Group #29591899 — subgroups

Subgroups — #29591899

Order 15616.

246 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 79 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	#21447696	C₂	1	1	yes	maximal, characteristic
2	#21444644	C₂	1	2	yes	maximal, complemented
3	#21443885	C₂	1	1	yes	maximal, characteristic, complemented
4	#21443787	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1313	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₉₇₆	C₄	1	1	yes	characteristic
7	C₄×D₄₈₈	C₂²	1	2	yes	—
8	C₈×D₂₄₄	—	2	2	no	—
9	#16839590	C₄	1	1	yes	characteristic
10	#16839564	C₄	1	2	yes	—
11	#16839513	C₂²	1	2	yes	—
12	#16839491	—	2	2	no	—
13	C₆₁×#346	C₂²	1	1	yes	characteristic
14	C₆₁×#346	C₂²	1	1	yes	characteristic
15	C₆₁×#322	C₂²	1	1	yes	characteristic
16	C₂×D₄₈₈	C₂×C₄	1	2	yes	—
17	C₂×C₈×D₆₁	—	4	2	no	—
18	C₄×D₂₄₄	—	2	2	no	—
19	D₉₇₆	C₈	1	4	yes	complemented
20	#14519711	C₂×C₄	1	2	yes	—
21	C₄⋊#92418	—	2	2	no	—
22	#14519699	—	2	2	no	—
23	#14519689	—	4	2	no	—
24	C₂×C₁₆×C₆₁	—	2	1	no	—
25	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₆₁	D₄	1	2	yes	—
28	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₂×D₂₄₄	—	2	2	no	—
30	C₂×C₄×D₆₁	—	4	2	no	—
31	C₂×C₈×C₆₁	—	2	2	no	—
32	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
34	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
35	#11758198	—	4	2	no	—
36	C₄⋊Dic₆₁	—	2	2	no	—
37	C₂×#92418	—	4	2	no	—
38	D₄₈₈	—	2	2	no	—
39	D₄₈₈	C₂×C₈	1	4	yes	complemented
40	C₈×D₆₁	—	8	4	no	—
41	C₁₆×C₆₁	—	2	1	no	—
42	C₁₆×C₆₁	—	4	1	no	—
43	C₁₆×C₆₁	C₂×C₈	1	2	yes	complemented
44	C₂²×D₆₁	—	4	2	no	—
45	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
47	C₂×Dic₆₁	—	4	2	no	—
48	D₂₄₄	—	2	4	no	—
49	D₂₄₄	—	4	2	no	—
50	C₄×D₆₁	—	8	2	no	—
51	C₄×D₆₁	—	8	2	no	—
52	C₈×C₆₁	—	2	1	no	—
53	C₈×C₆₁	—	4	2	no	—
54	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₆₁	#131	1	1	yes	characteristic
56	#92418	—	8	2	no	—
57	#12067	—	61	1	no	maximal, Sylow
58	C₂²×C₆₁	#442	1	1	yes	characteristic
59	C₂×D₆₁	—	4	4	no	—
60	C₂×D₆₁	—	8	2	no	—
61	C₂×D₆₁	—	8	2	no	—
62	C₄×C₆₁	—	2	2	no	—
63	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
64	C₄×C₆₁	#444	1	1	yes	characteristic
65	Dic₆₁	—	8	2	no	—
66	#2177	—	61	1	no	—
67	#1580	—	61	2	no	—
68	#1368	—	61	1	no	—
69	#1333	—	61	2	no	—
70	#1313	D₆₁	1	1	yes	characteristic, complemented
71	C₂×C₆₁	#1580	1	1	yes	characteristic
72	C₂×C₆₁	AGL(1,ℤ/16)	1	2	yes	—
73	D₆₁	—	8	4	no	—
74	C₂×D₁₆	—	61	1	no	—
75	C₄×D₈	—	61	2	no	—
76	C₈×D₄	—	122	2	no	—
77	#367	—	61	1	no	—
78	#357	—	61	2	no	—
79	#346	C₂×D₆₁	1	1	yes	characteristic
80	#346	C₂×D₆₁	1	1	yes	characteristic
81	#334	—	61	2	no	—
82	D₄⋊C₈	—	122	2	no	—
83	#322	C₂×D₆₁	1	1	yes	characteristic
84	C₆₁	#12067	1	1	yes	characteristic, complemented, Sylow
85	C₂×D₈	—	61	2	no	—
86	C₂²×C₈	—	244	2	no	—
87	C₄×D₄	—	122	2	no	—
88	D₁₆	—	61	4	no	—
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
91	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
92	#107	—	61	2	no	—
93	C₄⋊C₈	—	122	2	no	—
94	#102	—	122	2	no	—
95	C₂²⋊C₈	—	244	2	no	—
96	C₄×C₈	D₂₄₄	1	2	yes	—
97	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
98	C₂×D₄	—	122	2	no	—
99	C₂²×C₄	—	244	2	no	—
100	D₈	—	61	4	no	—
101	D₈	—	122	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	488	4	no	—
104	C₂×C₈	—	244	2	no	—
105	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
106	C₂×C₈	#11758222	1	1	yes	characteristic
107	C₄⋊C₄	—	122	2	no	—
108	C₂²⋊C₄	—	244	2	no	—
109	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	4	1	no	—
112	C₁₆	C₈×D₆₁	1	2	yes	complemented
113	C₂³	—	244	2	no	—
114	D₄	—	122	4	no	—
115	D₄	—	244	2	no	—
116	C₂×C₄	—	488	2	no	—
117	C₂×C₄	—	488	2	no	—
118	C₂×C₄	—	244	2	no	—
119	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
120	C₂×C₄	#14519805	1	2	yes	—
121	C₈	—	2	1	no	—
122	C₈	—	4	2	no	—
123	C₈	—	488	2	no	—
124	C₈	C₂×C₈×D₆₁	1	1	yes	characteristic
125	C₈	#14519783	1	1	yes	characteristic
126	C₂²	—	488	2	no	—
127	C₂²	—	488	2	no	—
128	C₂²	—	244	4	no	—
129	C₂²	#16839875	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	488	2	no	—
132	C₄	C₈×D₂₄₄	1	1	yes	characteristic
133	C₄	#16839885	1	1	yes	characteristic
134	C₂	—	488	4	no	—
135	C₂	#21444644	1	1	yes	characteristic
136	C₂	#21447719	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21447696

C₂

yes

maximal, characteristic

#21444644

C₂

yes

maximal, complemented

#21443885

C₂

yes

maximal, characteristic, complemented

#21443787

C₂

yes

maximal, complemented

C₆₁×#1313

C₂

yes

maximal, characteristic, complemented

C₂×D₉₇₆

C₄

yes

characteristic

C₄×D₄₈₈

C₂²

yes

—

C₈×D₂₄₄

—

#16839590

C₄

yes

characteristic

#16839564

C₄

yes

—

#16839513

C₂²

yes

—

#16839491

—

C₆₁×#346

C₂²

yes

characteristic

C₆₁×#346

C₂²

yes

characteristic

C₆₁×#322

C₂²

yes

characteristic

C₂×D₄₈₈

C₂×C₄

yes

—

C₂×C₈×D₆₁

—

C₄×D₂₄₄

—

D₉₇₆

C₈

yes

complemented

#14519711

C₂×C₄

yes

—

C₄⋊#92418

—

#14519699

—

#14519689

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×D₂₄₄

—

C₂×C₄×D₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

#11758198

—

C₄⋊Dic₆₁

—

C₂×#92418

—

D₄₈₈

—

D₄₈₈

C₂×C₈

yes

complemented

C₈×D₆₁

—

C₁₆×C₆₁

—

C₁₆×C₆₁

—

C₁₆×C₆₁

C₂×C₈

yes

complemented

C₂²×D₆₁

—

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂²×C₈

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₈×D₄

yes

characteristic

C₄×C₆₁

#444

yes

characteristic

Dic₆₁

—

—

—

—

—

D₆₁

yes

characteristic, complemented

C₂×C₆₁

#1580

yes

characteristic

C₂×C₆₁

AGL(1,ℤ/16)

yes

—

D₆₁

—

C₂×D₁₆

—

C₄×D₈

—

C₈×D₄

—

122

—

#367

—

#357

—

#346

C₂×D₆₁

yes

characteristic

#346

C₂×D₆₁

yes

characteristic

#334

—

D₄⋊C₈

—

122

—

#322

C₂×D₆₁

yes

characteristic

C₆₁

#12067

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

—

244

—

C₄×D₄

—

122

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₆₁

yes

characteristic

C₂×C₁₆

C₄×D₆₁

yes

characteristic

#107

—

C₄⋊C₈

—

122

—

#102

—

122

—

C₂²⋊C₈

—

244

—

C₄×C₈

D₂₄₄

yes

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×D₄

—

122

—

C₂²×C₄

—

244

—

100

D₈

—

101

D₈

—

122

—

102

C₂×C₈

—

103

C₂×C₈

—

488

—

104

C₂×C₈

—

244

—

105

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

106

C₂×C₈

#11758222

yes

characteristic

107

C₄⋊C₄

—

122

—

108

C₂²⋊C₄

—

244

—

109

C₄²

C₂×D₂₄₄

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

C₈×D₆₁

yes

complemented

113

C₂³

—

244

—

114

D₄

—

122

—

115

D₄

—

244

—

116

C₂×C₄

—

488

—

117

C₂×C₄

—

488

—

118

C₂×C₄

—

244

—

119

C₂×C₄

C₄×D₂₄₄

yes

characteristic

120

C₂×C₄

#14519805

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

488

—

124

C₈

C₂×C₈×D₆₁

yes

characteristic

125

C₈

#14519783

yes

characteristic

126

C₂²

—

488

—

127

C₂²

—

488

—

128

C₂²

—

244

—

129

C₂²

#16839875

yes

characteristic

130

C₄

—

131

C₄

—

488

—

132

C₄

C₈×D₂₄₄

yes

characteristic

133

C₄

#16839885

yes

characteristic

134

C₂

—

488

—

135

C₂

#21444644

yes

characteristic

136

C₂

#21447719

yes

—

	subgroup	G/H
center	C₂²	#16839875
commutator	C₈×C₆₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1313	C₂
socle	C₂²×C₆₁	#442
prime radical	C₂×D₉₇₆	C₄
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	#1313	D₆₁
hypocenter	C₆₁	#12067
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

subgroup

G/H

center

C₂²

#16839875

commutator

C₈×C₆₁

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1313

C₂

socle

C₂²×C₆₁

#442

prime radical

C₂×D₉₇₆

C₄

cosocle

C₄×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

N	H ≅ G/N	type
2. #21444644	C₂	semidirect
3. #21443885	C₂	semidirect
4. #21443787	C₂	semidirect
5. C₆₁×#1313	C₂	semidirect
19. D₉₇₆	C₈	semidirect
39. D₄₈₈	C₂×C₈	semidirect
43. C₁₆×C₆₁	C₂×C₈	semidirect
70. #1313	D₆₁	semidirect
84. C₆₁	#12067	semidirect
112. C₁₆	C₈×D₆₁	semidirect

H ≅ G/N

type

2. #21444644

C₂

semidirect

3. #21443885

C₂

semidirect

4. #21443787

C₂

semidirect

5. C₆₁×#1313

C₂

semidirect

19. D₉₇₆

C₈

semidirect

39. D₄₈₈

C₂×C₈

semidirect

43. C₁₆×C₆₁

C₂×C₈

semidirect

70. #1313

D₆₁

semidirect

84. C₆₁

#12067

semidirect

112. C₁₆

C₈×D₆₁

semidirect

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

subgroup	index
#21447696	2
#21444644	2
#21443885	2
#21443787	2
C₆₁×#1313	2
#12067	61

subgroup

index

C₆₁×#1313

Subgroups — #29591899

Conjugacy classes of subgroups

Complements

Maximal subgroups