Group #14521613 — subgroups

Subgroups — #14521613

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×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₂₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#11758524	C₂	1	2	yes	maximal, complemented
4	#11758517	C₂	1	2	yes	maximal, complemented
5	#11758514	C₂	1	4	yes	maximal, complemented
6	#11758507	C₂	1	4	yes	maximal, complemented
7	#11758503	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₃₁	C₂²	1	2	yes	—
9	#41×C₃₁	—	2	4	no	—
10	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₄×C₃₁	C₂²	1	2	yes	—
12	C₂×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₃₁	C₂²	1	2	yes	complemented
15	C₂×C₂²⋊D₃₁	C₂²	1	8	yes	complemented
16	C₂×D₁₂₄	—	2	4	no	—
17	C₂×D₁₂₄	C₂²	1	2	yes	—
18	C₂×D₁₂₄	C₂²	1	2	yes	—
19	C₂²⋊Dic₃₁	C₂²	1	2	yes	—
20	(C₂×D₃₁)⋊C₄	C₂²	1	8	yes	complemented
21	C₄⋊Dic₃₁	C₂²	1	2	yes	complemented
22	C₄×Dic₃₁	C₂²	1	2	yes	complemented
23	C₂³×C₃₁	C₂³	1	2	yes	—
24	C₂³×C₃₁	C₂³	1	1	yes	characteristic
25	C₂²×D₃₁	—	2	4	no	—
26	C₂²×D₃₁	—	4	4	no	—
27	C₂²×D₃₁	C₂³	1	4	yes	—
28	Q₈×C₃₁	—	2	2	no	—
29	D₄×C₃₁	—	2	4	no	—
30	D₄×C₃₁	—	2	2	no	—
31	C₂×C₄×C₃₁	—	2	4	no	—
32	C₂×C₄×C₃₁	D₄	1	4	yes	complemented
33	C₂×C₄×C₃₁	C₂³	1	2	yes	—
34	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
36	C₂²⋊D₃₁	—	4	8	no	—
37	C₂×Dic₃₁	C₂³	1	4	yes	—
38	D₁₂₄	—	2	4	no	—
39	D₁₂₄	—	2	4	no	—
40	C₂²×C₃₁	—	2	2	no	—
41	C₂²×C₃₁	—	2	2	no	—
42	C₂²×C₃₁	—	2	4	no	—
43	C₂²×C₃₁	C₂×D₄	1	2	yes	—
44	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
45	C₂×D₃₁	—	2	4	no	—
46	C₂×D₃₁	—	2	8	no	—
47	C₂×D₃₁	—	4	8	no	—
48	C₄×C₃₁	—	2	2	no	—
49	C₄×C₃₁	C₂×D₄	1	2	yes	—
50	C₄×C₃₁	C₂×D₄	1	2	yes	—
51	Dic₃₁	—	2	4	no	—
52	#535	—	31	1	no	maximal, Sylow
53	C₂×C₃₁	—	2	2	no	—
54	C₂×C₃₁	—	4	2	no	—
55	C₂×C₃₁	C₂²×D₄	1	1	yes	characteristic
56	C₂×C₃₁	H(5,𝔽₂)	1	2	yes	—
57	D₃₁	—	4	4	no	—
58	C₂×#41	D₃₁	1	1	yes	characteristic, complemented
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	—	31	1	no	—
65	C₃₁	#535	1	1	yes	characteristic, complemented, Sylow
66	C₂⁴	—	31	2	no	—
67	#41	—	2	4	no	—
68	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic, complemented
69	C₂×D₄	—	31	8	no	—
70	C₂×D₄	—	31	2	no	—
71	C₂×D₄	—	31	2	no	—
72	C₂×D₄	—	62	4	no	—
73	C₂×D₄	C₂×D₃₁	1	2	yes	—
74	C₂×D₄	C₂×D₃₁	1	1	yes	characteristic, complemented
75	C₂²×C₄	C₂×D₃₁	1	1	yes	characteristic
76	C₂²×C₄	C₂×D₃₁	1	2	yes	complemented
77	C₄⋊C₄	—	31	2	no	—
78	C₂²⋊C₄	—	31	2	no	—
79	C₂²⋊C₄	—	31	8	no	—
80	C₄²	—	31	2	no	—
81	C₂³	—	31	4	no	—
82	C₂³	—	62	4	no	—
83	C₂³	—	124	4	no	—
84	C₂³	C₂²×D₃₁	1	2	yes	—
85	C₂³	C₂²×D₃₁	1	1	yes	characteristic
86	Q₈	—	2	2	no	—
87	D₄	—	2	4	no	—
88	D₄	—	2	2	no	—
89	D₄	—	62	4	no	—
90	D₄	—	62	4	no	—
91	D₄	—	124	8	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	31	4	no	—
94	C₂×C₄	C₂²⋊D₃₁	1	4	yes	complemented
95	C₂×C₄	C₂²×D₃₁	1	2	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	—	2	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	—	62	8	no	—
102	C₂²	—	62	4	no	—
103	C₂²	—	124	8	no	—
104	C₂²	C₂×C₂²⋊D₃₁	1	2	yes	—
105	C₂²	C₂³×D₃₁	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	62	4	no	—
108	C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
109	C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	4	2	no	—
112	C₂	—	124	4	no	—
113	C₂	#11758566	1	2	yes	—
114	C₂	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#41×C₃₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₂₄

C₂

yes

maximal, characteristic, complemented

#11758524

C₂

yes

maximal, complemented

#11758517

C₂

yes

maximal, complemented

#11758514

C₂

yes

maximal, complemented

#11758507

C₂

yes

maximal, complemented

#11758503

C₂

yes

maximal, characteristic, complemented

C₂³×D₃₁

C₂²

yes

—

#41×C₃₁

—

C₂×Q₈×C₃₁

C₂²

yes

characteristic, complemented

C₂×D₄×C₃₁

C₂²

yes

—

C₂×D₄×C₃₁

C₂²

yes

characteristic, complemented

C₂²×C₄×C₃₁

C₂²

yes

characteristic

C₂²×C₄×C₃₁

C₂²

yes

complemented

C₂×C₂²⋊D₃₁

C₂²

yes

complemented

C₂×D₁₂₄

—

C₂×D₁₂₄

C₂²

yes

—

C₂×D₁₂₄

C₂²

yes

—

C₂²⋊Dic₃₁

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

complemented

C₄⋊Dic₃₁

C₂²

yes

complemented

C₄×Dic₃₁

C₂²

yes

complemented

C₂³×C₃₁

C₂³

yes

—

C₂³×C₃₁

C₂³

yes

characteristic

C₂²×D₃₁

—

C₂²×D₃₁

—

C₂²×D₃₁

C₂³

yes

—

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

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

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×D₄

yes

—

C₂²×C₃₁

C₂⁴

yes

characteristic

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

—

C₂×C₃₁

C₂²×D₄

yes

characteristic

C₂×C₃₁

H(5,𝔽₂)

yes

—

D₃₁

—

C₂×#41

D₃₁

yes

characteristic, complemented

C₂²×D₄

—

—

—

—

—

—

C₃₁

#535

yes

characteristic, complemented, Sylow

C₂⁴

—

#41

—

C₂×Q₈

C₂×D₃₁

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₃₁

yes

—

C₂×D₄

C₂×D₃₁

yes

characteristic, complemented

C₂²×C₄

C₂×D₃₁

yes

characteristic

C₂²×C₄

C₂×D₃₁

yes

complemented

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

124

—

C₂³

C₂²×D₃₁

yes

—

C₂³

C₂²×D₃₁

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

124

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊D₃₁

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×C₂²⋊D₃₁

yes

—

105

C₂²

C₂³×D₃₁

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

C₂×C₂²⋊D₃₁

yes

—

109

C₄

C₂×C₂²⋊D₃₁

yes

—

110

C₂

—

111

C₂

—

112

C₂

—

124

—

113

C₂

#11758566

yes

—

114

C₂

C₂²×C₂²⋊D₃₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₂×#41×C₃₁	C₂
hypercenter	C₂×#41	D₃₁
hypocenter	C₃₁	#535
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁴

Fitting

C₂×#41×C₃₁

C₂

hypercenter

C₂×#41

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ C₂×#41

N	H ≅ G/N	type
1. C₂×#41×C₃₁	C₂	semidirect
2. C₂²×D₁₂₄	C₂	semidirect
3. #11758524	C₂	semidirect
4. #11758517	C₂	semidirect
5. #11758514	C₂	semidirect
6. #11758507	C₂	semidirect
7. #11758503	C₂	semidirect
10. C₂×Q₈×C₃₁	C₂²	semidirect
12. C₂×D₄×C₃₁	C₂²	semidirect
14. C₂²×C₄×C₃₁	C₂²	semidirect
15. C₂×C₂²⋊D₃₁	C₂²	semidirect
20. (C₂×D₃₁)⋊C₄	C₂²	semidirect
21. C₄⋊Dic₃₁	C₂²	semidirect
22. C₄×Dic₃₁	C₂²	semidirect
32. C₂×C₄×C₃₁	D₄	semidirect
58. C₂×#41	D₃₁	semidirect
65. C₃₁	#535	semidirect
68. C₂×Q₈	C₂×D₃₁	semidirect
74. C₂×D₄	C₂×D₃₁	semidirect
76. C₂²×C₄	C₂×D₃₁	semidirect
94. C₂×C₄	C₂²⋊D₃₁	semidirect

H ≅ G/N

type

1. C₂×#41×C₃₁

C₂

semidirect

2. C₂²×D₁₂₄

C₂

semidirect

3. #11758524

C₂

semidirect

4. #11758517

C₂

semidirect

5. #11758514

C₂

semidirect

6. #11758507

C₂

semidirect

7. #11758503

C₂

semidirect

10. C₂×Q₈×C₃₁

C₂²

semidirect

12. C₂×D₄×C₃₁

C₂²

semidirect

14. C₂²×C₄×C₃₁

C₂²

semidirect

15. C₂×C₂²⋊D₃₁

C₂²

semidirect

20. (C₂×D₃₁)⋊C₄

C₂²

semidirect

21. C₄⋊Dic₃₁

C₂²

semidirect

22. C₄×Dic₃₁

C₂²

semidirect

32. C₂×C₄×C₃₁

D₄

semidirect

58. C₂×#41

D₃₁

semidirect

65. C₃₁

#535

semidirect

68. C₂×Q₈

C₂×D₃₁

semidirect

74. C₂×D₄

C₂×D₃₁

semidirect

76. C₂²×C₄

C₂×D₃₁

semidirect

94. C₂×C₄

C₂²⋊D₃₁

semidirect

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

subgroup	index
C₂×#41×C₃₁	2
C₂²×D₁₂₄	2
#11758524	2
#11758517	2
#11758514	2
#11758507	2
#11758503	2
#535	31

subgroup

index

C₂×#41×C₃₁

C₂²×D₁₂₄

Subgroups — #14521613

Conjugacy classes of subgroups

Complements

Maximal subgroups