Group #30665596 — subgroups

Subgroups — #30665596

Order 16768.

176 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 61 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	#22553710	C₂	1	1	yes	maximal, characteristic
2	#22553708	C₂	1	1	yes	maximal, characteristic
3	#22553708	C₂	1	1	yes	maximal, characteristic
4	#22553691	C₂	1	1	yes	maximal, characteristic
5	#22553664	C₂	1	1	yes	maximal, characteristic
6	#22553034	C₂	1	1	yes	maximal, characteristic
7	#487×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523503	—	2	2	no	—
9	#16862246	C₂²	1	1	yes	characteristic
10	#16862246	C₂²	1	1	yes	characteristic
11	#16862178	—	2	2	no	—
12	#16862177	—	2	2	no	—
13	#16862177	—	2	2	no	—
14	#16862169	—	2	2	no	—
15	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	#103×C₁₃₁	—	2	2	no	—
20	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₆₂	—	2	1	no	—
22	C₂×Dic₂₆₂	—	2	1	no	—
23	#14523503	—	4	4	no	—
24	C₂×C₁₃₁⋊C₈	D₄	1	2	yes	—
25	C₂×C₁₃₁⋊C₈	D₄	1	2	yes	—
26	C₄⋊Dic₁₃₁	—	2	1	no	—
27	C₄⋊Dic₁₃₁	—	2	1	no	—
28	C₄⋊Dic₁₃₁	—	2	1	no	—
29	C₄⋊Dic₁₃₁	—	2	1	no	—
30	C₂×Q₈×C₁₃₁	—	2	1	no	—
31	C₂×Q₈×C₁₃₁	—	2	1	no	—
32	C₂×C₈×C₁₃₁	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₃₁	—	2	1	no	—
34	C₄⋊C₄×C₁₃₁	—	2	1	no	—
35	C₄⋊C₄×C₁₃₁	—	2	1	no	—
36	C₄⋊C₄×C₁₃₁	—	2	1	no	—
37	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
38	Dic₂₆₂	—	4	2	no	—
39	Dic₂₆₂	—	4	2	no	—
40	C₂×Dic₁₃₁	—	4	1	no	—
41	C₂×Dic₁₃₁	—	4	1	no	—
42	C₁₃₁⋊C₈	—	2	2	no	—
43	C₁₃₁⋊C₈	SD₁₆	1	4	yes	—
44	Q₈×C₁₃₁	—	4	2	no	—
45	Q₈×C₁₃₁	—	4	2	no	—
46	C₂×C₄×C₁₃₁	—	4	1	no	—
47	C₂×C₄×C₁₃₁	—	4	1	no	—
48	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₃₁	—	2	2	no	—
52	C₂²×C₁₃₁	#127	1	1	yes	characteristic
53	C₄×C₁₃₁	—	8	1	no	—
54	C₄×C₁₃₁	—	8	1	no	—
55	C₄×C₁₃₁	C₂×SD₁₆	1	2	yes	—
56	C₄×C₁₃₁	#137	1	2	yes	—
57	C₄×C₁₃₁	#137	1	2	yes	—
58	Dic₁₃₁	—	8	1	no	—
59	Dic₁₃₁	—	8	1	no	—
60	C₂×C₁₃₁	#492	1	1	yes	characteristic
61	C₂×C₁₃₁	#497	1	1	yes	characteristic
62	C₂×C₁₃₁	#497	1	1	yes	characteristic
63	C₁₃₁	#1716	1	1	yes	characteristic, complemented, Sylow
64	#1716	—	131	1	no	maximal, Sylow
65	#498	—	131	1	no	—
66	#498	—	131	1	no	—
67	#498	—	131	1	no	—
68	#494	—	131	1	no	—
69	#487	—	131	1	no	—
70	#487	D₁₃₁	1	1	yes	characteristic
71	#322	—	131	1	no	—
72	C₂×Q₁₆	—	262	2	no	—
73	C₄⋊Q₈	—	131	1	no	—
74	C₄⋊Q₈	—	131	1	no	—
75	C₄⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
77	#106	—	262	2	no	—
78	#106	—	262	2	no	—
79	#106	—	262	2	no	—
80	#103	—	2	2	no	—
81	#103	—	262	2	no	—
82	C₄×C₈	—	131	1	no	—
83	C₄×C₈	—	131	1	no	—
84	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	262	1	no	—
88	C₂×Q₈	—	262	1	no	—
89	Q₁₆	—	524	4	no	—
90	C₂×C₈	—	131	2	no	—
91	C₂×C₈	—	131	2	no	—
92	C₂×C₈	C₂²⋊D₁₃₁	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	262	1	no	—
98	C₄⋊C₄	—	262	1	no	—
99	C₄⋊C₄	—	262	1	no	—
100	C₄⋊C₄	—	262	1	no	—
101	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	524	2	no	—
105	Q₈	—	524	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	524	1	no	—
109	C₂×C₄	—	524	1	no	—
110	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	262	2	no	—
115	C₈	—	131	4	no	—
116	C₂²	#16862242	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	1048	1	no	—
120	C₄	—	1048	1	no	—
121	C₄	SD₁₆×D₁₃₁	1	2	yes	—
122	C₄	#16862276	1	2	yes	—
123	C₄	#16862278	1	2	yes	—
124	C₂	#22553683	1	1	yes	characteristic
125	C₂	#22553705	1	1	yes	characteristic
126	C₂	#22553707	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22553710

C₂

yes

maximal, characteristic

#22553708

C₂

yes

maximal, characteristic

#22553708

C₂

yes

maximal, characteristic

#22553691

C₂

yes

maximal, characteristic

#22553664

C₂

yes

maximal, characteristic

#22553034

C₂

yes

maximal, characteristic

#487×C₁₃₁

C₂

yes

maximal, characteristic

C₂×#14523503

—

#16862246

C₂²

yes

characteristic

#16862246

C₂²

yes

—

—

—

—

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₁

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₁

C₂²

yes

characteristic

#103×C₁₃₁

—

C₄×C₈×C₁₃₁

C₂²

yes

characteristic

C₂×Dic₂₆₂

—

C₂×Dic₂₆₂

—

#14523503

—

C₂×C₁₃₁⋊C₈

D₄

yes

—

C₂×C₁₃₁⋊C₈

D₄

yes

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×C₈×C₁₃₁

D₄

yes

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄²×C₁₃₁

C₂³

yes

characteristic

Dic₂₆₂

—

Dic₂₆₂

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

SD₁₆

yes

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₈×C₁₃₁

—

C₂²×C₁₃₁

#127

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×SD₁₆

yes

—

C₄×C₁₃₁

#137

yes

—

C₄×C₁₃₁

#137

yes

—

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

#1716

—

131

maximal, Sylow

#498

—

131

—

#498

—

131

—

#498

—

131

—

#494

—

131

—

#487

—

131

—

#487

D₁₃₁

yes

characteristic

#322

—

131

—

C₂×Q₁₆

—

262

—

C₄⋊Q₈

—

131

—

C₄⋊Q₈

—

131

—

C₄⋊Q₈

C₂×D₁₃₁

yes

characteristic

C₄⋊Q₈

C₂×D₁₃₁

yes

characteristic

#106

—

262

—

#106

—

262

—

#106

—

262

—

#103

—

#103

—

262

—

C₄×C₈

—

131

—

C₄×C₈

—

131

—

C₄×C₈

C₂×D₁₃₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

262

—

C₂×Q₈

—

262

—

Q₁₆

—

524

—

C₂×C₈

—

131

—

C₂×C₈

—

131

—

C₂×C₈

C₂²⋊D₁₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

262

—

C₄⋊C₄

—

262

—

C₄⋊C₄

—

262

—

100

C₄⋊C₄

—

262

—

101

C₄²

C₂²×D₁₃₁

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

524

—

105

Q₈

—

524

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

524

—

109

C₂×C₄

—

524

—

110

C₂×C₄

D₄×D₁₃₁

yes

characteristic

111

C₂×C₄

D₄×D₁₃₁

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

characteristic

113

C₈

—

114

C₈

—

262

—

115

C₈

—

131

—

116

C₂²

#16862242

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

1048

—

120

C₄

—

1048

—

121

C₄

SD₁₆×D₁₃₁

yes

—

122

C₄

#16862276

yes

—

123

C₄

#16862278

yes

—

124

C₂

#22553683

yes

characteristic

125

C₂

#22553705

yes

characteristic

126

C₂

#22553707

yes

characteristic

	subgroup	G/H
center	C₂²	#16862242
commutator, cosocle	C₄²×C₁₃₁	C₂³
Frattini	C₄²	C₂²×D₁₃₁
Fitting	#487×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	#127
hypercenter	#487	D₁₃₁
hypocenter	C₁₃₁	#1716
LCS	G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#16862242

commutator, cosocle

C₄²×C₁₃₁

C₂³

Frattini

C₄²

C₂²×D₁₃₁

Fitting

#487×C₁₃₁

C₂

C₂²×C₁₃₁

D₁₃₁

C₁₃₁

G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
63. C₁₃₁	#1716	semidirect

H ≅ G/N

type

63. C₁₃₁

#1716

semidirect

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

subgroup	index
#22553710	2
#22553708	2
#22553691	2
#22553664	2
#22553034	2
#487×C₁₃₁	2
#1716	131

subgroup

index

#487×C₁₃₁

131

Subgroups — #30665596

Conjugacy classes of subgroups

Complements

Maximal subgroups