Subgroups — #89507

Order 448.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#6395	C₂	1	2	yes	maximal, complemented
5	#6391	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₇	—	2	1	no	—
8	C₂²×Dic₇	—	2	1	no	—
9	C₂×D₂₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₈	C₂²	1	2	yes	—
11	(C₂×D₇)⋊C₄	—	2	1	no	—
12	C₄⋊Dic₇	—	2	1	no	—
13	C₄⋊Dic₇	C₂²	1	2	yes	—
14	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
15	#481	—	7	1	no	maximal, Sylow
16	C₂³×C₇	D₄	1	1	yes	characteristic
17	C₂²×D₇	—	2	1	no	—
18	C₂×C₄×C₇	—	2	2	no	—
19	C₂×C₄×C₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂²⋊D₇	—	4	2	no	—
22	C₂×Dic₇	—	2	2	no	—
23	C₂×Dic₇	—	2	1	no	—
24	C₂×Dic₇	—	4	2	no	—
25	D₂₈	—	2	2	no	—
26	C₇×C₈	—	2	2	no	—
27	#121	—	7	1	no	—
28	C₂×C₄⋊C₄	—	7	1	no	—
29	#106	—	7	2	no	—
30	#102	—	7	2	no	—
31	C₂²⋊C₈	D₇	1	1	yes	characteristic, complemented
32	C₂²×C₇	—	2	1	no	—
33	C₂²×C₇	—	2	1	no	—
34	C₂²×C₇	SD₁₆	1	2	yes	—
35	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
36	C₂×D₇	—	4	1	no	—
37	C₂×D₇	—	4	1	no	—
38	C₂×D₇	—	4	1	no	—
39	C₄×C₇	—	2	1	no	—
40	C₄×C₇	#41	1	2	yes	—
41	Dic₇	—	4	1	no	—
42	Dic₇	—	4	2	no	—
43	C₂×D₄	—	7	1	no	—
44	C₂×D₄	—	14	1	no	—
45	C₂²×C₄	—	14	1	no	—
46	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
47	C₂×C₈	C₂×D₇	1	2	yes	—
48	C₄⋊C₄	—	7	2	no	—
49	C₄⋊C₄	—	7	1	no	—
50	C₄⋊C₄	—	14	1	no	—
51	C₂²⋊C₄	—	14	1	no	—
52	C₂×C₇	—	2	2	no	—
53	C₂×C₇	#123	1	1	yes	characteristic
54	C₂×C₇	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
56	D₇	—	8	1	no	—
57	C₂³	—	14	1	no	—
58	C₂³	D₂₈	1	1	yes	characteristic
59	D₄	—	14	2	no	—
60	D₄	—	28	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	14	1	no	—
63	C₂×C₄	—	14	2	no	—
64	C₂×C₄	—	28	2	no	—
65	C₂×C₄	D₂₈	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₇	#481	1	1	yes	characteristic, complemented, Sylow
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	28	1	no	—
72	C₂²	—	28	1	no	—
73	C₂²	—	28	1	no	—
74	C₂²	#1132	1	2	yes	—
75	C₂²	C₂×D₂₈	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	28	1	no	—
78	C₄	—	28	2	no	—
79	C₄	#1159	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	56	1	no	—
82	C₂	#6449	1	1	yes	characteristic
83	C₂	C₂×#1132	1	1	yes	characteristic
84	C₂	#6471	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₈
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×C₂²⋊C₈	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	#6493	C₂
hypercenter	C₂²⋊C₈	D₇
hypocenter	C₇	#481
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₇	C₂	semidirect
3. C₇×C₂²⋊C₈	C₂	semidirect
4. #6395	C₂	semidirect
5. #6391	C₂	semidirect
31. C₂²⋊C₈	D₇	semidirect
68. C₇	#481	semidirect

Maximal subgroups

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

subgroup	index
#6493	2
C₂×C₄⋊Dic₇	2
C₇×C₂²⋊C₈	2
#6395	2
#6391	2
#481	7