Group #14521714 — subgroups

Subgroups — #14521714

Order 1984.

280 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 115 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×Q₈×D₃₁	C₂	1	2	yes	maximal, complemented
2	C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11758523	C₂	1	2	yes	maximal, complemented
4	#11758473	C₂	1	4	yes	maximal, complemented
5	#11758470	C₂	1	2	yes	maximal
6	#11758465	C₂	1	2	yes	maximal, complemented
7	C₄×D₁₂₄	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Q₈×C₃₁	C₂²	1	2	yes	—
10	Q₈×D₃₁	—	2	8	no	—
11	C₂×D₁₂₄	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₃₁	C₂²	1	2	yes	—
13	C₂×C₄×D₃₁	C₂²	1	4	yes	—
14	C₂×Dic₆₂	C₂²	1	4	yes	—
15	C₂×Dic₆₂	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₄²×C₃₁	C₂²	1	1	yes	characteristic
19	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
20	(C₂×D₃₁)⋊C₄	C₂²	1	4	yes	—
21	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
23	Dic₃₁⋊C₄	C₂²	1	2	yes	—
24	Dic₃₁⋊C₄	C₂²	1	4	yes	—
25	C₄×Dic₃₁	C₂²	1	2	yes	—
26	C₂²×D₃₁	C₂³	1	2	yes	—
27	Q₈×C₃₁	—	2	4	no	—
28	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
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	C₂×Dic₃₁	C₂³	1	2	yes	—
33	C₂×Dic₃₁	C₂³	1	2	yes	—
34	C₂×Dic₃₁	C₂³	1	2	yes	—
35	D₁₂₄	Q₈	1	4	yes	complemented
36	C₄×D₃₁	—	2	4	no	—
37	C₄×D₃₁	—	2	8	no	—
38	Dic₆₂	—	2	8	no	—
39	Dic₆₂	D₄	1	4	yes	complemented
40	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
41	C₂×D₃₁	—	2	2	no	—
42	C₂×D₃₁	—	2	2	no	—
43	C₂×D₃₁	C₂×Q₈	1	4	yes	complemented
44	C₄×C₃₁	—	2	4	no	—
45	C₄×C₃₁	—	2	1	no	—
46	C₄×C₃₁	C₂×D₄	1	2	yes	—
47	C₄×C₃₁	C₂×Q₈	1	2	yes	complemented
48	Dic₃₁	—	2	2	no	—
49	Dic₃₁	—	2	2	no	—
50	Dic₃₁	C₂×D₄	1	4	yes	—
51	D₄×Q₈	—	31	1	no	maximal, Sylow
52	C₂×C₃₁	C₂²×D₄	1	1	yes	characteristic
53	C₂×C₃₁	C₂²×Q₈	1	1	yes	characteristic
54	C₂×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
55	D₃₁	—	2	4	no	—
56	C₂²×Q₈	—	31	2	no	—
57	C₄⋊Q₈	—	31	2	no	—
58	C₄⋊Q₈	D₃₁	1	1	yes	characteristic, complemented
59	C₂²⋊Q₈	—	31	2	no	—
60	C₂²⋊Q₈	—	31	4	no	—
61	C₄×Q₈	—	31	1	no	—
62	C₄×D₄	—	31	2	no	—
63	C₄×D₄	—	31	1	no	—
64	C₃₁	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
65	C₂×Q₈	—	31	1	no	—
66	C₂×Q₈	—	31	4	no	—
67	C₂×Q₈	—	62	8	no	—
68	C₂×Q₈	C₂×D₃₁	1	2	yes	—
69	C₂×D₄	—	31	1	no	—
70	C₂²×C₄	—	31	4	no	—
71	C₂²×C₄	—	31	2	no	—
72	C₄⋊C₄	—	31	2	no	—
73	C₄⋊C₄	—	31	4	no	—
74	C₄⋊C₄	—	31	1	no	—
75	C₄⋊C₄	—	31	1	no	—
76	C₄⋊C₄	C₂×D₃₁	1	2	yes	—
77	C₄⋊C₄	C₂×D₃₁	1	2	yes	—
78	C₂²⋊C₄	—	31	4	no	—
79	C₂²⋊C₄	—	31	2	no	—
80	C₄²	—	31	2	no	—
81	C₄²	C₂×D₃₁	1	1	yes	characteristic
82	C₂³	—	31	2	no	—
83	Q₈	—	2	4	no	—
84	Q₈	—	31	4	no	—
85	Q₈	—	62	8	no	—
86	D₄	—	31	4	no	—
87	C₂×C₄	—	31	2	no	—
88	C₂×C₄	—	31	2	no	—
89	C₂×C₄	—	31	2	no	—
90	C₂×C₄	—	62	8	no	—
91	C₂×C₄	—	62	4	no	—
92	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
93	C₂×C₄	C₂²×D₃₁	1	4	yes	—
94	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
96	C₂²	—	31	4	no	—
97	C₂²	—	62	2	no	—
98	C₂²	—	62	2	no	—
99	C₂²	C₂³×D₃₁	1	1	yes	characteristic
100	C₄	—	2	4	no	—
101	C₄	—	2	1	no	—
102	C₄	—	31	4	no	—
103	C₄	—	62	2	no	—
104	C₄	—	62	2	no	—
105	C₄	D₄×D₃₁	1	2	yes	—
106	C₄	Q₈×D₃₁	1	2	yes	complemented
107	C₂	—	62	4	no	—
108	C₂	C₂×D₄×D₃₁	1	1	yes	characteristic
109	C₂	C₂×Q₈×D₃₁	1	1	yes	characteristic
110	C₂	#11758564	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₃₁

C₂

yes

maximal, complemented

C₃₁×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

#11758523

C₂

yes

maximal, complemented

#11758473

C₂

yes

maximal, complemented

#11758470

C₂

yes

maximal

#11758465

C₂

yes

maximal, complemented

C₄×D₁₂₄

C₂

yes

maximal, characteristic

C₄×Dic₆₂

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₃₁

C₂²

yes

—

Q₈×D₃₁

—

C₂×D₁₂₄

C₂²

yes

characteristic

C₂×C₄×D₃₁

C₂²

yes

—

C₂×C₄×D₃₁

C₂²

yes

—

C₂×Dic₆₂

C₂²

yes

—

C₂×Dic₆₂

C₂²

yes

characteristic

C₄⋊C₄×C₃₁

C₂²

yes

—

C₄⋊C₄×C₃₁

C₂²

yes

—

C₄²×C₃₁

C₂²

yes

characteristic

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

characteristic

C₄⋊Dic₃₁

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

—

C₂²×D₃₁

C₂³

yes

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

D₁₂₄

Q₈

yes

complemented

C₄×D₃₁

—

C₄×D₃₁

—

Dic₆₂

—

Dic₆₂

D₄

yes

complemented

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

C₂×Q₈

yes

complemented

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×D₄

yes

—

C₄×C₃₁

C₂×Q₈

yes

complemented

Dic₃₁

—

Dic₃₁

—

Dic₃₁

C₂×D₄

yes

—

D₄×Q₈

—

maximal, Sylow

C₂×C₃₁

C₂²×D₄

yes

characteristic

C₂×C₃₁

C₂²×Q₈

yes

characteristic

C₂×C₃₁

2₋¹⁺⁴

yes

characteristic

D₃₁

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₃₁

yes

characteristic, complemented

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₃₁

D₄×Q₈

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₃₁

yes

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₁

yes

—

C₄⋊C₄

C₂×D₃₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₃₁

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₄

C₂²×D₃₁

yes

—

C₂×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₄

C₂²×D₃₁

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂³×D₃₁

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

D₄×D₃₁

yes

—

106

C₄

Q₈×D₃₁

yes

complemented

107

C₂

—

108

C₂

C₂×D₄×D₃₁

yes

characteristic

109

C₂

C₂×Q₈×D₃₁

yes

characteristic

110

C₂

#11758564

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×C₄⋊Q₈	C₂
prime radical	C₂×D₁₂₄	C₂²
hypercenter	C₄⋊Q₈	D₃₁
hypocenter	C₃₁	D₄×Q₈
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁴

Fitting

C₃₁×C₄⋊Q₈

C₂

prime radical

C₂×D₁₂₄

C₂²

hypercenter

C₄⋊Q₈

D₃₁

hypocenter

C₃₁

D₄×Q₈

LCS

G ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×Q₈×D₃₁	C₂	semidirect
2. C₃₁×C₄⋊Q₈	C₂	semidirect
3. #11758523	C₂	semidirect
4. #11758473	C₂	semidirect
6. #11758465	C₂	semidirect
8. C₄×Dic₆₂	C₂	semidirect
35. D₁₂₄	Q₈	semidirect
39. Dic₆₂	D₄	semidirect
43. C₂×D₃₁	C₂×Q₈	semidirect
47. C₄×C₃₁	C₂×Q₈	semidirect
58. C₄⋊Q₈	D₃₁	semidirect
64. C₃₁	D₄×Q₈	semidirect
106. C₄	Q₈×D₃₁	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₃₁

C₂

semidirect

2. C₃₁×C₄⋊Q₈

C₂

semidirect

3. #11758523

C₂

semidirect

4. #11758473

C₂

semidirect

6. #11758465

C₂

semidirect

8. C₄×Dic₆₂

C₂

semidirect

35. D₁₂₄

Q₈

semidirect

39. Dic₆₂

D₄

semidirect

43. C₂×D₃₁

C₂×Q₈

semidirect

47. C₄×C₃₁

C₂×Q₈

semidirect

58. C₄⋊Q₈

D₃₁

semidirect

64. C₃₁

D₄×Q₈

semidirect

106. C₄

Q₈×D₃₁

semidirect

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

subgroup	index
C₂×Q₈×D₃₁	2
C₃₁×C₄⋊Q₈	2
#11758523	2
#11758473	2
#11758470	2
#11758465	2
C₄×D₁₂₄	2
C₄×Dic₆₂	2
D₄×Q₈	31

subgroup

index

C₂×Q₈×D₃₁

C₃₁×C₄⋊Q₈

C₄×D₁₂₄

C₄×Dic₆₂

D₄×Q₈

Subgroups — #14521714

Conjugacy classes of subgroups

Complements

Maximal subgroups