Subgroups — #89773

Order 448.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1136	C₂	1	1	yes	maximal, characteristic, complemented
4	#6383	C₂	1	4	yes	maximal, complemented
5	C₂³×D₇	—	2	1	no	—
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂×D₂₈	—	2	1	no	—
8	C₂×D₂₈	—	2	2	no	—
9	C₂×D₂₈	C₄	1	4	yes	complemented
10	C₂×D₂₈	C₂²	1	2	yes	—
11	C₇×C₄⋊C₄	C₂²	1	2	yes	complemented
12	C₇×C₂²⋊C₄	—	2	1	no	—
13	C₄²×C₇	—	2	1	no	—
14	#1136	—	2	2	no	—
15	C₂×C₇⋊C₈	C₂²	1	2	yes	complemented
16	#418	—	7	1	no	maximal, Sylow
17	C₂³×C₇	D₄	1	1	yes	characteristic
18	C₂²×D₇	—	2	4	no	—
19	C₂²×D₇	—	2	2	no	—
20	C₂²×D₇	—	4	4	no	—
21	C₂×C₄×C₇	—	2	2	no	—
22	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₇	D₄	1	2	yes	—
24	C₂×C₄×C₇	D₄	1	1	yes	characteristic
25	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
26	D₂₈	—	2	4	no	—
27	D₂₈	—	2	2	no	—
28	D₂₈	C₂×C₄	1	4	yes	—
29	C₇⋊C₈	—	2	2	no	—
30	C₂²×D₄	—	7	1	no	—
31	C₂×M₁₆	—	7	1	no	—
32	#117	D₇	1	1	yes	characteristic, complemented
33	#102	—	7	4	no	—
34	C₂²×C₇	—	2	2	no	—
35	C₂²×C₇	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
37	C₂×D₇	—	2	4	no	—
38	C₂×D₇	—	4	8	no	—
39	C₂×D₇	—	4	4	no	—
40	C₄×C₇	—	4	2	no	—
41	C₄×C₇	C₂²⋊C₄	1	2	yes	—
42	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
43	C₄×C₇	C₂×D₄	1	1	yes	characteristic
44	C₂⁴	—	14	1	no	—
45	C₂×D₄	—	7	4	no	—
46	C₂×D₄	—	7	2	no	—
47	C₂×D₄	—	14	2	no	—
48	C₂×D₄	—	14	1	no	—
49	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
50	M₁₆	—	14	2	no	—
51	C₂×C₈	—	7	2	no	—
52	C₄⋊C₄	C₂×D₇	1	2	yes	complemented
53	C₂²⋊C₄	—	2	1	no	—
54	C₄²	—	2	1	no	—
55	C₂×C₇	—	2	2	no	—
56	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₇	AGL(1,ℤ/8)	1	2	yes	—
58	D₇	—	4	4	no	—
59	C₂³	—	14	4	no	—
60	C₂³	—	14	2	no	—
61	C₂³	—	28	4	no	—
62	C₂³	C₂²⋊D₇	1	1	yes	characteristic
63	D₄	—	7	4	no	—
64	D₄	—	14	4	no	—
65	D₄	—	14	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	C₄×D₇	1	2	yes	—
68	C₂×C₄	D₂₈	1	2	yes	—
69	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
71	C₈	—	14	2	no	—
72	C₇	#418	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	2	2	no	—
74	C₂²	—	14	4	no	—
75	C₂²	—	28	8	no	—
76	C₂²	—	28	4	no	—
77	C₂²	(C₂×D₇)⋊C₄	1	2	yes	—
78	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
79	C₄	—	4	2	no	—
80	C₄	(C₂×D₇)⋊C₄	1	2	yes	—
81	C₄	C₂×C₄×D₇	1	1	yes	characteristic
82	C₄	C₂×D₂₈	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	28	4	no	—
85	C₂	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
86	C₂	#6512	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#117	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂²×D₂₈	C₂
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#117	D₇
hypocenter	C₇	#418
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₇×#117	C₂	semidirect
3. C₂×#1136	C₂	semidirect
4. #6383	C₂	semidirect
9. C₂×D₂₈	C₄	semidirect
11. C₇×C₄⋊C₄	C₂²	semidirect
15. C₂×C₇⋊C₈	C₂²	semidirect
32. #117	D₇	semidirect
52. C₄⋊C₄	C₂×D₇	semidirect
72. C₇	#418	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₂₈	2
C₇×#117	2
C₂×#1136	2
#6383	2
#418	7