Group #16845880 — subgroups

Subgroups — #16845880

Order 3968.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂×#11758395	C₂	1	2	yes	maximal
2	C₂×#11758396	C₂	1	2	yes	maximal
3	C₂×#11758391	C₂	1	1	yes	maximal, characteristic
4	#14520919	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#382	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃₁	—	2	2	no	—
10	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
12	C₃₁×#95	—	2	2	no	—
13	#11758396	—	2	4	no	—
14	#11758395	—	2	4	no	—
15	#11758391	C₄	1	4	yes	complemented
16	C₂²×C₄×C₃₁	—	2	1	no	—
17	C₂²×C₄×C₃₁	—	2	1	no	—
18	C₂²×C₄×C₃₁	D₄	1	2	yes	—
19	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	C₂²×Dic₃₁	—	2	2	no	—
21	C₄⋊C₄×C₃₁	—	2	2	no	—
22	C₄⋊C₄×C₃₁	—	2	2	no	—
23	C₄²×C₃₁	C₂×C₄	1	2	yes	—
24	C₄²×C₃₁	Q₈	1	2	yes	—
25	C₄⋊Dic₃₁	—	2	2	no	—
26	C₄⋊Dic₃₁	—	2	2	no	—
27	C₄⋊Dic₃₁	—	4	4	no	—
28	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	—
29	C₂×C₃₁⋊C₈	D₄	1	4	yes	—
30	C₂×C₃₁⋊C₈	Q₈	1	4	yes	—
31	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₃₁	—	2	2	no	—
33	C₂×C₄×C₃₁	—	2	2	no	—
34	C₂×C₄×C₃₁	—	4	1	no	—
35	C₂×C₄×C₃₁	—	4	1	no	—
36	C₂×C₄×C₃₁	—	4	2	no	—
37	C₂×C₄×C₃₁	—	4	2	no	—
38	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
39	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
43	C₂×Dic₃₁	—	4	2	no	—
44	C₂×Dic₃₁	—	4	4	no	—
45	C₃₁⋊C₈	C₄⋊C₄	1	8	yes	complemented
46	#1847	—	31	1	no	maximal, Sylow
47	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₃₁	#127	1	1	yes	characteristic
49	C₂²×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
50	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
51	C₂²×C₃₁	#137	1	2	yes	—
52	C₄×C₃₁	—	2	4	no	—
53	C₄×C₃₁	—	8	1	no	—
54	C₄×C₃₁	—	8	1	no	—
55	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
56	C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
57	Dic₃₁	—	8	2	no	—
58	C₂×#107	—	31	2	no	—
59	C₂×#106	—	31	2	no	—
60	C₂×#97	—	31	1	no	—
61	#384	—	31	1	no	—
62	#382	D₃₁	1	1	yes	characteristic
63	C₂×C₃₁	#384	1	1	yes	characteristic
64	C₂×C₃₁	#428	1	2	yes	—
65	C₂×C₃₁	#496	1	1	yes	characteristic
66	C₂×C₃₁	#497	1	1	yes	characteristic
67	C₂×C₃₁	#501	1	2	yes	—
68	C₂²×C₈	—	31	2	no	—
69	C₂×C₄⋊C₄	—	31	2	no	—
70	C₂×C₄⋊C₄	—	62	2	no	—
71	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
72	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
73	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
74	#107	—	62	4	no	—
75	#106	—	62	4	no	—
76	#97	—	31	4	no	—
77	#95	—	2	2	no	—
78	C₃₁	#1847	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	62	2	no	—
82	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
83	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
84	C₂×C₈	—	31	4	no	—
85	C₂×C₈	—	31	4	no	—
86	C₂×C₈	—	31	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	62	2	no	—
90	C₄⋊C₄	—	62	2	no	—
91	C₄⋊C₄	—	124	4	no	—
92	C₄²	Dic₆₂	1	2	yes	—
93	C₄²	C₄×D₃₁	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	124	2	no	—
102	C₂×C₄	—	124	4	no	—
103	C₂×C₄	Dic₃₁⋊C₄	1	4	yes	—
104	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
106	C₂×C₄	D₄×D₃₁	1	2	yes	—
107	C₂×C₄	Q₈×D₃₁	1	2	yes	—
108	C₈	—	31	8	no	—
109	C₂²	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
110	C₂²	#11758517	1	1	yes	characteristic
111	C₂²	#11758521	1	1	yes	characteristic
112	C₂²	#11758526	1	2	yes	—
113	C₂²	#11758528	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	248	2	no	—
118	C₄	#11758465	1	2	yes	—
119	C₄	C₄⋊C₄×D₃₁	1	2	yes	—
120	C₂	#14520918	1	1	yes	characteristic
121	C₂	#14521092	1	2	yes	—
122	C₂	#14521439	1	1	yes	characteristic
123	C₂	#14521444	1	1	yes	characteristic
124	C₂	#14521459	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758395

C₂

yes

maximal

C₂×#11758396

C₂

yes

maximal

C₂×#11758391

C₂

yes

maximal, characteristic

#14520919

C₂

yes

maximal, characteristic

C₃₁×#382

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₃₁×#95

—

#11758396

—

#11758395

—

#11758391

C₄

yes

complemented

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

Q₈

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₂×C₃₁⋊C₈

Q₈

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

C₄⋊C₄

yes

complemented

#1847

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#127

yes

characteristic

C₂²×C₃₁

C₄⋊Q₈

yes

characteristic

C₂²×C₃₁

AGL(1,ℤ/8)

yes

—

C₂²×C₃₁

#137

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×C₄⋊C₄

yes

—

C₄×C₃₁

C₄⋊Q₈

yes

—

Dic₃₁

—

C₂×#107

—

C₂×#106

—

C₂×#97

—

#384

—

#382

D₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

characteristic

C₂×C₄⋊C₄

C₂×D₃₁

yes

characteristic

C₂×C₄²

C₂×D₃₁

yes

—

—

—

—

C₃₁

#1847

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

124

—

C₄²

Dic₆₂

yes

—

C₄²

C₄×D₃₁

yes

—

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

124

—

102

C₂×C₄

—

124

—

103

C₂×C₄

Dic₃₁⋊C₄

yes

—

104

C₂×C₄

C₂×Dic₆₂

yes

characteristic

105

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

106

C₂×C₄

D₄×D₃₁

yes

—

107

C₂×C₄

Q₈×D₃₁

yes

—

108

C₈

—

109

C₂²

C₂×Dic₃₁⋊C₄

yes

characteristic

110

C₂²

#11758517

yes

characteristic

111

C₂²

#11758521

yes

characteristic

112

C₂²

#11758526

yes

—

113

C₂²

#11758528

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

248

—

118

C₄

#11758465

yes

—

119

C₄

C₄⋊C₄×D₃₁

yes

—

120

C₂

#14520918

yes

characteristic

121

C₂

#14521092

yes

—

122

C₂

#14521439

yes

characteristic

123

C₂

#14521444

yes

characteristic

124

C₂

#14521459

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#382	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#382	D₃₁
hypocenter	C₃₁	#1847
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#382

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
15. #11758391	C₄	semidirect
45. C₃₁⋊C₈	C₄⋊C₄	semidirect
78. C₃₁	#1847	semidirect

H ≅ G/N

type

15. #11758391

C₄

semidirect

45. C₃₁⋊C₈

C₄⋊C₄

semidirect

78. C₃₁

#1847

semidirect

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

subgroup	index
C₂×#11758395	2
C₂×#11758396	2
C₂×#11758391	2
#14520919	2
C₃₁×#382	2
#1847	31

subgroup

index

C₃₁×#382

Subgroups — #16845880

Conjugacy classes of subgroups

Complements

Maximal subgroups