Group #29594684 — subgroups

Subgroups — #29594684

Order 15616.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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	#21448069	C₂	1	1	yes	maximal, characteristic
2	#21448044	C₂	1	2	yes	maximal, complemented
3	#21445412	C₂	1	2	yes	maximal, complemented
4	#21443735	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₉₇₆	—	2	2	no	—
7	#16840139	C₂²	1	2	yes	—
8	#16840136	—	2	2	no	—
9	#16840110	—	2	2	no	—
10	#16839562	—	2	4	no	—
11	C₄×#11758185	C₂²	1	1	yes	characteristic
12	C₈×#92418	C₂²	1	1	yes	characteristic
13	C₆₁×#500	C₂²	1	2	yes	—
14	C₆₁×#367	—	2	2	no	—
15	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#11758201	—	4	4	no	—
17	C₂×D₄₈₈	—	2	2	no	—
18	C₂×D₄₈₈	—	2	2	no	—
19	#14519774	—	2	2	no	—
20	D₉₇₆	—	4	4	no	—
21	C₂×#11758185	D₄	1	2	yes	—
22	#14519697	—	4	4	no	—
23	C₄×#92418	D₄	1	2	yes	—
24	C₄⋊Q₈×C₆₁	—	2	2	no	—
25	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
26	#107×C₆₁	—	2	2	no	—
27	#107×C₆₁	—	2	2	no	—
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	2	no	—
32	C₂×D₂₄₄	—	4	2	no	—
33	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₆₁	—	4	2	no	—
36	C₄⋊C₄×C₆₁	—	4	2	no	—
37	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
38	#11758201	—	8	8	no	—
39	C₂×#92418	D₈	1	4	yes	—
40	D₄₈₈	—	4	4	no	—
41	D₄₈₈	—	4	4	no	—
42	C₁₆×C₆₁	SD₁₆	1	4	yes	complemented
43	#11758185	—	2	2	no	—
44	C₂²×D₆₁	—	8	2	no	—
45	Q₈×C₆₁	—	8	4	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	4	no	—
52	D₂₄₄	—	8	4	no	—
53	C₈×C₆₁	C₂×SD₁₆	1	2	yes	—
54	C₈×C₆₁	#135	1	2	yes	—
55	#92418	—	2	4	no	—
56	#12506	—	61	1	no	maximal, Sylow
57	C₂²×C₆₁	#493	1	1	yes	characteristic
58	C₂×D₆₁	—	16	2	no	—
59	C₂×D₆₁	—	16	2	no	—
60	C₂×D₆₁	—	16	2	no	—
61	C₄×C₆₁	—	16	2	no	—
62	C₄×C₆₁	#492	1	1	yes	characteristic
63	C₄×C₆₁	#495	1	1	yes	characteristic
64	C₄×C₆₁	#509	1	4	yes	—
65	#2254	D₆₁	1	1	yes	characteristic, complemented
66	#2246	—	61	1	no	—
67	#2240	—	61	2	no	—
68	#1706	—	61	2	no	—
69	#1311	—	61	1	no	—
70	C₂×C₆₁	#1706	1	1	yes	characteristic
71	C₂×C₆₁	#2250	1	2	yes	—
72	D₆₁	—	32	2	no	—
73	C₂×D₁₆	—	122	2	no	—
74	#500	C₂×D₆₁	1	2	yes	—
75	#493	—	61	2	no	—
76	#492	—	122	2	no	—
77	#486	—	122	2	no	—
78	#367	—	2	2	no	—
79	#357	—	122	4	no	—
80	C₄×C₁₆	—	61	1	no	—
81	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
82	C₈²	—	61	1	no	—
83	C₆₁	#12506	1	1	yes	characteristic, complemented, Sylow
84	C₂×SD₁₆	—	244	4	no	—
85	C₂×D₈	—	122	2	no	—
86	C₂×D₈	—	122	2	no	—
87	C₄⋊Q₈	—	2	2	no	—
88	#127	—	122	2	no	—
89	D₁₆	—	244	4	no	—
90	C₂×C₁₆	—	61	2	no	—
91	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
92	#107	—	2	2	no	—
93	#107	—	2	2	no	—
94	#102	—	244	4	no	—
95	C₄×C₈	—	61	2	no	—
96	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
97	C₂×Q₈	—	4	2	no	—
98	C₂×D₄	—	244	2	no	—
99	C₂×D₄	—	244	2	no	—
100	C₂×D₄	—	244	2	no	—
101	SD₁₆	—	488	8	no	—
102	D₈	—	244	4	no	—
103	D₈	—	244	4	no	—
104	C₂×C₈	—	61	4	no	—
105	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
106	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄²	D₄×D₆₁	1	1	yes	characteristic
110	C₁₆	—	122	2	no	—
111	C₁₆	#11758201	1	4	yes	complemented
112	C₂³	—	488	2	no	—
113	Q₈	—	8	4	no	—
114	D₄	—	488	4	no	—
115	D₄	—	488	4	no	—
116	D₄	—	488	2	no	—
117	D₄	—	488	2	no	—
118	C₂×C₄	—	8	2	no	—
119	C₂×C₄	#14519773	1	1	yes	characteristic
120	C₂×C₄	D₈×D₆₁	1	2	yes	—
121	C₈	—	122	4	no	—
122	C₈	C₂×#11758201	1	2	yes	—
123	C₈	#14519800	1	2	yes	—
124	C₂²	—	976	2	no	—
125	C₂²	—	976	2	no	—
126	C₂²	—	976	2	no	—
127	C₂²	#16840137	1	1	yes	characteristic
128	C₄	—	16	2	no	—
129	C₄	#16840136	1	1	yes	characteristic
130	C₄	#16840147	1	1	yes	characteristic
131	C₄	#16840199	1	4	yes	—
132	C₂	—	1952	2	no	—
133	C₂	#21445412	1	1	yes	characteristic
134	C₂	#21448084	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21448069

C₂

yes

maximal, characteristic

#21448044

C₂

yes

maximal, complemented

#21445412

C₂

yes

maximal, complemented

#21443735

C₂

yes

maximal, characteristic, complemented

C₆₁×#2254

C₂

yes

maximal, characteristic, complemented

C₂×D₉₇₆

—

#16840139

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₈×#92418

C₂²

yes

characteristic

C₆₁×#500

C₂²

yes

—

C₆₁×#367

—

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×#11758201

—

C₂×D₄₈₈

—

C₂×D₄₈₈

—

#14519774

—

D₉₇₆

—

C₂×#11758185

D₄

yes

—

#14519697

—

C₄×#92418

D₄

yes

—

C₄⋊Q₈×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

—

#107×C₆₁

—

#107×C₆₁

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

#11758201

—

C₂×#92418

D₈

yes

—

D₄₈₈

—

D₄₈₈

—

C₁₆×C₆₁

SD₁₆

yes

complemented

#11758185

—

C₂²×D₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₈×C₆₁

C₂×SD₁₆

yes

—

C₈×C₆₁

#135

yes

—

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

#2254

D₆₁

yes

characteristic, complemented

—

—

—

—

C₂×C₆₁

#1706

yes

characteristic

C₂×C₆₁

#2250

yes

—

D₆₁

—

C₂×D₁₆

—

122

—

#500

C₂×D₆₁

yes

—

#493

—

#492

—

122

—

#486

—

122

—

#367

—

#357

—

122

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

C₈²

—

C₆₁

#12506

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

244

—

C₂×D₈

—

122

—

C₂×D₈

—

122

—

C₄⋊Q₈

—

#127

—

122

—

D₁₆

—

244

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₁

yes

—

#107

—

#107

—

#102

—

244

—

C₄×C₈

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

100

C₂×D₄

—

244

—

101

SD₁₆

—

488

—

102

D₈

—

244

—

103

D₈

—

244

—

104

C₂×C₈

—

105

C₂×C₈

D₄×D₆₁

yes

characteristic

106

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

D₄×D₆₁

yes

characteristic

110

C₁₆

—

122

—

111

C₁₆

#11758201

yes

complemented

112

C₂³

—

488

—

113

Q₈

—

114

D₄

—

488

—

115

D₄

—

488

—

116

D₄

—

488

—

117

D₄

—

488

—

118

C₂×C₄

—

119

C₂×C₄

#14519773

yes

characteristic

120

C₂×C₄

D₈×D₆₁

yes

—

121

C₈

—

122

—

122

C₈

C₂×#11758201

yes

—

123

C₈

#14519800

yes

—

124

C₂²

—

976

—

125

C₂²

—

976

—

126

C₂²

—

976

—

127

C₂²

#16840137

yes

characteristic

128

C₄

—

129

C₄

#16840136

yes

characteristic

130

C₄

#16840147

yes

characteristic

131

C₄

#16840199

yes

—

132

C₂

—

1952

—

133

C₂

#21445412

yes

characteristic

134

C₂

#21448084

yes

—

	subgroup	G/H
center	C₂²	#16840137
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#2254	C₂
socle	C₂²×C₆₁	#493
prime radical	#21448069	C₂
hypercenter	#2254	D₆₁
hypocenter	C₆₁	#12506
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#16840137

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#2254

C₂

C₂²×C₆₁

C₂

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

N	H ≅ G/N	type
2. #21448044	C₂	semidirect
3. #21445412	C₂	semidirect
4. #21443735	C₂	semidirect
5. C₆₁×#2254	C₂	semidirect
42. C₁₆×C₆₁	SD₁₆	semidirect
65. #2254	D₆₁	semidirect
83. C₆₁	#12506	semidirect
111. C₁₆	#11758201	semidirect

H ≅ G/N

type

2. #21448044

C₂

semidirect

3. #21445412

C₂

semidirect

4. #21443735

C₂

semidirect

5. C₆₁×#2254

C₂

semidirect

42. C₁₆×C₆₁

SD₁₆

semidirect

65. #2254

D₆₁

semidirect

83. C₆₁

#12506

semidirect

111. C₁₆

#11758201

semidirect

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

subgroup	index
#21448069	2
#21448044	2
#21445412	2
#21443735	2
C₆₁×#2254	2
#12506	61

subgroup

index

C₆₁×#2254

Subgroups — #29594684

Conjugacy classes of subgroups

Complements

Maximal subgroups