Subgroups — C₇×#19690 #13196906

Order 1792.

656 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3231	C₂	1	2	yes	maximal
2	C₇×#3185	C₂	1	2	yes	maximal
3	C₇×#3170	C₂	1	1	yes	maximal, characteristic
4	C₇×#3160	C₂	1	1	yes	maximal, characteristic
5	C₇×#1723	C₂	1	2	yes	maximal, complemented
6	C₇×#1723	C₂	1	2	yes	maximal, complemented
7	C₇×#1718	C₂	1	2	yes	maximal, complemented
8	C₇×#1716	C₂	1	2	yes	maximal, complemented
9	C₇×#1451	C₂	1	1	yes	maximal, characteristic
10	C₇×#533	C₂²	1	2	yes	—
11	C₇×#533	C₂²	1	2	yes	—
12	C₇×#500	C₂²	1	1	yes	characteristic
13	C₇×#500	C₂²	1	1	yes	characteristic
14	C₇×#500	C₂²	1	1	yes	characteristic
15	C₇×#500	C₂²	1	2	yes	—
16	C₇×#499	—	2	2	no	—
17	C₇×#498	C₂²	1	2	yes	—
18	C₇×#498	C₂²	1	2	yes	—
19	C₇×#498	C₂²	1	2	yes	—
20	C₇×#494	C₂²	1	1	yes	characteristic
21	C₇×#494	C₂²	1	2	yes	—
22	C₇×#487	C₂²	1	2	yes	—
23	C₇×#487	C₂²	1	2	yes	—
24	C₇×#487	C₂²	1	2	yes	—
25	C₇×#487	C₂²	1	2	yes	—
26	C₇×#487	C₂²	1	2	yes	—
27	C₇×#485	—	2	4	no	—
28	C₇×#479	—	2	4	no	—
29	C₇×#470	—	2	4	no	—
30	C₇×#467	—	2	2	no	—
31	C₇×#441	—	2	4	no	—
32	C₄×C₇×Q₁₆	—	2	2	no	—
33	C₇×#431	C₂²	1	1	yes	characteristic
34	C₇×#427	—	2	2	no	—
35	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
37	C₇×#322	C₂²	1	2	yes	—
38	C₇×#322	C₂²	1	2	yes	—
39	#19690	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₇×Q₁₆	—	2	2	no	—
41	C₂×C₇×Q₁₆	—	2	4	no	—
42	C₂×C₇×M₁₆	D₄	1	2	yes	—
43	C₂²×C₇×C₈	D₄	1	2	yes	—
44	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
45	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
46	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₇×#125	—	2	2	no	—
49	C₇×#125	—	2	2	no	—
50	C₇×C₂²⋊Q₈	—	2	2	no	—
51	C₇×C₂²⋊Q₈	—	2	2	no	—
52	C₇×C₂²⋊Q₈	—	2	2	no	—
53	C₇×C₂²⋊Q₈	—	2	2	no	—
54	C₄×C₇×Q₈	—	2	2	no	—
55	C₄×C₇×Q₈	—	2	2	no	—
56	C₄×C₇×Q₈	—	2	2	no	—
57	C₄×C₇×Q₈	—	2	2	no	—
58	C₇×#117	—	2	2	no	—
59	C₇×#117	—	2	2	no	—
60	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
61	C₇×#107	—	2	2	no	—
62	C₇×#107	—	2	4	no	—
63	C₇×#107	—	2	2	no	—
64	C₇×#107	—	2	2	no	—
65	C₇×#106	—	2	4	no	—
66	C₇×#106	—	2	4	no	—
67	C₇×#106	—	2	4	no	—
68	C₇×C₄⋊C₈	D₄	1	2	yes	—
69	C₇×#103	—	2	4	no	—
70	C₇×#103	—	2	4	no	—
71	C₇×#103	—	2	4	no	—
72	C₇×#103	—	2	4	no	—
73	C₇×#103	—	2	4	no	—
74	C₇×C₂²⋊C₈	D₄	1	2	yes	—
75	C₇×#97	D₄	1	2	yes	—
76	C₄×C₇×C₈	D₄	1	2	yes	—
77	C₄×C₇×C₈	C₂³	1	2	yes	—
78	C₄×C₇×C₈	C₂³	1	2	yes	—
79	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
80	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
81	#3231	C₂×C₇	1	2	yes	—
82	#3185	C₂×C₇	1	2	yes	—
83	#3170	C₂×C₇	1	1	yes	characteristic
84	#3160	C₂×C₇	1	1	yes	characteristic
85	#1723	C₂×C₇	1	2	yes	complemented
86	#1723	C₂×C₇	1	2	yes	complemented
87	#1718	C₂×C₇	1	2	yes	complemented
88	#1716	C₂×C₇	1	2	yes	complemented
89	#1451	C₂×C₇	1	1	yes	characteristic
90	C₂×C₇×Q₈	—	2	2	no	—
91	C₂×C₇×Q₈	—	2	2	no	—
92	C₂×C₇×Q₈	—	2	2	no	—
93	C₂×C₇×Q₈	—	2	2	no	—
94	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
97	C₇×Q₁₆	—	4	8	no	—
98	C₇×Q₁₆	—	4	4	no	—
99	C₇×M₁₆	—	4	2	no	—
100	C₂×C₇×C₈	—	2	4	no	—
101	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
102	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
103	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
104	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
105	C₇×C₄⋊C₄	—	2	2	no	—
106	C₇×C₄⋊C₄	—	2	2	no	—
107	C₇×C₄⋊C₄	—	2	2	no	—
108	C₇×C₄⋊C₄	—	2	2	no	—
109	C₇×C₄⋊C₄	—	2	2	no	—
110	C₇×C₄⋊C₄	—	2	2	no	—
111	C₇×C₄⋊C₄	—	2	2	no	—
112	C₇×C₄⋊C₄	—	2	2	no	—
113	C₇×C₄⋊C₄	—	4	2	no	—
114	C₇×C₄⋊C₄	—	4	2	no	—
115	C₇×C₄⋊C₄	—	4	2	no	—
116	C₇×C₄⋊C₄	—	4	2	no	—
117	C₇×C₂²⋊C₄	—	4	2	no	—
118	C₇×C₂²⋊C₄	—	4	2	no	—
119	C₄²×C₇	—	4	2	no	—
120	C₄²×C₇	—	4	2	no	—
121	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₇	C₂⁴	1	1	yes	characteristic
125	#533	C₂²×C₇	1	2	yes	—
126	#533	C₂²×C₇	1	2	yes	—
127	#500	C₂²×C₇	1	1	yes	characteristic
128	#500	C₂²×C₇	1	1	yes	characteristic
129	#500	C₂²×C₇	1	1	yes	characteristic
130	#500	C₂²×C₇	1	2	yes	—
131	#499	—	2	2	no	—
132	#498	C₂²×C₇	1	2	yes	—
133	#498	C₂²×C₇	1	2	yes	—
134	#498	C₂²×C₇	1	2	yes	—
135	#494	C₂²×C₇	1	1	yes	characteristic
136	#494	C₂²×C₇	1	2	yes	—
137	#487	C₂²×C₇	1	2	yes	—
138	#487	C₂²×C₇	1	2	yes	—
139	#487	C₂²×C₇	1	2	yes	—
140	#487	C₂²×C₇	1	2	yes	—
141	#487	C₂²×C₇	1	2	yes	—
142	#485	—	2	4	no	—
143	#479	—	2	4	no	—
144	#470	—	2	4	no	—
145	#467	—	2	2	no	—
146	#441	—	2	4	no	—
147	C₄×Q₁₆	—	2	2	no	—
148	#431	C₂²×C₇	1	1	yes	characteristic
149	#427	—	2	2	no	—
150	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
152	#322	C₂²×C₇	1	2	yes	—
153	#322	C₂²×C₇	1	2	yes	—
154	C₂³×C₇	#127	1	1	yes	characteristic
155	C₇×Q₈	—	4	4	no	—
156	C₇×Q₈	—	4	4	no	—
157	C₇×Q₈	—	4	4	no	—
158	C₇×Q₈	—	4	4	no	—
159	C₂×C₄×C₇	—	2	2	no	—
160	C₂×C₄×C₇	—	2	2	no	—
161	C₂×C₄×C₇	—	2	2	no	—
162	C₂×C₄×C₇	—	4	2	no	—
163	C₂×C₄×C₇	—	4	2	no	—
164	C₂×C₄×C₇	—	4	2	no	—
165	C₂×C₄×C₇	—	4	2	no	—
166	C₂×C₄×C₇	#127	1	1	yes	characteristic
167	C₂×C₄×C₇	#127	1	1	yes	characteristic
168	C₂×C₄×C₇	#127	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
172	C₇×C₈	—	2	2	no	—
173	C₇×C₈	—	2	4	no	—
174	C₇×C₈	—	2	4	no	—
175	C₇×C₈	#137	1	4	yes	—
176	C₂×Q₁₆	—	2	2	no	—
177	C₂×Q₁₆	—	2	4	no	—
178	C₂×M₁₆	C₇×D₄	1	2	yes	—
179	C₂²×C₈	C₇×D₄	1	2	yes	—
180	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
181	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
182	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
183	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
184	#125	—	2	2	no	—
185	#125	—	2	2	no	—
186	C₂²⋊Q₈	—	2	2	no	—
187	C₂²⋊Q₈	—	2	2	no	—
188	C₂²⋊Q₈	—	2	2	no	—
189	C₂²⋊Q₈	—	2	2	no	—
190	C₄×Q₈	—	2	2	no	—
191	C₄×Q₈	—	2	2	no	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	—	2	2	no	—
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
197	#107	—	2	2	no	—
198	#107	—	2	2	no	—
199	#107	—	2	2	no	—
200	#107	—	2	4	no	—
201	#106	—	2	4	no	—
202	#106	—	2	4	no	—
203	#106	—	2	4	no	—
204	C₄⋊C₈	C₇×D₄	1	2	yes	—
205	#103	—	2	4	no	—
206	#103	—	2	4	no	—
207	#103	—	2	4	no	—
208	#103	—	2	4	no	—
209	#103	—	2	4	no	—
210	C₂²⋊C₈	C₇×D₄	1	2	yes	—
211	#97	C₇×D₄	1	2	yes	—
212	C₄×C₈	C₇×D₄	1	2	yes	—
213	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
214	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
215	C₄×C₈	C₂³×C₇	1	2	yes	—
216	C₄×C₈	C₂³×C₇	1	2	yes	—
217	C₂²×C₇	—	2	1	no	—
218	C₂²×C₇	—	2	1	no	—
219	C₂²×C₇	—	2	1	no	—
220	C₂²×C₇	C₂×#127	1	1	yes	characteristic
221	C₄×C₇	—	2	1	no	—
222	C₄×C₇	—	2	1	no	—
223	C₄×C₇	—	2	1	no	—
224	C₄×C₇	—	8	2	no	—
225	C₄×C₇	—	8	2	no	—
226	C₄×C₇	—	8	2	no	—
227	C₄×C₇	—	8	2	no	—
228	C₄×C₇	C₂×#137	1	2	yes	—
229	C₄×C₇	#575	1	2	yes	—
230	C₄×C₇	#578	1	2	yes	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
238	Q₁₆	—	4	8	no	—
239	Q₁₆	—	4	4	no	—
240	M₁₆	—	4	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
243	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
244	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
245	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₄²	—	4	2	no	—
261	C₄²	—	4	2	no	—
262	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
263	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
264	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
265	C₄²	C₂⁴×C₇	1	1	yes	characteristic
266	C₂×C₇	—	4	1	no	—
267	C₂×C₇	#3150	1	1	yes	characteristic
268	C₂×C₇	#3152	1	1	yes	characteristic
269	C₂×C₇	#3156	1	1	yes	characteristic
270	C₂³	C₇×#127	1	1	yes	characteristic
271	Q₈	—	4	4	no	—
272	Q₈	—	4	4	no	—
273	Q₈	—	4	4	no	—
274	Q₈	—	4	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	C₇×#127	1	1	yes	characteristic
283	C₂×C₄	C₇×#127	1	1	yes	characteristic
284	C₂×C₄	C₇×#127	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
286	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
287	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	—	2	4	no	—
290	C₈	—	2	4	no	—
291	C₈	C₇×#137	1	4	yes	—
292	C₇	#19690	1	1	yes	characteristic, direct product factor, Sylow
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	C₂×C₇×#127	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	2	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	C₂×C₇×#137	1	2	yes	—
305	C₄	C₇×#575	1	2	yes	—
306	C₄	C₇×#578	1	2	yes	—
307	C₂	—	4	1	no	—
308	C₂	C₇×#3150	1	1	yes	characteristic
309	C₂	C₇×#3152	1	1	yes	characteristic
310	C₂	C₇×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂³×C₇	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₇×#1723	C₂	semidirect
6. C₇×#1723	C₂	semidirect
7. C₇×#1718	C₂	semidirect
8. C₇×#1716	C₂	semidirect
39. #19690	C₇	direct (N × H)
85. #1723	C₂×C₇	semidirect
86. #1723	C₂×C₇	semidirect
87. #1718	C₂×C₇	semidirect
88. #1716	C₂×C₇	semidirect
292. C₇	#19690	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×#3231	2
C₇×#3185	2
C₇×#3170	2
C₇×#3160	2
C₇×#1723	2
C₇×#1718	2
C₇×#1716	2
C₇×#1451	2
#19690	7

Subgroups — C7×#19690 #13196906

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19690 #13196906