Group #29596374 — subgroups

Subgroups — #29596374

Order 15616.

342 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 163 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	#21447575	C₂	1	2	yes	maximal
2	#21443901	C₂	1	4	yes	maximal
3	C₄²×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519717	C₂²	1	4	yes	—
5	C₄⋊#11758185	—	2	4	no	—
6	C₂×C₄×C₈×C₆₁	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₄³×C₆₁	C₄	1	1	yes	characteristic
11	#14519717	—	2	8	no	—
12	C₂×#11758185	—	2	4	no	—
13	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₆₁	—	2	2	no	—
17	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₆₁	—	2	4	no	—
20	C₄×C₈×C₆₁	—	2	4	no	—
21	C₄×C₈×C₆₁	C₈	1	2	yes	—
22	C₄×C₈×C₆₁	C₈	1	2	yes	—
23	C₄×C₈×C₆₁	D₄	1	2	yes	—
24	C₄×C₈×C₆₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₆₁	—	2	2	no	—
26	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
27	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₈×C₆₁	—	2	2	no	—
29	C₂×C₈×C₆₁	—	2	2	no	—
30	C₂×C₈×C₆₁	—	2	2	no	—
31	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₆₁	#41	1	4	yes	—
36	C₄²×C₆₁	—	2	8	no	—
37	C₄²×C₆₁	—	2	2	no	—
38	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
39	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₆₁	M₁₆	1	4	yes	—
42	#11758185	—	4	4	no	—
43	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₄×C₆₁	—	2	8	no	—
45	C₂×C₄×C₆₁	—	2	4	no	—
46	C₂×C₄×C₆₁	—	2	4	no	—
47	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
48	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
49	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	#117	1	2	yes	—
51	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
53	C₈×C₆₁	—	2	2	no	—
54	C₈×C₆₁	—	2	4	no	—
55	C₈×C₆₁	#124	1	2	yes	—
56	C₈×C₆₁	#125	1	2	yes	—
57	#12883	—	61	1	no	maximal, Sylow
58	C₂²×C₆₁	—	2	2	no	—
59	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
60	C₂²×C₆₁	#432	1	2	yes	—
61	C₄×C₆₁	—	2	8	no	—
62	C₄×C₆₁	—	2	2	no	—
63	C₄×C₆₁	#364	1	4	yes	—
64	C₄×C₆₁	#382	1	1	yes	characteristic
65	C₄×C₆₁	#423	1	1	yes	characteristic
66	C₄×C₆₁	#431	1	2	yes	—
67	#2150	—	61	2	no	—
68	C₄²×C₈	D₆₁	1	1	yes	characteristic
69	#1376	—	61	4	no	—
70	C₂×C₆₁	—	2	2	no	—
71	C₂×C₆₁	#1831	1	1	yes	characteristic
72	C₂×C₆₁	C₂×#364	1	2	yes	—
73	C₂×M₃₂	—	61	4	no	—
74	C₂×C₄×C₈	Dic₆₁	1	2	yes	—
75	C₂×C₄×C₈	Dic₆₁	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
77	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
78	C₄³	Dic₆₁	1	1	yes	characteristic
79	C₄⋊C₁₆	—	122	4	no	—
80	C₆₁	#12883	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
82	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
83	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
84	C₂×C₄²	—	2	2	no	—
85	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
86	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
87	M₃₂	—	122	8	no	—
88	C₂×C₁₆	—	122	4	no	—
89	C₄×C₈	—	2	4	no	—
90	C₄×C₈	—	2	4	no	—
91	C₄×C₈	#92418	1	2	yes	—
92	C₄×C₈	#92418	1	2	yes	—
93	C₄×C₈	Dic₁₂₂	1	2	yes	—
94	C₄×C₈	D₂₄₄	1	2	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	#11758194	1	2	yes	—
97	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	C₂×#92418	1	4	yes	—
102	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
103	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
104	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
105	C₂×C₈	#11758222	1	4	yes	—
106	C₄²	—	2	8	no	—
107	C₄²	—	2	2	no	—
108	C₄²	C₂×#92418	1	1	yes	characteristic
109	C₄²	C₂×#92418	1	1	yes	characteristic
110	C₄²	#11758194	1	4	yes	—
111	C₄²	C₄⋊Dic₆₁	1	2	yes	—
112	C₁₆	—	244	4	no	—
113	C₂³	C₂×#11758194	1	1	yes	characteristic
114	C₂×C₄	—	2	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	C₄⋊#92418	1	2	yes	—
118	C₂×C₄	C₄⋊#92418	1	2	yes	—
119	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
120	C₂×C₄	#14519736	1	2	yes	—
121	C₂×C₄	C₂²×#92418	1	1	yes	characteristic
122	C₂×C₄	C₂×#11758194	1	2	yes	—
123	C₈	—	2	2	no	—
124	C₈	—	2	4	no	—
125	C₈	#14519765	1	2	yes	—
126	C₈	#14519769	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	C₂×C₄⋊#92418	1	1	yes	characteristic
129	C₂²	#16839819	1	2	yes	—
130	C₄	—	2	8	no	—
131	C₄	—	2	2	no	—
132	C₄	#16839583	1	4	yes	—
133	C₄	#16839628	1	1	yes	characteristic
134	C₄	#16839786	1	1	yes	characteristic
135	C₄	#16839815	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	#21445936	1	1	yes	characteristic
138	C₂	C₂×#16839583	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21447575

C₂

yes

maximal

#21443901

C₂

yes

maximal

C₄²×C₈×C₆₁

C₂

yes

maximal, characteristic

C₂×#14519717

C₂²

yes

—

C₄⋊#11758185

—

C₂×C₄×C₈×C₆₁

C₄

yes

—

C₂×C₄×C₈×C₆₁

C₄

yes

characteristic

C₂×C₄×C₈×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₄³×C₆₁

C₄

yes

characteristic

#14519717

—

C₂×#11758185

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

C₈

yes

—

C₄×C₈×C₆₁

C₈

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

M₁₆

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

M₁₆

yes

—

#11758185

—

C₂³×C₆₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₈

yes

—

C₂×C₄×C₆₁

C₄⋊C₈

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#124

yes

—

C₈×C₆₁

#125

yes

—

#12883

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₆₁

#432

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#364

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

#2150

—

C₄²×C₈

D₆₁

yes

characteristic

#1376

—

C₂×C₆₁

—

C₂×C₆₁

#1831

yes

characteristic

C₂×C₆₁

C₂×#364

yes

—

C₂×M₃₂

—

C₂×C₄×C₈

Dic₆₁

yes

—

C₂×C₄×C₈

Dic₆₁

yes

characteristic

C₂×C₄×C₈

C₂×D₆₁

yes

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄³

Dic₆₁

yes

characteristic

C₄⋊C₁₆

—

122

—

C₆₁

#12883

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₆₁

yes

—

C₂²×C₈

C₂×Dic₆₁

yes

characteristic

C₂²×C₈

C₂²×D₆₁

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₆₁

yes

—

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

M₃₂

—

122

—

C₂×C₁₆

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#92418

yes

—

C₄×C₈

#92418

yes

—

C₄×C₈

Dic₁₂₂

yes

—

C₄×C₈

D₂₄₄

yes

—

C₂²×C₄

—

C₂²×C₄

#11758194

yes

—

C₂²×C₄

C₂²×Dic₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂×#92418

yes

—

102

C₂×C₈

C₄⋊Dic₆₁

yes

—

103

C₂×C₈

C₂×Dic₁₂₂

yes

characteristic

104

C₂×C₈

C₂×D₂₄₄

yes

characteristic

105

C₂×C₈

#11758222

yes

—

106

C₄²

—

107

C₄²

—

108

C₄²

C₂×#92418

yes

characteristic

109

C₄²

C₂×#92418

yes

characteristic

110

C₄²

#11758194

yes

—

111

C₄²

C₄⋊Dic₆₁

yes

—

112

C₁₆

—

244

—

113

C₂³

C₂×#11758194

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₄⋊#92418

yes

—

118

C₂×C₄

C₄⋊#92418

yes

—

119

C₂×C₄

C₂×C₄⋊Dic₆₁

yes

characteristic

120

C₂×C₄

#14519736

yes

—

121

C₂×C₄

C₂²×#92418

yes

characteristic

122

C₂×C₄

C₂×#11758194

yes

—

123

C₈

—

124

C₈

—

125

C₈

#14519765

yes

—

126

C₈

#14519769

yes

—

127

C₂²

—

128

C₂²

C₂×C₄⋊#92418

yes

characteristic

129

C₂²

#16839819

yes

—

130

C₄

—

131

C₄

—

132

C₄

#16839583

yes

—

133

C₄

#16839628

yes

characteristic

134

C₄

#16839786

yes

characteristic

135

C₄

#16839815

yes

—

136

C₂

—

137

C₂

#21445936

yes

characteristic

138

C₂

C₂×#16839583

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₄²×C₈×C₆₁	C₂
socle	C₂²×C₆₁	C₂×C₄⋊C₈
prime radical	C₂³×C₆₁	C₂×M₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₄²×C₈	D₆₁
hypocenter	C₆₁	#12883
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

subgroup

G/H

center

C₂×C₈

C₂×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₄²×C₈×C₆₁

C₂

socle

C₂²×C₆₁

C₂×C₄⋊C₈

prime radical

C₂³×C₆₁

C₂×M₁₆

cosocle

C₂²×C₈×C₆₁

C₂³

hypercenter

C₄²×C₈

D₆₁

hypocenter

C₆₁

#12883

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
80. C₆₁	#12883	semidirect

H ≅ G/N

type

80. C₆₁

#12883

semidirect

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

subgroup	index
#21447575	2
#21443901	2
C₄²×C₈×C₆₁	2
#12883	61

subgroup

index

#21447575

#21443901

C₄²×C₈×C₆₁

#12883

Subgroups — #29596374

Conjugacy classes of subgroups

Complements

Maximal subgroups