Group #29594698 — subgroups

Subgroups — #29594698

Order 15616.

306 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 95 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	#21448042	C₂	1	4	yes	maximal, complemented
2	#21445437	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×#11758185	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1712	C₂	1	1	yes	maximal, characteristic
5	#16840167	—	2	1	no	—
6	#16840167	C₂²	1	2	yes	—
7	#16840142	—	2	1	no	—
8	#16839563	—	2	8	no	—
9	C₄×#11758185	C₂²	1	2	yes	—
10	C₆₁×#493	—	2	1	no	—
11	C₆₁×#493	—	2	1	no	—
12	C₆₁×#493	C₂²	1	2	yes	—
13	C₈²×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₄₄	—	4	2	no	—
15	#14519777	—	2	2	no	—
16	C₂×#11758185	D₄	1	4	yes	—
17	#14519711	—	2	4	no	—
18	#14519711	—	4	2	no	—
19	C₂×D₈×C₆₁	—	2	4	no	—
20	C₂×D₈×C₆₁	—	4	2	no	—
21	C₂×D₈×C₆₁	—	4	2	no	—
22	#127×C₆₁	—	2	2	no	—
23	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
24	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
26	C₂×D₄×C₆₁	—	4	2	no	—
27	C₂×D₄×C₆₁	—	4	2	no	—
28	C₂×D₄×C₆₁	—	4	2	no	—
29	C₂×Dic₁₂₂	—	4	2	no	—
30	D₈×C₆₁	—	4	8	no	—
31	D₈×C₆₁	—	8	4	no	—
32	D₈×C₆₁	—	8	4	no	—
33	C₂×C₈×C₆₁	D₈	1	2	yes	—
34	C₂×C₈×C₆₁	D₈	1	2	yes	—
35	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
36	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₆₁	—	4	2	no	—
38	C₄⋊Dic₆₁	—	4	2	no	—
39	Dic₂₄₄	—	8	4	no	—
40	#11758185	—	2	4	no	—
41	C₂³×C₆₁	—	8	2	no	—
42	D₄×C₆₁	—	8	4	no	—
43	D₄×C₆₁	—	8	4	no	—
44	D₄×C₆₁	—	8	4	no	—
45	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₆₁	—	8	2	no	—
49	Dic₁₂₂	—	8	4	no	—
50	C₈×C₆₁	D₁₆	1	4	yes	—
51	C₈×C₆₁	SD₃₂	1	4	yes	—
52	C₈×C₆₁	#135	1	4	yes	—
53	#12509	—	61	1	no	maximal, Sylow
54	C₂²×C₆₁	—	16	2	no	—
55	C₂²×C₆₁	—	16	2	no	—
56	C₂²×C₆₁	—	16	2	no	—
57	C₂²×C₆₁	#493	1	1	yes	characteristic
58	C₄×C₆₁	#495	1	2	yes	—
59	C₄×C₆₁	C₂×D₁₆	1	2	yes	—
60	C₄×C₆₁	C₂×SD₃₂	1	2	yes	—
61	Dic₆₁	—	16	2	no	—
62	#2240	—	61	4	no	—
63	#1713	—	61	1	no	—
64	#1712	D₆₁	1	1	yes	characteristic
65	C₈×C₁₆	—	61	1	no	—
66	C₂×C₆₁	—	32	2	no	—
67	C₂×C₆₁	#1709	1	1	yes	characteristic
68	C₂×C₆₁	#2246	1	1	yes	characteristic
69	C₂×C₆₁	#2248	1	1	yes	characteristic
70	#500	—	61	2	no	—
71	#500	—	122	1	no	—
72	#494	—	122	1	no	—
73	#493	—	2	1	no	—
74	#493	—	2	1	no	—
75	#493	C₂×D₆₁	1	2	yes	—
76	#357	—	122	8	no	—
77	C₄×C₁₆	—	61	2	no	—
78	C₈²	C₂×D₆₁	1	1	yes	characteristic
79	C₆₁	#12509	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	—	244	2	no	—
81	C₂×D₈	—	2	4	no	—
82	C₂×D₈	—	4	2	no	—
83	C₂×D₈	—	4	2	no	—
84	C₄⋊Q₈	—	122	2	no	—
85	#127	—	2	2	no	—
86	C₂×C₁₆	—	61	4	no	—
87	#107	—	122	4	no	—
88	#107	—	244	2	no	—
89	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
90	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
91	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
92	C₂×Q₈	—	244	2	no	—
93	C₂×D₄	—	4	2	no	—
94	C₂×D₄	—	4	2	no	—
95	C₂×D₄	—	4	2	no	—
96	Q₁₆	—	488	4	no	—
97	D₈	—	4	8	no	—
98	D₈	—	8	4	no	—
99	D₈	—	8	4	no	—
100	C₂×C₈	#11758199	1	2	yes	—
101	C₂×C₈	#11758199	1	2	yes	—
102	C₂×C₈	D₄×D₆₁	1	2	yes	—
103	C₄⋊C₄	—	244	2	no	—
104	C₄⋊C₄	—	244	2	no	—
105	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
106	C₁₆	—	122	4	no	—
107	C₂³	—	8	2	no	—
108	Q₈	—	488	4	no	—
109	D₄	—	8	4	no	—
110	D₄	—	8	4	no	—
111	D₄	—	8	4	no	—
112	C₂×C₄	—	488	2	no	—
113	C₂×C₄	#14519773	1	1	yes	characteristic
114	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
115	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
116	C₈	#14519718	1	4	yes	—
117	C₈	#14519721	1	4	yes	—
118	C₈	#14519796	1	4	yes	—
119	C₂²	—	16	2	no	—
120	C₂²	—	16	2	no	—
121	C₂²	—	16	2	no	—
122	C₂²	#16840138	1	1	yes	characteristic
123	C₄	—	976	2	no	—
124	C₄	#16840145	1	2	yes	—
125	C₄	C₂×#14519718	1	2	yes	—
126	C₄	C₂×#14519721	1	2	yes	—
127	C₂	—	32	2	no	—
128	C₂	#21445424	1	1	yes	characteristic
129	C₂	#21448068	1	1	yes	characteristic
130	C₂	#21448074	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21448042

C₂

yes

maximal, complemented

#21445437

C₂

yes

maximal, characteristic, complemented

C₈×#11758185

C₂

yes

maximal, characteristic, complemented

C₆₁×#1712

C₂

yes

maximal, characteristic

#16840167

—

#16840167

C₂²

yes

—

#16840142

—

#16839563

—

C₄×#11758185

C₂²

yes

—

C₆₁×#493

—

C₆₁×#493

—

C₆₁×#493

C₂²

yes

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

#14519777

—

C₂×#11758185

D₄

yes

—

#14519711

—

#14519711

—

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

—

#127×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×Dic₁₂₂

—

D₈×C₆₁

—

D₈×C₆₁

—

D₈×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

Dic₂₄₄

—

#11758185

—

C₂³×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

D₁₆

yes

—

C₈×C₆₁

SD₃₂

yes

—

C₈×C₆₁

#135

yes

—

#12509

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#493

yes

characteristic

C₄×C₆₁

#495

yes

—

C₄×C₆₁

C₂×D₁₆

yes

—

C₄×C₆₁

C₂×SD₃₂

yes

—

Dic₆₁

—

#2240

—

#1713

—

#1712

D₆₁

yes

characteristic

C₈×C₁₆

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

122

—

#494

—

122

—

#493

—

#493

—

#493

C₂×D₆₁

yes

—

#357

—

122

—

C₄×C₁₆

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12509

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

244

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

122

—

#127

—

C₂×C₁₆

—

#107

—

122

—

#107

—

244

—

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

244

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

488

—

D₈

—

D₈

—

D₈

—

100

C₂×C₈

#11758199

yes

—

101

C₂×C₈

#11758199

yes

—

102

C₂×C₈

D₄×D₆₁

yes

—

103

C₄⋊C₄

—

244

—

104

C₄⋊C₄

—

244

—

105

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

106

C₁₆

—

122

—

107

C₂³

—

108

Q₈

—

488

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

488

—

113

C₂×C₄

#14519773

yes

characteristic

114

C₂×C₄

C₂×#11758199

yes

characteristic

115

C₂×C₄

C₂×#11758199

yes

characteristic

116

C₈

#14519718

yes

—

117

C₈

#14519721

yes

—

118

C₈

#14519796

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#16840138

yes

characteristic

123

C₄

—

976

—

124

C₄

#16840145

yes

—

125

C₄

C₂×#14519718

yes

—

126

C₄

C₂×#14519721

yes

—

127

C₂

—

128

C₂

#21445424

yes

characteristic

129

C₂

#21448068

yes

characteristic

130

C₂

#21448074

yes

characteristic

	subgroup	G/H
center	C₂²	#16840138
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting, prime radical	C₆₁×#1712	C₂
socle	C₂²×C₆₁	#493
hypercenter	#1712	D₆₁
hypocenter	C₆₁	#12509
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, prime radical

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. #21448042	C₂	semidirect
2. #21445437	C₂	semidirect
3. C₈×#11758185	C₂	semidirect
79. C₆₁	#12509	semidirect

H ≅ G/N

type

1. #21448042

C₂

semidirect

2. #21445437

C₂

semidirect

3. C₈×#11758185

C₂

semidirect

79. C₆₁

#12509

semidirect

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

subgroup	index
#21448042	2
#21445437	2
C₈×#11758185	2
C₆₁×#1712	2
#12509	61

subgroup

index

C₆₁×#1712

Subgroups — #29594698

Conjugacy classes of subgroups

Complements

Maximal subgroups