Subgroups — C₇×#12881 #13190097

Order 1792.

298 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#364	C₂	1	2	yes	maximal
2	C₇×#1726	C₂	1	1	yes	maximal, characteristic
3	C₇×#1377	C₂	1	2	yes	maximal
4	C₇×#1375	C₂	1	2	yes	maximal
5	C₂×C₇×M₃₂	C₂²	1	4	yes	—
6	C₂×C₇×#97	C₄	1	1	yes	characteristic
7	C₂×C₇×#97	C₂²	1	1	yes	characteristic
8	C₂×C₇×#97	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×#376	C₄	1	1	yes	characteristic
13	C₇×#364	—	2	4	no	—
14	#12881	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
22	C₇×M₃₂	—	2	4	no	—
23	C₇×M₃₂	—	2	4	no	—
24	C₂×C₇×C₁₆	—	2	4	no	—
25	C₇×#97	—	2	2	no	—
26	C₇×#97	—	2	2	no	—
27	C₇×#97	—	2	2	no	—
28	C₄×C₇×C₈	—	2	2	no	—
29	C₄×C₇×C₈	C₈	1	4	yes	—
30	C₄×C₇×C₈	D₄	1	2	yes	—
31	C₄×C₇×C₈	Q₈	1	2	yes	—
32	C₇×#95	—	2	2	no	—
33	C₂×#364	C₂×C₇	1	2	yes	—
34	#1726	C₂×C₇	1	1	yes	characteristic
35	#1377	C₂×C₇	1	2	yes	—
36	#1375	C₂×C₇	1	2	yes	—
37	C₂²×C₄×C₇	—	2	1	no	—
38	C₂²×C₄×C₇	—	2	1	no	—
39	C₂²×C₄×C₇	M₁₆	1	2	yes	—
40	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₇×C₈	—	2	1	no	—
42	C₂×C₇×C₈	—	2	1	no	—
43	C₂×C₇×C₈	—	2	1	no	—
44	C₂×C₇×C₈	—	2	1	no	—
45	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
47	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
48	C₂×C₇×C₈	M₁₆	1	2	yes	—
49	C₂×C₇×C₈	M₁₆	1	2	yes	—
50	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₇×C₈	#41	1	2	yes	—
53	C₂×C₇×C₈	#41	1	2	yes	—
54	C₄²×C₇	—	2	1	no	—
55	C₄²×C₇	—	2	1	no	—
56	C₄²×C₇	—	2	1	no	—
57	C₄²×C₇	—	2	1	no	—
58	C₄²×C₇	C₄⋊C₄	1	2	yes	—
59	C₄²×C₇	C₂×C₈	1	2	yes	—
60	C₇×C₁₆	—	4	4	no	—
61	C₂×M₃₂	C₂²×C₇	1	4	yes	—
62	C₂×#97	C₄×C₇	1	1	yes	characteristic
63	C₂×#97	C₂²×C₇	1	1	yes	characteristic
64	C₂×#97	C₂²×C₇	1	1	yes	characteristic
65	C₂×C₄×C₈	C₄×C₇	1	1	yes	characteristic
66	C₂×C₄×C₈	C₄×C₇	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
68	#376	C₄×C₇	1	1	yes	characteristic
69	#364	—	2	4	no	—
70	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇	—	2	2	no	—
72	C₂×C₄×C₇	—	2	2	no	—
73	C₂×C₄×C₇	—	2	2	no	—
74	C₂×C₄×C₇	—	4	2	no	—
75	C₂×C₄×C₇	—	4	2	no	—
76	C₂×C₄×C₇	—	4	2	no	—
77	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇	#117	1	1	yes	characteristic
79	C₂×C₄×C₇	#117	1	1	yes	characteristic
80	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
83	C₇×C₈	—	2	2	no	—
84	C₇×C₈	—	2	2	no	—
85	C₇×C₈	C₄⋊C₈	1	4	yes	—
86	C₇×C₈	#124	1	2	yes	—
87	C₇×C₈	#125	1	2	yes	—
88	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
89	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
90	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
91	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
92	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
93	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
94	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
95	M₃₂	—	2	4	no	—
96	M₃₂	—	2	4	no	—
97	C₂×C₁₆	—	2	4	no	—
98	#97	—	2	2	no	—
99	#97	—	2	2	no	—
100	#97	—	2	2	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	C₇×C₈	1	4	yes	—
103	C₄×C₈	C₇×D₄	1	2	yes	—
104	C₄×C₈	C₇×Q₈	1	2	yes	—
105	#95	—	2	2	no	—
106	C₂²×C₇	—	2	2	no	—
107	C₂²×C₇	#423	1	1	yes	characteristic
108	C₂²×C₇	#431	1	1	yes	characteristic
109	C₂²×C₇	#432	1	1	yes	characteristic
110	C₄×C₇	—	4	2	no	—
111	C₄×C₇	—	4	2	no	—
112	C₄×C₇	—	4	2	no	—
113	C₄×C₇	#382	1	1	yes	characteristic
114	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₄×C₇	#431	1	1	yes	characteristic
116	C₄×C₇	#432	1	1	yes	characteristic
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	C₇×M₁₆	1	2	yes	—
120	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	C₇×C₄⋊C₄	1	2	yes	—
126	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
127	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
128	C₂×C₈	C₇×M₁₆	1	2	yes	—
129	C₂×C₈	C₇×M₁₆	1	2	yes	—
130	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
131	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
132	C₂×C₈	C₇×#41	1	2	yes	—
133	C₂×C₈	C₇×#41	1	2	yes	—
134	C₄²	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	C₇×C₄⋊C₄	1	2	yes	—
139	C₄²	C₂×C₇×C₈	1	2	yes	—
140	C₁₆	—	4	4	no	—
141	C₂×C₇	—	2	1	no	—
142	C₂×C₇	—	2	1	no	—
143	C₂×C₇	#1831	1	1	yes	characteristic
144	C₂×C₇	#2166	1	2	yes	—
145	C₂³	C₂×C₇×M₁₆	1	1	yes	characteristic
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄	C₇×#117	1	1	yes	characteristic
154	C₂×C₄	C₇×#117	1	1	yes	characteristic
155	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
156	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
157	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
158	C₈	—	2	2	no	—
159	C₈	—	2	2	no	—
160	C₈	C₇×C₄⋊C₈	1	4	yes	—
161	C₈	C₇×#124	1	2	yes	—
162	C₈	C₇×#125	1	2	yes	—
163	C₇	#12881	1	1	yes	characteristic, direct product factor, Sylow
164	C₂²	—	2	2	no	—
165	C₂²	C₇×#423	1	1	yes	characteristic
166	C₂²	C₇×#431	1	1	yes	characteristic
167	C₂²	C₇×#432	1	1	yes	characteristic
168	C₄	—	4	2	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	2	no	—
171	C₄	C₇×#382	1	1	yes	characteristic
172	C₄	C₂×C₇×C₄⋊C₈	1	1	yes	characteristic
173	C₄	C₇×#431	1	1	yes	characteristic
174	C₄	C₇×#432	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	C₇×#1831	1	1	yes	characteristic
178	C₂	C₇×#2166	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	#117
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂²×C₇	#432
prime radical	C₂³×C₇	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#97 ◁ G

Complements

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

N	H ≅ G/N	type
14. #12881	C₇	direct (N × H)
163. C₇	#12881	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×#364	2
C₇×#1726	2
C₇×#1377	2
C₇×#1375	2
#12881	7

Subgroups — C7×#12881 #13190097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12881 #13190097