Group #26029384 — subgroups

Subgroups — #26029384

Order 11392.

394 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 221 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	Q₈×#10623494	C₂	1	2	yes	maximal, complemented
2	#19151676	C₂	1	6	yes	maximal, complemented
3	#19151651	C₂	1	3	yes	maximal, complemented
4	#19151622	C₂	1	3	yes	maximal, complemented
5	C₄×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₂²	1	6	yes	—
7	C₄⋊#10623494	C₂²	1	3	yes	—
8	C₄⋊#10623494	C₂²	1	6	yes	—
9	C₄⋊#10623494	C₂²	1	1	yes	characteristic
10	C₂²⋊#10623494	C₂²	1	6	yes	—
11	C₄×#10623494	C₂²	1	6	yes	—
12	C₂×#41×C₈₉	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₈₉	C₄	1	1	yes	characteristic
14	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₈₉	C₄	1	3	yes	—
16	C₄×D₄×C₈₉	C₂²	1	3	yes	—
17	#117×C₈₉	C₄	1	3	yes	—
18	C₂×C₄²×C₈₉	C₂²	1	3	yes	—
19	#13083558	—	4	6	no	—
20	C₂×#10623494	C₂³	1	6	yes	—
21	C₂×#10623494	C₂³	1	2	yes	—
22	#41×C₈₉	—	2	4	no	—
23	C₂×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₈₉	C₂×C₄	1	3	yes	—
25	C₂²×C₄×C₈₉	C₂×C₄	1	3	yes	—
26	C₂²×C₄×C₈₉	C₂×C₄	1	3	yes	—
27	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
28	C₄⋊C₄×C₈₉	C₂×C₄	1	3	yes	—
29	C₄⋊C₄×C₈₉	C₂×C₄	1	3	yes	—
30	C₂²⋊C₄×C₈₉	C₂×C₄	1	6	yes	—
31	C₄²×C₈₉	C₂×C₄	1	3	yes	—
32	C₄²×C₈₉	C₂×C₄	1	3	yes	—
33	C₄²×C₈₉	C₂³	1	1	yes	characteristic
34	C₄²×C₈₉	C₂³	1	3	yes	—
35	C₂³×C₈₉	C₂²×C₄	1	3	yes	—
36	Q₈×C₈₉	M₁₆	1	4	yes	complemented
37	D₄×C₈₉	—	2	6	no	—
38	C₂×C₄×C₈₉	—	2	6	no	—
39	C₂×C₄×C₈₉	—	2	6	no	—
40	C₂×C₄×C₈₉	—	2	3	no	—
41	C₂×C₄×C₈₉	—	2	3	no	—
42	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₈₉	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₈₉	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
48	#10623494	—	2	6	no	—
49	#10623494	—	2	2	no	—
50	C₂²×C₈₉	—	2	3	no	—
51	C₂²×C₈₉	—	2	3	no	—
52	C₂²×C₈₉	—	2	3	no	—
53	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₈₉	—	2	3	no	—
55	C₄×C₈₉	—	2	1	no	—
56	C₄×C₈₉	—	2	3	no	—
57	C₄×C₈₉	C₂×M₁₆	1	6	yes	—
58	C₄×C₈₉	#131	1	2	yes	—
59	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
60	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
61	C₂×C₈₉	—	4	3	no	—
62	C₂×C₈₉	#520	1	1	yes	characteristic
63	C₂×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
64	C₂×C₈₉	C₂×#131	1	1	yes	characteristic
65	#2991	—	89	1	no	maximal, Sylow
66	C₈₉	#2991	1	1	yes	characteristic, complemented, Sylow
67	C₄×#41	D₈₉	1	1	yes	characteristic
68	C₈×Q₈	—	89	2	no	—
69	#436	—	89	6	no	—
70	#431	—	89	3	no	—
71	#423	—	89	3	no	—
72	C₂×#41	Dic₈₉	1	1	yes	characteristic
73	C₂×M₁₆	—	89	6	no	—
74	C₄×Q₈	Dic₈₉	1	1	yes	characteristic
75	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
76	C₄×D₄	Dic₈₉	1	3	yes	—
77	C₄×D₄	C₂×D₈₉	1	3	yes	—
78	#117	Dic₈₉	1	3	yes	—
79	C₂×C₄²	C₂×D₈₉	1	3	yes	—
80	C₄⋊C₈	—	89	3	no	—
81	C₄⋊C₈	—	89	6	no	—
82	C₄⋊C₈	—	89	1	no	—
83	C₂²⋊C₈	—	89	6	no	—
84	C₄×C₈	—	89	6	no	—
85	#41	—	2	4	no	—
86	C₂×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
87	C₂×D₄	C₂×Dic₈₉	1	3	yes	—
88	C₂²×C₄	C₂×Dic₈₉	1	3	yes	—
89	C₂²×C₄	C₂×Dic₈₉	1	3	yes	—
90	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
91	M₁₆	—	356	6	no	—
92	C₂×C₈	—	89	2	no	—
93	C₂×C₈	—	89	6	no	—
94	C₄⋊C₄	C₂×Dic₈₉	1	3	yes	—
95	C₄⋊C₄	C₂×Dic₈₉	1	3	yes	—
96	C₂²⋊C₄	C₂×Dic₈₉	1	6	yes	—
97	C₄²	C₂×Dic₈₉	1	3	yes	—
98	C₄²	C₂×Dic₈₉	1	3	yes	—
99	C₄²	C₂²×D₈₉	1	3	yes	—
100	C₄²	C₂²×D₈₉	1	1	yes	characteristic
101	C₂³	C₂²×Dic₈₉	1	3	yes	—
102	Q₈	#13083558	1	4	yes	complemented
103	D₄	—	2	6	no	—
104	C₂×C₄	—	2	6	no	—
105	C₂×C₄	—	2	6	no	—
106	C₂×C₄	—	2	3	no	—
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
109	C₂×C₄	C₂²×Dic₈₉	1	3	yes	—
110	C₂×C₄	C₂²×Dic₈₉	1	3	yes	—
111	C₂×C₄	C₂²×Dic₈₉	1	3	yes	—
112	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
113	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
114	C₈	—	178	6	no	—
115	C₈	—	178	2	no	—
116	C₂²	—	2	3	no	—
117	C₂²	—	2	3	no	—
118	C₂²	—	2	3	no	—
119	C₂²	C₂³×Dic₈₉	1	1	yes	characteristic
120	C₄	—	2	3	no	—
121	C₄	—	2	1	no	—
122	C₄	—	2	3	no	—
123	C₄	C₂×#13083558	1	6	yes	—
124	C₄	#15668019	1	2	yes	—
125	C₄	#15668058	1	1	yes	characteristic
126	C₄	#15668059	1	1	yes	characteristic
127	C₂	—	4	3	no	—
128	C₂	#19152081	1	1	yes	characteristic
129	C₂	C₂²×#13083558	1	1	yes	characteristic
130	C₂	C₂×#15668019	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#10623494

C₂

yes

maximal, complemented

#19151676

C₂

yes

maximal, complemented

#19151651

C₂

yes

maximal, complemented

#19151622

C₂

yes

maximal, complemented

C₄×#41×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

characteristic

C₂²⋊#10623494

C₂²

yes

—

C₄×#10623494

C₂²

yes

—

C₂×#41×C₈₉

C₄

yes

characteristic

C₄×Q₈×C₈₉

C₄

yes

characteristic

C₄×Q₈×C₈₉

C₂²

yes

characteristic

C₄×D₄×C₈₉

C₄

yes

—

C₄×D₄×C₈₉

C₂²

yes

—

#117×C₈₉

C₄

yes

—

C₂×C₄²×C₈₉

C₂²

yes

—

#13083558

—

C₂×#10623494

C₂³

yes

—

C₂×#10623494

C₂³

yes

—

#41×C₈₉

—

C₂×Q₈×C₈₉

C₂×C₄

yes

characteristic

C₂×D₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂²⋊C₄×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂²×C₄

yes

—

Q₈×C₈₉

M₁₆

yes

complemented

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×M₁₆

yes

—

C₄×C₈₉

#131

yes

—

C₄×C₈₉

H(5,𝔽₂)

yes

characteristic

C₄×C₈₉

2₋¹⁺⁴

yes

characteristic

C₂×C₈₉

—

C₂×C₈₉

#520

yes

characteristic

C₂×C₈₉

C₂²×M₁₆

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₄×#41

D₈₉

yes

characteristic

C₈×Q₈

—

#436

—

#431

—

#423

—

C₂×#41

Dic₈₉

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

Dic₈₉

yes

characteristic

C₄×Q₈

C₂×D₈₉

yes

characteristic

C₄×D₄

Dic₈₉

yes

—

C₄×D₄

C₂×D₈₉

yes

—

#117

Dic₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#41

—

C₂×Q₈

C₂×Dic₈₉

yes

characteristic

C₂×D₄

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

M₁₆

—

356

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₈₉

yes

—

C₄⋊C₄

C₂×Dic₈₉

yes

—

C₂²⋊C₄

C₂×Dic₈₉

yes

—

C₄²

C₂×Dic₈₉

yes

—

C₄²

C₂×Dic₈₉

yes

—

C₄²

C₂²×D₈₉

yes

—

100

C₄²

C₂²×D₈₉

yes

characteristic

101

C₂³

C₂²×Dic₈₉

yes

—

102

Q₈

#13083558

yes

complemented

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

109

C₂×C₄

C₂²×Dic₈₉

yes

—

110

C₂×C₄

C₂²×Dic₈₉

yes

—

111

C₂×C₄

C₂²×Dic₈₉

yes

—

112

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

113

C₂×C₄

C₂³×D₈₉

yes

characteristic

114

C₈

—

178

—

115

C₈

—

178

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂³×Dic₈₉

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

C₂×#13083558

yes

—

124

C₄

#15668019

yes

—

125

C₄

#15668058

yes

characteristic

126

C₄

#15668059

yes

characteristic

127

C₂

—

128

C₂

#19152081

yes

characteristic

129

C₂

C₂²×#13083558

yes

characteristic

130

C₂

C₂×#15668019

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	C₄×#41×C₈₉	C₂
prime radical	C₂×#41×C₈₉	C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₄×#41	D₈₉
hypocenter	C₈₉	#2991
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂³×C₄

Fitting

C₄×#41×C₈₉

C₂

prime radical

C₂×#41×C₈₉

C₄

cosocle

C₂×C₄×C₈₉

C₂⁴

hypercenter

C₄×#41

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
1. Q₈×#10623494	C₂	semidirect
2. #19151676	C₂	semidirect
3. #19151651	C₂	semidirect
4. #19151622	C₂	semidirect
36. Q₈×C₈₉	M₁₆	semidirect
66. C₈₉	#2991	semidirect
102. Q₈	#13083558	semidirect

H ≅ G/N

type

1. Q₈×#10623494

C₂

semidirect

2. #19151676

C₂

semidirect

3. #19151651

C₂

semidirect

4. #19151622

C₂

semidirect

36. Q₈×C₈₉

M₁₆

semidirect

66. C₈₉

#2991

semidirect

102. Q₈

#13083558

semidirect

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

subgroup	index
Q₈×#10623494	2
#19151676	2
#19151651	2
#19151622	2
C₄×#41×C₈₉	2
#2991	89

subgroup

index

C₄×#41×C₈₉

Subgroups — #26029384

Conjugacy classes of subgroups

Complements

Maximal subgroups