Group #29594736 — subgroups

Subgroups — #29594736

Order 15616.

330 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 99 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	#21448075	C₂	1	2	yes	maximal, complemented
2	#21448072	C₂	1	1	yes	maximal, characteristic, complemented
3	#21448069	C₂	1	1	yes	maximal, characteristic
4	#21445425	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₈₈	—	2	2	no	—
7	C₂×#14519722	—	2	4	no	—
8	C₂×D₉₇₆	—	2	2	no	—
9	#16840142	C₂²	1	2	yes	—
10	#16840139	C₂²	1	2	yes	—
11	#16840109	—	2	4	no	—
12	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₈²×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₄₄	—	2	2	no	—
16	C₂×Dic₂₄₄	—	2	2	no	—
17	C₂×D₄₈₈	—	2	2	no	—
18	C₂×D₄₈₈	—	2	2	no	—
19	#14519777	—	2	2	no	—
20	#14519774	—	2	2	no	—
21	Dic₄₈₈	—	4	4	no	—
22	#14519722	—	4	8	no	—
23	D₉₇₆	—	4	4	no	—
24	#14519699	—	4	8	no	—
25	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
26	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
27	C₄×C₈×C₆₁	D₄	1	2	yes	—
28	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₂×D₂₄₄	—	4	2	no	—
30	C₂×D₂₄₄	—	4	4	no	—
31	C₂×Dic₁₂₂	—	4	2	no	—
32	C₂×C₈×C₆₁	D₈	1	4	yes	—
33	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₄⋊Dic₆₁	—	4	4	no	—
37	Dic₂₄₄	—	4	4	no	—
38	Dic₂₄₄	—	4	4	no	—
39	D₄₈₈	—	4	4	no	—
40	D₄₈₈	—	4	4	no	—
41	C₁₆×C₆₁	D₈	1	4	yes	—
42	C₁₆×C₆₁	D₈	1	4	yes	—
43	C₂²×D₆₁	—	8	2	no	—
44	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
46	C₂×Dic₆₁	—	8	2	no	—
47	D₂₄₄	—	8	2	no	—
48	D₂₄₄	—	8	2	no	—
49	D₂₄₄	—	8	8	no	—
50	Dic₁₂₂	—	8	2	no	—
51	Dic₁₂₂	—	8	2	no	—
52	C₈×C₆₁	—	2	4	no	—
53	C₈×C₆₁	C₂×D₈	1	2	yes	—
54	C₈×C₆₁	C₂×D₈	1	2	yes	—
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	#493	1	1	yes	characteristic
60	C₄×C₆₁	#493	1	1	yes	characteristic
61	C₄×C₆₁	#508	1	4	yes	—
62	Dic₆₁	—	16	2	no	—
63	#2248	—	61	2	no	—
64	#2247	—	61	1	no	—
65	#2246	—	61	1	no	—
66	#1709	—	61	2	no	—
67	C₈×C₁₆	D₆₁	1	1	yes	characteristic, complemented
68	C₂×C₆₁	#1712	1	1	yes	characteristic
69	C₂×C₆₁	#2249	1	2	yes	—
70	D₆₁	—	32	2	no	—
71	C₂×Q₃₂	—	122	2	no	—
72	C₂×SD₃₂	—	122	4	no	—
73	C₂×D₁₆	—	122	2	no	—
74	#494	—	61	2	no	—
75	#493	—	61	2	no	—
76	#486	—	122	4	no	—
77	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
78	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
79	C₈²	C₂×D₆₁	1	1	yes	characteristic
80	C₆₁	#12518	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₁₆	—	122	2	no	—
82	C₂×Q₁₆	—	122	2	no	—
83	C₂×D₈	—	122	2	no	—
84	C₂×D₈	—	122	2	no	—
85	C₄⋊Q₈	—	122	2	no	—
86	#127	—	122	2	no	—
87	Q₃₂	—	244	4	no	—
88	SD₃₂	—	244	8	no	—
89	D₁₆	—	244	4	no	—
90	C₂×C₁₆	D₂₄₄	1	2	yes	—
91	C₂×C₁₆	D₂₄₄	1	2	yes	—
92	#102	—	244	8	no	—
93	C₄×C₈	D₂₄₄	1	2	yes	—
94	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
95	C₂×Q₈	—	244	2	no	—
96	C₂×D₄	—	244	2	no	—
97	C₂×D₄	—	244	4	no	—
98	Q₁₆	—	244	4	no	—
99	Q₁₆	—	244	4	no	—
100	D₈	—	244	4	no	—
101	D₈	—	244	4	no	—
102	C₂×C₈	D₄₈₈	1	4	yes	—
103	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
104	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
105	C₄⋊C₄	—	244	4	no	—
106	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
107	C₁₆	D₄₈₈	1	4	yes	—
108	C₁₆	D₄₈₈	1	4	yes	—
109	C₂³	—	488	2	no	—
110	Q₈	—	488	2	no	—
111	Q₈	—	488	2	no	—
112	D₄	—	488	8	no	—
113	D₄	—	488	2	no	—
114	D₄	—	488	2	no	—
115	C₂×C₄	—	488	2	no	—
116	C₂×C₄	#14519774	1	1	yes	characteristic
117	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
118	C₈	—	2	4	no	—
119	C₈	C₂×D₄₈₈	1	2	yes	—
120	C₈	C₂×D₄₈₈	1	2	yes	—
121	C₂²	—	976	2	no	—
122	C₂²	—	976	4	no	—
123	C₂²	#16840139	1	1	yes	characteristic
124	C₄	—	976	2	no	—
125	C₄	#16840139	1	1	yes	characteristic
126	C₄	#16840139	1	1	yes	characteristic
127	C₄	#16840194	1	4	yes	—
128	C₂	—	1952	2	no	—
129	C₂	#21445434	1	1	yes	characteristic
130	C₂	#21448080	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21448075

C₂

yes

maximal, complemented

#21448072

C₂

yes

maximal, characteristic, complemented

#21448069

C₂

yes

maximal, characteristic

#21445425

C₂

yes

maximal, complemented

C₈×C₁₆×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₄₈₈

—

C₂×#14519722

—

C₂×D₉₇₆

—

#16840142

C₂²

yes

—

#16840139

C₂²

yes

—

#16840109

—

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₈²×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×D₄₈₈

—

C₂×D₄₈₈

—

#14519777

—

#14519774

—

Dic₄₈₈

—

#14519722

—

D₉₇₆

—

#14519699

—

C₂×C₁₆×C₆₁

D₄

yes

—

C₂×C₁₆×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

Dic₂₄₄

—

Dic₂₄₄

—

D₄₈₈

—

D₄₈₈

—

C₁₆×C₆₁

D₈

yes

—

C₁₆×C₆₁

D₈

yes

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂×D₈

yes

—

C₈×C₆₁

C₂×D₈

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

—

—

—

—

C₈×C₁₆

D₆₁

yes

characteristic, complemented

C₂×C₆₁

#1712

yes

characteristic

C₂×C₆₁

#2249

yes

—

D₆₁

—

C₂×Q₃₂

—

122

—

C₂×SD₃₂

—

122

—

C₂×D₁₆

—

122

—

#494

—

#493

—

#486

—

122

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

C₄×C₁₆

C₂×D₆₁

yes

characteristic

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12518

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

122

—

C₂×D₈

—

122

—

C₂×D₈

—

122

—

C₄⋊Q₈

—

122

—

#127

—

122

—

Q₃₂

—

244

—

SD₃₂

—

244

—

D₁₆

—

244

—

C₂×C₁₆

D₂₄₄

yes

—

C₂×C₁₆

D₂₄₄

yes

—

#102

—

244

—

C₄×C₈

D₂₄₄

yes

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

244

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

Q₁₆

—

244

—

Q₁₆

—

244

—

100

D₈

—

244

—

101

D₈

—

244

—

102

C₂×C₈

D₄₈₈

yes

—

103

C₂×C₈

C₂×D₂₄₄

yes

characteristic

104

C₂×C₈

C₂×D₂₄₄

yes

characteristic

105

C₄⋊C₄

—

244

—

106

C₄²

C₂×D₂₄₄

yes

characteristic

107

C₁₆

D₄₈₈

yes

—

108

C₁₆

D₄₈₈

yes

—

109

C₂³

—

488

—

110

Q₈

—

488

—

111

Q₈

—

488

—

112

D₄

—

488

—

113

D₄

—

488

—

114

D₄

—

488

—

115

C₂×C₄

—

488

—

116

C₂×C₄

#14519774

yes

characteristic

117

C₂×C₄

C₂×D₄₈₈

yes

—

118

C₈

—

119

C₈

C₂×D₄₈₈

yes

—

120

C₈

C₂×D₄₈₈

yes

—

121

C₂²

—

976

—

122

C₂²

—

976

—

123

C₂²

#16840139

yes

characteristic

124

C₄

—

976

—

125

C₄

#16840139

yes

characteristic

126

C₄

#16840139

yes

characteristic

127

C₄

#16840194

yes

—

128

C₂

—

1952

—

129

C₂

#21445434

yes

characteristic

130

C₂

#21448080

yes

—

	subgroup	G/H
center	C₂²	#16840139
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₈×C₁₆×C₆₁	C₂
socle	C₂²×C₆₁	#493
prime radical	#21448069	C₂
hypercenter	C₈×C₁₆	D₆₁
hypocenter	C₆₁	#12518
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#16840139

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₈×C₁₆×C₆₁

C₂

C₂²×C₆₁

C₂

C₈×C₁₆

D₆₁

hypocenter

C₆₁

#12518

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #21448075	C₂	semidirect
2. #21448072	C₂	semidirect
4. #21445425	C₂	semidirect
5. C₈×C₁₆×C₆₁	C₂	semidirect
67. C₈×C₁₆	D₆₁	semidirect
80. C₆₁	#12518	semidirect

H ≅ G/N

type

1. #21448075

C₂

semidirect

2. #21448072

C₂

semidirect

4. #21445425

C₂

semidirect

5. C₈×C₁₆×C₆₁

C₂

semidirect

67. C₈×C₁₆

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
#21448075	2
#21448072	2
#21448069	2
#21445425	2
C₈×C₁₆×C₆₁	2
#12518	61

subgroup

index

C₈×C₁₆×C₆₁

Subgroups — #29594736

Conjugacy classes of subgroups

Complements

Maximal subgroups