Group #16859286 — subgroups

Subgroups — C₂×#14521779 #16859286

Order 3968.

1112 conjugacy classes of subgroups; only the 581 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 581 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	#14521779	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11758465	C₂	1	4	yes	maximal
3	C₂×#11758521	C₂	1	4	yes	maximal
4	C₂×C₄×Dic₆₂	C₂	1	2	yes	maximal
5	C₂×#11758464	C₂	1	4	yes	maximal
6	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
8	C₂²×Dic₆₂	C₂²	1	4	yes	—
9	C₂²×Dic₆₂	C₂²	1	2	yes	—
10	C₃₁×C₄⋊Q₈	C₂²	1	8	yes	—
11	C₂×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
12	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	#11758521	C₂²	1	32	yes	—
14	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
15	C₂×Dic₃₁⋊C₄	C₂²	1	8	yes	—
16	C₂×Dic₃₁⋊C₄	C₂²	1	4	yes	—
17	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
18	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
19	#11758465	C₂²	1	32	yes	—
20	#11758464	C₂²	1	32	yes	—
21	C₄×Dic₆₂	C₂²	1	16	yes	—
22	C₂×Q₈×C₃₁	—	4	4	no	—
23	C₂×Q₈×C₃₁	C₂³	1	8	yes	—
24	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
25	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
27	C₂²×Dic₃₁	C₂³	1	4	yes	—
28	C₂²×Dic₃₁	C₂³	1	4	yes	—
29	C₂×Dic₆₂	—	4	8	no	—
30	C₂×Dic₆₂	Q₈	1	16	yes	complemented
31	C₂×Dic₆₂	C₂³	1	16	yes	—
32	C₂×Dic₆₂	C₂³	1	8	yes	—
33	C₄⋊C₄×C₃₁	C₂³	1	16	yes	—
34	C₄²×C₃₁	C₂³	1	4	yes	—
35	C₄⋊Dic₃₁	C₂³	1	8	yes	—
36	Dic₃₁⋊C₄	C₂³	1	16	yes	—
37	Dic₃₁⋊C₄	C₂³	1	32	yes	—
38	C₄×Dic₃₁	C₂³	1	16	yes	—
39	C₄×Dic₃₁	C₂³	1	16	yes	—
40	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
41	Q₈×C₃₁	—	4	8	no	—
42	C₂×C₄×C₃₁	—	2	8	no	—
43	C₂×C₄×C₃₁	—	2	2	no	—
44	C₂×C₄×C₃₁	C₂×Q₈	1	8	yes	—
45	C₂×C₄×C₃₁	C₂⁴	1	8	yes	—
46	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
47	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
48	C₂×Dic₃₁	—	2	8	no	—
49	C₂×Dic₃₁	C₂×Q₈	1	16	yes	—
50	C₂×Dic₃₁	C₂⁴	1	8	yes	—
51	C₂×Dic₃₁	C₂⁴	1	8	yes	—
52	Dic₆₂	—	4	16	no	—
53	Dic₆₂	C₂×Q₈	1	32	yes	complemented
54	C₂×Q₈²	—	31	1	no	maximal, Sylow
55	C₂²×C₃₁	C₂²×Q₈	1	4	yes	—
56	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
57	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
58	C₄×C₃₁	—	2	8	no	—
59	C₄×C₃₁	—	2	2	no	—
60	C₄×C₃₁	C₂²×Q₈	1	8	yes	—
61	Dic₃₁	—	2	8	no	—
62	Dic₃₁	C₂²×Q₈	1	16	yes	—
63	Q₈²	—	31	16	no	—
64	C₂×C₄⋊Q₈	—	31	4	no	—
65	C₂×C₄⋊Q₈	—	31	4	no	—
66	C₂×C₄⋊Q₈	D₃₁	1	1	yes	characteristic
67	C₂×C₄×Q₈	—	31	4	no	—
68	C₂×C₄×Q₈	—	31	2	no	—
69	C₂×C₃₁	Q₈²	1	4	yes	complemented
70	C₂×C₃₁	C₂³×Q₈	1	2	yes	—
71	C₂×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
72	C₂²×Q₈	—	31	2	no	—
73	C₂²×Q₈	—	31	4	no	—
74	C₂²×Q₈	C₂×D₃₁	1	2	yes	—
75	C₄⋊Q₈	—	31	32	no	—
76	C₄⋊Q₈	—	31	32	no	—
77	C₄⋊Q₈	C₂×D₃₁	1	8	yes	—
78	C₄×Q₈	—	31	32	no	—
79	C₄×Q₈	—	31	16	no	—
80	C₂×C₄⋊C₄	—	31	8	no	—
81	C₂×C₄⋊C₄	—	31	4	no	—
82	C₂×C₄⋊C₄	—	31	2	no	—
83	C₂×C₄⋊C₄	C₂×D₃₁	1	4	yes	—
84	C₂×C₄²	—	31	4	no	—
85	C₂×C₄²	—	31	4	no	—
86	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
87	C₃₁	C₂×Q₈²	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₈	—	4	4	no	—
89	C₂×Q₈	—	31	16	no	—
90	C₂×Q₈	—	31	8	no	—
91	C₂×Q₈	—	31	16	no	—
92	C₂×Q₈	—	124	8	no	—
93	C₂×Q₈	C₂²×D₃₁	1	8	yes	—
94	C₂²×C₄	—	31	4	no	—
95	C₂²×C₄	—	31	4	no	—
96	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
97	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
99	C₄⋊C₄	—	31	32	no	—
100	C₄⋊C₄	—	31	16	no	—
101	C₄⋊C₄	—	31	8	no	—
102	C₄⋊C₄	C₂²×D₃₁	1	16	yes	—
103	C₄²	—	31	16	no	—
104	C₄²	—	31	16	no	—
105	C₄²	C₂²×D₃₁	1	4	yes	—
106	C₂³	C₂³×D₃₁	1	1	yes	characteristic
107	Q₈	—	4	8	no	—
108	Q₈	—	31	32	no	—
109	Q₈	—	124	16	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	8	no	—
112	C₂×C₄	—	31	16	no	—
113	C₂×C₄	—	31	8	no	—
114	C₂×C₄	—	31	8	no	—
115	C₂×C₄	—	62	8	no	—
116	C₂×C₄	Q₈×D₃₁	1	8	yes	—
117	C₂×C₄	C₂³×D₃₁	1	8	yes	—
118	C₂×C₄	C₂³×D₃₁	1	2	yes	—
119	C₂×C₄	C₂³×D₃₁	1	4	yes	—
120	C₂²	#11758561	1	2	yes	—
121	C₂²	C₂×Q₈×D₃₁	1	4	yes	—
122	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	8	no	—
125	C₄	—	31	16	no	—
126	C₄	—	62	8	no	—
127	C₄	C₂×Q₈×D₃₁	1	8	yes	—
128	C₂	#14521779	1	4	yes	direct product factor
129	C₂	C₂²×Q₈×D₃₁	1	2	yes	—
130	C₂	C₂×#11758561	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521779

C₂

yes

maximal, direct product factor

C₂×#11758465

C₂

yes

maximal

C₂×#11758521

C₂

yes

maximal

C₂×C₄×Dic₆₂

C₂

yes

maximal

C₂×#11758464

C₂

yes

maximal

C₂×C₃₁×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

C₂²

yes

—

C₂²×Dic₆₂

C₂²

yes

—

C₂²×Dic₆₂

C₂²

yes

—

C₃₁×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

#11758521

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×Dic₃₁⋊C₄

C₂²

yes

—

C₂×Dic₃₁⋊C₄

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

—

#11758465

C₂²

yes

—

#11758464

C₂²

yes

—

C₄×Dic₆₂

C₂²

yes

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂×Dic₆₂

—

C₂×Dic₆₂

Q₈

yes

complemented

C₂×Dic₆₂

C₂³

yes

—

C₂×Dic₆₂

C₂³

yes

—

C₄⋊C₄×C₃₁

C₂³

yes

—

C₄²×C₃₁

C₂³

yes

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

C₂³

yes

—

Dic₃₁⋊C₄

C₂³

yes

—

C₄×Dic₃₁

C₂³

yes

—

C₄×Dic₃₁

C₂³

yes

—

C₂³×C₃₁

C₂⁴

yes

characteristic

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂×Q₈

yes

—

C₂×Dic₃₁

C₂⁴

yes

—

C₂×Dic₃₁

C₂⁴

yes

—

Dic₆₂

—

Dic₆₂

C₂×Q₈

yes

complemented

C₂×Q₈²

—

maximal, Sylow

C₂²×C₃₁

C₂²×Q₈

yes

—

C₂²×C₃₁

H(5,𝔽₂)

yes

—

C₂²×C₃₁

C₂⁵

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂²×Q₈

yes

—

Dic₃₁

—

Dic₃₁

C₂²×Q₈

yes

—

Q₈²

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₃₁

yes

characteristic

C₂×C₄×Q₈

—

C₂×C₄×Q₈

—

C₂×C₃₁

Q₈²

yes

complemented

C₂×C₃₁

C₂³×Q₈

yes

—

C₂×C₃₁

C₂×H(5,𝔽₂)

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₃₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₁

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₃₁

C₂×Q₈²

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

124

—

C₂×Q₈

C₂²×D₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

C₂²×D₃₁

yes

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂²×D₃₁

yes

—

106

C₂³

C₂³×D₃₁

yes

characteristic

107

Q₈

—

108

Q₈

—

109

Q₈

—

124

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

Q₈×D₃₁

yes

—

117

C₂×C₄

C₂³×D₃₁

yes

—

118

C₂×C₄

C₂³×D₃₁

yes

—

119

C₂×C₄

C₂³×D₃₁

yes

—

120

C₂²

#11758561

yes

—

121

C₂²

C₂×Q₈×D₃₁

yes

—

122

C₂²

C₂⁴×D₃₁

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

C₂×Q₈×D₃₁

yes

—

128

C₂

#14521779

yes

direct product factor

129

C₂

C₂²×Q₈×D₃₁

yes

—

130

C₂

C₂×#11758561

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₃₁
commutator, cosocle	C₂²×C₃₁	C₂⁵
Frattini	C₂²	C₂⁴×D₃₁
Fitting	C₂×C₃₁×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃₁	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₃₁
hypocenter	C₃₁	C₂×Q₈²
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂³×D₃₁

commutator, cosocle

C₂²×C₃₁

C₂⁵

Frattini

C₂²

C₂⁴×D₃₁

Fitting

C₂×C₃₁×C₄⋊Q₈

C₂

socle, prime radical

C₂³×C₃₁

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₃₁

hypocenter

C₃₁

C₂×Q₈²

LCS

G ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. #14521779	C₂	direct (N × H)
30. C₂×Dic₆₂	Q₈	semidirect
53. Dic₆₂	C₂×Q₈	semidirect
69. C₂×C₃₁	Q₈²	semidirect
87. C₃₁	C₂×Q₈²	semidirect
128. C₂	#14521779	direct (N × H)

H ≅ G/N

type

1. #14521779

C₂

direct (N × H)

30. C₂×Dic₆₂

Q₈

semidirect

53. Dic₆₂

C₂×Q₈

semidirect

69. C₂×C₃₁

Q₈²

semidirect

87. C₃₁

C₂×Q₈²

semidirect

128. C₂

#14521779

direct (N × H)

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

subgroup	index
#14521779	2
C₂×#11758465	2
C₂×#11758521	2
C₂×C₄×Dic₆₂	2
C₂×#11758464	2
C₂×C₃₁×C₄⋊Q₈	2
C₂×Q₈²	31

subgroup

index

C₂×C₄×Dic₆₂

C₂×C₃₁×C₄⋊Q₈

C₂×Q₈²

Subgroups — C₂×#14521779 #16859286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14521779 #16859286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521779 #16859286