Group #31789398 — subgroups

Subgroups — #31789398

Order 17536.

440 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 249 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	#22578170	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984095	C₂	1	1	yes	maximal, characteristic
3	#22577451	C₂	1	6	yes	maximal, complemented
4	C₂×#17984023	C₂	1	6	yes	maximal, complemented
5	#568×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#17984095	—	2	4	no	—
7	#17984095	C₄	1	8	yes	—
8	C₂×#14555494	C₂²	1	6	yes	—
9	C₂×#14555494	C₂²	1	3	yes	—
10	C₂²×#11764040	C₂²	1	3	yes	—
11	#17984023	C₂²	1	4	yes	—
12	#17984023	C₂²	1	12	yes	—
13	C₂×#41×C₁₃₇	C₂²	1	1	yes	characteristic
14	#131×C₁₃₇	—	2	4	no	—
15	C₂×M₁₆×C₁₃₇	C₂²	1	6	yes	—
16	#14555494	—	2	6	no	—
17	#14555494	C₂×C₄	1	12	yes	—
18	#14555494	C₂³	1	4	yes	—
19	C₂×#11764040	—	2	6	no	—
20	C₂×#11764040	C₂×C₄	1	12	yes	—
21	C₂×#11764040	C₂³	1	1	yes	characteristic
22	C₂×#11764040	C₂³	1	3	yes	—
23	#41×C₁₃₇	C₂×C₄	1	4	yes	—
24	#41×C₁₃₇	D₄	1	4	yes	—
25	C₂×Q₈×C₁₃₇	Q₈	1	1	yes	characteristic
26	C₂×D₄×C₁₃₇	Q₈	1	3	yes	—
27	C₂²×C₄×C₁₃₇	C₂³	1	3	yes	—
28	M₁₆×C₁₃₇	—	2	6	no	—
29	M₁₆×C₁₃₇	C₂³	1	4	yes	—
30	C₂×C₈×C₁₃₇	—	2	6	no	—
31	#11764040	—	2	4	no	—
32	#11764040	C₂²×C₄	1	2	yes	—
33	#11764040	C₂²×C₄	1	6	yes	—
34	C₂³×C₁₃₇	C₂×Q₈	1	3	yes	—
35	Q₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
36	D₄×C₁₃₇	C₄⋊C₄	1	12	yes	—
37	C₂×C₄×C₁₃₇	C₂²×C₄	1	6	yes	—
38	C₂×C₄×C₁₃₇	C₂×D₄	1	6	yes	—
39	C₂×C₄×C₁₃₇	C₂×Q₈	1	3	yes	—
40	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
41	C₈×C₁₃₇	—	2	4	no	—
42	C₂²×C₁₃₇	—	4	3	no	—
43	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	6	yes	—
44	C₂²×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₁₃₇	C₂×C₄⋊C₄	1	6	yes	—
46	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
47	C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
48	C₂×C₁₃₇	—	2	1	no	—
49	C₂×C₁₃₇	—	2	6	no	—
50	C₂×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
51	C₁₃₇	#2916	1	1	yes	characteristic, complemented, Sylow
52	#2916	—	137	1	no	maximal, Sylow
53	#568	—	137	1	no	—
54	#568	D₁₃₇	1	1	yes	characteristic
55	C₂×#131	—	137	1	no	—
56	#430	—	137	6	no	—
57	C₂×#108	—	137	6	no	—
58	C₂×#41	C₂×D₁₃₇	1	1	yes	characteristic
59	#131	—	2	4	no	—
60	#131	—	274	4	no	—
61	#131	—	137	8	no	—
62	C₂×M₁₆	—	137	6	no	—
63	C₂×M₁₆	—	137	3	no	—
64	C₂×M₁₆	C₂×D₁₃₇	1	6	yes	—
65	C₂²×C₈	—	137	3	no	—
66	#108	—	137	4	no	—
67	#108	—	137	12	no	—
68	#41	C₄×D₁₃₇	1	4	yes	—
69	#41	C₂²⋊D₁₃₇	1	4	yes	—
70	C₂×Q₈	Dic₂₇₄	1	1	yes	characteristic
71	C₂×D₄	Dic₂₇₄	1	3	yes	—
72	C₂²×C₄	C₂²×D₁₃₇	1	3	yes	—
73	M₁₆	—	2	6	no	—
74	M₁₆	—	274	6	no	—
75	M₁₆	—	137	4	no	—
76	M₁₆	—	137	12	no	—
77	M₁₆	C₂²×D₁₃₇	1	4	yes	—
78	C₂×C₈	—	2	6	no	—
79	C₂×C₈	—	274	6	no	—
80	C₂×C₈	—	137	12	no	—
81	C₂×C₈	—	137	1	no	—
82	C₂×C₈	—	137	3	no	—
83	C₂³	C₂×Dic₂₇₄	1	3	yes	—
84	Q₈	#14555489	1	4	yes	—
85	D₄	#14555489	1	12	yes	—
86	C₂×C₄	C₂×C₄×D₁₃₇	1	6	yes	—
87	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	6	yes	—
88	C₂×C₄	C₂×Dic₂₇₄	1	3	yes	—
89	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	—	274	4	no	—
92	C₈	—	137	2	no	—
93	C₈	—	137	6	no	—
94	C₂²	—	4	3	no	—
95	C₂²	C₂×#14555489	1	6	yes	—
96	C₂²	C₂²×Dic₂₇₄	1	1	yes	characteristic
97	C₄	C₂×#14555489	1	6	yes	—
98	C₄	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
99	C₄	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	2	6	no	—
102	C₂	C₂²×#14555489	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578170

C₂

yes

maximal, characteristic

C₂×#17984095

C₂

yes

maximal, characteristic

#22577451

C₂

yes

maximal, complemented

C₂×#17984023

C₂

yes

maximal, complemented

#568×C₁₃₇

C₂

yes

maximal, characteristic

#17984095

—

#17984095

C₄

yes

—

C₂×#14555494

C₂²

yes

—

C₂×#14555494

C₂²

yes

—

C₂²×#11764040

C₂²

yes

—

#17984023

C₂²

yes

—

#17984023

C₂²

yes

—

C₂×#41×C₁₃₇

C₂²

yes

characteristic

#131×C₁₃₇

—

C₂×M₁₆×C₁₃₇

C₂²

yes

—

#14555494

—

#14555494

C₂×C₄

yes

—

#14555494

C₂³

yes

—

C₂×#11764040

—

C₂×#11764040

C₂×C₄

yes

—

C₂×#11764040

C₂³

yes

characteristic

C₂×#11764040

C₂³

yes

—

#41×C₁₃₇

C₂×C₄

yes

—

#41×C₁₃₇

D₄

yes

—

C₂×Q₈×C₁₃₇

Q₈

yes

characteristic

C₂×D₄×C₁₃₇

Q₈

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

—

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

C₂³

yes

—

C₂×C₈×C₁₃₇

—

#11764040

—

#11764040

C₂²×C₄

yes

—

#11764040

C₂²×C₄

yes

—

C₂³×C₁₃₇

C₂×Q₈

yes

—

Q₈×C₁₃₇

C₄⋊C₄

yes

—

D₄×C₁₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×Q₈

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

—

C₂²×C₁₃₇

C₂²×Q₈

yes

characteristic

C₄×C₁₃₇

C₂×C₄⋊C₄

yes

—

C₄×C₁₃₇

C₂³×C₄

yes

characteristic

C₄×C₁₃₇

C₂²×D₄

yes

characteristic

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂²×C₄⋊C₄

yes

characteristic

C₁₃₇

#2916

yes

characteristic, complemented, Sylow

#2916

—

137

maximal, Sylow

#568

—

137

—

#568

D₁₃₇

yes

characteristic

C₂×#131

—

137

—

#430

—

137

—

C₂×#108

—

137

—

C₂×#41

C₂×D₁₃₇

yes

characteristic

#131

—

#131

—

274

—

#131

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

C₂×D₁₃₇

yes

—

C₂²×C₈

—

137

—

#108

—

137

—

#108

—

137

—

#41

C₄×D₁₃₇

yes

—

#41

C₂²⋊D₁₃₇

yes

—

C₂×Q₈

Dic₂₇₄

yes

characteristic

C₂×D₄

Dic₂₇₄

yes

—

C₂²×C₄

C₂²×D₁₃₇

yes

—

M₁₆

—

M₁₆

—

274

—

M₁₆

—

137

—

M₁₆

—

137

—

M₁₆

C₂²×D₁₃₇

yes

—

C₂×C₈

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂³

C₂×Dic₂₇₄

yes

—

Q₈

#14555489

yes

—

D₄

#14555489

yes

—

C₂×C₄

C₂×C₄×D₁₃₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

—

C₂×C₄

C₂×Dic₂₇₄

yes

—

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

C₈

—

C₈

—

274

—

C₈

—

137

—

C₈

—

137

—

C₂²

—

C₂²

C₂×#14555489

yes

—

C₂²

C₂²×Dic₂₇₄

yes

characteristic

C₄

C₂×#14555489

yes

—

C₄

C₂²×C₄×D₁₃₇

yes

characteristic

C₄

C₂²×C₂²⋊D₁₃₇

yes

characteristic

100

C₂

—

101

C₂

—

102

C₂

C₂²×#14555489

yes

characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₁₃₇
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#568×C₁₃₇	C₂
socle	C₂×C₁₃₇	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₁₃₇	C₂²
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#568	D₁₃₇
hypocenter	C₁₃₇	#2916
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

subgroup

G/H

center

C₄

C₂²×C₂²⋊D₁₃₇

commutator

C₄×C₁₃₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₃₇

Fitting

#568×C₁₃₇

C₂

socle

C₂×C₁₃₇

C₂²×C₄⋊C₄

prime radical

C₂×#41×C₁₃₇

C₂²

cosocle

C₂×C₄×C₁₃₇

C₂⁴

D₁₃₇

C₁₃₇

G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

N	H ≅ G/N	type
3. #22577451	C₂	semidirect
4. C₂×#17984023	C₂	semidirect
51. C₁₃₇	#2916	semidirect

H ≅ G/N

type

3. #22577451

C₂

semidirect

4. C₂×#17984023

C₂

semidirect

51. C₁₃₇

#2916

semidirect

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

subgroup	index
#22578170	2
C₂×#17984095	2
#22577451	2
C₂×#17984023	2
#568×C₁₃₇	2
#2916	137

subgroup

index

#568×C₁₃₇

137

Subgroups — #31789398

Conjugacy classes of subgroups

Complements

Maximal subgroups