Group #26021084 — subgroups

Subgroups — #26021084

Order 11392.

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

normal subgroups only

This group has 320 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	#19152148	C₂	1	1	yes	maximal, characteristic
2	#19151622	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15667940	C₂	1	4	yes	maximal, complemented
4	#376×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13083574	C₂²	1	2	yes	—
6	C₂×#13083558	C₂²	1	2	yes	—
7	#15668012	C₄	1	1	yes	characteristic, complemented
8	#15668009	C₄	1	1	yes	characteristic, complemented
9	#15668000	C₄	1	2	yes	complemented
10	#15667986	—	2	2	no	—
11	C₄×D₃₅₆	—	2	2	no	—
12	C₄⋊#10623494	—	2	2	no	—
13	#15667940	—	2	8	no	—
14	C₂×C₄²×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	#95×C₈₉	—	2	2	no	—
17	#13083574	—	2	4	no	—
18	#13083574	—	2	4	no	—
19	C₂×Dic₁₇₈	C₂×C₄	1	2	yes	—
20	C₂×D₃₅₆	—	2	2	no	—
21	C₂×D₃₅₆	C₂×C₄	1	2	yes	—
22	C₂×C₂²⋊D₈₉	—	2	2	no	—
23	C₂×C₄×D₈₉	—	2	2	no	—
24	#13083558	—	2	4	no	—
25	C₂×#10623494	—	2	2	no	—
26	C₄⋊Dic₈₉	—	2	2	no	—
27	#13083551	—	2	4	no	—
28	C₂²×C₄×C₈₉	—	2	2	no	—
29	C₂²×C₄×C₈₉	D₄	1	2	yes	—
30	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
31	C₄²×C₈₉	—	2	2	no	—
32	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₈₉	D₄	1	2	yes	—
35	C₄²×C₈₉	D₄	1	4	yes	complemented
36	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×D₈₉	—	2	2	no	—
38	C₂×C₄×C₈₉	—	2	4	no	—
39	C₂×C₄×C₈₉	—	4	2	no	—
40	C₂×C₄×C₈₉	—	4	4	no	—
41	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₉	#41	1	2	yes	—
47	C₂²⋊D₈₉	—	4	4	no	—
48	C₂×Dic₈₉	—	2	2	no	—
49	D₃₅₆	—	2	4	no	—
50	D₃₅₆	—	4	4	no	—
51	C₄×D₈₉	—	4	2	no	—
52	C₄×D₈₉	—	4	2	no	—
53	Dic₁₇₈	—	2	4	no	—
54	#10623494	—	4	2	no	—
55	C₂²×C₈₉	—	2	2	no	—
56	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₈₉	#121	1	2	yes	—
58	C₂×D₈₉	—	4	2	no	—
59	C₂×D₈₉	—	4	4	no	—
60	C₄×C₈₉	—	2	2	no	—
61	C₄×C₈₉	—	2	2	no	—
62	C₄×C₈₉	—	4	4	no	—
63	C₄×C₈₉	C₄×D₄	1	2	yes	—
64	C₄×C₈₉	#124	1	1	yes	characteristic
65	C₄×C₈₉	#127	1	1	yes	characteristic
66	Dic₈₉	—	4	2	no	—
67	C₂×C₈₉	—	2	2	no	—
68	C₂×C₈₉	#390	1	1	yes	characteristic
69	C₂×C₈₉	#421	1	2	yes	—
70	D₈₉	—	8	2	no	—
71	#1963	—	89	1	no	maximal, Sylow
72	C₈₉	#1963	1	1	yes	characteristic, complemented, Sylow
73	#532	—	89	1	no	—
74	#423	—	89	1	no	—
75	C₂×#104	—	89	4	no	—
76	#376	D₈₉	1	1	yes	characteristic, complemented
77	C₂×#41	—	89	2	no	—
78	C₂×M₁₆	—	89	2	no	—
79	C₄⋊Q₈	—	89	1	no	—
80	#127	—	89	1	no	—
81	#124	—	89	2	no	—
82	#121	—	178	2	no	—
83	C₄×D₄	—	178	2	no	—
84	C₂×C₄²	C₂×D₈₉	1	2	yes	—
85	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
86	C₄⋊C₈	—	178	2	no	—
87	#104	—	178	8	no	—
88	#95	—	2	2	no	—
89	#41	—	178	4	no	—
90	#41	—	178	4	no	—
91	C₂×Q₈	—	89	2	no	—
92	C₂×D₄	—	89	2	no	—
93	C₂×D₄	—	178	2	no	—
94	C₂×D₄	—	178	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	178	2	no	—
97	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
98	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
99	M₁₆	—	178	4	no	—
100	C₂×C₈	—	178	2	no	—
101	C₄⋊C₄	—	178	2	no	—
102	C₂²⋊C₄	—	178	4	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₄×D₈₉	1	1	yes	characteristic
105	C₄²	C₄×D₈₉	1	1	yes	characteristic
106	C₄²	D₃₅₆	1	2	yes	—
107	C₄²	D₃₅₆	1	4	yes	complemented
108	C₂³	—	178	2	no	—
109	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
110	Q₈	—	178	4	no	—
111	D₄	—	356	4	no	—
112	D₄	—	356	4	no	—
113	D₄	—	178	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	356	2	no	—
118	C₂×C₄	—	356	2	no	—
119	C₂×C₄	—	178	2	no	—
120	C₂×C₄	#13083551	1	2	yes	—
121	C₂×C₄	#13083551	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
123	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
124	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
125	C₂×C₄	#13083574	1	2	yes	—
126	C₈	—	356	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	356	4	no	—
129	C₂²	—	356	2	no	—
130	C₂²	C₂×#13083551	1	1	yes	characteristic
131	C₂²	#15667986	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	4	no	—
135	C₄	—	356	2	no	—
136	C₄	C₄×D₃₅₆	1	2	yes	—
137	C₄	#15668000	1	1	yes	characteristic
138	C₄	#15668009	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	712	2	no	—
141	C₂	#19151500	1	1	yes	characteristic
142	C₂	#19151616	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152148

C₂

yes

maximal, characteristic

#19151622

C₂

yes

maximal, characteristic, complemented

C₂×#15667940

C₂

yes

maximal, complemented

#376×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

—

C₂×#13083558

C₂²

yes

—

#15668012

C₄

yes

characteristic, complemented

#15668009

C₄

yes

characteristic, complemented

#15668000

C₄

yes

complemented

#15667986

—

C₄×D₃₅₆

—

C₄⋊#10623494

—

#15667940

—

C₂×C₄²×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#95×C₈₉

—

#13083574

—

#13083574

—

C₂×Dic₁₇₈

C₂×C₄

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂×C₄

yes

—

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

#13083551

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

D₄

yes

complemented

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

#121

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄×D₄

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#390

yes

characteristic

C₂×C₈₉

#421

yes

—

D₈₉

—

#1963

—

maximal, Sylow

C₈₉

#1963

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic, complemented

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

178

—

C₄×D₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

178

—

#104

—

178

—

#95

—

#41

—

178

—

#41

—

178

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

178

—

100

C₂×C₈

—

178

—

101

C₄⋊C₄

—

178

—

102

C₂²⋊C₄

—

178

—

103

C₄²

—

104

C₄²

C₄×D₈₉

yes

characteristic

105

C₄²

C₄×D₈₉

yes

characteristic

106

C₄²

D₃₅₆

yes

—

107

C₄²

D₃₅₆

yes

complemented

108

C₂³

—

178

—

109

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

110

Q₈

—

178

—

111

D₄

—

356

—

112

D₄

—

356

—

113

D₄

—

178

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

356

—

118

C₂×C₄

—

356

—

119

C₂×C₄

—

178

—

120

C₂×C₄

#13083551

yes

—

121

C₂×C₄

#13083551

yes

—

122

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

123

C₂×C₄

C₂×D₃₅₆

yes

—

124

C₂×C₄

C₂×D₃₅₆

yes

characteristic

125

C₂×C₄

#13083574

yes

—

126

C₈

—

356

—

127

C₂²

—

128

C₂²

—

356

—

129

C₂²

—

356

—

130

C₂²

C₂×#13083551

yes

characteristic

131

C₂²

#15667986

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

356

—

136

C₄

C₄×D₃₅₆

yes

—

137

C₄

#15668000

yes

characteristic

138

C₄

#15668009

yes

characteristic

139

C₂

—

140

C₂

—

712

—

141

C₂

#19151500

yes

characteristic

142

C₂

#19151616

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#376×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₄
prime radical	#19152148	C₂
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#376	D₈₉
hypocenter	C₈₉	#1963
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₃₅₆

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#376×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₂²⋊C₄

prime radical

#19152148

C₂

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
2. #19151622	C₂	semidirect
3. C₂×#15667940	C₂	semidirect
4. #376×C₈₉	C₂	semidirect
7. #15668012	C₄	semidirect
8. #15668009	C₄	semidirect
9. #15668000	C₄	semidirect
35. C₄²×C₈₉	D₄	semidirect
72. C₈₉	#1963	semidirect
76. #376	D₈₉	semidirect
107. C₄²	D₃₅₆	semidirect

H ≅ G/N

type

2. #19151622

C₂

semidirect

3. C₂×#15667940

C₂

semidirect

4. #376×C₈₉

C₂

semidirect

7. #15668012

C₄

semidirect

8. #15668009

C₄

semidirect

9. #15668000

C₄

semidirect

35. C₄²×C₈₉

D₄

semidirect

72. C₈₉

#1963

semidirect

76. #376

D₈₉

semidirect

107. C₄²

D₃₅₆

semidirect

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

subgroup	index
#19152148	2
#19151622	2
C₂×#15667940	2
#376×C₈₉	2
#1963	89

subgroup

index

#376×C₈₉

Subgroups — #26021084

Conjugacy classes of subgroups

Complements

Maximal subgroups