Group #30607301 — subgroups

Subgroups — #30607301

Order 15616.

322 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 136 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	C₂×C₄⋊#11758187	C₂	1	2	yes	maximal
2	#21460858	C₂	1	4	yes	maximal
3	#21445507	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11758187	C₂²	1	4	yes	—
5	C₄⋊#11758187	—	2	4	no	—
6	C₂×#14519687	C₄	1	2	yes	—
7	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
9	#16839608	C₄	1	1	yes	characteristic
10	C₂×#97×C₆₁	C₄	1	1	yes	characteristic
11	C₂×#11758187	—	2	4	no	—
12	C₂×#11758187	—	2	8	no	—
13	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
14	C₂²×#92418	C₂³	1	1	yes	characteristic
15	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
16	#14519687	—	2	4	no	—
17	C₄×#92418	D₄	1	2	yes	—
18	C₄×#92418	Q₈	1	2	yes	—
19	C₈×Dic₆₁	—	2	4	no	—
20	#14519684	C₈	1	4	yes	—
21	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
23	#97×C₆₁	—	2	2	no	—
24	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
25	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
26	C₂²×Dic₆₁	C₂×C₈	1	4	yes	—
27	C₂×C₈×C₆₁	—	2	2	no	—
28	C₂×C₈×C₆₁	—	2	2	no	—
29	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
30	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
31	C₄²×C₆₁	M₁₆	1	2	yes	—
32	C₄×Dic₆₁	—	2	2	no	—
33	C₄×Dic₆₁	—	2	2	no	—
34	C₂×#92418	—	2	2	no	—
35	C₂×#92418	C₄⋊C₄	1	2	yes	—
36	C₂×#92418	C₂×D₄	1	1	yes	characteristic
37	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
38	C₂×#92418	#41	1	4	yes	—
39	#11758187	—	4	4	no	—
40	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
41	C₂×C₄×C₆₁	—	2	4	no	—
42	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
43	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	#117	1	2	yes	—
45	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
46	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
47	C₂×Dic₆₁	—	2	4	no	—
48	C₂×Dic₆₁	—	2	8	no	—
49	C₈×C₆₁	—	2	4	no	—
50	#92418	—	2	2	no	—
51	#92418	#124	1	2	yes	—
52	#92418	#125	1	2	yes	—
53	#12880	—	61	1	no	maximal, Sylow
54	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
55	C₂²×C₆₁	#431	1	2	yes	—
56	C₂²×C₆₁	#504	1	4	yes	—
57	C₄×C₆₁	—	2	4	no	—
58	C₄×C₆₁	#382	1	1	yes	characteristic
59	C₄×C₆₁	#423	1	1	yes	characteristic
60	C₄×C₆₁	#432	1	2	yes	—
61	Dic₆₁	—	4	4	no	—
62	C₂×C₄⋊C₁₆	—	61	2	no	—
63	#1726	—	61	1	no	—
64	#1374	—	61	4	no	—
65	C₂×C₆₁	#1831	1	1	yes	characteristic
66	C₂×C₆₁	#2151	1	2	yes	—
67	C₂×C₆₁	#2164	1	4	yes	—
68	C₂²×C₁₆	—	61	4	no	—
69	C₂×#97	—	61	2	no	—
70	C₂×#97	#6923	1	1	yes	characteristic
71	C₂×C₄×C₈	—	61	1	no	—
72	C₂×C₄×C₈	—	61	2	no	—
73	#376	—	61	1	no	—
74	C₄⋊C₁₆	—	122	4	no	—
75	C₆₁	#12880	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	—	61	1	no	—
77	C₂²×C₈	—	61	1	no	—
78	C₂²×C₈	C₂×#6923	1	2	yes	—
79	C₂×C₄²	—	61	2	no	—
80	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
81	C₂×C₁₆	—	122	4	no	—
82	C₂×C₁₆	—	122	8	no	—
83	#97	—	2	2	no	—
84	#97	—	122	4	no	—
85	C₄×C₈	—	61	2	no	—
86	C₄×C₈	—	61	2	no	—
87	C₄×C₈	—	122	4	no	—
88	#95	—	61	4	no	—
89	C₂²×C₄	—	61	4	no	—
90	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
91	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	61	1	no	—
95	C₂×C₈	—	61	1	no	—
96	C₂×C₈	—	61	4	no	—
97	C₂×C₈	—	61	2	no	—
98	C₂×C₈	—	122	2	no	—
99	C₂×C₈	#11758213	1	4	yes	—
100	C₄²	—	122	2	no	—
101	C₄²	—	122	2	no	—
102	C₄²	C₄⋊#6923	1	2	yes	—
103	C₄²	#11758217	1	2	yes	—
104	C₁₆	—	244	4	no	—
105	C₂³	C₂×D₆₁⋊C₈	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	122	4	no	—
108	C₂×C₄	—	122	8	no	—
109	C₂×C₄	#14519844	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
111	C₂×C₄	#14519856	1	2	yes	—
112	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
113	C₂×C₄	C₂×#11758213	1	2	yes	—
114	C₈	—	2	4	no	—
115	C₈	—	61	2	no	—
116	C₈	—	61	2	no	—
117	C₈	—	122	2	no	—
118	C₂²	C₂×#14519844	1	1	yes	characteristic
119	C₂²	#16840781	1	2	yes	—
120	C₂²	#16840821	1	4	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	244	4	no	—
123	C₄	#16840706	1	1	yes	characteristic
124	C₄	#16840766	1	1	yes	characteristic
125	C₄	#16840783	1	2	yes	—
126	C₂	#21461736	1	1	yes	characteristic
127	C₂	#21462156	1	2	yes	—
128	C₂	#21462187	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11758187

C₂

yes

maximal

#21460858

C₂

yes

maximal

#21445507

C₂

yes

maximal, characteristic

C₂²×#11758187

C₂²

yes

—

C₄⋊#11758187

—

C₂×#14519687

C₄

yes

—

C₂×C₄×#92418

C₂²

yes

characteristic

C₂×C₈×Dic₆₁

C₂²

yes

—

#16839608

C₄

yes

characteristic

C₂×#97×C₆₁

C₄

yes

characteristic

C₂×#11758187

—

C₂×#11758187

—

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂³

yes

characteristic

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

#14519687

—

C₄×#92418

D₄

yes

—

C₄×#92418

Q₈

yes

—

C₈×Dic₆₁

—

#14519684

C₈

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

#97×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₈

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

M₁₆

yes

—

C₄²×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

M₁₆

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

C₄⋊C₄

yes

—

C₂×#92418

C₂×D₄

yes

characteristic

C₂×#92418

C₂×Q₈

yes

characteristic

C₂×#92418

#41

yes

—

#11758187

—

C₂³×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₈

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

#92418

—

#92418

#124

yes

—

#92418

#125

yes

—

#12880

—

maximal, Sylow

C₂²×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₆₁

#431

yes

—

C₂²×C₆₁

#504

yes

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

C₂×C₄⋊C₁₆

—

#1726

—

#1374

—

C₂×C₆₁

#1831

yes

characteristic

C₂×C₆₁

#2151

yes

—

C₂×C₆₁

#2164

yes

—

C₂²×C₁₆

—

C₂×#97

—

C₂×#97

#6923

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

—

C₄⋊C₁₆

—

122

—

C₆₁

#12880

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#6923

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×#6923

yes

characteristic

C₂×C₁₆

—

122

—

C₂×C₁₆

—

122

—

#97

—

#97

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

122

—

#95

—

C₂²×C₄

—

C₂²×C₄

D₆₁⋊C₈

yes

—

C₂²×C₄

C₂²×#6923

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

#11758213

yes

—

100

C₄²

—

122

—

101

C₄²

—

122

—

102

C₄²

C₄⋊#6923

yes

—

103

C₄²

#11758217

yes

—

104

C₁₆

—

244

—

105

C₂³

C₂×D₆₁⋊C₈

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

122

—

108

C₂×C₄

—

122

—

109

C₂×C₄

#14519844

yes

—

110

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

111

C₂×C₄

#14519856

yes

—

112

C₂×C₄

C₂×#11758217

yes

characteristic

113

C₂×C₄

C₂×#11758213

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

122

—

118

C₂²

C₂×#14519844

yes

characteristic

119

C₂²

#16840781

yes

—

120

C₂²

#16840821

yes

—

121

C₄

—

122

C₄

—

244

—

123

C₄

#16840706

yes

characteristic

124

C₄

#16840766

yes

characteristic

125

C₄

#16840783

yes

—

126

C₂

#21461736

yes

characteristic

127

C₂

#21462156

yes

—

128

C₂

#21462187

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator, socle, prime radical	C₂³×C₆₁	C₂²×C₈
Fitting	C₂×#97×C₆₁	C₄
cosocle	C₂²×#92418	C₂³
hypercenter	C₂×#97	#6923
hypocenter	C₆₁	#12880
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×#6923

commutator, socle, prime radical

C₂³×C₆₁

C₂²×C₈

Fitting

C₂×#97×C₆₁

C₄

cosocle

C₂²×#92418

C₂³

C₂×#97

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂²×C₄ ◁ C₂×#97

N	H ≅ G/N	type
75. C₆₁	#12880	semidirect

H ≅ G/N

type

75. C₆₁

#12880

semidirect

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

subgroup	index
C₂×C₄⋊#11758187	2
#21460858	2
#21445507	2
#12880	61

subgroup

index

Subgroups — #30607301

Conjugacy classes of subgroups

Complements

Maximal subgroups