Group #26029319 — subgroups

Subgroups — #26029319

Order 11392.

410 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 205 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	#19151676	C₂	1	4	yes	maximal, complemented
2	D₄×#10623494	C₂	1	4	yes	maximal, complemented
3	#19151651	C₂	1	2	yes	maximal, complemented
4	#19151565	C₂	1	4	yes	maximal, complemented
5	#532×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₂²	1	4	yes	—
7	C₂²×#10623494	C₂²	1	4	yes	—
8	C₄⋊#10623494	C₂²	1	2	yes	—
9	C₄⋊#10623494	C₂²	1	2	yes	—
10	C₂²⋊#10623494	C₂²	1	4	yes	—
11	C₂²⋊#10623494	C₂²	1	8	yes	—
12	C₄×#10623494	C₂²	1	4	yes	—
13	C₂×#41×C₈₉	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₈₉	C₄	1	1	yes	characteristic
15	#127×C₈₉	C₄	1	1	yes	characteristic
16	#124×C₈₉	C₄	1	2	yes	—
17	#121×C₈₉	C₄	1	4	yes	—
18	C₄×D₄×C₈₉	C₂²	1	4	yes	—
19	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
20	#13083558	—	4	4	no	—
21	C₂×#10623494	—	4	4	no	—
22	C₂×#10623494	C₂³	1	4	yes	—
23	C₂×#10623494	C₂³	1	4	yes	—
24	#41×C₈₉	—	4	2	no	—
25	#41×C₈₉	—	4	2	no	—
26	C₂×Q₈×C₈₉	C₂×C₄	1	2	yes	—
27	C₂×D₄×C₈₉	C₂×C₄	1	4	yes	—
28	C₂×D₄×C₈₉	C₂×C₄	1	4	yes	—
29	C₂×D₄×C₈₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
32	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
34	C₂²⋊C₄×C₈₉	C₂×C₄	1	8	yes	—
35	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₈₉	C₂³	1	2	yes	—
38	C₂³×C₈₉	C₂²×C₄	1	4	yes	—
39	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
40	Q₈×C₈₉	—	4	2	no	—
41	D₄×C₈₉	—	2	8	no	—
42	D₄×C₈₉	—	4	2	no	—
43	D₄×C₈₉	—	4	4	no	—
44	C₂×C₄×C₈₉	—	2	4	no	—
45	C₂×C₄×C₈₉	—	2	1	no	—
46	C₂×C₄×C₈₉	—	2	1	no	—
47	C₂×C₄×C₈₉	—	2	4	no	—
48	C₂×C₄×C₈₉	—	2	4	no	—
49	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₈₉	C₂²×C₄	1	4	yes	—
52	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
54	#10623494	—	2	4	no	—
55	#10623494	—	2	4	no	—
56	C₂²×C₈₉	—	2	1	no	—
57	C₂²×C₈₉	—	2	4	no	—
58	C₂²×C₈₉	—	2	2	no	—
59	C₂²×C₈₉	—	2	8	no	—
60	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₈₉	—	2	1	no	—
62	C₄×C₈₉	—	2	4	no	—
63	C₄×C₈₉	—	2	2	no	—
64	C₄×C₈₉	#131	1	4	yes	—
65	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
66	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
67	C₂×C₈₉	—	4	1	no	—
68	C₂×C₈₉	—	4	4	no	—
69	C₂×C₈₉	#518	1	1	yes	characteristic
70	C₂×C₈₉	C₂×#131	1	2	yes	—
71	#2979	—	89	1	no	maximal, Sylow
72	C₈₉	#2979	1	1	yes	characteristic, complemented, Sylow
73	#532	D₈₉	1	1	yes	characteristic
74	#436	—	89	4	no	—
75	C₈×D₄	—	89	4	no	—
76	#431	—	89	2	no	—
77	#408	—	89	4	no	—
78	C₂×#41	C₂×D₈₉	1	2	yes	—
79	C₂×M₁₆	—	89	4	no	—
80	C₂²×C₈	—	89	4	no	—
81	C₄⋊Q₈	Dic₈₉	1	1	yes	characteristic
82	#127	Dic₈₉	1	1	yes	characteristic
83	#124	Dic₈₉	1	2	yes	—
84	#121	Dic₈₉	1	4	yes	—
85	C₄×D₄	C₂×D₈₉	1	4	yes	—
86	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
87	C₄⋊C₈	—	89	2	no	—
88	C₄⋊C₈	—	89	2	no	—
89	C₂²⋊C₈	—	89	4	no	—
90	C₂²⋊C₈	—	89	8	no	—
91	C₄×C₈	—	89	4	no	—
92	#41	—	4	2	no	—
93	#41	—	4	2	no	—
94	C₂×Q₈	C₂×Dic₈₉	1	2	yes	—
95	C₂×D₄	C₂×Dic₈₉	1	4	yes	—
96	C₂×D₄	C₂×Dic₈₉	1	4	yes	—
97	C₂×D₄	C₂×Dic₈₉	1	2	yes	—
98	C₂²×C₄	C₂×Dic₈₉	1	2	yes	—
99	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
101	M₁₆	—	356	4	no	—
102	C₂×C₈	—	89	4	no	—
103	C₂×C₈	—	89	4	no	—
104	C₂×C₈	—	356	4	no	—
105	C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
106	C₂²⋊C₄	C₂×Dic₈₉	1	8	yes	—
107	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
108	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
109	C₄²	C₂²×D₈₉	1	2	yes	—
110	C₂³	C₂²×Dic₈₉	1	4	yes	—
111	C₂³	C₂²×Dic₈₉	1	1	yes	characteristic
112	Q₈	—	4	2	no	—
113	D₄	—	2	8	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
122	C₂×C₄	C₂²×Dic₈₉	1	2	yes	—
123	C₂×C₄	C₂²×Dic₈₉	1	4	yes	—
124	C₂×C₄	C₂²×Dic₈₉	1	2	yes	—
125	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
126	C₈	—	178	4	no	—
127	C₈	—	178	4	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	8	no	—
132	C₂²	C₂³×Dic₈₉	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	4	no	—
135	C₄	—	2	2	no	—
136	C₄	#15668019	1	4	yes	—
137	C₄	#15668057	1	1	yes	characteristic
138	C₄	#15668057	1	1	yes	characteristic
139	C₂	—	4	1	no	—
140	C₂	—	4	4	no	—
141	C₂	#19152073	1	1	yes	characteristic
142	C₂	C₂×#15668019	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151676

C₂

yes

maximal, complemented

D₄×#10623494

C₂

yes

maximal, complemented

#19151651

C₂

yes

maximal, complemented

#19151565

C₂

yes

maximal, complemented

#532×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

C₂²×#10623494

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

—

C₂²⋊#10623494

C₂²

yes

—

C₂²⋊#10623494

C₂²

yes

—

C₄×#10623494

C₂²

yes

—

C₂×#41×C₈₉

C₂²

yes

—

C₄⋊Q₈×C₈₉

C₄

yes

characteristic

#127×C₈₉

C₄

yes

characteristic

#124×C₈₉

C₄

yes

—

#121×C₈₉

C₄

yes

—

C₄×D₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

—

—

C₂³

yes

—

C₂×#10623494

C₂³

yes

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

C₂×C₄

yes

—

C₂×D₄×C₈₉

C₂×C₄

yes

—

C₂×D₄×C₈₉

C₂×C₄

yes

—

C₂×D₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂²⋊C₄×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂²×C₄

yes

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×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

C₂⁴

yes

characteristic

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#131

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#518

yes

characteristic

C₂×C₈₉

C₂×#131

yes

—

#2979

—

maximal, Sylow

C₈₉

#2979

yes

characteristic, complemented, Sylow

#532

D₈₉

yes

characteristic

#436

—

C₈×D₄

—

#431

—

#408

—

C₂×#41

C₂×D₈₉

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

Dic₈₉

yes

characteristic

#127

Dic₈₉

yes

characteristic

#124

Dic₈₉

yes

—

#121

Dic₈₉

yes

—

C₄×D₄

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₈₉

yes

—

C₂×D₄

C₂×Dic₈₉

yes

—

C₂×D₄

C₂×Dic₈₉

yes

—

C₂×D₄

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

100

C₂²×C₄

C₂²×D₈₉

yes

—

101

M₁₆

—

356

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

356

—

105

C₄⋊C₄

C₂×Dic₈₉

yes

—

106

C₂²⋊C₄

C₂×Dic₈₉

yes

—

107

C₄²

C₂×Dic₈₉

yes

characteristic

108

C₄²

C₂×Dic₈₉

yes

characteristic

109

C₄²

C₂²×D₈₉

yes

—

110

C₂³

C₂²×Dic₈₉

yes

—

111

C₂³

C₂²×Dic₈₉

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

122

C₂×C₄

C₂²×Dic₈₉

yes

—

123

C₂×C₄

C₂²×Dic₈₉

yes

—

124

C₂×C₄

C₂²×Dic₈₉

yes

—

125

C₂×C₄

C₂³×D₈₉

yes

characteristic

126

C₈

—

178

—

127

C₈

—

178

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×Dic₈₉

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#15668019

yes

—

137

C₄

#15668057

yes

characteristic

138

C₄

#15668057

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

#19152073

yes

characteristic

142

C₂

C₂×#15668019

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting, prime radical	#532×C₈₉	C₂
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#532	D₈₉
hypocenter	C₈₉	#2979
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂³×C₄

Fitting, prime radical

#532×C₈₉

C₂

cosocle

C₂×C₄×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. #19151676	C₂	semidirect
2. D₄×#10623494	C₂	semidirect
3. #19151651	C₂	semidirect
4. #19151565	C₂	semidirect
72. C₈₉	#2979	semidirect

H ≅ G/N

type

1. #19151676

C₂

semidirect

2. D₄×#10623494

C₂

semidirect

3. #19151651

C₂

semidirect

4. #19151565

C₂

semidirect

72. C₈₉

#2979

semidirect

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

subgroup	index
#19151676	2
D₄×#10623494	2
#19151651	2
#19151565	2
#532×C₈₉	2
#2979	89

subgroup

index

#532×C₈₉

Subgroups — #26029319

Conjugacy classes of subgroups

Complements

Maximal subgroups