Subgroups — C₇×#7896 #13185112

Order 1792.

722 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₇×#2285	C₂	1	1	yes	maximal, characteristic
4	C₇×#1279	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×M₁₆	—	2	1	no	—
6	C₂×C₇×#117	—	2	1	no	—
7	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₇×#431	—	2	2	no	—
9	C₇×#431	C₄	1	4	yes	complemented
10	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₇×C₂²⋊C₈	—	2	1	no	—
12	C₄×C₇×M₁₆	—	2	2	no	—
13	C₄×C₇×M₁₆	C₄	1	4	yes	complemented
14	C₂×C₇×#97	—	2	1	no	—
15	C₂×C₄×C₇×C₈	—	2	1	no	—
16	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
17	C₇×#384	C₄	1	2	yes	—
18	C₇×#382	—	2	1	no	—
19	C₇×#380	—	2	1	no	—
20	C₇×#380	C₄	1	2	yes	—
21	C₄×C₇×C₄⋊C₄	—	2	1	no	—
22	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
23	#7896	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
27	C₂×C₇×M₁₆	—	2	8	no	—
28	C₂×C₇×M₁₆	—	2	4	no	—
29	C₂²×C₇×C₈	—	2	2	no	—
30	C₂²×C₇×C₈	—	2	2	no	—
31	C₇×#117	—	4	4	no	—
32	C₂×C₇×C₄⋊C₄	—	2	2	no	—
33	C₂×C₇×C₄⋊C₄	—	2	2	no	—
34	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
35	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
36	C₂×C₄²×C₇	—	2	2	no	—
37	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇	D₄	1	2	yes	—
43	C₂×C₄²×C₇	Q₈	1	2	yes	—
44	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
45	C₇×C₄⋊C₈	—	2	2	no	—
46	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
47	C₇×C₂²⋊C₈	—	2	4	no	—
48	C₇×#97	—	2	4	no	—
49	C₄×C₇×C₈	—	2	4	no	—
50	C₄×C₇×C₈	—	2	2	no	—
51	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
52	C₇×#95	—	2	2	no	—
53	C₇×#95	—	2	2	no	—
54	C₂×#431	C₂×C₇	1	1	yes	characteristic
55	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic
56	#2285	C₂×C₇	1	1	yes	characteristic
57	#1279	C₂×C₇	1	4	yes	complemented
58	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	—	2	4	no	—
60	C₂²×C₄×C₇	—	2	4	no	—
61	C₂²×C₄×C₇	—	2	2	no	—
62	C₂²×C₄×C₇	—	2	2	no	—
63	C₂²×C₄×C₇	—	2	4	no	—
64	C₂²×C₄×C₇	C₄²	1	2	yes	—
65	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
76	C₇×M₁₆	—	2	16	no	—
77	C₂×C₇×C₈	—	2	8	no	—
78	C₂×C₇×C₈	—	2	4	no	—
79	C₂×C₇×C₈	—	2	4	no	—
80	C₂×C₇×C₈	—	4	2	no	—
81	C₂×C₇×C₈	—	4	2	no	—
82	C₇×C₄⋊C₄	—	2	8	no	—
83	C₇×C₄⋊C₄	—	4	4	no	—
84	C₇×C₄⋊C₄	—	4	4	no	—
85	C₇×C₂²⋊C₄	—	4	4	no	—
86	C₄²×C₇	—	4	4	no	—
87	C₄²×C₇	C₄²	1	2	yes	—
88	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₇	C₄⋊C₄	1	2	yes	—
91	C₄²×C₇	C₄⋊C₄	1	2	yes	—
92	C₄²×C₇	C₄⋊C₄	1	2	yes	—
93	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
97	C₂²×M₁₆	—	2	1	no	—
98	C₂×#117	—	2	1	no	—
99	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
100	#431	—	2	2	no	—
101	#431	C₄×C₇	1	4	yes	complemented
102	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
103	C₂×C₂²⋊C₈	—	2	1	no	—
104	C₄×M₁₆	—	2	2	no	—
105	C₄×M₁₆	C₄×C₇	1	4	yes	complemented
106	C₂×#97	—	2	1	no	—
107	C₂×C₄×C₈	—	2	1	no	—
108	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
109	#384	C₄×C₇	1	2	yes	—
110	#382	—	2	1	no	—
111	#380	—	2	1	no	—
112	#380	C₄×C₇	1	2	yes	—
113	C₄×C₄⋊C₄	—	2	1	no	—
114	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
115	C₂³×C₇	—	2	2	no	—
116	C₂³×C₇	—	2	1	no	—
117	C₂³×C₇	—	2	1	no	—
118	C₂³×C₇	—	2	2	no	—
119	C₂³×C₇	#95	1	2	yes	—
120	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	—	2	1	no	—
123	C₂×C₄×C₇	—	2	1	no	—
124	C₂×C₄×C₇	—	2	2	no	—
125	C₂×C₄×C₇	—	2	4	no	—
126	C₂×C₄×C₇	—	2	4	no	—
127	C₂×C₄×C₇	—	2	4	no	—
128	C₂×C₄×C₇	—	2	4	no	—
129	C₂×C₄×C₇	—	2	4	no	—
130	C₂×C₄×C₇	—	2	4	no	—
131	C₂×C₄×C₇	—	4	4	no	—
132	C₂×C₄×C₇	—	4	4	no	—
133	C₂×C₄×C₇	—	4	4	no	—
134	C₂×C₄×C₇	#95	1	2	yes	—
135	C₂×C₄×C₇	#95	1	2	yes	—
136	C₂×C₄×C₇	#95	1	2	yes	—
137	C₂×C₄×C₇	#104	1	4	yes	—
138	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₇	#131	1	4	yes	—
145	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
146	C₂×C₄×C₇	#137	1	2	yes	—
147	C₇×C₈	—	2	8	no	—
148	C₇×C₈	—	4	4	no	—
149	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
150	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
151	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
152	C₂×M₁₆	—	2	8	no	—
153	C₂×M₁₆	—	2	4	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	#117	—	4	4	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
160	C₂×C₂²⋊C₄	—	2	2	no	—
161	C₂×C₄²	—	2	2	no	—
162	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
163	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
164	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
165	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
166	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
167	C₂×C₄²	C₇×D₄	1	2	yes	—
168	C₂×C₄²	C₇×Q₈	1	2	yes	—
169	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
172	C₂²⋊C₈	—	2	4	no	—
173	#97	—	2	4	no	—
174	C₄×C₈	—	2	4	no	—
175	C₄×C₈	—	2	2	no	—
176	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
177	#95	—	2	2	no	—
178	#95	—	2	2	no	—
179	C₂²×C₇	—	2	2	no	—
180	C₂²×C₇	—	2	2	no	—
181	C₂²×C₇	—	2	2	no	—
182	C₂²×C₇	—	4	2	no	—
183	C₂²×C₇	—	4	2	no	—
184	C₂²×C₇	C₂×#95	1	1	yes	characteristic
185	C₂²×C₇	#408	1	2	yes	—
186	C₂²×C₇	#418	1	1	yes	characteristic
187	C₂²×C₇	#419	1	1	yes	characteristic
188	C₂²×C₇	C₂×#104	1	2	yes	—
189	C₄×C₇	—	2	2	no	—
190	C₄×C₇	—	2	1	no	—
191	C₄×C₇	—	2	1	no	—
192	C₄×C₇	—	2	2	no	—
193	C₄×C₇	—	8	4	no	—
194	C₄×C₇	#339	1	4	yes	—
195	C₄×C₇	#405	1	2	yes	—
196	C₄×C₇	#417	1	2	yes	—
197	C₄×C₇	#424	1	2	yes	—
198	C₄×C₇	#428	1	2	yes	—
199	C₂⁴	C₇×C₂²⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	C₄²×C₇	1	2	yes	—
206	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
208	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
209	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
210	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
212	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
213	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
217	M₁₆	—	2	16	no	—
218	C₂×C₈	—	2	8	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₄⋊C₄	—	2	8	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₂²⋊C₄	—	4	4	no	—
227	C₄²	—	4	4	no	—
228	C₄²	C₄²×C₇	1	2	yes	—
229	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
230	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
231	C₄²	C₇×C₄⋊C₄	1	2	yes	—
232	C₄²	C₇×C₄⋊C₄	1	2	yes	—
233	C₄²	C₇×C₄⋊C₄	1	2	yes	—
234	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
235	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
236	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
237	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
238	C₂×C₇	—	4	2	no	—
239	C₂×C₇	#1484	1	2	yes	—
240	C₂×C₇	#1485	1	2	yes	—
241	C₂×C₇	#1729	1	1	yes	characteristic
242	C₂×C₇	C₂×#339	1	1	yes	characteristic
243	C₂×C₇	#1736	1	1	yes	characteristic
244	C₂³	—	2	2	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	2	no	—
248	C₂³	C₇×#95	1	2	yes	—
249	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	C₇×#95	1	2	yes	—
264	C₂×C₄	C₇×#95	1	2	yes	—
265	C₂×C₄	C₇×#95	1	2	yes	—
266	C₂×C₄	C₇×#104	1	4	yes	—
267	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
268	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
272	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₇×#131	1	4	yes	—
274	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
275	C₂×C₄	C₇×#137	1	2	yes	—
276	C₈	—	2	8	no	—
277	C₈	—	4	4	no	—
278	C₇	#7896	1	1	yes	characteristic, direct product factor, Sylow
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	C₂×C₇×#95	1	1	yes	characteristic
285	C₂²	C₇×#408	1	2	yes	—
286	C₂²	C₇×#418	1	1	yes	characteristic
287	C₂²	C₇×#419	1	1	yes	characteristic
288	C₂²	C₂×C₇×#104	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	8	4	no	—
294	C₄	C₇×#339	1	4	yes	—
295	C₄	C₇×#405	1	2	yes	—
296	C₄	C₇×#417	1	2	yes	—
297	C₄	C₇×#424	1	2	yes	—
298	C₄	C₇×#428	1	2	yes	—
299	C₂	—	4	2	no	—
300	C₂	C₇×#1484	1	2	yes	—
301	C₂	C₇×#1485	1	2	yes	—
302	C₂	C₇×#1729	1	1	yes	characteristic
303	C₂	C₂×C₇×#339	1	1	yes	characteristic
304	C₂	C₇×#1736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂³×C₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1279	C₂	semidirect
9. C₇×#431	C₄	semidirect
13. C₄×C₇×M₁₆	C₄	semidirect
23. #7896	C₇	direct (N × H)
57. #1279	C₂×C₇	semidirect
101. #431	C₄×C₇	semidirect
105. C₄×M₁₆	C₄×C₇	semidirect
278. C₇	#7896	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#431	2
C₂×C₄×C₇×M₁₆	2
C₇×#2285	2
C₇×#1279	2
#7896	7

Subgroups — C7×#7896 #13185112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7896 #13185112