Group #22589780 — subgroups

Subgroups — #22589780

Order 9088.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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×#14556062	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986000	C₂	1	2	yes	maximal
3	#17985767	C₂	1	2	yes	maximal, complemented
4	#17985764	C₂	1	1	yes	maximal, characteristic, complemented
5	#408×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11765849	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
8	#14556062	—	2	2	no	—
9	#14556052	—	2	2	no	—
10	#14556049	—	2	2	no	—
11	C₂×#41×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈×C₇₁	—	2	2	no	—
15	C₂²×Dic₇₁	—	2	2	no	—
16	#11765849	—	2	2	no	—
17	#11765849	—	2	2	no	—
18	C₂×C₇₁⋊C₈	—	2	2	no	—
19	C₄⋊Dic₇₁	—	2	2	no	—
20	C₄⋊Dic₇₁	—	2	2	no	—
21	#41×C₇₁	—	2	2	no	—
22	#41×C₇₁	—	2	2	no	—
23	C₂×Q₈×C₇₁	D₄	1	1	yes	characteristic
24	C₂×D₄×C₇₁	—	2	1	no	—
25	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₁	—	2	1	no	—
27	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
28	M₁₆×C₇₁	—	2	2	no	—
29	C₂×C₈×C₇₁	—	2	2	no	—
30	C₂×C₈×C₇₁	—	2	1	no	—
31	C₂×C₈×C₇₁	D₄	1	2	yes	—
32	C₂³×C₇₁	—	2	1	no	—
33	C₂³×C₇₁	#41	1	1	yes	characteristic
34	Q₈×C₇₁	—	2	2	no	—
35	D₄×C₇₁	—	2	2	no	—
36	D₄×C₇₁	—	4	2	no	—
37	C₂×C₄×C₇₁	—	2	1	no	—
38	C₂×C₄×C₇₁	—	4	1	no	—
39	C₂×C₄×C₇₁	—	4	1	no	—
40	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₁	#41	1	2	yes	—
43	C₂×C₄×C₇₁	#41	1	2	yes	—
44	C₂×Dic₇₁	—	4	2	no	—
45	C₂×Dic₇₁	—	8	2	no	—
46	C₈×C₇₁	—	2	2	no	—
47	C₈×C₇₁	—	4	1	no	—
48	C₇₁⋊C₈	—	4	2	no	—
49	C₂²×C₇₁	—	2	1	no	—
50	C₂²×C₇₁	—	2	1	no	—
51	C₂²×C₇₁	—	4	1	no	—
52	C₂²×C₇₁	—	4	1	no	—
53	C₂²×C₇₁	—	4	1	no	—
54	C₂²×C₇₁	#121	1	1	yes	characteristic
55	C₂²×C₇₁	#126	1	2	yes	—
56	C₄×C₇₁	—	4	1	no	—
57	C₄×C₇₁	#123	1	1	yes	characteristic
58	C₄×C₇₁	#123	1	2	yes	—
59	C₄×C₇₁	#124	1	1	yes	characteristic
60	Dic₇₁	—	8	2	no	—
61	C₂×C₇₁	—	2	2	no	—
62	C₂×C₇₁	—	8	1	no	—
63	C₂×C₇₁	#397	1	1	yes	characteristic
64	C₂×C₇₁	#472	1	1	yes	characteristic
65	C₂×C₇₁	#473	1	1	yes	characteristic
66	#2078	—	71	1	no	maximal, Sylow
67	C₇₁	#2078	1	1	yes	characteristic, complemented, Sylow
68	C₂×#107	—	71	1	no	—
69	#417	—	71	2	no	—
70	#408	D₇₁	1	1	yes	characteristic
71	#343	—	71	2	no	—
72	#341	—	71	1	no	—
73	C₂×#41	C₂×D₇₁	1	1	yes	characteristic
74	C₂×M₁₆	—	71	2	no	—
75	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	71	2	no	—
78	#107	—	142	2	no	—
79	#103	—	142	2	no	—
80	#102	—	142	2	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	#41	—	2	2	no	—
83	#41	—	2	2	no	—
84	C₂×Q₈	D₂₈₄	1	1	yes	characteristic
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	D₂₈₄	1	1	yes	characteristic
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	142	2	no	—
89	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	142	2	no	—
92	M₁₆	—	142	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	142	2	no	—
96	C₂×C₈	C₂²⋊D₇₁	1	2	yes	—
97	C₄⋊C₄	—	142	2	no	—
98	C₄⋊C₄	—	142	2	no	—
99	C₂³	—	2	1	no	—
100	C₂³	#11765865	1	1	yes	characteristic
101	Q₈	—	2	2	no	—
102	D₄	—	2	2	no	—
103	D₄	—	4	2	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	284	2	no	—
108	C₂×C₄	—	568	2	no	—
109	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
110	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
111	C₂×C₄	#11765864	1	2	yes	—
112	C₂×C₄	#11765864	1	2	yes	—
113	C₈	—	2	2	no	—
114	C₈	—	4	1	no	—
115	C₈	—	284	2	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	1	no	—
121	C₂²	#14556102	1	1	yes	characteristic
122	C₂²	#14556121	1	2	yes	—
123	C₄	—	4	1	no	—
124	C₄	—	568	2	no	—
125	C₄	#14556109	1	1	yes	characteristic
126	C₄	#14556112	1	2	yes	—
127	C₄	#14556115	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	—	8	1	no	—
130	C₂	#17985924	1	1	yes	characteristic
131	C₂	#17986269	1	1	yes	characteristic
132	C₂	#17986274	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556062

C₂

yes

maximal, characteristic, complemented

#17986000

C₂

yes

maximal

#17985767

C₂

yes

maximal, complemented

#17985764

C₂

yes

maximal, characteristic, complemented

#408×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

#14556062

—

#14556052

—

#14556049

—

C₂×#41×C₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂²⋊C₈×C₇₁

—

C₂²×Dic₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

#41×C₇₁

—

#41×C₇₁

—

C₂×Q₈×C₇₁

D₄

yes

characteristic

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

—

C₂³×C₇₁

—

C₂³×C₇₁

#41

yes

characteristic

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#121

yes

characteristic

C₂²×C₇₁

#126

yes

—

C₄×C₇₁

—

C₄×C₇₁

#123

yes

characteristic

C₄×C₇₁

#123

yes

—

C₄×C₇₁

#124

yes

characteristic

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×#107

—

#417

—

#408

D₇₁

yes

characteristic

#343

—

#341

—

C₂×#41

C₂×D₇₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄⋊C₄

—

#107

—

142

—

#103

—

142

—

#102

—

142

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

D₂₈₄

yes

characteristic

C₂×D₄

—

C₂×D₄

D₂₈₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

142

—

M₁₆

—

142

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

C₂²⋊D₇₁

yes

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₂³

—

100

C₂³

#11765865

yes

characteristic

101

Q₈

—

102

D₄

—

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

284

—

108

C₂×C₄

—

568

—

109

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

110

C₂×C₄

C₂×D₂₈₄

yes

characteristic

111

C₂×C₄

#11765864

yes

—

112

C₂×C₄

#11765864

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

284

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#14556102

yes

characteristic

122

C₂²

#14556121

yes

—

123

C₄

—

124

C₄

—

568

—

125

C₄

#14556109

yes

characteristic

126

C₄

#14556112

yes

—

127

C₄

#14556115

yes

characteristic

128

C₂

—

129

C₂

—

130

C₂

#17985924

yes

characteristic

131

C₂

#17986269

yes

characteristic

132

C₂

#17986274

yes

characteristic

	subgroup	G/H
center	C₂²	#14556102
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#408×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#408	D₇₁
hypocenter	C₇₁	#2078
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₂²

#14556102

commutator

C₂²×C₄×C₇₁

C₂³

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#408×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
1. C₂×#14556062	C₂	semidirect
3. #17985767	C₂	semidirect
4. #17985764	C₂	semidirect
67. C₇₁	#2078	semidirect

H ≅ G/N

type

1. C₂×#14556062

C₂

semidirect

3. #17985767

C₂

semidirect

4. #17985764

C₂

semidirect

67. C₇₁

#2078

semidirect

Subgroups — #22589780

Conjugacy classes of subgroups

Complements