Group #30594136 — subgroups

Subgroups — #30594136

Order 15616.

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

normal subgroups only

This group has 230 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	#21460860	C₂	1	2	yes	maximal
2	#21446640	C₂	1	1	yes	maximal, characteristic
3	C₂×#14519851	—	2	2	no	—
4	C₂×#14519701	—	2	2	no	—
5	C₂×#14519688	C₂²	1	1	yes	characteristic
6	#16839628	C₄	1	1	yes	characteristic
7	C₂×C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
8	#14519851	—	4	2	no	—
9	#14519851	—	4	2	no	—
10	C₂×#11758187	—	2	4	no	—
11	C₂²×#92418	D₄	1	2	yes	—
12	C₂×C₄⋊Dic₆₁	—	2	1	no	—
13	#14519701	—	4	2	no	—
14	#14519701	—	4	2	no	—
15	#14519688	—	4	1	no	—
16	#14519684	—	4	1	no	—
17	C₂²×Q₈×C₆₁	—	2	1	no	—
18	C₄⋊Q₈×C₆₁	—	2	1	no	—
19	C₄⋊Q₈×C₆₁	—	4	1	no	—
20	C₄⋊Q₈×C₆₁	C₈	1	2	yes	—
21	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
22	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×Q₈×C₆₁	—	2	2	no	—
24	C₂×Q₈×C₆₁	—	4	1	no	—
25	C₂×Q₈×C₆₁	—	4	2	no	—
26	C₂×Q₈×C₆₁	—	4	2	no	—
27	C₂²×C₄×C₆₁	—	2	1	no	—
28	C₂²×C₄×C₆₁	—	4	1	no	—
29	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×Dic₆₁	—	4	1	no	—
31	C₄⋊C₄×C₆₁	—	4	2	no	—
32	C₄⋊C₄×C₆₁	—	4	2	no	—
33	C₄²×C₆₁	—	2	1	no	—
34	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
35	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
36	C₄⋊Dic₆₁	—	4	2	no	—
37	C₂×#92418	—	2	2	no	—
38	C₂×#92418	—	2	2	no	—
39	C₂×#92418	D₈	1	2	yes	—
40	C₂×#92418	SD₁₆	1	2	yes	—
41	#11758187	—	4	4	no	—
42	C₂³×C₆₁	#100	1	1	yes	characteristic
43	Q₈×C₆₁	—	4	4	no	—
44	Q₈×C₆₁	—	8	2	no	—
45	C₂×C₄×C₆₁	—	2	1	no	—
46	C₂×C₄×C₆₁	—	2	1	no	—
47	C₂×C₄×C₆₁	—	4	2	no	—
48	C₂×C₄×C₆₁	—	4	2	no	—
49	C₂×C₄×C₆₁	—	8	1	no	—
50	C₂×C₄×C₆₁	—	8	1	no	—
51	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#99	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#102	1	2	yes	—
54	C₂×C₄×C₆₁	#104	1	2	yes	—
55	C₂×Dic₆₁	—	8	2	no	—
56	C₂×Dic₆₁	—	8	1	no	—
57	#92418	—	2	2	no	—
58	#92418	—	2	2	no	—
59	#7117	—	61	1	no	maximal, Sylow
60	C₂²×C₆₁	—	2	2	no	—
61	C₂²×C₆₁	#323	1	1	yes	characteristic
62	C₂²×C₆₁	#329	1	1	yes	characteristic
63	C₂²×C₆₁	#331	1	1	yes	characteristic
64	C₄×C₆₁	—	4	1	no	—
65	C₄×C₆₁	—	4	1	no	—
66	C₄×C₆₁	—	8	2	no	—
67	C₄×C₆₁	D₄⋊C₈	1	2	yes	—
68	C₄×C₆₁	#327	1	2	yes	—
69	Dic₆₁	—	16	1	no	—
70	#1960	—	61	1	no	—
71	#1375	—	61	2	no	—
72	C₂×C₆₁	—	2	1	no	—
73	C₂×C₆₁	—	2	1	no	—
74	C₂×C₆₁	#1270	1	1	yes	characteristic
75	C₂×C₆₁	#1356	1	2	yes	—
76	C₂×C₄⋊Q₈	#6923	1	1	yes	characteristic
77	C₂×M₃₂	—	122	2	no	—
78	C₂×#103	—	122	2	no	—
79	C₂×#97	—	61	1	no	—
80	#382	—	61	1	no	—
81	C₆₁	#7117	1	1	yes	characteristic, complemented, Sylow
82	C₂²×Q₈	—	2	1	no	—
83	C₂²×C₈	—	61	2	no	—
84	C₄⋊Q₈	—	2	1	no	—
85	C₄⋊Q₈	—	4	1	no	—
86	C₄⋊Q₈	#92420	1	2	yes	—
87	C₂×C₄⋊C₄	—	4	1	no	—
88	C₂×C₄⋊C₄	—	122	1	no	—
89	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
90	M₃₂	—	244	2	no	—
91	M₃₂	—	244	2	no	—
92	C₂×C₁₆	—	122	4	no	—
93	#103	—	244	2	no	—
94	#103	—	244	2	no	—
95	#97	—	244	1	no	—
96	#95	—	244	1	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	4	1	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	4	1	no	—
103	C₂²×C₄	—	244	1	no	—
104	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
105	C₂×C₈	—	61	2	no	—
106	C₂×C₈	—	61	2	no	—
107	C₂×C₈	—	122	2	no	—
108	C₂×C₈	—	122	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	244	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	C₂×#92420	1	1	yes	characteristic
114	C₄²	#11758217	1	1	yes	characteristic
115	C₁₆	—	244	4	no	—
116	C₂³	#14519837	1	1	yes	characteristic
117	Q₈	—	4	4	no	—
118	Q₈	—	8	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	8	1	no	—
124	C₂×C₄	—	8	1	no	—
125	C₂×C₄	—	488	2	no	—
126	C₂×C₄	—	488	1	no	—
127	C₂×C₄	C₂²⋊#92420	1	1	yes	characteristic
128	C₂×C₄	#14519835	1	1	yes	characteristic
129	C₂×C₄	#14519839	1	2	yes	—
130	C₂×C₄	#14519841	1	2	yes	—
131	C₈	—	122	2	no	—
132	C₈	—	122	2	no	—
133	C₂²	—	2	2	no	—
134	C₂²	#16840605	1	1	yes	characteristic
135	C₂²	#16840617	1	1	yes	characteristic
136	C₂²	#16840620	1	1	yes	characteristic
137	C₄	—	4	1	no	—
138	C₄	—	4	1	no	—
139	C₄	—	8	2	no	—
140	C₄	—	976	1	no	—
141	C₄	D₄⋊#92420	1	2	yes	—
142	C₄	#16840613	1	2	yes	—
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	#21460635	1	1	yes	characteristic
146	C₂	#21460824	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21460860

C₂

yes

maximal

#21446640

C₂

yes

maximal, characteristic

C₂×#14519851

—

C₂×#14519701

—

C₂×#14519688

C₂²

yes

characteristic

#16839628

C₄

yes

characteristic

C₂×C₄⋊Q₈×C₆₁

C₄

yes

—

—

—

C₂²×#92418

D₄

yes

—

C₂×C₄⋊Dic₆₁

—

—

—

—

—

C₂²×Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₈

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×Dic₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

M₁₆

yes

characteristic

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

D₈

yes

—

C₂×#92418

SD₁₆

yes

—

#11758187

—

C₂³×C₆₁

#100

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

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

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

D₄⋊C₈

yes

—

C₄×C₆₁

#327

yes

—

Dic₆₁

—

#1960

—

#1375

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1270

yes

characteristic

C₂×C₆₁

#1356

yes

—

C₂×C₄⋊Q₈

#6923

yes

characteristic

C₂×M₃₂

—

122

—

C₂×#103

—

122

—

C₂×#97

—

#382

—

C₆₁

#7117

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#92420

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

C₂×#6923

yes

characteristic

M₃₂

—

244

—

M₃₂

—

244

—

C₂×C₁₆

—

122

—

#103

—

244

—

#103

—

244

—

#97

—

244

—

#95

—

244

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

244

—

104

C₂²×C₄

C₂²⋊#6923

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

122

—

108

C₂×C₈

—

122

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

244

—

112

C₄²

—

113

C₄²

C₂×#92420

yes

characteristic

114

C₄²

#11758217

yes

characteristic

115

C₁₆

—

244

—

116

C₂³

#14519837

yes

characteristic

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

488

—

126

C₂×C₄

—

488

—

127

C₂×C₄

C₂²⋊#92420

yes

characteristic

128

C₂×C₄

#14519835

yes

characteristic

129

C₂×C₄

#14519839

yes

—

130

C₂×C₄

#14519841

yes

—

131

C₈

—

122

—

132

C₈

—

122

—

133

C₂²

—

134

C₂²

#16840605

yes

characteristic

135

C₂²

#16840617

yes

characteristic

136

C₂²

#16840620

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

976

—

141

C₄

D₄⋊#92420

yes

—

142

C₄

#16840613

yes

—

143

C₂

—

144

C₂

—

145

C₂

#21460635

yes

characteristic

146

C₂

#21460824

yes

—

	subgroup	G/H
center	C₂²	#16840617
commutator	C₄²×C₆₁	C₂×C₈
Frattini	C₂×C₄²	C₂×#6923
Fitting	C₂×C₄⋊Q₈×C₆₁	C₄
socle	C₂²×C₆₁	#329
prime radical	C₂³×C₆₁	#100
cosocle	C₂×#14519688	C₂²
hypercenter	C₂×C₄⋊Q₈	#6923
hypocenter	C₆₁	#7117
LCS	G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

#16840617

commutator

C₄²×C₆₁

C₂×C₈

Frattini

C₂×C₄²

C₂×#6923

Fitting

C₂×C₄⋊Q₈×C₆₁

C₄

C₂²×C₆₁

C₂³×C₆₁

C₂²

C₂×C₄⋊Q₈

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
81. C₆₁	#7117	semidirect

H ≅ G/N

type

81. C₆₁

#7117

semidirect

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

subgroup	index
#21460860	2
#21446640	2
#7117	61

subgroup

index

#21460860

#21446640

#7117

Subgroups — #30594136

Conjugacy classes of subgroups

Complements

Maximal subgroups