Subgroups — C₇×#7899 #13185115

Order 1792.

638 conjugacy classes of subgroups.

The subgroup lattice has 294 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×C₇×#423	C₂	1	1	yes	maximal, characteristic
3	C₇×#2283	C₂	1	1	yes	maximal, characteristic
4	C₇×#1279	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×M₁₆	—	2	1	no	—
6	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₇×#431	—	2	2	no	—
8	C₇×#431	C₄	1	4	yes	complemented
9	C₇×#423	—	2	2	no	—
10	C₇×#423	C₄	1	4	yes	complemented
11	C₂×C₇×C₄⋊C₈	—	2	1	no	—
12	C₂×C₇×C₄⋊C₈	—	2	1	no	—
13	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₇×C₂²⋊C₈	—	2	1	no	—
15	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
16	C₇×#383	—	2	1	no	—
17	C₇×#382	C₄	1	2	yes	—
18	C₇×#381	—	2	1	no	—
19	C₇×#380	C₄	1	2	yes	—
20	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
21	C₄×C₇×C₂²⋊C₄	—	2	1	no	—
22	C₇×#376	—	2	1	no	—
23	#7899	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	4	no	—
28	C₂×C₇×M₁₆	—	4	4	no	—
29	C₂²×C₇×C₈	—	2	2	no	—
30	C₂²×C₇×C₈	—	2	2	no	—
31	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
32	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
33	C₂×C₄²×C₇	—	2	2	no	—
34	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇	D₄	1	2	yes	—
40	C₂×C₄²×C₇	Q₈	1	2	yes	—
41	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
42	C₇×C₄⋊C₈	—	2	4	no	—
43	C₇×C₄⋊C₈	—	2	4	no	—
44	C₇×C₄⋊C₈	—	2	2	no	—
45	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
46	C₇×C₂²⋊C₈	—	2	4	no	—
47	C₄×C₇×C₈	—	2	2	no	—
48	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
49	C₇×#95	—	2	2	no	—
50	C₇×#95	—	2	2	no	—
51	C₇×#95	—	2	2	no	—
52	C₇×#95	—	2	2	no	—
53	C₂×#431	C₂×C₇	1	1	yes	characteristic
54	C₂×#423	C₂×C₇	1	1	yes	characteristic
55	#2283	C₂×C₇	1	1	yes	characteristic
56	#1279	C₂×C₇	1	4	yes	complemented
57	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇	—	2	4	no	—
59	C₂²×C₄×C₇	—	2	2	no	—
60	C₂²×C₄×C₇	—	2	2	no	—
61	C₂²×C₄×C₇	—	2	4	no	—
62	C₂²×C₄×C₇	—	4	2	no	—
63	C₂²×C₄×C₇	C₄²	1	2	yes	—
64	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊C₄	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
75	C₇×M₁₆	—	4	8	no	—
76	C₂×C₇×C₈	—	2	4	no	—
77	C₂×C₇×C₈	—	2	4	no	—
78	C₂×C₇×C₈	—	4	4	no	—
79	C₂×C₇×C₈	—	4	4	no	—
80	C₇×C₄⋊C₄	—	2	8	no	—
81	C₇×C₂²⋊C₄	—	4	4	no	—
82	C₄²×C₇	—	4	2	no	—
83	C₄²×C₇	—	4	2	no	—
84	C₄²×C₇	C₄²	1	2	yes	—
85	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₇	C₂²⋊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	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
94	C₂²×M₁₆	—	2	1	no	—
95	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
96	#431	—	2	2	no	—
97	#431	C₄×C₇	1	4	yes	complemented
98	#423	—	2	2	no	—
99	#423	C₄×C₇	1	4	yes	complemented
100	C₂×C₄⋊C₈	—	2	1	no	—
101	C₂×C₄⋊C₈	—	2	1	no	—
102	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
103	C₂×C₂²⋊C₈	—	2	1	no	—
104	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
105	#383	—	2	1	no	—
106	#382	C₄×C₇	1	2	yes	—
107	#381	—	2	1	no	—
108	#380	C₄×C₇	1	2	yes	—
109	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
110	C₄×C₂²⋊C₄	—	2	1	no	—
111	#376	—	2	1	no	—
112	C₂³×C₇	—	2	2	no	—
113	C₂³×C₇	—	2	2	no	—
114	C₂³×C₇	—	2	2	no	—
115	C₂³×C₇	#95	1	2	yes	—
116	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
123	C₂×C₄×C₇	—	2	2	no	—
124	C₂×C₄×C₇	—	2	4	no	—
125	C₂×C₄×C₇	—	2	4	no	—
126	C₂×C₄×C₇	—	4	2	no	—
127	C₂×C₄×C₇	—	4	2	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	#95	1	2	yes	—
132	C₂×C₄×C₇	#95	1	2	yes	—
133	C₂×C₄×C₇	#95	1	2	yes	—
134	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂×C₄⋊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	C₂×C₄×C₇	#131	1	4	yes	—
141	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
142	C₂×C₄×C₇	#137	1	2	yes	—
143	C₇×C₈	—	4	4	no	—
144	C₇×C₈	—	4	4	no	—
145	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
146	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
147	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	—	4	4	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
153	C₂×C₂²⋊C₄	—	2	2	no	—
154	C₂×C₄²	—	2	2	no	—
155	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
156	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
157	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
158	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
159	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
160	C₂×C₄²	C₇×D₄	1	2	yes	—
161	C₂×C₄²	C₇×Q₈	1	2	yes	—
162	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
163	C₄⋊C₈	—	2	4	no	—
164	C₄⋊C₈	—	2	4	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
167	C₂²⋊C₈	—	2	4	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
170	#95	—	2	2	no	—
171	#95	—	2	2	no	—
172	#95	—	2	2	no	—
173	#95	—	2	2	no	—
174	C₂²×C₇	—	2	2	no	—
175	C₂²×C₇	—	2	2	no	—
176	C₂²×C₇	—	2	2	no	—
177	C₂²×C₇	—	4	4	no	—
178	C₂²×C₇	C₂×#95	1	1	yes	characteristic
179	C₂²×C₇	#408	1	2	yes	—
180	C₂²×C₇	#417	1	2	yes	—
181	C₂²×C₇	#421	1	2	yes	—
182	C₄×C₇	—	2	2	no	—
183	C₄×C₇	—	2	1	no	—
184	C₄×C₇	—	2	2	no	—
185	C₄×C₇	—	2	1	no	—
186	C₄×C₇	—	2	2	no	—
187	C₄×C₇	—	8	4	no	—
188	C₄×C₇	#405	1	2	yes	—
189	C₄×C₇	#418	1	1	yes	characteristic
190	C₄×C₇	#419	1	1	yes	characteristic
191	C₄×C₇	#424	1	2	yes	—
192	C₄×C₇	#428	1	2	yes	—
193	C₂⁴	C₇×C₂²⋊C₄	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	4	2	no	—
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	2	yes	—
204	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
205	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
206	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
207	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
208	C₂²×C₄	C₂²×C₄×C₇	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	M₁₆	—	4	8	no	—
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	C₄⋊C₄	—	2	8	no	—
217	C₂²⋊C₄	—	4	4	no	—
218	C₄²	—	4	2	no	—
219	C₄²	—	4	2	no	—
220	C₄²	C₄²×C₇	1	2	yes	—
221	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
222	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
223	C₄²	C₇×C₄⋊C₄	1	2	yes	—
224	C₄²	C₇×C₄⋊C₄	1	2	yes	—
225	C₄²	C₇×C₄⋊C₄	1	2	yes	—
226	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
227	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
228	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
229	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
230	C₂×C₇	—	4	2	no	—
231	C₂×C₇	#1495	1	4	yes	—
232	C₂×C₇	#1729	1	1	yes	characteristic
233	C₂×C₇	#1733	1	1	yes	characteristic
234	C₂×C₇	#1736	1	1	yes	characteristic
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	C₇×#95	1	2	yes	—
239	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	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	C₂×C₄²×C₇	1	1	yes	characteristic
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	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₇×#131	1	4	yes	—
264	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
265	C₂×C₄	C₇×#137	1	2	yes	—
266	C₈	—	4	4	no	—
267	C₈	—	4	4	no	—
268	C₇	#7899	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	4	4	no	—
273	C₂²	C₂×C₇×#95	1	1	yes	characteristic
274	C₂²	C₇×#408	1	2	yes	—
275	C₂²	C₇×#417	1	2	yes	—
276	C₂²	C₇×#421	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	2	no	—
282	C₄	—	8	4	no	—
283	C₄	C₇×#405	1	2	yes	—
284	C₄	C₇×#418	1	1	yes	characteristic
285	C₄	C₇×#419	1	1	yes	characteristic
286	C₄	C₇×#424	1	2	yes	—
287	C₄	C₇×#428	1	2	yes	—
288	C₂	—	4	2	no	—
289	C₂	C₇×#1495	1	4	yes	—
290	C₂	C₇×#1729	1	1	yes	characteristic
291	C₂	C₇×#1733	1	1	yes	characteristic
292	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
8. C₇×#431	C₄	semidirect
10. C₇×#423	C₄	semidirect
23. #7899	C₇	direct (N × H)
56. #1279	C₂×C₇	semidirect
97. #431	C₄×C₇	semidirect
99. #423	C₄×C₇	semidirect
268. C₇	#7899	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×C₇×#423	2
C₇×#2283	2
C₇×#1279	2
#7899	7

Subgroups — C7×#7899 #13185115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7899 #13185115