Group #22588389 — subgroups

Subgroups — #22588389

Order 9088.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#14556136	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556063	C₂	1	2	yes	maximal, complemented
3	#17985981	C₂	1	2	yes	maximal
4	C₇₁⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
5	#338×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11765865	C₂²	1	1	yes	characteristic
7	#14556136	—	2	4	no	—
8	C₂×#11765849	C₂²	1	2	yes	—
9	C₂×#11765848	C₄	1	2	yes	—
10	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₇₁	C₄	1	2	yes	—
12	#14556063	—	2	4	no	—
13	#14556046	—	2	2	no	—
14	#14556044	—	2	2	no	—
15	C₂×M₁₆×C₇₁	C₂²	1	2	yes	—
16	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	#11765865	—	2	2	no	—
18	#11765865	—	2	2	no	—
19	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
20	C₂×D₂₈₄	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₇₁	Q₈	1	2	yes	—
22	C₂×C₄×D₇₁	C₂×C₄	1	2	yes	—
23	#11765849	—	2	2	no	—
24	#11765849	—	2	2	no	—
25	#11765848	—	2	4	no	—
26	C₂×C₇₁⋊C₈	—	2	2	no	—
27	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₇₁⋊C₈	D₄	1	4	yes	—
30	C₈×D₇₁	—	2	4	no	—
31	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
32	M₁₆×C₇₁	—	2	2	no	—
33	M₁₆×C₇₁	—	2	2	no	—
34	M₁₆×C₇₁	—	2	2	no	—
35	C₂×C₈×C₇₁	—	2	2	no	—
36	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
37	C₂³×C₇₁	C₂×Q₈	1	1	yes	characteristic
38	C₂²×D₇₁	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₁	#41	1	2	yes	—
43	C₂²⋊D₇₁	—	2	4	no	—
44	C₂×Dic₇₁	C₄⋊C₄	1	2	yes	—
45	D₂₈₄	—	2	2	no	—
46	C₄×D₇₁	—	2	2	no	—
47	C₄×D₇₁	—	2	2	no	—
48	Dic₁₄₂	—	2	2	no	—
49	C₈×C₇₁	—	2	2	no	—
50	C₈×C₇₁	—	4	2	no	—
51	C₇₁⋊C₈	—	4	2	no	—
52	C₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
53	C₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₇₁	—	4	1	no	—
55	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₇₁	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₇₁	—	2	2	no	—
58	C₂×D₇₁	—	4	2	no	—
59	C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₇₁	#117	1	1	yes	characteristic
61	C₄×C₇₁	#121	1	2	yes	—
62	Dic₇₁	—	2	2	no	—
63	C₂×C₇₁	—	2	1	no	—
64	C₂×C₇₁	—	2	2	no	—
65	C₂×C₇₁	#380	1	1	yes	characteristic
66	D₇₁	—	4	2	no	—
67	#1813	—	71	1	no	maximal, Sylow
68	C₇₁	#1813	1	1	yes	characteristic, complemented, Sylow
69	C₂×#131	—	71	1	no	—
70	C₂×#108	—	71	2	no	—
71	#413	—	71	2	no	—
72	#338	—	71	1	no	—
73	#338	D₇₁	1	1	yes	characteristic, complemented
74	C₂×#41	—	71	1	no	—
75	#131	—	142	4	no	—
76	C₂×M₁₆	—	71	2	no	—
77	C₂×M₁₆	—	71	2	no	—
78	C₂×M₁₆	C₂×D₇₁	1	2	yes	—
79	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
80	C₂²×C₈	—	71	1	no	—
81	C₂²×C₈	—	71	2	no	—
82	#108	—	142	4	no	—
83	#101	—	142	2	no	—
84	#100	—	142	2	no	—
85	#41	—	142	2	no	—
86	#41	—	142	2	no	—
87	C₂×Q₈	—	71	1	no	—
88	C₂×D₄	—	71	2	no	—
89	C₂×D₄	—	71	1	no	—
90	C₂²×C₄	—	71	2	no	—
91	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	142	4	no	—
96	M₁₆	—	142	2	no	—
97	M₁₆	—	142	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	71	1	no	—
100	C₂×C₈	—	71	1	no	—
101	C₂×C₈	—	71	4	no	—
102	C₂×C₈	—	142	4	no	—
103	C₂×C₈	—	142	2	no	—
104	C₂×C₈	C₄×D₇₁	1	2	yes	—
105	C₂³	—	71	2	no	—
106	C₂³	Q₈×D₇₁	1	1	yes	characteristic
107	Q₈	—	142	2	no	—
108	D₄	—	142	4	no	—
109	D₄	—	142	2	no	—
110	C₂×C₄	—	71	2	no	—
111	C₂×C₄	—	142	2	no	—
112	C₂×C₄	—	142	2	no	—
113	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
114	C₂×C₄	D₄×D₇₁	1	2	yes	—
115	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
116	C₂×C₄	#11765865	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	284	2	no	—
120	C₈	—	71	2	no	—
121	C₈	—	71	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	284	2	no	—
124	C₂²	—	142	2	no	—
125	C₂²	C₄⋊C₄×D₇₁	1	1	yes	characteristic
126	C₂²	#14556106	1	2	yes	—
127	C₄	—	142	2	no	—
128	C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
129	C₄	#14556085	1	1	yes	characteristic
130	C₄	#14556101	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	284	2	no	—
134	C₂	#17985858	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556136

C₂

yes

maximal, characteristic

C₂×#14556063

C₂

yes

maximal, complemented

#17985981

C₂

yes

maximal

C₇₁⋊#338

C₂

yes

maximal, characteristic, complemented

#338×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765865

C₂²

yes

characteristic

#14556136

—

C₂×#11765849

C₂²

yes

—

C₂×#11765848

C₄

yes

—

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₇₁

C₄

yes

—

#14556063

—

#14556046

—

#14556044

—

C₂×M₁₆×C₇₁

C₂²

yes

—

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

#11765865

—

#11765865

—

C₂×Dic₁₄₂

D₄

yes

characteristic

C₂×D₂₈₄

D₄

yes

characteristic

C₂×C₂²⋊D₇₁

Q₈

yes

—

C₂×C₄×D₇₁

C₂×C₄

yes

—

#11765849

—

#11765849

—

#11765848

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

D₄

yes

—

C₈×D₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂³×C₇₁

C₂×Q₈

yes

characteristic

C₂²×D₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₂²⋊D₇₁

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₂²⋊C₄

yes

—

C₇₁⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

C₂²⋊Q₈

yes

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

C₂×C₂²⋊C₄

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#380

yes

characteristic

D₇₁

—

#1813

—

maximal, Sylow

C₇₁

#1813

yes

characteristic, complemented, Sylow

—

—

—

—

D₇₁

yes

characteristic, complemented

C₂×#41

—

#131

—

142

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#108

—

142

—

#101

—

142

—

#100

—

142

—

#41

—

142

—

#41

—

142

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

142

—

M₁₆

—

142

—

M₁₆

—

142

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

142

—

103

C₂×C₈

—

142

—

104

C₂×C₈

C₄×D₇₁

yes

—

105

C₂³

—

106

C₂³

Q₈×D₇₁

yes

characteristic

107

Q₈

—

142

—

108

D₄

—

142

—

109

D₄

—

142

—

110

C₂×C₄

—

111

C₂×C₄

—

142

—

112

C₂×C₄

—

142

—

113

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

114

C₂×C₄

D₄×D₇₁

yes

—

115

C₂×C₄

D₄×D₇₁

yes

characteristic

116

C₂×C₄

#11765865

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

284

—

120

C₈

—

121

C₈

—

122

C₂²

—

123

C₂²

—

284

—

124

C₂²

—

142

—

125

C₂²

C₄⋊C₄×D₇₁

yes

characteristic

126

C₂²

#14556106

yes

—

127

C₄

—

142

—

128

C₄

C₂²⋊C₄×D₇₁

yes

characteristic

129

C₄

#14556085

yes

characteristic

130

C₄

#14556101

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

284

—

134

C₂

#17985858

yes

characteristic

	subgroup	G/H
center	C₄	#14556085
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#338×C₇₁	C₂
socle	C₂×C₇₁	#380
hypercenter	#338	D₇₁
hypocenter	C₇₁	#1813
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#14556085

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#338×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#14556063	C₂	semidirect
4. C₇₁⋊#338	C₂	semidirect
5. #338×C₇₁	C₂	semidirect
68. C₇₁	#1813	semidirect
73. #338	D₇₁	semidirect

H ≅ G/N

type

2. C₂×#14556063

C₂

semidirect

4. C₇₁⋊#338

C₂

semidirect

5. #338×C₇₁

C₂

semidirect

68. C₇₁

#1813

semidirect

73. #338

D₇₁

semidirect

Subgroups — #22588389

Conjugacy classes of subgroups

Complements