Group #26021699 — subgroups

Subgroups — #26021699

Order 11392.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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×#15667944	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151600	C₂	1	2	yes	maximal
3	#19151564	C₂	1	1	yes	maximal, characteristic
4	#19151366	C₂	1	2	yes	maximal, complemented
5	#341×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083574	C₂²	1	1	yes	characteristic
7	C₂×#13083558	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
9	#15667944	—	2	2	no	—
10	#15667937	—	2	2	no	—
11	#15667934	—	2	2	no	—
12	#15667924	—	2	2	no	—
13	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	#13083574	—	2	2	no	—
16	#13083574	—	2	2	no	—
17	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
18	C₂×D₃₅₆	D₄	1	1	yes	characteristic
19	C₂×C₂²⋊D₈₉	—	2	1	no	—
20	C₂²×Dic₈₉	—	2	2	no	—
21	C₂×C₄×D₈₉	—	2	1	no	—
22	#13083558	—	2	2	no	—
23	C₂×#10623494	—	2	1	no	—
24	C₄⋊Dic₈₉	—	2	2	no	—
25	C₄⋊Dic₈₉	—	2	2	no	—
26	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₈₉	—	2	2	no	—
28	M₁₆×C₈₉	—	2	2	no	—
29	C₂×C₈×C₈₉	—	2	2	no	—
30	C₂×C₈×C₈₉	—	2	2	no	—
31	C₂×C₈×C₈₉	D₄	1	2	yes	—
32	C₂³×C₈₉	#41	1	1	yes	characteristic
33	C₂²×D₈₉	—	2	1	no	—
34	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	#41	1	2	yes	—
37	C₂×C₄×C₈₉	#41	1	2	yes	—
38	C₂²⋊D₈₉	—	4	2	no	—
39	C₂×Dic₈₉	—	2	1	no	—
40	C₂×Dic₈₉	—	4	2	no	—
41	C₂×Dic₈₉	—	8	2	no	—
42	D₃₅₆	—	2	2	no	—
43	C₄×D₈₉	—	4	1	no	—
44	C₄×D₈₉	—	4	1	no	—
45	Dic₁₇₈	—	2	2	no	—
46	C₈×C₈₉	—	2	2	no	—
47	C₈×C₈₉	—	4	2	no	—
48	#10623494	—	4	1	no	—
49	C₂²×C₈₉	—	2	2	no	—
50	C₂²×C₈₉	#121	1	1	yes	characteristic
51	C₂²×C₈₉	#126	1	2	yes	—
52	C₂×D₈₉	—	4	1	no	—
53	C₂×D₈₉	—	4	2	no	—
54	C₄×C₈₉	#123	1	2	yes	—
55	C₄×C₈₉	#123	1	1	yes	characteristic
56	C₄×C₈₉	#124	1	1	yes	characteristic
57	Dic₈₉	—	4	1	no	—
58	Dic₈₉	—	8	2	no	—
59	C₂×C₈₉	—	2	2	no	—
60	C₂×C₈₉	#397	1	1	yes	characteristic
61	C₂×C₈₉	#471	1	2	yes	—
62	D₈₉	—	8	1	no	—
63	#2079	—	89	1	no	maximal, Sylow
64	C₈₉	#2079	1	1	yes	characteristic, complemented, Sylow
65	C₂×#106	—	89	1	no	—
66	#417	—	89	2	no	—
67	#408	—	89	1	no	—
68	#343	—	89	2	no	—
69	#341	D₈₉	1	1	yes	characteristic, complemented
70	C₂×#41	—	89	1	no	—
71	C₂×M₁₆	—	89	1	no	—
72	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
73	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	89	2	no	—
75	#106	—	178	2	no	—
76	#103	—	178	2	no	—
77	#102	—	178	2	no	—
78	C₂²⋊C₈	—	178	2	no	—
79	#41	—	178	2	no	—
80	#41	—	178	2	no	—
81	C₂×Q₈	—	89	1	no	—
82	C₂×D₄	—	89	1	no	—
83	C₂×D₄	—	178	1	no	—
84	C₂²×C₄	—	178	1	no	—
85	C₂²×C₄	—	178	2	no	—
86	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	178	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	178	1	no	—
93	C₂×C₈	D₃₅₆	1	2	yes	—
94	C₄⋊C₄	—	178	2	no	—
95	C₄⋊C₄	—	178	2	no	—
96	C₂³	—	178	1	no	—
97	C₂³	#13083575	1	1	yes	characteristic
98	Q₈	—	178	2	no	—
99	D₄	—	356	2	no	—
100	D₄	—	178	2	no	—
101	C₂×C₄	—	356	1	no	—
102	C₂×C₄	—	356	1	no	—
103	C₂×C₄	—	356	2	no	—
104	C₂×C₄	—	178	1	no	—
105	C₂×C₄	—	712	2	no	—
106	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
107	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
108	C₂×C₄	#13083573	1	2	yes	—
109	C₂×C₄	#13083574	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	4	2	no	—
112	C₈	—	356	1	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	356	1	no	—
115	C₂²	—	356	2	no	—
116	C₂²	#15667987	1	1	yes	characteristic
117	C₂²	#15668007	1	2	yes	—
118	C₄	—	356	1	no	—
119	C₄	—	712	2	no	—
120	C₄	#15667994	1	2	yes	—
121	C₄	#15667996	1	1	yes	characteristic
122	C₄	#15668000	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	712	1	no	—
125	C₂	#19151525	1	1	yes	characteristic
126	C₂	#19151862	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667944

C₂

yes

maximal, characteristic, complemented

#19151600

C₂

yes

maximal

#19151564

C₂

yes

maximal, characteristic

#19151366

C₂

yes

maximal, complemented

#341×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

characteristic

C₂×#13083558

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

—

—

—

—

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#13083574

—

#13083574

—

C₂×Dic₁₇₈

D₄

yes

characteristic

C₂×D₃₅₆

D₄

yes

characteristic

C₂×C₂²⋊D₈₉

—

C₂²×Dic₈₉

—

C₂×C₄×D₈₉

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₂³×C₈₉

#41

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#121

yes

characteristic

C₂²×C₈₉

#126

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

#123

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#397

yes

characteristic

C₂×C₈₉

#471

yes

—

D₈₉

—

#2079

—

maximal, Sylow

C₈₉

#2079

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

#106

—

178

—

#103

—

178

—

#102

—

178

—

C₂²⋊C₈

—

178

—

#41

—

178

—

#41

—

178

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

D₃₅₆

yes

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₂³

—

178

—

C₂³

#13083575

yes

characteristic

Q₈

—

178

—

D₄

—

356

—

100

D₄

—

178

—

101

C₂×C₄

—

356

—

102

C₂×C₄

—

356

—

103

C₂×C₄

—

356

—

104

C₂×C₄

—

178

—

105

C₂×C₄

—

712

—

106

C₂×C₄

D₄×D₈₉

yes

characteristic

107

C₂×C₄

C₂×D₃₅₆

yes

characteristic

108

C₂×C₄

#13083573

yes

—

109

C₂×C₄

#13083574

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

356

—

113

C₂²

—

114

C₂²

—

356

—

115

C₂²

—

356

—

116

C₂²

#15667987

yes

characteristic

117

C₂²

#15668007

yes

—

118

C₄

—

356

—

119

C₄

—

712

—

120

C₄

#15667994

yes

—

121

C₄

#15667996

yes

characteristic

122

C₄

#15668000

yes

characteristic

123

C₂

—

124

C₂

—

712

—

125

C₂

#19151525

yes

characteristic

126

C₂

#19151862

yes

—

	subgroup	G/H
center	C₂²	#15667987
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#341×C₈₉	C₂
socle	C₂²×C₈₉	#121
prime radical	C₂×#13083574	C₂²
hypercenter	#341	D₈₉
hypocenter	C₈₉	#2079
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#15667987

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#341×C₈₉

C₂

C₂²×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
1. C₂×#15667944	C₂	semidirect
4. #19151366	C₂	semidirect
5. #341×C₈₉	C₂	semidirect
64. C₈₉	#2079	semidirect
69. #341	D₈₉	semidirect

H ≅ G/N

type

1. C₂×#15667944

C₂

semidirect

4. #19151366

C₂

semidirect

5. #341×C₈₉

C₂

semidirect

64. C₈₉

#2079

semidirect

69. #341

D₈₉

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#15667944	2
#19151600	2
#19151564	2
#19151366	2
#341×C₈₉	2
#2079	89

subgroup

index

#341×C₈₉

Subgroups — #26021699

Conjugacy classes of subgroups

Complements

Maximal subgroups