Group #30664239 — subgroups

Subgroups — M₁₆×C₁₃₁⋊C₈ #30664239

Order 16768.

224 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 153 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	C₂×C₄×C₁₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22553033	C₂	1	2	yes	maximal, complemented
4	C₈×C₁₃₁⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄×M₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃₁⋊C₈	C₄	1	2	yes	—
7	C₄⋊(C₁₃₁⋊C₈)	C₄	1	2	yes	—
8	C₂²⋊(C₁₃₁⋊C₈)	C₄	1	2	yes	—
9	C₄×C₁₃₁⋊C₈	C₄	1	2	yes	—
10	C₄×C₁₃₁⋊C₈	C₂²	1	2	yes	—
11	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₃₁	C₄	1	2	yes	—
14	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
15	#97×C₁₃₁	C₄	1	2	yes	—
16	C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
17	C₂×C₁₃₁⋊C₈	—	2	4	no	—
18	C₂×C₁₃₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₁₃₁⋊C₈	C₂×C₄	1	4	yes	—
20	C₂×C₁₃₁⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
24	M₁₆×C₁₃₁	C₈	1	8	yes	complemented
25	C₂×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
26	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
30	C₁₃₁⋊C₈	—	2	2	no	—
31	C₁₃₁⋊C₈	—	2	4	no	—
32	C₁₃₁⋊C₈	M₁₆	1	4	yes	direct product factor
33	C₂³×C₁₃₁	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₁₃₁	—	2	2	no	—
35	C₂×C₄×C₁₃₁	—	2	2	no	—
36	C₂×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
38	C₂×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
39	C₂×C₄×C₁₃₁	C₂×C₈	1	4	yes	—
40	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
43	C₈×C₁₃₁	C₂×C₈	1	8	yes	complemented
44	C₂²×C₁₃₁	—	2	1	no	—
45	C₂²×C₁₃₁	—	2	1	no	—
46	C₂²×C₁₃₁	C₄×C₈	1	2	yes	—
47	C₂²×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₃₁	—	2	1	no	—
49	C₄×C₁₃₁	—	2	1	no	—
50	C₄×C₁₃₁	C₄×C₈	1	2	yes	—
51	C₄×C₁₃₁	C₂²×C₈	1	2	yes	—
52	C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
53	C₄×C₁₃₁	#131	1	2	yes	—
54	C₂×C₁₃₁	—	2	2	no	—
55	C₂×C₁₃₁	C₂×C₄×C₈	1	1	yes	characteristic
56	C₂×C₁₃₁	C₄×M₁₆	1	1	yes	characteristic
57	C₂×C₁₃₁	#405	1	1	yes	characteristic
58	C₁₃₁	C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
59	C₈×M₁₆	—	131	1	no	maximal, Sylow
60	#431	—	131	1	no	—
61	C₄×M₁₆	D₁₃₁	1	1	yes	characteristic
62	C₂×C₄×C₈	—	131	1	no	—
63	#322	—	131	2	no	—
64	C₈²	—	131	2	no	—
65	C₂×M₁₆	Dic₁₃₁	1	2	yes	—
66	C₂²×C₈	—	131	2	no	—
67	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
68	C₄⋊C₈	—	131	2	no	—
69	C₂²⋊C₈	—	131	2	no	—
70	#97	Dic₁₃₁	1	2	yes	—
71	C₄×C₈	—	131	2	no	—
72	C₄×C₈	—	131	1	no	—
73	C₄×C₈	—	131	2	no	—
74	C₄×C₈	—	131	1	no	—
75	C₄×C₈	C₂×D₁₃₁	1	2	yes	—
76	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
77	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
79	M₁₆	C₁₃₁⋊C₈	1	8	yes	direct product factor
80	C₂×C₈	—	262	4	no	—
81	C₂×C₈	—	131	2	no	—
82	C₂×C₈	—	131	4	no	—
83	C₂×C₈	—	131	2	no	—
84	C₂×C₈	C₂×Dic₁₃₁	1	4	yes	—
85	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
86	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
87	C₄²	C₂×Dic₁₃₁	1	1	yes	characteristic
88	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
89	C₂³	C₄×Dic₁₃₁	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	C₄×Dic₁₃₁	1	1	yes	characteristic
93	C₂×C₄	C₄×Dic₁₃₁	1	1	yes	characteristic
94	C₂×C₄	C₄×Dic₁₃₁	1	1	yes	characteristic
95	C₂×C₄	C₂×C₁₃₁⋊C₈	1	4	yes	—
96	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
97	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
98	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
99	C₈	—	262	2	no	—
100	C₈	—	262	4	no	—
101	C₈	—	131	4	no	—
102	C₈	C₂×C₁₃₁⋊C₈	1	8	yes	complemented
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	C₄×C₁₃₁⋊C₈	1	2	yes	—
106	C₂²	C₂×C₄×Dic₁₃₁	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	C₄×C₁₃₁⋊C₈	1	2	yes	—
110	C₄	C₂²×C₁₃₁⋊C₈	1	2	yes	—
111	C₄	M₁₆×D₁₃₁	1	2	yes	—
112	C₄	#16862254	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	C₂×C₄×C₁₃₁⋊C₈	1	1	yes	characteristic
115	C₂	M₁₆×Dic₁₃₁	1	1	yes	characteristic
116	C₂	#22553270	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22553365

C₂

yes

maximal, characteristic

C₂×C₄×C₁₃₁⋊C₈

C₂

yes

maximal, characteristic

#22553033

C₂

yes

maximal, complemented

C₈×C₁₃₁⋊C₈

C₂

yes

maximal, complemented

C₄×M₁₆×C₁₃₁

C₂

yes

maximal, characteristic

C₂²×C₁₃₁⋊C₈

C₄

yes

—

C₄⋊(C₁₃₁⋊C₈)

C₄

yes

—

C₂²⋊(C₁₃₁⋊C₈)

C₄

yes

—

C₄×C₁₃₁⋊C₈

C₄

yes

—

C₄×C₁₃₁⋊C₈

C₂²

yes

—

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₂×M₁₆×C₁₃₁

C₄

yes

—

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

#97×C₁₃₁

C₄

yes

—

C₄×C₈×C₁₃₁

C₂²

yes

—

C₂×C₁₃₁⋊C₈

—

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

M₁₆×C₁₃₁

C₈

yes

complemented

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

M₁₆

yes

direct product factor

C₂³×C₁₃₁

C₄²

yes

characteristic

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

C₂×C₈

yes

—

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₈×C₁₃₁

C₂×C₈

yes

complemented

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

C₄×C₈

yes

—

C₂²×C₁₃₁

C₂×C₄²

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₄×C₈

yes

—

C₄×C₁₃₁

C₂²×C₈

yes

—

C₄×C₁₃₁

C₂×M₁₆

yes

—

C₄×C₁₃₁

#131

yes

—

C₂×C₁₃₁

—

C₂×C₁₃₁

C₂×C₄×C₈

yes

characteristic

C₂×C₁₃₁

C₄×M₁₆

yes

characteristic

C₂×C₁₃₁

#405

yes

characteristic

C₁₃₁

C₈×M₁₆

yes

characteristic, complemented, Sylow

C₈×M₁₆

—

131

maximal, Sylow

#431

—

131

—

C₄×M₁₆

D₁₃₁

yes

characteristic

C₂×C₄×C₈

—

131

—

#322

—

131

—

C₈²

—

131

—

C₂×M₁₆

Dic₁₃₁

yes

—

C₂²×C₈

—

131

—

C₂×C₄²

C₂×D₁₃₁

yes

characteristic

C₄⋊C₈

—

131

—

C₂²⋊C₈

—

131

—

#97

Dic₁₃₁

yes

—

C₄×C₈

—

131

—

C₄×C₈

—

131

—

C₄×C₈

—

131

—

C₄×C₈

—

131

—

C₄×C₈

C₂×D₁₃₁

yes

—

C₂²×C₄

C₄×D₁₃₁

yes

characteristic

C₂²×C₄

C₄×D₁₃₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₃₁

yes

characteristic

M₁₆

C₁₃₁⋊C₈

yes

direct product factor

C₂×C₈

—

262

—

C₂×C₈

—

131

—

C₂×C₈

—

131

—

C₂×C₈

—

131

—

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

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₃₁

yes

characteristic

C₂×C₄

C₄×Dic₁₃₁

yes

characteristic

C₂×C₄

C₄×Dic₁₃₁

yes

characteristic

C₂×C₄

C₂×C₁₃₁⋊C₈

yes

—

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

C₂×C₄

C₂²×Dic₁₃₁

yes

characteristic

C₈

—

262

—

100

C₈

—

262

—

101

C₈

—

131

—

102

C₈

C₂×C₁₃₁⋊C₈

yes

complemented

103

C₂²

—

104

C₂²

—

105

C₂²

C₄×C₁₃₁⋊C₈

yes

—

106

C₂²

C₂×C₄×Dic₁₃₁

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

C₄×C₁₃₁⋊C₈

yes

—

110

C₄

C₂²×C₁₃₁⋊C₈

yes

—

111

C₄

M₁₆×D₁₃₁

yes

—

112

C₄

#16862254

yes

—

113

C₂

—

114

C₂

C₂×C₄×C₁₃₁⋊C₈

yes

characteristic

115

C₂

M₁₆×Dic₁₃₁

yes

characteristic

116

C₂

#22553270

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₃₁
commutator	C₂×C₁₃₁	C₂×C₄×C₈
Fitting	C₄×M₁₆×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×C₄²
prime radical	C₂³×C₁₃₁	C₄²
cosocle	C₄²×C₁₃₁	C₂³
hypercenter	C₄×M₁₆	D₁₃₁
hypocenter	C₁₃₁	C₈×M₁₆
LCS	G ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

subgroup

G/H

center, Frattini

C₄²

C₂²×D₁₃₁

commutator

C₂×C₁₃₁

C₂×C₄×C₈

Fitting

C₄×M₁₆×C₁₃₁

C₂

socle

C₂²×C₁₃₁

C₂×C₄²

prime radical

C₂³×C₁₃₁

C₄²

cosocle

C₄²×C₁₃₁

C₂³

hypercenter

C₄×M₁₆

D₁₃₁

hypocenter

C₁₃₁

C₈×M₁₆

LCS

G ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
3. #22553033	C₂	semidirect
4. C₈×C₁₃₁⋊C₈	C₂	semidirect
24. M₁₆×C₁₃₁	C₈	semidirect
32. C₁₃₁⋊C₈	M₁₆	direct (N × H)
43. C₈×C₁₃₁	C₂×C₈	semidirect
58. C₁₃₁	C₈×M₁₆	semidirect
79. M₁₆	C₁₃₁⋊C₈	direct (N × H)
102. C₈	C₂×C₁₃₁⋊C₈	semidirect

H ≅ G/N

type

3. #22553033

C₂

semidirect

4. C₈×C₁₃₁⋊C₈

C₂

semidirect

24. M₁₆×C₁₃₁

C₈

semidirect

32. C₁₃₁⋊C₈

M₁₆

direct (N × H)

43. C₈×C₁₃₁

C₂×C₈

semidirect

58. C₁₃₁

C₈×M₁₆

semidirect

79. M₁₆

C₁₃₁⋊C₈

direct (N × H)

102. C₈

C₂×C₁₃₁⋊C₈

semidirect

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

subgroup	index
#22553365	2
C₂×C₄×C₁₃₁⋊C₈	2
#22553033	2
C₈×C₁₃₁⋊C₈	2
C₄×M₁₆×C₁₃₁	2
C₈×M₁₆	131

subgroup

index

#22553365

C₂×C₄×C₁₃₁⋊C₈

#22553033

C₈×C₁₃₁⋊C₈

C₄×M₁₆×C₁₃₁

C₈×M₁₆

131

Subgroups — M₁₆×C₁₃₁⋊C₈ #30664239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M16×C131⋊C8 #30664239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×C₁₃₁⋊C₈ #30664239