Group #31797157 — subgroups

Subgroups — #31797157

Order 17536.

138 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 64 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	C₄⋊#14555513	C₂	1	2	yes	maximal, complemented
2	#22578298	C₂	1	2	yes	maximal
3	#22578292	C₂	1	1	yes	maximal, characteristic, complemented
4	#22578289	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577360	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555513	C₂²	1	2	yes	—
7	C₂×#14555513	C₂²	1	2	yes	—
8	C₂×#14555493	C₄	1	1	yes	characteristic
9	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
10	#17984045	C₄	1	1	yes	characteristic
11	#17983998	C₄	1	1	yes	characteristic
12	C₄×#11764040	C₂²	1	1	yes	characteristic
13	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
14	#97×C₁₃₇	C₄	1	1	yes	characteristic
15	#14555513	—	2	2	no	—
16	#14555513	—	2	2	no	—
17	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
18	#14555493	—	2	2	no	—
19	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
20	C₂×#11764040	C₂³	1	1	yes	characteristic
21	C₈×D₁₃₇	—	2	2	no	—
22	#14555489	C₈	1	2	yes	—
23	#14555487	C₈	1	2	yes	—
24	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₁₃₇	C₂×C₈	1	1	yes	characteristic
29	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
30	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
31	C₄×D₁₃₇	—	2	1	no	—
32	C₄×D₁₃₇	—	2	1	no	—
33	#11764040	—	2	1	no	—
34	#11764040	#41	1	2	yes	—
35	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₈×C₁₃₇	—	2	1	no	—
38	C₈×C₁₃₇	M₁₆	1	2	yes	—
39	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
40	C₂×D₁₃₇	—	2	1	no	—
41	C₂×D₁₃₇	—	2	2	no	—
42	C₄×C₁₃₇	—	2	2	no	—
43	C₄×C₁₃₇	#117	1	1	yes	characteristic
44	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
45	Dic₁₃₇	—	2	1	no	—
46	Dic₁₃₇	—	2	2	no	—
47	C₂×C₁₃₇	#431	1	1	yes	characteristic
48	C₂×C₁₃₇	#504	1	2	yes	—
49	D₁₃₇	—	4	1	no	—
50	C₁₃₇	#2164	1	1	yes	characteristic, complemented, Sylow
51	#2164	—	137	1	no	maximal, Sylow
52	#405	—	137	1	no	—
53	C₄⋊C₁₆	—	137	2	no	—
54	C₂²⋊C₁₆	—	137	2	no	—
55	#346	—	137	1	no	—
56	C₄×C₁₆	—	137	1	no	—
57	C₂×M₁₆	—	137	1	no	—
58	C₂²×C₈	—	137	1	no	—
59	#117	—	137	1	no	—
60	C₂×C₁₆	—	137	2	no	—
61	C₂×C₁₆	—	137	2	no	—
62	#97	—	137	1	no	—
63	#97	#10587739	1	1	yes	characteristic
64	C₄×C₈	—	137	1	no	—
65	C₄×C₈	—	137	1	no	—
66	C₂²×C₄	—	137	1	no	—
67	M₁₆	—	274	2	no	—
68	C₂×C₈	—	274	2	no	—
69	C₂×C₈	—	137	1	no	—
70	C₂×C₈	—	137	1	no	—
71	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
72	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
73	C₄⋊C₄	—	137	2	no	—
74	C₂²⋊C₄	—	137	2	no	—
75	C₄²	—	137	1	no	—
76	C₄²	C₂×#10587739	1	1	yes	characteristic
77	C₁₆	—	274	2	no	—
78	C₁₆	—	274	2	no	—
79	C₂³	—	137	1	no	—
80	C₂×C₄	—	274	1	no	—
81	C₂×C₄	—	274	1	no	—
82	C₂×C₄	—	137	2	no	—
83	C₂×C₄	—	137	1	no	—
84	C₂×C₄	#14555518	1	2	yes	—
85	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
86	C₈	—	2	1	no	—
87	C₈	—	274	1	no	—
88	C₈	—	137	2	no	—
89	C₈	#14555520	1	2	yes	—
90	C₂²	—	274	2	no	—
91	C₂²	—	274	1	no	—
92	C₂²	C₂×#14555518	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	274	2	no	—
95	C₄	—	274	1	no	—
96	C₄	#17984167	1	1	yes	characteristic
97	C₄	C₂×#14555520	1	1	yes	characteristic
98	C₂	—	548	1	no	—
99	C₂	#22578422	1	1	yes	characteristic
100	C₂	#22578462	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#14555513

C₂

yes

maximal, complemented

#22578298

C₂

yes

maximal

#22578292

C₂

yes

maximal, characteristic, complemented

#22578289

C₂

yes

maximal, characteristic, complemented

#22577360

C₂

yes

maximal, characteristic

C₂×#14555513

C₂²

yes

—

C₂×#14555513

C₂²

yes

—

C₂×#14555493

C₄

yes

characteristic

C₂×C₈×D₁₃₇

C₂²

yes

characteristic

#17984045

C₄

yes

characteristic

#17983998

C₄

yes

characteristic

C₄×#11764040

C₂²

yes

characteristic

C₈×Dic₁₃₇

C₂²

yes

characteristic

#97×C₁₃₇

C₄

yes

characteristic

#14555513

—

#14555513

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

characteristic

#14555493

—

C₂×#11764040

C₂×C₄

yes

characteristic

C₂×#11764040

C₂³

yes

characteristic

C₈×D₁₃₇

—

#14555489

C₈

yes

—

#14555487

C₈

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×D₁₃₇

C₂×C₈

yes

characteristic

C₂×Dic₁₃₇

C₂×C₈

yes

—

C₂×Dic₁₃₇

C₂×C₈

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

—

#11764040

—

#11764040

#41

yes

—

C₂×C₄×C₁₃₇

C₂×C₈

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

M₁₆

yes

—

C₂²×C₁₃₇

C₂²×C₈

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#117

yes

characteristic

C₄×C₁₃₇

C₂×M₁₆

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

#431

yes

characteristic

C₂×C₁₃₇

#504

yes

—

D₁₃₇

—

C₁₃₇

#2164

yes

characteristic, complemented, Sylow

#2164

—

137

maximal, Sylow

#405

—

137

—

C₄⋊C₁₆

—

137

—

C₂²⋊C₁₆

—

137

—

#346

—

137

—

C₄×C₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

—

137

—

#117

—

137

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

#97

—

137

—

#97

#10587739

yes

characteristic

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₂²×C₄

—

137

—

M₁₆

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

characteristic

C₁₆

—

274

—

C₁₆

—

274

—

C₂³

—

137

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

#14555518

yes

—

C₂×C₄

C₂²×#10587739

yes

characteristic

C₈

—

C₈

—

274

—

C₈

—

137

—

C₈

#14555520

yes

—

C₂²

—

274

—

C₂²

—

274

—

C₂²

C₂×#14555518

yes

characteristic

C₄

—

C₄

—

274

—

C₄

—

274

—

C₄

yes

C₄

yes

C₂

—

548

—

C₂

#22578422

yes

characteristic

100

C₂

#22578462

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10587739
commutator, socle	C₂²×C₁₃₇	C₂²×C₈
Fitting	#97×C₁₃₇	C₄
prime radical	C₂²×D₁₃₇	C₂×C₈
cosocle	C₂×#11764040	C₂³
hypercenter	#97	#10587739
hypocenter	C₁₃₇	#2164
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10587739

commutator, socle

C₂²×C₁₃₇

C₂²×C₈

Fitting

#97×C₁₃₇

C₄

prime radical

C₂²×D₁₃₇

C₂×C₈

C₂³

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
1. C₄⋊#14555513	C₂	semidirect
3. #22578292	C₂	semidirect
4. #22578289	C₂	semidirect
50. C₁₃₇	#2164	semidirect

H ≅ G/N

type

1. C₄⋊#14555513

C₂

semidirect

3. #22578292

C₂

semidirect

4. #22578289

C₂

semidirect

50. C₁₃₇

#2164

semidirect

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

subgroup	index
C₄⋊#14555513	2
#22578298	2
#22578292	2
#22578289	2
#22577360	2
#2164	137

subgroup

index

137

Subgroups — #31797157

Conjugacy classes of subgroups

Complements

Maximal subgroups