Subgroups — C₇×#7916 #13185132

Order 1792.

622 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#383	C₂	1	1	yes	maximal, characteristic
4	C₇×#1280	C₂	1	4	yes	maximal
5	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₇×#432	—	2	2	no	—
7	C₇×#432	C₄	1	4	yes	complemented
8	C₇×#431	—	2	2	no	—
9	C₇×#431	C₄	1	4	yes	complemented
10	C₂×C₇×C₄⋊C₈	—	2	1	no	—
11	C₂×C₇×C₄⋊C₈	—	2	1	no	—
12	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
13	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
14	C₂×C₇×#97	—	2	1	no	—
15	C₂×C₄×C₇×C₈	—	2	1	no	—
16	C₇×#383	—	2	1	no	—
17	C₇×#383	—	2	1	no	—
18	C₇×#383	—	2	1	no	—
19	C₇×#383	—	2	1	no	—
20	C₂×C₇×#95	C₄	1	2	yes	—
21	C₂×C₇×#95	C₄	1	2	yes	—
22	C₂×C₇×#95	C₂²	1	2	yes	—
23	#7916	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₇	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₂²×C₇×C₈	—	2	2	no	—
29	C₂²×C₇×C₈	—	2	2	no	—
30	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₇	D₄	1	2	yes	—
33	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₇	Q₈	1	2	yes	—
38	C₇×C₄⋊C₈	—	2	4	no	—
39	C₇×C₄⋊C₈	—	2	4	no	—
40	C₇×C₂²⋊C₈	—	2	2	no	—
41	C₇×C₂²⋊C₈	—	2	2	no	—
42	C₇×C₂²⋊C₈	C₂×C₄	1	4	yes	—
43	C₇×C₂²⋊C₈	C₂×C₄	1	4	yes	—
44	C₇×#97	—	2	4	no	—
45	C₄×C₇×C₈	—	2	4	no	—
46	C₇×#95	—	2	2	no	—
47	C₇×#95	—	2	2	no	—
48	C₇×#95	—	2	2	no	—
49	C₇×#95	—	2	2	no	—
50	C₇×#95	—	2	2	no	—
51	C₇×#95	—	2	2	no	—
52	C₂×#432	C₂×C₇	1	1	yes	characteristic
53	C₂×#431	C₂×C₇	1	1	yes	characteristic
54	C₂×#383	C₂×C₇	1	1	yes	characteristic
55	#1280	C₂×C₇	1	4	yes	—
56	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	—	2	4	no	—
58	C₂²×C₄×C₇	—	2	4	no	—
59	C₂²×C₄×C₇	—	2	8	no	—
60	C₂²×C₄×C₇	—	4	2	no	—
61	C₂²×C₄×C₇	—	4	2	no	—
62	C₂²×C₄×C₇	—	4	4	no	—
63	C₂²×C₄×C₇	—	4	4	no	—
64	C₂²×C₄×C₇	C₄²	1	2	yes	—
65	C₂²×C₄×C₇	C₄²	1	2	yes	—
66	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
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	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₇×C₈	—	2	4	no	—
79	C₂×C₇×C₈	—	2	4	no	—
80	C₂×C₇×C₈	—	4	4	no	—
81	C₂×C₇×C₈	—	4	4	no	—
82	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₇	C₄⋊C₄	1	2	yes	—
87	C₄²×C₇	C₄⋊C₄	1	2	yes	—
88	C₄²×C₇	C₄⋊C₄	1	2	yes	—
89	C₄²×C₇	C₄⋊C₄	1	2	yes	—
90	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
91	#432	—	2	2	no	—
92	#432	C₄×C₇	1	4	yes	complemented
93	#431	—	2	2	no	—
94	#431	C₄×C₇	1	4	yes	complemented
95	C₂×C₄⋊C₈	—	2	1	no	—
96	C₂×C₄⋊C₈	—	2	1	no	—
97	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
98	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
99	C₂×#97	—	2	1	no	—
100	C₂×C₄×C₈	—	2	1	no	—
101	#383	—	2	1	no	—
102	#383	—	2	1	no	—
103	#383	—	2	1	no	—
104	#383	—	2	1	no	—
105	C₂×#95	C₄×C₇	1	2	yes	—
106	C₂×#95	C₄×C₇	1	2	yes	—
107	C₂×#95	C₂²×C₇	1	2	yes	—
108	C₂³×C₇	—	2	2	no	—
109	C₂³×C₇	—	2	2	no	—
110	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
111	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂³×C₇	#131	1	4	yes	—
114	C₂×C₄×C₇	—	2	4	no	—
115	C₂×C₄×C₇	—	2	2	no	—
116	C₂×C₄×C₇	—	2	2	no	—
117	C₂×C₄×C₇	—	2	2	no	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	—	2	2	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	2	4	no	—
122	C₂×C₄×C₇	—	4	2	no	—
123	C₂×C₄×C₇	—	4	2	no	—
124	C₂×C₄×C₇	—	4	1	no	—
125	C₂×C₄×C₇	—	4	1	no	—
126	C₂×C₄×C₇	—	4	1	no	—
127	C₂×C₄×C₇	—	4	1	no	—
128	C₂×C₄×C₇	—	4	4	no	—
129	C₂×C₄×C₇	—	4	4	no	—
130	C₂×C₄×C₇	—	4	4	no	—
131	C₂×C₄×C₇	—	8	8	no	—
132	C₂×C₄×C₇	#95	1	2	yes	—
133	C₂×C₄×C₇	#95	1	2	yes	—
134	C₂×C₄×C₇	#95	1	2	yes	—
135	C₂×C₄×C₇	#95	1	2	yes	—
136	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₇×C₈	—	4	4	no	—
141	C₇×C₈	—	4	4	no	—
142	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
143	C₂³×C₄	C₂×C₄×C₇	1	4	yes	—
144	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
145	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
149	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
150	C₂×C₄²	C₇×D₄	1	2	yes	—
151	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
152	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
153	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
154	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
155	C₂×C₄²	C₇×Q₈	1	2	yes	—
156	C₄⋊C₈	—	2	4	no	—
157	C₄⋊C₈	—	2	4	no	—
158	C₂²⋊C₈	—	2	2	no	—
159	C₂²⋊C₈	—	2	2	no	—
160	C₂²⋊C₈	C₂×C₄×C₇	1	4	yes	—
161	C₂²⋊C₈	C₂×C₄×C₇	1	4	yes	—
162	#97	—	2	4	no	—
163	C₄×C₈	—	2	4	no	—
164	#95	—	2	2	no	—
165	#95	—	2	2	no	—
166	#95	—	2	2	no	—
167	#95	—	2	2	no	—
168	#95	—	2	2	no	—
169	#95	—	2	2	no	—
170	C₂²×C₇	—	2	8	no	—
171	C₂²×C₇	—	2	1	no	—
172	C₂²×C₇	—	2	1	no	—
173	C₂²×C₇	—	2	1	no	—
174	C₂²×C₇	—	2	1	no	—
175	C₂²×C₇	C₂×#95	1	1	yes	characteristic
176	C₂²×C₇	#405	1	2	yes	—
177	C₂²×C₇	#408	1	2	yes	—
178	C₂²×C₇	#410	1	2	yes	—
179	C₂²×C₇	#413	1	2	yes	—
180	C₂²×C₇	#424	1	2	yes	—
181	C₄×C₇	—	2	2	no	—
182	C₄×C₇	—	2	2	no	—
183	C₄×C₇	—	2	1	no	—
184	C₄×C₇	—	2	1	no	—
185	C₄×C₇	—	2	1	no	—
186	C₄×C₇	—	2	1	no	—
187	C₄×C₇	—	2	2	no	—
188	C₄×C₇	—	2	2	no	—
189	C₄×C₇	—	8	4	no	—
190	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	4	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	C₄²×C₇	1	2	yes	—
199	C₂²×C₄	C₄²×C₇	1	2	yes	—
200	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
201	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
202	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
204	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
205	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
206	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	4	no	—
215	C₂×C₈	—	4	4	no	—
216	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
217	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
218	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
219	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
220	C₄²	C₇×C₄⋊C₄	1	2	yes	—
221	C₄²	C₇×C₄⋊C₄	1	2	yes	—
222	C₄²	C₇×C₄⋊C₄	1	2	yes	—
223	C₄²	C₇×C₄⋊C₄	1	2	yes	—
224	C₂×C₇	—	2	2	no	—
225	C₂×C₇	—	2	2	no	—
226	C₂×C₇	#1473	1	4	yes	—
227	C₂×C₇	#1729	1	1	yes	characteristic
228	C₂×C₇	#1740	1	1	yes	characteristic
229	C₂×C₇	#1742	1	1	yes	characteristic
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	C₂×C₄²×C₇	1	1	yes	characteristic
233	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
234	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
235	C₂³	C₇×#131	1	4	yes	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	8	8	no	—
254	C₂×C₄	C₇×#95	1	2	yes	—
255	C₂×C₄	C₇×#95	1	2	yes	—
256	C₂×C₄	C₇×#95	1	2	yes	—
257	C₂×C₄	C₇×#95	1	2	yes	—
258	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
261	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
262	C₈	—	4	4	no	—
263	C₈	—	4	4	no	—
264	C₇	#7916	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²	—	2	8	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	C₂×C₇×#95	1	1	yes	characteristic
271	C₂²	C₇×#405	1	2	yes	—
272	C₂²	C₇×#408	1	2	yes	—
273	C₂²	C₇×#410	1	2	yes	—
274	C₂²	C₇×#413	1	2	yes	—
275	C₂²	C₇×#424	1	2	yes	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	8	4	no	—
285	C₂	—	2	2	no	—
286	C₂	—	2	2	no	—
287	C₂	C₇×#1473	1	4	yes	—
288	C₂	C₇×#1729	1	1	yes	characteristic
289	C₂	C₇×#1740	1	1	yes	characteristic
290	C₂	C₇×#1742	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
7. C₇×#432	C₄	semidirect
9. C₇×#431	C₄	semidirect
23. #7916	C₇	direct (N × H)
92. #432	C₄×C₇	semidirect
94. #431	C₄×C₇	semidirect
264. C₇	#7916	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×#432	2
C₂×C₇×#431	2
C₂×C₇×#383	2
C₇×#1280	2
#7916	7

Subgroups — C7×#7916 #13185132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7916 #13185132