Group #16845388 — subgroups

Subgroups — #16845388

Order 3968.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#14521454	C₂	1	1	yes	maximal, characteristic
2	#14521454	C₂	1	1	yes	maximal, characteristic
3	#14521447	C₂	1	1	yes	maximal, characteristic
4	#14521429	C₂	1	1	yes	maximal, characteristic
5	#14521400	C₂	1	1	yes	maximal, characteristic
6	#14521400	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#322	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₂₄	—	2	2	no	—
9	#11758446	C₂²	1	1	yes	characteristic
10	#11758446	C₂²	1	1	yes	characteristic
11	#11758446	C₂²	1	1	yes	characteristic
12	#11758446	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	#11758406	—	2	2	no	—
17	#11758406	—	2	2	no	—
18	#11758405	—	2	2	no	—
19	#11758404	—	2	2	no	—
20	#11758404	—	2	2	no	—
21	C₂×Dic₆₂	—	2	1	no	—
22	C₂×Dic₆₂	—	2	1	no	—
23	C₂×Dic₆₂	—	2	1	no	—
24	C₂×Dic₆₂	—	2	1	no	—
25	C₂×C₈×C₃₁	D₄	1	2	yes	—
26	C₂×C₈×C₃₁	D₄	1	2	yes	—
27	C₂×C₈×C₃₁	D₄	1	2	yes	—
28	C₄²×C₃₁	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₃₁	—	2	1	no	—
30	C₄⋊Dic₃₁	—	2	1	no	—
31	C₄⋊Dic₃₁	—	2	1	no	—
32	C₄⋊Dic₃₁	—	2	1	no	—
33	C₄⋊Dic₃₁	—	2	1	no	—
34	C₄⋊Dic₃₁	—	2	1	no	—
35	C₄⋊Dic₃₁	—	2	1	no	—
36	C₄⋊Dic₃₁	—	2	1	no	—
37	Dic₁₂₄	—	4	4	no	—
38	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₃₁	—	4	1	no	—
42	C₂×Dic₃₁	—	4	1	no	—
43	C₂×Dic₃₁	—	4	1	no	—
44	C₂×Dic₃₁	—	4	1	no	—
45	Dic₆₂	—	4	2	no	—
46	Dic₆₂	—	4	2	no	—
47	Dic₆₂	—	4	2	no	—
48	Dic₆₂	—	4	2	no	—
49	C₈×C₃₁	—	2	2	no	—
50	C₈×C₃₁	—	2	2	no	—
51	C₈×C₃₁	Q₁₆	1	4	yes	complemented
52	#1723	—	31	1	no	maximal, Sylow
53	C₂²×C₃₁	#127	1	1	yes	characteristic
54	C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
55	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
56	C₄×C₃₁	#137	1	2	yes	—
57	Dic₃₁	—	8	1	no	—
58	Dic₃₁	—	8	1	no	—
59	Dic₃₁	—	8	1	no	—
60	Dic₃₁	—	8	1	no	—
61	#500	—	31	1	no	—
62	#500	—	31	1	no	—
63	#498	—	31	1	no	—
64	#494	—	31	1	no	—
65	#487	—	31	1	no	—
66	#487	—	31	1	no	—
67	#322	D₃₁	1	1	yes	characteristic
68	C₂×C₃₁	#494	1	1	yes	characteristic
69	C₂×C₃₁	#496	1	1	yes	characteristic
70	C₂×C₃₁	#497	1	1	yes	characteristic
71	C₂×Q₁₆	—	62	2	no	—
72	C₄⋊Q₈	—	31	1	no	—
73	C₄⋊Q₈	—	31	1	no	—
74	C₄⋊Q₈	—	31	1	no	—
75	C₄⋊Q₈	—	31	1	no	—
76	#107	—	62	2	no	—
77	#107	—	62	2	no	—
78	#106	—	62	2	no	—
79	#103	—	62	2	no	—
80	#103	—	62	2	no	—
81	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
82	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
83	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
84	C₃₁	#1723	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₈	—	62	1	no	—
86	C₂×Q₈	—	62	1	no	—
87	C₂×Q₈	—	62	1	no	—
88	C₂×Q₈	—	62	1	no	—
89	Q₁₆	—	124	4	no	—
90	C₂×C₈	D₁₂₄	1	2	yes	—
91	C₂×C₈	D₁₂₄	1	2	yes	—
92	C₂×C₈	D₁₂₄	1	2	yes	—
93	C₄⋊C₄	—	62	1	no	—
94	C₄⋊C₄	—	62	1	no	—
95	C₄⋊C₄	—	62	1	no	—
96	C₄⋊C₄	—	62	1	no	—
97	C₄⋊C₄	—	62	1	no	—
98	C₄⋊C₄	—	62	1	no	—
99	C₄⋊C₄	—	62	1	no	—
100	C₄⋊C₄	—	62	1	no	—
101	C₄²	C₂²×D₃₁	1	1	yes	characteristic
102	Q₈	—	124	2	no	—
103	Q₈	—	124	2	no	—
104	Q₈	—	124	2	no	—
105	Q₈	—	124	2	no	—
106	C₂×C₄	—	124	1	no	—
107	C₂×C₄	—	124	1	no	—
108	C₂×C₄	—	124	1	no	—
109	C₂×C₄	—	124	1	no	—
110	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
111	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
112	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	Dic₁₂₄	1	4	yes	complemented
116	C₂²	#11758451	1	1	yes	characteristic
117	C₄	—	248	1	no	—
118	C₄	—	248	1	no	—
119	C₄	—	248	1	no	—
120	C₄	—	248	1	no	—
121	C₄	C₂×Dic₁₂₄	1	2	yes	—
122	C₄	#11758485	1	2	yes	—
123	C₄	#11758486	1	2	yes	—
124	C₂	#14521429	1	1	yes	characteristic
125	C₂	#14521438	1	1	yes	characteristic
126	C₂	#14521443	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521454

C₂

yes

maximal, characteristic

#14521454

C₂

yes

maximal, characteristic

#14521447

C₂

yes

maximal, characteristic

#14521429

C₂

yes

maximal, characteristic

#14521400

C₂

yes

maximal, characteristic

#14521400

C₂

yes

maximal, characteristic

C₃₁×#322

C₂

yes

maximal, characteristic

C₂×Dic₁₂₄

—

#11758446

C₂²

yes

characteristic

#11758446

C₂²

yes

characteristic

#11758446

C₂²

yes

characteristic

#11758446

C₂²

yes

characteristic

C₄×C₈×C₃₁

C₂²

yes

characteristic

C₄×C₈×C₃₁

C₂²

yes

characteristic

C₄×C₈×C₃₁

C₂²

yes

—

—

—

—

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×C₈×C₃₁

D₄

yes

—

C₂×C₈×C₃₁

D₄

yes

—

C₂×C₈×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

Dic₁₂₄

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

Q₁₆

yes

—

C₂²×C₃₁

yes

C₄×C₃₁

C₂×Q₁₆

yes

—

C₄×C₃₁

AGL(1,ℤ/8)

yes

—

C₄×C₃₁

#137

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

—

—

—

—

—

—

D₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

—

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₄×C₈

C₂×D₃₁

yes

characteristic

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#1723

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

124

—

C₂×C₈

D₁₂₄

yes

—

C₂×C₈

D₁₂₄

yes

—

C₂×C₈

D₁₂₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

C₂²×D₃₁

yes

characteristic

102

Q₈

—

124

—

103

Q₈

—

124

—

104

Q₈

—

124

—

105

Q₈

—

124

—

106

C₂×C₄

—

124

—

107

C₂×C₄

—

124

—

108

C₂×C₄

—

124

—

109

C₂×C₄

—

124

—

110

C₂×C₄

C₂×D₁₂₄

yes

characteristic

111

C₂×C₄

C₂×D₁₂₄

yes

characteristic

112

C₂×C₄

C₂×D₁₂₄

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

Dic₁₂₄

yes

complemented

116

C₂²

#11758451

yes

characteristic

117

C₄

—

248

—

118

C₄

—

248

—

119

C₄

—

248

—

120

C₄

—

248

—

121

C₄

C₂×Dic₁₂₄

yes

—

122

C₄

#11758485

yes

—

123

C₄

#11758486

yes

—

124

C₂

#14521429

yes

characteristic

125

C₂

#14521438

yes

characteristic

126

C₂

#14521443

yes

characteristic

	subgroup	G/H
center	C₂²	#11758451
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#322	C₂
socle, prime radical	C₂²×C₃₁	#127
hypercenter	#322	D₃₁
hypocenter	C₃₁	#1723
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#11758451

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#322

C₂

C₂²×C₃₁

D₃₁

C₃₁

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
51. C₈×C₃₁	Q₁₆	semidirect
84. C₃₁	#1723	semidirect
115. C₈	Dic₁₂₄	semidirect

H ≅ G/N

type

51. C₈×C₃₁

Q₁₆

semidirect

84. C₃₁

#1723

semidirect

115. C₈

Dic₁₂₄

semidirect

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

subgroup	index
#14521454	2
#14521447	2
#14521429	2
#14521400	2
C₃₁×#322	2
#1723	31

subgroup

index

C₃₁×#322

Subgroups — #16845388

Conjugacy classes of subgroups

Complements

Maximal subgroups