Group #29594737 — subgroups

Subgroups — #29594737

Order 15616.

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

normal subgroups only

This group has 330 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	#21448076	C₂	1	2	yes	maximal, complemented
2	#21448069	C₂	1	1	yes	maximal, characteristic
3	#21445426	C₂	1	2	yes	maximal, complemented
4	C₁₆×#92418	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2247	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14519720	—	2	4	no	—
7	C₂×D₉₇₆	—	2	2	no	—
8	#16840139	C₂²	1	2	yes	—
9	#16840110	—	2	4	no	—
10	C₄×#11758185	C₂²	1	1	yes	characteristic
11	C₈×#92418	C₂²	1	1	yes	characteristic
12	C₂×Q₃₂×C₆₁	—	2	2	no	—
13	C₆₁×#494	C₂²	1	2	yes	—
14	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×D₄₈₈	—	2	2	no	—
16	C₂×D₄₈₈	—	2	2	no	—
17	#14519774	—	2	2	no	—
18	#14519720	—	4	8	no	—
19	D₉₇₆	—	4	4	no	—
20	C₂×#11758185	D₄	1	2	yes	—
21	#14519697	—	4	8	no	—
22	C₄×#92418	D₄	1	2	yes	—
23	C₂×Q₁₆×C₆₁	—	2	2	no	—
24	C₂×Q₁₆×C₆₁	—	2	2	no	—
25	C₄⋊Q₈×C₆₁	—	2	2	no	—
26	Q₃₂×C₆₁	—	4	4	no	—
27	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
28	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₆₁	—	4	2	no	—
30	C₂×D₂₄₄	—	4	2	no	—
31	C₂×D₂₄₄	—	4	4	no	—
32	Q₁₆×C₆₁	—	4	4	no	—
33	Q₁₆×C₆₁	—	4	4	no	—
34	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₆₁	—	4	4	no	—
37	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂×#92418	D₈	1	4	yes	—
39	D₄₈₈	—	4	4	no	—
40	D₄₈₈	—	4	4	no	—
41	C₁₆×C₆₁	D₈	1	4	yes	—
42	#11758185	D₈	1	4	yes	—
43	C₂²×D₆₁	—	8	2	no	—
44	Q₈×C₆₁	—	8	2	no	—
45	Q₈×C₆₁	—	8	2	no	—
46	C₂×C₄×C₆₁	—	8	2	no	—
47	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
49	D₂₄₄	—	8	2	no	—
50	D₂₄₄	—	8	2	no	—
51	D₂₄₄	—	8	8	no	—
52	C₈×C₆₁	C₂×D₈	1	2	yes	—
53	C₈×C₆₁	C₂×D₈	1	2	yes	—
54	#92418	—	2	4	no	—
55	#12518	—	61	1	no	maximal, Sylow
56	C₂²×C₆₁	#493	1	1	yes	characteristic
57	C₂×D₆₁	—	16	2	no	—
58	C₂×D₆₁	—	16	4	no	—
59	C₄×C₆₁	—	16	2	no	—
60	C₄×C₆₁	#493	1	1	yes	characteristic
61	C₄×C₆₁	#493	1	1	yes	characteristic
62	C₄×C₆₁	#508	1	4	yes	—
63	#2248	—	61	2	no	—
64	#2247	D₆₁	1	1	yes	characteristic, complemented
65	#2246	—	61	1	no	—
66	#1709	—	61	2	no	—
67	C₈×C₁₆	—	61	1	no	—
68	C₂×C₆₁	#1712	1	1	yes	characteristic
69	C₂×C₆₁	#2249	1	2	yes	—
70	D₆₁	—	32	2	no	—
71	C₂×Q₃₂	—	2	2	no	—
72	C₂×SD₃₂	—	122	4	no	—
73	C₂×D₁₆	—	122	2	no	—
74	#494	C₂×D₆₁	1	2	yes	—
75	#493	—	61	2	no	—
76	#486	—	122	4	no	—
77	C₄×C₁₆	—	61	1	no	—
78	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
79	C₈²	—	61	1	no	—
80	C₆₁	#12518	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	—	2	2	no	—
83	C₂×D₈	—	122	2	no	—
84	C₂×D₈	—	122	2	no	—
85	C₄⋊Q₈	—	2	2	no	—
86	#127	—	122	2	no	—
87	Q₃₂	—	4	4	no	—
88	SD₃₂	—	244	8	no	—
89	D₁₆	—	244	4	no	—
90	C₂×C₁₆	—	61	2	no	—
91	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
92	#102	—	244	8	no	—
93	C₄×C₈	—	61	2	no	—
94	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
95	C₂×Q₈	—	4	2	no	—
96	C₂×D₄	—	244	2	no	—
97	C₂×D₄	—	244	4	no	—
98	Q₁₆	—	4	4	no	—
99	Q₁₆	—	4	4	no	—
100	D₈	—	244	4	no	—
101	D₈	—	244	4	no	—
102	C₂×C₈	—	61	4	no	—
103	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
104	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
105	C₄⋊C₄	—	4	4	no	—
106	C₄²	D₄×D₆₁	1	1	yes	characteristic
107	C₁₆	—	61	4	no	—
108	C₁₆	#11758199	1	4	yes	—
109	C₂³	—	488	2	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	8	2	no	—
112	D₄	—	488	8	no	—
113	D₄	—	488	2	no	—
114	D₄	—	488	2	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	#14519773	1	1	yes	characteristic
117	C₂×C₄	D₈×D₆₁	1	2	yes	—
118	C₈	—	122	4	no	—
119	C₈	D₈×D₆₁	1	2	yes	—
120	C₈	C₂×#11758199	1	2	yes	—
121	C₂²	—	976	2	no	—
122	C₂²	—	976	4	no	—
123	C₂²	#16840137	1	1	yes	characteristic
124	C₄	—	16	2	no	—
125	C₄	#16840137	1	1	yes	characteristic
126	C₄	#16840138	1	1	yes	characteristic
127	C₄	#16840195	1	4	yes	—
128	C₂	—	1952	2	no	—
129	C₂	#21445433	1	1	yes	characteristic
130	C₂	#21448081	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21448076

C₂

yes

maximal, complemented

#21448069

C₂

yes

maximal, characteristic

#21445426

C₂

yes

maximal, complemented

C₁₆×#92418

C₂

yes

maximal, characteristic, complemented

C₆₁×#2247

C₂

yes

maximal, characteristic, complemented

C₂×#14519720

—

C₂×D₉₇₆

—

#16840139

C₂²

yes

—

#16840110

—

C₄×#11758185

C₂²

yes

characteristic

C₈×#92418

C₂²

yes

characteristic

C₂×Q₃₂×C₆₁

—

C₆₁×#494

C₂²

yes

—

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×D₄₈₈

—

C₂×D₄₈₈

—

#14519774

—

#14519720

—

D₉₇₆

—

C₂×#11758185

D₄

yes

—

#14519697

—

C₄×#92418

D₄

yes

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₄⋊Q₈×C₆₁

—

Q₃₂×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₂×#92418

D₈

yes

—

D₄₈₈

—

D₄₈₈

—

C₁₆×C₆₁

D₈

yes

—

#11758185

D₈

yes

—

C₂²×D₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₈×C₆₁

C₂×D₈

yes

—

C₈×C₆₁

C₂×D₈

yes

—

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

#2248

—

#2247

D₆₁

yes

characteristic, complemented

#2246

—

#1709

—

C₈×C₁₆

—

C₂×C₆₁

#1712

yes

characteristic

C₂×C₆₁

#2249

yes

—

D₆₁

—

C₂×Q₃₂

—

C₂×SD₃₂

—

122

—

C₂×D₁₆

—

122

—

#494

C₂×D₆₁

yes

—

#493

—

#486

—

122

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

C₈²

—

C₆₁

#12518

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

122

—

C₂×D₈

—

122

—

C₄⋊Q₈

—

#127

—

122

—

Q₃₂

—

SD₃₂

—

244

—

D₁₆

—

244

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₁

yes

—

#102

—

244

—

C₄×C₈

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

Q₁₆

—

Q₁₆

—

100

D₈

—

244

—

101

D₈

—

244

—

102

C₂×C₈

—

103

C₂×C₈

D₄×D₆₁

yes

characteristic

104

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

105

C₄⋊C₄

—

106

C₄²

D₄×D₆₁

yes

characteristic

107

C₁₆

—

108

C₁₆

#11758199

yes

—

109

C₂³

—

488

—

110

Q₈

—

111

Q₈

—

112

D₄

—

488

—

113

D₄

—

488

—

114

D₄

—

488

—

115

C₂×C₄

—

116

C₂×C₄

#14519773

yes

characteristic

117

C₂×C₄

D₈×D₆₁

yes

—

118

C₈

—

122

—

119

C₈

D₈×D₆₁

yes

—

120

C₈

C₂×#11758199

yes

—

121

C₂²

—

976

—

122

C₂²

—

976

—

123

C₂²

#16840137

yes

characteristic

124

C₄

—

125

C₄

#16840137

yes

characteristic

126

C₄

#16840138

yes

characteristic

127

C₄

#16840195

yes

—

128

C₂

—

1952

—

129

C₂

#21445433

yes

characteristic

130

C₂

#21448081

yes

—

	subgroup	G/H
center	C₂²	#16840137
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#2247	C₂
socle	C₂²×C₆₁	#493
prime radical	#21448069	C₂
hypercenter	#2247	D₆₁
hypocenter	C₆₁	#12518
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#16840137

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#2247

C₂

C₂²×C₆₁

C₂

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
1. #21448076	C₂	semidirect
3. #21445426	C₂	semidirect
4. C₁₆×#92418	C₂	semidirect
5. C₆₁×#2247	C₂	semidirect
64. #2247	D₆₁	semidirect
80. C₆₁	#12518	semidirect

H ≅ G/N

type

1. #21448076

C₂

semidirect

3. #21445426

C₂

semidirect

4. C₁₆×#92418

C₂

semidirect

5. C₆₁×#2247

C₂

semidirect

64. #2247

D₆₁

semidirect

80. C₆₁

#12518

semidirect

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

subgroup	index
#21448076	2
#21448069	2
#21445426	2
C₁₆×#92418	2
C₆₁×#2247	2
#12518	61

subgroup

index

#21448076

#21448069

#21445426

C₁₆×#92418

C₆₁×#2247

#12518

Subgroups — #29594737

Conjugacy classes of subgroups

Complements

Maximal subgroups