Subgroups — C₇×#19687 #13196903

Order 1792.

764 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3228	C₂	1	2	yes	maximal, complemented
2	C₇×#3182	C₂	1	2	yes	maximal, complemented
3	C₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3147	C₂	1	1	yes	maximal, characteristic
5	C₇×#1723	C₂	1	2	yes	maximal, complemented
6	C₇×#1722	C₂	1	2	yes	maximal, complemented
7	C₇×#1717	C₂	1	2	yes	maximal, complemented
8	C₇×#1714	C₂	1	2	yes	maximal, complemented
9	C₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×#135	—	2	2	no	—
11	C₇×#533	C₂²	1	2	yes	—
12	C₇×#532	C₂²	1	2	yes	—
13	C₇×#500	C₂²	1	1	yes	characteristic
14	C₇×#500	C₂²	1	2	yes	—
15	C₇×#500	C₂²	1	2	yes	—
16	C₇×#498	C₂²	1	2	yes	—
17	C₇×#495	—	2	2	no	—
18	C₇×#494	C₂²	1	1	yes	characteristic
19	C₇×#494	C₂²	1	1	yes	characteristic
20	C₇×#493	C₂²	1	1	yes	characteristic
21	C₇×#492	C₂²	1	2	yes	—
22	C₇×#492	C₂²	1	2	yes	—
23	C₇×#487	C₂²	1	2	yes	—
24	C₇×#487	C₂²	1	2	yes	—
25	C₇×#486	C₂²	1	2	yes	—
26	C₇×#486	C₂²	1	2	yes	—
27	C₇×#486	C₂²	1	2	yes	—
28	C₇×#482	—	2	4	no	—
29	C₇×#478	—	2	4	no	—
30	C₇×#468	—	2	4	no	—
31	C₇×#467	—	2	2	no	—
32	C₇×#440	—	2	4	no	—
33	C₄×C₇×Q₁₆	—	2	2	no	—
34	C₇×#431	C₂²	1	1	yes	characteristic
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	#19687	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₇×#41	—	2	2	no	—
41	C₇×#135	—	4	8	no	—
42	C₂×C₇×Q₁₆	—	2	2	no	—
43	C₂×C₇×Q₁₆	—	2	2	no	—
44	C₂×C₇×SD₁₆	—	2	4	no	—
45	C₂×C₇×SD₁₆	—	2	4	no	—
46	C₂×C₇×D₈	—	2	2	no	—
47	C₂×C₇×M₁₆	D₄	1	2	yes	—
48	C₂²×C₇×C₈	D₄	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×#127	C₂³	1	2	yes	—
53	C₇×#125	—	2	2	no	—
54	C₇×#124	—	2	2	no	—
55	C₇×C₂²⋊Q₈	—	2	2	no	—
56	C₇×C₂²⋊Q₈	—	2	2	no	—
57	C₇×#121	—	2	2	no	—
58	C₇×#121	—	2	2	no	—
59	C₄×C₇×Q₈	—	2	2	no	—
60	C₄×C₇×Q₈	—	2	2	no	—
61	C₄×C₇×D₄	—	2	2	no	—
62	C₄×C₇×D₄	—	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	4	no	—
68	C₇×#106	—	2	4	no	—
69	C₇×C₄⋊C₈	D₄	1	2	yes	—
70	C₇×#103	—	2	4	no	—
71	C₇×#103	—	2	4	no	—
72	C₇×#102	—	2	4	no	—
73	C₇×#102	—	2	4	no	—
74	C₇×#102	—	2	4	no	—
75	C₇×C₂²⋊C₈	D₄	1	2	yes	—
76	C₇×#97	D₄	1	2	yes	—
77	C₄×C₇×C₈	D₄	1	2	yes	—
78	C₄×C₇×C₈	C₂³	1	2	yes	—
79	C₄×C₇×C₈	C₂³	1	2	yes	—
80	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
81	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
82	#3228	C₂×C₇	1	2	yes	complemented
83	#3182	C₂×C₇	1	2	yes	complemented
84	#3170	C₂×C₇	1	1	yes	characteristic, complemented
85	#3147	C₂×C₇	1	1	yes	characteristic
86	#1723	C₂×C₇	1	2	yes	complemented
87	#1722	C₂×C₇	1	2	yes	complemented
88	#1717	C₂×C₇	1	2	yes	complemented
89	#1714	C₂×C₇	1	2	yes	complemented
90	#1451	C₂×C₇	1	1	yes	characteristic, complemented
91	C₇×#41	—	4	8	no	—
92	C₂×C₇×Q₈	—	2	2	no	—
93	C₂×C₇×Q₈	—	2	2	no	—
94	C₂×C₇×Q₈	—	2	2	no	—
95	C₂×C₇×D₄	—	2	2	no	—
96	C₂×C₇×D₄	—	2	2	no	—
97	C₂×C₇×D₄	—	2	2	no	—
98	C₂×C₇×D₄	—	4	2	no	—
99	C₂²×C₄×C₇	—	4	2	no	—
100	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
103	C₇×Q₁₆	—	4	4	no	—
104	C₇×Q₁₆	—	4	4	no	—
105	C₇×SD₁₆	—	4	8	no	—
106	C₇×SD₁₆	—	4	8	no	—
107	C₇×D₈	—	4	4	no	—
108	C₇×M₁₆	—	4	2	no	—
109	C₂×C₇×C₈	—	2	4	no	—
110	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
111	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
112	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
113	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
114	C₇×C₄⋊C₄	—	2	2	no	—
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	—	4	2	no	—
121	C₇×C₄⋊C₄	—	4	2	no	—
122	C₇×C₂²⋊C₄	—	4	2	no	—
123	C₇×C₂²⋊C₄	—	4	2	no	—
124	C₇×C₂²⋊C₄	—	4	2	no	—
125	C₄²×C₇	—	4	2	no	—
126	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₇	C₂⁴	1	1	yes	characteristic
130	C₂×#135	—	2	2	no	—
131	#533	C₂²×C₇	1	2	yes	—
132	#532	C₂²×C₇	1	2	yes	—
133	#500	C₂²×C₇	1	1	yes	characteristic
134	#500	C₂²×C₇	1	2	yes	—
135	#500	C₂²×C₇	1	2	yes	—
136	#498	C₂²×C₇	1	2	yes	—
137	#495	—	2	2	no	—
138	#494	C₂²×C₇	1	1	yes	characteristic
139	#494	C₂²×C₇	1	1	yes	characteristic
140	#493	C₂²×C₇	1	1	yes	characteristic
141	#492	C₂²×C₇	1	2	yes	—
142	#492	C₂²×C₇	1	2	yes	—
143	#487	C₂²×C₇	1	2	yes	—
144	#487	C₂²×C₇	1	2	yes	—
145	#486	C₂²×C₇	1	2	yes	—
146	#486	C₂²×C₇	1	2	yes	—
147	#486	C₂²×C₇	1	2	yes	—
148	#482	—	2	4	no	—
149	#478	—	2	4	no	—
150	#468	—	2	4	no	—
151	#467	—	2	2	no	—
152	#440	—	2	4	no	—
153	C₄×Q₁₆	—	2	2	no	—
154	#431	C₂²×C₇	1	1	yes	characteristic
155	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
157	#322	C₂²×C₇	1	2	yes	—
158	#322	C₂²×C₇	1	2	yes	—
159	C₂³×C₇	—	4	2	no	—
160	C₂³×C₇	#127	1	1	yes	characteristic
161	C₇×Q₈	—	4	4	no	—
162	C₇×Q₈	—	4	4	no	—
163	C₇×Q₈	—	4	4	no	—
164	C₇×D₄	—	4	4	no	—
165	C₇×D₄	—	4	4	no	—
166	C₇×D₄	—	4	4	no	—
167	C₇×D₄	—	8	4	no	—
168	C₂×C₄×C₇	—	2	2	no	—
169	C₂×C₄×C₇	—	2	2	no	—
170	C₂×C₄×C₇	—	2	2	no	—
171	C₂×C₄×C₇	—	4	2	no	—
172	C₂×C₄×C₇	—	4	2	no	—
173	C₂×C₄×C₇	—	4	2	no	—
174	C₂×C₄×C₇	—	8	4	no	—
175	C₂×C₄×C₇	#127	1	1	yes	characteristic
176	C₂×C₄×C₇	#127	1	1	yes	characteristic
177	C₂×C₄×C₇	#127	1	1	yes	characteristic
178	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
181	C₇×C₈	—	2	2	no	—
182	C₇×C₈	—	2	4	no	—
183	C₇×C₈	—	2	4	no	—
184	C₇×C₈	AGL(1,ℤ/8)	1	4	yes	—
185	C₂×#41	—	2	2	no	—
186	#135	—	4	8	no	—
187	C₂×Q₁₆	—	2	2	no	—
188	C₂×Q₁₆	—	2	2	no	—
189	C₂×SD₁₆	—	2	4	no	—
190	C₂×SD₁₆	—	2	4	no	—
191	C₂×D₈	—	2	2	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	#127	C₂³×C₇	1	2	yes	—
198	#125	—	2	2	no	—
199	#124	—	2	2	no	—
200	C₂²⋊Q₈	—	2	2	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	#121	—	2	2	no	—
203	#121	—	2	2	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×D₄	—	2	2	no	—
207	C₄×D₄	—	2	2	no	—
208	#117	—	2	2	no	—
209	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
210	#107	—	2	2	no	—
211	#107	—	2	4	no	—
212	#107	—	2	4	no	—
213	#106	—	2	4	no	—
214	C₄⋊C₈	C₇×D₄	1	2	yes	—
215	#103	—	2	4	no	—
216	#103	—	2	4	no	—
217	#102	—	2	4	no	—
218	#102	—	2	4	no	—
219	#102	—	2	4	no	—
220	C₂²⋊C₈	C₇×D₄	1	2	yes	—
221	#97	C₇×D₄	1	2	yes	—
222	C₄×C₈	C₇×D₄	1	2	yes	—
223	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
224	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
225	C₄×C₈	C₂³×C₇	1	2	yes	—
226	C₄×C₈	C₂³×C₇	1	2	yes	—
227	C₂²×C₇	—	2	1	no	—
228	C₂²×C₇	—	2	1	no	—
229	C₂²×C₇	—	2	1	no	—
230	C₂²×C₇	—	8	2	no	—
231	C₂²×C₇	—	8	2	no	—
232	C₂²×C₇	—	8	2	no	—
233	C₂²×C₇	C₂×#127	1	1	yes	characteristic
234	C₄×C₇	—	2	1	no	—
235	C₄×C₇	—	2	1	no	—
236	C₄×C₇	—	2	1	no	—
237	C₄×C₇	—	8	2	no	—
238	C₄×C₇	—	8	2	no	—
239	C₄×C₇	—	8	2	no	—
240	C₄×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
241	C₄×C₇	#575	1	2	yes	—
242	C₄×C₇	#578	1	2	yes	—
243	#41	—	4	8	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
255	Q₁₆	—	4	4	no	—
256	Q₁₆	—	4	4	no	—
257	SD₁₆	—	4	8	no	—
258	SD₁₆	—	4	8	no	—
259	D₈	—	4	4	no	—
260	M₁₆	—	4	2	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
263	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
264	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
265	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
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	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
279	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
280	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
281	C₄²	C₂⁴×C₇	1	1	yes	characteristic
282	C₂×C₇	—	4	1	no	—
283	C₂×C₇	—	16	2	no	—
284	C₂×C₇	#3150	1	1	yes	characteristic
285	C₂×C₇	#3151	1	1	yes	characteristic
286	C₂×C₇	#3156	1	1	yes	characteristic
287	C₂³	—	4	2	no	—
288	C₂³	C₇×#127	1	1	yes	characteristic
289	Q₈	—	4	4	no	—
290	Q₈	—	4	4	no	—
291	Q₈	—	4	4	no	—
292	D₄	—	4	4	no	—
293	D₄	—	4	4	no	—
294	D₄	—	4	4	no	—
295	D₄	—	8	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	8	4	no	—
303	C₂×C₄	C₇×#127	1	1	yes	characteristic
304	C₂×C₄	C₇×#127	1	1	yes	characteristic
305	C₂×C₄	C₇×#127	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	2	4	no	—
311	C₈	—	2	4	no	—
312	C₈	C₇×AGL(1,ℤ/8)	1	4	yes	—
313	C₇	#19687	1	1	yes	characteristic, direct product factor, Sylow
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	C₂×C₇×#127	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	2	no	—
327	C₄	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
328	C₄	C₇×#575	1	2	yes	—
329	C₄	C₇×#578	1	2	yes	—
330	C₂	—	4	1	no	—
331	C₂	—	16	2	no	—
332	C₂	C₇×#3150	1	1	yes	characteristic
333	C₂	C₇×#3151	1	1	yes	characteristic
334	C₂	C₇×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#3147	C₂
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
1. C₇×#3228	C₂	semidirect
2. C₇×#3182	C₂	semidirect
3. C₇×#3170	C₂	semidirect
5. C₇×#1723	C₂	semidirect
6. C₇×#1722	C₂	semidirect
7. C₇×#1717	C₂	semidirect
8. C₇×#1714	C₂	semidirect
9. C₇×#1451	C₂	semidirect
39. #19687	C₇	direct (N × H)
82. #3228	C₂×C₇	semidirect
83. #3182	C₂×C₇	semidirect
84. #3170	C₂×C₇	semidirect
86. #1723	C₂×C₇	semidirect
87. #1722	C₂×C₇	semidirect
88. #1717	C₂×C₇	semidirect
89. #1714	C₂×C₇	semidirect
90. #1451	C₂×C₇	semidirect
313. C₇	#19687	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×#3228	2
C₇×#3182	2
C₇×#3170	2
C₇×#3147	2
C₇×#1723	2
C₇×#1722	2
C₇×#1717	2
C₇×#1714	2
C₇×#1451	2
#19687	7

Subgroups — C7×#19687 #13196903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19687 #13196903