Group #29594700 — subgroups

Subgroups — #29594700

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	#21448044	C₂	1	4	yes	maximal, complemented
2	#21445434	C₂	1	1	yes	maximal, characteristic
3	C₈×#11758185	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	#16840139	—	2	1	no	—
6	#16840139	—	2	1	no	—
7	#16840139	C₂²	1	2	yes	—
8	#16839562	—	2	8	no	—
9	C₄×#11758185	C₂²	1	2	yes	—
10	C₆₁×#500	—	2	1	no	—
11	C₆₁×#500	C₂²	1	2	yes	—
12	C₆₁×#494	—	2	1	no	—
13	C₈²×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×D₄₈₈	—	2	4	no	—
15	C₂×D₄₈₈	—	4	2	no	—
16	C₂×D₄₈₈	—	4	2	no	—
17	#14519774	—	2	2	no	—
18	C₂×#11758185	D₄	1	4	yes	—
19	C₂×Q₁₆×C₆₁	—	4	2	no	—
20	C₄⋊Q₈×C₆₁	—	2	2	no	—
21	#107×C₆₁	—	2	4	no	—
22	#107×C₆₁	—	4	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×Q₈×C₆₁	—	4	2	no	—
27	C₂×D₂₄₄	—	4	2	no	—
28	C₂×D₂₄₄	—	4	2	no	—
29	C₂×D₂₄₄	—	4	2	no	—
30	Q₁₆×C₆₁	—	8	4	no	—
31	C₂×C₈×C₆₁	D₈	1	2	yes	—
32	C₂×C₈×C₆₁	D₈	1	2	yes	—
33	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
34	C₄⋊C₄×C₆₁	—	4	2	no	—
35	C₄⋊C₄×C₆₁	—	4	2	no	—
36	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
37	D₄₈₈	—	4	8	no	—
38	D₄₈₈	—	8	4	no	—
39	D₄₈₈	—	8	4	no	—
40	#11758185	—	2	4	no	—
41	C₂²×D₆₁	—	8	2	no	—
42	Q₈×C₆₁	—	8	4	no	—
43	C₂×C₄×C₆₁	—	8	2	no	—
44	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
47	D₂₄₄	—	8	4	no	—
48	D₂₄₄	—	8	4	no	—
49	D₂₄₄	—	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	#493	1	1	yes	characteristic
55	C₂×D₆₁	—	16	2	no	—
56	C₂×D₆₁	—	16	2	no	—
57	C₂×D₆₁	—	16	2	no	—
58	C₄×C₆₁	—	16	2	no	—
59	C₄×C₆₁	#495	1	2	yes	—
60	C₄×C₆₁	C₂×D₁₆	1	2	yes	—
61	C₄×C₆₁	C₂×SD₃₂	1	2	yes	—
62	#2240	—	61	4	no	—
63	#1713	D₆₁	1	1	yes	characteristic, complemented
64	#1712	—	61	1	no	—
65	C₈×C₁₆	—	61	1	no	—
66	C₂×C₆₁	#1709	1	1	yes	characteristic
67	C₂×C₆₁	#2246	1	1	yes	characteristic
68	C₂×C₆₁	#2248	1	1	yes	characteristic
69	D₆₁	—	32	2	no	—
70	#500	—	2	1	no	—
71	#500	C₂×D₆₁	1	2	yes	—
72	#494	—	2	1	no	—
73	#493	—	61	2	no	—
74	#493	—	122	1	no	—
75	#493	—	122	1	no	—
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	—	4	2	no	—
81	C₂×D₈	—	122	4	no	—
82	C₂×D₈	—	244	2	no	—
83	C₂×D₈	—	244	2	no	—
84	C₄⋊Q₈	—	2	2	no	—
85	#127	—	122	2	no	—
86	C₂×C₁₆	—	61	4	no	—
87	#107	—	2	4	no	—
88	#107	—	4	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	—	4	2	no	—
93	C₂×D₄	—	244	2	no	—
94	C₂×D₄	—	244	2	no	—
95	C₂×D₄	—	244	2	no	—
96	Q₁₆	—	8	4	no	—
97	D₈	—	488	4	no	—
98	D₈	—	488	4	no	—
99	D₈	—	244	8	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	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
106	C₁₆	—	122	4	no	—
107	C₂³	—	488	2	no	—
108	Q₈	—	8	4	no	—
109	D₄	—	488	4	no	—
110	D₄	—	488	4	no	—
111	D₄	—	488	4	no	—
112	C₂×C₄	—	8	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	#14519720	1	4	yes	—
118	C₈	#14519800	1	4	yes	—
119	C₂²	—	976	2	no	—
120	C₂²	—	976	2	no	—
121	C₂²	—	976	2	no	—
122	C₂²	#16840138	1	1	yes	characteristic
123	C₄	—	16	2	no	—
124	C₄	#16840147	1	2	yes	—
125	C₄	C₂×#14519718	1	2	yes	—
126	C₄	C₂×#14519720	1	2	yes	—
127	C₂	—	1952	2	no	—
128	C₂	#21445426	1	1	yes	characteristic
129	C₂	#21448068	1	1	yes	characteristic
130	C₂	#21448076	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21448044

C₂

yes

maximal, complemented

#21445434

C₂

yes

maximal, characteristic

C₈×#11758185

C₂

yes

maximal, characteristic, complemented

C₆₁×#1713

C₂

yes

maximal, characteristic, complemented

#16840139

—

#16840139

—

#16840139

C₂²

yes

—

#16839562

—

C₄×#11758185

C₂²

yes

—

C₆₁×#500

—

C₆₁×#500

C₂²

yes

—

C₆₁×#494

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×D₄₈₈

—

C₂×D₄₈₈

—

C₂×D₄₈₈

—

#14519774

—

C₂×#11758185

D₄

yes

—

C₂×Q₁₆×C₆₁

—

C₄⋊Q₈×C₆₁

—

#107×C₆₁

—

#107×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

D₄₈₈

—

D₄₈₈

—

D₄₈₈

—

#11758185

—

C₂²×D₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₈×C₆₁

D₁₆

yes

—

C₈×C₆₁

SD₃₂

yes

—

C₈×C₆₁

#135

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#495

yes

—

C₄×C₆₁

C₂×D₁₆

yes

—

C₄×C₆₁

C₂×SD₃₂

yes

—

#2240

—

#1713

D₆₁

yes

characteristic, complemented

#1712

—

C₈×C₁₆

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#500

—

#500

C₂×D₆₁

yes

—

#494

—

#493

—

#493

—

122

—

#493

—

122

—

#357

—

122

—

C₄×C₁₆

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12509

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

122

—

C₂×D₈

—

244

—

C₂×D₈

—

244

—

C₄⋊Q₈

—

#127

—

122

—

C₂×C₁₆

—

#107

—

#107

—

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

Q₁₆

—

D₈

—

488

—

D₈

—

488

—

D₈

—

244

—

100

C₂×C₈

#11758199

yes

—

101

C₂×C₈

#11758199

yes

—

102

C₂×C₈

D₄×D₆₁

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

106

C₁₆

—

122

—

107

C₂³

—

488

—

108

Q₈

—

109

D₄

—

488

—

110

D₄

—

488

—

111

D₄

—

488

—

112

C₂×C₄

—

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

#14519720

yes

—

118

C₈

#14519800

yes

—

119

C₂²

—

976

—

120

C₂²

—

976

—

121

C₂²

—

976

—

122

C₂²

#16840138

yes

characteristic

123

C₄

—

124

C₄

#16840147

yes

—

125

C₄

C₂×#14519718

yes

—

126

C₄

C₂×#14519720

yes

—

127

C₂

—

1952

—

128

C₂

#21445426

yes

characteristic

129

C₂

#21448068

yes

characteristic

130

C₂

#21448076

yes

characteristic

	subgroup	G/H
center	C₂²	#16840138
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1713	C₂
socle	C₂²×C₆₁	#493
prime radical	#21445434	C₂
hypercenter	#1713	D₆₁
hypocenter	C₆₁	#12509
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#16840138

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1713

C₂

C₂²×C₆₁

C₂

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
1. #21448044	C₂	semidirect
3. C₈×#11758185	C₂	semidirect
4. C₆₁×#1713	C₂	semidirect
63. #1713	D₆₁	semidirect
79. C₆₁	#12509	semidirect

H ≅ G/N

type

1. #21448044

C₂

semidirect

3. C₈×#11758185

C₂

semidirect

4. C₆₁×#1713

C₂

semidirect

63. #1713

D₆₁

semidirect

79. C₆₁

#12509

semidirect

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

subgroup	index
#21448044	2
#21445434	2
C₈×#11758185	2
C₆₁×#1713	2
#12509	61

subgroup

index

C₆₁×#1713

Subgroups — #29594700

Conjugacy classes of subgroups

Complements

Maximal subgroups