Subgroups — C₇×#3344 #11742653

Order 896.

344 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#554	C₂	1	1	yes	maximal, characteristic
2	C₇×#552	C₂	1	1	yes	maximal, characteristic
3	C₇×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#485	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#484	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#483	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#483	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#479	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#475	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#470	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#467	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#441	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#435	C₂	1	1	yes	maximal, characteristic
14	C₇×#428	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₇×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₇×#126	—	2	1	no	—
21	C₇×#125	—	2	1	no	—
22	C₇×#125	—	2	1	no	—
23	C₇×#125	C₂²	1	1	yes	characteristic
24	C₇×#123	—	2	1	no	—
25	C₇×C₂²⋊Q₈	—	2	1	no	—
26	C₇×C₂²⋊Q₈	—	2	1	no	—
27	C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
28	C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
29	C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
30	C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
31	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
32	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
33	C₄×C₇×D₄	—	2	1	no	—
34	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
35	C₇×#117	—	2	1	no	—
36	C₇×#117	C₂²	1	1	yes	characteristic
37	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
38	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
39	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
40	C₇×#107	C₂²	1	1	yes	characteristic
41	C₇×#107	C₂²	1	1	yes	characteristic
42	C₇×#107	C₂²	1	1	yes	characteristic
43	C₇×#106	C₂²	1	1	yes	characteristic
44	C₇×#106	C₂²	1	1	yes	characteristic
45	C₇×#106	C₂²	1	1	yes	characteristic
46	C₇×#106	C₂²	1	1	yes	characteristic
47	C₇×#106	C₂²	1	1	yes	characteristic
48	C₇×#106	C₂²	1	1	yes	characteristic
49	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
50	C₇×#103	C₂²	1	1	yes	characteristic
51	C₇×#103	C₂²	1	1	yes	characteristic
52	C₇×#103	C₂²	1	1	yes	characteristic
53	C₇×#103	C₂²	1	1	yes	characteristic
54	C₇×#103	C₂²	1	1	yes	characteristic
55	C₇×#103	C₂²	1	1	yes	characteristic
56	C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
57	C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
58	C₇×#97	C₂²	1	1	yes	characteristic
59	#3344	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×C₇×Q₈	—	2	1	no	—
61	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
62	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
63	C₂×C₇×D₄	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	—	2	1	no	—
65	C₂²×C₄×C₇	—	2	1	no	—
66	C₂²×C₄×C₇	—	2	1	no	—
67	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
69	C₇×Q₁₆	—	2	2	no	—
70	C₇×M₁₆	—	2	2	no	—
71	C₂×C₇×C₈	—	2	2	no	—
72	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
74	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
75	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
76	C₇×C₄⋊C₄	—	2	1	no	—
77	C₇×C₄⋊C₄	—	2	1	no	—
78	C₇×C₄⋊C₄	—	2	1	no	—
79	C₇×C₄⋊C₄	—	2	1	no	—
80	C₇×C₄⋊C₄	—	2	1	no	—
81	C₇×C₄⋊C₄	—	2	1	no	—
82	C₇×C₄⋊C₄	—	2	1	no	—
83	C₇×C₄⋊C₄	—	2	1	no	—
84	C₇×C₄⋊C₄	—	2	1	no	—
85	C₇×C₄⋊C₄	—	2	1	no	—
86	C₇×C₄⋊C₄	—	2	1	no	—
87	C₇×C₄⋊C₄	—	2	1	no	—
88	C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
89	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
90	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
92	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
93	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
94	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
95	C₇×C₂²⋊C₄	—	2	1	no	—
96	C₇×C₂²⋊C₄	—	2	1	no	—
97	C₇×C₂²⋊C₄	—	2	1	no	—
98	C₇×C₂²⋊C₄	—	2	1	no	—
99	C₇×C₂²⋊C₄	—	2	1	no	—
100	C₇×C₂²⋊C₄	D₄	1	1	yes	characteristic
101	C₇×C₂²⋊C₄	D₄	1	1	yes	characteristic
102	C₄²×C₇	—	2	1	no	—
103	C₄²×C₇	—	2	1	no	—
104	C₄²×C₇	—	2	1	no	—
105	C₄²×C₇	C₂³	1	1	yes	characteristic
106	#554	C₂×C₇	1	1	yes	characteristic
107	#552	C₂×C₇	1	1	yes	characteristic
108	#501	C₂×C₇	1	1	yes	characteristic, complemented
109	#485	C₂×C₇	1	1	yes	characteristic, complemented
110	#484	C₂×C₇	1	1	yes	characteristic, complemented
111	#483	C₂×C₇	1	1	yes	characteristic, complemented
112	#483	C₂×C₇	1	1	yes	characteristic, complemented
113	#479	C₂×C₇	1	1	yes	characteristic, complemented
114	#475	C₂×C₇	1	1	yes	characteristic, complemented
115	#470	C₂×C₇	1	1	yes	characteristic, complemented
116	#467	C₂×C₇	1	1	yes	characteristic, complemented
117	#441	C₂×C₇	1	1	yes	characteristic, complemented
118	#435	C₂×C₇	1	1	yes	characteristic
119	#428	C₂×C₇	1	1	yes	characteristic, complemented
120	C₂×#106	C₂×C₇	1	1	yes	characteristic, complemented
121	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
122	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
123	C₇×Q₈	—	2	1	no	—
124	C₇×Q₈	—	2	1	no	—
125	C₇×Q₈	—	2	1	no	—
126	C₇×Q₈	—	2	1	no	—
127	C₇×Q₈	—	8	1	no	—
128	C₇×D₄	—	2	2	no	—
129	C₂×C₄×C₇	—	2	1	no	—
130	C₂×C₄×C₇	—	2	1	no	—
131	C₂×C₄×C₇	—	2	1	no	—
132	C₂×C₄×C₇	—	2	1	no	—
133	C₂×C₄×C₇	—	2	1	no	—
134	C₂×C₄×C₇	—	2	1	no	—
135	C₂×C₄×C₇	—	2	1	no	—
136	C₂×C₄×C₇	—	2	1	no	—
137	C₂×C₄×C₇	—	2	1	no	—
138	C₂×C₄×C₇	—	2	1	no	—
139	C₂×C₄×C₇	—	2	1	no	—
140	C₂×C₄×C₇	—	2	1	no	—
141	C₂×C₄×C₇	—	4	2	no	—
142	C₂×C₄×C₇	—	8	1	no	—
143	C₂×C₄×C₇	—	8	1	no	—
144	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
148	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
149	C₇×C₈	—	2	1	no	—
150	C₇×C₈	—	2	1	no	—
151	C₇×C₈	—	2	1	no	—
152	C₇×C₈	#41	1	2	yes	—
153	C₂×Q₁₆	C₂²×C₇	1	1	yes	characteristic
154	C₂×M₁₆	C₂²×C₇	1	1	yes	characteristic
155	C₂²×C₈	C₂²×C₇	1	1	yes	characteristic
156	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
157	#126	—	2	1	no	—
158	#125	—	2	1	no	—
159	#125	—	2	1	no	—
160	#125	C₂²×C₇	1	1	yes	characteristic
161	#123	—	2	1	no	—
162	C₂²⋊Q₈	—	2	1	no	—
163	C₂²⋊Q₈	—	2	1	no	—
164	C₂²⋊Q₈	C₂²×C₇	1	1	yes	characteristic
165	C₂²⋊Q₈	C₂²×C₇	1	1	yes	characteristic
166	C₂²⋊Q₈	C₂²×C₇	1	1	yes	characteristic
167	C₂²⋊Q₈	C₂²×C₇	1	1	yes	characteristic
168	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
169	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	C₂²×C₇	1	1	yes	characteristic
172	#117	—	2	1	no	—
173	#117	C₂²×C₇	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
177	#107	C₂²×C₇	1	1	yes	characteristic
178	#107	C₂²×C₇	1	1	yes	characteristic
179	#107	C₂²×C₇	1	1	yes	characteristic
180	#106	C₂²×C₇	1	1	yes	characteristic
181	#106	C₂²×C₇	1	1	yes	characteristic
182	#106	C₂²×C₇	1	1	yes	characteristic
183	#106	C₂²×C₇	1	1	yes	characteristic
184	#106	C₂²×C₇	1	1	yes	characteristic
185	#106	C₂²×C₇	1	1	yes	characteristic
186	C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
187	#103	C₂²×C₇	1	1	yes	characteristic
188	#103	C₂²×C₇	1	1	yes	characteristic
189	#103	C₂²×C₇	1	1	yes	characteristic
190	#103	C₂²×C₇	1	1	yes	characteristic
191	#103	C₂²×C₇	1	1	yes	characteristic
192	#103	C₂²×C₇	1	1	yes	characteristic
193	C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
194	C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
195	#97	C₂²×C₇	1	1	yes	characteristic
196	C₂²×C₇	—	2	1	no	—
197	C₂²×C₇	—	2	1	no	—
198	C₂²×C₇	—	2	1	no	—
199	C₂²×C₇	—	2	1	no	—
200	C₂²×C₇	—	2	1	no	—
201	C₂²×C₇	#137	1	2	yes	—
202	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
203	C₄×C₇	—	2	1	no	—
204	C₄×C₇	—	2	1	no	—
205	C₄×C₇	—	2	1	no	—
206	C₄×C₇	—	2	1	no	—
207	C₄×C₇	—	4	1	no	—
208	C₄×C₇	—	4	1	no	—
209	C₄×C₇	—	4	1	no	—
210	C₄×C₇	—	4	1	no	—
211	C₄×C₇	—	4	1	no	—
212	C₄×C₇	—	4	1	no	—
213	C₄×C₇	—	4	1	no	—
214	C₄×C₇	—	4	1	no	—
215	C₄×C₇	C₂×#41	1	1	yes	characteristic
216	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
217	C₂×Q₈	—	2	1	no	—
218	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
219	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
220	C₂×D₄	C₇×D₄	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
225	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
226	Q₁₆	—	2	2	no	—
227	M₁₆	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
230	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
231	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
232	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
246	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
247	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
248	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
249	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
250	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
251	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	2	1	no	—
256	C₂²⋊C₄	—	2	1	no	—
257	C₂²⋊C₄	C₇×D₄	1	1	yes	characteristic
258	C₂²⋊C₄	C₇×D₄	1	1	yes	characteristic
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	C₂³×C₇	1	1	yes	characteristic
263	C₂×C₇	—	2	2	no	—
264	C₂×C₇	—	4	1	no	—
265	C₂×C₇	#547	1	1	yes	characteristic
266	C₂×C₇	C₂×#137	1	1	yes	characteristic
267	C₂×C₇	#577	1	1	yes	characteristic
268	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
269	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
270	Q₈	—	2	1	no	—
271	Q₈	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	8	1	no	—
275	D₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
292	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
293	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
294	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
295	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	C₇×#41	1	2	yes	—
300	C₇	#3344	1	1	yes	characteristic, direct product factor, Sylow
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	C₇×#137	1	2	yes	—
307	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	C₂×C₇×#41	1	1	yes	characteristic
321	C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	1	no	—
324	C₂	C₇×#547	1	1	yes	characteristic
325	C₂	C₂×C₇×#137	1	1	yes	characteristic
326	C₂	C₇×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
prime radical	C₂×C₇×D₄	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#501	C₂	semidirect
4. C₇×#485	C₂	semidirect
5. C₇×#484	C₂	semidirect
6. C₇×#483	C₂	semidirect
7. C₇×#483	C₂	semidirect
8. C₇×#479	C₂	semidirect
9. C₇×#475	C₂	semidirect
10. C₇×#470	C₂	semidirect
11. C₇×#467	C₂	semidirect
12. C₇×#441	C₂	semidirect
14. C₇×#428	C₂	semidirect
15. C₂×C₇×#106	C₂	semidirect
59. #3344	C₇	direct (N × H)
108. #501	C₂×C₇	semidirect
109. #485	C₂×C₇	semidirect
110. #484	C₂×C₇	semidirect
111. #483	C₂×C₇	semidirect
112. #483	C₂×C₇	semidirect
113. #479	C₂×C₇	semidirect
114. #475	C₂×C₇	semidirect
115. #470	C₂×C₇	semidirect
116. #467	C₂×C₇	semidirect
117. #441	C₂×C₇	semidirect
119. #428	C₂×C₇	semidirect
120. C₂×#106	C₂×C₇	semidirect
300. C₇	#3344	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×#554	2
C₇×#552	2
C₇×#501	2
C₇×#485	2
C₇×#484	2
C₇×#483	2
C₇×#479	2
C₇×#475	2
C₇×#470	2
C₇×#467	2
C₇×#441	2
C₇×#435	2
C₇×#428	2
C₂×C₇×#106	2
#3344	7

Subgroups — C7×#3344 #11742653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3344 #11742653