Subgroups — C₇×#60401 #13237617

Order 1792.

4854 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×H(7,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3485	C₂	1	3	yes	maximal, complemented
3	C₇×#3028	C₂	1	4	yes	maximal, complemented
4	C₇×#3026	C₂	1	8	yes	maximal, complemented
5	C₇×#3025	C₂	1	4	yes	maximal, complemented
6	C₂×C₇×#458	C₂	1	8	yes	maximal, complemented
7	C₇×#2883	C₂	1	3	yes	maximal, complemented
8	C₇×#585	—	2	12	no	—
9	C₇×#585	C₂²	1	4	yes	—
10	C₂×C₇×H(5,𝔽₂)	—	2	12	no	—
11	C₂×C₇×H(5,𝔽₂)	C₂²	1	8	yes	—
12	C₂×C₇×H(5,𝔽₂)	C₂²	1	3	yes	—
13	C₂²×C₇×#41	C₂²	1	1	yes	characteristic, complemented
14	C₇×#547	—	2	12	no	—
15	C₇×#546	—	2	12	no	—
16	C₇×#536	C₂²	1	6	yes	complemented
17	C₇×#534	C₂²	1	12	yes	complemented
18	C₇×#525	C₂²	1	6	yes	complemented
19	C₂×C₇×#123	C₂²	1	12	yes	complemented
20	C₇×#518	C₂²	1	3	yes	complemented
21	C₇×#458	C₂²	1	16	yes	complemented
22	C₇×#458	C₂²	1	32	yes	complemented
23	C₇×B(4,𝔽₂)	C₂²	1	16	yes	complemented
24	C₇×#410	C₂²	1	12	yes	complemented
25	C₂×C₇×#99	C₂²	1	24	yes	complemented
26	#60401	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₇×2₋¹⁺⁴	—	2	16	no	—
28	C₇×2₋¹⁺⁴	—	2	12	no	—
29	C₇×H(5,𝔽₂)	—	2	48	no	—
30	C₇×H(5,𝔽₂)	—	2	12	no	—
31	C₇×H(5,𝔽₂)	—	2	48	no	—
32	C₇×H(5,𝔽₂)	—	2	24	no	—
33	C₇×H(5,𝔽₂)	C₂³	1	16	yes	—
34	C₂×C₇×#41	—	2	48	no	—
35	C₂×C₇×#41	—	2	12	no	—
36	C₂×C₇×#41	—	2	24	no	—
37	C₂×C₇×#41	—	2	24	no	—
38	C₂×C₇×#41	D₄	1	12	yes	complemented
39	C₂×C₇×#41	C₂³	1	6	yes	—
40	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic, complemented
41	C₂²×C₇×D₄	—	2	16	no	—
42	C₂²×C₇×D₄	—	2	24	no	—
43	C₂²×C₇×D₄	D₄	1	12	yes	complemented
44	C₂²×C₇×D₄	C₂³	1	12	yes	—
45	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
46	C₂²×C₇×D₄	C₂³	1	2	yes	complemented
47	C₂³×C₄×C₇	C₂³	1	2	yes	complemented
48	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic, complemented
49	C₇×C₄⋊Q₈	—	2	6	no	—
50	C₇×#124	—	2	6	no	—
51	C₇×#123	—	2	24	no	—
52	C₇×#123	—	2	12	no	—
53	C₇×#123	C₂³	1	12	yes	complemented
54	C₇×#123	C₂³	1	24	yes	complemented
55	C₇×C₂²⋊Q₈	—	2	6	no	—
56	C₇×C₂²⋊Q₈	—	2	12	no	—
57	C₇×#121	—	2	6	no	—
58	C₇×#121	—	2	24	no	—
59	C₇×#120	—	2	12	no	—
60	C₇×#120	C₂³	1	12	yes	complemented
61	C₄×C₇×D₄	—	2	12	no	—
62	C₄×C₇×D₄	—	2	12	no	—
63	C₇×#117	C₂³	1	6	yes	complemented
64	C₂×C₇×C₄⋊C₄	—	2	12	no	—
65	C₂×C₇×C₂²⋊C₄	—	2	12	no	—
66	C₂×C₇×C₂²⋊C₄	C₂³	1	12	yes	complemented
67	C₇×#99	C₂³	1	48	yes	complemented
68	H(7,𝔽₂)	C₂×C₇	1	1	yes	characteristic, complemented
69	#3485	C₂×C₇	1	3	yes	complemented
70	#3028	C₂×C₇	1	4	yes	complemented
71	#3026	C₂×C₇	1	8	yes	complemented
72	#3025	C₂×C₇	1	4	yes	complemented
73	C₂×#458	C₂×C₇	1	8	yes	complemented
74	#2883	C₂×C₇	1	3	yes	complemented
75	C₂⁴×C₇	—	2	8	no	—
76	C₂⁴×C₇	C₂×D₄	1	12	yes	—
77	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
78	C₂⁴×C₇	C₂⁴	1	2	yes	—
79	C₇×#41	—	2	96	no	—
80	C₇×#41	—	2	48	no	—
81	C₇×#41	—	2	16	no	—
82	C₇×#41	—	2	48	no	—
83	C₇×#41	—	2	48	no	—
84	C₇×#41	—	2	24	no	—
85	C₇×#41	—	4	16	no	—
86	C₂×C₇×Q₈	—	2	24	no	—
87	C₂×C₇×Q₈	—	2	8	no	—
88	C₂×C₇×Q₈	—	2	6	no	—
89	C₂×C₇×Q₈	C₂×D₄	1	6	yes	—
90	C₂×C₇×D₄	—	2	96	no	—
91	C₂×C₇×D₄	—	2	24	no	—
92	C₂×C₇×D₄	—	2	24	no	—
93	C₂×C₇×D₄	—	2	48	no	—
94	C₂×C₇×D₄	—	2	48	no	—
95	C₂×C₇×D₄	—	2	48	no	—
96	C₂×C₇×D₄	—	2	12	no	—
97	C₂×C₇×D₄	—	2	12	no	—
98	C₂×C₇×D₄	C₂×D₄	1	24	yes	—
99	C₂×C₇×D₄	C₂×D₄	1	6	yes	—
100	C₂×C₇×D₄	C₂×D₄	1	12	yes	—
101	C₂×C₇×D₄	C₂⁴	1	12	yes	—
102	C₂²×C₄×C₇	—	2	24	no	—
103	C₂²×C₄×C₇	—	2	16	no	—
104	C₂²×C₄×C₇	—	2	12	no	—
105	C₂²×C₄×C₇	—	2	12	no	—
106	C₂²×C₄×C₇	—	2	6	no	—
107	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
108	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
109	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
110	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
111	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
112	C₇×C₄⋊C₄	—	2	12	no	—
113	C₇×C₄⋊C₄	—	2	24	no	—
114	C₇×C₄⋊C₄	—	2	6	no	—
115	C₇×C₂²⋊C₄	—	2	12	no	—
116	C₇×C₂²⋊C₄	—	2	24	no	—
117	C₇×C₂²⋊C₄	—	2	12	no	—
118	C₇×C₂²⋊C₄	—	2	12	no	—
119	C₇×C₂²⋊C₄	C₂⁴	1	12	yes	—
120	C₄²×C₇	—	2	6	no	—
121	#585	—	2	12	no	—
122	#585	C₂²×C₇	1	4	yes	—
123	C₂×H(5,𝔽₂)	—	2	12	no	—
124	C₂×H(5,𝔽₂)	C₂²×C₇	1	8	yes	—
125	C₂×H(5,𝔽₂)	C₂²×C₇	1	3	yes	—
126	C₂²×#41	C₂²×C₇	1	1	yes	characteristic, complemented
127	#547	—	2	12	no	—
128	#546	—	2	12	no	—
129	#536	C₂²×C₇	1	6	yes	complemented
130	#534	C₂²×C₇	1	12	yes	complemented
131	#525	C₂²×C₇	1	6	yes	complemented
132	C₂×#123	C₂²×C₇	1	12	yes	complemented
133	#518	C₂²×C₇	1	3	yes	complemented
134	#458	C₂²×C₇	1	16	yes	complemented
135	#458	C₂²×C₇	1	32	yes	complemented
136	B(4,𝔽₂)	C₂²×C₇	1	16	yes	complemented
137	#410	C₂²×C₇	1	12	yes	complemented
138	C₂×#99	C₂²×C₇	1	24	yes	complemented
139	C₂³×C₇	—	2	16	no	—
140	C₂³×C₇	—	2	24	no	—
141	C₂³×C₇	—	2	3	no	—
142	C₂³×C₇	—	8	2	no	—
143	C₂³×C₇	—	8	1	no	—
144	C₂³×C₇	—	8	12	no	—
145	C₂³×C₇	—	16	8	no	—
146	C₂³×C₇	C₂²×D₄	1	12	yes	—
147	C₂³×C₇	C₂²×D₄	1	12	yes	—
148	C₂³×C₇	C₂⁵	1	1	yes	characteristic
149	C₇×Q₈	—	2	16	no	—
150	C₇×Q₈	—	2	12	no	—
151	C₇×Q₈	—	4	4	no	—
152	C₇×D₄	—	2	48	no	—
153	C₇×D₄	—	2	48	no	—
154	C₇×D₄	—	2	24	no	—
155	C₇×D₄	—	2	12	no	—
156	C₇×D₄	—	4	8	no	—
157	C₇×D₄	—	8	12	no	—
158	C₇×D₄	#120	1	16	yes	—
159	C₂×C₄×C₇	—	2	48	no	—
160	C₂×C₄×C₇	—	2	24	no	—
161	C₂×C₄×C₇	—	2	24	no	—
162	C₂×C₄×C₇	—	2	12	no	—
163	C₂×C₄×C₇	—	2	3	no	—
164	C₂×C₄×C₇	—	2	6	no	—
165	C₂×C₄×C₇	—	4	8	no	—
166	C₂×C₄×C₇	—	4	4	no	—
167	C₂×C₄×C₇	—	8	12	no	—
168	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
169	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
170	2₋¹⁺⁴	—	2	16	no	—
171	2₋¹⁺⁴	—	2	12	no	—
172	H(5,𝔽₂)	—	2	48	no	—
173	H(5,𝔽₂)	—	2	12	no	—
174	H(5,𝔽₂)	—	2	48	no	—
175	H(5,𝔽₂)	—	2	24	no	—
176	H(5,𝔽₂)	C₂³×C₇	1	16	yes	—
177	C₂×#41	—	2	48	no	—
178	C₂×#41	—	2	12	no	—
179	C₂×#41	—	2	24	no	—
180	C₂×#41	—	2	24	no	—
181	C₂×#41	C₇×D₄	1	12	yes	complemented
182	C₂×#41	C₂³×C₇	1	6	yes	—
183	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic, complemented
184	C₂²×D₄	—	2	16	no	—
185	C₂²×D₄	—	2	24	no	—
186	C₂²×D₄	C₇×D₄	1	12	yes	complemented
187	C₂²×D₄	C₂³×C₇	1	12	yes	—
188	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
189	C₂²×D₄	C₂³×C₇	1	2	yes	complemented
190	C₂³×C₄	C₂³×C₇	1	2	yes	complemented
191	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic, complemented
192	C₄⋊Q₈	—	2	6	no	—
193	#124	—	2	6	no	—
194	#123	—	2	24	no	—
195	#123	—	2	12	no	—
196	#123	C₂³×C₇	1	12	yes	complemented
197	#123	C₂³×C₇	1	24	yes	complemented
198	C₂²⋊Q₈	—	2	6	no	—
199	C₂²⋊Q₈	—	2	12	no	—
200	#121	—	2	6	no	—
201	#121	—	2	24	no	—
202	#120	—	2	12	no	—
203	#120	C₂³×C₇	1	12	yes	complemented
204	C₄×D₄	—	2	12	no	—
205	C₄×D₄	—	2	12	no	—
206	#117	C₂³×C₇	1	6	yes	complemented
207	C₂×C₄⋊C₄	—	2	12	no	—
208	C₂×C₂²⋊C₄	—	2	12	no	—
209	C₂×C₂²⋊C₄	C₂³×C₇	1	12	yes	complemented
210	#99	C₂³×C₇	1	48	yes	complemented
211	C₂²×C₇	—	2	12	no	—
212	C₂²×C₇	—	4	12	no	—
213	C₂²×C₇	—	4	1	no	—
214	C₂²×C₇	—	4	12	no	—
215	C₂²×C₇	—	4	6	no	—
216	C₂²×C₇	—	4	1	no	—
217	C₂²×C₇	—	8	24	no	—
218	C₂²×C₇	—	8	16	no	—
219	C₂²×C₇	C₂×#120	1	8	yes	—
220	C₂²×C₇	C₂³×D₄	1	3	yes	—
221	C₄×C₇	—	2	12	no	—
222	C₄×C₇	—	4	12	no	—
223	C₄×C₇	—	4	1	no	—
224	C₄×C₇	—	4	2	no	—
225	C₄×C₇	C₂×#120	1	4	yes	—
226	C₂⁴	—	2	8	no	—
227	C₂⁴	C₂×C₇×D₄	1	12	yes	—
228	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
229	C₂⁴	C₂⁴×C₇	1	2	yes	—
230	#41	—	2	96	no	—
231	#41	—	2	48	no	—
232	#41	—	2	16	no	—
233	#41	—	2	48	no	—
234	#41	—	2	48	no	—
235	#41	—	2	24	no	—
236	#41	—	4	16	no	—
237	C₂×Q₈	—	2	24	no	—
238	C₂×Q₈	—	2	8	no	—
239	C₂×Q₈	—	2	6	no	—
240	C₂×Q₈	C₂×C₇×D₄	1	6	yes	—
241	C₂×D₄	—	2	96	no	—
242	C₂×D₄	—	2	24	no	—
243	C₂×D₄	—	2	24	no	—
244	C₂×D₄	—	2	48	no	—
245	C₂×D₄	—	2	48	no	—
246	C₂×D₄	—	2	48	no	—
247	C₂×D₄	—	2	12	no	—
248	C₂×D₄	—	2	12	no	—
249	C₂×D₄	C₂×C₇×D₄	1	24	yes	—
250	C₂×D₄	C₂×C₇×D₄	1	6	yes	—
251	C₂×D₄	C₂×C₇×D₄	1	12	yes	—
252	C₂×D₄	C₂⁴×C₇	1	12	yes	—
253	C₂²×C₄	—	2	24	no	—
254	C₂²×C₄	—	2	16	no	—
255	C₂²×C₄	—	2	12	no	—
256	C₂²×C₄	—	2	12	no	—
257	C₂²×C₄	—	2	6	no	—
258	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
259	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
260	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
261	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
262	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
263	C₄⋊C₄	—	2	12	no	—
264	C₄⋊C₄	—	2	24	no	—
265	C₄⋊C₄	—	2	6	no	—
266	C₂²⋊C₄	—	2	12	no	—
267	C₂²⋊C₄	—	2	24	no	—
268	C₂²⋊C₄	—	2	12	no	—
269	C₂²⋊C₄	—	2	12	no	—
270	C₂²⋊C₄	C₂⁴×C₇	1	12	yes	—
271	C₄²	—	2	6	no	—
272	C₂×C₇	—	2	3	no	—
273	C₂×C₇	—	2	8	no	—
274	C₂×C₇	—	4	12	no	—
275	C₂×C₇	—	8	1	no	—
276	C₂×C₇	C₂²×#120	1	1	yes	characteristic
277	C₂³	—	2	16	no	—
278	C₂³	—	2	24	no	—
279	C₂³	—	2	3	no	—
280	C₂³	—	8	2	no	—
281	C₂³	—	8	1	no	—
282	C₂³	—	8	12	no	—
283	C₂³	—	16	8	no	—
284	C₂³	C₂²×C₇×D₄	1	12	yes	—
285	C₂³	C₂²×C₇×D₄	1	12	yes	—
286	C₂³	C₂⁵×C₇	1	1	yes	characteristic
287	Q₈	—	2	16	no	—
288	Q₈	—	2	12	no	—
289	Q₈	—	4	4	no	—
290	D₄	—	2	48	no	—
291	D₄	—	2	24	no	—
292	D₄	—	2	48	no	—
293	D₄	—	2	12	no	—
294	D₄	—	4	8	no	—
295	D₄	—	8	12	no	—
296	D₄	C₇×#120	1	16	yes	—
297	C₂×C₄	—	2	48	no	—
298	C₂×C₄	—	2	24	no	—
299	C₂×C₄	—	2	24	no	—
300	C₂×C₄	—	2	12	no	—
301	C₂×C₄	—	2	3	no	—
302	C₂×C₄	—	2	6	no	—
303	C₂×C₄	—	4	8	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	8	12	no	—
306	C₂×C₄	C₂²×C₇×D₄	1	12	yes	—
307	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
308	C₇	#60401	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²	—	2	12	no	—
310	C₂²	—	4	12	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	12	no	—
313	C₂²	—	4	6	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	8	24	no	—
316	C₂²	—	8	16	no	—
317	C₂²	C₂×C₇×#120	1	8	yes	—
318	C₂²	C₂³×C₇×D₄	1	3	yes	—
319	C₄	—	2	12	no	—
320	C₄	—	4	12	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	2	no	—
323	C₄	C₂×C₇×#120	1	4	yes	—
324	C₂	—	2	3	no	—
325	C₂	—	2	8	no	—
326	C₂	—	4	12	no	—
327	C₂	—	8	1	no	—
328	C₂	C₂²×C₇×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	C₂²×#120
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×H(7,𝔽₂)	C₂	semidirect
2. C₇×#3485	C₂	semidirect
3. C₇×#3028	C₂	semidirect
4. C₇×#3026	C₂	semidirect
5. C₇×#3025	C₂	semidirect
6. C₂×C₇×#458	C₂	semidirect
7. C₇×#2883	C₂	semidirect
13. C₂²×C₇×#41	C₂²	semidirect
16. C₇×#536	C₂²	semidirect
17. C₇×#534	C₂²	semidirect
18. C₇×#525	C₂²	semidirect
19. C₂×C₇×#123	C₂²	semidirect
20. C₇×#518	C₂²	semidirect
21. C₇×#458	C₂²	semidirect
22. C₇×#458	C₂²	semidirect
23. C₇×B(4,𝔽₂)	C₂²	semidirect
24. C₇×#410	C₂²	semidirect
25. C₂×C₇×#99	C₂²	semidirect
26. #60401	C₇	direct (N × H)
38. C₂×C₇×#41	D₄	semidirect
40. C₂²×C₇×Q₈	C₂³	semidirect
43. C₂²×C₇×D₄	D₄	semidirect
46. C₂²×C₇×D₄	C₂³	semidirect
47. C₂³×C₄×C₇	C₂³	semidirect
48. C₂³×C₄×C₇	C₂³	semidirect
53. C₇×#123	C₂³	semidirect
54. C₇×#123	C₂³	semidirect
60. C₇×#120	C₂³	semidirect
63. C₇×#117	C₂³	semidirect
66. C₂×C₇×C₂²⋊C₄	C₂³	semidirect
67. C₇×#99	C₂³	semidirect
68. H(7,𝔽₂)	C₂×C₇	semidirect
69. #3485	C₂×C₇	semidirect
70. #3028	C₂×C₇	semidirect
71. #3026	C₂×C₇	semidirect
72. #3025	C₂×C₇	semidirect
73. C₂×#458	C₂×C₇	semidirect
74. #2883	C₂×C₇	semidirect
126. C₂²×#41	C₂²×C₇	semidirect
129. #536	C₂²×C₇	semidirect
130. #534	C₂²×C₇	semidirect
131. #525	C₂²×C₇	semidirect
132. C₂×#123	C₂²×C₇	semidirect
133. #518	C₂²×C₇	semidirect
134. #458	C₂²×C₇	semidirect
135. #458	C₂²×C₇	semidirect
136. B(4,𝔽₂)	C₂²×C₇	semidirect
137. #410	C₂²×C₇	semidirect
138. C₂×#99	C₂²×C₇	semidirect
181. C₂×#41	C₇×D₄	semidirect
183. C₂²×Q₈	C₂³×C₇	semidirect
186. C₂²×D₄	C₇×D₄	semidirect
189. C₂²×D₄	C₂³×C₇	semidirect
190. C₂³×C₄	C₂³×C₇	semidirect
191. C₂³×C₄	C₂³×C₇	semidirect
196. #123	C₂³×C₇	semidirect
197. #123	C₂³×C₇	semidirect
203. #120	C₂³×C₇	semidirect
206. #117	C₂³×C₇	semidirect
209. C₂×C₂²⋊C₄	C₂³×C₇	semidirect
210. #99	C₂³×C₇	semidirect
308. C₇	#60401	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×H(7,𝔽₂)	2
C₇×#3485	2
C₇×#3028	2
C₇×#3026	2
C₇×#3025	2
C₂×C₇×#458	2
C₇×#2883	2
#60401	7

Subgroups — C7×#60401 #13237617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60401 #13237617