Group #30594837 — subgroups

Subgroups — #30594837

Order 15616.

204 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 40 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	#21460759	C₂	1	2	yes	maximal
2	#21446639	C₂	1	1	yes	maximal, characteristic
3	#16840607	—	2	2	no	—
4	C₂×#14519777	C₄	1	1	yes	characteristic
5	C₂×#14519702	—	2	2	no	—
6	#16839628	C₂²	1	1	yes	characteristic
7	C₂×#97×C₆₁	C₄	1	1	yes	characteristic
8	C₂²⋊#92420	—	4	2	no	—
9	C₂²×Dic₁₂₂	—	2	1	no	—
10	#14519777	—	2	1	no	—
11	#14519777	—	4	1	no	—
12	#14519777	C₈	1	2	yes	complemented
13	C₂×C₄⋊Dic₆₁	—	4	1	no	—
14	C₂×C₄⋊Dic₆₁	D₄	1	2	yes	—
15	#14519702	—	4	4	no	—
16	#14519684	—	4	1	no	—
17	C₂²×C₈×C₆₁	—	2	1	no	—
18	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
19	#97×C₆₁	—	4	1	no	—
20	C₂²×C₄×C₆₁	—	2	1	no	—
21	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₂²×Dic₆₁	—	2	2	no	—
23	C₂²×Dic₆₁	—	4	1	no	—
24	C₂×Dic₁₂₂	—	2	2	no	—
25	C₂×Dic₁₂₂	—	4	2	no	—
26	C₂×Dic₁₂₂	—	4	2	no	—
27	C₂×Dic₁₂₂	—	4	1	no	—
28	C₂×#92420	—	8	2	no	—
29	C₂×C₈×C₆₁	—	2	2	no	—
30	C₂×C₈×C₆₁	—	4	1	no	—
31	C₂×C₈×C₆₁	—	4	1	no	—
32	C₄²×C₆₁	—	2	1	no	—
33	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
34	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
35	C₄⋊Dic₆₁	—	4	2	no	—
36	C₄⋊Dic₆₁	—	4	2	no	—
37	C₄⋊Dic₆₁	—	4	2	no	—
38	C₂³×C₆₁	#99	1	1	yes	characteristic
39	C₂×C₄×C₆₁	—	2	1	no	—
40	C₂×C₄×C₆₁	—	2	1	no	—
41	C₂×C₄×C₆₁	—	2	1	no	—
42	C₂×C₄×C₆₁	—	2	1	no	—
43	C₂×C₄×C₆₁	—	4	2	no	—
44	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₆₁	#100	1	1	yes	characteristic
46	C₂×Dic₆₁	—	4	2	no	—
47	C₂×Dic₆₁	—	4	2	no	—
48	C₂×Dic₆₁	—	8	1	no	—
49	C₂×Dic₆₁	—	8	1	no	—
50	C₂×Dic₆₁	—	8	2	no	—
51	Dic₁₂₂	—	4	4	no	—
52	Dic₁₂₂	—	8	2	no	—
53	#92420	—	16	2	no	—
54	C₈×C₆₁	—	4	2	no	—
55	#7353	—	61	1	no	maximal, Sylow
56	C₂²×C₆₁	—	2	2	no	—
57	C₂²×C₆₁	#323	1	1	yes	characteristic
58	C₂²×C₆₁	#353	1	1	yes	characteristic
59	C₂²×C₆₁	#355	1	1	yes	characteristic
60	C₄×C₆₁	—	2	1	no	—
61	C₄×C₆₁	—	2	1	no	—
62	C₄×C₆₁	—	4	1	no	—
63	C₄×C₆₁	—	4	1	no	—
64	Dic₆₁	—	8	2	no	—
65	Dic₆₁	—	8	2	no	—
66	#1960	—	61	1	no	—
67	#1324	—	61	2	no	—
68	C₂×C₆₁	—	2	1	no	—
69	C₂×C₆₁	—	2	1	no	—
70	C₂×C₆₁	#1318	1	1	yes	characteristic
71	C₂×C₆₁	#1407	1	2	yes	—
72	C₂×C₄⋊Q₈	—	61	1	no	—
73	C₂×#103	—	122	2	no	—
74	C₂×#97	#6923	1	1	yes	characteristic
75	#382	—	61	1	no	—
76	#324	—	122	2	no	—
77	C₆₁	#7353	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₈	—	122	1	no	—
79	C₂²×C₈	—	2	1	no	—
80	C₄⋊Q₈	—	61	2	no	—
81	C₄⋊Q₈	—	122	1	no	—
82	C₄⋊Q₈	—	244	1	no	—
83	C₂×C₄⋊C₄	—	61	2	no	—
84	C₂×C₄⋊C₄	—	244	1	no	—
85	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
86	#103	—	244	4	no	—
87	C₂²⋊C₈	—	244	2	no	—
88	#97	—	4	1	no	—
89	#95	—	244	1	no	—
90	C₂×Q₈	—	122	2	no	—
91	C₂×Q₈	—	244	2	no	—
92	C₂×Q₈	—	244	2	no	—
93	C₂×Q₈	—	244	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	122	2	no	—
96	C₂²×C₄	—	244	1	no	—
97	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	4	1	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	488	2	no	—
102	C₄⋊C₄	—	244	2	no	—
103	C₄⋊C₄	—	244	2	no	—
104	C₄⋊C₄	—	244	2	no	—
105	C₄²	—	2	1	no	—
106	C₄²	D₆₁⋊C₈	1	1	yes	characteristic
107	C₄²	#11758213	1	1	yes	characteristic
108	C₂³	#14519836	1	1	yes	characteristic
109	Q₈	—	488	2	no	—
110	Q₈	—	244	4	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	488	1	no	—
117	C₂×C₄	—	488	1	no	—
118	C₂×C₄	—	488	2	no	—
119	C₂×C₄	—	244	2	no	—
120	C₂×C₄	—	244	2	no	—
121	C₂×C₄	#14519834	1	1	yes	characteristic
122	C₂×C₄	#14519837	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	976	2	no	—
125	C₂²	—	2	2	no	—
126	C₂²	#16840606	1	1	yes	characteristic
127	C₂²	#16840667	1	1	yes	characteristic
128	C₂²	#16840671	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	488	2	no	—
134	C₄	—	488	2	no	—
135	C₂	—	2	1	no	—
136	C₂	—	2	1	no	—
137	C₂	#21460748	1	1	yes	characteristic
138	C₂	#21460949	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21460759

C₂

yes

maximal

#21446639

C₂

yes

maximal, characteristic

#16840607

—

C₂×#14519777

C₄

yes

characteristic

C₂×#14519702

—

#16839628

C₂²

yes

characteristic

C₂×#97×C₆₁

C₄

yes

characteristic

C₂²⋊#92420

—

C₂²×Dic₁₂₂

—

#14519777

—

#14519777

—

#14519777

C₈

yes

complemented

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

D₄

yes

—

#14519702

—

#14519684

—

C₂²×C₈×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

#97×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×#92420

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

M₁₆

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂³×C₆₁

#99

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₆₁

#100

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

#92420

—

C₈×C₆₁

—

#7353

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

Dic₆₁

—

#1960

—

#1324

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1318

yes

characteristic

C₂×C₆₁

#1407

yes

—

C₂×C₄⋊Q₈

—

C₂×#103

—

122

—

C₂×#97

yes

—

—

122

—

C₆₁

#7353

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

122

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

—

244

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

244

—

C₂×C₄²

C₂×#6923

yes

characteristic

#103

—

244

—

C₂²⋊C₈

—

244

—

#97

—

#95

—

244

—

C₂×Q₈

—

122

—

C₂×Q₈

—

244

—

C₂×Q₈

—

244

—

C₂×Q₈

—

244

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

—

244

—

C₂²×C₄

C₂²⋊#6923

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

488

—

102

C₄⋊C₄

—

244

—

103

C₄⋊C₄

—

244

—

104

C₄⋊C₄

—

244

—

105

C₄²

—

106

C₄²

D₆₁⋊C₈

yes

characteristic

107

C₄²

#11758213

yes

characteristic

108

C₂³

#14519836

yes

characteristic

109

Q₈

—

488

—

110

Q₈

—

244

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

488

—

117

C₂×C₄

—

488

—

118

C₂×C₄

—

488

—

119

C₂×C₄

—

244

—

120

C₂×C₄

—

244

—

121

C₂×C₄

#14519834

yes

characteristic

122

C₂×C₄

#14519837

yes

characteristic

123

C₈

—

124

C₈

—

976

—

125

C₂²

—

126

C₂²

#16840606

yes

characteristic

127

C₂²

#16840667

yes

characteristic

128

C₂²

#16840671

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

488

—

134

C₄

—

488

—

135

C₂

—

136

C₂

—

137

C₂

#21460748

yes

characteristic

138

C₂

#21460949

yes

—

	subgroup	G/H
center	C₂²	#16840667
commutator	C₄²×C₆₁	C₂×C₈
Frattini	C₂×C₄²	C₂×#6923
Fitting	C₂×#97×C₆₁	C₄
socle	C₂²×C₆₁	#353
prime radical	C₂³×C₆₁	#99
cosocle	#16839628	C₂²
hypercenter	C₂×#97	#6923
hypocenter	C₆₁	#7353
LCS	G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂²

#16840667

commutator

C₄²×C₆₁

C₂×C₈

Frattini

C₂×C₄²

C₂×#6923

Fitting

C₂×#97×C₆₁

C₄

C₂²×C₆₁

C₂³×C₆₁

C₂²

C₂×#97

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
12. #14519777	C₈	semidirect
77. C₆₁	#7353	semidirect

H ≅ G/N

type

12. #14519777

C₈

semidirect

77. C₆₁

#7353

semidirect

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

subgroup	index
#21460759	2
#21446639	2
#7353	61

subgroup

index

#21460759

#21446639

#7353

Subgroups — #30594837

Conjugacy classes of subgroups

Complements

Maximal subgroups