Group #16845072 — subgroups

Subgroups — #16845072

Order 3968.

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

normal subgroups only

This group has 188 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	#14521426	C₂	1	1	yes	maximal, characteristic
2	#14521250	C₂	1	2	yes	maximal, complemented
3	C₈×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#14520792	C₂	1	2	yes	maximal, complemented
5	C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₄₈	—	2	1	no	—
7	C₂×D₂₄₈	—	2	1	no	—
8	C₂×#92470	—	2	2	no	—
9	#11758451	C₂²	1	2	yes	—
10	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
11	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	#11758409	—	2	2	no	—
14	Dic₃₁⋊C₈	—	2	1	no	—
15	C₈×Dic₃₁	—	2	1	no	—
16	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×D₁₂₄	—	2	2	no	—
18	C₂×D₁₂₄	—	2	2	no	—
19	C₂×D₁₂₄	—	2	2	no	—
20	C₂×Dic₆₂	D₄	1	1	yes	characteristic
21	C₂×C₈×C₃₁	—	2	2	no	—
22	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
23	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
24	C₄²×C₃₁	C₂³	1	1	yes	characteristic
25	C₄⋊Dic₃₁	D₄	1	1	yes	characteristic
26	Dic₃₁⋊C₄	—	2	1	no	—
27	C₄×Dic₃₁	—	2	1	no	—
28	C₂×C₃₁⋊C₈	—	2	1	no	—
29	D₂₄₈	—	4	2	no	—
30	D₂₄₈	—	4	2	no	—
31	#92470	—	4	4	no	—
32	C₂²×D₃₁	—	4	2	no	—
33	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
36	C₂×Dic₃₁	—	2	1	no	—
37	C₂×Dic₃₁	—	2	1	no	—
38	D₁₂₄	—	4	2	no	—
39	D₁₂₄	—	4	2	no	—
40	D₁₂₄	—	4	2	no	—
41	D₁₂₄	—	4	2	no	—
42	D₁₂₄	—	8	2	no	—
43	Dic₆₂	—	2	1	no	—
44	Dic₆₂	D₈	1	2	yes	complemented
45	C₈×C₃₁	—	2	1	no	—
46	C₈×C₃₁	—	4	2	no	—
47	C₈×C₃₁	SD₁₆	1	2	yes	complemented
48	C₃₁⋊C₈	—	4	1	no	—
49	#1668	—	31	1	no	maximal, Sylow
50	C₂²×C₃₁	#121	1	1	yes	characteristic
51	C₂×D₃₁	—	8	2	no	—
52	C₂×D₃₁	—	8	2	no	—
53	C₂×D₃₁	—	8	2	no	—
54	C₄×C₃₁	—	2	1	no	—
55	C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
56	C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₄×C₃₁	#135	1	1	yes	characteristic
58	C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
59	Dic₃₁	—	2	2	no	—
60	Dic₃₁	—	4	1	no	—
61	#493	—	31	1	no	—
62	#460	—	31	2	no	—
63	C₈×Q₈	—	31	1	no	—
64	#335	D₃₁	1	1	yes	characteristic, complemented
65	#331	—	31	2	no	—
66	C₂×C₃₁	#460	1	1	yes	characteristic
67	C₂×C₃₁	#466	1	1	yes	characteristic
68	C₂×C₃₁	#472	1	1	yes	characteristic
69	D₃₁	—	16	2	no	—
70	C₂×SD₁₆	—	62	2	no	—
71	C₂×D₈	—	62	1	no	—
72	C₂×D₈	—	62	1	no	—
73	#127	—	31	2	no	—
74	C₄×Q₈	—	31	1	no	—
75	C₄⋊C₈	—	31	1	no	—
76	C₄⋊C₈	—	62	1	no	—
77	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
78	#102	—	62	2	no	—
79	C₄×C₈	—	62	1	no	—
80	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
81	C₃₁	#1668	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₈	—	31	1	no	—
83	C₂×D₄	—	62	2	no	—
84	C₂×D₄	—	62	2	no	—
85	C₂×D₄	—	62	2	no	—
86	SD₁₆	—	124	4	no	—
87	D₈	—	124	2	no	—
88	D₈	—	124	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	62	1	no	—
91	C₂×C₈	D₁₂₄	1	1	yes	characteristic
92	C₂×C₈	D₁₂₄	1	1	yes	characteristic
93	C₄⋊C₄	—	31	1	no	—
94	C₄⋊C₄	—	62	1	no	—
95	C₄²	—	62	1	no	—
96	C₄²	C₂²×D₃₁	1	1	yes	characteristic
97	C₂³	—	124	2	no	—
98	Q₈	—	31	2	no	—
99	Q₈	—	62	1	no	—
100	D₄	—	124	2	no	—
101	D₄	—	124	2	no	—
102	D₄	—	124	2	no	—
103	D₄	—	124	2	no	—
104	D₄	—	248	2	no	—
105	C₂×C₄	—	62	1	no	—
106	C₂×C₄	—	62	1	no	—
107	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
108	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
109	C₂×C₄	#92499	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	4	2	no	—
112	C₈	—	124	1	no	—
113	C₈	#92470	1	2	yes	complemented
114	C₂²	—	248	2	no	—
115	C₂²	—	248	2	no	—
116	C₂²	—	248	2	no	—
117	C₂²	#11758472	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	62	2	no	—
120	C₄	—	124	1	no	—
121	C₄	C₂×#92470	1	1	yes	characteristic
122	C₄	#11758485	1	1	yes	characteristic
123	C₄	D₈×D₃₁	1	1	yes	characteristic
124	C₄	#11758496	1	1	yes	characteristic
125	C₂	—	496	2	no	—
126	C₂	#14521250	1	1	yes	characteristic
127	C₂	#14521286	1	1	yes	characteristic
128	C₂	#14521320	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521426

C₂

yes

maximal, characteristic

#14521250

C₂

yes

maximal, complemented

C₈×Dic₆₂

C₂

yes

maximal, characteristic, complemented

#14520792

C₂

yes

maximal, complemented

C₃₁×#335

C₂

yes

maximal, characteristic, complemented

C₂×D₂₄₈

—

C₂×D₂₄₈

—

C₂×#92470

—

#11758451

C₂²

yes

—

C₄×Dic₆₂

C₂²

yes

characteristic

C₃₁×C₄⋊C₈

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

characteristic

#11758409

—

Dic₃₁⋊C₈

—

C₈×Dic₃₁

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

characteristic

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×Dic₆₂

D₄

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₄⋊Dic₃₁

D₄

yes

characteristic

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₂×C₃₁⋊C₈

—

D₂₄₈

—

D₂₄₈

—

#92470

—

C₂²×D₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

Dic₆₂

—

Dic₆₂

D₈

yes

complemented

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

SD₁₆

yes

complemented

C₃₁⋊C₈

—

—

C₂²×C₃₁

yes

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×D₈

yes

characteristic

C₄×C₃₁

C₂×SD₁₆

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

Dic₃₁

—

#493

—

#460

—

C₈×Q₈

—

#335

D₃₁

yes

characteristic, complemented

#331

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

D₃₁

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

#127

—

C₄×Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₁

yes

—

#102

—

C₄×C₈

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#1668

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

124

—

D₈

—

124

—

D₈

—

124

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₂₄

yes

characteristic

C₂×C₈

D₁₂₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂²×D₃₁

yes

characteristic

C₂³

—

124

—

Q₈

—

Q₈

—

100

D₄

—

124

—

101

D₄

—

124

—

102

D₄

—

124

—

103

D₄

—

124

—

104

D₄

—

248

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×D₁₂₄

yes

characteristic

108

C₂×C₄

D₄×D₃₁

yes

characteristic

109

C₂×C₄

#92499

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

124

—

113

C₈

#92470

yes

complemented

114

C₂²

—

248

—

115

C₂²

—

248

—

116

C₂²

—

248

—

117

C₂²

#11758472

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

124

—

121

C₄

C₂×#92470

yes

characteristic

122

C₄

#11758485

yes

characteristic

123

C₄

D₈×D₃₁

yes

characteristic

124

C₄

#11758496

yes

characteristic

125

C₂

—

496

—

126

C₂

#14521250

yes

characteristic

127

C₂

#14521286

yes

characteristic

128

C₂

#14521320

yes

characteristic

	subgroup	G/H
center	C₂²	#11758472
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#335	C₂
socle	C₂²×C₃₁	#121
prime radical	#14521426	C₂
hypercenter	#335	D₃₁
hypocenter	C₃₁	#1668
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#11758472

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#335

C₂

C₂²×C₃₁

C₂

D₃₁

C₃₁

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
2. #14521250	C₂	semidirect
3. C₈×Dic₆₂	C₂	semidirect
4. #14520792	C₂	semidirect
5. C₃₁×#335	C₂	semidirect
44. Dic₆₂	D₈	semidirect
47. C₈×C₃₁	SD₁₆	semidirect
64. #335	D₃₁	semidirect
81. C₃₁	#1668	semidirect
113. C₈	#92470	semidirect

H ≅ G/N

type

2. #14521250

C₂

semidirect

3. C₈×Dic₆₂

C₂

semidirect

4. #14520792

C₂

semidirect

5. C₃₁×#335

C₂

semidirect

44. Dic₆₂

D₈

semidirect

47. C₈×C₃₁

SD₁₆

semidirect

64. #335

D₃₁

semidirect

81. C₃₁

#1668

semidirect

113. C₈

#92470

semidirect

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

subgroup	index
#14521426	2
#14521250	2
C₈×Dic₆₂	2
#14520792	2
C₃₁×#335	2
#1668	31

subgroup

index

C₈×Dic₆₂

C₃₁×#335

Subgroups — #16845072

Conjugacy classes of subgroups

Complements

Maximal subgroups