Subgroups — C₇×#1342 #11725708

Order 896.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#470	C₂	1	1	yes	maximal, characteristic
2	C₇×#349	C₂	1	1	yes	maximal, characteristic
3	C₇×#338	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×Q₁₆	C₄	1	1	yes	characteristic
5	C₂×C₇×M₁₆	—	2	1	no	—
6	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
7	C₇×C₂²⋊Q₈	—	2	1	no	—
8	C₇×M₃₂	—	2	1	no	—
9	C₇×#106	C₄	1	1	yes	characteristic
10	C₇×#103	—	2	1	no	—
11	#1342	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂×C₇×Q₈	—	2	1	no	—
13	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
14	C₇×Q₁₆	—	4	1	no	—
15	C₇×M₁₆	—	4	1	no	—
16	C₇×M₁₆	—	4	1	no	—
17	C₇×M₁₆	—	4	1	no	—
18	C₂×C₇×C₈	—	2	2	no	—
19	C₂×C₇×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₇×C₈	D₄	1	1	yes	characteristic
21	C₇×C₄⋊C₄	—	2	1	no	—
22	C₇×C₄⋊C₄	—	4	1	no	—
23	C₇×C₂²⋊C₄	—	4	1	no	—
24	C₇×C₁₆	—	4	1	no	—
25	#470	C₂×C₇	1	1	yes	characteristic
26	#349	C₂×C₇	1	1	yes	characteristic
27	#338	C₂×C₇	1	1	yes	characteristic
28	C₂³×C₇	SD₁₆	1	1	yes	characteristic
29	C₇×Q₈	—	4	1	no	—
30	C₇×Q₈	—	4	1	no	—
31	C₂×C₄×C₇	—	2	1	no	—
32	C₂×C₄×C₇	—	2	1	no	—
33	C₂×C₄×C₇	—	4	1	no	—
34	C₂×C₄×C₇	—	4	1	no	—
35	C₂×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₇	D₈	1	1	yes	characteristic
37	C₇×C₈	—	2	1	no	—
38	C₇×C₈	—	2	1	no	—
39	C₇×C₈	—	4	2	no	—
40	C₂×Q₁₆	C₄×C₇	1	1	yes	characteristic
41	C₂×M₁₆	—	2	1	no	—
42	C₂×M₁₆	C₂²×C₇	1	1	yes	characteristic
43	C₂²⋊Q₈	—	2	1	no	—
44	M₃₂	—	2	1	no	—
45	#106	C₄×C₇	1	1	yes	characteristic
46	#103	—	2	1	no	—
47	C₂²×C₇	—	2	1	no	—
48	C₂²×C₇	—	4	1	no	—
49	C₂²×C₇	#102	1	1	yes	characteristic
50	C₄×C₇	—	2	1	no	—
51	C₄×C₇	—	8	1	no	—
52	C₄×C₇	—	8	1	no	—
53	C₄×C₇	#99	1	1	yes	characteristic
54	C₄×C₇	#104	1	1	yes	characteristic
55	C₂×Q₈	—	2	1	no	—
56	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
57	Q₁₆	—	4	1	no	—
58	M₁₆	—	4	1	no	—
59	M₁₆	—	4	1	no	—
60	M₁₆	—	4	1	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	C₂×C₄×C₇	1	1	yes	characteristic
63	C₂×C₈	C₇×D₄	1	1	yes	characteristic
64	C₄⋊C₄	—	2	1	no	—
65	C₄⋊C₄	—	4	1	no	—
66	C₂²⋊C₄	—	4	1	no	—
67	C₁₆	—	4	1	no	—
68	C₂×C₇	—	2	1	no	—
69	C₂×C₇	—	4	1	no	—
70	C₂×C₇	#327	1	1	yes	characteristic
71	C₂³	C₇×SD₁₆	1	1	yes	characteristic
72	Q₈	—	4	1	no	—
73	Q₈	—	4	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄	C₇×D₈	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	2	1	no	—
82	C₈	—	4	2	no	—
83	C₇	#1342	1	1	yes	characteristic, direct product factor, Sylow
84	C₂²	—	2	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	C₇×#102	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	8	1	no	—
89	C₄	—	8	1	no	—
90	C₄	C₇×#99	1	1	yes	characteristic
91	C₄	C₇×#104	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	4	1	no	—
94	C₂	C₇×#327	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#327
commutator	C₂×C₈	C₂×C₄×C₇
Frattini, cosocle	C₂×M₁₆	C₂²×C₇
prime radical	C₂³×C₇	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
11. #1342	C₇	direct (N × H)
83. C₇	#1342	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×#470	2
C₇×#349	2
C₇×#338	2
#1342	7

Subgroups — C7×#1342 #11725708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#1342 #11725708