Group #30594161 — subgroups

Subgroups — #30594161

Order 15616.

210 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 56 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	#21460643	C₂	1	2	yes	maximal
2	#21446799	C₂	1	1	yes	maximal, characteristic
3	#16840778	—	2	2	no	—
4	#16840319	C₄	1	1	yes	characteristic
5	#16839608	C₂²	1	1	yes	characteristic
6	#16839523	—	2	2	no	—
7	C₆₁×#423	C₄	1	1	yes	characteristic
8	C₄⋊#92420	—	4	2	no	—
9	C₂²⋊#92420	—	4	2	no	—
10	C₄×#92420	—	2	4	no	—
11	#14519777	—	2	1	no	—
12	#14519769	C₈	1	2	yes	complemented
13	#14519756	—	4	1	no	—
14	C₄×Dic₁₂₂	—	4	1	no	—
15	#14519736	—	2	1	no	—
16	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
17	#14519684	—	4	1	no	—
18	C₂×M₁₆×C₆₁	—	2	1	no	—
19	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₄⋊C₈×C₆₁	—	4	1	no	—
21	C₂²×C₄×C₆₁	—	2	1	no	—
22	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂²×Dic₆₁	—	2	2	no	—
24	C₂×Dic₁₂₂	—	4	1	no	—
25	C₂×#92420	—	4	4	no	—
26	M₁₆×C₆₁	—	4	2	no	—
27	C₂×C₈×C₆₁	—	4	1	no	—
28	C₄²×C₆₁	—	2	1	no	—
29	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
30	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
31	C₂²⋊Dic₆₁	—	4	1	no	—
32	C₄⋊Dic₆₁	—	2	2	no	—
33	C₄⋊Dic₆₁	—	4	1	no	—
34	#11758196	—	4	2	no	—
35	C₄×Dic₆₁	—	2	2	no	—
36	C₄×Dic₆₁	—	4	1	no	—
37	C₄×Dic₆₁	SD₁₆	1	2	yes	—
38	C₄×Dic₆₁	Q₁₆	1	2	yes	—
39	C₂³×C₆₁	#99	1	1	yes	characteristic
40	C₂×C₄×C₆₁	—	2	1	no	—
41	C₂×C₄×C₆₁	—	2	1	no	—
42	C₂×C₄×C₆₁	—	8	1	no	—
43	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₄×C₆₁	#100	1	1	yes	characteristic
45	C₂×C₄×C₆₁	#103	1	2	yes	—
46	C₂×C₄×C₆₁	#104	1	2	yes	—
47	C₂×Dic₆₁	—	2	2	no	—
48	C₂×Dic₆₁	—	2	2	no	—
49	C₂×Dic₆₁	—	4	2	no	—
50	C₂×Dic₆₁	—	8	2	no	—
51	Dic₁₂₂	—	8	2	no	—
52	#92420	—	8	4	no	—
53	C₈×C₆₁	—	8	1	no	—
54	#7129	—	61	1	no	maximal, Sylow
55	C₂²×C₆₁	—	4	1	no	—
56	C₂²×C₆₁	#323	1	1	yes	characteristic
57	C₂²×C₆₁	#328	1	2	yes	—
58	C₄×C₆₁	—	4	1	no	—
59	C₄×C₆₁	—	4	1	no	—
60	C₄×C₆₁	Q₈⋊C₈	1	2	yes	—
61	C₄×C₆₁	#329	1	1	yes	characteristic
62	C₄×C₆₁	#333	1	1	yes	characteristic
63	Dic₆₁	—	4	2	no	—
64	Dic₆₁	—	4	2	no	—
65	Dic₆₁	—	8	2	no	—
66	#1995	—	61	1	no	—
67	#1274	—	61	2	no	—
68	C₂×C₆₁	—	2	2	no	—
69	C₂×C₆₁	—	2	1	no	—
70	C₂×C₆₁	#1272	1	1	yes	characteristic
71	#533	—	61	1	no	—
72	#431	—	122	2	no	—
73	#423	#6923	1	1	yes	characteristic
74	#376	—	61	1	no	—
75	#339	—	122	2	no	—
76	C₆₁	#7129	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₁₆	—	2	1	no	—
78	C₄⋊Q₈	—	122	1	no	—
79	#125	—	61	2	no	—
80	C₂²⋊Q₈	—	244	1	no	—
81	C₄×Q₈	—	244	1	no	—
82	#117	—	122	1	no	—
83	C₂×C₄²	—	61	2	no	—
84	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
85	C₄⋊C₈	—	4	1	no	—
86	C₄⋊C₈	—	244	2	no	—
87	C₂²⋊C₈	—	244	2	no	—
88	C₄×C₈	—	122	4	no	—
89	#95	—	244	1	no	—
90	C₂×Q₈	—	244	1	no	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	122	2	no	—
93	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
94	M₁₆	—	4	2	no	—
95	C₂×C₈	—	4	1	no	—
96	C₂×C₈	—	244	4	no	—
97	C₄⋊C₄	—	122	2	no	—
98	C₄⋊C₄	—	244	2	no	—
99	C₄⋊C₄	—	244	1	no	—
100	C₂²⋊C₄	—	244	1	no	—
101	C₄²	—	2	1	no	—
102	C₄²	—	61	2	no	—
103	C₄²	—	61	2	no	—
104	C₄²	—	122	2	no	—
105	C₄²	—	244	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	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	8	1	no	—
113	C₂×C₄	—	122	2	no	—
114	C₂×C₄	—	122	2	no	—
115	C₂×C₄	—	488	2	no	—
116	C₂×C₄	—	244	2	no	—
117	C₂×C₄	#14519834	1	1	yes	characteristic
118	C₂×C₄	#14519837	1	1	yes	characteristic
119	C₂×C₄	#14519840	1	2	yes	—
120	C₂×C₄	#14519841	1	2	yes	—
121	C₈	—	8	1	no	—
122	C₈	—	488	4	no	—
123	C₂²	—	4	1	no	—
124	C₂²	#16840606	1	1	yes	characteristic
125	C₂²	#16840615	1	2	yes	—
126	C₄	—	4	1	no	—
127	C₄	—	4	1	no	—
128	C₄	—	488	2	no	—
129	C₄	—	244	2	no	—
130	C₄	—	244	2	no	—
131	C₄	#16840612	1	2	yes	—
132	C₄	#16840617	1	1	yes	characteristic
133	C₄	#16840622	1	1	yes	characteristic
134	C₂	—	2	1	no	—
135	C₂	—	2	2	no	—
136	C₂	#21460640	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21460643

C₂

yes

maximal

#21446799

C₂

yes

maximal, characteristic

#16840778

—

#16840319

C₄

yes

characteristic

#16839608

C₂²

yes

characteristic

#16839523

—

C₆₁×#423

C₄

yes

—

—

—

—

C₈

yes

complemented

#14519756

—

C₄×Dic₁₂₂

—

#14519736

—

C₂×C₄×Dic₆₁

D₄

yes

—

#14519684

—

C₂×M₁₆×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×#92420

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

M₁₆

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

SD₁₆

yes

—

C₄×Dic₆₁

Q₁₆

yes

—

C₂³×C₆₁

#99

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

#104

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

#92420

—

C₈×C₆₁

—

#7129

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

#323

yes

characteristic

C₂²×C₆₁

#328

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

Q₈⋊C₈

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#1995

—

#1274

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

—

—

122

—

yes

—

—

122

—

C₆₁

#7129

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

122

—

#125

—

C₂²⋊Q₈

—

244

—

C₄×Q₈

—

244

—

#117

—

122

—

C₂×C₄²

—

C₂×C₄²

C₂×#6923

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

244

—

C₂²⋊C₈

—

244

—

C₄×C₈

—

122

—

#95

—

244

—

C₂×Q₈

—

244

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²⋊#6923

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

244

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

100

C₂²⋊C₄

—

244

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

122

—

105

C₄²

—

244

—

106

C₄²

D₆₁⋊C₈

yes

characteristic

107

C₄²

#11758213

yes

characteristic

108

C₂³

#14519836

yes

characteristic

109

Q₈

—

488

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

122

—

114

C₂×C₄

—

122

—

115

C₂×C₄

—

488

—

116

C₂×C₄

—

244

—

117

C₂×C₄

#14519834

yes

characteristic

118

C₂×C₄

#14519837

yes

characteristic

119

C₂×C₄

#14519840

yes

—

120

C₂×C₄

#14519841

yes

—

121

C₈

—

122

C₈

—

488

—

123

C₂²

—

124

C₂²

#16840606

yes

characteristic

125

C₂²

#16840615

yes

—

126

C₄

—

127

C₄

—

128

C₄

—

488

—

129

C₄

—

244

—

130

C₄

—

244

—

131

C₄

#16840612

yes

—

132

C₄

#16840617

yes

characteristic

133

C₄

#16840622

yes

characteristic

134

C₂

—

135

C₂

—

136

C₂

#21460640

yes

characteristic

	subgroup	G/H
center	C₄	#16840617
commutator	C₄²×C₆₁	C₂×C₈
Frattini	C₂×C₄²	C₂×#6923
Fitting	C₆₁×#423	C₄
socle	C₂×C₆₁	#1272
prime radical	C₂³×C₆₁	#99
cosocle	#16839608	C₂²
hypercenter	#423	#6923
hypocenter	C₆₁	#7129
LCS	G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₄

#16840617

commutator

C₄²×C₆₁

C₂×C₈

Frattini

C₂×C₄²

C₂×#6923

Fitting

C₆₁×#423

C₄

C₂×C₆₁

C₂³×C₆₁

C₂²

C₆₁

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
12. #14519769	C₈	semidirect
76. C₆₁	#7129	semidirect

H ≅ G/N

type

12. #14519769

C₈

semidirect

76. C₆₁

#7129

semidirect

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

subgroup	index
#21460643	2
#21446799	2
#7129	61

subgroup

index

#21460643

#21446799

#7129

Subgroups — #30594161

Conjugacy classes of subgroups

Complements

Maximal subgroups