Group #22598854 — subgroups

Subgroups — #22598854

Order 9088.

356 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986556	C₂	1	2	yes	maximal
2	#17986404	C₂	1	2	yes	maximal, complemented
3	#17986401	C₂	1	4	yes	maximal, complemented
4	#17986397	C₂	1	2	yes	maximal, complemented
5	#17986037	C₂	1	4	yes	maximal
6	C₂×C₄×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
7	#14556128	C₂²	1	4	yes	—
8	#14556120	C₂²	1	4	yes	—
9	#14556107	—	2	4	no	—
10	C₄×Dic₁₄₂	—	2	4	no	—
11	#14556087	C₂²	1	4	yes	—
12	#14556062	C₂²	1	8	yes	—
13	#14556060	C₂²	1	8	yes	—
14	C₂²×C₈×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₇₁	C₂²	1	2	yes	—
18	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₄₂	—	2	4	no	—
20	C₄⋊Dic₇₁	—	2	4	no	—
21	C₄⋊Dic₇₁	C₂³	1	8	yes	—
22	Dic₇₁⋊C₄	—	2	8	no	—
23	C₂²⋊Dic₇₁	—	2	4	no	—
24	C₄×Dic₇₁	—	2	4	no	—
25	C₂²×C₄×C₇₁	D₄	1	2	yes	—
26	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₇₁	Q₈	1	8	yes	complemented
28	C₂×C₈×C₇₁	C₂³	1	2	yes	—
29	C₂×C₈×C₇₁	C₂³	1	2	yes	—
30	C₄²×C₇₁	D₄	1	2	yes	—
31	C₄²×C₇₁	C₂³	1	1	yes	characteristic
32	C₄²×C₇₁	C₂³	1	1	yes	characteristic
33	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	—	2	4	no	—
35	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	C₂×Q₈	1	4	yes	—
39	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
40	C₂×Dic₇₁	—	2	8	no	—
41	Dic₁₄₂	—	4	8	no	—
42	C₈×C₇₁	C₂×Q₈	1	4	yes	—
43	C₈×C₇₁	C₂×Q₈	1	4	yes	—
44	C₂²×C₇₁	—	2	2	no	—
45	C₂²×C₇₁	C₄⋊Q₈	1	2	yes	—
46	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
47	C₄×C₇₁	—	2	2	no	—
48	C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
49	C₄×C₇₁	#135	1	4	yes	—
50	C₄×C₇₁	C₂²×Q₈	1	2	yes	—
51	Dic₇₁	—	4	8	no	—
52	C₂×C₇₁	—	2	2	no	—
53	C₂×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₇₁	C₂×#135	1	2	yes	—
55	#3160	—	71	1	no	maximal, Sylow
56	C₇₁	#3160	1	1	yes	characteristic, complemented, Sylow
57	#533	—	71	2	no	—
58	#500	—	71	2	no	—
59	#499	—	71	4	no	—
60	#498	—	71	2	no	—
61	#427	—	71	4	no	—
62	C₂×C₄×C₈	D₇₁	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₇₁	1	2	yes	—
64	C₄⋊Q₈	—	71	4	no	—
65	#125	—	71	4	no	—
66	C₂²⋊Q₈	—	142	4	no	—
67	C₄×Q₈	—	142	4	no	—
68	#117	—	71	4	no	—
69	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
70	#107	—	71	8	no	—
71	#106	—	71	8	no	—
72	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
73	C₄×C₈	C₂×D₇₁	1	2	yes	—
74	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
75	C₂×Q₈	—	142	4	no	—
76	C₂²×C₄	D₂₈₄	1	2	yes	—
77	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
78	C₂×C₈	Dic₁₄₂	1	8	yes	complemented
79	C₂×C₈	C₂²×D₇₁	1	2	yes	—
80	C₂×C₈	C₂²×D₇₁	1	2	yes	—
81	C₄⋊C₄	—	71	8	no	—
82	C₄⋊C₄	—	142	8	no	—
83	C₄⋊C₄	—	142	4	no	—
84	C₂²⋊C₄	—	142	4	no	—
85	C₄²	—	142	4	no	—
86	C₄²	D₂₈₄	1	2	yes	—
87	C₄²	C₂²×D₇₁	1	1	yes	characteristic
88	C₄²	C₂²×D₇₁	1	1	yes	characteristic
89	C₂³	C₂×D₂₈₄	1	1	yes	characteristic
90	Q₈	—	284	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	142	8	no	—
93	C₂×C₄	C₂×D₂₈₄	1	2	yes	—
94	C₂×C₄	C₂×D₂₈₄	1	2	yes	—
95	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
96	C₂×C₄	C₂×Dic₁₄₂	1	4	yes	—
97	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
98	C₈	C₂×Dic₁₄₂	1	4	yes	—
99	C₈	C₂×Dic₁₄₂	1	4	yes	—
100	C₂²	—	2	2	no	—
101	C₂²	#14556128	1	2	yes	—
102	C₂²	C₂²×D₂₈₄	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	284	8	no	—
105	C₄	#14556128	1	2	yes	—
106	C₄	#14556150	1	4	yes	—
107	C₄	C₂²×Dic₁₄₂	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₂×#14556128	1	1	yes	characteristic
110	C₂	C₂×#14556150	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17986556

C₂

yes

maximal

#17986404

C₂

yes

maximal, complemented

#17986401

C₂

yes

maximal, complemented

#17986397

C₂

yes

maximal, complemented

#17986037

C₂

yes

maximal

C₂×C₄×C₈×C₇₁

C₂

yes

maximal, characteristic

#14556128

C₂²

yes

—

#14556120

C₂²

yes

—

#14556107

—

C₄×Dic₁₄₂

—

#14556087

C₂²

yes

—

#14556062

C₂²

yes

—

#14556060

C₂²

yes

—

C₂²×C₈×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂³

yes

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

Q₈

yes

complemented

C₂×C₈×C₇₁

C₂³

yes

—

C₂×C₈×C₇₁

C₂³

yes

—

C₄²×C₇₁

D₄

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

C₂×Q₈

yes

—

C₈×C₇₁

C₂×Q₈

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄⋊Q₈

yes

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

C₄⋊Q₈

yes

—

C₄×C₇₁

#135

yes

—

C₄×C₇₁

C₂²×Q₈

yes

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₇₁

C₂×#135

yes

—

#3160

—

maximal, Sylow

C₇₁

#3160

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₂×C₄×C₈

D₇₁

yes

characteristic

C₂²×C₈

C₂×D₇₁

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

142

—

C₄×Q₈

—

142

—

#117

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₄×C₈

C₂×D₇₁

yes

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

142

—

C₂²×C₄

D₂₈₄

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

Dic₁₄₂

yes

complemented

C₂×C₈

C₂²×D₇₁

yes

—

C₂×C₈

C₂²×D₇₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₂²⋊C₄

—

142

—

C₄²

—

142

—

C₄²

D₂₈₄

yes

—

C₄²

C₂²×D₇₁

yes

characteristic

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

C₂×D₂₈₄

yes

characteristic

Q₈

—

284

—

C₂×C₄

—

C₂×C₄

—

142

—

C₂×C₄

C₂×D₂₈₄

yes

—

C₂×C₄

C₂×D₂₈₄

yes

—

C₂×C₄

C₂×D₂₈₄

yes

characteristic

C₂×C₄

C₂×Dic₁₄₂

yes

—

C₂×C₄

C₂³×D₇₁

yes

characteristic

C₈

C₂×Dic₁₄₂

yes

—

C₈

C₂×Dic₁₄₂

yes

—

100

C₂²

—

101

C₂²

#14556128

yes

—

102

C₂²

C₂²×D₂₈₄

yes

characteristic

103

C₄

—

104

C₄

—

284

—

105

C₄

#14556128

yes

—

106

C₄

#14556150

yes

—

107

C₄

C₂²×Dic₁₄₂

yes

—

108

C₂

—

109

C₂

C₂×#14556128

yes

characteristic

110

C₂

C₂×#14556150

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	C₂×C₄×C₈×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	C₂×C₄×C₈	D₇₁
hypocenter	C₇₁	#3160
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₂₈₄

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

C₂×C₄×C₈×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

hypercenter

C₂×C₄×C₈

D₇₁

hypocenter

C₇₁

#3160

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #17986404	C₂	semidirect
3. #17986401	C₂	semidirect
4. #17986397	C₂	semidirect
27. C₂×C₈×C₇₁	Q₈	semidirect
56. C₇₁	#3160	semidirect
78. C₂×C₈	Dic₁₄₂	semidirect

H ≅ G/N

type

2. #17986404

C₂

semidirect

3. #17986401

C₂

semidirect

4. #17986397

C₂

semidirect

27. C₂×C₈×C₇₁

Q₈

semidirect

56. C₇₁

#3160

semidirect

78. C₂×C₈

Dic₁₄₂

semidirect

Subgroups — #22598854

Conjugacy classes of subgroups

Complements