Group #16845507 — subgroups

Subgroups — C₄×#11758424 #16845507

Order 3968.

296 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 129 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×#11758424	C₂	1	2	yes	maximal
2	C₄×#92474	C₂	1	2	yes	maximal
3	#14520818	C₂	1	2	yes	maximal
4	C₂×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₄×Q₈×C₃₁	—	2	2	no	—
7	C₄×Q₈×C₃₁	—	2	2	no	—
8	C₂×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
9	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₃₁	C₄	1	2	yes	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×#92474	C₂²	1	4	yes	—
13	#11758424	C₄	1	8	yes	direct product factor
14	#11758391	—	2	2	no	—
15	C₄×C₃₁⋊C₈	—	2	2	no	—
16	C₂×Q₈×C₃₁	—	2	2	no	—
17	C₂×Q₈×C₃₁	—	2	2	no	—
18	C₂×Q₈×C₃₁	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₃₁	—	2	4	no	—
25	C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₄²×C₃₁	—	2	2	no	—
27	C₄²×C₃₁	—	2	2	no	—
28	C₄²×C₃₁	D₄	1	2	yes	—
29	C₄²×C₃₁	D₄	1	2	yes	—
30	#92474	—	2	4	no	—
31	#92474	C₂×C₄	1	8	yes	—
32	C₂×C₃₁⋊C₈	—	2	4	no	—
33	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
34	Q₈×C₃₁	—	2	8	no	—
35	C₂×C₄×C₃₁	—	2	2	no	—
36	C₂×C₄×C₃₁	—	2	2	no	—
37	C₂×C₄×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	—	2	2	no	—
39	C₂×C₄×C₃₁	—	4	1	no	—
40	C₂×C₄×C₃₁	—	4	2	no	—
41	C₂×C₄×C₃₁	C₄²	1	4	yes	—
42	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₃₁	#41	1	2	yes	—
48	C₃₁⋊C₈	—	2	8	no	—
49	C₄×#101	—	31	1	no	maximal, Sylow
50	C₂²×C₃₁	—	2	1	no	—
51	C₂²×C₃₁	—	2	1	no	—
52	C₂²×C₃₁	C₂×C₄²	1	1	yes	characteristic
53	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₃₁	#117	1	1	yes	characteristic
55	C₄×C₃₁	—	2	4	no	—
56	C₄×C₃₁	—	2	1	no	—
57	C₄×C₃₁	—	2	2	no	—
58	C₄×C₃₁	—	4	2	no	—
59	C₄×C₃₁	#101	1	2	yes	complemented
60	C₄×C₃₁	C₄×D₄	1	4	yes	—
61	C₂×C₄×Q₈	D₃₁	1	1	yes	characteristic
62	C₂×#101	—	31	2	no	—
63	C₄×M₁₆	—	31	2	no	—
64	#343	—	31	2	no	—
65	C₂×C₃₁	—	2	1	no	—
66	C₂×C₃₁	—	2	1	no	—
67	C₂×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₃₁	C₂×#101	1	1	yes	characteristic
69	C₂×C₃₁	#413	1	1	yes	characteristic
70	C₂²×Q₈	C₂×D₃₁	1	1	yes	characteristic
71	C₂×M₁₆	—	31	4	no	—
72	C₄×Q₈	—	2	2	no	—
73	C₄×Q₈	—	2	2	no	—
74	C₂×C₄⋊C₄	Dic₃₁	1	2	yes	—
75	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
76	C₂×C₄²	Dic₃₁	1	2	yes	—
77	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
78	#101	—	31	8	no	—
79	#97	—	62	2	no	—
80	C₄×C₈	—	62	2	no	—
81	C₃₁	C₄×#101	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	C₄×D₃₁	1	4	yes	—
85	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
86	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
88	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
90	M₁₆	—	31	8	no	—
91	M₁₆	—	62	4	no	—
92	C₂×C₈	—	62	4	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄²	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²⋊D₃₁	1	2	yes	—
98	C₄²	C₂²⋊D₃₁	1	2	yes	—
99	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
100	Q₈	—	2	8	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	C₄×Dic₃₁	1	4	yes	—
108	C₂×C₄	C₂²⋊Dic₃₁	1	1	yes	characteristic
109	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
110	C₂×C₄	C₂²⋊Dic₃₁	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
112	C₂×C₄	#92495	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
114	C₈	—	62	8	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	C₂×C₄×Dic₃₁	1	1	yes	characteristic
118	C₂²	#11758503	1	1	yes	characteristic
119	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
120	C₄	—	2	4	no	—
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	—	4	2	no	—
124	C₄	#11758424	1	2	yes	direct product factor
125	C₄	C₄×C₂²⋊D₃₁	1	4	yes	—
126	C₂	—	2	1	no	—
127	C₂	—	2	1	no	—
128	C₂	C₄×C₂²⋊Dic₃₁	1	1	yes	characteristic
129	C₂	C₂×#11758424	1	1	yes	characteristic
130	C₂	#14521033	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758424

C₂

yes

maximal

C₄×#92474

C₂

yes

maximal

#14520818

C₂

yes

maximal

C₂×C₄×Q₈×C₃₁

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

C₂²

yes

characteristic

C₄×Q₈×C₃₁

—

C₄×Q₈×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₄

yes

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₄

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

—

#11758424

C₄

yes

direct product factor

#11758391

—

C₄×C₃₁⋊C₈

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

D₄

yes

—

#92474

—

#92474

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄²

yes

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₃₁⋊C₈

—

C₄×#101

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₄²

yes

characteristic

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

#117

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#101

yes

complemented

C₄×C₃₁

C₄×D₄

yes

—

C₂×C₄×Q₈

D₃₁

yes

characteristic

C₂×#101

—

C₄×M₁₆

—

#343

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₄×C₂²⋊C₄

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×Q₈

C₂×D₃₁

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₃₁

yes

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

characteristic

C₂×C₄²

Dic₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#101

—

#97

—

C₄×C₈

—

C₃₁

C₄×#101

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄×D₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₃₁

yes

—

C₄²

C₂²⋊D₃₁

yes

—

C₂³

C₂²×Dic₃₁

yes

characteristic

100

Q₈

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄×Dic₃₁

yes

—

108

C₂×C₄

C₂²⋊Dic₃₁

yes

characteristic

109

C₂×C₄

C₂²⋊Dic₃₁

yes

—

110

C₂×C₄

C₂²⋊Dic₃₁

yes

characteristic

111

C₂×C₄

C₂×C₄×D₃₁

yes

—

112

C₂×C₄

#92495

yes

—

113

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

114

C₈

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂×C₄×Dic₃₁

yes

characteristic

118

C₂²

#11758503

yes

characteristic

119

C₂²

C₂×C₂²⋊Dic₃₁

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#11758424

yes

direct product factor

125

C₄

C₄×C₂²⋊D₃₁

yes

—

126

C₂

—

127

C₂

—

128

C₂

C₄×C₂²⋊Dic₃₁

yes

characteristic

129

C₂

C₂×#11758424

yes

characteristic

130

C₂

#14521033

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₃₁
commutator	C₂²×C₃₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₄×Q₈×C₃₁	C₂
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×C₄×Q₈	D₃₁
hypocenter	C₃₁	C₄×#101
LCS	G ▷ C₂²×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₃₁

commutator

C₂²×C₃₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₄×Q₈×C₃₁

C₂

socle

C₂²×C₃₁

C₂×C₂²⋊C₄

prime radical

C₂³×C₃₁

C₂²×C₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×C₄×Q₈

D₃₁

hypocenter

C₃₁

C₄×#101

LCS

G ▷ C₂²×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
13. #11758424	C₄	direct (N × H)
59. C₄×C₃₁	#101	semidirect
81. C₃₁	C₄×#101	semidirect
124. C₄	#11758424	direct (N × H)

H ≅ G/N

type

13. #11758424

C₄

direct (N × H)

59. C₄×C₃₁

#101

semidirect

81. C₃₁

C₄×#101

semidirect

124. C₄

#11758424

direct (N × H)

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

subgroup	index
C₂×#11758424	2
C₄×#92474	2
#14520818	2
C₂×C₄×Q₈×C₃₁	2
C₄×#101	31

subgroup

index

C₂×C₄×Q₈×C₃₁

Subgroups — C₄×#11758424 #16845507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#11758424 #16845507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11758424 #16845507