Group #11772285 — subgroups

Subgroups — #11772285

Order 1152.

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

normal subgroups only

This group has 178 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	#10596481	C₂	1	2	yes	maximal
2	C₂×#63767	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#157	C₄	1	1	yes	characteristic
4	C₂×#3787	C₂²	1	1	yes	characteristic
5	#64218	—	2	2	no	—
6	#64214	—	2	2	no	—
7	#64213	C₄	1	4	yes	complemented
8	C₂×C₈×C₃⋊D₃	C₄	1	1	yes	characteristic
9	#63767	—	2	2	no	—
10	C₂²×#157	—	2	1	no	—
11	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
12	C₂×#622	D₄	1	1	yes	characteristic
13	C₂×#622	Q₈	1	1	yes	characteristic
14	C₄×#157	—	2	1	no	—
15	C₄×#157	C₈	1	2	yes	—
16	#3787	C₂×C₄	1	2	yes	—
17	#3787	D₄	1	2	yes	—
18	#3771	—	2	4	no	—
19	C₂×C₃²×C₈	C₈	1	1	yes	characteristic
20	C₂×C₃⋊(C₃⋊C₈)	C₈	1	1	yes	characteristic
21	C₈×C₃⋊D₃	—	2	1	no	—
22	C₈×C₃⋊D₃	—	2	1	no	—
23	#1375	—	9	1	no	maximal, Sylow
24	C₂×D₃×C₈	—	12	1	no	—
25	C₂²×C₃⋊D₃	M₁₆	1	1	yes	characteristic
26	C₂×#157	—	2	2	no	—
27	C₂×#157	—	4	2	no	—
28	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
29	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₃⋊D₃	C₄²	1	1	yes	characteristic
31	C₄×C₃⋊D₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
33	C₄×C₃⋊D₃	M₁₆	1	1	yes	characteristic
34	#622	C₂²⋊C₄	1	2	yes	—
35	#622	C₄⋊C₄	1	2	yes	—
36	C₃²×C₈	—	2	1	no	—
37	C₃⋊(C₃⋊C₈)	—	2	1	no	—
38	C₂×M₃₂	—	9	2	no	—
39	C₂×#97	—	9	1	no	—
40	C₂×C₄×D₃	—	12	1	no	—
41	C₂×C₃×C₈	—	4	1	no	—
42	C₂×C₃⋊C₈	—	12	1	no	—
43	D₃×C₈	—	12	2	no	—
44	D₃×C₈	—	12	2	no	—
45	C₂²×C₃²	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₃⋊D₃	—	2	2	no	—
47	C₂×C₃⋊D₃	#97	1	1	yes	characteristic
48	C₂×C₃⋊D₃	C₂²⋊C₈	1	1	yes	characteristic
49	#157	—	4	2	no	—
50	C₃²×C₄	C₄×C₈	1	1	yes	characteristic
51	C₃²×C₄	C₂²⋊C₈	1	1	yes	characteristic
52	C₃⋊Dic₃	#95	1	1	yes	characteristic
53	C₃⋊Dic₃	C₄⋊C₈	1	1	yes	characteristic
54	C₂²×C₈	—	9	1	no	—
55	C₂²×C₈	—	9	1	no	—
56	C₂×C₄²	—	9	1	no	—
57	M₃₂	—	18	2	no	—
58	M₃₂	—	18	2	no	—
59	C₂×C₁₆	—	9	4	no	—
60	#97	—	18	2	no	—
61	C₂²×D₃	—	12	1	no	—
62	C₂×C₃×C₄	—	4	1	no	—
63	C₂×Dic₃	—	12	1	no	—
64	C₄×D₃	—	12	1	no	—
65	C₄×D₃	—	12	1	no	—
66	C₄×D₃	—	12	1	no	—
67	C₄×D₃	—	12	1	no	—
68	C₃×C₈	—	4	2	no	—
69	C₃⋊C₈	—	12	2	no	—
70	C₂×C₃²	—	2	1	no	—
71	C₂×C₃²	#336	1	1	yes	characteristic
72	C₃⋊D₃	—	2	1	no	—
73	C₃⋊D₃	#347	1	2	yes	—
74	C₂²×C₄	—	9	1	no	—
75	C₂²×C₄	—	18	1	no	—
76	C₂×C₈	—	9	2	no	—
77	C₂×C₈	—	9	2	no	—
78	C₂×C₈	—	9	1	no	—
79	C₂×C₈	—	9	1	no	—
80	C₂×C₈	—	9	1	no	—
81	C₂×C₈	—	18	1	no	—
82	C₂×C₈	—	18	1	no	—
83	C₂×C₈	AGL(1,𝔽₉)	1	1	yes	characteristic
84	C₄²	—	9	2	no	—
85	C₄²	—	18	1	no	—
86	C₁₆	—	18	4	no	—
87	C₂²×C₃	—	4	1	no	—
88	C₂×D₃	—	12	1	no	—
89	C₂×D₃	—	12	1	no	—
90	C₂×D₃	—	12	4	no	—
91	C₃×C₄	—	4	1	no	—
92	C₃×C₄	—	4	1	no	—
93	Dic₃	—	12	1	no	—
94	Dic₃	—	12	1	no	—
95	C₃²	#1375	1	1	yes	characteristic, complemented, Sylow
96	C₂³	—	9	1	no	—
97	C₂×C₄	—	9	1	no	—
98	C₂×C₄	—	9	1	no	—
99	C₂×C₄	—	9	1	no	—
100	C₂×C₄	—	9	1	no	—
101	C₂×C₄	—	9	1	no	—
102	C₂×C₄	—	18	2	no	—
103	C₂×C₄	—	36	2	no	—
104	C₂×C₄	C₂×AGL(1,𝔽₉)	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	9	2	no	—
107	C₈	—	9	2	no	—
108	C₈	—	18	1	no	—
109	C₂×C₃	—	4	2	no	—
110	C₂×C₃	—	4	1	no	—
111	D₃	—	12	2	no	—
112	D₃	—	12	2	no	—
113	C₂²	—	9	1	no	—
114	C₂²	—	9	1	no	—
115	C₂²	—	18	2	no	—
116	C₂²	C₄⋊AGL(1,𝔽₉)	1	1	yes	characteristic
117	C₄	—	9	1	no	—
118	C₄	—	9	1	no	—
119	C₄	—	36	2	no	—
120	C₄	C₄×AGL(1,𝔽₉)	1	1	yes	characteristic
121	C₄	C₂²⋊AGL(1,𝔽₉)	1	1	yes	characteristic
122	C₃	—	4	1	no	—
123	C₂	—	2	1	no	—
124	C₂	—	9	2	no	—
125	C₂	—	18	1	no	—
126	C₂	#10593267	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10596481

C₂

yes

maximal

C₂×#63767

C₂

yes

maximal, characteristic

C₂×C₄×#157

C₄

yes

characteristic

C₂×#3787

C₂²

yes

—

—

C₄

yes

complemented

C₂×C₈×C₃⋊D₃

C₄

yes

characteristic

#63767

—

C₂²×#157

—

C₂×C₄×C₃⋊D₃

C₂×C₄

yes

characteristic

C₂×#622

D₄

yes

characteristic

C₂×#622

Q₈

yes

characteristic

C₄×#157

—

C₄×#157

C₈

yes

—

#3787

C₂×C₄

yes

—

#3787

D₄

yes

—

#3771

—

C₂×C₃²×C₈

C₈

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₈

yes

characteristic

C₈×C₃⋊D₃

—

C₈×C₃⋊D₃

—

#1375

—

maximal, Sylow

C₂×D₃×C₈

—

C₂²×C₃⋊D₃

M₁₆

yes

characteristic

C₂×#157

—

C₂×#157

—

C₂×C₃²×C₄

C₂×C₈

yes

characteristic

C₂×C₃⋊Dic₃

C₄⋊C₄

yes

characteristic

C₄×C₃⋊D₃

C₄²

yes

characteristic

C₄×C₃⋊D₃

C₂²⋊C₄

yes

characteristic

C₄×C₃⋊D₃

C₂×C₈

yes

characteristic

C₄×C₃⋊D₃

M₁₆

yes

characteristic

#622

C₂²⋊C₄

yes

—

#622

C₄⋊C₄

yes

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

C₂×M₃₂

—

C₂×#97

—

C₂×C₄×D₃

—

C₂×C₃×C₈

—

C₂×C₃⋊C₈

—

D₃×C₈

—

D₃×C₈

—

C₂²×C₃²

C₄⋊C₈

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

#97

yes

characteristic

C₂×C₃⋊D₃

C₂²⋊C₈

yes

characteristic

#157

—

C₃²×C₄

C₄×C₈

yes

characteristic

C₃²×C₄

C₂²⋊C₈

yes

characteristic

C₃⋊Dic₃

#95

yes

characteristic

C₃⋊Dic₃

C₄⋊C₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#97

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#336

yes

characteristic

C₃⋊D₃

—

C₃⋊D₃

#347

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

AGL(1,𝔽₉)

yes

characteristic

C₄²

—

C₄²

—

C₁₆

—

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

C₃²

#1375

yes

characteristic, complemented, Sylow

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×AGL(1,𝔽₉)

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₂×C₃

—

110

C₂×C₃

—

111

D₃

—

112

D₃

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₄⋊AGL(1,𝔽₉)

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₄×AGL(1,𝔽₉)

yes

characteristic

121

C₄

C₂²⋊AGL(1,𝔽₉)

yes

characteristic

122

C₃

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

#10593267

yes

characteristic

	subgroup	G/H
center	C₄	C₄×AGL(1,𝔽₉)
commutator	C₃²×C₄	C₄×C₈
Frattini	C₂×C₄	C₂×AGL(1,𝔽₉)
Fitting	C₂×C₃²×C₈	C₈
socle	C₂×C₃²	#336
prime radical	C₂²×C₃⋊D₃	M₁₆
cosocle	C₂×#3787	C₂²
hypercenter	C₂×C₈	AGL(1,𝔽₉)
hypocenter	C₃²	#1375
LCS	G ▷ C₃²×C₄ ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₈

subgroup

G/H

center

C₄

C₄×AGL(1,𝔽₉)

commutator

C₃²×C₄

C₄×C₈

Frattini

C₂×C₄

C₂×AGL(1,𝔽₉)

Fitting

C₂×C₃²×C₈

C₈

socle

C₂×C₃²

#336

prime radical

C₂²×C₃⋊D₃

M₁₆

C₂²

C₂×C₈

C₃²

G ▷ C₃²×C₄ ▷ C₃²

C₁ ◁ C₄ ◁ C₂×C₈

N	H ≅ G/N	type
7. #64213	C₄	semidirect
95. C₃²	#1375	semidirect

H ≅ G/N

type

7. #64213

C₄

semidirect

95. C₃²

#1375

semidirect

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

subgroup	index
#10596481	2
C₂×#63767	2
#1375	9

subgroup

index

#10596481

C₂×#63767

#1375

Subgroups — #11772285

Conjugacy classes of subgroups

Complements

Maximal subgroups