Group #31795993 — subgroups

Subgroups — #31795993

Order 17536.

212 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 106 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	#22578421	C₂	1	2	yes	maximal
2	#22578244	C₂	1	2	yes	maximal, complemented
3	C₈×#11764047	C₂	1	2	yes	maximal, complemented
4	C₄×C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#17984155	C₄	1	4	yes	—
6	#17984145	C₄	1	4	yes	—
7	C₄×#11764047	C₂²	1	4	yes	—
8	C₂×C₈×D₁₃₇	C₄	1	1	yes	characteristic
9	C₂×C₈×D₁₃₇	C₄	1	1	yes	characteristic
10	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
11	C₄×#11764040	C₄	1	1	yes	characteristic
12	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
13	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
15	C₂×#11764047	C₂×C₄	1	8	yes	—
16	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
17	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
19	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
20	C₈×D₁₃₇	C₈	1	4	yes	complemented
21	C₈×D₁₃₇	C₈	1	4	yes	complemented
22	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
23	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	#11764047	—	2	8	no	—
28	C₂²×D₁₃₇	C₄²	1	1	yes	characteristic
29	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
30	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
31	C₄×D₁₃₇	—	2	4	no	—
32	C₄×D₁₃₇	C₂×C₈	1	2	yes	—
33	C₄×D₁₃₇	C₂×C₈	1	2	yes	—
34	#11764040	C₂×C₈	1	2	yes	—
35	#11764040	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₁₃₇	C₄²	1	2	yes	—
37	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₃₇	C₂×C₈	1	2	yes	—
39	C₈×C₁₃₇	C₂×C₈	1	2	yes	—
40	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
41	C₂×D₁₃₇	—	2	2	no	—
42	C₂×D₁₃₇	C₄×C₈	1	2	yes	—
43	C₄×C₁₃₇	—	2	2	no	—
44	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
45	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
46	Dic₁₃₇	C₄×C₈	1	2	yes	—
47	Dic₁₃₇	#131	1	4	yes	—
48	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
49	C₂×C₁₃₇	#405	1	2	yes	—
50	D₁₃₇	—	2	2	no	—
51	C₁₃₇	#1450	1	1	yes	characteristic, complemented, Sylow
52	#1450	—	137	1	no	maximal, Sylow
53	#431	—	137	2	no	—
54	C₂×C₄×C₈	—	137	1	no	—
55	#322	—	137	2	no	—
56	C₈²	—	137	2	no	—
57	C₂²×C₈	—	137	1	no	—
58	C₂²×C₈	—	137	1	no	—
59	C₂×C₄²	—	137	1	no	—
60	C₄⋊C₈	—	137	4	no	—
61	C₂²⋊C₈	—	137	4	no	—
62	C₄×C₈	—	137	4	no	—
63	C₄×C₈	—	137	1	no	—
64	C₄×C₈	—	137	1	no	—
65	C₄×C₈	—	137	1	no	—
66	C₄×C₈	#10587739	1	1	yes	characteristic
67	C₂²×C₄	—	137	2	no	—
68	C₂²×C₄	—	137	1	no	—
69	C₂×C₈	—	137	8	no	—
70	C₂×C₈	—	137	1	no	—
71	C₂×C₈	—	137	1	no	—
72	C₂×C₈	—	137	4	no	—
73	C₂×C₈	—	137	4	no	—
74	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
75	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
76	C₄²	—	137	2	no	—
77	C₄²	—	137	1	no	—
78	C₄²	C₂×#10587739	1	1	yes	characteristic
79	C₂³	—	137	1	no	—
80	C₂×C₄	—	274	4	no	—
81	C₂×C₄	—	137	2	no	—
82	C₂×C₄	—	137	1	no	—
83	C₂×C₄	—	137	2	no	—
84	C₂×C₄	—	137	2	no	—
85	C₂×C₄	C₄×#10587739	1	2	yes	—
86	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
87	C₈	—	274	8	no	—
88	C₈	—	137	2	no	—
89	C₈	—	137	2	no	—
90	C₈	#14555518	1	2	yes	—
91	C₈	#14555518	1	2	yes	—
92	C₂²	—	274	2	no	—
93	C₂²	—	137	2	no	—
94	C₂²	C₂×C₄×#10587739	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	137	2	no	—
97	C₄	—	137	4	no	—
98	C₄	C₂×#14555518	1	1	yes	characteristic
99	C₄	C₂×#14555518	1	1	yes	characteristic
100	C₂	—	274	2	no	—
101	C₂	C₄×#14555518	1	1	yes	characteristic
102	C₂	#22578366	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22578421

C₂

yes

maximal

#22578244

C₂

yes

maximal, complemented

C₈×#11764047

C₂

yes

maximal, complemented

C₄×C₈×D₁₃₇

C₂

yes

maximal, characteristic

#17984155

C₄

yes

—

#17984145

C₄

yes

—

C₄×#11764047

C₂²

yes

—

C₂×C₈×D₁₃₇

C₄

yes

characteristic

C₂×C₈×D₁₃₇

C₄

yes

characteristic

C₄²×D₁₃₇

C₂²

yes

characteristic

C₄×#11764040

C₄

yes

characteristic

C₈×Dic₁₃₇

C₂²

yes

characteristic

C₈×Dic₁₃₇

C₂²

yes

characteristic

C₄×C₈×C₁₃₇

C₄

yes

characteristic

C₂×#11764047

C₂×C₄

yes

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

characteristic

C₂×#11764040

C₂×C₄

yes

characteristic

C₂×#11764040

C₂×C₄

yes

characteristic

C₈×D₁₃₇

C₈

yes

complemented

C₈×D₁₃₇

C₈

yes

complemented

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

#11764047

—

C₂²×D₁₃₇

C₄²

yes

characteristic

C₂×Dic₁₃₇

C₄²

yes

characteristic

C₂×Dic₁₃₇

C₂²×C₄

yes

—

C₄×D₁₃₇

—

C₄×D₁₃₇

C₂×C₈

yes

—

C₄×D₁₃₇

C₂×C₈

yes

—

#11764040

C₂×C₈

yes

—

#11764040

C₂×C₈

yes

—

C₂×C₄×C₁₃₇

C₄²

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

C₂×C₈

yes

—

C₈×C₁₃₇

C₂×C₈

yes

—

C₂²×C₁₃₇

C₂×C₄²

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

C₄×C₈

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂²×C₈

yes

characteristic

C₄×C₁₃₇

C₂²×C₈

yes

characteristic

Dic₁₃₇

C₄×C₈

yes

—

Dic₁₃₇

#131

yes

—

C₂×C₁₃₇

C₂×C₄×C₈

yes

characteristic

C₂×C₁₃₇

#405

yes

—

D₁₃₇

—

C₁₃₇

#1450

yes

characteristic, complemented, Sylow

#1450

—

137

maximal, Sylow

#431

—

137

—

C₂×C₄×C₈

—

137

—

#322

—

137

—

C₈²

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

—

137

—

C₂×C₄²

—

137

—

C₄⋊C₈

—

137

—

C₂²⋊C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

#10587739

yes

characteristic

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

characteristic

C₂³

—

137

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₄×#10587739

yes

—

C₂×C₄

C₂²×#10587739

yes

characteristic

C₈

—

274

—

C₈

—

137

—

C₈

—

137

—

C₈

#14555518

yes

—

C₈

#14555518

yes

—

C₂²

—

274

—

C₂²

—

137

—

C₂²

C₂×C₄×#10587739

yes

characteristic

C₄

—

C₄

—

137

—

C₄

—

137

—

C₄

yes

C₄

yes

100

C₂

—

274

—

101

C₂

C₄×#14555518

yes

characteristic

102

C₂

#22578366

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×#10587739
commutator	C₂×C₁₃₇	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₄×C₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₄²
prime radical	C₂²×D₁₃₇	C₄²
cosocle	C₄×Dic₁₃₇	C₂³
hypercenter	C₄×C₈	#10587739
hypocenter	C₁₃₇	#1450
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂×#10587739

commutator

C₂×C₁₃₇

C₂×C₄×C₈

Frattini

C₂×C₄

C₂²×#10587739

Fitting

C₄×C₈×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₂×C₄²

prime radical

C₂²×D₁₃₇

C₄²

cosocle

C₄×Dic₁₃₇

C₂³

C₄×C₈

C₁₃₇

G ▷ C₂×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
2. #22578244	C₂	semidirect
3. C₈×#11764047	C₂	semidirect
20. C₈×D₁₃₇	C₈	semidirect
21. C₈×D₁₃₇	C₈	semidirect
51. C₁₃₇	#1450	semidirect

H ≅ G/N

type

2. #22578244

C₂

semidirect

3. C₈×#11764047

C₂

semidirect

20. C₈×D₁₃₇

C₈

semidirect

21. C₈×D₁₃₇

C₈

semidirect

51. C₁₃₇

#1450

semidirect

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

subgroup	index
#22578421	2
#22578244	2
C₈×#11764047	2
C₄×C₈×D₁₃₇	2
#1450	137

subgroup

index

C₄×C₈×D₁₃₇

137

Subgroups — #31795993

Conjugacy classes of subgroups

Complements

Maximal subgroups