Group #29594726 — subgroups

Subgroups — #29594726

Order 15616.

310 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 310 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	#21448072	C₂	1	1	yes	maximal, characteristic
2	#21448071	C₂	1	2	yes	maximal
3	#21445435	C₂	1	2	yes	maximal
4	C₁₆×#92418	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2247	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₈₈	—	2	2	no	—
7	C₂×#14519723	—	2	4	no	—
8	#16840165	—	2	4	no	—
9	#16840142	C₂²	1	2	yes	—
10	C₄×#11758185	C₂²	1	1	yes	characteristic
11	C₈×#92418	C₂²	1	1	yes	characteristic
12	C₂×Q₃₂×C₆₁	—	2	2	no	—
13	C₆₁×#494	C₂²	1	2	yes	—
14	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₄₄	—	2	2	no	—
16	C₂×Dic₂₄₄	—	2	2	no	—
17	#14519777	—	2	2	no	—
18	Dic₄₈₈	—	4	4	no	—
19	#14519723	—	4	8	no	—
20	C₂×#11758185	D₄	1	2	yes	—
21	#14519710	—	4	8	no	—
22	C₄×#92418	D₄	1	2	yes	—
23	C₂×Q₁₆×C₆₁	—	2	2	no	—
24	C₂×Q₁₆×C₆₁	—	2	2	no	—
25	C₄⋊Q₈×C₆₁	—	2	2	no	—
26	Q₃₂×C₆₁	—	4	4	no	—
27	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
28	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₆₁	—	4	2	no	—
30	C₂×Dic₁₂₂	—	4	2	no	—
31	Q₁₆×C₆₁	—	4	4	no	—
32	Q₁₆×C₆₁	—	4	4	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	4	no	—
36	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₆₁	—	4	4	no	—
38	C₂×#92418	D₈	1	4	yes	—
39	Dic₂₄₄	—	4	4	no	—
40	Dic₂₄₄	—	4	4	no	—
41	C₁₆×C₆₁	D₈	1	4	yes	—
42	#11758185	D₈	1	4	yes	—
43	Q₈×C₆₁	—	8	2	no	—
44	Q₈×C₆₁	—	8	2	no	—
45	C₂×C₄×C₆₁	—	8	2	no	—
46	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
48	C₂×Dic₆₁	—	8	2	no	—
49	Dic₁₂₂	—	8	2	no	—
50	Dic₁₂₂	—	8	2	no	—
51	C₈×C₆₁	C₂×D₈	1	2	yes	—
52	C₈×C₆₁	C₂×D₈	1	2	yes	—
53	#92418	Q₃₂	1	8	yes	complemented
54	#12516	—	61	1	no	maximal, Sylow
55	C₂²×C₆₁	#493	1	1	yes	characteristic
56	C₄×C₆₁	—	16	2	no	—
57	C₄×C₆₁	#493	1	1	yes	characteristic
58	C₄×C₆₁	#493	1	1	yes	characteristic
59	C₄×C₆₁	C₂×Q₃₂	1	4	yes	—
60	Dic₆₁	—	16	2	no	—
61	#2247	—	61	2	no	—
62	#2247	—	61	1	no	—
63	#2247	D₆₁	1	1	yes	characteristic
64	#1713	—	61	2	no	—
65	C₈×C₁₆	—	61	1	no	—
66	C₂×C₆₁	#1712	1	1	yes	characteristic
67	C₂×C₆₁	#2247	1	2	yes	—
68	C₂×Q₃₂	—	2	2	no	—
69	C₂×Q₃₂	—	122	4	no	—
70	C₂×Q₃₂	—	122	2	no	—
71	#500	—	122	4	no	—
72	#494	—	61	2	no	—
73	#494	C₂×D₆₁	1	2	yes	—
74	C₄×C₁₆	—	61	1	no	—
75	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
76	C₈²	—	61	1	no	—
77	C₆₁	#12516	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	2	2	no	—
79	C₂×Q₁₆	—	2	2	no	—
80	C₂×Q₁₆	—	122	2	no	—
81	C₂×Q₁₆	—	122	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	122	2	no	—
84	Q₃₂	—	4	4	no	—
85	Q₃₂	—	244	8	no	—
86	Q₃₂	—	244	4	no	—
87	C₂×C₁₆	—	61	2	no	—
88	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
89	#107	—	244	8	no	—
90	C₄×C₈	—	61	2	no	—
91	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
92	C₂×Q₈	—	4	2	no	—
93	C₂×Q₈	—	244	2	no	—
94	Q₁₆	—	4	4	no	—
95	Q₁₆	—	4	4	no	—
96	Q₁₆	—	244	4	no	—
97	Q₁₆	—	244	4	no	—
98	C₂×C₈	—	61	4	no	—
99	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
101	C₄⋊C₄	—	4	4	no	—
102	C₄⋊C₄	—	244	4	no	—
103	C₄²	D₄×D₆₁	1	1	yes	characteristic
104	C₁₆	—	61	4	no	—
105	C₁₆	#11758199	1	4	yes	—
106	Q₈	—	8	2	no	—
107	Q₈	—	8	2	no	—
108	Q₈	—	488	2	no	—
109	Q₈	—	488	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	488	2	no	—
112	C₂×C₄	#14519773	1	1	yes	characteristic
113	C₂×C₄	D₈×D₆₁	1	2	yes	—
114	C₈	—	61	8	no	—
115	C₈	D₈×D₆₁	1	2	yes	—
116	C₈	C₂×#11758199	1	2	yes	—
117	C₂²	#16840137	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	976	2	no	—
120	C₄	#16840137	1	1	yes	characteristic
121	C₄	#16840138	1	1	yes	characteristic
122	C₄	Q₃₂×D₆₁	1	4	yes	—
123	C₂	#21445433	1	1	yes	characteristic
124	C₂	#21448070	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21448072

C₂

yes

maximal, characteristic

#21448071

C₂

yes

maximal

#21445435

C₂

yes

maximal

C₁₆×#92418

C₂

yes

maximal, characteristic

C₆₁×#2247

C₂

yes

maximal, characteristic

C₂×Dic₄₈₈

—

C₂×#14519723

—

#16840165

—

#16840142

C₂²

yes

—

C₄×#11758185

C₂²

yes

characteristic

C₈×#92418

C₂²

yes

characteristic

C₂×Q₃₂×C₆₁

—

C₆₁×#494

C₂²

yes

—

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

#14519777

—

Dic₄₈₈

—

#14519723

—

C₂×#11758185

D₄

yes

—

#14519710

—

C₄×#92418

D₄

yes

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₄⋊Q₈×C₆₁

—

Q₃₂×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Dic₁₂₂

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

C₂×#92418

D₈

yes

—

Dic₂₄₄

—

Dic₂₄₄

—

C₁₆×C₆₁

D₈

yes

—

#11758185

D₈

yes

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

C₂×D₈

yes

—

C₈×C₆₁

C₂×D₈

yes

—

#92418

Q₃₂

yes

—

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₂×Q₃₂

yes

—

Dic₆₁

—

#2247

—

#2247

—

#2247

D₆₁

yes

characteristic

#1713

—

C₈×C₁₆

—

C₂×C₆₁

#1712

yes

characteristic

C₂×C₆₁

#2247

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

122

—

C₂×Q₃₂

—

122

—

#500

—

122

—

#494

—

#494

C₂×D₆₁

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

C₈²

—

C₆₁

#12516

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

Q₃₂

—

Q₃₂

—

244

—

Q₃₂

—

244

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₁

yes

—

#107

—

244

—

C₄×C₈

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

244

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

244

—

Q₁₆

—

244

—

C₂×C₈

—

C₂×C₈

D₄×D₆₁

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

244

—

103

C₄²

D₄×D₆₁

yes

characteristic

104

C₁₆

—

105

C₁₆

#11758199

yes

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

488

—

109

Q₈

—

488

—

110

C₂×C₄

—

111

C₂×C₄

—

488

—

112

C₂×C₄

#14519773

yes

characteristic

113

C₂×C₄

D₈×D₆₁

yes

—

114

C₈

—

115

C₈

D₈×D₆₁

yes

—

116

C₈

C₂×#11758199

yes

—

117

C₂²

#16840137

yes

characteristic

118

C₄

—

119

C₄

—

976

—

120

C₄

#16840137

yes

characteristic

121

C₄

#16840138

yes

characteristic

122

C₄

Q₃₂×D₆₁

yes

—

123

C₂

#21445433

yes

characteristic

124

C₂

#21448070

yes

—

	subgroup	G/H
center	C₂²	#16840137
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#2247	C₂
socle, prime radical	C₂²×C₆₁	#493
hypercenter	#2247	D₆₁
hypocenter	C₆₁	#12516
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#16840137

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#2247

C₂

C₂²×C₆₁

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
53. #92418	Q₃₂	semidirect
77. C₆₁	#12516	semidirect

H ≅ G/N

type

53. #92418

Q₃₂

semidirect

77. C₆₁

#12516

semidirect

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

subgroup	index
#21448072	2
#21448071	2
#21445435	2
C₁₆×#92418	2
C₆₁×#2247	2
#12516	61

subgroup

index

#21448072

#21448071

#21445435

C₁₆×#92418

C₆₁×#2247

#12516

Subgroups — #29594726

Conjugacy classes of subgroups

Complements

Maximal subgroups