Group #31824178 — subgroups

Subgroups — #31824178

Order 17984.

236 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22582879	C₂	1	1	yes	maximal, characteristic, complemented
2	#22582865	C₂	1	4	yes	maximal, complemented
3	#22582853	C₂	1	2	yes	maximal, complemented
4	#22582850	C₂	1	2	yes	maximal, complemented
5	C₄×D₁₁₂₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#22582838	C₂	1	2	yes	maximal, complemented
7	#22582837	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₂₈₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₅₆₂	C₂²	1	2	yes	—
10	C₂×D₁₁₂₄	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₂₈₁	C₂²	1	4	yes	—
12	C₂×C₄×D₂₈₁	C₂²	1	2	yes	—
13	C₄⋊Dic₂₈₁	C₂²	1	2	yes	—
14	C₄⋊Dic₂₈₁	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₂₈₁	C₂²	1	2	yes	—
16	C₂²⋊Dic₂₈₁	C₂²	1	4	yes	—
17	#17985115	C₂²	1	4	yes	—
18	#17985115	C₂²	1	2	yes	—
19	C₄×Dic₂₈₁	C₂²	1	4	yes	—
20	C₂×D₄×C₂₈₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₂₈₁	C₂²	1	2	yes	—
22	C₄⋊C₄×C₂₈₁	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₂₈₁	C₂²	1	2	yes	—
24	C₄²×C₂₈₁	C₂²	1	1	yes	characteristic
25	C₂²×D₂₈₁	C₂³	1	2	yes	—
26	Dic₅₆₂	—	4	2	no	—
27	D₁₁₂₄	—	2	2	no	—
28	C₂²⋊D₂₈₁	—	4	4	no	—
29	C₂×Dic₂₈₁	C₂³	1	2	yes	—
30	C₂×Dic₂₈₁	C₂³	1	4	yes	—
31	C₄×D₂₈₁	—	2	4	no	—
32	C₂³×C₂₈₁	C₂³	1	2	yes	—
33	D₄×C₂₈₁	—	2	2	no	—
34	C₂×C₄×C₂₈₁	—	2	4	no	—
35	C₂×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₂₈₁	C₂³	1	2	yes	—
37	C₂×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
39	C₂²×C₂₈₁	—	2	2	no	—
40	C₂²×C₂₈₁	—	2	2	no	—
41	C₂²×C₂₈₁	—	2	2	no	—
42	C₂²×C₂₈₁	C₂⁴	1	1	yes	characteristic
43	C₂×D₂₈₁	—	2	2	no	—
44	C₂×D₂₈₁	—	2	2	no	—
45	C₂×D₂₈₁	—	2	2	no	—
46	C₄×C₂₈₁	—	2	2	no	—
47	C₄×C₂₈₁	—	2	1	no	—
48	C₄×C₂₈₁	#41	1	2	yes	—
49	C₄×C₂₈₁	#41	1	2	yes	—
50	Dic₂₈₁	—	2	2	no	—
51	Dic₂₈₁	—	2	4	no	—
52	C₂×C₂₈₁	—	4	2	no	—
53	C₂×C₂₈₁	C₂×#41	1	1	yes	characteristic
54	C₂×C₂₈₁	C₂×#41	1	1	yes	characteristic
55	C₂×C₂₈₁	H(5,𝔽₂)	1	1	yes	characteristic
56	D₂₈₁	—	4	2	no	—
57	C₂₈₁	#555	1	1	yes	characteristic, complemented, Sylow
58	#555	—	281	1	no	maximal, Sylow
59	C₄⋊Q₈	—	281	1	no	—
60	#124	—	281	4	no	—
61	#121	—	281	2	no	—
62	#121	—	281	2	no	—
63	C₄×D₄	—	281	1	no	—
64	C₄×D₄	D₂₈₁	1	1	yes	characteristic, complemented
65	#117	—	281	2	no	—
66	#117	—	281	2	no	—
67	C₂×Q₈	—	281	2	no	—
68	C₂×D₄	—	281	4	no	—
69	C₂×D₄	—	281	1	no	—
70	C₂×D₄	C₂×D₂₈₁	1	1	yes	characteristic
71	C₂²×C₄	—	281	2	no	—
72	C₂²×C₄	C₂×D₂₈₁	1	2	yes	—
73	C₄⋊C₄	—	281	1	no	—
74	C₄⋊C₄	—	281	2	no	—
75	C₄⋊C₄	—	281	2	no	—
76	C₄⋊C₄	C₂×D₂₈₁	1	1	yes	characteristic
77	C₂²⋊C₄	—	281	4	no	—
78	C₂²⋊C₄	—	281	4	no	—
79	C₂²⋊C₄	—	281	2	no	—
80	C₂²⋊C₄	C₂×D₂₈₁	1	2	yes	—
81	C₄²	—	281	4	no	—
82	C₄²	C₂×D₂₈₁	1	1	yes	characteristic
83	C₂³	—	281	2	no	—
84	C₂³	C₂²×D₂₈₁	1	2	yes	—
85	Q₈	—	1124	2	no	—
86	D₄	—	2	2	no	—
87	D₄	—	562	2	no	—
88	D₄	—	1124	4	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	281	2	no	—
91	C₂×C₄	—	281	4	no	—
92	C₂×C₄	—	562	4	no	—
93	C₂×C₄	C₂²×D₂₈₁	1	2	yes	—
94	C₂×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
97	C₂²	—	2	2	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	562	2	no	—
101	C₂²	—	562	2	no	—
102	C₂²	—	562	2	no	—
103	C₂²	C₂³×D₂₈₁	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	562	2	no	—
107	C₄	—	562	4	no	—
108	C₄	#17985137	1	2	yes	—
109	C₄	#17985138	1	2	yes	—
110	C₂	—	4	2	no	—
111	C₂	—	1124	2	no	—
112	C₂	C₂×#17985137	1	1	yes	characteristic
113	C₂	C₂×#17985138	1	1	yes	characteristic
114	C₂	#22582925	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582879

C₂

yes

maximal, characteristic, complemented

#22582865

C₂

yes

maximal, complemented

#22582853

C₂

yes

maximal, complemented

#22582850

C₂

yes

maximal, complemented

C₄×D₁₁₂₄

C₂

yes

maximal, characteristic, complemented

#22582838

C₂

yes

maximal, complemented

#22582837

C₂

yes

maximal, complemented

C₄×D₄×C₂₈₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₅₆₂

C₂²

yes

—

C₂×D₁₁₂₄

C₂²

yes

characteristic

C₂×C₂²⋊D₂₈₁

C₂²

yes

—

C₂×C₄×D₂₈₁

C₂²

yes

—

C₄⋊Dic₂₈₁

C₂²

yes

—

C₄⋊Dic₂₈₁

C₂²

yes

characteristic

C₄⋊Dic₂₈₁

C₂²

yes

—

C₂²⋊Dic₂₈₁

C₂²

yes

—

#17985115

C₂²

yes

—

#17985115

C₂²

yes

—

C₄×Dic₂₈₁

C₂²

yes

—

C₂×D₄×C₂₈₁

C₂²

yes

characteristic

C₂²×C₄×C₂₈₁

C₂²

yes

—

C₄⋊C₄×C₂₈₁

C₂²

yes

characteristic

C₂²⋊C₄×C₂₈₁

C₂²

yes

—

C₄²×C₂₈₁

C₂²

yes

characteristic

C₂²×D₂₈₁

C₂³

yes

—

Dic₅₆₂

—

D₁₁₂₄

—

C₂²⋊D₂₈₁

—

C₂×Dic₂₈₁

C₂³

yes

—

C₂×Dic₂₈₁

C₂³

yes

—

C₄×D₂₈₁

—

C₂³×C₂₈₁

C₂³

yes

—

D₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂³

yes

characteristic

C₂×C₄×C₂₈₁

C₂³

yes

—

C₂×C₄×C₂₈₁

C₂³

yes

characteristic

C₂×C₄×C₂₈₁

C₂³

yes

characteristic

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

C₂⁴

yes

characteristic

C₂×D₂₈₁

—

C₂×D₂₈₁

—

C₂×D₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

#41

yes

—

C₄×C₂₈₁

#41

yes

—

Dic₂₈₁

—

Dic₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

C₂×#41

yes

characteristic

C₂×C₂₈₁

C₂×#41

yes

characteristic

C₂×C₂₈₁

H(5,𝔽₂)

yes

characteristic

D₂₈₁

—

C₂₈₁

#555

yes

characteristic, complemented, Sylow

#555

—

281

maximal, Sylow

C₄⋊Q₈

—

281

—

#124

—

281

—

#121

—

281

—

#121

—

281

—

C₄×D₄

—

281

—

C₄×D₄

D₂₈₁

yes

characteristic, complemented

#117

—

281

—

#117

—

281

—

C₂×Q₈

—

281

—

C₂×D₄

—

281

—

C₂×D₄

—

281

—

C₂×D₄

C₂×D₂₈₁

yes

characteristic

C₂²×C₄

—

281

—

C₂²×C₄

C₂×D₂₈₁

yes

—

C₄⋊C₄

—

281

—

C₄⋊C₄

—

281

—

C₄⋊C₄

—

281

—

C₄⋊C₄

C₂×D₂₈₁

yes

characteristic

C₂²⋊C₄

—

281

—

C₂²⋊C₄

—

281

—

C₂²⋊C₄

—

281

—

C₂²⋊C₄

C₂×D₂₈₁

yes

—

C₄²

—

281

—

C₄²

C₂×D₂₈₁

yes

characteristic

C₂³

—

281

—

C₂³

C₂²×D₂₈₁

yes

—

Q₈

—

1124

—

D₄

—

D₄

—

562

—

D₄

—

1124

—

C₂×C₄

—

C₂×C₄

—

281

—

C₂×C₄

—

281

—

C₂×C₄

—

562

—

C₂×C₄

C₂²×D₂₈₁

yes

—

C₂×C₄

C₂²×D₂₈₁

yes

characteristic

C₂×C₄

C₂²×D₂₈₁

yes

characteristic

C₂×C₄

C₂²×D₂₈₁

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

562

—

101

C₂²

—

562

—

102

C₂²

—

562

—

103

C₂²

C₂³×D₂₈₁

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

562

—

107

C₄

—

562

—

108

C₄

#17985137

yes

—

109

C₄

#17985138

yes

—

110

C₂

—

111

C₂

—

1124

—

112

C₂

C₂×#17985137

yes

characteristic

113

C₂

C₂×#17985138

yes

characteristic

114

C₂

#22582925

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₈₁
commutator, socle, cosocle	C₂²×C₂₈₁	C₂⁴
Fitting	C₄×D₄×C₂₈₁	C₂
hypercenter	C₄×D₄	D₂₈₁
hypocenter	C₂₈₁	#555
LCS	G ▷ C₂²×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₈₁

commutator, socle, cosocle

C₂²×C₂₈₁

C₂⁴

Fitting

C₄×D₄×C₂₈₁

C₂

hypercenter

C₄×D₄

D₂₈₁

C₂₈₁

G ▷ C₂²×C₂₈₁ ▷ C₂₈₁

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #22582879	C₂	semidirect
2. #22582865	C₂	semidirect
3. #22582853	C₂	semidirect
4. #22582850	C₂	semidirect
5. C₄×D₁₁₂₄	C₂	semidirect
6. #22582838	C₂	semidirect
7. #22582837	C₂	semidirect
8. C₄×D₄×C₂₈₁	C₂	semidirect
57. C₂₈₁	#555	semidirect
64. C₄×D₄	D₂₈₁	semidirect

H ≅ G/N

type

1. #22582879

C₂

semidirect

2. #22582865

C₂

semidirect

3. #22582853

C₂

semidirect

4. #22582850

C₂

semidirect

5. C₄×D₁₁₂₄

C₂

semidirect

6. #22582838

C₂

semidirect

7. #22582837

C₂

semidirect

8. C₄×D₄×C₂₈₁

C₂

semidirect

57. C₂₈₁

#555

semidirect

64. C₄×D₄

D₂₈₁

semidirect

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

subgroup	index
#22582879	2
#22582865	2
#22582853	2
#22582850	2
C₄×D₁₁₂₄	2
#22582838	2
#22582837	2
C₄×D₄×C₂₈₁	2
#555	281

subgroup

index

C₄×D₁₁₂₄

C₄×D₄×C₂₈₁

281

Subgroups — #31824178

Conjugacy classes of subgroups

Complements

Maximal subgroups