Group #31798006 — subgroups

Subgroups — #31798006

Order 17536.

146 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 53 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	#22578524	C₂	1	2	yes	maximal, complemented
2	#22578508	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578503	C₂	1	2	yes	maximal, complemented
4	#10587739⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#22578445	C₂	1	1	yes	maximal, characteristic
6	#17984185	C₂²	1	2	yes	—
7	#17984184	—	2	2	no	—
8	#17984183	C₂²	1	1	yes	characteristic
9	#17984183	C₂²	1	1	yes	characteristic
10	C₂×#14555522	—	2	2	no	—
11	#17984175	C₂²	1	2	yes	—
12	#17984159	C₄	1	1	yes	characteristic
13	#17984152	C₄	1	2	yes	—
14	C₈×#10587739	C₂²	1	1	yes	characteristic
15	#17984107	C₄	1	1	yes	characteristic
16	#14555522	—	2	2	no	—
17	#14555522	—	2	1	no	—
18	#14555522	D₄	1	1	yes	characteristic
19	#14555522	D₄	1	1	yes	characteristic
20	#14555520	C₂×C₄	1	2	yes	—
21	#14555519	—	2	2	no	—
22	#14555518	C₂³	1	1	yes	characteristic
23	C₂×#11764047	—	2	2	no	—
24	C₄×#10587739	C₂×C₄	1	1	yes	characteristic
25	#14555509	C₂×C₄	1	2	yes	—
26	Dic₅₄₈	C₈	1	1	yes	characteristic
27	#14555498	C₈	1	2	yes	—
28	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
29	D₈×C₁₃₇	C₈	1	1	yes	characteristic
30	C₂×#10587739	—	2	1	no	—
31	#11764047	—	2	2	no	—
32	#11764047	C₂×D₄	1	1	yes	characteristic
33	#11764047	#41	1	1	yes	characteristic
34	Dic₂₇₄	C₂×C₈	1	2	yes	—
35	C₂²⋊D₁₃₇	—	2	2	no	—
36	C₂×Dic₁₃₇	—	2	2	no	—
37	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
38	#11764040	C₂×C₈	1	1	yes	characteristic
39	D₄×C₁₃₇	C₂×C₈	1	2	yes	—
40	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₁₃₇	—	2	2	no	—
42	C₂×D₁₃₇	#131	1	1	yes	characteristic
43	#10587739	—	2	2	no	—
44	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
45	Dic₁₃₇	—	2	2	no	—
46	Dic₁₃₇	C₄×D₄	1	1	yes	characteristic
47	C₂×C₁₃₇	—	4	2	no	—
48	C₂×C₁₃₇	C₈×D₄	1	1	yes	characteristic
49	D₁₃₇	—	2	1	no	—
50	C₁₃₇	#2170	1	1	yes	characteristic, complemented, Sylow
51	#2170	—	137	1	no	maximal, Sylow
52	#504	—	137	2	no	—
53	#443	—	137	1	no	—
54	#364	—	137	1	no	—
55	#350	—	137	2	no	—
56	C₄×C₁₆	—	137	1	no	—
57	#135	—	137	1	no	—
58	#131	—	137	2	no	—
59	M₃₂	—	274	2	no	—
60	M₃₂	—	137	2	no	—
61	C₂×C₁₆	—	274	2	no	—
62	C₂×C₁₆	—	137	1	no	—
63	C₂×C₁₆	—	137	1	no	—
64	#108	—	137	1	no	—
65	#104	—	137	2	no	—
66	C₄×C₈	—	137	1	no	—
67	#41	—	137	2	no	—
68	Q₁₆	—	137	1	no	—
69	SD₁₆	—	137	2	no	—
70	D₈	#11764049	1	1	yes	characteristic
71	M₁₆	—	274	2	no	—
72	M₁₆	—	137	2	no	—
73	C₂×C₈	—	274	2	no	—
74	C₂×C₈	—	137	1	no	—
75	C₂×C₈	—	137	1	no	—
76	C₄²	—	137	1	no	—
77	C₁₆	—	274	2	no	—
78	C₁₆	—	274	1	no	—
79	C₁₆	—	137	1	no	—
80	C₁₆	—	137	1	no	—
81	Q₈	—	137	2	no	—
82	D₄	—	274	2	no	—
83	D₄	C₂×#11764049	1	2	yes	—
84	C₂×C₄	—	274	1	no	—
85	C₂×C₄	—	274	2	no	—
86	C₂×C₄	—	137	1	no	—
87	C₈	—	274	2	no	—
88	C₈	—	137	1	no	—
89	C₈	—	137	1	no	—
90	C₈	—	137	1	no	—
91	C₈	C₂×#11764049	1	1	yes	characteristic
92	C₂²	—	2	2	no	—
93	C₂²	—	137	1	no	—
94	C₄	—	274	2	no	—
95	C₄	—	274	2	no	—
96	C₄	—	137	1	no	—
97	C₄	C₂²×#11764049	1	1	yes	characteristic
98	C₂	—	4	2	no	—
99	C₂	—	274	1	no	—
100	C₂	D₄×#11764049	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578524

C₂

yes

maximal, complemented

#22578508

C₂

yes

maximal, characteristic, complemented

#22578503

C₂

yes

maximal, complemented

#10587739⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#22578445

C₂

yes

maximal, characteristic

#17984185

C₂²

yes

—

#17984184

—

#17984183

C₂²

yes

characteristic

#17984183

C₂²

yes

characteristic

C₂×#14555522

—

#17984175

C₂²

yes

—

#17984159

C₄

yes

characteristic

#17984152

C₄

yes

—

C₈×#10587739

C₂²

yes

characteristic

#17984107

C₄

yes

—

—

D₄

yes

characteristic

#14555522

D₄

yes

characteristic

#14555520

C₂×C₄

yes

—

#14555519

—

#14555518

C₂³

yes

characteristic

C₂×#11764047

—

C₄×#10587739

C₂×C₄

yes

characteristic

#14555509

C₂×C₄

yes

—

Dic₅₄₈

C₈

yes

characteristic

#14555498

C₈

yes

—

C₈×D₁₃₇

C₂×C₄

yes

characteristic

D₈×C₁₃₇

C₈

yes

—

—

C₂×D₄

yes

yes

Dic₂₇₄

C₂×C₈

yes

—

C₂²⋊D₁₃₇

—

C₂×Dic₁₃₇

—

C₄×D₁₃₇

C₂²×C₄

yes

characteristic

#11764040

C₂×C₈

yes

characteristic

D₄×C₁₃₇

C₂×C₈

yes

—

C₈×C₁₃₇

C₂×C₈

yes

characteristic

C₂²×C₁₃₇

—

C₂×D₁₃₇

#131

yes

characteristic

#10587739

—

C₄×C₁₃₇

C₂²×C₈

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

C₄×D₄

yes

characteristic

C₂×C₁₃₇

—

C₂×C₁₃₇

C₈×D₄

yes

characteristic

D₁₃₇

—

C₁₃₇

#2170

yes

characteristic, complemented, Sylow

#2170

—

137

maximal, Sylow

#504

—

137

—

#443

—

137

—

#364

—

137

—

#350

—

137

—

C₄×C₁₆

—

137

—

#135

—

137

—

#131

—

137

—

M₃₂

—

274

—

M₃₂

—

137

—

C₂×C₁₆

—

274

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

#108

—

137

—

#104

—

137

—

C₄×C₈

—

137

—

#41

—

137

—

Q₁₆

—

137

—

SD₁₆

—

137

—

D₈

#11764049

yes

characteristic

M₁₆

—

274

—

M₁₆

—

137

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₄²

—

137

—

C₁₆

—

274

—

C₁₆

—

274

—

C₁₆

—

137

—

C₁₆

—

137

—

Q₈

—

137

—

D₄

—

274

—

D₄

C₂×#11764049

yes

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₈

—

274

—

C₈

—

137

—

C₈

—

137

—

C₈

—

137

—

C₈

C₂×#11764049

yes

characteristic

C₂²

—

C₂²

—

137

—

C₄

—

274

—

C₄

—

274

—

C₄

—

137

—

C₄

C₂²×#11764049

yes

characteristic

C₂

—

C₂

—

274

—

100

C₂

D₄×#11764049

yes

characteristic

	subgroup	G/H
center	C₂	D₄×#11764049
commutator	C₄×C₁₃₇	C₂²×C₈
Frattini	C₄	C₂²×#11764049
Fitting	D₈×C₁₃₇	C₈
socle	C₂×C₁₃₇	C₈×D₄
prime radical	#17984107	C₄
cosocle	#14555518	C₂³
hypercenter	D₈	#11764049
hypocenter	C₁₃₇	#2170
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

subgroup

G/H

center

C₂

D₄×#11764049

commutator

C₄×C₁₃₇

C₂²×C₈

Frattini

C₄

C₂²×#11764049

Fitting

D₈×C₁₃₇

C₈

socle

C₂×C₁₃₇

C₈×D₄

C₄

C₂³

D₈

C₁₃₇

G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂ ◁ C₄ ◁ D₈

N	H ≅ G/N	type
1. #22578524	C₂	semidirect
2. #22578508	C₂	semidirect
3. #22578503	C₂	semidirect
4. #10587739⋊C₁₆	C₂	semidirect
50. C₁₃₇	#2170	semidirect

H ≅ G/N

type

1. #22578524

C₂

semidirect

2. #22578508

C₂

semidirect

3. #22578503

C₂

semidirect

4. #10587739⋊C₁₆

C₂

semidirect

50. C₁₃₇

#2170

semidirect

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

subgroup	index
#22578524	2
#22578508	2
#22578503	2
#10587739⋊C₁₆	2
#22578445	2
#2170	137

subgroup

index

#10587739⋊C₁₆

137

Subgroups — #31798006

Conjugacy classes of subgroups

Complements

Maximal subgroups