Group #29594683 — subgroups

Subgroups — #29594683

Order 15616.

302 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 91 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	#21448101	C₂	1	1	yes	maximal, characteristic, complemented
2	#21448069	C₂	1	1	yes	maximal, characteristic
3	#21448043	C₂	1	2	yes	maximal, complemented
4	#21445411	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₉₇₆	—	2	2	no	—
7	#16840167	C₂²	1	2	yes	—
8	#16840139	C₂²	1	2	yes	—
9	#16840135	—	2	2	no	—
10	#16840109	—	2	2	no	—
11	#16839590	—	2	2	no	—
12	#16839564	—	2	4	no	—
13	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₈²×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#11758190	—	4	4	no	—
17	C₂×D₄₈₈	—	2	2	no	—
18	C₂×D₄₈₈	—	2	2	no	—
19	#14519777	—	2	2	no	—
20	#14519774	—	2	2	no	—
21	D₉₇₆	—	4	4	no	—
22	#14519711	—	2	2	no	—
23	#14519711	—	2	2	no	—
24	#14519699	—	4	4	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	2	no	—
31	C₂×D₂₄₄	—	4	2	no	—
32	C₂×Dic₁₂₂	—	4	2	no	—
33	C₂×C₈×C₆₁	D₈	1	4	yes	—
34	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₆₁	—	4	2	no	—
38	C₄⋊Dic₆₁	—	4	2	no	—
39	D₄₈₈	—	4	4	no	—
40	D₄₈₈	—	4	4	no	—
41	#11758190	—	8	8	no	—
42	C₁₆×C₆₁	—	2	2	no	—
43	C₁₆×C₆₁	SD₁₆	1	4	yes	complemented
44	C₂²×D₆₁	—	8	2	no	—
45	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
47	C₂×Dic₆₁	—	8	2	no	—
48	D₂₄₄	—	8	2	no	—
49	D₂₄₄	—	8	2	no	—
50	D₂₄₄	—	8	4	no	—
51	D₂₄₄	—	8	4	no	—
52	Dic₁₂₂	—	8	4	no	—
53	C₈×C₆₁	—	2	4	no	—
54	C₈×C₆₁	C₂×SD₁₆	1	2	yes	—
55	C₈×C₆₁	#135	1	2	yes	—
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	#492	1	1	yes	characteristic
62	C₄×C₆₁	#495	1	1	yes	characteristic
63	C₄×C₆₁	#509	1	4	yes	—
64	Dic₆₁	—	16	2	no	—
65	#2254	—	61	1	no	—
66	#2246	—	61	1	no	—
67	#2240	—	61	2	no	—
68	#1706	—	61	2	no	—
69	#1311	D₆₁	1	1	yes	characteristic, complemented
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	—	61	2	no	—
75	#493	—	61	2	no	—
76	#492	—	122	2	no	—
77	#486	—	122	2	no	—
78	#367	—	122	2	no	—
79	#357	—	122	4	no	—
80	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
81	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
82	C₈²	C₂×D₆₁	1	1	yes	characteristic
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	—	122	2	no	—
88	#127	—	122	2	no	—
89	D₁₆	—	244	4	no	—
90	C₂×C₁₆	D₂₄₄	1	2	yes	—
91	C₂×C₁₆	D₂₄₄	1	2	yes	—
92	#107	—	122	2	no	—
93	#107	—	122	2	no	—
94	#102	—	244	4	no	—
95	C₄×C₈	D₂₄₄	1	2	yes	—
96	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
97	C₂×Q₈	—	244	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	D₄₈₈	1	4	yes	—
105	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
106	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
107	C₄⋊C₄	—	244	2	no	—
108	C₄⋊C₄	—	244	2	no	—
109	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
110	C₁₆	—	2	2	no	—
111	C₁₆	#11758190	1	4	yes	complemented
112	C₂³	—	488	2	no	—
113	Q₈	—	488	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	—	488	2	no	—
119	C₂×C₄	#14519774	1	1	yes	characteristic
120	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
121	C₈	—	2	4	no	—
122	C₈	C₂×#11758190	1	2	yes	—
123	C₈	#14519799	1	2	yes	—
124	C₂²	—	976	2	no	—
125	C₂²	—	976	2	no	—
126	C₂²	—	976	2	no	—
127	C₂²	#16840139	1	1	yes	characteristic
128	C₄	—	976	2	no	—
129	C₄	#16840135	1	1	yes	characteristic
130	C₄	#16840146	1	1	yes	characteristic
131	C₄	#16840200	1	4	yes	—
132	C₂	—	1952	2	no	—
133	C₂	#21445411	1	1	yes	characteristic
134	C₂	#21448085	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21448101

C₂

yes

maximal, characteristic, complemented

#21448069

C₂

yes

maximal, characteristic

#21448043

C₂

yes

maximal, complemented

#21445411

C₂

yes

maximal, complemented

C₆₁×#1311

C₂

yes

maximal, characteristic, complemented

C₂×D₉₇₆

—

#16840167

C₂²

yes

—

#16840139

C₂²

yes

—

—

—

—

—

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₈²×C₆₁

C₂²

yes

characteristic

C₂×#11758190

—

C₂×D₄₈₈

—

C₂×D₄₈₈

—

#14519777

—

#14519774

—

D₉₇₆

—

#14519711

—

#14519711

—

#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×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

—

C₄⋊Dic₆₁

—

D₄₈₈

—

D₄₈₈

—

#11758190

—

C₁₆×C₆₁

—

C₁₆×C₆₁

SD₁₆

yes

complemented

C₂²×D₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂×SD₁₆

yes

—

C₈×C₆₁

#135

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

—

—

—

—

D₆₁

yes

characteristic, complemented

C₂×C₆₁

#1706

yes

characteristic

C₂×C₆₁

#2250

yes

—

D₆₁

—

C₂×D₁₆

—

122

—

#500

—

#493

—

#492

—

122

—

#486

—

122

—

#367

—

122

—

#357

—

122

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

C₄×C₁₆

C₂×D₆₁

yes

characteristic

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12506

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

244

—

C₂×D₈

—

122

—

C₂×D₈

—

122

—

C₄⋊Q₈

—

122

—

#127

—

122

—

D₁₆

—

244

—

C₂×C₁₆

D₂₄₄

yes

—

C₂×C₁₆

D₂₄₄

yes

—

#107

—

122

—

#107

—

122

—

#102

—

244

—

C₄×C₈

D₂₄₄

yes

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

244

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

100

C₂×D₄

—

244

—

101

SD₁₆

—

488

—

102

D₈

—

244

—

103

D₈

—

244

—

104

C₂×C₈

D₄₈₈

yes

—

105

C₂×C₈

C₂×D₂₄₄

yes

characteristic

106

C₂×C₈

C₂×D₂₄₄

yes

characteristic

107

C₄⋊C₄

—

244

—

108

C₄⋊C₄

—

244

—

109

C₄²

C₂×D₂₄₄

yes

characteristic

110

C₁₆

—

111

C₁₆

#11758190

yes

complemented

112

C₂³

—

488

—

113

Q₈

—

488

—

114

D₄

—

488

—

115

D₄

—

488

—

116

D₄

—

488

—

117

D₄

—

488

—

118

C₂×C₄

—

488

—

119

C₂×C₄

#14519774

yes

characteristic

120

C₂×C₄

C₂×D₄₈₈

yes

—

121

C₈

—

122

C₈

C₂×#11758190

yes

—

123

C₈

#14519799

yes

—

124

C₂²

—

976

—

125

C₂²

—

976

—

126

C₂²

—

976

—

127

C₂²

#16840139

yes

characteristic

128

C₄

—

976

—

129

C₄

#16840135

yes

characteristic

130

C₄

#16840146

yes

characteristic

131

C₄

#16840200

yes

—

132

C₂

—

1952

—

133

C₂

#21445411

yes

characteristic

134

C₂

#21448085

yes

—

	subgroup	G/H
center	C₂²	#16840139
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1311	C₂
socle	C₂²×C₆₁	#493
prime radical	#21448069	C₂
hypercenter	#1311	D₆₁
hypocenter	C₆₁	#12506
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#16840139

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1311

C₂

C₂²×C₆₁

C₂

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #21448101	C₂	semidirect
3. #21448043	C₂	semidirect
4. #21445411	C₂	semidirect
5. C₆₁×#1311	C₂	semidirect
43. C₁₆×C₆₁	SD₁₆	semidirect
69. #1311	D₆₁	semidirect
83. C₆₁	#12506	semidirect
111. C₁₆	#11758190	semidirect

H ≅ G/N

type

1. #21448101

C₂

semidirect

3. #21448043

C₂

semidirect

4. #21445411

C₂

semidirect

5. C₆₁×#1311

C₂

semidirect

43. C₁₆×C₆₁

SD₁₆

semidirect

69. #1311

D₆₁

semidirect

83. C₆₁

#12506

semidirect

111. C₁₆

#11758190

semidirect

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

subgroup	index
#21448101	2
#21448069	2
#21448043	2
#21445411	2
C₆₁×#1311	2
#12506	61

subgroup

index

C₆₁×#1311

Subgroups — #29594683

Conjugacy classes of subgroups

Complements

Maximal subgroups