Group #26029383 — subgroups

Subgroups — #26029383

Order 11392.

394 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₄×#13083575	C₂	1	1	yes	maximal, characteristic
2	Q₈×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151677	C₂	1	3	yes	maximal, complemented
4	#19151675	C₂	1	3	yes	maximal, complemented
5	#19151649	C₂	1	3	yes	maximal, complemented
6	#19151620	C₂	1	3	yes	maximal, complemented
7	C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13083575	C₄	1	1	yes	characteristic
9	C₂×#13083557	C₂²	1	6	yes	—
10	Q₈×Dic₈₉	C₄	1	1	yes	characteristic
11	C₄×D₃₅₆	C₂²	1	3	yes	—
12	#15667974	C₄	1	3	yes	—
13	#15667970	C₄	1	3	yes	—
14	C₄²×D₈₉	C₂²	1	3	yes	—
15	C₄⋊#10623494	C₂²	1	3	yes	—
16	#15667941	C₂²	1	3	yes	—
17	#15667941	C₂²	1	1	yes	characteristic
18	#15667924	C₂²	1	3	yes	—
19	#15667924	C₂²	1	3	yes	—
20	C₄×#10623494	C₂²	1	3	yes	—
21	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
23	C₄×C₈×C₈₉	C₂²	1	3	yes	—
24	#13083575	—	2	4	no	—
25	C₂×D₃₅₆	C₂×C₄	1	3	yes	—
26	C₂×C₄×D₈₉	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₈₉	C₂×C₄	1	3	yes	—
28	C₂×C₄×D₈₉	C₂³	1	3	yes	—
29	#13083557	—	4	6	no	—
30	C₂×#10623494	C₂³	1	3	yes	—
31	C₂×#10623494	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₈₉	C₂×C₄	1	3	yes	—
33	#13083551	C₂×C₄	1	6	yes	—
34	C₄×Dic₈₉	C₂×C₄	1	3	yes	—
35	C₄×Dic₈₉	C₂×C₄	1	3	yes	—
36	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₈₉	C₂³	1	3	yes	—
39	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₈₉	C₂×C₄	1	3	yes	—
41	C₄²×C₈₉	C₂³	1	3	yes	—
42	C₂²×D₈₉	C₂²×C₄	1	3	yes	—
43	Q₈×C₈₉	M₁₆	1	4	yes	complemented
44	C₂×C₄×C₈₉	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₈₉	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₈₉	C₂²×C₄	1	3	yes	—
49	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
50	D₃₅₆	—	2	6	no	—
51	C₄×D₈₉	—	2	6	no	—
52	C₄×D₈₉	—	2	3	no	—
53	C₄×D₈₉	—	2	3	no	—
54	C₄×D₈₉	—	2	6	no	—
55	C₈×C₈₉	—	2	3	no	—
56	C₈×C₈₉	—	2	1	no	—
57	#10623494	—	2	3	no	—
58	#10623494	—	2	1	no	—
59	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
60	C₂×D₈₉	—	2	3	no	—
61	C₂×D₈₉	—	2	3	no	—
62	C₂×D₈₉	—	2	3	no	—
63	C₄×C₈₉	—	2	3	no	—
64	C₄×C₈₉	C₂×M₁₆	1	6	yes	—
65	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
66	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
67	Dic₈₉	—	2	3	no	—
68	Dic₈₉	—	2	1	no	—
69	Dic₈₉	#131	1	2	yes	—
70	C₂×C₈₉	#520	1	1	yes	characteristic
71	C₂×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
72	C₂×C₈₉	C₂×#131	1	1	yes	characteristic
73	D₈₉	—	4	3	no	—
74	#2991	—	89	1	no	maximal, Sylow
75	C₈₉	#2991	1	1	yes	characteristic, complemented, Sylow
76	C₄×#41	—	89	1	no	—
77	C₈×Q₈	—	89	1	no	—
78	C₈×Q₈	D₈₉	1	1	yes	characteristic, complemented
79	#436	—	89	3	no	—
80	#436	—	89	3	no	—
81	#431	—	89	3	no	—
82	#423	—	89	3	no	—
83	C₂×#41	—	89	1	no	—
84	C₂×M₁₆	—	89	6	no	—
85	C₄×Q₈	—	89	1	no	—
86	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
87	C₄×D₄	—	89	3	no	—
88	C₄×D₄	—	89	3	no	—
89	#117	—	89	3	no	—
90	C₂×C₄²	—	89	3	no	—
91	C₄⋊C₈	—	89	3	no	—
92	C₄⋊C₈	—	89	1	no	—
93	C₄⋊C₈	—	89	3	no	—
94	C₄⋊C₈	C₂×D₈₉	1	3	yes	—
95	C₂²⋊C₈	—	89	3	no	—
96	C₂²⋊C₈	—	89	3	no	—
97	C₄×C₈	—	89	3	no	—
98	C₄×C₈	C₂×D₈₉	1	3	yes	—
99	#41	—	178	4	no	—
100	C₂×Q₈	C₄×D₈₉	1	1	yes	characteristic
101	C₂×D₄	—	89	3	no	—
102	C₂²×C₄	—	89	3	no	—
103	C₂²×C₄	—	89	3	no	—
104	C₂²×C₄	—	89	3	no	—
105	M₁₆	—	356	6	no	—
106	C₂×C₈	—	89	1	no	—
107	C₂×C₈	—	89	3	no	—
108	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₈₉	1	3	yes	—
110	C₄⋊C₄	—	89	3	no	—
111	C₄⋊C₄	C₄×D₈₉	1	3	yes	—
112	C₂²⋊C₄	—	89	6	no	—
113	C₄²	—	89	3	no	—
114	C₄²	—	89	3	no	—
115	C₄²	—	89	1	no	—
116	C₄²	C₂²×D₈₉	1	3	yes	—
117	C₂³	—	89	3	no	—
118	Q₈	#13083557	1	4	yes	complemented
119	D₄	—	178	6	no	—
120	C₂×C₄	—	89	1	no	—
121	C₂×C₄	—	89	1	no	—
122	C₂×C₄	—	89	3	no	—
123	C₂×C₄	—	178	6	no	—
124	C₂×C₄	—	178	6	no	—
125	C₂×C₄	—	178	3	no	—
126	C₂×C₄	—	178	3	no	—
127	C₂×C₄	C₂×C₄×D₈₉	1	3	yes	—
128	C₂×C₄	C₂×C₄×D₈₉	1	3	yes	—
129	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
130	C₈	—	2	3	no	—
131	C₈	—	2	1	no	—
132	C₈	—	178	3	no	—
133	C₈	—	178	1	no	—
134	C₂²	—	178	3	no	—
135	C₂²	—	178	3	no	—
136	C₂²	—	178	3	no	—
137	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	89	2	no	—
140	C₄	—	178	3	no	—
141	C₄	—	178	1	no	—
142	C₄	C₂×#13083557	1	6	yes	—
143	C₄	#15668058	1	1	yes	characteristic
144	C₄	#15668060	1	1	yes	characteristic
145	C₂	—	356	3	no	—
146	C₂	#19152080	1	1	yes	characteristic
147	C₂	C₂²×#13083557	1	1	yes	characteristic
148	C₂	#131×D₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13083575

C₂

yes

maximal, characteristic

Q₈×#10623494

C₂

yes

maximal, characteristic, complemented

#19151677

C₂

yes

maximal, complemented

#19151675

C₂

yes

maximal, complemented

#19151649

C₂

yes

maximal, complemented

#19151620

C₂

yes

maximal, complemented

C₈×Q₈×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083575

C₄

yes

characteristic

C₂×#13083557

C₂²

yes

—

Q₈×Dic₈₉

C₄

yes

characteristic

C₄×D₃₅₆

C₂²

yes

—

#15667974

C₄

yes

—

#15667970

C₄

yes

—

C₄²×D₈₉

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

—

#15667941

C₂²

yes

—

#15667941

C₂²

yes

characteristic

#15667924

C₂²

yes

—

#15667924

C₂²

yes

—

C₄×#10623494

C₂²

yes

—

C₄×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

—

#13083575

—

C₂×D₃₅₆

C₂×C₄

yes

—

C₂×C₄×D₈₉

C₂×C₄

yes

—

C₂×C₄×D₈₉

C₂×C₄

yes

—

C₂×C₄×D₈₉

C₂³

yes

—

#13083557

—

C₂×#10623494

C₂³

yes

—

C₂×#10623494

C₂³

yes

characteristic

C₄⋊Dic₈₉

C₂×C₄

yes

—

#13083551

C₂×C₄

yes

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

C₂³

yes

characteristic

C₂×Q₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂²×D₈₉

C₂²×C₄

yes

—

Q₈×C₈₉

M₁₆

yes

complemented

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

C₂²×C₄

yes

characteristic

C₂×Dic₈₉

C₂²×C₄

yes

—

C₂×Dic₈₉

C₂²×C₄

yes

characteristic

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×M₁₆

yes

—

C₄×C₈₉

H(5,𝔽₂)

yes

characteristic

C₄×C₈₉

2₋¹⁺⁴

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

#131

yes

—

C₂×C₈₉

#520

yes

characteristic

C₂×C₈₉

C₂²×M₁₆

yes

characteristic

C₂×C₈₉

C₂×#131

yes

characteristic

D₈₉

—

#2991

—

maximal, Sylow

C₈₉

#2991

yes

characteristic, complemented, Sylow

C₄×#41

—

C₈×Q₈

—

C₈×Q₈

D₈₉

yes

characteristic, complemented

—

—

—

—

C₂×#41

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

C₂×D₈₉

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₈₉

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₉

yes

—

#41

—

178

—

100

C₂×Q₈

C₄×D₈₉

yes

characteristic

101

C₂×D₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

M₁₆

—

356

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²×D₈₉

yes

characteristic

109

C₂×C₈

C₂²×D₈₉

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₄×D₈₉

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₂²×D₈₉

yes

—

117

C₂³

—

118

Q₈

#13083557

yes

complemented

119

D₄

—

178

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

178

—

124

C₂×C₄

—

178

—

125

C₂×C₄

—

178

—

126

C₂×C₄

—

178

—

127

C₂×C₄

C₂×C₄×D₈₉

yes

—

128

C₂×C₄

C₂×C₄×D₈₉

yes

—

129

C₂×C₄

C₂³×D₈₉

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

178

—

133

C₈

—

178

—

134

C₂²

—

178

—

135

C₂²

—

178

—

136

C₂²

—

178

—

137

C₂²

C₂²×C₄×D₈₉

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

178

—

141

C₄

—

178

—

142

C₄

C₂×#13083557

yes

—

143

C₄

#15668058

yes

characteristic

144

C₄

#15668060

yes

characteristic

145

C₂

—

356

—

146

C₂

#19152080

yes

characteristic

147

C₂

C₂²×#13083557

yes

characteristic

148

C₂

#131×D₈₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	C₈×Q₈×C₈₉	C₂
prime radical	C₂×#13083575	C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₈×Q₈	D₈₉
hypocenter	C₈₉	#2991
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂³×C₄

Fitting

C₈×Q₈×C₈₉

C₂

prime radical

C₂×#13083575

C₄

cosocle

C₂×C₄×C₈₉

C₂⁴

hypercenter

C₈×Q₈

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₄ ◁ C₈×Q₈

N	H ≅ G/N	type
2. Q₈×#10623494	C₂	semidirect
3. #19151677	C₂	semidirect
4. #19151675	C₂	semidirect
5. #19151649	C₂	semidirect
6. #19151620	C₂	semidirect
7. C₈×Q₈×C₈₉	C₂	semidirect
43. Q₈×C₈₉	M₁₆	semidirect
75. C₈₉	#2991	semidirect
78. C₈×Q₈	D₈₉	semidirect
118. Q₈	#13083557	semidirect

H ≅ G/N

type

2. Q₈×#10623494

C₂

semidirect

3. #19151677

C₂

semidirect

4. #19151675

C₂

semidirect

5. #19151649

C₂

semidirect

6. #19151620

C₂

semidirect

7. C₈×Q₈×C₈₉

C₂

semidirect

43. Q₈×C₈₉

M₁₆

semidirect

75. C₈₉

#2991

semidirect

78. C₈×Q₈

D₈₉

semidirect

118. Q₈

#13083557

semidirect

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

subgroup	index
C₄×#13083575	2
Q₈×#10623494	2
#19151677	2
#19151675	2
#19151649	2
#19151620	2
C₈×Q₈×C₈₉	2
#2991	89

subgroup

index

C₈×Q₈×C₈₉

Subgroups — #26029383

Conjugacy classes of subgroups

Complements

Maximal subgroups