Group #30664261 — subgroups

Subgroups — #30664261

Order 16768.

178 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 101 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	#22553365	C₂	1	1	yes	maximal, characteristic
2	#22553365	C₂	1	1	yes	maximal, characteristic
3	#22553034	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553034	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553033	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₁₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#431×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	C₄⋊(C₁₃₁⋊C₈)	C₄	1	2	yes	—
9	C₄⋊(C₁₃₁⋊C₈)	C₄	1	2	yes	—
10	C₂²⋊(C₁₃₁⋊C₈)	C₄	1	2	yes	—
11	C₂²⋊(C₁₃₁⋊C₈)	C₄	1	2	yes	—
12	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₁₃₁	C₄	1	2	yes	—
18	C₂²⋊C₈×C₁₃₁	C₄	1	2	yes	—
19	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₁₃₁⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₁₃₁⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₁₃₁⋊C₈	C₂×C₄	1	2	yes	—
24	C₂×C₁₃₁⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
30	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
34	C₁₃₁⋊C₈	—	2	2	no	—
35	C₁₃₁⋊C₈	—	2	2	no	—
36	C₁₃₁⋊C₈	—	2	2	no	—
37	C₁₃₁⋊C₈	—	2	2	no	—
38	C₂³×C₁₃₁	C₄²	1	1	yes	characteristic
39	C₂×C₄×C₁₃₁	—	2	2	no	—
40	C₂×C₄×C₁₃₁	—	2	2	no	—
41	C₂×C₄×C₁₃₁	—	2	2	no	—
42	C₂×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
43	C₂×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
44	C₂×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
45	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
48	C₈×C₁₃₁	—	2	2	no	—
49	C₈×C₁₃₁	—	2	2	no	—
50	C₂²×C₁₃₁	—	2	1	no	—
51	C₂²×C₁₃₁	—	2	1	no	—
52	C₂²×C₁₃₁	—	2	1	no	—
53	C₂²×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
54	C₄×C₁₃₁	—	2	1	no	—
55	C₄×C₁₃₁	—	2	1	no	—
56	C₄×C₁₃₁	—	2	1	no	—
57	C₄×C₁₃₁	#131	1	2	yes	—
58	C₄×C₁₃₁	#131	1	2	yes	—
59	C₄×C₁₃₁	#131	1	2	yes	—
60	C₂×C₁₃₁	—	4	1	no	—
61	C₂×C₁₃₁	#405	1	1	yes	characteristic
62	C₂×C₁₃₁	#405	1	1	yes	characteristic
63	C₂×C₁₃₁	#405	1	1	yes	characteristic
64	C₁₃₁	#1454	1	1	yes	characteristic, complemented, Sylow
65	#1454	—	131	1	no	maximal, Sylow
66	#431	—	131	1	no	—
67	#431	—	131	1	no	—
68	#431	D₁₃₁	1	1	yes	characteristic
69	#322	—	131	1	no	—
70	#322	—	131	1	no	—
71	#322	—	131	1	no	—
72	C₈²	—	131	1	no	—
73	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
74	C₄⋊C₈	—	131	2	no	—
75	C₄⋊C₈	—	131	2	no	—
76	C₄⋊C₈	Dic₁₃₁	1	2	yes	—
77	C₂²⋊C₈	—	131	2	no	—
78	C₂²⋊C₈	—	131	2	no	—
79	C₂²⋊C₈	Dic₁₃₁	1	2	yes	—
80	C₄×C₈	—	131	1	no	—
81	C₄×C₈	—	131	1	no	—
82	C₄×C₈	—	131	1	no	—
83	C₄×C₈	—	131	1	no	—
84	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
85	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
86	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
89	C₂×C₈	—	131	2	no	—
90	C₂×C₈	—	131	2	no	—
91	C₂×C₈	—	131	2	no	—
92	C₂×C₈	—	131	2	no	—
93	C₂×C₈	C₂×Dic₁₃₁	1	2	yes	—
94	C₂×C₈	C₂×Dic₁₃₁	1	2	yes	—
95	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
96	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
97	C₄²	C₂×Dic₁₃₁	1	1	yes	characteristic
98	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
99	C₂³	C₄×Dic₁₃₁	1	1	yes	characteristic
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	C₄×Dic₁₃₁	1	1	yes	characteristic
104	C₂×C₄	C₄×Dic₁₃₁	1	1	yes	characteristic
105	C₂×C₄	C₄×Dic₁₃₁	1	1	yes	characteristic
106	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
107	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
108	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	2	no	—
111	C₈	—	262	2	no	—
112	C₈	—	262	2	no	—
113	C₈	—	262	2	no	—
114	C₈	—	262	2	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	C₂×C₄×Dic₁₃₁	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	#16862252	1	2	yes	—
123	C₄	#16862253	1	2	yes	—
124	C₄	#16862254	1	2	yes	—
125	C₂	—	4	1	no	—
126	C₂	#22553267	1	1	yes	characteristic
127	C₂	#22553268	1	1	yes	characteristic
128	C₂	#22553270	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22553365

C₂

yes

maximal, characteristic

#22553365

C₂

yes

maximal, characteristic

#22553034

C₂

yes

maximal, characteristic, complemented

#22553034

C₂

yes

maximal, characteristic, complemented

#22553033

C₂

yes

maximal, characteristic, complemented

C₈×C₁₃₁⋊C₈

C₂

yes

maximal, characteristic, complemented

#431×C₁₃₁

C₂

yes

maximal, characteristic

C₄⋊(C₁₃₁⋊C₈)

C₄

yes

—

C₄⋊(C₁₃₁⋊C₈)

C₄

yes

—

C₂²⋊(C₁₃₁⋊C₈)

C₄

yes

—

C₂²⋊(C₁₃₁⋊C₈)

C₄

yes

—

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₁

C₄

yes

—

C₂²⋊C₈×C₁₃₁

C₄

yes

—

C₄×C₈×C₁₃₁

C₂²

yes

characteristic

C₄×C₈×C₁₃₁

C₂²

yes

characteristic

C₂×C₁₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₃₁⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₁

C₂×C₄

yes

—

C₂×C₈×C₁₃₁

C₂×C₄

yes

—

C₄²×C₁₃₁

C₂×C₄

yes

characteristic

C₄²×C₁₃₁

C₂×C₄

yes

characteristic

C₄²×C₁₃₁

C₂×C₄

yes

characteristic

C₄²×C₁₃₁

C₂³

yes

characteristic

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

—

C₂³×C₁₃₁

C₄²

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₄²

yes

characteristic

C₂×C₄×C₁₃₁

C₄²

yes

characteristic

C₂×C₄×C₁₃₁

C₄²

yes

characteristic

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

C₂×C₄²

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#131

yes

—

C₄×C₁₃₁

#131

yes

—

C₄×C₁₃₁

#131

yes

—

C₂×C₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

#1454

—

131

maximal, Sylow

#431

—

131

—

#431

—

131

—

#431

D₁₃₁

yes

characteristic

#322

—

131

—

#322

—

131

—

#322

—

131

—

C₈²

—

131

—

C₂×C₄²

C₂×D₁₃₁

yes

characteristic

C₄⋊C₈

—

131

—

C₄⋊C₈

—

131

—

C₄⋊C₈

Dic₁₃₁

yes

—

C₂²⋊C₈

—

131

—

C₂²⋊C₈

—

131

—

C₂²⋊C₈

Dic₁₃₁

yes

—

C₄×C₈

—

131

—

C₄×C₈

—

131

—

C₄×C₈

—

131

—

C₄×C₈

—

131

—

C₄×C₈

C₂×D₁₃₁

yes

characteristic

C₄×C₈

C₂×D₁₃₁

yes

characteristic

C₂²×C₄

C₄×D₁₃₁

yes

characteristic

C₂²×C₄

C₄×D₁₃₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₃₁

yes

characteristic

C₂×C₈

—

131

—

C₂×C₈

—

131

—

C₂×C₈

—

131

—

C₂×C₈

—

131

—

C₂×C₈

C₂×Dic₁₃₁

yes

—

C₂×C₈

C₂×Dic₁₃₁

yes

—

C₄²

C₄×D₁₃₁

yes

characteristic

C₄²

C₄×D₁₃₁

yes

characteristic

C₄²

C₂×Dic₁₃₁

yes

characteristic

C₄²

C₂²×D₁₃₁

yes

characteristic

C₂³

C₄×Dic₁₃₁

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₄×Dic₁₃₁

yes

characteristic

104

C₂×C₄

C₄×Dic₁₃₁

yes

characteristic

105

C₂×C₄

C₄×Dic₁₃₁

yes

characteristic

106

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

107

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

108

C₂×C₄

C₂²×Dic₁₃₁

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

262

—

112

C₈

—

262

—

113

C₈

—

262

—

114

C₈

—

262

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂×C₄×Dic₁₃₁

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#16862252

yes

—

123

C₄

#16862253

yes

—

124

C₄

#16862254

yes

—

125

C₂

—

126

C₂

#22553267

yes

characteristic

127

C₂

#22553268

yes

characteristic

128

C₂

#22553270

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂×C₄²
Fitting	#431×C₁₃₁	C₂
prime radical	C₂³×C₁₃₁	C₄²
cosocle	C₄²×C₁₃₁	C₂³
hypercenter	#431	D₁₃₁
hypocenter	C₁₃₁	#1454
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₁₃₁

commutator, socle

C₂²×C₁₃₁

C₂×C₄²

Fitting

#431×C₁₃₁

C₂

prime radical

C₂³×C₁₃₁

C₄²

cosocle

C₄²×C₁₃₁

C₂³

D₁₃₁

C₁₃₁

G ▷ C₂²×C₁₃₁ ▷ C₁₃₁

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
3. #22553034	C₂	semidirect
4. #22553034	C₂	semidirect
5. #22553033	C₂	semidirect
6. C₈×C₁₃₁⋊C₈	C₂	semidirect
64. C₁₃₁	#1454	semidirect

H ≅ G/N

type

3. #22553034

C₂

semidirect

4. #22553034

C₂

semidirect

5. #22553033

C₂

semidirect

6. C₈×C₁₃₁⋊C₈

C₂

semidirect

64. C₁₃₁

#1454

semidirect

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

subgroup	index
#22553365	2
#22553034	2
#22553033	2
C₈×C₁₃₁⋊C₈	2
#431×C₁₃₁	2
#1454	131

subgroup

index

#22553365

#22553034

#22553033

C₈×C₁₃₁⋊C₈

#431×C₁₃₁

#1454

131

Subgroups — #30664261

Conjugacy classes of subgroups

Complements

Maximal subgroups