Group #16847430 — subgroups

Subgroups — #16847430

Order 3968.

220 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×Dic₁₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758400	C₂	1	1	yes	maximal, characteristic
3	#14520852	C₂	1	4	yes	maximal, complemented
4	C₃₁×#427	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₆₂	—	2	1	no	—
6	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#117	—	2	1	no	—
8	C₂×Dic₁₂₄	—	2	1	no	—
9	C₂×Dic₁₂₄	—	2	2	no	—
10	C₂×Dic₁₂₄	C₄	1	4	yes	complemented
11	C₂×Dic₁₂₄	C₂²	1	2	yes	—
12	C₃₁×#107	C₂²	1	2	yes	—
13	C₃₁×#106	—	2	1	no	—
14	#11758400	—	2	2	no	—
15	C₂×C₃₁⋊C₁₆	C₂²	1	2	yes	—
16	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
17	C₂²×Dic₃₁	—	4	1	no	—
18	C₂×Dic₆₂	—	2	4	no	—
19	C₂×Dic₆₂	—	2	2	no	—
20	C₂×Dic₆₂	—	4	1	no	—
21	C₂×Dic₆₂	—	4	2	no	—
22	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
24	C₂×C₈×C₃₁	D₄	1	2	yes	—
25	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₂²⋊C₄×C₃₁	—	4	1	no	—
28	C₄²×C₃₁	—	4	1	no	—
29	Dic₁₂₄	—	2	4	no	—
30	Dic₁₂₄	—	2	2	no	—
31	Dic₁₂₄	C₂×C₄	1	4	yes	—
32	C₃₁⋊C₁₆	—	2	2	no	—
33	C₂³×C₃₁	D₈	1	1	yes	characteristic
34	C₂×C₄×C₃₁	—	4	2	no	—
35	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₃₁	D₈	1	1	yes	characteristic
37	C₂×C₄×C₃₁	SD₁₆	1	2	yes	—
38	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₃₁	—	4	4	no	—
40	C₂×Dic₃₁	—	4	2	no	—
41	Dic₆₂	—	2	4	no	—
42	Dic₆₂	—	4	4	no	—
43	Dic₆₂	—	4	2	no	—
44	C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
45	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
47	#2141	—	31	1	no	maximal, Sylow
48	C₂²×C₃₁	—	2	2	no	—
49	C₂²×C₃₁	#102	1	2	yes	—
50	C₂²×C₃₁	C₂×D₈	1	1	yes	characteristic
51	C₄×C₃₁	—	8	2	no	—
52	C₄×C₃₁	#102	1	2	yes	—
53	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
55	Dic₃₁	—	4	4	no	—
56	C₂²×Q₁₆	—	31	1	no	—
57	C₂×M₃₂	—	31	1	no	—
58	#427	D₃₁	1	1	yes	characteristic
59	#358	—	31	4	no	—
60	C₂×C₃₁	—	2	2	no	—
61	C₂×C₃₁	C₂×#102	1	1	yes	characteristic
62	C₂×C₃₁	#510	1	2	yes	—
63	C₂²×Q₈	—	62	1	no	—
64	C₂×Q₁₆	—	31	4	no	—
65	C₂×Q₁₆	—	31	2	no	—
66	C₂×Q₁₆	—	62	2	no	—
67	C₂×Q₁₆	—	62	1	no	—
68	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
69	#117	—	2	1	no	—
70	M₃₂	—	62	2	no	—
71	C₂×C₁₆	—	31	2	no	—
72	#107	C₂×D₃₁	1	2	yes	—
73	#106	—	2	1	no	—
74	C₃₁	#2141	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₈	—	62	4	no	—
76	C₂×Q₈	—	62	2	no	—
77	C₂×Q₈	—	124	1	no	—
78	C₂×Q₈	—	124	2	no	—
79	C₂²×C₄	—	124	1	no	—
80	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
81	Q₁₆	—	31	4	no	—
82	Q₁₆	—	62	4	no	—
83	Q₁₆	—	62	2	no	—
84	C₂×C₈	C₄×D₃₁	1	2	yes	—
85	C₂×C₈	D₁₂₄	1	2	yes	—
86	C₂×C₈	C₂²⋊D₃₁	1	1	yes	characteristic
87	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
88	C₄⋊C₄	—	2	2	no	—
89	C₂²⋊C₄	—	4	1	no	—
90	C₄²	—	4	1	no	—
91	C₁₆	—	62	2	no	—
92	C₂³	#92479	1	1	yes	characteristic
93	Q₈	—	62	4	no	—
94	Q₈	—	124	4	no	—
95	Q₈	—	124	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	124	4	no	—
98	C₂×C₄	—	124	2	no	—
99	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
100	C₂×C₄	#92479	1	1	yes	characteristic
101	C₂×C₄	#92481	1	2	yes	—
102	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
103	C₈	(C₂×D₃₁)⋊C₄	1	2	yes	—
104	C₈	C₂×C₄×D₃₁	1	1	yes	characteristic
105	C₈	C₂×D₁₂₄	1	1	yes	characteristic
106	C₂²	—	2	2	no	—
107	C₂²	#11758397	1	2	yes	—
108	C₂²	C₂×#92479	1	1	yes	characteristic
109	C₄	—	8	2	no	—
110	C₄	—	124	4	no	—
111	C₄	#11758397	1	2	yes	—
112	C₄	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
113	C₄	C₂×#92481	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	C₂×#11758397	1	1	yes	characteristic
116	C₂	#14521489	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₂₄

C₂

yes

maximal, characteristic

C₂×#11758400

C₂

yes

maximal, characteristic

#14520852

C₂

yes

maximal, complemented

C₃₁×#427

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

—

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#117

—

C₂×Dic₁₂₄

—

C₂×Dic₁₂₄

—

C₂×Dic₁₂₄

C₄

yes

complemented

C₂×Dic₁₂₄

C₂²

yes

—

C₃₁×#107

C₂²

yes

—

C₃₁×#106

—

#11758400

—

C₂×C₃₁⋊C₁₆

C₂²

yes

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

—

C₂×C₈×C₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

—

Dic₁₂₄

—

Dic₁₂₄

—

Dic₁₂₄

C₂×C₄

yes

—

C₃₁⋊C₁₆

—

C₂³×C₃₁

D₈

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

D₈

yes

characteristic

C₂×C₄×C₃₁

SD₁₆

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

C₂²⋊C₄

yes

—

C₈×C₃₁

C₂²×C₄

yes

characteristic

C₈×C₃₁

C₂×D₄

yes

characteristic

#2141

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

#102

yes

—

C₂²×C₃₁

C₂×D₈

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

#102

yes

—

C₄×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₁

C₂×SD₁₆

yes

characteristic

Dic₃₁

—

C₂²×Q₁₆

—

C₂×M₃₂

—

#427

D₃₁

yes

characteristic

#358

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#102

yes

characteristic

C₂×C₃₁

#510

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

#107

C₂×D₃₁

yes

—

#106

—

C₃₁

#2141

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

124

—

C₂×Q₈

—

124

—

C₂²×C₄

—

124

—

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

C₄×D₃₁

yes

—

C₂×C₈

D₁₂₄

yes

—

C₂×C₈

C₂²⋊D₃₁

yes

characteristic

C₂×C₈

C₂²×D₃₁

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₂³

#92479

yes

characteristic

Q₈

—

Q₈

—

124

—

Q₈

—

124

—

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

—

124

—

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

100

C₂×C₄

#92479

yes

characteristic

101

C₂×C₄

#92481

yes

—

102

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

103

C₈

(C₂×D₃₁)⋊C₄

yes

—

104

C₈

C₂×C₄×D₃₁

yes

characteristic

105

C₈

C₂×D₁₂₄

yes

characteristic

106

C₂²

—

107

C₂²

#11758397

yes

—

108

C₂²

C₂×#92479

yes

characteristic

109

C₄

—

110

C₄

—

124

—

111

C₄

#11758397

yes

—

112

C₄

C₂×(C₂×D₃₁)⋊C₄

yes

characteristic

113

C₄

C₂×#92481

yes

characteristic

114

C₂

—

115

C₂

C₂×#11758397

yes

characteristic

116

C₂

#14521489

yes

—

	subgroup	G/H
center	C₂²	C₂×#92479
commutator	C₈×C₃₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃₁
Fitting	C₃₁×#427	C₂
socle	C₂²×C₃₁	C₂×D₈
prime radical	C₂³×C₃₁	D₈
cosocle	C₂×C₈×C₃₁	C₂³
hypercenter	#427	D₃₁
hypocenter	C₃₁	#2141
LCS	G ▷ C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₂²

C₂×#92479

commutator

C₈×C₃₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₃₁

Fitting

C₃₁×#427

C₂

socle

C₂²×C₃₁

C₂×D₈

prime radical

C₂³×C₃₁

D₈

cosocle

C₂×C₈×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
3. #14520852	C₂	semidirect
10. C₂×Dic₁₂₄	C₄	semidirect
74. C₃₁	#2141	semidirect

H ≅ G/N

type

3. #14520852

C₂

semidirect

10. C₂×Dic₁₂₄

C₄

semidirect

74. C₃₁

#2141

semidirect

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

subgroup	index
C₂²×Dic₁₂₄	2
C₂×#11758400	2
#14520852	2
C₃₁×#427	2
#2141	31

subgroup

index

C₂²×Dic₁₂₄

C₂×#11758400

#14520852

C₃₁×#427

#2141

Subgroups — #16847430

Conjugacy classes of subgroups

Complements

Maximal subgroups