Group #30606095 — subgroups

Subgroups — #30606095

Order 15616.

198 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 62 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	#21461466	C₂	1	2	yes	maximal
2	#21460843	C₂	1	1	yes	maximal, characteristic
3	#21460779	C₂	1	2	yes	maximal
4	#21460735	C₂	1	1	yes	maximal, characteristic
5	#21447690	C₂	1	1	yes	maximal, characteristic
6	#16840813	—	2	2	no	—
7	#16840651	C₂²	1	1	yes	characteristic
8	#16840623	C₂²	1	2	yes	—
9	Q₈⋊#92420	—	2	2	no	—
10	#16840603	C₂²	1	1	yes	characteristic
11	Q₁₆×Dic₆₁	C₂²	1	2	yes	—
12	#16839590	C₄	1	1	yes	characteristic
13	#16839566	C₄	1	2	yes	—
14	#16839537	C₂²	1	1	yes	characteristic
15	C₂×Q₃₂×C₆₁	C₄	1	1	yes	characteristic
16	C₂×#11758187	—	2	1	no	—
17	#14519844	—	4	2	no	—
18	C₄⋊#92420	—	2	2	no	—
19	#14519832	—	4	2	no	—
20	C₄×#92420	D₄	1	2	yes	—
21	Q₈×Dic₆₁	—	2	2	no	—
22	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
23	#14519711	C₂×C₄	1	2	yes	—
24	#14519701	—	2	2	no	—
25	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
26	C₂×Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
27	Q₃₂×C₆₁	—	2	2	no	—
28	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂×Q₈×C₆₁	—	2	2	no	—
30	C₂×#92420	—	2	2	no	—
31	C₂×#92420	—	4	2	no	—
32	Q₁₆×C₆₁	—	2	2	no	—
33	Q₁₆×C₆₁	—	2	2	no	—
34	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	C₄⋊Dic₆₁	—	2	2	no	—
36	C₄⋊Dic₆₁	—	4	2	no	—
37	C₄×Dic₆₁	—	4	2	no	—
38	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂×#92418	#41	1	1	yes	characteristic
40	#11758187	—	4	1	no	—
41	C₁₆×C₆₁	M₁₆	1	2	yes	—
42	#11758185	—	2	1	no	—
43	Q₈×C₆₁	—	4	2	no	—
44	Q₈×C₆₁	—	4	2	no	—
45	C₂×C₄×C₆₁	—	4	2	no	—
46	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
47	C₂×Dic₆₁	—	4	2	no	—
48	C₂×Dic₆₁	AGL(1,ℤ/8)	1	2	yes	—
49	#92420	—	4	2	no	—
50	#92420	—	8	2	no	—
51	C₈×C₆₁	C₂×M₁₆	1	1	yes	characteristic
52	C₈×C₆₁	#131	1	1	yes	characteristic
53	#92418	—	2	1	no	—
54	#12088	—	61	1	no	maximal, Sylow
55	C₂²×C₆₁	#442	1	1	yes	characteristic
56	C₄×C₆₁	—	8	2	no	—
57	C₄×C₆₁	#436	1	1	yes	characteristic
58	C₄×C₆₁	#444	1	1	yes	characteristic
59	Dic₆₁	—	2	2	no	—
60	Dic₆₁	—	8	2	no	—
61	#2176	—	61	1	no	—
62	#1593	—	61	2	no	—
63	#1368	—	61	1	no	—
64	#1334	—	61	2	no	—
65	#1313	—	61	1	no	—
66	C₂×C₆₁	#1594	1	1	yes	characteristic
67	C₂×C₆₁	#2180	1	2	yes	—
68	C₂×Q₃₂	#6923	1	1	yes	characteristic
69	#446	—	122	2	no	—
70	C₄×Q₁₆	—	61	2	no	—
71	#367	—	61	1	no	—
72	#358	—	61	2	no	—
73	#346	—	61	1	no	—
74	#346	—	61	1	no	—
75	#334	—	61	2	no	—
76	Q₈⋊C₈	—	122	2	no	—
77	#322	—	61	1	no	—
78	C₆₁	#12088	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	C₂×#6923	1	2	yes	—
80	C₄×Q₈	—	122	2	no	—
81	Q₃₂	—	2	2	no	—
82	C₂×C₁₆	—	61	1	no	—
83	C₂×C₁₆	—	122	1	no	—
84	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
85	#107	—	61	2	no	—
86	C₄⋊C₈	—	122	2	no	—
87	C₄⋊C₈	—	244	2	no	—
88	#103	—	122	2	no	—
89	#97	—	244	2	no	—
90	C₄×C₈	—	61	1	no	—
91	C₄×C₈	—	61	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	—	2	2	no	—
95	C₂×C₈	—	61	1	no	—
96	C₂×C₈	—	122	2	no	—
97	C₂×C₈	—	244	2	no	—
98	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
99	C₄⋊C₄	—	122	2	no	—
100	C₄⋊C₄	—	244	2	no	—
101	C₄²	—	61	1	no	—
102	C₄²	—	244	2	no	—
103	C₁₆	—	122	1	no	—
104	C₁₆	—	244	1	no	—
105	C₁₆	#11758217	1	2	yes	—
106	Q₈	—	4	2	no	—
107	Q₈	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	61	2	no	—
110	C₂×C₄	—	244	2	no	—
111	C₂×C₄	D₄×#6923	1	1	yes	characteristic
112	C₈	—	122	1	no	—
113	C₈	—	488	2	no	—
114	C₈	—	244	2	no	—
115	C₈	C₂×#11758217	1	1	yes	characteristic
116	C₈	#14519864	1	1	yes	characteristic
117	C₂²	#16840803	1	1	yes	characteristic
118	C₄	—	8	2	no	—
119	C₄	—	122	2	no	—
120	C₄	—	488	2	no	—
121	C₄	#16840797	1	1	yes	characteristic
122	C₄	#16840809	1	1	yes	characteristic
123	C₂	#21461469	1	1	yes	characteristic
124	C₂	#21462226	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21461466

C₂

yes

maximal

#21460843

C₂

yes

maximal, characteristic

#21460779

C₂

yes

maximal

#21460735

C₂

yes

maximal, characteristic

#21447690

C₂

yes

maximal, characteristic

#16840813

—

#16840651

C₂²

yes

characteristic

#16840623

C₂²

yes

—

Q₈⋊#92420

—

#16840603

C₂²

yes

characteristic

Q₁₆×Dic₆₁

C₂²

yes

—

#16839590

C₄

yes

characteristic

#16839566

C₄

yes

—

#16839537

C₂²

yes

characteristic

C₂×Q₃₂×C₆₁

C₄

yes

—

—

—

—

D₄

yes

—

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

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×#92420

—

C₂×#92420

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×D₄

yes

yes

—

C₁₆×C₆₁

M₁₆

yes

—

#11758185

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

AGL(1,ℤ/8)

yes

—

#92420

—

#92420

—

C₈×C₆₁

C₂×M₁₆

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

—

—

—

—

—

C₂×C₆₁

#1594

yes

characteristic

C₂×C₆₁

#2180

yes

—

C₂×Q₃₂

#6923

yes

characteristic

#446

—

122

—

C₄×Q₁₆

—

—

—

—

—

—

Q₈⋊C₈

—

122

—

#322

—

C₆₁

#12088

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×#6923

yes

—

C₄×Q₈

—

122

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

C₂×#6923

yes

characteristic

#107

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

244

—

#103

—

122

—

#97

—

244

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

C₂²×#6923

yes

characteristic

C₄⋊C₄

—

122

—

100

C₄⋊C₄

—

244

—

101

C₄²

—

102

C₄²

—

244

—

103

C₁₆

—

122

—

104

C₁₆

—

244

—

105

C₁₆

#11758217

yes

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

244

—

111

C₂×C₄

D₄×#6923

yes

characteristic

112

C₈

—

122

—

113

C₈

—

488

—

114

C₈

—

244

—

115

C₈

C₂×#11758217

yes

characteristic

116

C₈

#14519864

yes

characteristic

117

C₂²

#16840803

yes

characteristic

118

C₄

—

119

C₄

—

122

—

120

C₄

—

488

—

121

C₄

#16840797

yes

characteristic

122

C₄

#16840809

yes

characteristic

123

C₂

#21461469

yes

characteristic

124

C₂

#21462226

yes

—

	subgroup	G/H
center	C₂²	#16840803
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₂×Q₃₂×C₆₁	C₄
socle, prime radical	C₂²×C₆₁	#442
cosocle	C₈×Dic₆₁	C₂³
hypercenter	C₂×Q₃₂	#6923
hypocenter	C₆₁	#12088
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

#16840803

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×#6923

Fitting

C₂×Q₃₂×C₆₁

C₄

socle, prime radical

C₂²×C₆₁

#442

cosocle

C₈×Dic₆₁

C₂³

C₂×Q₃₂

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
78. C₆₁	#12088	semidirect

H ≅ G/N

type

78. C₆₁

#12088

semidirect

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

subgroup	index
#21461466	2
#21460843	2
#21460779	2
#21460735	2
#21447690	2
#12088	61

subgroup

index

Subgroups — #30606095

Conjugacy classes of subgroups

Complements

Maximal subgroups