Group #26021414 — subgroups

Subgroups — #26021414

Order 11392.

246 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 67 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	#19152229	C₂	1	1	yes	maximal, characteristic
2	#19151608	C₂	1	2	yes	maximal
3	#19151503	C₂	1	1	yes	maximal, characteristic
4	#19151359	C₂	1	2	yes	maximal
5	#343×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
7	#15668011	—	2	2	no	—
8	#15668003	—	2	2	no	—
9	#15667991	—	2	2	no	—
10	#15667971	C₂²	1	2	yes	—
11	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
12	#15667937	—	2	2	no	—
13	#15667937	—	2	2	no	—
14	#15667919	—	2	2	no	—
15	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₈	—	2	2	no	—
18	C₂×Dic₁₇₈	—	2	2	no	—
19	C₂×Dic₁₇₈	—	2	1	no	—
20	C₂×Dic₁₇₈	—	2	2	no	—
21	C₂×Dic₁₇₈	Q₈	1	2	yes	complemented
22	C₂²×Dic₈₉	—	2	1	no	—
23	C₂²×Dic₈₉	—	2	1	no	—
24	C₄⋊Dic₈₉	—	2	2	no	—
25	C₄⋊Dic₈₉	D₄	1	2	yes	—
26	C₄⋊Dic₈₉	D₄	1	2	yes	—
27	#13083553	—	2	4	no	—
28	C₂²⋊Dic₈₉	—	2	2	no	—
29	C₄×Dic₈₉	—	2	2	no	—
30	C₄×Dic₈₉	—	2	2	no	—
31	C₂²×C₄×C₈₉	D₄	1	2	yes	—
32	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
33	M₁₆×C₈₉	—	2	2	no	—
34	M₁₆×C₈₉	—	2	2	no	—
35	C₂×C₈×C₈₉	—	2	2	no	—
36	C₄⋊C₄×C₈₉	—	2	2	no	—
37	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	—	2	2	no	—
39	C₂×C₄×C₈₉	—	4	2	no	—
40	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
43	C₂×C₄×C₈₉	#41	1	2	yes	—
44	C₂×Dic₈₉	—	2	2	no	—
45	C₂×Dic₈₉	—	2	2	no	—
46	C₂×Dic₈₉	—	2	2	no	—
47	C₂×Dic₈₉	—	2	2	no	—
48	C₂×Dic₈₉	—	4	1	no	—
49	C₂×Dic₈₉	—	4	1	no	—
50	C₂×Dic₈₉	—	4	1	no	—
51	C₂×Dic₈₉	—	4	1	no	—
52	Dic₁₇₈	—	2	4	no	—
53	Dic₁₇₈	—	4	2	no	—
54	Dic₁₇₈	—	8	2	no	—
55	C₈×C₈₉	—	4	2	no	—
56	C₂²×C₈₉	—	2	1	no	—
57	C₂²×C₈₉	—	2	1	no	—
58	C₂²×C₈₉	#120	1	1	yes	characteristic
59	C₂²×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂²×C₈₉	#124	1	1	yes	characteristic
61	C₄×C₈₉	—	4	2	no	—
62	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
63	C₄×C₈₉	#124	1	2	yes	—
64	Dic₈₉	—	4	2	no	—
65	Dic₈₉	—	4	2	no	—
66	Dic₈₉	—	4	2	no	—
67	Dic₈₉	—	4	2	no	—
68	C₂×C₈₉	—	2	1	no	—
69	C₂×C₈₉	—	2	1	no	—
70	C₂×C₈₉	#393	1	1	yes	characteristic
71	C₂×C₈₉	#455	1	1	yes	characteristic
72	C₂×C₈₉	#456	1	1	yes	characteristic
73	#2028	—	89	1	no	maximal, Sylow
74	C₈₉	#2028	1	1	yes	characteristic, complemented, Sylow
75	#544	—	89	1	no	—
76	#419	—	89	2	no	—
77	#391	—	89	1	no	—
78	#343	D₈₉	1	1	yes	characteristic
79	#339	—	89	2	no	—
80	C₂²×Q₈	—	89	1	no	—
81	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
82	C₄⋊Q₈	—	178	2	no	—
83	#125	—	178	2	no	—
84	C₂²⋊Q₈	—	178	2	no	—
85	#117	—	89	2	no	—
86	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
87	C₂×C₄²	—	89	1	no	—
88	#103	—	178	2	no	—
89	#103	—	178	2	no	—
90	#95	—	178	2	no	—
91	C₂×Q₈	—	89	2	no	—
92	C₂×Q₈	—	178	2	no	—
93	C₂×Q₈	—	178	2	no	—
94	C₂×Q₈	—	178	2	no	—
95	C₂×Q₈	—	178	1	no	—
96	C₂²×C₄	—	178	1	no	—
97	C₂²×C₄	—	178	1	no	—
98	C₂²×C₄	D₃₅₆	1	2	yes	—
99	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
100	M₁₆	—	2	2	no	—
101	M₁₆	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	89	2	no	—
105	C₄⋊C₄	—	89	2	no	—
106	C₄⋊C₄	—	178	2	no	—
107	C₄⋊C₄	—	178	4	no	—
108	C₂²⋊C₄	—	178	2	no	—
109	C₄²	—	178	2	no	—
110	C₄²	—	178	2	no	—
111	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
112	Q₈	—	356	2	no	—
113	Q₈	—	178	4	no	—
114	Q₈	—	712	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	356	1	no	—
118	C₂×C₄	—	356	1	no	—
119	C₂×C₄	—	356	1	no	—
120	C₂×C₄	—	356	1	no	—
121	C₂×C₄	—	178	2	no	—
122	C₂×C₄	—	178	2	no	—
123	C₂×C₄	—	178	2	no	—
124	C₂×C₄	—	178	2	no	—
125	C₂×C₄	D₄×D₈₉	1	2	yes	—
126	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
127	C₂×C₄	#13083573	1	1	yes	characteristic
128	C₂×C₄	#13083574	1	2	yes	—
129	C₈	—	4	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	#15667982	1	1	yes	characteristic
133	C₂²	#15667992	1	1	yes	characteristic
134	C₂²	#15668000	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	356	2	no	—
137	C₄	—	356	2	no	—
138	C₄	—	356	2	no	—
139	C₄	—	356	2	no	—
140	C₄	#15667990	1	2	yes	—
141	C₄	#15667998	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#19151511	1	1	yes	characteristic
145	C₂	#19151778	1	1	yes	characteristic
146	C₂	#19151783	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152229

C₂

yes

maximal, characteristic

#19151608

C₂

yes

maximal

#19151503

C₂

yes

maximal, characteristic

#19151359

C₂

yes

maximal

#343×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

—

—

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

—

—

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

Q₈

yes

complemented

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

—

C₄⋊Dic₈₉

D₄

yes

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#120

yes

characteristic

C₂²×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₈₉

#124

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

#124

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic

#339

—

C₂²×Q₈

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₄⋊Q₈

—

178

—

#125

—

178

—

C₂²⋊Q₈

—

178

—

#117

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₂×C₄²

—

#103

—

178

—

#103

—

178

—

#95

—

178

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

D₃₅₆

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

178

—

107

C₄⋊C₄

—

178

—

108

C₂²⋊C₄

—

178

—

109

C₄²

—

178

—

110

C₄²

—

178

—

111

C₂³

C₂×D₃₅₆

yes

characteristic

112

Q₈

—

356

—

113

Q₈

—

178

—

114

Q₈

—

712

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

356

—

118

C₂×C₄

—

356

—

119

C₂×C₄

—

356

—

120

C₂×C₄

—

356

—

121

C₂×C₄

—

178

—

122

C₂×C₄

—

178

—

123

C₂×C₄

—

178

—

124

C₂×C₄

—

178

—

125

C₂×C₄

D₄×D₈₉

yes

—

126

C₂×C₄

Q₈×D₈₉

yes

characteristic

127

C₂×C₄

#13083573

yes

characteristic

128

C₂×C₄

#13083574

yes

—

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

#15667982

yes

characteristic

133

C₂²

#15667992

yes

characteristic

134

C₂²

#15668000

yes

characteristic

135

C₄

—

136

C₄

—

356

—

137

C₄

—

356

—

138

C₄

—

356

—

139

C₄

—

356

—

140

C₄

#15667990

yes

—

141

C₄

#15667998

yes

—

142

C₂

—

143

C₂

—

144

C₂

#19151511

yes

characteristic

145

C₂

#19151778

yes

characteristic

146

C₂

#19151783

yes

characteristic

	subgroup	G/H
center	C₂²	#15667982
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#343×C₈₉	C₂
socle	C₂²×C₈₉	#120
prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#343	D₈₉
hypocenter	C₈₉	#2028
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

#15667982

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#343×C₈₉

C₂

socle

C₂²×C₈₉

#120

prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
21. C₂×Dic₁₇₈	Q₈	semidirect
74. C₈₉	#2028	semidirect

H ≅ G/N

type

21. C₂×Dic₁₇₈

Q₈

semidirect

74. C₈₉

#2028

semidirect

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

subgroup	index
#19152229	2
#19151608	2
#19151503	2
#19151359	2
#343×C₈₉	2
#2028	89

subgroup

index

#343×C₈₉

Subgroups — #26021414

Conjugacy classes of subgroups

Complements

Maximal subgroups