Subgroups — #89331

Order 448.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#120	C₂	1	1	yes	maximal, characteristic
2	#6408	C₂	1	1	yes	maximal, characteristic, complemented
3	#6407	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×C₇	C₄	1	1	yes	characteristic, complemented
5	C₂×C₇×D₄	—	2	1	no	—
6	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
7	C₇×C₂²⋊C₄	—	2	1	no	—
8	C₇×C₂²⋊C₄	C₄	1	1	yes	characteristic, complemented
9	C₂²⋊Dic₇	—	2	1	no	—
10	#1136	—	2	1	no	—
11	AΣL(1,𝔽₁₆)	—	7	1	no	maximal, Sylow
12	C₂³×C₇	—	2	1	no	—
13	C₂³×C₇	—	4	2	no	—
14	C₂³×C₇	—	4	1	no	—
15	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₇	D₄	1	1	yes	characteristic
17	C₇×D₄	—	4	1	no	—
18	C₇×D₄	—	4	2	no	—
19	C₂×C₄×C₇	—	2	1	no	—
20	C₂×C₄×C₇	D₄	1	1	yes	characteristic
21	C₂×Dic₇	—	4	1	no	—
22	C₇⋊C₈	—	4	1	no	—
23	#120	D₇	1	1	yes	characteristic
24	#100	—	7	1	no	—
25	#99	—	7	1	no	—
26	C₂²×C₇	—	2	2	no	—
27	C₂²×C₇	—	2	1	no	—
28	C₂²×C₇	—	2	1	no	—
29	C₂²×C₇	—	4	1	no	—
30	C₂²×C₇	—	4	1	no	—
31	C₂²×C₇	—	4	1	no	—
32	C₂²×C₇	—	4	2	no	—
33	C₂²×C₇	—	4	2	no	—
34	C₂²×C₇	—	4	1	no	—
35	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₄×C₇	—	2	1	no	—
37	C₄×C₇	—	4	1	no	—
38	Dic₇	—	8	1	no	—
39	C₂⁴	Dic₇	1	1	yes	characteristic, complemented
40	C₂×D₄	—	2	1	no	—
41	C₂×D₄	C₂×D₇	1	1	yes	characteristic
42	M₁₆	—	14	1	no	—
43	C₂²⋊C₄	—	2	1	no	—
44	C₂²⋊C₄	—	14	1	no	—
45	C₂²⋊C₄	Dic₇	1	1	yes	characteristic, complemented
46	C₂×C₇	—	2	1	no	—
47	C₂×C₇	—	4	1	no	—
48	C₂×C₇	—	4	2	no	—
49	C₂×C₇	—	4	1	no	—
50	C₂×C₇	#99	1	1	yes	characteristic
51	C₂³	—	2	1	no	—
52	C₂³	—	4	2	no	—
53	C₂³	—	4	1	no	—
54	C₂³	C₂×Dic₇	1	1	yes	characteristic
55	C₂³	C₂²⋊D₇	1	1	yes	characteristic
56	D₄	—	4	1	no	—
57	D₄	—	4	2	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	28	1	no	—
60	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
61	C₈	—	28	1	no	—
62	C₇	AΣL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
63	C₂²	—	2	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	4	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	2	no	—
70	C₂²	—	4	2	no	—
71	C₂²	—	4	1	no	—
72	C₂²	C₂²⋊Dic₇	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	4	1	no	—
75	C₄	—	56	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	—	4	2	no	—
79	C₂	—	4	1	no	—
80	C₂	#6408	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6408
commutator	C₂³×C₇	C₂×C₄
Frattini	C₂×D₄	C₂×D₇
Fitting, prime radical	C₇×#120	C₂
socle	C₂×C₇	#99
cosocle	C₂×C₇×D₄	C₂²
hypercenter	#120	D₇
hypocenter	C₇	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₇ ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #120

Complements

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

N	H ≅ G/N	type
2. #6408	C₂	semidirect
3. #6407	C₂	semidirect
4. C₂⁴×C₇	C₄	semidirect
8. C₇×C₂²⋊C₄	C₄	semidirect
39. C₂⁴	Dic₇	semidirect
45. C₂²⋊C₄	Dic₇	semidirect
62. C₇	AΣL(1,𝔽₁₆)	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#120	2
#6408	2
#6407	2
AΣL(1,𝔽₁₆)	7