Group #30607315 — subgroups

Subgroups — #30607315

Order 15616.

302 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 120 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	#21462151	C₂	1	2	yes	maximal
2	#21460863	C₂	1	4	yes	maximal
3	#21445506	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519851	C₂²	1	4	yes	—
5	C₄⋊#11758187	—	2	4	no	—
6	C₂×#14519688	C₄	1	2	yes	—
7	C₂×#14519688	C₄	1	1	yes	characteristic
8	C₂×C₄×#92418	C₂²	1	2	yes	—
9	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
10	C₆₁×#376	C₄	1	1	yes	characteristic
11	#14519851	—	2	8	no	—
12	C₂×#11758187	—	2	4	no	—
13	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
14	C₂²×#92418	C₂×C₄	1	2	yes	—
15	C₂²×#92418	C₂³	1	1	yes	characteristic
16	#14519688	—	2	4	no	—
17	#14519688	C₈	1	4	yes	—
18	C₄×#92418	—	2	4	no	—
19	C₄×#92418	D₄	1	2	yes	—
20	C₄×#92418	Q₈	1	2	yes	—
21	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
23	#95×C₆₁	—	2	2	no	—
24	C₂²×C₄×C₆₁	—	2	2	no	—
25	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
26	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
27	C₄²×C₆₁	—	2	2	no	—
28	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
29	C₄²×C₆₁	C₂×C₈	1	2	yes	—
30	C₄²×C₆₁	M₁₆	1	4	yes	—
31	C₂×#92418	—	2	2	no	—
32	C₂×#92418	—	2	2	no	—
33	C₂×#92418	—	2	2	no	—
34	C₂×#92418	C₄⋊C₄	1	2	yes	—
35	C₂×#92418	C₂×C₈	1	4	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	C₂×M₁₆	1	1	yes	characteristic
41	C₂×C₄×C₆₁	—	2	4	no	—
42	C₂×C₄×C₆₁	—	4	2	no	—
43	C₂×C₄×C₆₁	—	4	4	no	—
44	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
45	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	#117	1	2	yes	—
47	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
49	#92418	—	2	2	no	—
50	#92418	—	2	4	no	—
51	#92418	#124	1	2	yes	—
52	#92418	#125	1	2	yes	—
53	#12884	—	61	1	no	maximal, Sylow
54	C₂²×C₆₁	—	2	2	no	—
55	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
56	C₂²×C₆₁	#432	1	2	yes	—
57	C₄×C₆₁	—	2	4	no	—
58	C₄×C₆₁	—	4	4	no	—
59	C₄×C₆₁	#382	1	1	yes	characteristic
60	C₄×C₆₁	#423	1	1	yes	characteristic
61	C₄×C₆₁	#431	1	2	yes	—
62	#2150	—	61	2	no	—
63	#1726	—	61	1	no	—
64	#1376	—	61	4	no	—
65	C₂×C₆₁	—	2	2	no	—
66	C₂×C₆₁	#1831	1	1	yes	characteristic
67	C₂×C₆₁	#2153	1	2	yes	—
68	C₂×M₃₂	—	61	4	no	—
69	C₂×#97	—	61	1	no	—
70	C₂×#97	—	61	2	no	—
71	C₂×C₄×C₈	—	61	1	no	—
72	C₂×C₄×C₈	—	61	2	no	—
73	#376	#6923	1	1	yes	characteristic
74	C₄⋊C₁₆	—	122	4	no	—
75	C₆₁	#12884	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	—	61	1	no	—
77	C₂²×C₈	—	61	1	no	—
78	C₂²×C₈	—	61	2	no	—
79	C₂×C₄²	C₂×#6923	1	2	yes	—
80	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
81	M₃₂	—	122	8	no	—
82	C₂×C₁₆	—	122	4	no	—
83	#97	—	61	4	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	—	2	2	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	#11758217	1	2	yes	—
91	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
92	C₂×C₈	—	61	1	no	—
93	C₂×C₈	—	61	1	no	—
94	C₂×C₈	—	61	4	no	—
95	C₂×C₈	—	61	2	no	—
96	C₂×C₈	—	61	4	no	—
97	C₂×C₈	—	122	2	no	—
98	C₂×C₈	—	122	2	no	—
99	C₂×C₈	—	122	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	D₆₁⋊C₈	1	2	yes	—
102	C₄²	#11758213	1	4	yes	—
103	C₄²	C₄⋊#6923	1	2	yes	—
104	C₁₆	—	244	4	no	—
105	C₂³	C₂×#11758217	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	4	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×D₆₁⋊C₈	1	1	yes	characteristic
113	C₂×C₄	C₂×#11758213	1	2	yes	—
114	C₈	—	61	2	no	—
115	C₈	—	61	2	no	—
116	C₈	—	122	2	no	—
117	C₈	—	122	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	C₂×#14519844	1	1	yes	characteristic
120	C₂²	#16840783	1	2	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	#16840706	1	1	yes	characteristic
124	C₄	#16840766	1	1	yes	characteristic
125	C₄	#16840781	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	#21461736	1	1	yes	characteristic
128	C₂	#21462162	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21462151

C₂

yes

maximal

#21460863

C₂

yes

maximal

#21445506

C₂

yes

maximal, characteristic

C₂×#14519851

C₂²

yes

—

C₄⋊#11758187

—

C₂×#14519688

C₄

yes

—

C₂×#14519688

C₄

yes

characteristic

C₂×C₄×#92418

C₂²

yes

—

C₂×C₄×#92418

C₂²

yes

characteristic

C₆₁×#376

C₄

yes

characteristic

#14519851

—

C₂×#11758187

—

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂×C₄

yes

—

C₂²×#92418

C₂³

yes

characteristic

#14519688

—

#14519688

C₈

yes

—

C₄×#92418

—

C₄×#92418

D₄

yes

—

C₄×#92418

Q₈

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

M₁₆

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

—

C₄²×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂×C₈

yes

—

C₄²×C₆₁

M₁₆

yes

—

—

—

—

C₄⋊C₄

yes

—

C₂×#92418

C₂×C₈

yes

—

C₂×#92418

C₂×D₄

yes

characteristic

C₂×#92418

C₂×Q₈

yes

characteristic

C₂×#92418

#41

yes

—

#11758187

—

C₂³×C₆₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

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

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

—

—

yes

—

#92418

#125

yes

—

#12884

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₆₁

#432

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

#2150

—

#1726

—

#1376

—

C₂×C₆₁

—

C₂×C₆₁

#1831

yes

characteristic

C₂×C₆₁

#2153

yes

—

C₂×M₃₂

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

#6923

yes

characteristic

C₄⋊C₁₆

—

122

—

C₆₁

#12884

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×#6923

yes

—

C₂×C₄²

C₂×#6923

yes

characteristic

M₃₂

—

122

—

C₂×C₁₆

—

122

—

#97

—

#97

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

122

—

#95

—

C₂²×C₄

—

C₂²×C₄

#11758217

yes

—

C₂²×C₄

C₂²×#6923

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

100

C₄²

—

101

C₄²

D₆₁⋊C₈

yes

—

102

C₄²

#11758213

yes

—

103

C₄²

C₄⋊#6923

yes

—

104

C₁₆

—

244

—

105

C₂³

C₂×#11758217

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#14519844

yes

—

110

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

111

C₂×C₄

#14519856

yes

—

112

C₂×C₄

C₂×D₆₁⋊C₈

yes

characteristic

113

C₂×C₄

C₂×#11758213

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

122

—

117

C₈

—

122

—

118

C₂²

—

119

C₂²

C₂×#14519844

yes

characteristic

120

C₂²

#16840783

yes

—

121

C₄

—

122

C₄

—

123

C₄

#16840706

yes

characteristic

124

C₄

#16840766

yes

characteristic

125

C₄

#16840781

yes

—

126

C₂

—

127

C₂

#21461736

yes

characteristic

128

C₂

#21462162

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₆₁×#376	C₄
socle	C₂²×C₆₁	C₂×C₄⋊C₈
prime radical	C₂³×C₆₁	C₂×M₁₆
cosocle	C₂²×#92418	C₂³
hypercenter	#376	#6923
hypocenter	C₆₁	#12884
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6923

Fitting

C₆₁×#376

C₄

socle

C₂²×C₆₁

C₂×C₄⋊C₈

prime radical

C₂³×C₆₁

C₂×M₁₆

cosocle

C₂²×#92418

C₂³

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
75. C₆₁	#12884	semidirect

H ≅ G/N

type

75. C₆₁

#12884

semidirect

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

subgroup	index
#21462151	2
#21460863	2
#21445506	2
#12884	61

subgroup

index

Subgroups — #30607315

Conjugacy classes of subgroups

Complements

Maximal subgroups