Group #30605953 — subgroups

Subgroups — #30605953

Order 15616.

212 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 102 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	Q₈×#11758187	C₂	1	2	yes	maximal
2	#21460841	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11758187	C₂	1	2	yes	maximal
4	#21460732	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	C₄×#11758187	—	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	2	no	—
17	C₂×#11758187	—	2	1	no	—
18	C₂×#11758187	D₄	1	2	yes	—
19	C₄⋊#92418	—	2	2	no	—
20	C₄⋊#92418	C₂×C₄	1	2	yes	—
21	C₄×#92418	—	2	2	no	—
22	C₄×#92418	C₂×C₄	1	1	yes	characteristic
23	C₄×#92418	C₂³	1	1	yes	characteristic
24	C₂×Q₁₆×C₆₁	C₈	1	1	yes	characteristic
25	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
26	#107×C₆₁	C₈	1	1	yes	characteristic
27	#103×C₆₁	C₈	1	2	yes	—
28	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂×Q₈×C₆₁	C₂×C₈	1	2	yes	—
30	Q₁₆×C₆₁	C₁₆	1	4	yes	complemented
31	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
33	C₄⋊C₄×C₆₁	—	2	2	no	—
34	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
35	C₄²×C₆₁	—	2	2	no	—
36	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×#92418	—	2	2	no	—
38	C₂×#92418	—	2	1	no	—
39	C₂×#92418	C₂×D₄	1	1	yes	characteristic
40	C₂×#92418	#41	1	1	yes	characteristic
41	#11758187	—	2	2	no	—
42	#11758187	—	4	1	no	—
43	#11758187	—	4	2	no	—
44	Q₈×C₆₁	—	2	2	no	—
45	Q₈×C₆₁	C₂×C₁₆	1	4	yes	—
46	C₂×C₄×C₆₁	—	2	2	no	—
47	C₂×C₄×C₆₁	—	2	2	no	—
48	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
51	C₈×C₆₁	—	2	1	no	—
52	C₈×C₆₁	C₂×C₁₆	1	2	yes	—
53	#92418	—	2	1	no	—
54	#92418	—	4	1	no	—
55	#92418	—	4	2	no	—
56	#92418	#137	1	2	yes	—
57	#12059	—	61	1	no	maximal, Sylow
58	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
59	C₄×C₆₁	—	2	4	no	—
60	C₄×C₆₁	—	2	1	no	—
61	C₄×C₆₁	—	4	2	no	—
62	C₄×C₆₁	#441	1	1	yes	characteristic
63	C₄×C₆₁	#444	1	1	yes	characteristic
64	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
65	C₄×C₆₁	#504	1	1	yes	characteristic
66	Q₈×C₁₆	—	61	2	no	—
67	C₈×Q₁₆	—	61	1	no	—
68	#1367	—	61	1	no	—
69	Q₈⋊C₁₆	—	61	2	no	—
70	#1312	—	61	1	no	—
71	C₂×C₆₁	#1579	1	1	yes	characteristic
72	C₂×C₆₁	D₄×C₁₆	1	1	yes	characteristic
73	C₂×C₆₁	#2171	1	1	yes	characteristic
74	C₈×Q₈	—	61	2	no	—
75	C₄×Q₁₆	#6923	1	1	yes	characteristic
76	C₄⋊C₁₆	—	61	2	no	—
77	C₄⋊C₁₆	—	122	2	no	—
78	C₄×C₁₆	—	61	1	no	—
79	C₄×C₁₆	—	61	1	no	—
80	C₄×C₁₆	—	122	2	no	—
81	#335	—	61	1	no	—
82	Q₈⋊C₈	—	61	2	no	—
83	C₈²	—	61	1	no	—
84	C₆₁	#12059	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	#92420	1	1	yes	characteristic
86	C₄×Q₈	C₂×#6923	1	2	yes	—
87	C₂×C₁₆	—	61	2	no	—
88	C₂×C₁₆	—	122	1	no	—
89	C₂×C₁₆	—	122	2	no	—
90	#107	#92420	1	1	yes	characteristic
91	C₄⋊C₈	—	61	2	no	—
92	C₄⋊C₈	—	122	2	no	—
93	#103	#92420	1	2	yes	—
94	C₄×C₈	—	61	1	no	—
95	C₄×C₈	—	61	1	no	—
96	C₄×C₈	—	122	2	no	—
97	C₄×C₈	C₂×#6923	1	1	yes	characteristic
98	C₂×Q₈	C₂×#92420	1	2	yes	—
99	Q₁₆	#11758187	1	4	yes	complemented
100	C₂×C₈	—	61	1	no	—
101	C₂×C₈	—	61	1	no	—
102	C₂×C₈	—	122	2	no	—
103	C₂×C₈	—	122	1	no	—
104	C₂×C₈	C₂×#92420	1	1	yes	characteristic
105	C₂×C₈	C₂×#92420	1	1	yes	characteristic
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	C₂×#92420	1	2	yes	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂²×#6923	1	1	yes	characteristic
110	C₁₆	—	122	2	no	—
111	C₁₆	—	244	2	no	—
112	C₁₆	—	244	1	no	—
113	Q₈	—	2	2	no	—
114	Q₈	C₂×#11758187	1	4	yes	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	D₄×#6923	1	1	yes	characteristic
118	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
119	C₂×C₄	#14519864	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	61	2	no	—
122	C₈	—	122	1	no	—
123	C₈	—	244	1	no	—
124	C₈	—	244	2	no	—
125	C₈	C₂×#11758187	1	2	yes	—
126	C₂²	D₄×#92420	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	2	1	no	—
129	C₄	—	4	2	no	—
130	C₄	#16840802	1	1	yes	characteristic
131	C₄	#16840808	1	1	yes	characteristic
132	C₄	C₂²×#11758187	1	1	yes	characteristic
133	C₄	#16840820	1	1	yes	characteristic
134	C₂	#21461412	1	1	yes	characteristic
135	C₂	D₄×#11758187	1	1	yes	characteristic
136	C₂	#21462209	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11758187

C₂

yes

maximal

#21460841

C₂

yes

maximal, characteristic

Q₈⋊#11758187

C₂

yes

maximal

#21460732

C₂

yes

maximal, characteristic

Q₁₆×#92418

C₂

yes

maximal, characteristic

C₄⋊#11758187

—

C₄⋊#11758187

C₂²

yes

—

C₄×#11758187

—

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

—

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

C₁₆

yes

complemented

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

—

—

—

Q₈×C₆₁

—

Q₈×C₆₁

C₂×C₁₆

yes

—

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

C₂×C₁₆

yes

—

—

—

—

yes

—

#12059

—

maximal, Sylow

C₂²×C₆₁

C₈×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₂²×C₁₆

yes

characteristic

C₄×C₆₁

#504

yes

characteristic

Q₈×C₁₆

—

C₈×Q₁₆

—

#1367

—

Q₈⋊C₁₆

—

#1312

—

C₂×C₆₁

#1579

yes

characteristic

C₂×C₆₁

D₄×C₁₆

yes

characteristic

C₂×C₆₁

#2171

yes

characteristic

C₈×Q₈

—

C₄×Q₁₆

#6923

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

122

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

122

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₆₁

#12059

yes

characteristic, complemented, Sylow

C₂×Q₁₆

#92420

yes

characteristic

C₄×Q₈

C₂×#6923

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

—

122

—

#107

#92420

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

122

—

#103

#92420

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

122

—

C₄×C₈

C₂×#6923

yes

characteristic

C₂×Q₈

C₂×#92420

yes

—

Q₁₆

#11758187

yes

complemented

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

122

—

103

C₂×C₈

—

122

—

104

C₂×C₈

C₂×#92420

yes

characteristic

105

C₂×C₈

C₂×#92420

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×#92420

yes

—

108

C₄²

—

109

C₄²

C₂²×#6923

yes

characteristic

110

C₁₆

—

122

—

111

C₁₆

—

244

—

112

C₁₆

—

244

—

113

Q₈

—

114

Q₈

C₂×#11758187

yes

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

D₄×#6923

yes

characteristic

118

C₂×C₄

C₂²×#92420

yes

characteristic

119

C₂×C₄

#14519864

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

122

—

123

C₈

—

244

—

124

C₈

—

244

—

125

C₈

C₂×#11758187

yes

—

126

C₂²

D₄×#92420

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#16840802

yes

characteristic

131

C₄

#16840808

yes

characteristic

132

C₄

C₂²×#11758187

yes

characteristic

133

C₄

#16840820

yes

characteristic

134

C₂

#21461412

yes

characteristic

135

C₂

D₄×#11758187

yes

characteristic

136

C₂

#21462209

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6923
commutator	C₄×C₆₁	C₂²×C₁₆
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	#12059
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×#6923

commutator

C₄×C₆₁

C₂²×C₁₆

Frattini

C₄²

C₂²×#6923

Fitting

C₄×Q₁₆×C₆₁

C₄

socle, prime radical

C₂²×C₆₁

C₈×D₄

C₂³

C₄×Q₁₆

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
30. Q₁₆×C₆₁	C₁₆	semidirect
84. C₆₁	#12059	semidirect
99. Q₁₆	#11758187	semidirect

H ≅ G/N

type

30. Q₁₆×C₆₁

C₁₆

semidirect

84. C₆₁

#12059

semidirect

99. Q₁₆

#11758187

semidirect

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

subgroup	index
Q₈×#11758187	2
#21460841	2
Q₈⋊#11758187	2
#21460732	2
Q₁₆×#92418	2
#12059	61

subgroup

index

Q₁₆×#92418

Subgroups — #30605953

Conjugacy classes of subgroups

Complements

Maximal subgroups