Group #26022300 — subgroups

Subgroups — #26022300

Order 11392.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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×#15667958	C₂	1	1	yes	maximal, characteristic
2	#19151976	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
4	#19151416	C₂	1	1	yes	maximal, characteristic
5	#19151403	C₂	1	1	yes	maximal, characteristic
6	#19151402	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
9	C₂×#13083564	—	2	1	no	—
10	#15668010	—	2	1	no	—
11	Q₈×Dic₈₉	—	2	1	no	—
12	#15667958	—	2	2	no	—
13	C₂×#13083549	C₂²	1	1	yes	characteristic
14	#15667946	C₂²	1	1	yes	characteristic
15	#15667936	—	2	1	no	—
16	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
19	Q₃₂×C₈₉	—	2	2	no	—
20	C₂×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₇₈	—	2	1	no	—
22	Dic₃₅₆	—	2	1	no	—
23	Dic₃₅₆	—	2	1	no	—
24	#13083564	—	4	2	no	—
25	C₂×#10623494	D₄	1	1	yes	characteristic
26	C₄⋊Dic₈₉	—	2	1	no	—
27	C₄⋊Dic₈₉	—	4	1	no	—
28	#13083553	—	4	1	no	—
29	#13083553	—	4	1	no	—
30	C₄×Dic₈₉	—	4	1	no	—
31	C₄×Dic₈₉	D₄	1	1	yes	characteristic
32	#13083549	—	2	1	no	—
33	C₂×Q₈×C₈₉	—	2	1	no	—
34	C₂×Q₈×C₈₉	—	2	1	no	—
35	Q₁₆×C₈₉	—	2	1	no	—
36	Q₁₆×C₈₉	—	2	1	no	—
37	Q₁₆×C₈₉	—	2	1	no	—
38	Q₁₆×C₈₉	D₄	1	2	yes	—
39	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
40	C₁₆×C₈₉	—	2	1	no	—
41	Q₈×C₈₉	—	2	2	no	—
42	Q₈×C₈₉	—	4	1	no	—
43	Q₈×C₈₉	—	4	1	no	—
44	Q₈×C₈₉	—	4	1	no	—
45	C₂×C₄×C₈₉	—	4	1	no	—
46	C₂×C₄×C₈₉	—	4	1	no	—
47	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₈₉	—	4	1	no	—
49	C₂×Dic₈₉	—	4	1	no	—
50	C₂×Dic₈₉	D₈	1	1	yes	characteristic
51	C₂×Dic₈₉	D₈	1	1	yes	characteristic
52	Dic₁₇₈	—	4	1	no	—
53	Dic₁₇₈	—	4	1	no	—
54	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₈×C₈₉	#41	1	1	yes	characteristic
56	#10623494	—	2	1	no	—
57	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
58	C₄×C₈₉	—	4	2	no	—
59	C₄×C₈₉	—	8	1	no	—
60	C₄×C₈₉	#121	1	1	yes	characteristic
61	C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₈₉	—	2	1	no	—
63	Dic₈₉	—	8	1	no	—
64	Dic₈₉	—	8	1	no	—
65	Dic₈₉	Q₃₂	1	2	yes	complemented
66	C₂×C₈₉	#459	1	1	yes	characteristic
67	C₂×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
68	C₂×C₈₉	#510	1	1	yes	characteristic
69	#2209	—	89	1	no	maximal, Sylow
70	C₈₉	#2209	1	1	yes	characteristic, complemented, Sylow
71	C₂×Q₃₂	—	89	1	no	—
72	C₂×Q₃₂	D₈₉	1	1	yes	characteristic
73	#494	—	89	1	no	—
74	C₄×Q₁₆	—	89	1	no	—
75	C₄⋊C₁₆	—	89	1	no	—
76	#358	—	89	1	no	—
77	#358	—	89	1	no	—
78	C₂×Q₁₆	—	89	1	no	—
79	C₂×Q₁₆	—	178	1	no	—
80	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
81	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
82	C₄⋊Q₈	—	178	1	no	—
83	C₄×Q₈	—	178	1	no	—
84	Q₃₂	—	2	2	no	—
85	Q₃₂	—	178	2	no	—
86	C₂×C₁₆	—	89	1	no	—
87	C₂×C₁₆	C₂×D₈₉	1	1	yes	characteristic
88	#107	—	89	1	no	—
89	#103	—	178	1	no	—
90	C₄×C₈	—	89	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	178	1	no	—
94	Q₁₆	—	2	1	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	2	1	no	—
97	Q₁₆	—	356	2	no	—
98	Q₁₆	—	178	1	no	—
99	Q₁₆	—	178	1	no	—
100	Q₁₆	C₂²⋊D₈₉	1	2	yes	—
101	C₂×C₈	—	89	1	no	—
102	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
103	C₄⋊C₄	—	356	1	no	—
104	C₄⋊C₄	—	356	1	no	—
105	C₄⋊C₄	—	356	1	no	—
106	C₄⋊C₄	—	178	1	no	—
107	C₄²	—	89	1	no	—
108	C₄²	—	356	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	178	1	no	—
111	Q₈	—	2	2	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	4	1	no	—
115	Q₈	—	356	1	no	—
116	Q₈	—	356	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	89	1	no	—
120	C₂×C₄	—	89	1	no	—
121	C₂×C₄	—	356	1	no	—
122	C₂×C₄	—	356	1	no	—
123	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
124	C₈	—	178	1	no	—
125	C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
126	C₈	#13083573	1	1	yes	characteristic
127	C₂²	D₈×D₈₉	1	1	yes	characteristic
128	C₄	—	4	2	no	—
129	C₄	—	8	1	no	—
130	C₄	—	89	2	no	—
131	C₄	—	178	1	no	—
132	C₄	—	712	1	no	—
133	C₄	—	712	1	no	—
134	C₄	#15667984	1	1	yes	characteristic
135	C₄	#15668036	1	1	yes	characteristic
136	C₂	#19151791	1	1	yes	characteristic
137	C₂	Q₃₂×D₈₉	1	1	yes	characteristic
138	C₂	#19152040	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667958

C₂

yes

maximal, characteristic

#19151976

C₂

yes

maximal, characteristic

Q₁₆×Dic₈₉

C₂

yes

maximal, characteristic

#19151416

C₂

yes

maximal, characteristic

#19151403

C₂

yes

maximal, characteristic

#19151402

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₆

C₂²

yes

characteristic

C₂×#13083564

—

#15668010

—

Q₈×Dic₈₉

—

#15667958

—

C₂×#13083549

C₂²

yes

characteristic

#15667946

C₂²

yes

characteristic

#15667936

—

C₈×Dic₈₉

C₂²

yes

characteristic

C₂×Q₁₆×C₈₉

C₂²

yes

characteristic

C₂×Q₁₆×C₈₉

C₂²

yes

characteristic

Q₃₂×C₈₉

—

C₂×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

#13083564

—

C₂×#10623494

D₄

yes

characteristic

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

D₄

yes

characteristic

#13083549

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₁₆×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

D₈

yes

characteristic

C₂×Dic₈₉

D₈

yes

characteristic

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

C₂×D₄

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

C₂×D₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

Q₃₂

yes

complemented

C₂×C₈₉

#459

yes

characteristic

C₂×C₈₉

C₂×Q₃₂

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

D₈₉

yes

characteristic

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#358

—

#358

—

C₂×Q₁₆

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

C₂×D₈₉

yes

characteristic

C₄⋊Q₈

—

178

—

C₄×Q₈

—

178

—

Q₃₂

—

Q₃₂

—

178

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₈₉

yes

characteristic

#107

—

#103

—

178

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

178

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

356

—

Q₁₆

—

178

—

Q₁₆

—

178

—

100

Q₁₆

C₂²⋊D₈₉

yes

—

101

C₂×C₈

—

102

C₂×C₈

C₂²×D₈₉

yes

characteristic

103

C₄⋊C₄

—

356

—

104

C₄⋊C₄

—

356

—

105

C₄⋊C₄

—

356

—

106

C₄⋊C₄

—

178

—

107

C₄²

—

108

C₄²

—

356

—

109

C₁₆

—

110

C₁₆

—

178

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

356

—

116

Q₈

—

356

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

356

—

122

C₂×C₄

—

356

—

123

C₂×C₄

D₄×D₈₉

yes

characteristic

124

C₈

—

178

—

125

C₈

C₂×C₂²⋊D₈₉

yes

characteristic

126

C₈

#13083573

yes

characteristic

127

C₂²

D₈×D₈₉

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

178

—

132

C₄

—

712

—

133

C₄

—

712

—

134

C₄

#15667984

yes

characteristic

135

C₄

#15668036

yes

characteristic

136

C₂

#19151791

yes

characteristic

137

C₂

Q₃₂×D₈₉

yes

characteristic

138

C₂

#19152040

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂³
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	C₂×Q₃₂×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₈
hypercenter	C₂×Q₃₂	D₈₉
hypocenter	C₈₉	#2209
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₈₉

commutator, cosocle

C₂×C₈×C₈₉

C₂³

Frattini

C₂×C₈

C₂²×D₈₉

Fitting

C₂×Q₃₂×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

C₂×D₈

hypercenter

C₂×Q₃₂

D₈₉

hypocenter

C₈₉

#2209

LCS

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
65. Dic₈₉	Q₃₂	semidirect
70. C₈₉	#2209	semidirect

H ≅ G/N

type

65. Dic₈₉

Q₃₂

semidirect

70. C₈₉

#2209

semidirect

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

subgroup	index
C₂×#15667958	2
#19151976	2
Q₁₆×Dic₈₉	2
#19151416	2
#19151403	2
#19151402	2
C₂×Q₃₂×C₈₉	2
#2209	89

subgroup

index

Q₁₆×Dic₈₉

C₂×Q₃₂×C₈₉

Subgroups — #26022300

Conjugacy classes of subgroups

Complements

Maximal subgroups