Group #26018923 — subgroups

Subgroups — #26018923

Order 11392.

240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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×#15668012	C₂	1	1	yes	maximal, characteristic
2	#19151622	C₂	1	1	yes	maximal, characteristic
3	#19151343	C₂	1	4	yes	maximal, complemented
4	#431×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₇₈	—	2	1	no	—
6	C₂×#13083558	—	2	1	no	—
7	#15668012	—	2	1	no	—
8	#15668012	—	2	1	no	—
9	#15668012	C₄	1	2	yes	—
10	#15668012	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₈₉	—	2	1	no	—
12	C₂×C₄⋊Dic₈₉	C₄	1	2	yes	—
13	C₄⋊#10623494	—	2	1	no	—
14	C₄⋊#10623494	C₂²	1	2	yes	—
15	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₈₉	—	2	1	no	—
18	C₄×C₈×C₈₉	—	2	1	no	—
19	C₂×Dic₁₇₈	—	2	2	no	—
20	C₂×Dic₁₇₈	—	2	2	no	—
21	C₂×Dic₁₇₈	—	4	2	no	—
22	C₂×Dic₁₇₈	—	4	2	no	—
23	C₂²×Dic₈₉	—	2	2	no	—
24	#13083558	—	4	2	no	—
25	C₂×#10623494	—	2	2	no	—
26	C₄⋊Dic₈₉	—	2	2	no	—
27	C₄⋊Dic₈₉	—	2	2	no	—
28	C₄⋊Dic₈₉	—	2	2	no	—
29	C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₈₉	—	2	2	no	—
34	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₈₉	D₄	1	1	yes	characteristic
36	C₄²×C₈₉	D₄	1	1	yes	characteristic
37	C₄²×C₈₉	C₂³	1	1	yes	characteristic
38	C₂³×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₈₉	—	2	2	no	—
40	C₂×C₄×C₈₉	—	2	2	no	—
41	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₈₉	D₈	1	2	yes	—
45	C₂×C₄×C₈₉	SD₁₆	1	2	yes	—
46	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₈₉	—	2	4	no	—
50	C₂×Dic₈₉	—	4	4	no	—
51	Dic₁₇₈	—	4	4	no	—
52	Dic₁₇₈	—	4	2	no	—
53	Dic₁₇₈	—	4	2	no	—
54	C₈×C₈₉	—	4	2	no	—
55	#10623494	—	4	2	no	—
56	C₂²×C₈₉	—	2	1	no	—
57	C₂²×C₈₉	—	2	1	no	—
58	C₂²×C₈₉	#102	1	2	yes	—
59	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₈₉	—	2	1	no	—
61	C₄×C₈₉	—	2	1	no	—
62	C₄×C₈₉	#101	1	2	yes	—
63	C₄×C₈₉	#102	1	2	yes	—
64	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
65	C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×C₈₉	#137	1	2	yes	—
67	Dic₈₉	—	4	4	no	—
68	C₂×C₈₉	—	2	2	no	—
69	C₂×C₈₉	C₂×#101	1	1	yes	characteristic
70	C₂×C₈₉	C₂×#102	1	1	yes	characteristic
71	C₂×C₈₉	#419	1	1	yes	characteristic
72	#1546	—	89	1	no	maximal, Sylow
73	C₈₉	#1546	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄⋊Q₈	—	89	1	no	—
75	#431	D₈₉	1	1	yes	characteristic
76	#423	—	89	1	no	—
77	#332	—	89	4	no	—
78	C₂²×Q₈	—	178	1	no	—
79	C₂×M₁₆	—	178	1	no	—
80	C₄⋊Q₈	—	89	2	no	—
81	C₄⋊Q₈	—	89	2	no	—
82	C₄⋊Q₈	—	178	1	no	—
83	C₄⋊Q₈	—	178	1	no	—
84	C₂×C₄⋊C₄	—	89	2	no	—
85	C₂×C₄⋊C₄	—	178	1	no	—
86	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
87	C₄⋊C₈	—	89	2	no	—
88	C₄⋊C₈	—	178	1	no	—
89	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
90	C₂²⋊C₈	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₂×Q₈	—	356	2	no	—
93	C₂×Q₈	—	356	2	no	—
94	C₂×Q₈	—	178	2	no	—
95	C₂×Q₈	—	178	2	no	—
96	C₂²×C₄	—	178	2	no	—
97	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
98	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
100	M₁₆	—	356	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	178	2	no	—
103	C₄⋊C₄	—	89	4	no	—
104	C₄⋊C₄	—	178	2	no	—
105	C₄⋊C₄	—	178	2	no	—
106	C₄⋊C₄	—	178	2	no	—
107	C₄²	C₄×D₈₉	1	1	yes	characteristic
108	C₄²	D₃₅₆	1	1	yes	characteristic
109	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
110	C₄²	C₂²×D₈₉	1	1	yes	characteristic
111	C₂³	#13083551	1	1	yes	characteristic
112	Q₈	—	356	2	no	—
113	Q₈	—	356	2	no	—
114	Q₈	—	356	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	356	4	no	—
118	C₂×C₄	—	178	4	no	—
119	C₂×C₄	#13083551	1	1	yes	characteristic
120	C₂×C₄	#13083551	1	1	yes	characteristic
121	C₂×C₄	#13083551	1	1	yes	characteristic
122	C₂×C₄	D₇₁₂	1	2	yes	—
123	C₂×C₄	#13083563	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
126	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₈	—	356	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#15667934	1	2	yes	—
132	C₂²	C₂×#13083551	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	356	4	no	—
136	C₄	#15667930	1	2	yes	—
137	C₄	#15667934	1	2	yes	—
138	C₄	C₂×D₇₁₂	1	1	yes	characteristic
139	C₄	C₂×#13083563	1	1	yes	characteristic
140	C₄	#15668042	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₂×#15667930	1	1	yes	characteristic
143	C₂	C₂×#15667934	1	1	yes	characteristic
144	C₂	#19151606	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668012

C₂

yes

maximal, characteristic

#19151622

C₂

yes

maximal, characteristic

#19151343

C₂

yes

maximal, complemented

#431×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

—

—

—

—

C₄

yes

—

#15668012

C₂²

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₄

yes

—

C₄⋊#10623494

—

C₄⋊#10623494

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₂²⋊C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

D₈

yes

—

C₂×C₄×C₈₉

SD₁₆

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#102

yes

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#101

yes

—

C₄×C₈₉

#102

yes

—

C₄×C₈₉

C₂×D₈

yes

characteristic

C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₄×C₈₉

#137

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#431

D₈₉

yes

characteristic

#423

—

#332

—

C₂²×Q₈

—

178

—

C₂×M₁₆

—

178

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

178

—

C₄⋊C₈

C₂×D₈₉

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

356

—

C₂×Q₈

—

356

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

C₄×D₈₉

yes

characteristic

C₂²×C₄

D₃₅₆

yes

characteristic

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

100

M₁₆

—

356

—

101

C₂×C₈

—

102

C₂×C₈

—

178

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

178

—

105

C₄⋊C₄

—

178

—

106

C₄⋊C₄

—

178

—

107

C₄²

C₄×D₈₉

yes

characteristic

108

C₄²

D₃₅₆

yes

characteristic

109

C₄²

C₂²⋊D₈₉

yes

characteristic

110

C₄²

C₂²×D₈₉

yes

characteristic

111

C₂³

#13083551

yes

characteristic

112

Q₈

—

356

—

113

Q₈

—

356

—

114

Q₈

—

356

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

356

—

118

C₂×C₄

—

178

—

119

C₂×C₄

#13083551

yes

characteristic

120

C₂×C₄

#13083551

yes

characteristic

121

C₂×C₄

#13083551

yes

characteristic

122

C₂×C₄

D₇₁₂

yes

—

123

C₂×C₄

#13083563

yes

—

124

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

126

C₂×C₄

C₂×D₃₅₆

yes

characteristic

127

C₈

—

128

C₈

—

356

—

129

C₂²

—

130

C₂²

—

131

C₂²

#15667934

yes

—

132

C₂²

C₂×#13083551

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

356

—

136

C₄

#15667930

yes

—

137

C₄

#15667934

yes

—

138

C₄

C₂×D₇₁₂

yes

characteristic

139

C₄

C₂×#13083563

yes

characteristic

140

C₄

#15668042

yes

—

141

C₂

—

142

C₂

C₂×#15667930

yes

characteristic

143

C₂

C₂×#15667934

yes

characteristic

144

C₂

#19151606

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#13083551
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₄²	C₂²×D₈₉
Fitting	#431×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₈₉	C₂²⋊C₄
cosocle	C₄²×C₈₉	C₂³
hypercenter	#431	D₈₉
hypocenter	C₈₉	#1546
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂×#13083551

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₄²

C₂²×D₈₉

Fitting

#431×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₈₉

C₂²⋊C₄

cosocle

C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #19151343	C₂	semidirect
73. C₈₉	#1546	semidirect

H ≅ G/N

type

3. #19151343

C₂

semidirect

73. C₈₉

#1546

semidirect

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

subgroup	index
C₂×#15668012	2
#19151622	2
#19151343	2
#431×C₈₉	2
#1546	89

subgroup

index

#431×C₈₉

Subgroups — #26018923

Conjugacy classes of subgroups

Complements

Maximal subgroups