Group #22590380 — subgroups

Subgroups — #22590380

Order 9088.

174 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₂×#14556071	C₂	1	2	yes	maximal, complemented
2	#17986376	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985799	C₂	1	2	yes	maximal, complemented
5	C₄×Q₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₆₈	—	2	1	no	—
7	C₂×D₅₆₈	C₂²	1	2	yes	—
8	#14556125	—	2	1	no	—
9	#14556071	—	2	4	no	—
10	C₂×C₇₁⋊C₁₆	C₂²	1	2	yes	—
11	C₂×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
12	C₄×Q₈×C₇₁	—	2	1	no	—
13	#107×C₇₁	C₂²	1	1	yes	characteristic
14	#103×C₇₁	—	2	1	no	—
15	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×D₂₈₄	—	2	2	no	—
17	C₂×D₂₈₄	—	4	1	no	—
18	C₂×D₂₈₄	—	4	1	no	—
19	D₅₆₈	—	2	2	no	—
20	D₅₆₈	—	2	2	no	—
21	D₅₆₈	—	4	2	no	—
22	C₇₁⋊C₁₆	—	2	2	no	—
23	C₂×Q₈×C₇₁	—	2	1	no	—
24	Q₁₆×C₇₁	—	2	1	no	—
25	Q₁₆×C₇₁	D₄	1	2	yes	complemented
26	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₇₁	—	2	1	no	—
29	C₄⋊C₄×C₇₁	—	4	1	no	—
30	C₄²×C₇₁	—	4	1	no	—
31	C₄²×C₇₁	D₄	1	1	yes	characteristic
32	C₂²×D₇₁	—	4	2	no	—
33	Q₈×C₇₁	—	2	2	no	—
34	Q₈×C₇₁	—	4	1	no	—
35	C₂×C₄×C₇₁	—	4	1	no	—
36	C₂×C₄×C₇₁	—	4	1	no	—
37	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
38	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
39	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
40	D₂₈₄	—	4	2	no	—
41	D₂₈₄	—	4	2	no	—
42	D₂₈₄	—	8	2	no	—
43	D₂₈₄	—	8	2	no	—
44	C₈×C₇₁	—	2	1	no	—
45	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₈×C₇₁	#41	1	1	yes	characteristic
47	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
48	C₂×D₇₁	—	8	2	no	—
49	C₂×D₇₁	—	8	2	no	—
50	C₂×D₇₁	—	8	2	no	—
51	C₄×C₇₁	—	2	1	no	—
52	C₄×C₇₁	—	4	2	no	—
53	C₄×C₇₁	—	8	1	no	—
54	C₄×C₇₁	SD₃₂	1	2	yes	complemented
55	C₄×C₇₁	#121	1	1	yes	characteristic
56	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
57	C₂×C₇₁	#459	1	1	yes	characteristic
58	C₂×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
59	C₂×C₇₁	#509	1	1	yes	characteristic
60	D₇₁	—	16	2	no	—
61	#2207	—	71	1	no	maximal, Sylow
62	C₇₁	#2207	1	1	yes	characteristic, complemented, Sylow
63	C₂×SD₃₂	—	71	2	no	—
64	#493	—	71	1	no	—
65	C₄×Q₁₆	D₇₁	1	1	yes	characteristic, complemented
66	C₄⋊C₁₆	—	71	1	no	—
67	#357	—	71	2	no	—
68	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
69	C₂×D₈	—	71	2	no	—
70	C₂×D₈	—	142	1	no	—
71	#127	—	142	1	no	—
72	C₄×Q₈	—	2	1	no	—
73	SD₃₂	—	142	4	no	—
74	C₂×C₁₆	—	71	2	no	—
75	#107	C₂×D₇₁	1	1	yes	characteristic
76	#103	—	2	1	no	—
77	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
78	C₂×Q₈	—	2	1	no	—
79	C₂×D₄	—	284	1	no	—
80	C₂×D₄	—	284	1	no	—
81	C₂×D₄	—	142	2	no	—
82	Q₁₆	—	2	1	no	—
83	Q₁₆	D₂₈₄	1	2	yes	complemented
84	D₈	—	284	2	no	—
85	D₈	—	142	2	no	—
86	D₈	—	142	2	no	—
87	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
88	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
89	C₄⋊C₄	—	2	1	no	—
90	C₄⋊C₄	—	4	1	no	—
91	C₄²	—	4	1	no	—
92	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
93	C₁₆	—	142	2	no	—
94	C₂³	—	284	2	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	4	1	no	—
97	D₄	—	284	2	no	—
98	D₄	—	284	2	no	—
99	D₄	—	568	2	no	—
100	D₄	—	568	2	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	#11765850	1	1	yes	characteristic
104	C₂×C₄	#11765850	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
106	C₈	—	2	1	no	—
107	C₈	C₂×D₂₈₄	1	1	yes	characteristic
108	C₈	#11765865	1	1	yes	characteristic
109	C₂²	—	568	2	no	—
110	C₂²	—	568	2	no	—
111	C₂²	—	568	2	no	—
112	C₂²	C₂×#11765850	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	8	1	no	—
116	C₄	#14556071	1	2	yes	complemented
117	C₄	#14556102	1	1	yes	characteristic
118	C₄	#14556153	1	1	yes	characteristic
119	C₂	—	1136	2	no	—
120	C₂	#17986194	1	1	yes	characteristic
121	C₂	C₂×#14556071	1	1	yes	characteristic
122	C₂	#17986434	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556071

C₂

yes

maximal, complemented

#17986376

C₂

yes

maximal, characteristic

C₄⋊(C₇₁⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

#17985799

C₂

yes

maximal, complemented

C₄×Q₁₆×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×D₅₆₈

—

C₂×D₅₆₈

C₂²

yes

—

#14556125

—

#14556071

—

C₂×C₇₁⋊C₁₆

C₂²

yes

—

C₂×Q₁₆×C₇₁

C₂²

yes

characteristic

C₄×Q₈×C₇₁

—

#107×C₇₁

C₂²

yes

characteristic

#103×C₇₁

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

D₅₆₈

—

D₅₆₈

—

D₅₆₈

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

D₄

yes

complemented

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

D₄

yes

characteristic

C₂²×D₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

C₈×C₇₁

—

C₈×C₇₁

C₂×D₄

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

SD₃₂

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

C₂×SD₃₂

yes

characteristic

C₂×C₇₁

#509

yes

characteristic

D₇₁

—

#2207

—

maximal, Sylow

C₇₁

#2207

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

#493

—

C₄×Q₁₆

D₇₁

yes

characteristic, complemented

C₄⋊C₁₆

—

#357

—

C₂×Q₁₆

C₂×D₇₁

yes

characteristic

C₂×D₈

—

C₂×D₈

—

142

—

#127

—

142

—

C₄×Q₈

—

SD₃₂

—

142

—

C₂×C₁₆

—

#107

C₂×D₇₁

yes

characteristic

#103

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

284

—

C₂×D₄

—

284

—

C₂×D₄

—

142

—

Q₁₆

—

Q₁₆

D₂₈₄

yes

complemented

D₈

—

284

—

D₈

—

142

—

D₈

—

142

—

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂²⋊D₇₁

yes

characteristic

C₁₆

—

142

—

C₂³

—

284

—

Q₈

—

Q₈

—

D₄

—

284

—

D₄

—

284

—

D₄

—

568

—

100

D₄

—

568

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#11765850

yes

characteristic

104

C₂×C₄

#11765850

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

106

C₈

—

107

C₈

C₂×D₂₈₄

yes

characteristic

108

C₈

#11765865

yes

characteristic

109

C₂²

—

568

—

110

C₂²

—

568

—

111

C₂²

—

568

—

112

C₂²

C₂×#11765850

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#14556071

yes

complemented

117

C₄

#14556102

yes

characteristic

118

C₄

#14556153

yes

characteristic

119

C₂

—

1136

—

120

C₂

#17986194

yes

characteristic

121

C₂

C₂×#14556071

yes

characteristic

122

C₂

#17986434

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	C₄×Q₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	C₄×Q₁₆	D₇₁
hypocenter	C₇₁	#2207
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×#11765850

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

C₄×Q₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₈

hypercenter

C₄×Q₁₆

D₇₁

hypocenter

C₇₁

#2207

LCS

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

N	H ≅ G/N	type
1. C₂×#14556071	C₂	semidirect
3. C₄⋊(C₇₁⋊C₁₆)	C₂	semidirect
4. #17985799	C₂	semidirect
5. C₄×Q₁₆×C₇₁	C₂	semidirect
25. Q₁₆×C₇₁	D₄	semidirect
54. C₄×C₇₁	SD₃₂	semidirect
62. C₇₁	#2207	semidirect
65. C₄×Q₁₆	D₇₁	semidirect
83. Q₁₆	D₂₈₄	semidirect
116. C₄	#14556071	semidirect

H ≅ G/N

type

1. C₂×#14556071

C₂

semidirect

3. C₄⋊(C₇₁⋊C₁₆)

C₂

semidirect

4. #17985799

C₂

semidirect

5. C₄×Q₁₆×C₇₁

C₂

semidirect

25. Q₁₆×C₇₁

D₄

semidirect

54. C₄×C₇₁

SD₃₂

semidirect

62. C₇₁

#2207

semidirect

65. C₄×Q₁₆

D₇₁

semidirect

83. Q₁₆

D₂₈₄

semidirect

116. C₄

#14556071

semidirect

Subgroups — #22590380

Conjugacy classes of subgroups

Complements