Subgroups — #89353

Order 448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#1136	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
5	C₇×M₁₆	—	2	1	no	—
6	C₇×M₁₆	C₄	1	2	yes	—
7	C₂×C₇×C₈	—	2	1	no	—
8	#1136	—	2	1	no	—
9	#1136	C₄	1	2	yes	—
10	C₂×C₇⋊C₈	—	2	1	no	—
11	C₂×C₇⋊C₈	—	2	1	no	—
12	C₂×C₇⋊C₈	—	2	1	no	—
13	C₂×C₇⋊C₈	C₄	1	2	yes	—
14	#341	—	7	1	no	maximal, Sylow
15	C₂³×C₇	Q₈	1	1	yes	characteristic
16	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₇	D₄	1	1	yes	characteristic
22	C₇×C₈	—	2	2	no	—
23	C₇⋊C₈	—	2	2	no	—
24	C₇⋊C₈	—	2	2	no	—
25	C₂×M₁₆	—	7	1	no	—
26	C₂×M₁₆	D₇	1	1	yes	characteristic
27	C₂²×C₈	—	7	1	no	—
28	C₂²×C₇	—	2	2	no	—
29	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
32	C₄×C₇	C₄²	1	1	yes	characteristic
33	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
37	M₁₆	—	2	1	no	—
38	M₁₆	—	7	2	no	—
39	M₁₆	—	14	1	no	—
40	M₁₆	Dic₇	1	2	yes	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	7	2	no	—
43	C₂×C₈	—	14	1	no	—
44	C₂×C₈	—	14	1	no	—
45	C₂×C₈	—	14	1	no	—
46	C₂×C₇	—	2	1	no	—
47	C₂×C₇	—	2	1	no	—
48	C₂×C₇	#95	1	1	yes	characteristic
49	C₂×C₇	#108	1	2	yes	—
50	C₂³	Dic₁₄	1	1	yes	characteristic
51	C₂×C₄	C₄×D₇	1	1	yes	characteristic
52	C₂×C₄	C₄×D₇	1	1	yes	characteristic
53	C₂×C₄	D₂₈	1	1	yes	characteristic
54	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
57	C₈	—	2	2	no	—
58	C₈	—	14	2	no	—
59	C₈	—	14	2	no	—
60	C₇	#341	1	1	yes	characteristic, complemented, Sylow
61	C₂²	—	2	2	no	—
62	C₂²	Dic₇⋊C₄	1	1	yes	characteristic
63	C₂²	Dic₇⋊C₄	1	1	yes	characteristic
64	C₂²	C₄⋊Dic₇	1	1	yes	characteristic
65	C₄	C₄×Dic₇	1	1	yes	characteristic
66	C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
67	C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
68	C₄	C₂²⋊Dic₇	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	2	1	no	—
71	C₂	#6396	1	2	yes	—
72	C₂	C₇⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₇
commutator	C₄×C₇	C₄²
Frattini	C₂²×C₄	C₂×D₇
Fitting	C₂×C₇×M₁₆	C₂
socle	C₂²×C₇	C₄⋊C₄
prime radical	C₂³×C₇	Q₈
cosocle	C₂²×C₄×C₇	C₂²
hypercenter	C₂×M₁₆	D₇
hypocenter	C₇	#341
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
60. C₇	#341	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×M₁₆	2
C₂×#1136	2
C₂²×C₇⋊C₈	2
#341	7