Group #16854306 — subgroups

Subgroups — #16854306

Order 3968.

476 conjugacy classes of subgroups; only the 293 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 293 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	#14521532	C₂	1	2	yes	maximal
2	#14521091	C₂	1	6	yes	maximal, complemented
3	#14521087	C₂	1	3	yes	maximal, complemented
4	C₂×#11758405	C₂	1	3	yes	maximal, complemented
5	C₂×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#131	C₄	1	8	yes	complemented
8	C₂×M₁₆×C₃₁	C₂²	1	3	yes	—
9	C₂²×C₈×C₃₁	C₂²	1	3	yes	—
10	Q₈×Dic₃₁	—	2	2	no	—
11	D₄×Dic₃₁	—	2	6	no	—
12	#11758503	C₂²	1	6	yes	—
13	C₂×C₄⋊Dic₃₁	C₂²	1	6	yes	—
14	#11758406	C₂²	1	6	yes	—
15	#11758405	C₂²	1	6	yes	—
16	#11758405	C₂²	1	3	yes	—
17	#11758405	C₂²	1	1	yes	characteristic
18	#41×C₃₁	C₂×C₄	1	4	yes	—
19	#41×C₃₁	Q₈	1	4	yes	—
20	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
21	C₂×D₄×C₃₁	D₄	1	3	yes	—
22	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
23	C₂²×Dic₃₁	—	2	6	no	—
24	M₁₆×C₃₁	C₂×C₄	1	12	yes	complemented
25	C₂×C₈×C₃₁	C₂×C₄	1	12	yes	complemented
26	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₃₁	C₂³	1	3	yes	—
28	C₂²⋊Dic₃₁	—	2	6	no	—
29	C₄⋊Dic₃₁	—	2	6	no	—
30	C₄⋊Dic₃₁	C₂³	1	6	yes	—
31	C₄⋊Dic₃₁	C₂³	1	2	yes	—
32	C₄×Dic₃₁	—	2	6	no	—
33	C₂³×C₃₁	C₂×D₄	1	3	yes	—
34	Q₈×C₃₁	C₄⋊C₄	1	4	yes	—
35	D₄×C₃₁	C₄⋊C₄	1	12	yes	—
36	C₂×C₄×C₃₁	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₃₁	C₂×D₄	1	3	yes	—
38	C₂×C₄×C₃₁	C₂×Q₈	1	6	yes	—
39	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
40	C₂×Dic₃₁	—	2	2	no	—
41	C₂×Dic₃₁	—	2	6	no	—
42	C₂×Dic₃₁	—	8	6	no	—
43	C₈×C₃₁	C₂²×C₄	1	2	yes	—
44	C₈×C₃₁	C₂²×C₄	1	6	yes	—
45	#2912	—	31	1	no	maximal, Sylow
46	C₂²×C₃₁	—	2	6	no	—
47	C₂²×C₃₁	C₂×C₄⋊C₄	1	6	yes	—
48	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₃₁	C₂×C₄⋊C₄	1	6	yes	—
50	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
51	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
52	Dic₃₁	—	4	2	no	—
53	Dic₃₁	—	4	6	no	—
54	C₂×#131	D₃₁	1	1	yes	characteristic
55	#520	—	31	2	no	—
56	#428	—	31	6	no	—
57	#427	—	31	3	no	—
58	C₂×#106	—	31	3	no	—
59	C₂×C₃₁	—	2	6	no	—
60	C₂×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₃₁	#577	1	2	yes	—
62	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
63	#131	Dic₃₁	1	8	yes	complemented
64	C₂×M₁₆	C₂×D₃₁	1	3	yes	—
65	C₂²×C₈	C₂×D₃₁	1	3	yes	—
66	C₄×Q₈	—	62	2	no	—
67	C₄×D₄	—	62	6	no	—
68	#117	—	31	6	no	—
69	C₂×C₄⋊C₄	—	31	6	no	—
70	#107	—	31	6	no	—
71	#106	—	31	3	no	—
72	#106	—	31	6	no	—
73	#106	—	31	1	no	—
74	C₃₁	#2912	1	1	yes	characteristic, complemented, Sylow
75	#41	Dic₆₂	1	4	yes	—
76	#41	C₂×Dic₃₁	1	4	yes	—
77	C₂×Q₈	D₁₂₄	1	1	yes	characteristic
78	C₂×D₄	D₁₂₄	1	3	yes	—
79	C₂²×C₄	—	62	6	no	—
80	C₂²×C₄	C₂²×D₃₁	1	3	yes	—
81	M₁₆	C₂×Dic₃₁	1	12	yes	complemented
82	C₂×C₈	C₂×Dic₃₁	1	12	yes	complemented
83	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
84	C₂×C₈	C₂²×D₃₁	1	3	yes	—
85	C₄⋊C₄	—	31	6	no	—
86	C₄⋊C₄	—	31	2	no	—
87	C₄⋊C₄	—	62	6	no	—
88	C₂²⋊C₄	—	62	6	no	—
89	C₄²	—	62	6	no	—
90	C₂³	C₂×D₁₂₄	1	3	yes	—
91	Q₈	C₄⋊Dic₃₁	1	4	yes	—
92	D₄	C₄⋊Dic₃₁	1	12	yes	—
93	C₂×C₄	—	62	2	no	—
94	C₂×C₄	—	62	6	no	—
95	C₂×C₄	—	248	6	no	—
96	C₂×C₄	C₂×Dic₆₂	1	6	yes	—
97	C₂×C₄	C₂×D₁₂₄	1	3	yes	—
98	C₂×C₄	C₂²×Dic₃₁	1	6	yes	—
99	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
100	C₈	C₂²×Dic₃₁	1	2	yes	—
101	C₈	C₂²×Dic₃₁	1	6	yes	—
102	C₂²	—	2	6	no	—
103	C₂²	C₂×C₄⋊Dic₃₁	1	6	yes	—
104	C₂²	C₂²×D₁₂₄	1	1	yes	characteristic
105	C₄	—	124	6	no	—
106	C₄	—	124	2	no	—
107	C₄	C₂×C₄⋊Dic₃₁	1	6	yes	—
108	C₄	C₂²×Dic₆₂	1	1	yes	characteristic
109	C₄	C₂³×Dic₃₁	1	1	yes	characteristic
110	C₂	—	2	6	no	—
111	C₂	C₂²×C₄⋊Dic₃₁	1	1	yes	characteristic
112	C₂	#14521861	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521532

C₂

yes

maximal

#14521091

C₂

yes

maximal, complemented

#14521087

C₂

yes

maximal, complemented

C₂×#11758405

C₂

yes

maximal, complemented

C₂×C₃₁×#131

C₂

yes

maximal, characteristic

C₂×#41×C₃₁

C₂²

yes

characteristic

C₃₁×#131

C₄

yes

complemented

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂²×C₈×C₃₁

C₂²

yes

—

Q₈×Dic₃₁

—

D₄×Dic₃₁

—

#11758503

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

#11758406

C₂²

yes

—

#11758405

C₂²

yes

—

#11758405

C₂²

yes

—

#11758405

C₂²

yes

characteristic

#41×C₃₁

C₂×C₄

yes

—

#41×C₃₁

Q₈

yes

—

C₂×Q₈×C₃₁

D₄

yes

characteristic

C₂×D₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×Dic₃₁

—

M₁₆×C₃₁

C₂×C₄

yes

complemented

C₂×C₈×C₃₁

C₂×C₄

yes

complemented

C₂×C₈×C₃₁

C₂³

yes

characteristic

C₂×C₈×C₃₁

C₂³

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

C₄⋊Dic₃₁

C₂³

yes

—

C₄×Dic₃₁

—

C₂³×C₃₁

C₂×D₄

yes

—

Q₈×C₃₁

C₄⋊C₄

yes

—

D₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₈×C₃₁

C₂²×C₄

yes

—

C₈×C₃₁

C₂²×C₄

yes

—

#2912

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₄⋊C₄

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

C₂×C₄⋊C₄

yes

—

C₄×C₃₁

C₂³×C₄

yes

characteristic

C₄×C₃₁

C₂²×Q₈

yes

characteristic

Dic₃₁

—

Dic₃₁

—

C₂×#131

D₃₁

yes

—

—

—

—

C₂×C₃₁

—

C₂×C₃₁

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₃₁

#577

yes

—

C₂×#41

C₂×D₃₁

yes

characteristic

#131

Dic₃₁

yes

complemented

C₂×M₁₆

C₂×D₃₁

yes

—

C₂²×C₈

C₂×D₃₁

yes

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×C₄⋊C₄

—

—

—

—

—

C₃₁

#2912

yes

characteristic, complemented, Sylow

#41

Dic₆₂

yes

—

#41

C₂×Dic₃₁

yes

—

C₂×Q₈

D₁₂₄

yes

characteristic

C₂×D₄

D₁₂₄

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

—

M₁₆

C₂×Dic₃₁

yes

complemented

C₂×C₈

C₂×Dic₃₁

yes

complemented

C₂×C₈

C₂²×D₃₁

yes

characteristic

C₂×C₈

C₂²×D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

C₂×D₁₂₄

yes

—

Q₈

C₄⋊Dic₃₁

yes

—

D₄

C₄⋊Dic₃₁

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

248

—

C₂×C₄

C₂×Dic₆₂

yes

—

C₂×C₄

C₂×D₁₂₄

yes

—

C₂×C₄

C₂²×Dic₃₁

yes

—

C₂×C₄

C₂³×D₃₁

yes

characteristic

100

C₈

C₂²×Dic₃₁

yes

—

101

C₈

C₂²×Dic₃₁

yes

—

102

C₂²

—

103

C₂²

C₂×C₄⋊Dic₃₁

yes

—

104

C₂²

C₂²×D₁₂₄

yes

characteristic

105

C₄

—

124

—

106

C₄

—

124

—

107

C₄

C₂×C₄⋊Dic₃₁

yes

—

108

C₄

C₂²×Dic₆₂

yes

characteristic

109

C₄

C₂³×Dic₃₁

yes

characteristic

110

C₂

—

111

C₂

C₂²×C₄⋊Dic₃₁

yes

characteristic

112

C₂

#14521861

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₂₄
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₃₁×#131	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×#41×C₃₁	C₂²
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×#131	D₃₁
hypocenter	C₃₁	#2912
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

subgroup

G/H

center

C₂²

C₂²×D₁₂₄

commutator

C₄×C₃₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₂×C₃₁×#131

C₂

socle

C₂²×C₃₁

C₂²×D₄

prime radical

C₂×#41×C₃₁

C₂²

cosocle

C₂×C₄×C₃₁

C₂⁴

D₃₁

C₃₁

G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

N	H ≅ G/N	type
2. #14521091	C₂	semidirect
3. #14521087	C₂	semidirect
4. C₂×#11758405	C₂	semidirect
7. C₃₁×#131	C₄	semidirect
24. M₁₆×C₃₁	C₂×C₄	semidirect
25. C₂×C₈×C₃₁	C₂×C₄	semidirect
63. #131	Dic₃₁	semidirect
74. C₃₁	#2912	semidirect
81. M₁₆	C₂×Dic₃₁	semidirect
82. C₂×C₈	C₂×Dic₃₁	semidirect

H ≅ G/N

type

2. #14521091

C₂

semidirect

3. #14521087

C₂

semidirect

4. C₂×#11758405

C₂

semidirect

7. C₃₁×#131

C₄

semidirect

24. M₁₆×C₃₁

C₂×C₄

semidirect

25. C₂×C₈×C₃₁

C₂×C₄

semidirect

63. #131

Dic₃₁

semidirect

74. C₃₁

#2912

semidirect

81. M₁₆

C₂×Dic₃₁

semidirect

82. C₂×C₈

C₂×Dic₃₁

semidirect

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

subgroup	index
#14521532	2
#14521091	2
#14521087	2
C₂×#11758405	2
C₂×C₃₁×#131	2
#2912	31

subgroup

index

C₂×C₃₁×#131

Subgroups — #16854306

Conjugacy classes of subgroups

Complements

Maximal subgroups