Group #16856046 — subgroups

Subgroups — C₂×#14521414 #16856046

Order 3968.

488 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 149 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×#11758451	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758447	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521414	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11758409	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₂₄	—	2	2	no	—
7	C₂²×D₁₂₄	C₂²	1	2	yes	—
8	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
9	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	C₂×Dic₃₁⋊C₄	—	2	2	no	—
12	#11758451	—	2	2	no	—
13	#11758451	C₂²	1	4	yes	—
14	#11758447	—	2	2	no	—
15	#11758447	C₂²	1	4	yes	complemented
16	C₃₁×#97	C₂²	1	4	yes	complemented
17	#11758409	C₂²	1	16	yes	complemented
18	C₂³×D₃₁	—	2	2	no	—
19	C₂²×C₄×C₃₁	D₄	1	2	yes	—
20	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
21	C₂²×Dic₃₁	—	2	2	no	—
22	C₂×D₁₂₄	—	2	8	no	—
23	C₂×D₁₂₄	—	2	2	no	—
24	C₂×D₁₂₄	—	2	8	no	—
25	C₂×D₁₂₄	—	4	8	no	—
26	C₂×D₁₂₄	C₂³	1	4	yes	—
27	C₂×C₈×C₃₁	—	2	4	no	—
28	C₂×C₈×C₃₁	C₂³	1	4	yes	—
29	C₄²×C₃₁	D₄	1	2	yes	—
30	C₄²×C₃₁	C₂³	1	2	yes	—
31	C₄⋊Dic₃₁	—	2	2	no	—
32	C₄⋊Dic₃₁	C₂³	1	4	yes	—
33	Dic₃₁⋊C₄	—	2	8	no	—
34	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×D₃₁	—	2	4	no	—
36	C₂²×D₃₁	—	4	4	no	—
37	C₂²×D₃₁	—	4	8	no	—
38	C₂×C₄×C₃₁	—	2	4	no	—
39	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
40	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₃₁	#41	1	4	yes	—
42	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
43	C₂×Dic₃₁	—	2	4	no	—
44	C₂×Dic₃₁	—	4	4	no	—
45	D₁₂₄	—	2	8	no	—
46	D₁₂₄	—	4	16	no	—
47	D₁₂₄	—	4	4	no	—
48	C₈×C₃₁	—	2	4	no	—
49	C₂×#490	—	31	1	no	maximal, Sylow
50	C₂²×C₃₁	#124	1	2	yes	—
51	C₂²×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
52	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
53	C₂×D₃₁	—	4	4	no	—
54	C₂×D₃₁	—	4	8	no	—
55	C₂×D₃₁	—	8	8	no	—
56	C₄×C₃₁	—	2	4	no	—
57	C₄×C₃₁	#124	1	2	yes	—
58	C₄×C₃₁	C₂×#41	1	2	yes	—
59	Dic₃₁	—	4	4	no	—
60	C₂×#127	—	31	1	no	—
61	C₂×#125	—	31	1	no	—
62	#490	—	31	8	no	—
63	C₂×#102	—	31	4	no	—
64	C₂×#97	D₃₁	1	1	yes	characteristic, complemented
65	C₂×C₃₁	#490	1	4	yes	complemented
66	C₂×C₃₁	C₂×#124	1	1	yes	characteristic
67	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
68	D₃₁	—	8	4	no	—
69	C₂²×D₄	—	31	2	no	—
70	C₂²×D₄	—	62	2	no	—
71	C₂²×C₈	C₂×D₃₁	1	2	yes	—
72	#127	—	31	4	no	—
73	#127	—	62	2	no	—
74	#125	—	31	4	no	—
75	#125	—	62	2	no	—
76	C₂×C₄⋊C₄	—	31	2	no	—
77	C₂×C₄⋊C₄	—	62	2	no	—
78	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
79	#102	—	31	16	no	—
80	#97	C₂×D₃₁	1	4	yes	complemented
81	C₃₁	C₂×#490	1	1	yes	characteristic, complemented, Sylow
82	C₂⁴	—	62	2	no	—
83	C₂×D₄	—	31	4	no	—
84	C₂×D₄	—	62	8	no	—
85	C₂×D₄	—	62	2	no	—
86	C₂×D₄	—	62	8	no	—
87	C₂×D₄	—	124	8	no	—
88	C₂²×C₄	—	62	2	no	—
89	C₂²×C₄	D₁₂₄	1	2	yes	—
90	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	C₂²×D₃₁	1	4	yes	—
93	C₄⋊C₄	—	31	4	no	—
94	C₄⋊C₄	—	62	8	no	—
95	C₄⋊C₄	—	62	2	no	—
96	C₄²	D₁₂₄	1	2	yes	—
97	C₄²	C₂²×D₃₁	1	2	yes	—
98	C₂³	—	62	4	no	—
99	C₂³	—	124	4	no	—
100	C₂³	—	124	8	no	—
101	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
102	D₄	—	62	8	no	—
103	D₄	—	124	16	no	—
104	D₄	—	124	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	62	4	no	—
107	C₂×C₄	—	124	4	no	—
108	C₂×C₄	C₂×D₁₂₄	1	4	yes	—
109	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
110	C₂×C₄	#92495	1	4	yes	—
111	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
112	C₈	—	2	4	no	—
113	C₂²	—	124	4	no	—
114	C₂²	—	124	8	no	—
115	C₂²	—	248	8	no	—
116	C₂²	#11758452	1	2	yes	—
117	C₂²	#11758485	1	4	yes	—
118	C₂²	C₂²×D₁₂₄	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	124	4	no	—
121	C₄	#11758452	1	2	yes	—
122	C₄	C₂×#92495	1	2	yes	—
123	C₂	—	248	4	no	—
124	C₂	#14521414	1	4	yes	direct product factor
125	C₂	C₂×#11758452	1	1	yes	characteristic
126	C₂	C₂×#11758485	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758451

C₂

yes

maximal, characteristic

C₂×#11758447

C₂

yes

maximal, characteristic, complemented

#14521414

C₂

yes

maximal, direct product factor

C₂×#11758409

C₂

yes

maximal, complemented

C₂×C₃₁×#97

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₂₄

—

C₂²×D₁₂₄

C₂²

yes

—

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×Dic₃₁⋊C₄

—

#11758451

—

#11758451

C₂²

yes

—

#11758447

—

#11758447

C₂²

yes

complemented

C₃₁×#97

C₂²

yes

complemented

#11758409

C₂²

yes

complemented

C₂³×D₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×D₁₂₄

C₂³

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂²×D₃₁

—

C₂²×D₃₁

—

C₂²×D₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

C₈×C₃₁

—

C₂×#490

—

maximal, Sylow

C₂²×C₃₁

#124

yes

—

C₂²×C₃₁

AGL(1,ℤ/8)

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#124

yes

—

C₄×C₃₁

C₂×#41

yes

—

Dic₃₁

—

—

—

—

—

C₂×#97

D₃₁

yes

characteristic, complemented

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

D₃₁

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

C₂×D₃₁

yes

—

—

—

—

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#102

—

#97

C₂×D₃₁

yes

complemented

C₃₁

C₂×#490

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

124

—

C₂²×C₄

—

C₂²×C₄

D₁₂₄

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂²×D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₁₂₄

yes

—

C₄²

C₂²×D₃₁

yes

—

C₂³

—

C₂³

—

124

—

100

C₂³

—

124

—

101

C₂³

C₂×D₁₂₄

yes

characteristic

102

D₄

—

103

D₄

—

124

—

104

D₄

—

124

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

124

—

108

C₂×C₄

C₂×D₁₂₄

yes

—

109

C₂×C₄

C₂×D₁₂₄

yes

characteristic

110

C₂×C₄

#92495

yes

—

111

C₂×C₄

C₂³×D₃₁

yes

characteristic

112

C₈

—

113

C₂²

—

124

—

114

C₂²

—

124

—

115

C₂²

—

248

—

116

C₂²

#11758452

yes

—

117

C₂²

#11758485

yes

—

118

C₂²

C₂²×D₁₂₄

yes

characteristic

119

C₄

—

120

C₄

—

124

—

121

C₄

#11758452

yes

—

122

C₄

C₂×#92495

yes

—

123

C₂

—

248

—

124

C₂

#14521414

yes

direct product factor

125

C₂

C₂×#11758452

yes

characteristic

126

C₂

C₂×#11758485

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₂₄
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₃₁×#97	C₂
socle	C₂³×C₃₁	C₂×D₄
prime radical	C₂×#11758451	C₂
hypercenter	C₂×#97	D₃₁
hypocenter	C₃₁	C₂×#490
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₂₄

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₂×C₃₁×#97

C₂

socle

C₂³×C₃₁

C₂×D₄

prime radical

C₂×#11758451

C₂

hypercenter

C₂×#97

D₃₁

hypocenter

C₃₁

C₂×#490

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
2. C₂×#11758447	C₂	semidirect
3. #14521414	C₂	direct (N × H)
4. C₂×#11758409	C₂	semidirect
5. C₂×C₃₁×#97	C₂	semidirect
15. #11758447	C₂²	semidirect
16. C₃₁×#97	C₂²	semidirect
17. #11758409	C₂²	semidirect
64. C₂×#97	D₃₁	semidirect
65. C₂×C₃₁	#490	semidirect
80. #97	C₂×D₃₁	semidirect
81. C₃₁	C₂×#490	semidirect
124. C₂	#14521414	direct (N × H)

H ≅ G/N

type

2. C₂×#11758447

C₂

semidirect

3. #14521414

C₂

direct (N × H)

4. C₂×#11758409

C₂

semidirect

5. C₂×C₃₁×#97

C₂

semidirect

15. #11758447

C₂²

semidirect

16. C₃₁×#97

C₂²

semidirect

17. #11758409

C₂²

semidirect

64. C₂×#97

D₃₁

semidirect

65. C₂×C₃₁

#490

semidirect

80. #97

C₂×D₃₁

semidirect

81. C₃₁

C₂×#490

semidirect

124. C₂

#14521414

direct (N × H)

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

subgroup	index
C₂×#11758451	2
C₂×#11758447	2
#14521414	2
C₂×#11758409	2
C₂×C₃₁×#97	2
C₂×#490	31

subgroup

index

C₂×C₃₁×#97

Subgroups — C₂×#14521414 #16856046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14521414 #16856046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521414 #16856046