Group #30605934 — subgroups

Subgroups — #30605934

Order 15616.

206 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 88 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	#21462234	C₂	1	2	yes	maximal
2	#21460848	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11758187	C₂	1	2	yes	maximal
4	C₈×#11758187	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×#92418	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11758187	—	2	2	no	—
7	C₄⋊#11758187	C₂²	1	2	yes	—
8	#16840650	—	2	2	no	—
9	C₄×#11758187	C₂²	1	1	yes	characteristic
10	C₄×#11758187	C₂²	1	1	yes	characteristic
11	Q₈×#92418	C₂²	1	2	yes	—
12	#16839515	C₄	1	1	yes	characteristic
13	Q₈⋊#92418	C₄	1	2	yes	—
14	C₈×#92418	C₂²	1	1	yes	characteristic
15	C₄×Q₁₆×C₆₁	C₄	1	1	yes	characteristic
16	C₂×#11758187	—	2	1	no	—
17	C₂×#11758187	—	2	2	no	—
18	C₂×#11758187	D₄	1	1	yes	characteristic
19	C₂×#11758187	D₄	1	1	yes	characteristic
20	C₄⋊#92418	—	2	2	no	—
21	C₄⋊#92418	C₂×C₄	1	2	yes	—
22	C₄×#92418	—	2	2	no	—
23	C₄×#92418	C₂×C₄	1	1	yes	characteristic
24	C₄×#92418	C₂³	1	1	yes	characteristic
25	C₂×Q₁₆×C₆₁	C₈	1	1	yes	characteristic
26	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
27	#107×C₆₁	C₈	1	1	yes	characteristic
28	#103×C₆₁	C₈	1	2	yes	—
29	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×Q₈×C₆₁	C₂×C₈	1	2	yes	—
31	Q₁₆×C₆₁	—	2	2	no	—
32	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
34	C₄⋊C₄×C₆₁	—	2	2	no	—
35	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
36	C₄²×C₆₁	—	2	2	no	—
37	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×#92418	—	2	1	no	—
39	C₂×#92418	—	2	2	no	—
40	C₂×#92418	C₂×D₄	1	1	yes	characteristic
41	C₂×#92418	#41	1	1	yes	characteristic
42	#11758187	—	2	2	no	—
43	#11758187	—	2	1	no	—
44	#11758187	—	4	2	no	—
45	#11758187	Q₁₆	1	2	yes	complemented
46	Q₈×C₆₁	—	2	2	no	—
47	Q₈×C₆₁	—	2	2	no	—
48	C₂×C₄×C₆₁	—	2	2	no	—
49	C₂×C₄×C₆₁	—	2	2	no	—
50	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
53	C₈×C₆₁	—	2	1	no	—
54	C₈×C₆₁	M₃₂	1	2	yes	—
55	#92418	—	2	2	no	—
56	#92418	—	2	1	no	—
57	#92418	—	4	2	no	—
58	#92418	C₂×Q₁₆	1	1	yes	characteristic
59	#92418	#135	1	1	yes	characteristic
60	#12054	—	61	1	no	maximal, Sylow
61	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
62	C₄×C₆₁	—	2	1	no	—
63	C₄×C₆₁	—	4	2	no	—
64	C₄×C₆₁	—	4	2	no	—
65	C₄×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
66	C₄×C₆₁	#443	1	1	yes	characteristic
67	C₄×C₆₁	C₂×M₃₂	1	1	yes	characteristic
68	C₄×C₆₁	#504	1	1	yes	characteristic
69	#2183	—	61	2	no	—
70	C₈×Q₁₆	—	61	1	no	—
71	#1370	—	61	1	no	—
72	Q₈⋊C₁₆	—	61	2	no	—
73	C₈×C₁₆	—	61	1	no	—
74	C₂×C₆₁	C₈×Q₁₆	1	1	yes	characteristic
75	C₂×C₆₁	#2169	1	1	yes	characteristic
76	C₂×C₆₁	#2170	1	1	yes	characteristic
77	C₈×Q₈	—	61	2	no	—
78	C₄×Q₁₆	#6923	1	1	yes	characteristic
79	C₄⋊C₁₆	—	61	2	no	—
80	C₄⋊C₁₆	—	122	2	no	—
81	#346	—	122	2	no	—
82	C₄×C₁₆	—	61	1	no	—
83	C₄×C₁₆	—	61	1	no	—
84	#335	—	61	1	no	—
85	Q₈⋊C₈	—	61	2	no	—
86	C₈²	—	61	1	no	—
87	C₆₁	#12054	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	#92420	1	1	yes	characteristic
89	C₄×Q₈	C₂×#6923	1	2	yes	—
90	C₂×C₁₆	—	61	1	no	—
91	C₂×C₁₆	—	61	1	no	—
92	C₂×C₁₆	—	122	1	no	—
93	C₂×C₁₆	—	122	2	no	—
94	#107	#92420	1	1	yes	characteristic
95	C₄⋊C₈	—	61	2	no	—
96	C₄⋊C₈	—	122	2	no	—
97	#103	#92420	1	2	yes	—
98	C₄×C₈	—	61	1	no	—
99	C₄×C₈	—	61	1	no	—
100	C₄×C₈	—	122	2	no	—
101	C₄×C₈	C₂×#6923	1	1	yes	characteristic
102	C₂×Q₈	C₂×#92420	1	2	yes	—
103	Q₁₆	—	2	2	no	—
104	C₂×C₈	—	61	1	no	—
105	C₂×C₈	—	61	1	no	—
106	C₂×C₈	—	122	1	no	—
107	C₂×C₈	—	122	2	no	—
108	C₂×C₈	C₂×#92420	1	1	yes	characteristic
109	C₂×C₈	C₂×#92420	1	1	yes	characteristic
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂×#92420	1	2	yes	—
112	C₄²	—	2	2	no	—
113	C₄²	C₂²×#6923	1	1	yes	characteristic
114	C₁₆	—	61	2	no	—
115	C₁₆	—	122	1	no	—
116	C₁₆	—	122	2	no	—
117	C₁₆	—	244	2	no	—
118	Q₈	—	2	2	no	—
119	Q₈	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	D₄×#6923	1	1	yes	characteristic
123	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
124	C₂×C₄	#14519864	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	61	1	no	—
127	C₈	—	61	1	no	—
128	C₈	—	122	2	no	—
129	C₈	—	122	1	no	—
130	C₈	—	244	2	no	—
131	C₈	#14519851	1	2	yes	—
132	C₂²	D₄×#92420	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	4	2	no	—
135	C₄	—	4	2	no	—
136	C₄	Q₁₆×#6923	1	1	yes	characteristic
137	C₄	#16840806	1	1	yes	characteristic
138	C₄	C₂×#14519851	1	1	yes	characteristic
139	C₄	#16840820	1	1	yes	characteristic
140	C₂	Q₁₆×#92420	1	1	yes	characteristic
141	C₂	#21462203	1	1	yes	characteristic
142	C₂	#21462206	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462234

C₂

yes

maximal

#21460848

C₂

yes

maximal, characteristic

Q₈⋊#11758187

C₂

yes

maximal

C₈×#11758187

C₂

yes

maximal, characteristic

Q₁₆×#92418

C₂

yes

maximal, characteristic

C₄⋊#11758187

—

C₄⋊#11758187

C₂²

yes

—

#16840650

—

C₄×#11758187

C₂²

yes

characteristic

C₄×#11758187

C₂²

yes

characteristic

Q₈×#92418

C₂²

yes

—

#16839515

C₄

yes

characteristic

Q₈⋊#92418

C₄

yes

—

C₈×#92418

C₂²

yes

characteristic

C₄×Q₁₆×C₆₁

C₄

yes

—

—

D₄

yes

characteristic

C₂×#11758187

D₄

yes

characteristic

C₄⋊#92418

—

C₄⋊#92418

C₂×C₄

yes

—

C₄×#92418

—

C₄×#92418

C₂×C₄

yes

characteristic

C₄×#92418

C₂³

yes

characteristic

C₂×Q₁₆×C₆₁

C₈

yes

characteristic

C₄×Q₈×C₆₁

C₂×C₄

yes

—

#107×C₆₁

C₈

yes

characteristic

#103×C₆₁

C₈

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×Q₈×C₆₁

C₂×C₈

yes

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₈

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

—

—

C₂×D₄

yes

yes

—

—

—

Q₁₆

yes

complemented

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

#131

yes

characteristic

C₈×C₆₁

—

C₈×C₆₁

M₃₂

yes

—

—

—

—

C₂×Q₁₆

yes

yes

—

C₂²×C₆₁

C₈×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×Q₁₆

yes

characteristic

C₄×C₆₁

#443

yes

characteristic

C₄×C₆₁

C₂×M₃₂

yes

C₄×C₆₁

yes

—

C₈×Q₁₆

—

#1370

—

Q₈⋊C₁₆

—

C₈×C₁₆

—

C₂×C₆₁

C₈×Q₁₆

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₈×Q₈

—

C₄×Q₁₆

#6923

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

122

—

#346

—

122

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₆₁

#12054

yes

characteristic, complemented, Sylow

C₂×Q₁₆

#92420

yes

characteristic

C₄×Q₈

C₂×#6923

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

—

122

—

#107

#92420

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

122

—

#103

#92420

yes

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

122

—

101

C₄×C₈

C₂×#6923

yes

characteristic

102

C₂×Q₈

C₂×#92420

yes

—

103

Q₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

122

—

107

C₂×C₈

—

122

—

108

C₂×C₈

C₂×#92420

yes

characteristic

109

C₂×C₈

C₂×#92420

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂×#92420

yes

—

112

C₄²

—

113

C₄²

C₂²×#6923

yes

characteristic

114

C₁₆

—

115

C₁₆

—

122

—

116

C₁₆

—

122

—

117

C₁₆

—

244

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

D₄×#6923

yes

characteristic

123

C₂×C₄

C₂²×#92420

yes

characteristic

124

C₂×C₄

#14519864

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

122

—

129

C₈

—

122

—

130

C₈

—

244

—

131

C₈

#14519851

yes

—

132

C₂²

D₄×#92420

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

Q₁₆×#6923

yes

characteristic

137

C₄

#16840806

yes

characteristic

138

C₄

C₂×#14519851

yes

characteristic

139

C₄

#16840820

yes

characteristic

140

C₂

Q₁₆×#92420

yes

characteristic

141

C₂

#21462203

yes

characteristic

142

C₂

#21462206

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6923
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₄²	C₂²×#6923
Fitting	C₄×Q₁₆×C₆₁	C₄
socle, prime radical	C₂²×C₆₁	C₈×D₄
cosocle	C₄×#92418	C₂³
hypercenter	C₄×Q₁₆	#6923
hypocenter	C₆₁	#12054
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×#6923

commutator

C₂×C₄×C₆₁

C₂²×C₈

Frattini

C₄²

C₂²×#6923

Fitting

C₄×Q₁₆×C₆₁

C₄

socle, prime radical

C₂²×C₆₁

C₈×D₄

C₂³

C₄×Q₁₆

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
45. #11758187	Q₁₆	semidirect
87. C₆₁	#12054	semidirect

H ≅ G/N

type

45. #11758187

Q₁₆

semidirect

87. C₆₁

#12054

semidirect

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

subgroup	index
#21462234	2
#21460848	2
Q₈⋊#11758187	2
C₈×#11758187	2
Q₁₆×#92418	2
#12054	61

subgroup

index

Q₁₆×#92418

Subgroups — #30605934

Conjugacy classes of subgroups

Complements

Maximal subgroups