Group #14521618 — subgroups

Subgroups — #14521618

Order 1984.

334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 111 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×#92495	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₂₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758472	C₂	1	4	yes	maximal, complemented
5	#11758459	C₂	1	4	yes	maximal, complemented
6	#11758452	C₂	1	2	yes	maximal, complemented
7	#11758451	C₂	1	2	yes	maximal, complemented
8	C₂³×D₃₁	C₂²	1	2	yes	—
9	C₂²×C₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₃₁	C₂²	1	2	yes	—
11	#92495	—	2	4	no	—
12	C₂×D₁₂₄	—	2	4	no	—
13	C₂×D₁₂₄	C₂²	1	2	yes	—
14	C₂×D₁₂₄	C₂²	1	2	yes	—
15	C₂×D₁₂₄	C₂²	1	1	yes	characteristic, complemented
16	C₂×D₁₂₄	C₂²	1	8	yes	complemented
17	C₂×C₄×D₃₁	C₂²	1	2	yes	complemented
18	C₂×Dic₆₂	C₂²	1	1	yes	characteristic, complemented
19	C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
20	C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₄²×C₃₁	C₂²	1	2	yes	complemented
22	(C₂×D₃₁)⋊C₄	C₂²	1	8	yes	complemented
23	C₂³×C₃₁	C₂³	1	1	yes	characteristic
24	C₂²×D₃₁	—	2	4	no	—
25	C₂²×D₃₁	—	4	4	no	—
26	C₂²×D₃₁	C₂³	1	4	yes	—
27	C₂²×D₃₁	C₂³	1	2	yes	—
28	C₂×C₄×C₃₁	D₄	1	4	yes	complemented
29	C₂×C₄×C₃₁	C₂³	1	4	yes	—
30	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²⋊D₃₁	—	2	4	no	—
33	C₂×Dic₃₁	C₂³	1	2	yes	—
34	D₁₂₄	—	2	4	no	—
35	D₁₂₄	—	2	4	no	—
36	D₁₂₄	—	2	2	no	—
37	D₁₂₄	—	4	8	no	—
38	C₄×D₃₁	—	2	4	no	—
39	Dic₆₂	—	2	2	no	—
40	C₂²×C₃₁	—	2	2	no	—
41	C₂²×C₃₁	C₂×D₄	1	2	yes	—
42	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
43	C₂×D₃₁	—	2	2	no	—
44	C₂×D₃₁	—	2	4	no	—
45	C₂×D₃₁	—	2	4	no	—
46	C₂×D₃₁	—	2	8	no	—
47	C₂×D₃₁	—	4	8	no	—
48	C₄×C₃₁	—	2	4	no	—
49	C₄×C₃₁	C₂×D₄	1	2	yes	—
50	C₄×C₃₁	C₂×D₄	1	2	yes	—
51	Dic₃₁	—	2	2	no	—
52	#535	—	31	1	no	maximal, Sylow
53	C₂×C₃₁	—	2	2	no	—
54	C₂×C₃₁	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₃₁	H(5,𝔽₂)	1	2	yes	—
56	D₃₁	—	4	2	no	—
57	D₃₁	—	4	4	no	—
58	C₂×#41	—	31	1	no	—
59	C₂²×D₄	—	31	1	no	—
60	#127	—	31	2	no	—
61	#124	—	31	2	no	—
62	#121	—	31	4	no	—
63	#120	—	31	4	no	—
64	#117	D₃₁	1	1	yes	characteristic, complemented
65	C₃₁	#535	1	1	yes	characteristic, complemented, Sylow
66	C₂⁴	—	31	2	no	—
67	#41	—	62	4	no	—
68	C₂×Q₈	—	31	1	no	—
69	C₂×D₄	—	31	2	no	—
70	C₂×D₄	—	31	2	no	—
71	C₂×D₄	—	31	2	no	—
72	C₂×D₄	—	31	1	no	—
73	C₂×D₄	—	31	8	no	—
74	C₂×D₄	—	62	4	no	—
75	C₂²×C₄	—	31	2	no	—
76	C₂²×C₄	C₂×D₃₁	1	1	yes	characteristic
77	C₄⋊C₄	C₂×D₃₁	1	2	yes	complemented
78	C₂²⋊C₄	—	31	8	no	—
79	C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
80	C₄²	C₂×D₃₁	1	2	yes	complemented
81	C₂³	—	31	4	no	—
82	C₂³	—	31	2	no	—
83	C₂³	—	62	4	no	—
84	C₂³	—	124	4	no	—
85	C₂³	C₂²×D₃₁	1	1	yes	characteristic
86	Q₈	—	62	2	no	—
87	D₄	—	62	4	no	—
88	D₄	—	62	4	no	—
89	D₄	—	62	4	no	—
90	D₄	—	62	2	no	—
91	D₄	—	124	8	no	—
92	C₂×C₄	—	31	2	no	—
93	C₂×C₄	—	62	4	no	—
94	C₂×C₄	D₁₂₄	1	4	yes	complemented
95	C₂×C₄	C₂²×D₃₁	1	4	yes	—
96	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	62	2	no	—
100	C₂²	—	62	4	no	—
101	C₂²	—	62	4	no	—
102	C₂²	—	62	8	no	—
103	C₂²	—	124	8	no	—
104	C₂²	C₂×D₁₂₄	1	2	yes	—
105	C₂²	C₂³×D₃₁	1	1	yes	characteristic
106	C₄	—	2	4	no	—
107	C₄	—	62	2	no	—
108	C₄	C₂×D₁₂₄	1	2	yes	—
109	C₄	C₂×D₁₂₄	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	124	4	no	—
112	C₂	—	124	2	no	—
113	C₂	C₂²×D₁₂₄	1	1	yes	characteristic
114	C₂	#11758566	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#92495

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₂₄

C₂

yes

maximal, characteristic, complemented

C₃₁×#117

C₂

yes

maximal, characteristic, complemented

#11758472

C₂

yes

maximal, complemented

#11758459

C₂

yes

maximal, complemented

#11758452

C₂

yes

maximal, complemented

#11758451

C₂

yes

maximal, complemented

C₂³×D₃₁

C₂²

yes

—

C₂²×C₄×C₃₁

C₂²

yes

characteristic

C₂×C₂²⋊D₃₁

C₂²

yes

—

#92495

—

C₂×D₁₂₄

—

C₂×D₁₂₄

C₂²

yes

—

C₂×D₁₂₄

C₂²

yes

—

C₂×D₁₂₄

C₂²

yes

characteristic, complemented

C₂×D₁₂₄

C₂²

yes

complemented

C₂×C₄×D₃₁

C₂²

yes

complemented

C₂×Dic₆₂

C₂²

yes

characteristic, complemented

C₄⋊C₄×C₃₁

C₂²

yes

complemented

C₂²⋊C₄×C₃₁

C₂²

yes

—

C₄²×C₃₁

C₂²

yes

complemented

(C₂×D₃₁)⋊C₄

C₂²

yes

complemented

C₂³×C₃₁

C₂³

yes

characteristic

C₂²×D₃₁

—

C₂²×D₃₁

—

C₂²×D₃₁

C₂³

yes

—

C₂²×D₃₁

C₂³

yes

—

C₂×C₄×C₃₁

D₄

yes

complemented

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂²⋊D₃₁

—

C₂×Dic₃₁

C₂³

yes

—

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

C₄×D₃₁

—

Dic₆₂

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×D₄

yes

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×D₄

yes

—

C₄×C₃₁

C₂×D₄

yes

—

Dic₃₁

—

#535

—

maximal, Sylow

C₂×C₃₁

—

C₂×C₃₁

C₂²×D₄

yes

characteristic

C₂×C₃₁

H(5,𝔽₂)

yes

—

D₃₁

—

D₃₁

—

C₂×#41

—

C₂²×D₄

—

—

—

—

—

D₃₁

yes

characteristic, complemented

C₃₁

#535

yes

characteristic, complemented, Sylow

C₂⁴

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₃₁

yes

characteristic

C₄⋊C₄

C₂×D₃₁

yes

complemented

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₃₁

yes

—

C₄²

C₂×D₃₁

yes

complemented

C₂³

—

C₂³

—

C₂³

—

C₂³

—

124

—

C₂³

C₂²×D₃₁

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

124

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₁₂₄

yes

complemented

C₂×C₄

C₂²×D₃₁

yes

—

C₂×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₄

C₂²×D₃₁

yes

characteristic

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

124

—

104

C₂²

C₂×D₁₂₄

yes

—

105

C₂²

C₂³×D₃₁

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

C₂×D₁₂₄

yes

—

109

C₄

C₂×D₁₂₄

yes

—

110

C₂

—

111

C₂

—

124

—

112

C₂

—

124

—

113

C₂

C₂²×D₁₂₄

yes

characteristic

114

C₂

#11758566

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×#117	C₂
hypercenter	#117	D₃₁
hypocenter	C₃₁	#535
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁴

Fitting

C₃₁×#117

C₂

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₂×#92495	C₂	semidirect
2. C₂²×D₁₂₄	C₂	semidirect
3. C₃₁×#117	C₂	semidirect
4. #11758472	C₂	semidirect
5. #11758459	C₂	semidirect
6. #11758452	C₂	semidirect
7. #11758451	C₂	semidirect
15. C₂×D₁₂₄	C₂²	semidirect
16. C₂×D₁₂₄	C₂²	semidirect
17. C₂×C₄×D₃₁	C₂²	semidirect
18. C₂×Dic₆₂	C₂²	semidirect
19. C₄⋊C₄×C₃₁	C₂²	semidirect
21. C₄²×C₃₁	C₂²	semidirect
22. (C₂×D₃₁)⋊C₄	C₂²	semidirect
28. C₂×C₄×C₃₁	D₄	semidirect
64. #117	D₃₁	semidirect
65. C₃₁	#535	semidirect
77. C₄⋊C₄	C₂×D₃₁	semidirect
80. C₄²	C₂×D₃₁	semidirect
94. C₂×C₄	D₁₂₄	semidirect

H ≅ G/N

type

1. C₂×#92495

C₂

semidirect

2. C₂²×D₁₂₄

C₂

semidirect

3. C₃₁×#117

C₂

semidirect

4. #11758472

C₂

semidirect

5. #11758459

C₂

semidirect

6. #11758452

C₂

semidirect

7. #11758451

C₂

semidirect

15. C₂×D₁₂₄

C₂²

semidirect

16. C₂×D₁₂₄

C₂²

semidirect

17. C₂×C₄×D₃₁

C₂²

semidirect

18. C₂×Dic₆₂

C₂²

semidirect

19. C₄⋊C₄×C₃₁

C₂²

semidirect

21. C₄²×C₃₁

C₂²

semidirect

22. (C₂×D₃₁)⋊C₄

C₂²

semidirect

28. C₂×C₄×C₃₁

D₄

semidirect

64. #117

D₃₁

semidirect

65. C₃₁

#535

semidirect

77. C₄⋊C₄

C₂×D₃₁

semidirect

80. C₄²

C₂×D₃₁

semidirect

94. C₂×C₄

D₁₂₄

semidirect

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

subgroup	index
C₂×#92495	2
C₂²×D₁₂₄	2
C₃₁×#117	2
#11758472	2
#11758459	2
#11758452	2
#11758451	2
#535	31

subgroup

index

C₂×#92495

C₂²×D₁₂₄

C₃₁×#117

Subgroups — #14521618

Conjugacy classes of subgroups

Complements

Maximal subgroups