	no	—
285	#520	C₂²×C₇	1	2	yes	—
286	#520	C₂²×C₇	1	2	yes	—
287	#519	C₂²×C₇	1	2	yes	—
288	#518	—	2	2	no	—
289	#518	C₂²×C₇	1	2	yes	—
290	C₄×#41	—	2	2	no	—
291	C₄×#41	C₂²×C₇	1	4	yes	—
292	C₂×#117	—	2	2	no	—
293	C₂×#117	C₂²×C₇	1	1	yes	characteristic
294	C₂×#117	C₂²×C₇	1	1	yes	characteristic
295	C₂×#117	C₂²×C₇	1	1	yes	characteristic
296	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
297	#479	C₂²×C₇	1	8	yes	complemented
298	#479	C₂²×C₇	1	8	yes	complemented
299	#478	C₂²×C₇	1	8	yes	complemented
300	#478	C₂²×C₇	1	8	yes	complemented
301	#477	C₂²×C₇	1	8	yes	complemented
302	#476	C₂²×C₇	1	8	yes	complemented
303	#475	C₂²×C₇	1	8	yes	complemented
304	#474	C₂²×C₇	1	8	yes	complemented
305	#428	C₂²×C₇	1	4	yes	—
306	#428	C₂²×C₇	1	2	yes	complemented
307	#428	C₂²×C₇	1	4	yes	complemented
308	#428	C₂²×C₇	1	2	yes	complemented
309	#428	C₂²×C₇	1	2	yes	complemented
310	#428	C₂²×C₇	1	2	yes	complemented
311	#427	C₂²×C₇	1	4	yes	—
312	C₂×#107	C₂²×C₇	1	2	yes	complemented
313	C₂×#106	C₂²×C₇	1	2	yes	complemented
314	#424	C₂²×C₇	1	4	yes	—
315	#424	C₂²×C₇	1	4	yes	complemented
316	#423	C₂²×C₇	1	4	yes	complemented
317	#419	C₂²×C₇	1	2	yes	—
318	#418	C₂²×C₇	1	2	yes	—
319	#417	C₂²×C₇	1	2	yes	—
320	#417	C₂²×C₇	1	4	yes	—
321	#417	C₂²×C₇	1	2	yes	—
322	#417	C₂²×C₇	1	4	yes	—
323	#416	C₂²×C₇	1	4	yes	—
324	C₂×#103	C₂²×C₇	1	2	yes	—
325	C₂×#102	C₂²×C₇	1	2	yes	—
326	C₂³×C₇	—	2	2	no	—
327	C₂³×C₇	—	2	4	no	—
328	C₂³×C₇	—	4	1	no	—
329	C₂³×C₇	—	4	1	no	—
330	C₂³×C₇	—	8	1	no	—
331	C₂³×C₇	—	8	1	no	—
332	C₂³×C₇	C₂²⋊Q₈	1	4	yes	—
333	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
334	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
335	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
336	C₇×Q₈	—	2	4	no	—
337	C₇×Q₈	—	2	2	no	—
338	C₇×Q₈	—	8	8	no	—
339	C₇×Q₈	C₂²×Q₈	1	4	yes	—
340	C₇×D₄	—	2	8	no	—
341	C₇×D₄	—	2	8	no	—
342	C₇×D₄	—	2	4	no	—
343	C₇×D₄	—	2	2	no	—
344	C₇×D₄	C₂²×Q₈	1	4	yes	—
345	C₂×C₄×C₇	—	2	4	no	—
346	C₂×C₄×C₇	—	2	2	no	—
347	C₂×C₄×C₇	—	2	8	no	—
348	C₂×C₄×C₇	—	2	8	no	—
349	C₂×C₄×C₇	—	2	2	no	—
350	C₂×C₄×C₇	—	2	2	no	—
351	C₂×C₄×C₇	—	2	2	no	—
352	C₂×C₄×C₇	—	2	2	no	—
353	C₂×C₄×C₇	—	2	4	no	—
354	C₂×C₄×C₇	—	4	2	no	—
355	C₂×C₄×C₇	—	4	2	no	—
356	C₂×C₄×C₇	—	4	2	no	—
357	C₂×C₄×C₇	—	4	4	no	—
358	C₂×C₄×C₇	—	4	2	no	—
359	C₂×C₄×C₇	—	8	4	no	—
360	C₂×C₄×C₇	—	8	4	no	—
361	C₂×C₄×C₇	—	8	1	no	—
362	C₂×C₄×C₇	—	8	1	no	—
363	C₂×C₄×C₇	—	8	2	no	—
364	C₂×C₄×C₇	—	8	2	no	—
365	C₂×C₄×C₇	C₂²⋊Q₈	1	4	yes	—
366	C₂×C₄×C₇	C₂²⋊Q₈	1	4	yes	—
367	C₂×C₄×C₇	C₂²⋊Q₈	1	4	yes	—
368	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
369	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
370	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
371	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
372	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
373	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
374	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
375	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
376	C₂×C₄×C₇	C₂×#41	1	2	yes	—
377	C₂×C₄×C₇	C₂×#41	1	2	yes	—
378	C₂×C₄×C₇	C₂×#41	1	2	yes	—
379	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
380	C₇×C₈	—	2	4	no	—
381	C₇×C₈	—	2	4	no	—
382	C₂×#41	—	2	1	no	—
383	C₂×#41	—	2	1	no	—
384	C₂×#41	—	2	4	no	—
385	C₂×#41	C₇×Q₈	1	8	yes	complemented
386	C₂×#41	C₂³×C₇	1	2	yes	—
387	C₂×#41	C₂³×C₇	1	2	yes	—
388	C₂²×Q₈	—	2	2	no	—
389	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
390	C₂²×D₄	—	2	1	no	—
391	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
392	C₂³×C₄	—	2	1	no	—
393	C₂³×C₄	—	2	1	no	—
394	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
395	C₂×M₁₆	—	2	4	no	—
396	C₂×M₁₆	—	2	4	no	—
397	C₂×M₁₆	C₂³×C₇	1	2	yes	—
398	C₂×M₁₆	C₂³×C₇	1	2	yes	—
399	C₂×M₁₆	C₂³×C₇	1	2	yes	—
400	C₂×M₁₆	C₂³×C₇	1	2	yes	—
401	C₂²×C₈	C₂³×C₇	1	2	yes	—
402	C₂²×C₈	C₂³×C₇	1	2	yes	—
403	C₄⋊Q₈	—	2	4	no	—
404	C₄⋊Q₈	—	2	4	no	—
405	C₄⋊Q₈	—	2	4	no	—
406	C₄⋊Q₈	C₂³×C₇	1	8	yes	—
407	#125	—	2	4	no	—
408	#125	—	2	4	no	—
409	#125	—	2	4	no	—
410	#125	C₂³×C₇	1	8	yes	—
411	C₂²⋊Q₈	—	2	8	no	—
412	C₂²⋊Q₈	—	2	8	no	—
413	C₄×Q₈	—	2	2	no	—
414	C₄×Q₈	—	2	2	no	—
415	C₄×Q₈	C₂³×C₇	1	4	yes	—
416	C₄×Q₈	C₂³×C₇	1	4	yes	—
417	C₄×D₄	—	2	4	no	—
418	C₄×D₄	—	2	4	no	—
419	C₄×D₄	—	2	2	no	—
420	C₄×D₄	—	2	2	no	—
421	C₄×D₄	—	2	4	no	—
422	C₄×D₄	—	2	4	no	—
423	C₄×D₄	C₂³×C₇	1	4	yes	—
424	C₄×D₄	C₂³×C₇	1	4	yes	—
425	#117	—	2	1	no	—
426	#117	—	2	1	no	—
427	#117	—	2	1	no	—
428	#117	—	2	1	no	—
429	#117	—	2	2	no	—
430	#117	—	2	2	no	—
431	#117	—	2	4	no	—
432	#117	—	2	4	no	—
433	#117	C₇×D₄	1	2	yes	—
434	#117	C₇×D₄	1	2	yes	—
435	#117	C₂³×C₇	1	2	yes	—
436	#117	C₂³×C₇	1	2	yes	—
437	#117	C₂³×C₇	1	2	yes	—
438	#117	C₂³×C₇	1	2	yes	—
439	#117	C₂³×C₇	1	2	yes	—
440	#117	C₂³×C₇	1	2	yes	—
441	C₂×C₄⋊C₄	—	2	1	no	—
442	C₂×C₄⋊C₄	—	2	1	no	—
443	C₂×C₄⋊C₄	—	2	1	no	—
444	C₂×C₄⋊C₄	—	2	2	no	—
445	C₂×C₄⋊C₄	—	2	2	no	—
446	C₂×C₄⋊C₄	—	2	2	no	—
447	C₂×C₄⋊C₄	—	2	2	no	—
448	C₂×C₄⋊C₄	—	2	2	no	—
449	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
450	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
451	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
452	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
453	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
454	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
455	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
456	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
457	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
458	C₂×C₂²⋊C₄	—	2	1	no	—
459	C₂×C₂²⋊C₄	—	2	1	no	—
460	C₂×C₂²⋊C₄	—	2	4	no	—
461	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
462	C₂×C₄²	—	2	1	no	—
463	C₂×C₄²	—	2	1	no	—
464	C₂×C₄²	—	2	4	no	—
465	C₂×C₄²	C₂³×C₇	1	2	yes	—
466	#107	C₂³×C₇	1	8	yes	—
467	#107	C₂³×C₇	1	4	yes	—
468	#107	C₂³×C₇	1	4	yes	—
469	#106	C₂³×C₇	1	8	yes	—
470	#106	C₂³×C₇	1	4	yes	—
471	#106	C₂³×C₇	1	4	yes	—
472	C₄⋊C₈	C₂³×C₇	1	8	yes	—
473	C₄⋊C₈	C₂³×C₇	1	8	yes	—
474	#103	C₂³×C₇	1	8	yes	—
475	#103	C₂³×C₇	1	4	yes	—
476	#103	C₂³×C₇	1	4	yes	—
477	#102	C₂³×C₇	1	8	yes	—
478	#102	C₂³×C₇	1	4	yes	—
479	#102	C₂³×C₇	1	4	yes	—
480	C₂²×C₇	—	2	1	no	—
481	C₂²×C₇	—	2	1	no	—
482	C₂²×C₇	—	2	1	no	—
483	C₂²×C₇	—	2	1	no	—
484	C₂²×C₇	—	2	1	no	—
485	C₂²×C₇	—	2	1	no	—
486	C₂²×C₇	—	2	4	no	—
487	C₂²×C₇	—	4	4	no	—
488	C₂²×C₇	—	4	2	no	—
489	C₂²×C₇	—	4	2	no	—
490	C₂²×C₇	—	8	4	no	—
491	C₂²×C₇	C₂×C₂²⋊Q₈	1	2	yes	—
492	C₂²×C₇	C₂×C₂²⋊Q₈	1	2	yes	—
493	C₂²×C₇	C₂×C₂²⋊Q₈	1	2	yes	—
494	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
495	C₄×C₇	—	2	4	no	—
496	C₄×C₇	—	2	4	no	—
497	C₄×C₇	—	2	4	no	—
498	C₄×C₇	—	4	4	no	—
499	C₄×C₇	—	4	2	no	—
500	C₄×C₇	—	4	2	no	—
501	C₄×C₇	—	4	2	no	—
502	C₄×C₇	—	4	2	no	—
503	C₄×C₇	C₂×C₂²⋊Q₈	1	2	yes	—
504	C₄×C₇	C₂×C₂²⋊Q₈	1	2	yes	—
505	C₄×C₇	C₂×C₂²⋊Q₈	1	2	yes	—
506	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
507	C₄×C₇	C₂²×#41	1	1	yes	characteristic
508	C₂⁴	—	2	1	no	—
509	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
510	#41	—	2	8	no	—
511	#41	—	2	8	no	—
512	#41	—	2	4	no	—
513	#41	—	2	4	no	—
514	#41	C₂×C₇×Q₈	1	8	yes	—
515	#41	C₂×C₇×Q₈	1	8	yes	complemented
516	C₂×Q₈	—	2	1	no	—
517	C₂×Q₈	—	2	8	no	—
518	C₂×Q₈	—	2	2	no	—
519	C₂×Q₈	—	8	4	no	—
520	C₂×Q₈	C₂×C₇×Q₈	1	4	yes	complemented
521	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
522	C₂×D₄	—	2	8	no	—
523	C₂×D₄	—	2	2	no	—
524	C₂×D₄	—	2	2	no	—
525	C₂×D₄	—	2	1	no	—
526	C₂×D₄	—	2	2	no	—
527	C₂×D₄	C₂×C₇×Q₈	1	4	yes	complemented
528	C₂×D₄	C₂⁴×C₇	1	2	yes	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	2	8	no	—
531	C₂²×C₄	—	2	2	no	—
532	C₂²×C₄	—	2	2	no	—
533	C₂²×C₄	—	2	1	no	—
534	C₂²×C₄	—	2	1	no	—
535	C₂²×C₄	—	2	2	no	—
536	C₂²×C₄	—	2	1	no	—
537	C₂²×C₄	—	2	1	no	—
538	C₂²×C₄	—	2	2	no	—
539	C₂²×C₄	—	2	2	no	—
540	C₂²×C₄	—	2	4	no	—
541	C₂²×C₄	—	2	4	no	—
542	C₂²×C₄	—	8	2	no	—
543	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
544	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
545	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
546	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
547	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
548	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
549	C₂²×C₄	C₇×#41	1	4	yes	—
550	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
551	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
552	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
553	M₁₆	—	2	4	no	—
554	M₁₆	—	2	4	no	—
555	M₁₆	—	2	4	no	—
556	M₁₆	—	2	4	no	—
557	C₂×C₈	—	2	4	no	—
558	C₂×C₈	—	2	4	no	—
559	C₂×C₈	C₂⁴×C₇	1	4	yes	—
560	C₂×C₈	C₂⁴×C₇	1	4	yes	—
561	C₄⋊C₄	—	2	1	no	—
562	C₄⋊C₄	—	2	1	no	—
563	C₄⋊C₄	—	2	2	no	—
564	C₄⋊C₄	—	2	2	no	—
565	C₄⋊C₄	—	2	2	no	—
566	C₄⋊C₄	—	2	4	no	—
567	C₄⋊C₄	—	2	4	no	—
568	C₄⋊C₄	—	2	8	no	—
569	C₄⋊C₄	—	2	8	no	—
570	C₄⋊C₄	—	2	8	no	—
571	C₄⋊C₄	—	2	1	no	—
572	C₄⋊C₄	—	2	1	no	—
573	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
574	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
575	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
576	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
577	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
578	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
579	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
580	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
581	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
582	C₂²⋊C₄	—	2	2	no	—
583	C₂²⋊C₄	—	2	2	no	—
584	C₂²⋊C₄	—	2	2	no	—
585	C₂²⋊C₄	—	2	2	no	—
586	C₂²⋊C₄	—	2	4	no	—
587	C₂²⋊C₄	—	2	4	no	—
588	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
589	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
590	C₄²	—	2	2	no	—
591	C₄²	—	2	4	no	—
592	C₄²	—	2	4	no	—
593	C₄²	—	2	2	no	—
594	C₄²	—	2	2	no	—
595	C₄²	—	2	2	no	—
596	C₄²	C₂×C₇×D₄	1	4	yes	—
597	C₄²	C₂⁴×C₇	1	4	yes	—
598	C₂×C₇	—	2	2	no	—
599	C₂×C₇	—	2	2	no	—
600	C₂×C₇	—	2	2	no	—
601	C₂×C₇	—	4	4	no	—
602	C₂×C₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
603	C₂×C₇	#3585	1	1	yes	characteristic
604	C₂×C₇	#3586	1	1	yes	characteristic
605	C₂³	—	2	2	no	—
606	C₂³	—	2	4	no	—
607	C₂³	—	4	1	no	—
608	C₂³	—	4	1	no	—
609	C₂³	—	8	1	no	—
610	C₂³	—	8	1	no	—
611	C₂³	C₇×C₂²⋊Q₈	1	4	yes	—
612	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
613	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
614	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
615	Q₈	—	2	2	no	—
616	Q₈	—	2	4	no	—
617	Q₈	—	8	8	no	—
618	Q₈	C₂²×C₇×Q₈	1	4	yes	—
619	D₄	—	2	8	no	—
620	D₄	—	2	8	no	—
621	D₄	—	2	2	no	—
622	D₄	—	2	4	no	—
623	D₄	C₂²×C₇×Q₈	1	4	yes	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	4	no	—
626	C₂×C₄	—	2	8	no	—
627	C₂×C₄	—	2	8	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	2	no	—
631	C₂×C₄	—	2	2	no	—
632	C₂×C₄	—	2	4	no	—
633	C₂×C₄	—	4	2	no	—
634	C₂×C₄	—	4	2	no	—
635	C₂×C₄	—	4	2	no	—
636	C₂×C₄	—	4	4	no	—
637	C₂×C₄	—	4	2	no	—
638	C₂×C₄	—	8	4	no	—
639	C₂×C₄	—	8	4	no	—
640	C₂×C₄	—	8	1	no	—
641	C₂×C₄	—	8	1	no	—
642	C₂×C₄	—	8	2	no	—
643	C₂×C₄	—	8	2	no	—
644	C₂×C₄	C₇×C₂²⋊Q₈	1	4	yes	—
645	C₂×C₄	C₇×C₂²⋊Q₈	1	4	yes	—
646	C₂×C₄	C₇×C₂²⋊Q₈	1	4	yes	—
647	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
648	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
649	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
650	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
651	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
652	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
653	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
654	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
655	C₂×C₄	C₂×C₇×#41	1	2	yes	—
656	C₂×C₄	C₂×C₇×#41	1	2	yes	—
657	C₂×C₄	C₂×C₇×#41	1	2	yes	—
658	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
659	C₈	—	2	4	no	—
660	C₈	—	2	4	no	—
661	C₇	#60493	1	1	yes	characteristic, direct product factor, Sylow
662	C₂²	—	2	1	no	—
663	C₂²	—	2	1	no	—
664	C₂²	—	2	1	no	—
665	C₂²	—	2	1	no	—
666	C₂²	—	2	1	no	—
667	C₂²	—	2	1	no	—
668	C₂²	—	2	4	no	—
669	C₂²	—	4	4	no	—
670	C₂²	—	4	2	no	—
671	C₂²	—	4	2	no	—
672	C₂²	—	8	4	no	—
673	C₂²	C₂×C₇×C₂²⋊Q₈	1	2	yes	—
674	C₂²	C₂×C₇×C₂²⋊Q₈	1	2	yes	—
675	C₂²	C₂×C₇×C₂²⋊Q₈	1	2	yes	—
676	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
677	C₄	—	2	4	no	—
678	C₄	—	2	4	no	—
679	C₄	—	2	4	no	—
680	C₄	—	4	4	no	—
681	C₄	—	4	2	no	—
682	C₄	—	4	2	no	—
683	C₄	—	4	2	no	—
684	C₄	—	4	2	no	—
685	C₄	C₂×C₇×C₂²⋊Q₈	1	2	yes	—
686	C₄	C₂×C₇×C₂²⋊Q₈	1	2	yes	—
687	C₄	C₂×C₇×C₂²⋊Q₈	1	2	yes	—
688	C₄	C₂³×C₇×Q₈	1	1	yes	characteristic
689	C₄	C₂²×C₇×#41	1	1	yes	characteristic
690	C₂	—	2	2	no	—
691	C₂	—	2	2	no	—
692	C₂	—	2	2	no	—
693	C₂	—	4	4	no	—
694	C₂	C₂²×C₇×C₂²⋊Q₈	1	1	yes	characteristic
695	C₂	C₇×#3585	1	1	yes	characteristic
696	C₂	C₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂²×C₇×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₇×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#544	C₂	semidirect
3. C₇×#3084	C₂	semidirect
4. C₇×#3083	C₂	semidirect
5. C₇×#3082	C₂	semidirect
6. C₇×#3081	C₂	semidirect
7. C₇×#3080	C₂	semidirect
8. C₇×#3079	C₂	semidirect
9. C₇×#2911	C₂	semidirect
10. C₂×C₇×#428	C₂	semidirect
11. C₇×#2905	C₂	semidirect
20. C₇×#544	C₂²	semidirect
22. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
24. C₂×C₇×#125	C₂²	semidirect
40. C₇×#479	C₂²	semidirect
41. C₇×#479	C₂²	semidirect
42. C₇×#478	C₂²	semidirect
43. C₇×#478	C₂²	semidirect
44. C₇×#477	C₂²	semidirect
45. C₇×#476	C₂²	semidirect
46. C₇×#475	C₂²	semidirect
47. C₇×#474	C₂²	semidirect
49. C₇×#428	C₂²	semidirect
50. C₇×#428	C₂²	semidirect
51. C₇×#428	C₂²	semidirect
52. C₇×#428	C₂²	semidirect
53. C₇×#428	C₂²	semidirect
55. C₂×C₇×#107	C₂²	semidirect
56. C₂×C₇×#106	C₂²	semidirect
58. C₇×#424	C₂²	semidirect
59. C₇×#423	C₂²	semidirect
69. #60493	C₇	direct (N × H)
73. C₂×C₇×#41	Q₈	semidirect
168. C₂×#544	C₂×C₇	semidirect
170. #3084	C₂×C₇	semidirect
171. #3083	C₂×C₇	semidirect
172. #3082	C₂×C₇	semidirect
173. #3081	C₂×C₇	semidirect
174. #3080	C₂×C₇	semidirect
175. #3079	C₂×C₇	semidirect
176. #2911	C₂×C₇	semidirect
177. C₂×#428	C₂×C₇	semidirect
178. #2905	C₂×C₇	semidirect
188. C₇×#41	C₂×Q₈	semidirect
193. C₂×C₇×Q₈	C₂×Q₈	semidirect
200. C₂×C₇×D₄	C₂×Q₈	semidirect
277. #544	C₂²×C₇	semidirect
279. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
281. C₂×#125	C₂²×C₇	semidirect
297. #479	C₂²×C₇	semidirect
298. #479	C₂²×C₇	semidirect
299. #478	C₂²×C₇	semidirect
300. #478	C₂²×C₇	semidirect
301. #477	C₂²×C₇	semidirect
302. #476	C₂²×C₇	semidirect
303. #475	C₂²×C₇	semidirect
304. #474	C₂²×C₇	semidirect
306. #428	C₂²×C₇	semidirect
307. #428	C₂²×C₇	semidirect
308. #428	C₂²×C₇	semidirect
309. #428	C₂²×C₇	semidirect
310. #428	C₂²×C₇	semidirect
312. C₂×#107	C₂²×C₇	semidirect
313. C₂×#106	C₂²×C₇	semidirect
315. #424	C₂²×C₇	semidirect
316. #423	C₂²×C₇	semidirect
385. C₂×#41	C₇×Q₈	semidirect
515. #41	C₂×C₇×Q₈	semidirect
520. C₂×Q₈	C₂×C₇×Q₈	semidirect
527. C₂×D₄	C₂×C₇×Q₈	semidirect
661. C₇	#60493	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#544	2
C₇×#3428	2
C₇×#3084	2
C₇×#3083	2
C₇×#3082	2
C₇×#3081	2
C₇×#3080	2
C₇×#3079	2
C₇×#2911	2
C₂×C₇×#428	2
C₇×#2905	2
C₇×#2896	2
C₂×C₇×#417	2
#60493	7

Subgroups — C7×#60493 #13237709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60493 #13237709