Subgroups — #89489

Order 448.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6467	C₂	1	1	yes	maximal, characteristic
2	#6464	C₂	1	2	yes	maximal, complemented
3	C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	#6393	C₂	1	1	yes	maximal, characteristic, complemented
5	#6379	C₂	1	2	yes	maximal, complemented
6	#1157	C₂²	1	2	yes	—
7	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₄²×C₇	C₂²	1	1	yes	characteristic
10	#1136	C₂²	1	2	yes	—
11	Dic₂₈	C₄	1	1	yes	characteristic, complemented
12	D₅₆	C₄	1	1	yes	characteristic, complemented
13	#1132	C₄	1	2	yes	complemented
14	#1131	—	2	2	no	—
15	C₈×D₇	—	2	2	no	—
16	#444	—	7	1	no	maximal, Sylow
17	C₂×C₄×C₇	—	2	1	no	—
18	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂²⋊D₇	—	2	2	no	—
20	D₂₈	C₂×C₄	1	2	yes	—
21	C₄×D₇	—	2	2	no	—
22	Dic₁₄	C₂×C₄	1	2	yes	—
23	C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₇×C₈	D₄	1	2	yes	—
26	C₇⋊C₈	—	2	2	no	—
27	#135	—	7	1	no	—
28	#131	—	7	2	no	—
29	#108	—	7	1	no	—
30	#104	—	7	2	no	—
31	#97	D₇	1	1	yes	characteristic, complemented
32	C₂²×C₇	#41	1	1	yes	characteristic
33	C₂×D₇	—	2	2	no	—
34	C₄×C₇	—	4	1	no	—
35	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₇	C₂×D₄	1	1	yes	characteristic
37	Dic₇	—	2	2	no	—
38	#41	—	7	2	no	—
39	Q₁₆	—	7	1	no	—
40	SD₁₆	—	7	2	no	—
41	D₈	—	7	1	no	—
42	M₁₆	—	7	2	no	—
43	M₁₆	—	14	2	no	—
44	C₂×C₈	—	14	2	no	—
45	C₂×C₈	C₂×D₇	1	1	yes	characteristic
46	C₂×C₈	C₂×D₇	1	1	yes	characteristic
47	C₄²	C₂×D₇	1	1	yes	characteristic
48	C₂×C₇	—	2	1	no	—
49	C₂×C₇	C₄×D₄	1	1	yes	characteristic
50	D₇	—	4	2	no	—
51	Q₈	—	7	2	no	—
52	D₄	—	7	2	no	—
53	D₄	—	14	2	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	14	2	no	—
56	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
57	C₈	—	14	2	no	—
58	C₈	C₄×D₇	1	1	yes	characteristic
59	C₈	C₄×D₇	1	1	yes	characteristic
60	C₈	D₂₈	1	2	yes	—
61	C₇	#444	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	14	2	no	—
63	C₂²	#1157	1	1	yes	characteristic
64	C₄	—	4	1	no	—
65	C₄	—	14	2	no	—
66	C₄	C₂×C₄×D₇	1	1	yes	characteristic
67	C₄	C₂×D₂₈	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	28	2	no	—
70	C₂	C₄×D₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₂₈
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#97	C₂
socle	C₂×C₇	C₄×D₄
prime radical	#6467	C₂
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#97	D₇
hypocenter	C₇	#444
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #97

Complements

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

N	H ≅ G/N	type
2. #6464	C₂	semidirect
3. C₇×#97	C₂	semidirect
4. #6393	C₂	semidirect
5. #6379	C₂	semidirect
11. Dic₂₈	C₄	semidirect
12. D₅₆	C₄	semidirect
13. #1132	C₄	semidirect
31. #97	D₇	semidirect
61. C₇	#444	semidirect

Maximal subgroups

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

subgroup	index
#6467	2
#6464	2
C₇×#97	2
#6393	2
#6379	2
#444	7