Subgroups — C₇×#19648 #13196864

Order 1792.

664 conjugacy classes of subgroups.

The subgroup lattice has 332 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×#1718	C₂	1	2	yes	maximal, complemented
6	C₇×#1716	C₂	1	2	yes	maximal, complemented
7	C₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1710	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1707	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1707	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
12	C₇×#533	C₂²	1	2	yes	—
13	C₇×#533	C₂²	1	2	yes	—
14	C₇×#500	C₂²	1	1	yes	characteristic
15	C₇×#500	C₂²	1	1	yes	characteristic
16	C₇×#500	C₂²	1	1	yes	characteristic
17	C₇×#500	C₂²	1	2	yes	—
18	C₇×#499	—	2	2	no	—
19	C₇×#498	C₂²	1	2	yes	—
20	C₇×#498	C₂²	1	2	yes	—
21	C₇×#498	C₂²	1	2	yes	—
22	C₇×#494	C₂²	1	1	yes	characteristic
23	C₇×#494	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×#487	C₂²	1	2	yes	—
29	C₇×#485	—	2	4	no	—
30	C₇×#479	—	2	4	no	—
31	C₇×#470	—	2	4	no	—
32	C₇×#467	—	2	2	no	—
33	C₇×#441	—	2	4	no	—
34	C₄×C₇×Q₁₆	—	2	2	no	—
35	C₇×#431	C₂²	1	1	yes	characteristic
36	C₇×#427	—	2	2	no	—
37	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
39	C₇×#322	C₂²	1	1	yes	characteristic
40	C₇×#322	C₂²	1	1	yes	characteristic
41	C₇×C₈²	C₂²	1	1	yes	characteristic
42	C₇×C₈²	C₂²	1	1	yes	characteristic
43	#19648	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₇×Q₁₆	—	2	2	no	—
45	C₂×C₇×Q₁₆	—	2	4	no	—
46	C₂×C₇×M₁₆	D₄	1	2	yes	—
47	C₂²×C₇×C₈	D₄	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×C₄⋊Q₈	C₂³	1	2	yes	—
52	C₇×#125	—	2	2	no	—
53	C₇×#125	—	2	2	no	—
54	C₇×C₂²⋊Q₈	—	2	2	no	—
55	C₇×C₂²⋊Q₈	—	2	2	no	—
56	C₇×C₂²⋊Q₈	—	2	2	no	—
57	C₇×C₂²⋊Q₈	—	2	2	no	—
58	C₄×C₇×Q₈	—	2	2	no	—
59	C₄×C₇×Q₈	—	2	2	no	—
60	C₄×C₇×Q₈	—	2	2	no	—
61	C₄×C₇×Q₈	—	2	2	no	—
62	C₇×#117	—	2	2	no	—
63	C₇×#117	—	2	2	no	—
64	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
65	C₇×#107	—	2	2	no	—
66	C₇×#107	—	2	4	no	—
67	C₇×#107	—	2	2	no	—
68	C₇×#107	—	2	2	no	—
69	C₇×#106	—	2	4	no	—
70	C₇×#106	—	2	4	no	—
71	C₇×#106	—	2	4	no	—
72	C₇×C₄⋊C₈	D₄	1	2	yes	—
73	C₇×#103	—	2	4	no	—
74	C₇×#103	—	2	4	no	—
75	C₇×#103	—	2	4	no	—
76	C₇×#103	—	2	4	no	—
77	C₇×#103	—	2	4	no	—
78	C₇×C₂²⋊C₈	D₄	1	2	yes	—
79	C₇×#97	D₄	1	2	yes	—
80	C₄×C₇×C₈	D₄	1	2	yes	—
81	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
82	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
83	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
84	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
85	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
86	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
87	#3231	C₂×C₇	1	2	yes	—
88	#3185	C₂×C₇	1	2	yes	—
89	#3170	C₂×C₇	1	1	yes	characteristic
90	#3160	C₂×C₇	1	1	yes	characteristic
91	#1718	C₂×C₇	1	2	yes	complemented
92	#1716	C₂×C₇	1	2	yes	complemented
93	#1713	C₂×C₇	1	1	yes	characteristic, complemented
94	#1710	C₂×C₇	1	1	yes	characteristic, complemented
95	#1707	C₂×C₇	1	1	yes	characteristic, complemented
96	#1707	C₂×C₇	1	1	yes	characteristic, complemented
97	C₈×M₁₆	C₂×C₇	1	1	yes	characteristic
98	C₂×C₇×Q₈	—	2	2	no	—
99	C₂×C₇×Q₈	—	2	2	no	—
100	C₂×C₇×Q₈	—	2	2	no	—
101	C₂×C₇×Q₈	—	2	2	no	—
102	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
105	C₇×Q₁₆	—	4	8	no	—
106	C₇×Q₁₆	—	4	4	no	—
107	C₇×M₁₆	—	2	4	no	—
108	C₂×C₇×C₈	—	2	4	no	—
109	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
110	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
111	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
112	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
113	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
114	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
115	C₇×C₄⋊C₄	—	2	2	no	—
116	C₇×C₄⋊C₄	—	2	2	no	—
117	C₇×C₄⋊C₄	—	2	2	no	—
118	C₇×C₄⋊C₄	—	2	2	no	—
119	C₇×C₄⋊C₄	—	2	2	no	—
120	C₇×C₄⋊C₄	—	2	2	no	—
121	C₇×C₄⋊C₄	—	2	2	no	—
122	C₇×C₄⋊C₄	—	2	2	no	—
123	C₇×C₄⋊C₄	—	4	2	no	—
124	C₇×C₄⋊C₄	—	4	2	no	—
125	C₇×C₄⋊C₄	—	4	2	no	—
126	C₇×C₄⋊C₄	—	4	2	no	—
127	C₇×C₂²⋊C₄	—	4	2	no	—
128	C₇×C₂²⋊C₄	—	4	2	no	—
129	C₄²×C₇	—	4	2	no	—
130	C₄²×C₇	—	4	2	no	—
131	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₇	C₂⁴	1	1	yes	characteristic
135	#533	C₂²×C₇	1	2	yes	—
136	#533	C₂²×C₇	1	2	yes	—
137	#500	C₂²×C₇	1	1	yes	characteristic
138	#500	C₂²×C₇	1	1	yes	characteristic
139	#500	C₂²×C₇	1	1	yes	characteristic
140	#500	C₂²×C₇	1	2	yes	—
141	#499	—	2	2	no	—
142	#498	C₂²×C₇	1	2	yes	—
143	#498	C₂²×C₇	1	2	yes	—
144	#498	C₂²×C₇	1	2	yes	—
145	#494	C₂²×C₇	1	1	yes	characteristic
146	#494	C₂²×C₇	1	2	yes	—
147	#487	C₂²×C₇	1	2	yes	—
148	#487	C₂²×C₇	1	2	yes	—
149	#487	C₂²×C₇	1	2	yes	—
150	#487	C₂²×C₇	1	2	yes	—
151	#487	C₂²×C₇	1	2	yes	—
152	#485	—	2	4	no	—
153	#479	—	2	4	no	—
154	#470	—	2	4	no	—
155	#467	—	2	2	no	—
156	#441	—	2	4	no	—
157	C₄×Q₁₆	—	2	2	no	—
158	#431	C₂²×C₇	1	1	yes	characteristic
159	#427	—	2	2	no	—
160	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
162	#322	C₂²×C₇	1	1	yes	characteristic
163	#322	C₂²×C₇	1	1	yes	characteristic
164	C₈²	C₂²×C₇	1	1	yes	characteristic
165	C₈²	C₂²×C₇	1	1	yes	characteristic
166	C₂³×C₇	#127	1	1	yes	characteristic
167	C₇×Q₈	—	4	4	no	—
168	C₇×Q₈	—	4	4	no	—
169	C₇×Q₈	—	4	4	no	—
170	C₇×Q₈	—	4	4	no	—
171	C₂×C₄×C₇	—	2	2	no	—
172	C₂×C₄×C₇	—	2	2	no	—
173	C₂×C₄×C₇	—	2	2	no	—
174	C₂×C₄×C₇	—	4	2	no	—
175	C₂×C₄×C₇	—	4	2	no	—
176	C₂×C₄×C₇	—	4	2	no	—
177	C₂×C₄×C₇	—	4	2	no	—
178	C₂×C₄×C₇	#127	1	1	yes	characteristic
179	C₂×C₄×C₇	#127	1	1	yes	characteristic
180	C₂×C₄×C₇	#127	1	1	yes	characteristic
181	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
184	C₇×C₈	—	2	2	no	—
185	C₇×C₈	—	2	2	no	—
186	C₇×C₈	—	2	2	no	—
187	C₇×C₈	—	2	2	no	—
188	C₇×C₈	#135	1	4	yes	—
189	C₇×C₈	#137	1	4	yes	—
190	C₂×Q₁₆	—	2	2	no	—
191	C₂×Q₁₆	—	2	4	no	—
192	C₂×M₁₆	C₇×D₄	1	2	yes	—
193	C₂²×C₈	C₇×D₄	1	2	yes	—
194	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
195	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
196	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
197	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
198	#125	—	2	2	no	—
199	#125	—	2	2	no	—
200	C₂²⋊Q₈	—	2	2	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	C₂²⋊Q₈	—	2	2	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	—	2	2	no	—
208	#117	—	2	2	no	—
209	#117	—	2	2	no	—
210	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
211	#107	—	2	2	no	—
212	#107	—	2	2	no	—
213	#107	—	2	2	no	—
214	#107	—	2	4	no	—
215	#106	—	2	4	no	—
216	#106	—	2	4	no	—
217	#106	—	2	4	no	—
218	C₄⋊C₈	C₇×D₄	1	2	yes	—
219	#103	—	2	4	no	—
220	#103	—	2	4	no	—
221	#103	—	2	4	no	—
222	#103	—	2	4	no	—
223	#103	—	2	4	no	—
224	C₂²⋊C₈	C₇×D₄	1	2	yes	—
225	#97	C₇×D₄	1	2	yes	—
226	C₄×C₈	C₇×D₄	1	2	yes	—
227	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
229	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
232	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
233	C₂²×C₇	—	2	1	no	—
234	C₂²×C₇	—	2	1	no	—
235	C₂²×C₇	—	2	1	no	—
236	C₂²×C₇	C₂×#127	1	1	yes	characteristic
237	C₄×C₇	—	2	1	no	—
238	C₄×C₇	—	2	1	no	—
239	C₄×C₇	—	2	1	no	—
240	C₄×C₇	—	8	2	no	—
241	C₄×C₇	—	8	2	no	—
242	C₄×C₇	—	8	2	no	—
243	C₄×C₇	—	8	2	no	—
244	C₄×C₇	C₂×#135	1	2	yes	—
245	C₄×C₇	C₂×#137	1	2	yes	—
246	C₄×C₇	#578	1	2	yes	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
254	Q₁₆	—	4	8	no	—
255	Q₁₆	—	4	4	no	—
256	M₁₆	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
259	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
260	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
261	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
262	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
263	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₄²	—	4	2	no	—
279	C₄²	—	4	2	no	—
280	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
281	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
282	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
283	C₄²	C₂⁴×C₇	1	1	yes	characteristic
284	C₂×C₇	—	4	1	no	—
285	C₂×C₇	#3147	1	1	yes	characteristic
286	C₂×C₇	#3152	1	1	yes	characteristic
287	C₂×C₇	#3156	1	1	yes	characteristic
288	C₂³	C₇×#127	1	1	yes	characteristic
289	Q₈	—	4	4	no	—
290	Q₈	—	4	4	no	—
291	Q₈	—	4	4	no	—
292	Q₈	—	4	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	C₇×#127	1	1	yes	characteristic
301	C₂×C₄	C₇×#127	1	1	yes	characteristic
302	C₂×C₄	C₇×#127	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	2	no	—
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	C₇×#135	1	4	yes	—
311	C₈	C₇×#137	1	4	yes	—
312	C₇	#19648	1	1	yes	characteristic, direct product factor, Sylow
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	C₂×C₇×#127	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	C₂×C₇×#135	1	2	yes	—
325	C₄	C₂×C₇×#137	1	2	yes	—
326	C₄	C₇×#578	1	2	yes	—
327	C₂	—	4	1	no	—
328	C₂	C₇×#3147	1	1	yes	characteristic
329	C₂	C₇×#3152	1	1	yes	characteristic
330	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×#1718	C₂	semidirect
6. C₇×#1716	C₂	semidirect
7. C₇×#1713	C₂	semidirect
8. C₇×#1710	C₂	semidirect
9. C₇×#1707	C₂	semidirect
10. C₇×#1707	C₂	semidirect
43. #19648	C₇	direct (N × H)
91. #1718	C₂×C₇	semidirect
92. #1716	C₂×C₇	semidirect
93. #1713	C₂×C₇	semidirect
94. #1710	C₂×C₇	semidirect
95. #1707	C₂×C₇	semidirect
96. #1707	C₂×C₇	semidirect
312. C₇	#19648	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×#1718	2
C₇×#1716	2
C₇×#1713	2
C₇×#1710	2
C₇×#1707	2
C₇×C₈×M₁₆	2
#19648	7

Subgroups — C7×#19648 #13196864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19648 #13196864