Group #31780948 — subgroups

Subgroups — #31780948

Order 17536.

280 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 149 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×#17984021	C₂	1	4	yes	maximal
2	C₂×C₄×#11764040	C₂	1	1	yes	maximal, characteristic
3	#22577273	C₂	1	1	yes	maximal, characteristic
4	#384×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11764040	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₃₇	—	2	2	no	—
7	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
8	#17984021	—	2	8	no	—
9	C₄×#11764040	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
11	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₃₇	—	2	2	no	—
14	C₂×#11764040	C₂×C₄	1	4	yes	—
15	C₂×#11764040	D₄	1	4	yes	—
16	C₂×#11764040	Q₈	1	4	yes	—
17	C₄⋊Dic₁₃₇	—	2	4	no	—
18	C₄⋊Dic₁₃₇	—	4	4	no	—
19	C₂²×C₄×C₁₃₇	—	2	2	no	—
20	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
21	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₁₃₇	—	2	4	no	—
23	C₄⋊C₄×C₁₃₇	—	4	4	no	—
24	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
25	C₄²×C₁₃₇	Q₈	1	2	yes	—
26	C₂×Dic₁₃₇	—	4	2	no	—
27	C₂×Dic₁₃₇	—	4	4	no	—
28	#11764040	C₄⋊C₄	1	8	yes	complemented
29	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₃₇	—	4	2	no	—
31	C₂×C₄×C₁₃₇	—	4	4	no	—
32	C₂×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
33	C₂×C₄×C₁₃₇	D₈	1	4	yes	—
34	C₂×C₄×C₁₃₇	Q₁₆	1	4	yes	—
35	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₃₇	#127	1	1	yes	characteristic
41	C₂²×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₁₃₇	C₂×D₈	1	2	yes	—
43	C₂²×C₁₃₇	C₂×Q₁₆	1	2	yes	—
44	C₄×C₁₃₇	—	8	2	no	—
45	C₄×C₁₃₇	#107	1	8	yes	complemented
46	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
47	C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
48	Dic₁₃₇	—	8	2	no	—
49	C₂×C₁₃₇	#384	1	1	yes	characteristic
50	C₂×C₁₃₇	C₂×#107	1	2	yes	—
51	C₂×C₁₃₇	#493	1	1	yes	characteristic
52	C₂×C₁₃₇	#494	1	1	yes	characteristic
53	C₂×C₁₃₇	#500	1	2	yes	—
54	C₁₃₇	#1845	1	1	yes	characteristic, complemented, Sylow
55	#1845	—	137	1	no	maximal, Sylow
56	C₂×#107	—	137	4	no	—
57	C₂×C₄×C₈	—	137	1	no	—
58	#384	—	137	1	no	—
59	#384	D₁₃₇	1	1	yes	characteristic
60	C₂²×C₈	—	137	2	no	—
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	—	274	2	no	—
63	C₂×C₄⋊C₄	—	137	2	no	—
64	C₂×C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
65	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
66	#107	—	274	8	no	—
67	C₄×C₈	—	137	4	no	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	—	274	2	no	—
70	C₂²×C₄	C₂²⋊D₁₃₇	1	2	yes	—
71	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
72	C₂×C₈	—	137	4	no	—
73	C₂×C₈	—	137	4	no	—
74	C₂×C₈	—	137	4	no	—
75	C₄⋊C₄	—	2	4	no	—
76	C₄⋊C₄	—	4	4	no	—
77	C₄⋊C₄	—	274	4	no	—
78	C₄⋊C₄	—	548	4	no	—
79	C₄²	C₄×D₁₃₇	1	2	yes	—
80	C₄²	Dic₂₇₄	1	2	yes	—
81	C₂³	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	548	2	no	—
85	C₂×C₄	—	548	4	no	—
86	C₂×C₄	#14555489	1	4	yes	—
87	C₂×C₄	#14555495	1	4	yes	—
88	C₂×C₄	#14555500	1	4	yes	—
89	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
90	C₂×C₄	D₄×D₁₃₇	1	2	yes	—
91	C₂×C₄	Q₈×D₁₃₇	1	2	yes	—
92	C₂×C₄	C₂×Dic₂₇₄	1	1	yes	characteristic
93	C₈	—	137	8	no	—
94	C₂²	C₂×#14555489	1	1	yes	characteristic
95	C₂²	#17984084	1	1	yes	characteristic
96	C₂²	#17984086	1	1	yes	characteristic
97	C₂²	C₂×#14555495	1	2	yes	—
98	C₂²	C₂×#14555500	1	2	yes	—
99	C₄	—	8	2	no	—
100	C₄	—	1096	2	no	—
101	C₄	#17984021	1	8	yes	complemented
102	C₄	C₄⋊C₄×D₁₃₇	1	2	yes	—
103	C₄	#17984087	1	2	yes	—
104	C₂	#22577272	1	1	yes	characteristic
105	C₂	C₂×#17984021	1	2	yes	—
106	C₂	#22577779	1	1	yes	characteristic
107	C₂	#22577782	1	1	yes	characteristic
108	C₂	#22577806	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984021

C₂

yes

maximal

C₂×C₄×#11764040

C₂

yes

maximal, characteristic

#22577273

C₂

yes

maximal, characteristic

#384×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×#11764040

C₂²

yes

—

C₂×C₄⋊Dic₁₃₇

—

C₂×C₄⋊Dic₁₃₇

C₂²

yes

—

#17984021

—

C₄×#11764040

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₃₇

—

C₂×C₄⋊C₄×C₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₂²×Dic₁₃₇

—

C₂×#11764040

C₂×C₄

yes

—

C₂×#11764040

D₄

yes

—

C₂×#11764040

Q₈

yes

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

Q₈

yes

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

C₄⋊C₄

yes

complemented

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

D₈

yes

—

C₂×C₄×C₁₃₇

Q₁₆

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×Q₈

yes

—

C₂×C₄×C₁₃₇

C₂×Q₈

yes

characteristic

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₇

#127

yes

characteristic

C₂²×C₁₃₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₃₇

C₂×D₈

yes

—

C₂²×C₁₃₇

C₂×Q₁₆

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#107

yes

complemented

C₄×C₁₃₇

C₂×C₄⋊C₄

yes

—

C₄×C₁₃₇

C₄⋊Q₈

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

#384

yes

characteristic

C₂×C₁₃₇

C₂×#107

yes

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

—

C₁₃₇

#1845

yes

characteristic, complemented, Sylow

#1845

—

137

maximal, Sylow

C₂×#107

—

137

—

C₂×C₄×C₈

—

137

—

#384

—

137

—

#384

D₁₃₇

yes

characteristic

C₂²×C₈

—

137

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

274

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄⋊C₄

C₂×D₁₃₇

yes

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

#107

—

274

—

C₄×C₈

—

137

—

C₂²×C₄

—

C₂²×C₄

—

274

—

C₂²×C₄

C₂²⋊D₁₃₇

yes

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

548

—

C₄²

C₄×D₁₃₇

yes

—

C₄²

Dic₂₇₄

yes

—

C₂³

C₂×C₂²⋊D₁₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

548

—

C₂×C₄

—

548

—

C₂×C₄

#14555489

yes

—

C₂×C₄

#14555495

yes

—

C₂×C₄

#14555500

yes

—

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

C₂×C₄

D₄×D₁₃₇

yes

—

C₂×C₄

Q₈×D₁₃₇

yes

—

C₂×C₄

C₂×Dic₂₇₄

yes

characteristic

C₈

—

137

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

C₂×#14555500

yes

—

C₄

—

100

C₄

—

1096

—

101

C₄

#17984021

yes

complemented

102

C₄

C₄⋊C₄×D₁₃₇

yes

—

103

C₄

#17984087

yes

—

104

C₂

#22577272

yes

characteristic

105

C₂

C₂×#17984021

yes

—

106

C₂

#22577779

yes

characteristic

107

C₂

#22577782

yes

characteristic

108

C₂

#22577806

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	#384×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	C₂×D₄
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	#384	D₁₃₇
hypocenter	C₁₃₇	#1845
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₇

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₇

Fitting

#384×C₁₃₇

C₂

socle, prime radical

C₂³×C₁₃₇

C₂×D₄

cosocle

C₂²×C₄×C₁₃₇

C₂³

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×#11764040	C₄	semidirect
28. #11764040	C₄⋊C₄	semidirect
45. C₄×C₁₃₇	#107	semidirect
54. C₁₃₇	#1845	semidirect
101. C₄	#17984021	semidirect

H ≅ G/N

type

9. C₄×#11764040

C₄

semidirect

28. #11764040

C₄⋊C₄

semidirect

45. C₄×C₁₃₇

#107

semidirect

54. C₁₃₇

#1845

semidirect

101. C₄

#17984021

semidirect

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

subgroup	index
C₂×#17984021	2
C₂×C₄×#11764040	2
#22577273	2
#384×C₁₃₇	2
#1845	137

subgroup

index

#384×C₁₃₇

137

Subgroups — #31780948

Conjugacy classes of subgroups

Complements

Maximal subgroups