Group #11797036 — subgroups

Subgroups — #11797036

Order 1152.

198 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 71 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×#64214	C₂	1	1	yes	maximal, characteristic
2	C₂×#64213	C₂	1	1	yes	maximal, characteristic
3	#10593748	C₂	1	1	yes	maximal, characteristic
4	#10593265	C₂	1	2	yes	maximal, complemented
5	C₄⋊#3771	C₂	1	2	yes	maximal, complemented
6	C₂×#3788	C₄	1	1	yes	characteristic
7	C₂×#3787	C₂²	1	1	yes	characteristic
8	C₂×#3771	C₂²	1	1	yes	characteristic
9	C₂×#3771	C₂²	1	1	yes	characteristic
10	C₂×#3771	C₂²	1	2	yes	—
11	#64214	—	2	2	no	—
12	#64213	—	2	2	no	—
13	#64101	C₄	1	1	yes	characteristic
14	#63777	C₄	1	2	yes	—
15	C₄×#622	C₂²	1	2	yes	—
16	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
17	C₂×#622	C₂×C₄	1	2	yes	—
18	C₂×#622	C₂×C₄	1	1	yes	characteristic
19	C₂×#622	C₂³	1	1	yes	characteristic
20	#3788	—	2	2	no	—
21	#3787	D₄	1	2	yes	—
22	#3787	Q₈	1	2	yes	—
23	#3771	—	2	1	no	—
24	#3771	—	2	1	no	—
25	#3771	—	2	2	no	—
26	C₃²×C₄⋊C₄	C₈	1	1	yes	characteristic
27	#3752	C₈	1	2	yes	—
28	C₄⋊(C₃⋊Dic₃)	C₈	1	1	yes	characteristic
29	C₄×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
30	#2151	—	9	1	no	maximal, Sylow
31	#892	—	12	1	no	—
32	C₂²×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
34	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
35	C₂×C₃⋊Dic₃	C₂×C₈	1	2	yes	—
36	C₂×C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₃⋊D₃	C₄⋊C₄	1	2	yes	—
38	C₄×C₃⋊D₃	M₁₆	1	2	yes	—
39	#622	—	2	2	no	—
40	#622	C₄⋊C₄	1	2	yes	—
41	#622	C₂×D₄	1	1	yes	characteristic
42	#622	C₂×Q₈	1	1	yes	characteristic
43	C₂×M₃₂	—	9	1	no	—
44	C₂²×C₁₆	—	9	1	no	—
45	#405	—	9	1	no	—
46	C₄⋊C₁₆	—	9	2	no	—
47	C₄⋊C₁₆	—	9	2	no	—
48	C₂×C₄×D₃	—	12	1	no	—
49	C₃×C₄⋊C₄	—	4	1	no	—
50	(C₂×D₃)⋊C₄	—	12	2	no	—
51	C₄⋊Dic₃	—	12	1	no	—
52	C₄×Dic₃	—	12	2	no	—
53	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₃⋊D₃	—	2	1	no	—
55	C₂×C₃⋊D₃	—	2	1	no	—
56	C₂×C₃⋊D₃	C₄⋊C₈	1	2	yes	—
57	C₃²×C₄	—	2	2	no	—
58	C₃²×C₄	C₄⋊C₈	1	2	yes	—
59	C₃⋊Dic₃	—	2	2	no	—
60	C₃⋊Dic₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₃⋊Dic₃	C₂×M₁₆	1	1	yes	characteristic
62	C₂×M₁₆	—	9	1	no	—
63	C₂²×C₈	—	9	1	no	—
64	#117	—	9	1	no	—
65	M₃₂	—	18	2	no	—
66	C₂×C₁₆	—	9	1	no	—
67	C₂×C₁₆	—	9	1	no	—
68	C₂×C₁₆	—	9	2	no	—
69	C₂×C₁₆	—	18	2	no	—
70	#97	—	9	2	no	—
71	C₄×C₈	—	9	2	no	—
72	C₂²×D₃	—	12	1	no	—
73	C₂×C₃×C₄	—	4	1	no	—
74	C₂×C₃×C₄	—	4	2	no	—
75	C₂×Dic₃	—	12	1	no	—
76	C₂×Dic₃	—	12	2	no	—
77	C₄×D₃	—	12	2	no	—
78	C₄×D₃	—	12	2	no	—
79	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₂×C₃²	#504	1	1	yes	characteristic
81	C₂×C₃²	#504	1	1	yes	characteristic
82	C₃⋊D₃	—	2	2	no	—
83	C₂²×C₄	—	9	1	no	—
84	M₁₆	—	18	2	no	—
85	C₂×C₈	—	9	1	no	—
86	C₂×C₈	—	9	1	no	—
87	C₂×C₈	—	9	2	no	—
88	C₂×C₈	—	9	2	no	—
89	C₂×C₈	—	9	2	no	—
90	C₄⋊C₄	—	9	1	no	—
91	C₄⋊C₄	AGL(1,𝔽₉)	1	1	yes	characteristic
92	C₂²⋊C₄	—	9	2	no	—
93	C₄²	—	9	2	no	—
94	C₁₆	—	18	1	no	—
95	C₁₆	—	18	1	no	—
96	C₁₆	—	18	2	no	—
97	C₂²×C₃	—	4	1	no	—
98	C₂×D₃	—	12	2	no	—
99	C₂×D₃	—	24	1	no	—
100	C₂×D₃	—	24	1	no	—
101	C₃×C₄	—	4	2	no	—
102	C₃×C₄	—	8	2	no	—
103	Dic₃	—	12	1	no	—
104	Dic₃	—	12	1	no	—
105	Dic₃	—	24	2	no	—
106	C₃²	#2151	1	1	yes	characteristic, complemented, Sylow
107	C₂³	—	9	1	no	—
108	C₂×C₄	—	9	1	no	—
109	C₂×C₄	—	9	2	no	—
110	C₂×C₄	—	9	2	no	—
111	C₂×C₄	—	9	2	no	—
112	C₂×C₄	C₂×AGL(1,𝔽₉)	1	2	yes	—
113	C₂×C₄	C₂×AGL(1,𝔽₉)	1	1	yes	characteristic
114	C₈	—	9	1	no	—
115	C₈	—	9	2	no	—
116	C₈	—	9	1	no	—
117	C₈	—	18	2	no	—
118	C₂×C₃	—	4	1	no	—
119	C₂×C₃	—	4	1	no	—
120	C₂×C₃	—	4	1	no	—
121	D₃	—	24	2	no	—
122	C₂²	—	9	2	no	—
123	C₂²	—	18	1	no	—
124	C₂²	—	18	1	no	—
125	C₂²	C₂²×AGL(1,𝔽₉)	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	9	1	no	—
128	C₄	—	9	1	no	—
129	C₄	—	18	2	no	—
130	C₄	C₄⋊AGL(1,𝔽₉)	1	2	yes	—
131	C₃	—	4	1	no	—
132	C₂	—	18	2	no	—
133	C₂	C₂×C₄⋊AGL(1,𝔽₉)	1	1	yes	characteristic
134	C₂	#10596485	1	1	yes	characteristic
135	C₂	#10596487	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64214

C₂

yes

maximal, characteristic

C₂×#64213

C₂

yes

maximal, characteristic

#10593748

C₂

yes

maximal, characteristic

#10593265

C₂

yes

maximal, complemented

C₄⋊#3771

C₂

yes

maximal, complemented

C₂×#3788

C₄

yes

characteristic

C₂×#3787

C₂²

yes

characteristic

C₂×#3771

C₂²

yes

characteristic

C₂×#3771

C₂²

yes

characteristic

C₂×#3771

C₂²

yes

—

#64214

—

#64213

—

#64101

C₄

yes

characteristic

#63777

C₄

yes

—

C₄×#622

C₂²

yes

—

C₂×C₄×C₃⋊D₃

C₂×C₄

yes

characteristic

C₂×#622

C₂×C₄

yes

—

C₂×#622

C₂×C₄

yes

characteristic

C₂×#622

C₂³

yes

characteristic

#3788

—

#3787

D₄

yes

—

#3787

Q₈

yes

—

#3771

—

#3771

—

#3771

—

C₃²×C₄⋊C₄

C₈

yes

characteristic

#3752

C₈

yes

—

C₄⋊(C₃⋊Dic₃)

C₈

yes

characteristic

C₄×C₃⋊Dic₃

C₂×C₄

yes

—

#2151

—

maximal, Sylow

#892

—

C₂²×C₃⋊D₃

C₂×C₈

yes

characteristic

C₂×C₃²×C₄

C₂×C₈

yes

—

C₂×C₃²×C₄

C₂×C₈

yes

characteristic

C₂×C₃⋊Dic₃

C₂×C₈

yes

—

C₂×C₃⋊Dic₃

C₂²×C₄

yes

characteristic

C₄×C₃⋊D₃

C₄⋊C₄

yes

—

C₄×C₃⋊D₃

M₁₆

yes

—

#622

—

#622

C₄⋊C₄

yes

—

#622

C₂×D₄

yes

characteristic

#622

C₂×Q₈

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂×C₄×D₃

—

C₃×C₄⋊C₄

—

(C₂×D₃)⋊C₄

—

C₄⋊Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

C₂²×C₈

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

C₄⋊C₈

yes

—

C₃²×C₄

—

C₃²×C₄

C₄⋊C₈

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

C₂×C₄⋊C₄

yes

characteristic

C₃⋊Dic₃

C₂×M₁₆

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₂×C₃²

C₂×C₄⋊C₈

yes

C₂×C₃²

yes

C₂×C₃²

yes

C₃⋊D₃

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

AGL(1,𝔽₉)

yes

characteristic

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

100

C₂×D₃

—

101

C₃×C₄

—

102

C₃×C₄

—

103

Dic₃

—

104

Dic₃

—

105

Dic₃

—

106

C₃²

#2151

yes

characteristic, complemented, Sylow

107

C₂³

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂×AGL(1,𝔽₉)

yes

—

113

C₂×C₄

C₂×AGL(1,𝔽₉)

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₂×C₃

—

119

C₂×C₃

—

120

C₂×C₃

—

121

D₃

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂²×AGL(1,𝔽₉)

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

C₄⋊AGL(1,𝔽₉)

yes

—

131

C₃

—

132

C₂

—

133

C₂

C₂×C₄⋊AGL(1,𝔽₉)

yes

characteristic

134

C₂

#10596485

yes

characteristic

135

C₂

#10596487

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₉)
commutator, socle	C₂²×C₃²	C₂²×C₈
Fitting	C₃²×C₄⋊C₄	C₈
prime radical	C₂²×C₃⋊D₃	C₂×C₈
cosocle	C₂×#622	C₂³
hypercenter	C₄⋊C₄	AGL(1,𝔽₉)
hypocenter	C₃²	#2151
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂²×AGL(1,𝔽₉)

commutator, socle

C₂²×C₃²

C₂²×C₈

Fitting

C₃²×C₄⋊C₄

C₈

prime radical

C₂²×C₃⋊D₃

C₂×C₈

C₂³

C₄⋊C₄

C₃²

G ▷ C₂²×C₃² ▷ C₃²

C₁ ◁ C₂² ◁ C₄⋊C₄

N	H ≅ G/N	type
4. #10593265	C₂	semidirect
5. C₄⋊#3771	C₂	semidirect
106. C₃²	#2151	semidirect

H ≅ G/N

type

4. #10593265

C₂

semidirect

5. C₄⋊#3771

C₂

semidirect

106. C₃²

#2151

semidirect

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

subgroup	index
C₂×#64214	2
C₂×#64213	2
#10593748	2
#10593265	2
C₄⋊#3771	2
#2151	9

subgroup

index

Subgroups — #11797036

Conjugacy classes of subgroups

Complements

Maximal subgroups