Subgroups — C₇×#3183 #11742492

Order 896.

370 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#529	C₂	1	2	yes	maximal, complemented
2	C₇×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#488	C₂	1	2	yes	maximal, complemented
5	C₇×#470	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#469	C₂	1	1	yes	maximal, characteristic
7	C₇×#468	C₂	1	2	yes	maximal, complemented
8	C₇×#442	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#441	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#440	C₂	1	2	yes	maximal, complemented
11	C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₇×SD₁₆	C₂²	1	2	yes	—
14	C₂×C₇×D₈	C₂²	1	1	yes	characteristic
15	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
17	C₇×#126	—	2	2	no	—
18	C₇×#125	C₂²	1	2	yes	—
19	C₇×#124	C₂²	1	2	yes	—
20	C₇×#123	—	2	2	no	—
21	C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₇×#121	C₂²	1	2	yes	—
23	C₄×C₇×Q₈	C₂²	1	2	yes	—
24	C₄×C₇×D₄	—	2	2	no	—
25	C₄×C₇×D₄	C₂²	1	2	yes	—
26	C₇×#117	—	2	2	no	—
27	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
28	C₇×#107	C₂²	1	2	yes	—
29	C₇×#106	C₂²	1	1	yes	characteristic
30	C₇×#106	C₂²	1	1	yes	characteristic
31	C₇×#103	C₂²	1	2	yes	—
32	C₇×#103	C₂²	1	2	yes	—
33	C₇×#102	C₂²	1	2	yes	—
34	C₇×#102	C₂²	1	2	yes	—
35	C₇×#97	C₂²	1	1	yes	characteristic
36	C₇×#97	C₂²	1	1	yes	characteristic
37	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
38	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
39	#3183	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₇×Q₈	C₂³	1	2	yes	—
41	C₂×C₇×D₄	—	2	2	no	—
42	C₂×C₇×D₄	C₂³	1	2	yes	—
43	C₂²×C₄×C₇	—	2	2	no	—
44	C₂²×C₄×C₇	D₄	1	2	yes	—
45	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₇×Q₁₆	—	2	2	no	—
47	C₇×SD₁₆	—	2	4	no	—
48	C₇×D₈	—	2	2	no	—
49	C₇×M₁₆	—	2	2	no	—
50	C₇×M₁₆	—	2	2	no	—
51	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₇×C₄⋊C₄	—	2	2	no	—
56	C₇×C₄⋊C₄	—	2	2	no	—
57	C₇×C₄⋊C₄	—	2	2	no	—
58	C₇×C₄⋊C₄	—	2	2	no	—
59	C₇×C₄⋊C₄	C₂³	1	2	yes	—
60	C₇×C₄⋊C₄	C₂³	1	2	yes	—
61	C₇×C₂²⋊C₄	—	2	2	no	—
62	C₇×C₂²⋊C₄	—	2	2	no	—
63	C₇×C₂²⋊C₄	—	2	2	no	—
64	C₇×C₂²⋊C₄	—	2	2	no	—
65	C₇×C₂²⋊C₄	—	2	2	no	—
66	C₄²×C₇	—	2	2	no	—
67	C₄²×C₇	D₄	1	2	yes	—
68	C₄²×C₇	C₂³	1	1	yes	characteristic
69	C₄²×C₇	C₂³	1	1	yes	characteristic
70	#529	C₂×C₇	1	2	yes	complemented
71	#499	C₂×C₇	1	1	yes	characteristic, complemented
72	#495	C₂×C₇	1	1	yes	characteristic, complemented
73	#488	C₂×C₇	1	2	yes	complemented
74	#470	C₂×C₇	1	1	yes	characteristic, complemented
75	#469	C₂×C₇	1	1	yes	characteristic
76	#468	C₂×C₇	1	2	yes	complemented
77	#442	C₂×C₇	1	1	yes	characteristic, complemented
78	#441	C₂×C₇	1	1	yes	characteristic, complemented
79	#440	C₂×C₇	1	2	yes	complemented
80	C₄×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
81	C₂³×C₇	—	2	2	no	—
82	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
83	C₇×Q₈	—	2	2	no	—
84	C₇×Q₈	—	2	2	no	—
85	C₇×D₄	—	2	2	no	—
86	C₇×D₄	—	2	2	no	—
87	C₇×D₄	—	4	4	no	—
88	C₂×C₄×C₇	—	2	1	no	—
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	2	2	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	4	2	no	—
94	C₂×C₄×C₇	—	4	4	no	—
95	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
99	C₇×C₈	—	2	1	no	—
100	C₇×C₈	—	2	1	no	—
101	C₇×C₈	#41	1	2	yes	—
102	C₇×C₈	#41	1	2	yes	—
103	C₂×Q₁₆	C₂²×C₇	1	1	yes	characteristic
104	C₂×SD₁₆	C₂²×C₇	1	2	yes	—
105	C₂×D₈	C₂²×C₇	1	1	yes	characteristic
106	C₂×M₁₆	C₂²×C₇	1	1	yes	characteristic
107	C₂×M₁₆	C₂²×C₇	1	1	yes	characteristic
108	#126	—	2	2	no	—
109	#125	C₂²×C₇	1	2	yes	—
110	#124	C₂²×C₇	1	2	yes	—
111	#123	—	2	2	no	—
112	C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
113	#121	C₂²×C₇	1	2	yes	—
114	C₄×Q₈	C₂²×C₇	1	2	yes	—
115	C₄×D₄	—	2	2	no	—
116	C₄×D₄	C₂²×C₇	1	2	yes	—
117	#117	—	2	2	no	—
118	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
119	#107	C₂²×C₇	1	2	yes	—
120	#106	C₂²×C₇	1	1	yes	characteristic
121	#106	C₂²×C₇	1	1	yes	characteristic
122	#103	C₂²×C₇	1	2	yes	—
123	#103	C₂²×C₇	1	2	yes	—
124	#102	C₂²×C₇	1	2	yes	—
125	#102	C₂²×C₇	1	2	yes	—
126	#97	C₂²×C₇	1	1	yes	characteristic
127	#97	C₂²×C₇	1	1	yes	characteristic
128	C₄×C₈	C₂²×C₇	1	1	yes	characteristic
129	C₄×C₈	C₂²×C₇	1	1	yes	characteristic
130	C₂²×C₇	—	2	1	no	—
131	C₂²×C₇	—	2	2	no	—
132	C₂²×C₇	—	4	2	no	—
133	C₂²×C₇	—	4	2	no	—
134	C₂²×C₇	—	4	2	no	—
135	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₇	—	2	1	no	—
137	C₄×C₇	—	2	2	no	—
138	C₄×C₇	—	2	2	no	—
139	C₄×C₇	—	4	2	no	—
140	C₄×C₇	—	4	2	no	—
141	C₄×C₇	—	4	2	no	—
142	C₄×C₇	C₂×#41	1	1	yes	characteristic
143	C₄×C₇	C₂×#41	1	1	yes	characteristic
144	C₂×Q₈	C₂³×C₇	1	2	yes	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	C₂³×C₇	1	2	yes	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	C₇×D₄	1	2	yes	—
149	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
150	Q₁₆	—	2	2	no	—
151	SD₁₆	—	2	4	no	—
152	D₈	—	2	2	no	—
153	M₁₆	—	2	2	no	—
154	M₁₆	—	2	2	no	—
155	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
156	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
157	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
158	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	C₂³×C₇	1	2	yes	—
164	C₄⋊C₄	C₂³×C₇	1	2	yes	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₄²	—	2	2	no	—
171	C₄²	C₇×D₄	1	2	yes	—
172	C₄²	C₂³×C₇	1	1	yes	characteristic
173	C₄²	C₂³×C₇	1	1	yes	characteristic
174	C₂×C₇	—	4	1	no	—
175	C₂×C₇	—	8	2	no	—
176	C₂×C₇	#532	1	1	yes	characteristic
177	C₂×C₇	#575	1	2	yes	—
178	C₂³	—	2	2	no	—
179	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
180	Q₈	—	2	2	no	—
181	Q₈	—	2	2	no	—
182	D₄	—	2	2	no	—
183	D₄	—	2	2	no	—
184	D₄	—	4	4	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
193	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
194	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
195	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	C₇×#41	1	2	yes	—
199	C₈	C₇×#41	1	2	yes	—
200	C₇	#3183	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²	—	2	1	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	2	no	—
213	C₄	C₂×C₇×#41	1	1	yes	characteristic
214	C₄	C₂×C₇×#41	1	1	yes	characteristic
215	C₂	—	4	1	no	—
216	C₂	—	8	2	no	—
217	C₂	C₇×#532	1	1	yes	characteristic
218	C₂	C₇×#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
prime radical	C₇×#469	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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×#529	C₂	semidirect
2. C₇×#499	C₂	semidirect
3. C₇×#495	C₂	semidirect
4. C₇×#488	C₂	semidirect
5. C₇×#470	C₂	semidirect
7. C₇×#468	C₂	semidirect
8. C₇×#442	C₂	semidirect
9. C₇×#441	C₂	semidirect
10. C₇×#440	C₂	semidirect
11. C₄×C₇×M₁₆	C₂	semidirect
39. #3183	C₇	direct (N × H)
70. #529	C₂×C₇	semidirect
71. #499	C₂×C₇	semidirect
72. #495	C₂×C₇	semidirect
73. #488	C₂×C₇	semidirect
74. #470	C₂×C₇	semidirect
76. #468	C₂×C₇	semidirect
77. #442	C₂×C₇	semidirect
78. #441	C₂×C₇	semidirect
79. #440	C₂×C₇	semidirect
80. C₄×M₁₆	C₂×C₇	semidirect
200. C₇	#3183	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×#529	2
C₇×#499	2
C₇×#495	2
C₇×#488	2
C₇×#470	2
C₇×#469	2
C₇×#468	2
C₇×#442	2
C₇×#441	2
C₇×#440	2
C₄×C₇×M₁₆	2
#3183	7

Subgroups — C7×#3183 #11742492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3183 #11742492