Group #22589894 — subgroups

Subgroups — #22589894

Order 9088.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#17986048	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986026	C₂	1	2	yes	maximal, complemented
3	#17986017	C₂	1	2	yes	maximal
4	#17985967	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11765865	C₂²	1	1	yes	characteristic
7	C₂×#11765849	C₂²	1	2	yes	—
8	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	#14556064	—	2	2	no	—
10	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
11	#14556056	—	2	4	no	—
12	(C₂×D₇₁)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	#117×C₇₁	C₂²	1	2	yes	—
15	#11765865	—	2	2	no	—
16	#11765865	—	2	2	no	—
17	C₂×Dic₁₄₂	Q₈	1	1	yes	characteristic
18	C₂×D₂₈₄	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₇₁	—	2	1	no	—
20	C₂×C₄×D₇₁	—	2	1	no	—
21	#11765849	—	2	2	no	—
22	#11765849	—	2	2	no	—
23	C₂×C₇₁⋊C₈	—	2	1	no	—
24	C₂×C₇₁⋊C₈	—	2	2	no	—
25	C₂×C₇₁⋊C₈	—	2	2	no	—
26	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₇₁	—	2	2	no	—
28	C₂×C₈×C₇₁	D₄	1	2	yes	—
29	C₄⋊C₄×C₇₁	—	2	2	no	—
30	C₂²⋊C₄×C₇₁	—	2	2	no	—
31	C₄²×C₇₁	D₄	1	4	yes	—
32	C₂³×C₇₁	#41	1	1	yes	characteristic
33	C₂²×D₇₁	—	2	1	no	—
34	C₂×C₄×C₇₁	—	2	4	no	—
35	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₇₁	#41	1	2	yes	—
39	C₂²⋊D₇₁	—	4	2	no	—
40	C₂×Dic₇₁	—	2	1	no	—
41	D₂₈₄	—	2	2	no	—
42	C₄×D₇₁	—	4	1	no	—
43	C₄×D₇₁	—	4	1	no	—
44	Dic₁₄₂	—	2	2	no	—
45	C₈×C₇₁	—	2	2	no	—
46	C₇₁⋊C₈	—	4	1	no	—
47	C₇₁⋊C₈	—	4	2	no	—
48	C₂²×C₇₁	—	4	1	no	—
49	C₂²×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₇₁	#123	1	2	yes	—
51	C₂×D₇₁	—	4	2	no	—
52	C₂×D₇₁	—	4	1	no	—
53	C₄×C₇₁	—	4	4	no	—
54	C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₇₁	#123	1	1	yes	characteristic
56	C₄×C₇₁	#127	1	1	yes	characteristic
57	Dic₇₁	—	4	1	no	—
58	C₂×C₇₁	—	2	2	no	—
59	C₂×C₇₁	—	2	1	no	—
60	C₂×C₇₁	#399	1	1	yes	characteristic
61	D₇₁	—	8	1	no	—
62	#2098	—	71	1	no	maximal, Sylow
63	C₇₁	#2098	1	1	yes	characteristic, complemented, Sylow
64	#430	—	71	1	no	—
65	#424	—	71	2	no	—
66	#421	—	71	2	no	—
67	#408	—	71	1	no	—
68	H(3,ℤ/4)	D₇₁	1	1	yes	characteristic, complemented
69	C₂×#41	—	71	1	no	—
70	C₂×M₁₆	—	71	2	no	—
71	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
72	C₂²×C₈	—	71	1	no	—
73	#117	C₂×D₇₁	1	2	yes	—
74	#108	—	142	2	no	—
75	C₄⋊C₈	—	142	4	no	—
76	#104	—	142	4	no	—
77	C₂²⋊C₈	—	142	2	no	—
78	#41	—	142	2	no	—
79	#41	—	142	2	no	—
80	C₂×Q₈	—	71	1	no	—
81	C₂×D₄	—	71	1	no	—
82	C₂×D₄	—	142	1	no	—
83	C₂²×C₄	—	142	1	no	—
84	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	142	2	no	—
87	M₁₆	—	142	2	no	—
88	C₂×C₈	—	142	1	no	—
89	C₂×C₈	—	142	2	no	—
90	C₂×C₈	—	142	2	no	—
91	C₂×C₈	D₂₈₄	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₂₈₄	1	4	yes	—
95	C₂³	—	142	1	no	—
96	C₂³	#11765864	1	1	yes	characteristic
97	Q₈	—	142	2	no	—
98	D₄	—	284	2	no	—
99	D₄	—	142	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	284	1	no	—
102	C₂×C₄	—	284	1	no	—
103	C₂×C₄	—	142	1	no	—
104	C₂×C₄	C₂×D₂₈₄	1	2	yes	—
105	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
107	C₂×C₄	#11765864	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	284	1	no	—
110	C₈	—	284	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	284	2	no	—
113	C₂²	—	284	1	no	—
114	C₂²	#14556108	1	1	yes	characteristic
115	C₂²	#14556112	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	284	1	no	—
118	C₄	#14556108	1	2	yes	—
119	C₄	#14556112	1	1	yes	characteristic
120	C₄	#14556125	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	568	1	no	—
124	C₂	#17985936	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986048

C₂

yes

maximal, characteristic, complemented

#17986026

C₂

yes

maximal, complemented

#17986017

C₂

yes

maximal

#17985967

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765865

C₂²

yes

characteristic

C₂×#11765849

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

#14556064

—

C₄⋊(C₇₁⋊C₈)

—

#14556056

—

(C₂×D₇₁)⋊C₈

—

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

#117×C₇₁

C₂²

yes

—

#11765865

—

#11765865

—

C₂×Dic₁₄₂

Q₈

yes

characteristic

C₂×D₂₈₄

Q₈

yes

characteristic

C₂×C₂²⋊D₇₁

—

C₂×C₄×D₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

D₄

yes

—

C₂³×C₇₁

#41

yes

characteristic

C₂²×D₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₂²⋊D₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₁

#123

yes

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂²⋊Q₈

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#399

yes

characteristic

D₇₁

—

#2098

—

maximal, Sylow

C₇₁

#2098

yes

characteristic, complemented, Sylow

—

—

—

—

D₇₁

yes

characteristic, complemented

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

C₂²×C₈

—

#117

C₂×D₇₁

yes

—

#108

—

142

—

C₄⋊C₈

—

142

—

#104

—

142

—

C₂²⋊C₈

—

142

—

#41

—

142

—

#41

—

142

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

142

—

M₁₆

—

142

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

D₂₈₄

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₂₈₄

yes

—

C₂³

—

142

—

C₂³

#11765864

yes

characteristic

Q₈

—

142

—

D₄

—

284

—

D₄

—

142

—

100

C₂×C₄

—

101

C₂×C₄

—

284

—

102

C₂×C₄

—

284

—

103

C₂×C₄

—

142

—

104

C₂×C₄

C₂×D₂₈₄

yes

—

105

C₂×C₄

C₂×D₂₈₄

yes

characteristic

106

C₂×C₄

Q₈×D₇₁

yes

characteristic

107

C₂×C₄

#11765864

yes

—

108

C₈

—

109

C₈

—

284

—

110

C₈

—

284

—

111

C₂²

—

112

C₂²

—

284

—

113

C₂²

—

284

—

114

C₂²

#14556108

yes

characteristic

115

C₂²

#14556112

yes

—

116

C₄

—

117

C₄

—

284

—

118

C₄

#14556108

yes

—

119

C₄

#14556112

yes

characteristic

120

C₄

#14556125

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

568

—

124

C₂

#17985936

yes

characteristic

	subgroup	G/H
center	C₄	#14556125
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	H(3,ℤ/4)×C₇₁	C₂
socle	C₂×C₇₁	#399
hypercenter	H(3,ℤ/4)	D₇₁
hypocenter	C₇₁	#2098
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#14556125

commutator

C₂²×C₄×C₇₁

C₂³

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

H(3,ℤ/4)×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #17986048	C₂	semidirect
2. #17986026	C₂	semidirect
5. H(3,ℤ/4)×C₇₁	C₂	semidirect
63. C₇₁	#2098	semidirect
68. H(3,ℤ/4)	D₇₁	semidirect

H ≅ G/N

type

1. #17986048

C₂

semidirect

2. #17986026

C₂

semidirect

5. H(3,ℤ/4)×C₇₁

C₂

semidirect

63. C₇₁

#2098

semidirect

68. H(3,ℤ/4)

D₇₁

semidirect

Subgroups — #22589894

Conjugacy classes of subgroups

Complements