Group #26020121 — subgroups

Subgroups — #26020121

Order 11392.

272 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 149 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×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667923	C₂	1	2	yes	maximal
3	C₂×C₄×#10623494	C₂	1	1	yes	maximal, characteristic
4	#19151353	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10623494	C₂²	1	2	yes	—
7	C₂²×#10623494	C₂²	1	1	yes	characteristic
8	C₂²×#10623494	C₂²	1	1	yes	characteristic
9	C₄⋊#10623494	C₄	1	4	yes	complemented
10	#15667923	—	2	4	no	—
11	C₄×#10623494	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₄×C₈₉	C₄	1	2	yes	—
13	C₂×C₄²×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
15	#95×C₈₉	C₄	1	2	yes	—
16	C₂×#10623494	—	2	2	no	—
17	C₂×#10623494	—	2	1	no	—
18	C₂×#10623494	—	2	1	no	—
19	C₂×#10623494	—	2	2	no	—
20	C₂×#10623494	C₂×C₄	1	2	yes	—
21	C₂×#10623494	C₂×C₄	1	4	yes	—
22	C₂×#10623494	C₂×C₄	1	2	yes	—
23	C₂×#10623494	D₄	1	2	yes	—
24	C₂×#10623494	Q₈	1	2	yes	—
25	C₂²×C₄×C₈₉	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₈₉	—	2	4	no	—
30	C₄²×C₈₉	—	2	2	no	—
31	C₄²×C₈₉	—	2	2	no	—
32	C₄²×C₈₉	C₂×C₄	1	2	yes	—
33	C₄²×C₈₉	C₂×C₄	1	2	yes	—
34	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈₉	—	2	2	no	—
36	C₂×C₄×C₈₉	—	2	4	no	—
37	C₂×C₄×C₈₉	—	2	4	no	—
38	C₂×C₄×C₈₉	—	2	4	no	—
39	C₂×C₄×C₈₉	C₄²	1	2	yes	—
40	C₂×C₄×C₈₉	C₄²	1	2	yes	—
41	C₂×C₄×C₈₉	M₁₆	1	4	yes	—
42	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
47	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
48	#10623494	—	2	2	no	—
49	#10623494	—	2	4	no	—
50	#10623494	C₄⋊C₄	1	4	yes	complemented
51	C₂²×C₈₉	C₂×C₄²	1	1	yes	characteristic
52	C₂²×C₈₉	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₈₉	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₈₉	C₂×M₁₆	1	2	yes	—
55	C₂²×C₈₉	#131	1	2	yes	—
56	C₄×C₈₉	—	2	2	no	—
57	C₄×C₈₉	—	4	4	no	—
58	C₄×C₈₉	#97	1	4	yes	complemented
59	C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₈₉	#117	1	1	yes	characteristic
61	C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
62	C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
63	C₂×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₈₉	C₂×#97	1	1	yes	characteristic
65	C₂×C₈₉	#405	1	1	yes	characteristic
66	C₂×C₈₉	#436	1	2	yes	—
67	C₂×C₈₉	#446	1	2	yes	—
68	#1770	—	89	1	no	maximal, Sylow
69	C₈₉	#1770	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₄⋊C₈	—	89	1	no	—
71	C₂×#97	—	89	2	no	—
72	C₂×C₄×C₈	—	89	1	no	—
73	C₄×C₄⋊C₄	D₈₉	1	1	yes	characteristic
74	#336	—	89	2	no	—
75	C₂²×C₈	—	89	1	no	—
76	C₂²×C₈	—	89	1	no	—
77	C₂²×C₈	—	89	2	no	—
78	C₂×C₄⋊C₄	Dic₈₉	1	2	yes	—
79	C₂×C₄²	C₂×D₈₉	1	2	yes	—
80	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
81	C₄⋊C₈	—	89	4	no	—
82	#97	—	178	4	no	—
83	C₄×C₈	—	89	4	no	—
84	#95	Dic₈₉	1	2	yes	—
85	C₂²×C₄	C₂×Dic₈₉	1	4	yes	—
86	C₂²×C₄	C₂×Dic₈₉	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₈₉	1	1	yes	characteristic
88	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
89	C₂×C₈	—	89	2	no	—
90	C₂×C₈	—	89	2	no	—
91	C₂×C₈	—	89	2	no	—
92	C₂×C₈	—	89	2	no	—
93	C₂×C₈	—	89	4	no	—
94	C₂×C₈	—	178	2	no	—
95	C₂×C₈	—	178	1	no	—
96	C₂×C₈	—	178	1	no	—
97	C₂×C₈	—	178	2	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄²	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	C₄×D₈₉	1	2	yes	—
102	C₄²	C₄×D₈₉	1	2	yes	—
103	C₂³	C₂²×Dic₈₉	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	C₄×Dic₈₉	1	2	yes	—
109	C₂×C₄	C₄×Dic₈₉	1	2	yes	—
110	C₂×C₄	#13083558	1	4	yes	—
111	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
113	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
115	C₂×C₄	#13083573	1	1	yes	characteristic
116	C₂×C₄	#13083575	1	1	yes	characteristic
117	C₈	—	89	4	no	—
118	C₈	—	178	2	no	—
119	C₈	—	178	4	no	—
120	C₂²	C₂×C₄×Dic₈₉	1	1	yes	characteristic
121	C₂²	D₄×Dic₈₉	1	1	yes	characteristic
122	C₂²	Q₈×Dic₈₉	1	1	yes	characteristic
123	C₂²	C₂×#13083558	1	2	yes	—
124	C₂²	#15668019	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	#15667923	1	4	yes	complemented
128	C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
129	C₄	#15667970	1	1	yes	characteristic
130	C₄	#15667974	1	1	yes	characteristic
131	C₄	#15667977	1	1	yes	characteristic
132	C₂	C₄⋊C₄×Dic₈₉	1	1	yes	characteristic
133	C₂	C₂×#15667923	1	1	yes	characteristic
134	C₂	#19151554	1	1	yes	characteristic
135	C₂	#19151676	1	2	yes	—
136	C₂	#19151729	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#10623494

C₂

yes

maximal, characteristic

C₂×#15667923

C₂

yes

maximal

C₂×C₄×#10623494

C₂

yes

maximal, characteristic

#19151353

C₂

yes

maximal

C₄×C₄⋊C₄×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

C₂²

yes

—

C₂²×#10623494

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

characteristic

C₄⋊#10623494

C₄

yes

complemented

#15667923

—

C₄×#10623494

C₄

yes

complemented

C₂×C₄⋊C₄×C₈₉

C₄

yes

—

C₂×C₄²×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#95×C₈₉

C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

—

C₂×#10623494

D₄

yes

—

C₂×#10623494

Q₈

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂×C₄

yes

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄²

yes

—

C₂×C₄×C₈₉

C₄²

yes

—

C₂×C₄×C₈₉

M₁₆

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

—

C₄⋊C₄

yes

complemented

C₂²×C₈₉

C₂×C₄²

yes

characteristic

C₂²×C₈₉

C₄×D₄

yes

characteristic

C₂²×C₈₉

C₄×Q₈

yes

characteristic

C₂²×C₈₉

C₂×M₁₆

yes

—

C₂²×C₈₉

#131

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#97

yes

complemented

C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₉

#117

yes

characteristic

C₄×C₈₉

C₄×D₄

yes

characteristic

C₄×C₈₉

C₄×Q₈

yes

characteristic

C₂×C₈₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₈₉

C₂×#97

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#446

yes

—

#1770

—

maximal, Sylow

C₈₉

#1770

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

D₈₉

yes

characteristic

#336

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

Dic₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

#97

—

178

—

C₄×C₈

—

#95

Dic₈₉

yes

—

C₂²×C₄

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂×Dic₈₉

yes

characteristic

C₂²×C₄

C₂×Dic₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₄⋊C₄

—

C₄²

—

100

C₄²

—

101

C₄²

C₄×D₈₉

yes

—

102

C₄²

C₄×D₈₉

yes

—

103

C₂³

C₂²×Dic₈₉

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₄×Dic₈₉

yes

—

109

C₂×C₄

C₄×Dic₈₉

yes

—

110

C₂×C₄

#13083558

yes

—

111

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

112

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

113

C₂×C₄

D₄×D₈₉

yes

characteristic

114

C₂×C₄

Q₈×D₈₉

yes

characteristic

115

C₂×C₄

#13083573

yes

characteristic

116

C₂×C₄

#13083575

yes

characteristic

117

C₈

—

118

C₈

—

178

—

119

C₈

—

178

—

120

C₂²

C₂×C₄×Dic₈₉

yes

characteristic

121

C₂²

D₄×Dic₈₉

yes

characteristic

122

C₂²

Q₈×Dic₈₉

yes

characteristic

123

C₂²

C₂×#13083558

yes

—

124

C₂²

#15668019

yes

—

125

C₄

—

126

C₄

—

127

C₄

#15667923

yes

complemented

128

C₄

C₄⋊C₄×D₈₉

yes

characteristic

129

C₄

#15667970

yes

characteristic

130

C₄

#15667974

yes

characteristic

131

C₄

#15667977

yes

characteristic

132

C₂

C₄⋊C₄×Dic₈₉

yes

characteristic

133

C₂

C₂×#15667923

yes

characteristic

134

C₂

#19151554

yes

characteristic

135

C₂

#19151676

yes

—

136

C₂

#19151729

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂×C₄²
Fitting	C₄×C₄⋊C₄×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×C₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₈₉
hypocenter	C₈₉	#1770
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₈₉

commutator

C₂²×C₈₉

C₂×C₄²

Fitting

C₄×C₄⋊C₄×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂²×C₄

cosocle

C₂²×C₄×C₈₉

C₂³

hypercenter

C₄×C₄⋊C₄

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
9. C₄⋊#10623494	C₄	semidirect
11. C₄×#10623494	C₄	semidirect
50. #10623494	C₄⋊C₄	semidirect
58. C₄×C₈₉	#97	semidirect
69. C₈₉	#1770	semidirect
127. C₄	#15667923	semidirect

H ≅ G/N

type

9. C₄⋊#10623494

C₄

semidirect

11. C₄×#10623494

C₄

semidirect

50. #10623494

C₄⋊C₄

semidirect

58. C₄×C₈₉

#97

semidirect

69. C₈₉

#1770

semidirect

127. C₄

#15667923

semidirect

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

subgroup	index
C₂×C₄⋊#10623494	2
C₂×#15667923	2
C₂×C₄×#10623494	2
#19151353	2
C₄×C₄⋊C₄×C₈₉	2
#1770	89

subgroup

index

C₄×C₄⋊C₄×C₈₉

Subgroups — #26020121

Conjugacy classes of subgroups

Complements

Maximal subgroups