Subgroups — C₇×#19751 #13196967

Order 1792.

684 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3185	C₂	1	1	yes	maximal, characteristic
2	C₇×#3185	C₂	1	2	yes	maximal
3	C₇×#3165	C₂	1	2	yes	maximal
4	C₇×#3139	C₂	1	1	yes	maximal, characteristic
5	C₇×#1723	C₂	1	2	yes	maximal, complemented
6	C₇×#1718	C₂	1	2	yes	maximal, complemented
7	C₇×#1716	C₂	1	2	yes	maximal, complemented
8	C₇×#1707	C₂	1	2	yes	maximal, complemented
9	C₇×#1455	C₂	1	1	yes	maximal, characteristic
10	C₇×#533	C₂²	1	2	yes	—
11	C₇×#533	C₂²	1	1	yes	characteristic
12	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₇×#501	—	2	4	no	—
14	C₇×#500	C₂²	1	2	yes	—
15	C₇×#500	C₂²	1	2	yes	—
16	C₇×#498	C₂²	1	2	yes	—
17	C₇×#498	C₂²	1	2	yes	—
18	C₇×#498	C₂²	1	1	yes	characteristic
19	C₇×#498	C₂²	1	2	yes	—
20	C₇×#494	C₂²	1	2	yes	—
21	C₇×#494	C₂²	1	1	yes	characteristic
22	C₇×#491	—	2	2	no	—
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×#487	C₂²	1	2	yes	—
28	C₇×#470	—	2	4	no	—
29	C₇×#470	—	2	2	no	—
30	C₇×#441	—	2	2	no	—
31	C₇×#441	—	2	4	no	—
32	C₇×#428	—	2	4	no	—
33	C₇×#419	—	2	2	no	—
34	C₄×C₇×M₁₆	C₂²	1	2	yes	—
35	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
36	C₇×#322	C₂²	1	2	yes	—
37	C₇×#322	C₂²	1	1	yes	characteristic
38	C₇×C₈²	C₂²	1	1	yes	characteristic
39	#19751	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₇×Q₈	—	2	1	no	—
41	C₂×C₇×Q₁₆	—	2	2	no	—
42	C₂×C₇×Q₁₆	—	2	4	no	—
43	C₂×C₇×M₁₆	D₄	1	4	yes	—
44	C₂×C₇×M₁₆	D₄	1	2	yes	—
45	C₇×C₄⋊Q₈	—	2	2	no	—
46	C₇×C₄⋊Q₈	—	2	1	no	—
47	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
51	C₇×#125	—	2	1	no	—
52	C₇×#125	—	2	2	no	—
53	C₇×C₂²⋊Q₈	—	2	2	no	—
54	C₇×C₂²⋊Q₈	—	2	2	no	—
55	C₇×C₂²⋊Q₈	—	2	2	no	—
56	C₄×C₇×Q₈	—	2	2	no	—
57	C₄×C₇×Q₈	—	2	2	no	—
58	C₄×C₇×Q₈	—	2	2	no	—
59	C₇×#117	—	2	1	no	—
60	C₇×#117	—	2	2	no	—
61	C₂×C₇×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
63	C₇×#107	—	2	4	no	—
64	C₇×#107	—	2	4	no	—
65	C₇×#106	—	2	2	no	—
66	C₇×#106	—	2	4	no	—
67	C₇×#106	—	2	4	no	—
68	C₇×#106	—	2	4	no	—
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×#97	D₄	1	2	yes	—
75	C₇×#97	D₄	1	4	yes	—
76	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
77	C₄×C₇×C₈	C₂³	1	2	yes	—
78	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
79	C₄×C₇×C₈	C₂³	1	2	yes	—
80	#3185	C₂×C₇	1	2	yes	—
81	#3185	C₂×C₇	1	1	yes	characteristic
82	#3165	C₂×C₇	1	2	yes	—
83	#3139	C₂×C₇	1	1	yes	characteristic
84	#1723	C₂×C₇	1	2	yes	complemented
85	#1718	C₂×C₇	1	2	yes	complemented
86	#1716	C₂×C₇	1	2	yes	complemented
87	#1707	C₂×C₇	1	2	yes	complemented
88	#1455	C₂×C₇	1	1	yes	characteristic
89	C₂×C₇×Q₈	—	2	2	no	—
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	—	4	4	no	—
94	C₂×C₇×Q₈	—	4	1	no	—
95	C₂²×C₄×C₇	—	4	1	no	—
96	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
98	C₇×Q₁₆	—	4	4	no	—
99	C₇×Q₁₆	—	4	8	no	—
100	C₇×M₁₆	—	2	8	no	—
101	C₇×M₁₆	—	4	2	no	—
102	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
103	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
104	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
105	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
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	—	2	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×C₄⋊C₄	—	4	2	no	—
118	C₇×C₂²⋊C₄	—	4	1	no	—
119	C₇×C₂²⋊C₄	—	4	2	no	—
120	C₄²×C₇	—	4	2	no	—
121	C₄²×C₇	—	4	1	no	—
122	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₇	C₂×D₄	1	2	yes	—
124	C₄²×C₇	C₂⁴	1	1	yes	characteristic
125	#533	C₂²×C₇	1	1	yes	characteristic
126	#533	C₂²×C₇	1	2	yes	—
127	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
128	#501	—	2	4	no	—
129	#500	C₂²×C₇	1	2	yes	—
130	#500	C₂²×C₇	1	2	yes	—
131	#498	C₂²×C₇	1	2	yes	—
132	#498	C₂²×C₇	1	2	yes	—
133	#498	C₂²×C₇	1	2	yes	—
134	#498	C₂²×C₇	1	1	yes	characteristic
135	#494	C₂²×C₇	1	2	yes	—
136	#494	C₂²×C₇	1	1	yes	characteristic
137	#491	—	2	2	no	—
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	#487	C₂²×C₇	1	2	yes	—
143	#470	—	2	4	no	—
144	#470	—	2	2	no	—
145	#441	—	2	2	no	—
146	#441	—	2	4	no	—
147	#428	—	2	4	no	—
148	#419	—	2	2	no	—
149	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
150	C₄×M₁₆	C₂²×C₇	1	2	yes	—
151	#322	C₂²×C₇	1	2	yes	—
152	#322	C₂²×C₇	1	1	yes	characteristic
153	C₈²	C₂²×C₇	1	1	yes	characteristic
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	C₇×Q₈	—	8	2	no	—
160	C₂×C₄×C₇	—	2	4	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	—	8	2	no	—
167	C₂×C₄×C₇	#127	1	2	yes	—
168	C₂×C₄×C₇	#127	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
170	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
171	C₇×C₈	—	2	4	no	—
172	C₇×C₈	—	2	2	no	—
173	C₇×C₈	—	2	2	no	—
174	C₇×C₈	#137	1	8	yes	—
175	C₂²×Q₈	—	2	1	no	—
176	C₂×Q₁₆	—	2	2	no	—
177	C₂×Q₁₆	—	2	4	no	—
178	C₂×M₁₆	C₇×D₄	1	4	yes	—
179	C₂×M₁₆	C₇×D₄	1	2	yes	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	—	2	1	no	—
182	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
183	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
184	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
185	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
186	#125	—	2	1	no	—
187	#125	—	2	2	no	—
188	C₂²⋊Q₈	—	2	2	no	—
189	C₂²⋊Q₈	—	2	2	no	—
190	C₂²⋊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	1	no	—
195	#117	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
198	#107	—	2	4	no	—
199	#107	—	2	4	no	—
200	#106	—	2	4	no	—
201	#106	—	2	4	no	—
202	#106	—	2	2	no	—
203	#106	—	2	4	no	—
204	#103	—	2	4	no	—
205	#103	—	2	4	no	—
206	#103	—	2	4	no	—
207	#103	—	2	4	no	—
208	#103	—	2	4	no	—
209	#97	C₇×D₄	1	2	yes	—
210	#97	C₇×D₄	1	4	yes	—
211	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
212	C₄×C₈	C₂³×C₇	1	2	yes	—
213	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
214	C₄×C₈	C₂³×C₇	1	2	yes	—
215	C₂²×C₇	—	2	2	no	—
216	C₂²×C₇	—	2	1	no	—
217	C₂²×C₇	C₂×#127	1	1	yes	characteristic
218	C₄×C₇	—	2	2	no	—
219	C₄×C₇	—	2	1	no	—
220	C₄×C₇	—	8	2	no	—
221	C₄×C₇	—	8	2	no	—
222	C₄×C₇	—	8	2	no	—
223	C₄×C₇	—	8	2	no	—
224	C₄×C₇	C₂×#137	1	4	yes	—
225	C₄×C₇	#575	1	2	yes	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	4	4	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
234	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
235	Q₁₆	—	4	4	no	—
236	Q₁₆	—	4	8	no	—
237	M₁₆	—	2	8	no	—
238	M₁₆	—	4	2	no	—
239	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
240	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
241	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
242	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
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	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	1	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	4	2	no	—
258	C₄²	—	4	1	no	—
259	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
260	C₄²	C₂×C₇×D₄	1	2	yes	—
261	C₄²	C₂⁴×C₇	1	1	yes	characteristic
262	C₂×C₇	—	4	1	no	—
263	C₂×C₇	#3152	1	2	yes	—
264	C₂×C₇	#3153	1	1	yes	characteristic
265	C₂³	C₇×#127	1	1	yes	characteristic
266	Q₈	—	4	4	no	—
267	Q₈	—	4	4	no	—
268	Q₈	—	4	4	no	—
269	Q₈	—	4	4	no	—
270	Q₈	—	8	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	C₇×#127	1	2	yes	—
279	C₂×C₄	C₇×#127	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
281	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
282	C₈	—	2	4	no	—
283	C₈	—	2	2	no	—
284	C₈	—	2	2	no	—
285	C₈	C₇×#137	1	8	yes	—
286	C₇	#19751	1	1	yes	characteristic, direct product factor, Sylow
287	C₂²	—	2	2	no	—
288	C₂²	—	2	1	no	—
289	C₂²	C₂×C₇×#127	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	C₂×C₇×#137	1	4	yes	—
297	C₄	C₇×#575	1	2	yes	—
298	C₂	—	4	1	no	—
299	C₂	C₇×#3152	1	2	yes	—
300	C₂	C₇×#3153	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×#1718	C₂	semidirect
7. C₇×#1716	C₂	semidirect
8. C₇×#1707	C₂	semidirect
39. #19751	C₇	direct (N × H)
84. #1723	C₂×C₇	semidirect
85. #1718	C₂×C₇	semidirect
86. #1716	C₂×C₇	semidirect
87. #1707	C₂×C₇	semidirect
286. C₇	#19751	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×#3185	2
C₇×#3165	2
C₇×#3139	2
C₇×#1723	2
C₇×#1718	2
C₇×#1716	2
C₇×#1707	2
C₇×#1455	2
#19751	7

Subgroups — C7×#19751 #13196967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19751 #13196967