Subgroups — #89833

Order 448.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6548	C₂	1	1	yes	maximal, characteristic
2	C₇×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	#6505	C₂	1	2	yes	maximal, complemented
4	#6407	C₂	1	1	yes	maximal, characteristic, complemented
5	#6379	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
7	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊D₇	—	2	2	no	—
9	#1159	—	2	2	no	—
10	D₄×D₇	—	2	2	no	—
11	#1157	C₂²	1	2	yes	—
12	C₇×C₂²⋊C₄	—	2	2	no	—
13	C₄²×C₇	C₂²	1	1	yes	characteristic, complemented
14	#1144	—	2	2	no	—
15	#1143	—	2	2	no	—
16	#1136	C₂²	1	2	yes	complemented
17	#455	—	7	1	no	maximal, Sylow
18	C₂³×C₇	D₄	1	2	yes	—
19	C₂²×D₇	—	2	2	no	—
20	C₇×Q₈	—	2	2	no	—
21	C₇×D₄	—	2	2	no	—
22	C₂×C₄×C₇	—	2	1	no	—
23	C₂×C₄×C₇	—	2	1	no	—
24	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂²⋊D₇	—	2	4	no	—
26	C₂²⋊D₇	—	2	2	no	—
27	C₂×Dic₇	—	2	2	no	—
28	D₂₈	D₄	1	2	yes	—
29	C₄×D₇	—	2	2	no	—
30	Dic₁₄	D₄	1	2	yes	—
31	C₇⋊C₈	—	2	2	no	—
32	H(5,𝔽₂)	—	7	1	no	—
33	#137	—	7	2	no	—
34	#124	D₇	1	1	yes	characteristic, complemented
35	#104	—	7	2	no	—
36	#100	—	7	1	no	—
37	C₂²×C₇	—	2	2	no	—
38	C₂²×C₇	—	4	2	no	—
39	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
40	C₂×D₇	—	2	2	no	—
41	C₂×D₇	—	8	2	no	—
42	C₄×C₇	—	4	1	no	—
43	C₄×C₇	—	4	1	no	—
44	C₄×C₇	C₂×D₄	1	2	yes	—
45	Dic₇	—	2	2	no	—
46	#41	—	7	2	no	—
47	#41	—	14	2	no	—
48	C₂×Q₈	C₂×D₇	1	1	yes	characteristic, complemented
49	C₂×D₄	—	14	2	no	—
50	C₂×D₄	—	14	2	no	—
51	C₂×D₄	C₂×D₇	1	1	yes	characteristic
52	Q₁₆	—	14	2	no	—
53	SD₁₆	—	14	2	no	—
54	M₁₆	—	7	2	no	—
55	C₂²⋊C₄	—	2	2	no	—
56	C₄²	C₂×D₇	1	1	yes	characteristic, complemented
57	C₂×C₇	—	2	1	no	—
58	C₂×C₇	—	4	2	no	—
59	C₂×C₇	#120	1	1	yes	characteristic
60	D₇	—	4	2	no	—
61	C₂³	—	14	2	no	—
62	C₂³	C₂²⋊D₇	1	2	yes	—
63	Q₈	—	2	2	no	—
64	Q₈	—	7	2	no	—
65	D₄	—	2	2	no	—
66	D₄	—	7	2	no	—
67	D₄	—	14	4	no	—
68	D₄	—	14	2	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	14	2	no	—
72	C₂×C₄	—	14	2	no	—
73	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
74	C₈	—	14	2	no	—
75	C₇	#455	1	1	yes	characteristic, complemented, Sylow
76	C₂²	—	2	2	no	—
77	C₂²	—	4	2	no	—
78	C₂²	—	14	2	no	—
79	C₂²	—	56	2	no	—
80	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
81	C₄	—	4	1	no	—
82	C₄	—	4	1	no	—
83	C₄	—	14	2	no	—
84	C₄	D₄×D₇	1	2	yes	—
85	C₂	—	2	1	no	—
86	C₂	—	4	2	no	—
87	C₂	—	28	2	no	—
88	C₂	#6500	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6500
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#124	C₂
socle	C₂×C₇	#120
prime radical	#6548	C₂
hypercenter	#124	D₇
hypocenter	C₇	#455
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #124

Complements

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

N	H ≅ G/N	type
2. C₇×#124	C₂	semidirect
3. #6505	C₂	semidirect
4. #6407	C₂	semidirect
5. #6379	C₂	semidirect
6. C₂×C₇×Q₈	C₂²	semidirect
13. C₄²×C₇	C₂²	semidirect
16. #1136	C₂²	semidirect
34. #124	D₇	semidirect
48. C₂×Q₈	C₂×D₇	semidirect
56. C₄²	C₂×D₇	semidirect
75. C₇	#455	semidirect

Maximal subgroups

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

subgroup	index
#6548	2
C₇×#124	2
#6505	2
#6407	2
#6379	2
#455	7