Group #29596377 — subgroups

Subgroups — #29596377

Order 15616.

302 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 155 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	#21447575	C₂	1	2	yes	maximal
2	#21443901	C₂	1	4	yes	maximal
3	C₆₁×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519717	C₂²	1	4	yes	—
5	C₄⋊#11758185	—	2	4	no	—
6	C₂×#97×C₆₁	C₄	1	2	yes	—
7	C₂×#97×C₆₁	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₆₁×#376	C₄	1	1	yes	characteristic
11	#14519717	—	2	8	no	—
12	C₂×#11758185	—	2	4	no	—
13	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
18	#97×C₆₁	—	2	4	no	—
19	#97×C₆₁	C₈	1	4	yes	—
20	C₄×C₈×C₆₁	—	2	4	no	—
21	C₄×C₈×C₆₁	D₄	1	2	yes	—
22	C₄×C₈×C₆₁	Q₈	1	2	yes	—
23	#95×C₆₁	—	2	2	no	—
24	C₂²×C₄×C₆₁	—	2	2	no	—
25	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
26	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₆₁	—	2	2	no	—
28	C₂×C₈×C₆₁	—	2	2	no	—
29	C₂×C₈×C₆₁	—	2	2	no	—
30	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₆₁	#41	1	4	yes	—
35	C₄²×C₆₁	—	2	2	no	—
36	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
37	C₄²×C₆₁	C₂×C₈	1	2	yes	—
38	C₄²×C₆₁	M₁₆	1	4	yes	—
39	#11758185	—	4	4	no	—
40	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
41	C₂×C₄×C₆₁	—	2	4	no	—
42	C₂×C₄×C₆₁	—	4	2	no	—
43	C₂×C₄×C₆₁	—	4	4	no	—
44	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
45	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	#117	1	2	yes	—
47	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
49	C₈×C₆₁	—	2	2	no	—
50	C₈×C₆₁	—	2	4	no	—
51	C₈×C₆₁	#124	1	2	yes	—
52	C₈×C₆₁	#125	1	2	yes	—
53	#12884	—	61	1	no	maximal, Sylow
54	C₂²×C₆₁	—	2	2	no	—
55	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
56	C₂²×C₆₁	#432	1	2	yes	—
57	C₄×C₆₁	—	2	4	no	—
58	C₄×C₆₁	—	4	4	no	—
59	C₄×C₆₁	#382	1	1	yes	characteristic
60	C₄×C₆₁	#423	1	1	yes	characteristic
61	C₄×C₆₁	#431	1	2	yes	—
62	#2150	—	61	2	no	—
63	#1726	D₆₁	1	1	yes	characteristic
64	#1376	—	61	4	no	—
65	C₂×C₆₁	—	2	2	no	—
66	C₂×C₆₁	#1831	1	1	yes	characteristic
67	C₂×C₆₁	#2153	1	2	yes	—
68	C₂×M₃₂	—	61	4	no	—
69	C₂×#97	Dic₆₁	1	2	yes	—
70	C₂×#97	Dic₆₁	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
72	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
73	#376	Dic₆₁	1	1	yes	characteristic
74	C₄⋊C₁₆	—	122	4	no	—
75	C₆₁	#12884	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
77	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
78	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
79	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
80	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
81	M₃₂	—	122	8	no	—
82	C₂×C₁₆	—	122	4	no	—
83	#97	—	2	4	no	—
84	#97	#92418	1	4	yes	—
85	C₄×C₈	—	2	4	no	—
86	C₄×C₈	Dic₁₂₂	1	2	yes	—
87	C₄×C₈	D₂₄₄	1	2	yes	—
88	#95	—	2	2	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	#11758194	1	2	yes	—
91	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	C₂×#92418	1	4	yes	—
96	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
97	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
98	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
99	C₂×C₈	#11758222	1	4	yes	—
100	C₄²	—	2	2	no	—
101	C₄²	C₂×#92418	1	2	yes	—
102	C₄²	#11758194	1	4	yes	—
103	C₄²	C₄⋊Dic₆₁	1	2	yes	—
104	C₁₆	—	244	4	no	—
105	C₂³	C₂×#11758194	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	C₄⋊#92418	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
111	C₂×C₄	#14519736	1	2	yes	—
112	C₂×C₄	C₂²×#92418	1	1	yes	characteristic
113	C₂×C₄	C₂×#11758194	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	2	4	no	—
116	C₈	#14519765	1	2	yes	—
117	C₈	#14519769	1	2	yes	—
118	C₂²	—	2	2	no	—
119	C₂²	C₂×C₄⋊#92418	1	1	yes	characteristic
120	C₂²	#16839819	1	2	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	#16839628	1	1	yes	characteristic
124	C₄	#16839786	1	1	yes	characteristic
125	C₄	#16839815	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	#21445936	1	1	yes	characteristic
128	C₂	#21447585	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21447575

C₂

yes

maximal

#21443901

C₂

yes

maximal

C₆₁×#1726

C₂

yes

maximal, characteristic

C₂×#14519717

C₂²

yes

—

C₄⋊#11758185

—

C₂×#97×C₆₁

C₄

yes

—

C₂×#97×C₆₁

C₄

yes

characteristic

C₂×C₄×C₈×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#376

C₄

yes

characteristic

#14519717

—

C₂×#11758185

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

#97×C₆₁

—

#97×C₆₁

C₈

yes

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

M₁₆

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂×C₈

yes

—

C₄²×C₆₁

M₁₆

yes

—

#11758185

—

C₂³×C₆₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₈

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#124

yes

—

C₈×C₆₁

#125

yes

—

#12884

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₆₁

#432

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

#2150

—

#1726

D₆₁

yes

characteristic

#1376

—

C₂×C₆₁

—

C₂×C₆₁

#1831

yes

characteristic

C₂×C₆₁

#2153

yes

—

C₂×M₃₂

—

C₂×#97

Dic₆₁

yes

—

C₂×#97

Dic₆₁

yes

characteristic

C₂×C₄×C₈

C₂×D₆₁

yes

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

#376

Dic₆₁

yes

characteristic

C₄⋊C₁₆

—

122

—

C₆₁

#12884

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₆₁

yes

—

C₂²×C₈

C₂×Dic₆₁

yes

characteristic

C₂²×C₈

C₂²×D₆₁

yes

characteristic

C₂×C₄²

C₂×Dic₆₁

yes

—

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

M₃₂

—

122

—

C₂×C₁₆

—

122

—

#97

—

#97

#92418

yes

—

C₄×C₈

—

C₄×C₈

Dic₁₂₂

yes

—

C₄×C₈

D₂₄₄

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

#11758194

yes

—

C₂²×C₄

C₂²×Dic₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#92418

yes

—

C₂×C₈

C₄⋊Dic₆₁

yes

—

C₂×C₈

C₂×Dic₁₂₂

yes

characteristic

C₂×C₈

C₂×D₂₄₄

yes

characteristic

C₂×C₈

#11758222

yes

—

100

C₄²

—

101

C₄²

C₂×#92418

yes

—

102

C₄²

#11758194

yes

—

103

C₄²

C₄⋊Dic₆₁

yes

—

104

C₁₆

—

244

—

105

C₂³

C₂×#11758194

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊#92418

yes

—

110

C₂×C₄

C₂×C₄⋊Dic₆₁

yes

characteristic

111

C₂×C₄

#14519736

yes

—

112

C₂×C₄

C₂²×#92418

yes

characteristic

113

C₂×C₄

C₂×#11758194

yes

—

114

C₈

—

115

C₈

—

116

C₈

#14519765

yes

—

117

C₈

#14519769

yes

—

118

C₂²

—

119

C₂²

C₂×C₄⋊#92418

yes

characteristic

120

C₂²

#16839819

yes

—

121

C₄

—

122

C₄

—

123

C₄

#16839628

yes

characteristic

124

C₄

#16839786

yes

characteristic

125

C₄

#16839815

yes

—

126

C₂

—

127

C₂

#21445936

yes

characteristic

128

C₂

#21447585

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1726	C₂
socle	C₂²×C₆₁	C₂×C₄⋊C₈
prime radical	C₂³×C₆₁	C₂×M₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1726	D₆₁
hypocenter	C₆₁	#12884
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

subgroup

G/H

center

C₂×C₈

C₂×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1726

C₂

socle

C₂²×C₆₁

C₂×C₄⋊C₈

prime radical

C₂³×C₆₁

C₂×M₁₆

cosocle

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

N	H ≅ G/N	type
75. C₆₁	#12884	semidirect

H ≅ G/N

type

75. C₆₁

#12884

semidirect

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

subgroup	index
#21447575	2
#21443901	2
C₆₁×#1726	2
#12884	61

subgroup

index

#21447575

#21443901

C₆₁×#1726

#12884

Subgroups — #29596377

Conjugacy classes of subgroups

Complements

Maximal subgroups