Group #30605899 — subgroups

Subgroups — #30605899

Order 15616.

222 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 82 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	Q₁₆×#92420	C₂	1	2	yes	maximal
2	#21460852	C₂	1	1	yes	maximal, characteristic
3	#21460775	C₂	1	2	yes	maximal
4	#21460730	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	Q₈×#92420	—	2	2	no	—
7	#16840648	C₂²	1	1	yes	characteristic
8	#16840625	C₂²	1	2	yes	—
9	Q₈⋊#92420	—	2	2	no	—
10	C₈×#92420	C₂²	1	1	yes	characteristic
11	Q₁₆×Dic₆₁	C₂²	1	2	yes	—
12	#16839588	C₄	1	1	yes	characteristic
13	#16839566	C₄	1	2	yes	—
14	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₂×Q₃₂×C₆₁	C₄	1	1	yes	characteristic
16	C₂×#11758187	—	2	1	no	—
17	C₄⋊#92420	—	2	2	no	—
18	C₄⋊#92420	—	4	2	no	—
19	C₄×#92420	—	4	2	no	—
20	C₄×#92420	D₄	1	1	yes	characteristic
21	C₄×#92420	D₄	1	1	yes	characteristic
22	Q₈×Dic₆₁	—	2	2	no	—
23	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
24	#14519711	C₂×C₄	1	2	yes	—
25	#14519701	—	2	2	no	—
26	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
27	C₂×Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
28	Q₃₂×C₆₁	C₈	1	4	yes	complemented
29	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×Q₈×C₆₁	—	2	2	no	—
31	C₂×#92420	—	2	1	no	—
32	C₂×#92420	—	4	2	no	—
33	C₂×#92420	D₈	1	2	yes	—
34	Q₁₆×C₆₁	—	2	2	no	—
35	Q₁₆×C₆₁	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊Dic₆₁	—	2	2	no	—
38	C₄⋊Dic₆₁	—	4	2	no	—
39	C₄×Dic₆₁	—	4	2	no	—
40	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂×#92418	#41	1	1	yes	characteristic
42	#11758187	—	2	2	no	—
43	C₁₆×C₆₁	C₂×C₈	1	2	yes	—
44	#11758185	—	2	1	no	—
45	Q₈×C₆₁	—	2	4	no	—
46	Q₈×C₆₁	—	4	2	no	—
47	C₂×C₄×C₆₁	—	4	2	no	—
48	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
49	C₂×Dic₆₁	—	4	2	no	—
50	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
51	C₂×Dic₆₁	#135	1	1	yes	characteristic
52	#92420	—	2	2	no	—
53	#92420	—	4	1	no	—
54	#92420	—	8	2	no	—
55	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₆₁	#131	1	1	yes	characteristic
57	#92418	—	2	1	no	—
58	#12044	—	61	1	no	maximal, Sylow
59	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
60	C₄×C₆₁	—	4	4	no	—
61	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
62	C₄×C₆₁	#443	1	1	yes	characteristic
63	Dic₆₁	—	2	1	no	—
64	Dic₆₁	—	8	2	no	—
65	Dic₆₁	#510	1	2	yes	—
66	C₄×Q₃₂	—	61	1	no	—
67	C₈×Q₁₆	—	61	2	no	—
68	#1372	—	61	1	no	—
69	#1332	—	61	2	no	—
70	#1311	—	61	1	no	—
71	C₂×C₆₁	C₈×D₈	1	1	yes	characteristic
72	C₂×C₆₁	#2176	1	1	yes	characteristic
73	C₂×C₆₁	#2179	1	1	yes	characteristic
74	C₂×Q₃₂	#6923	1	1	yes	characteristic
75	C₈×Q₈	—	122	2	no	—
76	C₄×Q₁₆	—	61	2	no	—
77	#366	—	61	1	no	—
78	#358	—	61	2	no	—
79	C₄×C₁₆	—	61	1	no	—
80	C₄×C₁₆	—	61	1	no	—
81	#335	—	61	2	no	—
82	Q₈⋊C₈	—	122	2	no	—
83	C₈²	—	61	1	no	—
84	C₆₁	#12044	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	C₂×#6923	1	2	yes	—
86	C₄×Q₈	—	122	2	no	—
87	Q₃₂	#92420	1	4	yes	complemented
88	C₂×C₁₆	—	61	1	no	—
89	C₂×C₁₆	—	122	1	no	—
90	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
91	#107	—	61	2	no	—
92	C₄⋊C₈	—	122	2	no	—
93	C₄⋊C₈	—	244	2	no	—
94	#103	—	122	2	no	—
95	C₄×C₈	—	61	1	no	—
96	C₄×C₈	—	61	1	no	—
97	C₄×C₈	—	61	1	no	—
98	C₄×C₈	—	244	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	Q₁₆	—	2	2	no	—
101	Q₁₆	C₂×#92420	1	4	yes	—
102	C₂×C₈	—	61	2	no	—
103	C₂×C₈	—	61	1	no	—
104	C₂×C₈	—	122	1	no	—
105	C₂×C₈	—	244	2	no	—
106	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
107	C₄⋊C₄	—	122	2	no	—
108	C₄⋊C₄	—	244	2	no	—
109	C₄²	—	61	1	no	—
110	C₄²	—	244	2	no	—
111	C₁₆	—	122	2	no	—
112	C₁₆	—	122	1	no	—
113	C₁₆	C₂×#92420	1	2	yes	—
114	Q₈	—	2	4	no	—
115	Q₈	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	61	1	no	—
118	C₂×C₄	—	61	1	no	—
119	C₂×C₄	—	244	2	no	—
120	C₂×C₄	D₄×#6923	1	1	yes	characteristic
121	C₈	—	122	2	no	—
122	C₈	—	122	1	no	—
123	C₈	—	488	2	no	—
124	C₈	—	244	1	no	—
125	C₈	C₂²×#92420	1	1	yes	characteristic
126	C₈	#14519864	1	1	yes	characteristic
127	C₂²	D₈×#6923	1	1	yes	characteristic
128	C₄	—	4	4	no	—
129	C₄	—	61	2	no	—
130	C₄	—	122	1	no	—
131	C₄	—	488	2	no	—
132	C₄	D₄×#92420	1	1	yes	characteristic
133	C₄	#16840804	1	1	yes	characteristic
134	C₂	D₈×#92420	1	1	yes	characteristic
135	C₂	#21462216	1	1	yes	characteristic
136	C₂	#21462222	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₁₆×#92420

C₂

yes

maximal

#21460852

C₂

yes

maximal, characteristic

#21460775

C₂

yes

maximal

#21460730

C₂

yes

maximal, characteristic

Q₃₂×Dic₆₁

C₂

yes

maximal, characteristic

Q₈×#92420

—

#16840648

C₂²

yes

characteristic

#16840625

C₂²

yes

—

Q₈⋊#92420

—

C₈×#92420

C₂²

yes

characteristic

Q₁₆×Dic₆₁

C₂²

yes

—

#16839588

C₄

yes

characteristic

#16839566

C₄

yes

—

C₁₆×Dic₆₁

C₂²

yes

characteristic

C₂×Q₃₂×C₆₁

C₄

yes

—

—

—

—

D₄

yes

characteristic

C₄×#92420

D₄

yes

characteristic

Q₈×Dic₆₁

—

C₂×#11758185

C₂×C₄

yes

characteristic

#14519711

C₂×C₄

yes

—

#14519701

—

C₈×Dic₆₁

C₂³

yes

characteristic

C₂×Q₁₆×C₆₁

C₂×C₄

yes

—

Q₃₂×C₆₁

C₈

yes

complemented

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×#92420

—

C₂×#92420

—

C₂×#92420

D₈

yes

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×D₄

yes

yes

—

C₁₆×C₆₁

C₂×C₈

yes

—

#11758185

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₈

yes

C₂×Dic₆₁

yes

—

—

—

C₈×C₆₁

C₂²×C₈

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

C₄×D₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₈×D₄

yes

characteristic

C₄×C₆₁

#443

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#510

yes

—

C₄×Q₃₂

—

C₈×Q₁₆

—

#1372

—

#1332

—

#1311

—

C₂×C₆₁

C₈×D₈

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×Q₃₂

yes

C₈×Q₈

—

122

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

122

—

C₈²

—

C₆₁

#12044

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×#6923

yes

—

C₄×Q₈

—

122

—

Q₃₂

#92420

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

C₂×#6923

yes

characteristic

#107

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

244

—

#103

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

244

—

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

C₂×#92420

yes

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

122

—

105

C₂×C₈

—

244

—

106

C₂×C₈

C₂²×#6923

yes

characteristic

107

C₄⋊C₄

—

122

—

108

C₄⋊C₄

—

244

—

109

C₄²

—

110

C₄²

—

244

—

111

C₁₆

—

122

—

112

C₁₆

—

122

—

113

C₁₆

C₂×#92420

yes

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

244

—

120

C₂×C₄

D₄×#6923

yes

characteristic

121

C₈

—

122

—

122

C₈

—

122

—

123

C₈

—

488

—

124

C₈

—

244

—

125

C₈

C₂²×#92420

yes

characteristic

126

C₈

#14519864

yes

characteristic

127

C₂²

D₈×#6923

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

122

—

131

C₄

—

488

—

132

C₄

D₄×#92420

yes

characteristic

133

C₄

#16840804

yes

characteristic

134

C₂

D₈×#92420

yes

characteristic

135

C₂

#21462216

yes

characteristic

136

C₂

#21462222

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×#6923
commutator	C₈×C₆₁	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₂×Q₃₂×C₆₁	C₄
socle, prime radical	C₂²×C₆₁	C₄×D₈
cosocle	C₈×Dic₆₁	C₂³
hypercenter	C₂×Q₃₂	#6923
hypocenter	C₆₁	#12044
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×#6923

commutator

C₈×C₆₁

C₂²×C₈

Frattini

C₂×C₈

C₂²×#6923

Fitting

C₂×Q₃₂×C₆₁

C₄

socle, prime radical

C₂²×C₆₁

C₄×D₈

cosocle

C₈×Dic₆₁

C₂³

C₂×Q₃₂

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
28. Q₃₂×C₆₁	C₈	semidirect
84. C₆₁	#12044	semidirect
87. Q₃₂	#92420	semidirect

H ≅ G/N

type

28. Q₃₂×C₆₁

C₈

semidirect

84. C₆₁

#12044

semidirect

87. Q₃₂

#92420

semidirect

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

subgroup	index
Q₁₆×#92420	2
#21460852	2
#21460775	2
#21460730	2
Q₃₂×Dic₆₁	2
#12044	61

subgroup

index

Q₁₆×#92420

Q₃₂×Dic₆₁

Subgroups — #30605899

Conjugacy classes of subgroups

Complements

Maximal subgroups