Group #30665608 — subgroups

Subgroups — #30665608

Order 16768.

164 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22553715	C₂	1	1	yes	maximal, characteristic
2	#22553708	C₂	1	1	yes	maximal, characteristic
3	#22553664	C₂	1	1	yes	maximal, characteristic
4	#22553663	C₂	1	1	yes	maximal, characteristic
5	#22553662	C₂	1	1	yes	maximal, characteristic
6	#22553033	C₂	1	1	yes	maximal, characteristic
7	#487×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	#16862246	C₂²	1	1	yes	characteristic
9	#16862246	C₂²	1	1	yes	characteristic
10	#16862179	—	2	2	no	—
11	#16862177	—	2	2	no	—
12	#16862171	—	2	2	no	—
13	#16862170	—	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	C₂×C₁₃₁⋊C₈	D₄	1	2	yes	—
24	C₂×C₁₃₁⋊C₈	D₄	1	2	yes	—
25	C₄⋊Dic₁₃₁	—	2	1	no	—
26	C₄⋊Dic₁₃₁	—	2	1	no	—
27	C₄⋊Dic₁₃₁	—	2	1	no	—
28	C₄⋊Dic₁₃₁	—	2	1	no	—
29	C₂×Q₈×C₁₃₁	—	2	1	no	—
30	C₂×Q₈×C₁₃₁	—	2	1	no	—
31	C₂×C₈×C₁₃₁	D₄	1	2	yes	—
32	C₄⋊C₄×C₁₃₁	—	2	1	no	—
33	C₄⋊C₄×C₁₃₁	—	2	1	no	—
34	C₄⋊C₄×C₁₃₁	—	2	1	no	—
35	C₄⋊C₄×C₁₃₁	—	2	1	no	—
36	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
37	Dic₂₆₂	—	4	2	no	—
38	Dic₂₆₂	—	4	2	no	—
39	C₂×Dic₁₃₁	—	4	1	no	—
40	C₂×Dic₁₃₁	—	4	1	no	—
41	C₁₃₁⋊C₈	—	2	2	no	—
42	C₁₃₁⋊C₈	—	2	2	no	—
43	Q₈×C₁₃₁	—	4	2	no	—
44	Q₈×C₁₃₁	—	4	2	no	—
45	C₂×C₄×C₁₃₁	—	4	1	no	—
46	C₂×C₄×C₁₃₁	—	4	1	no	—
47	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
50	C₈×C₁₃₁	—	2	2	no	—
51	C₂²×C₁₃₁	#127	1	1	yes	characteristic
52	C₄×C₁₃₁	—	8	1	no	—
53	C₄×C₁₃₁	—	8	1	no	—
54	C₄×C₁₃₁	#135	1	2	yes	—
55	C₄×C₁₃₁	#137	1	2	yes	—
56	C₄×C₁₃₁	#137	1	2	yes	—
57	Dic₁₃₁	—	8	1	no	—
58	Dic₁₃₁	—	8	1	no	—
59	C₂×C₁₃₁	#495	1	1	yes	characteristic
60	C₂×C₁₃₁	#497	1	1	yes	characteristic
61	C₂×C₁₃₁	#497	1	1	yes	characteristic
62	C₁₃₁	#1718	1	1	yes	characteristic, complemented, Sylow
63	#1718	—	131	1	no	maximal, Sylow
64	#500	—	131	1	no	—
65	#498	—	131	1	no	—
66	#487	—	131	1	no	—
67	#487	—	131	1	no	—
68	#487	—	131	1	no	—
69	#487	D₁₃₁	1	1	yes	characteristic
70	#322	—	131	1	no	—
71	C₄⋊Q₈	—	131	1	no	—
72	C₄⋊Q₈	—	131	1	no	—
73	C₄⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
74	C₄⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
75	#107	—	262	2	no	—
76	#106	—	262	2	no	—
77	#103	—	2	2	no	—
78	#103	—	262	2	no	—
79	#103	—	262	2	no	—
80	#103	—	262	2	no	—
81	C₄×C₈	—	131	1	no	—
82	C₄×C₈	—	131	1	no	—
83	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
84	C₂×Q₈	—	2	1	no	—
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	262	1	no	—
87	C₂×Q₈	—	262	1	no	—
88	C₂×C₈	—	131	2	no	—
89	C₂×C₈	—	131	2	no	—
90	C₂×C₈	C₂²⋊D₁₃₁	1	2	yes	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	262	1	no	—
96	C₄⋊C₄	—	262	1	no	—
97	C₄⋊C₄	—	262	1	no	—
98	C₄⋊C₄	—	262	1	no	—
99	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
100	Q₈	—	4	2	no	—
101	Q₈	—	4	2	no	—
102	Q₈	—	524	2	no	—
103	Q₈	—	524	2	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	524	1	no	—
107	C₂×C₄	—	524	1	no	—
108	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
109	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
110	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	262	2	no	—
113	C₈	—	262	2	no	—
114	C₂²	#16862242	1	1	yes	characteristic
115	C₄	—	8	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	1048	1	no	—
118	C₄	—	1048	1	no	—
119	C₄	#16862266	1	2	yes	—
120	C₄	#16862275	1	2	yes	—
121	C₄	#16862278	1	2	yes	—
122	C₂	#22553694	1	1	yes	characteristic
123	C₂	#22553703	1	1	yes	characteristic
124	C₂	#22553705	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22553715

C₂

yes

maximal, characteristic

#22553708

C₂

yes

maximal, characteristic

#22553664

C₂

yes

maximal, characteristic

#22553663

C₂

yes

maximal, characteristic

#22553662

C₂

yes

maximal, characteristic

#22553033

C₂

yes

maximal, characteristic

#487×C₁₃₁

C₂

yes

maximal, characteristic

#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

—

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

—

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

#135

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

#1718

—

131

maximal, Sylow

#500

—

131

—

#498

—

131

—

#487

—

131

—

#487

—

131

—

#487

—

131

—

#487

D₁₃₁

yes

characteristic

#322

—

131

—

C₄⋊Q₈

—

131

—

C₄⋊Q₈

—

131

—

C₄⋊Q₈

C₂×D₁₃₁

yes

characteristic

C₄⋊Q₈

C₂×D₁₃₁

yes

characteristic

#107

—

262

—

#106

—

262

—

#103

—

#103

—

262

—

#103

—

262

—

#103

—

262

—

C₄×C₈

—

131

—

C₄×C₈

—

131

—

C₄×C₈

C₂×D₁₃₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

262

—

C₂×Q₈

—

262

—

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

—

C₄⋊C₄

—

262

—

C₄²

C₂²×D₁₃₁

yes

characteristic

100

Q₈

—

101

Q₈

—

102

Q₈

—

524

—

103

Q₈

—

524

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

524

—

107

C₂×C₄

—

524

—

108

C₂×C₄

D₄×D₁₃₁

yes

characteristic

109

C₂×C₄

D₄×D₁₃₁

yes

characteristic

110

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

characteristic

111

C₈

—

112

C₈

—

262

—

113

C₈

—

262

—

114

C₂²

#16862242

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

1048

—

118

C₄

—

1048

—

119

C₄

#16862266

yes

—

120

C₄

#16862275

yes

—

121

C₄

#16862278

yes

—

122

C₂

#22553694

yes

characteristic

123

C₂

#22553703

yes

characteristic

124

C₂

#22553705

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	#1718
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
62. C₁₃₁	#1718	semidirect

H ≅ G/N

type

62. C₁₃₁

#1718

semidirect

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

subgroup	index
#22553715	2
#22553708	2
#22553664	2
#22553663	2
#22553662	2
#22553033	2
#487×C₁₃₁	2
#1718	131

subgroup

index

#487×C₁₃₁

131

Subgroups — #30665608

Conjugacy classes of subgroups

Complements

Maximal subgroups