Group #29594739 — subgroups

Subgroups — #29594739

Order 15616.

360 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 87 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	#21448074	C₂	1	2	yes	maximal, complemented
2	#21448068	C₂	1	2	yes	maximal, complemented
3	#21445425	C₂	1	1	yes	maximal, characteristic, complemented
4	#21443736	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1712	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14519721	—	2	4	no	—
7	C₂×#14519718	—	2	4	no	—
8	#16840142	C₂²	1	1	yes	characteristic
9	#16840139	C₂²	1	1	yes	characteristic
10	#16840109	—	2	2	no	—
11	C₄×#11758185	C₂²	1	2	yes	—
12	C₆₁×#493	—	2	2	no	—
13	C₆₁×#493	C₂²	1	2	yes	—
14	C₈²×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₄₄	—	2	2	no	—
16	C₂×D₄₈₈	—	2	2	no	—
17	#14519777	—	2	1	no	—
18	#14519774	—	2	1	no	—
19	#14519721	—	4	8	no	—
20	#14519718	—	4	8	no	—
21	C₂×#11758185	D₄	1	4	yes	—
22	#14519699	—	4	4	no	—
23	C₂×D₈×C₆₁	—	2	2	no	—
24	C₂×D₈×C₆₁	—	2	2	no	—
25	C₂×D₈×C₆₁	—	4	4	no	—
26	#127×C₆₁	—	2	2	no	—
27	C₄×C₈×C₆₁	D₄	1	2	yes	—
28	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₂×D₄×C₆₁	—	4	2	no	—
30	C₂×D₄×C₆₁	—	4	4	no	—
31	C₂×D₂₄₄	—	4	1	no	—
32	C₂×D₂₄₄	—	4	2	no	—
33	C₂×Dic₁₂₂	—	4	1	no	—
34	D₈×C₆₁	—	4	4	no	—
35	D₈×C₆₁	—	4	4	no	—
36	D₈×C₆₁	—	8	8	no	—
37	C₂×C₈×C₆₁	D₈	1	4	yes	—
38	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
39	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₄⋊Dic₆₁	—	4	2	no	—
41	Dic₂₄₄	—	4	4	no	—
42	D₄₈₈	—	4	4	no	—
43	#11758185	D₈	1	8	yes	complemented
44	C₂³×C₆₁	—	8	2	no	—
45	C₂²×D₆₁	—	8	1	no	—
46	D₄×C₆₁	—	8	2	no	—
47	D₄×C₆₁	—	8	2	no	—
48	D₄×C₆₁	—	8	8	no	—
49	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
51	C₂×Dic₆₁	—	8	1	no	—
52	D₂₄₄	—	8	2	no	—
53	D₂₄₄	—	8	4	no	—
54	Dic₁₂₂	—	8	2	no	—
55	C₈×C₆₁	—	2	4	no	—
56	C₈×C₆₁	C₂×D₈	1	4	yes	—
57	#12519	—	61	1	no	maximal, Sylow
58	C₂²×C₆₁	—	16	2	no	—
59	C₂²×C₆₁	—	16	4	no	—
60	C₂²×C₆₁	#493	1	1	yes	characteristic
61	C₂×D₆₁	—	16	1	no	—
62	C₂×D₆₁	—	16	2	no	—
63	C₄×C₆₁	#493	1	2	yes	—
64	C₄×C₆₁	#509	1	4	yes	—
65	Dic₆₁	—	16	1	no	—
66	#2248	—	61	2	no	—
67	#2246	—	61	2	no	—
68	#1712	D₆₁	1	1	yes	characteristic, complemented
69	#1709	—	61	1	no	—
70	#1311	—	61	1	no	—
71	C₂×C₆₁	—	32	2	no	—
72	C₂×C₆₁	#1712	1	1	yes	characteristic
73	C₂×C₆₁	#2250	1	2	yes	—
74	D₆₁	—	32	1	no	—
75	C₂×SD₃₂	—	122	4	no	—
76	C₂×D₁₆	—	122	4	no	—
77	#494	—	61	1	no	—
78	#493	—	2	2	no	—
79	#493	—	61	1	no	—
80	#493	C₂×D₆₁	1	2	yes	—
81	#486	—	122	2	no	—
82	C₄×C₁₆	—	61	2	no	—
83	C₈²	C₂×D₆₁	1	1	yes	characteristic
84	C₆₁	#12519	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	—	122	2	no	—
86	C₂×D₈	—	2	2	no	—
87	C₂×D₈	—	2	2	no	—
88	C₂×D₈	—	4	4	no	—
89	C₂×D₈	—	122	2	no	—
90	C₄⋊Q₈	—	122	1	no	—
91	#127	—	2	2	no	—
92	#127	—	122	1	no	—
93	SD₃₂	—	244	8	no	—
94	D₁₆	—	244	8	no	—
95	C₂×C₁₆	—	61	4	no	—
96	#102	—	244	4	no	—
97	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
98	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
99	C₂×Q₈	—	244	1	no	—
100	C₂×D₄	—	4	2	no	—
101	C₂×D₄	—	4	4	no	—
102	C₂×D₄	—	244	1	no	—
103	C₂×D₄	—	244	2	no	—
104	Q₁₆	—	244	4	no	—
105	D₈	—	4	4	no	—
106	D₈	—	4	4	no	—
107	D₈	—	8	8	no	—
108	D₈	—	244	4	no	—
109	C₂×C₈	#11758199	1	4	yes	—
110	C₂×C₈	D₄×D₆₁	1	2	yes	—
111	C₄⋊C₄	—	244	2	no	—
112	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
113	C₁₆	—	61	8	no	—
114	C₂³	—	8	2	no	—
115	C₂³	—	488	1	no	—
116	Q₈	—	488	2	no	—
117	D₄	—	8	8	no	—
118	D₄	—	8	2	no	—
119	D₄	—	8	2	no	—
120	D₄	—	488	4	no	—
121	D₄	—	488	2	no	—
122	C₂×C₄	—	488	1	no	—
123	C₂×C₄	#14519773	1	1	yes	characteristic
124	C₂×C₄	C₂×#11758199	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	D₈×D₆₁	1	4	yes	—
127	C₂²	—	16	2	no	—
128	C₂²	—	16	4	no	—
129	C₂²	—	976	1	no	—
130	C₂²	—	976	2	no	—
131	C₂²	#16840138	1	1	yes	characteristic
132	C₄	—	976	1	no	—
133	C₄	#16840137	1	2	yes	—
134	C₄	#16840198	1	4	yes	—
135	C₂	—	32	2	no	—
136	C₂	—	1952	1	no	—
137	C₂	#21445433	1	1	yes	characteristic
138	C₂	#21448083	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21448074

C₂

yes

maximal, complemented

#21448068

C₂

yes

maximal, complemented

#21445425

C₂

yes

maximal, characteristic, complemented

#21443736

C₂

yes

maximal, characteristic, complemented

C₆₁×#1712

C₂

yes

maximal, characteristic, complemented

C₂×#14519721

—

C₂×#14519718

—

#16840142

C₂²

yes

characteristic

#16840139

C₂²

yes

characteristic

#16840109

—

C₄×#11758185

C₂²

yes

—

C₆₁×#493

—

C₆₁×#493

C₂²

yes

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

C₂×D₄₈₈

—

—

—

—

—

D₄

yes

—

#14519699

—

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

—

#127×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×Dic₁₂₂

—

D₈×C₆₁

—

D₈×C₆₁

—

D₈×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

Dic₂₄₄

—

D₄₈₈

—

#11758185

D₈

yes

complemented

C₂³×C₆₁

—

C₂²×D₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂×D₈

yes

—

#12519

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#493

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

#493

yes

—

C₄×C₆₁

#509

yes

—

Dic₆₁

—

#2248

—

#2246

—

#1712

D₆₁

yes

characteristic, complemented

#1709

—

#1311

—

C₂×C₆₁

—

C₂×C₆₁

#1712

yes

characteristic

C₂×C₆₁

#2250

yes

—

D₆₁

—

C₂×SD₃₂

—

122

—

C₂×D₁₆

—

122

—

—

—

—

C₂×D₆₁

yes

—

#486

—

122

—

C₄×C₁₆

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12519

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

122

—

C₄⋊Q₈

—

122

—

#127

—

#127

—

122

—

SD₃₂

—

244

—

D₁₆

—

244

—

C₂×C₁₆

—

#102

—

244

—

C₄×C₈

C₂²⋊D₆₁

yes

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

244

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

244

—

103

C₂×D₄

—

244

—

104

Q₁₆

—

244

—

105

D₈

—

106

D₈

—

107

D₈

—

108

D₈

—

244

—

109

C₂×C₈

#11758199

yes

—

110

C₂×C₈

D₄×D₆₁

yes

—

111

C₄⋊C₄

—

244

—

112

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

113

C₁₆

—

114

C₂³

—

115

C₂³

—

488

—

116

Q₈

—

488

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

488

—

121

D₄

—

488

—

122

C₂×C₄

—

488

—

123

C₂×C₄

#14519773

yes

characteristic

124

C₂×C₄

C₂×#11758199

yes

—

125

C₈

—

126

C₈

D₈×D₆₁

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

976

—

130

C₂²

—

976

—

131

C₂²

#16840138

yes

characteristic

132

C₄

—

976

—

133

C₄

#16840137

yes

—

134

C₄

#16840198

yes

—

135

C₂

—

136

C₂

—

1952

—

137

C₂

#21445433

yes

characteristic

138

C₂

#21448083

yes

—

	subgroup	G/H
center	C₂²	#16840138
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1712	C₂
socle	C₂²×C₆₁	#493
hypercenter	#1712	D₆₁
hypocenter	C₆₁	#12519
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#16840138

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1712

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #21448074	C₂	semidirect
2. #21448068	C₂	semidirect
3. #21445425	C₂	semidirect
4. #21443736	C₂	semidirect
5. C₆₁×#1712	C₂	semidirect
43. #11758185	D₈	semidirect
68. #1712	D₆₁	semidirect
84. C₆₁	#12519	semidirect

H ≅ G/N

type

1. #21448074

C₂

semidirect

2. #21448068

C₂

semidirect

3. #21445425

C₂

semidirect

4. #21443736

C₂

semidirect

5. C₆₁×#1712

C₂

semidirect

43. #11758185

D₈

semidirect

68. #1712

D₆₁

semidirect

84. C₆₁

#12519

semidirect

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

subgroup	index
#21448074	2
#21448068	2
#21445425	2
#21443736	2
C₆₁×#1712	2
#12519	61

subgroup

index

C₆₁×#1712

Subgroups — #29594739

Conjugacy classes of subgroups

Complements

Maximal subgroups