Group #31780694 — subgroups

Subgroups — #31780694

Order 17536.

220 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 105 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×#17984023	C₂	1	2	yes	maximal
2	M₁₆×Dic₁₃₇	C₂	1	2	yes	maximal
3	C₂×#17983998	C₂	1	1	yes	maximal, characteristic
4	#22577168	C₂	1	1	yes	maximal, characteristic
5	#341×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555494	C₂²	1	2	yes	—
7	C₂²×#11764040	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
9	#17984023	C₄	1	8	yes	complemented
10	#17983998	—	2	2	no	—
11	#17983998	C₄	1	4	yes	—
12	C₈×Dic₁₃₇	—	2	2	no	—
13	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₃₇	Q₈	1	2	yes	—
16	#14555494	—	2	2	no	—
17	#14555494	C₂×C₄	1	4	yes	—
18	C₂×#11764040	—	2	2	no	—
19	C₂×#11764040	C₂×C₄	1	4	yes	—
20	C₂×#11764040	C₂×C₄	1	2	yes	—
21	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₃₇	D₄	1	2	yes	—
23	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
24	M₁₆×C₁₃₇	—	2	2	no	—
25	M₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₁₃₇	—	2	2	no	—
27	C₂×C₈×C₁₃₇	—	2	2	no	—
28	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₃₇	—	4	2	no	—
30	C₂×Dic₁₃₇	C₄⋊C₄	1	4	yes	—
31	#11764040	—	2	4	no	—
32	#11764040	C₄²	1	4	yes	—
33	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₇	#41	1	2	yes	—
38	C₈×C₁₃₇	—	2	2	no	—
39	C₈×C₁₃₇	—	2	4	no	—
40	C₂²×C₁₃₇	—	2	2	no	—
41	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₁₃₇	C₄×Q₈	1	2	yes	—
43	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₁₃₇	#117	1	1	yes	characteristic
45	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
46	Dic₁₃₇	—	2	4	no	—
47	C₂×C₁₃₇	—	2	2	no	—
48	C₂×C₁₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₁₃₇	#430	1	2	yes	—
50	C₁₃₇	#1778	1	1	yes	characteristic, complemented, Sylow
51	#1778	—	137	1	no	maximal, Sylow
52	C₂×#108	—	137	2	no	—
53	C₄×M₁₆	—	137	2	no	—
54	C₂×#97	—	137	1	no	—
55	#341	—	137	1	no	—
56	#341	D₁₃₇	1	1	yes	characteristic
57	C₂×M₁₆	—	137	2	no	—
58	C₂×M₁₆	C₂×D₁₃₇	1	2	yes	—
59	C₂²×C₈	—	137	1	no	—
60	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
61	C₂×C₄²	—	137	1	no	—
62	#108	—	137	8	no	—
63	#97	—	274	2	no	—
64	#97	—	137	4	no	—
65	C₄×C₈	—	274	2	no	—
66	C₂²×C₄	—	137	2	no	—
67	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	274	2	no	—
70	M₁₆	—	137	4	no	—
71	M₁₆	C₄×D₁₃₇	1	4	yes	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	274	2	no	—
75	C₂×C₈	—	137	2	no	—
76	C₂×C₈	—	137	4	no	—
77	C₂×C₈	C₄×D₁₃₇	1	2	yes	—
78	C₄²	—	137	2	no	—
79	C₄²	—	137	2	no	—
80	C₂³	Q₈×D₁₃₇	1	1	yes	characteristic
81	C₂×C₄	—	548	2	no	—
82	C₂×C₄	—	137	4	no	—
83	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
84	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
85	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
86	C₂×C₄	#14555509	1	2	yes	—
87	C₈	—	2	2	no	—
88	C₈	—	2	4	no	—
89	C₈	—	274	4	no	—
90	C₈	—	137	4	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₄⋊C₄×D₁₃₇	1	1	yes	characteristic
93	C₂²	#17984053	1	2	yes	—
94	C₄	—	274	4	no	—
95	C₄	C₄²×D₁₃₇	1	1	yes	characteristic
96	C₄	#17984044	1	1	yes	characteristic
97	C₄	#17984048	1	2	yes	—
98	C₂	—	2	2	no	—
99	C₂	#22577254	1	1	yes	characteristic
100	C₂	#22577450	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984023

C₂

yes

maximal

M₁₆×Dic₁₃₇

C₂

yes

maximal

C₂×#17983998

C₂

yes

maximal, characteristic

#22577168

C₂

yes

maximal, characteristic

#341×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555494

C₂²

yes

—

C₂²×#11764040

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

#17984023

C₄

yes

complemented

#17983998

—

#17983998

C₄

yes

—

C₈×Dic₁₃₇

—

C₂×M₁₆×C₁₃₇

C₂²

yes

—

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

C₂²×Dic₁₃₇

Q₈

yes

—

#14555494

—

#14555494

C₂×C₄

yes

—

C₂×#11764040

—

C₂×#11764040

C₂×C₄

yes

—

C₂×#11764040

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

C₂×C₄

yes

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₄⋊C₄

yes

—

#11764040

—

#11764040

C₄²

yes

—

C₂³×C₁₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₇

C₄×Q₈

yes

—

C₄×C₁₃₇

C₂×C₄²

yes

characteristic

C₄×C₁₃₇

#117

yes

characteristic

C₄×C₁₃₇

C₄×D₄

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₃₇

#430

yes

—

C₁₃₇

#1778

yes

characteristic, complemented, Sylow

#1778

—

137

maximal, Sylow

C₂×#108

—

137

—

C₄×M₁₆

—

137

—

C₂×#97

—

137

—

#341

—

137

—

#341

D₁₃₇

yes

characteristic

C₂×M₁₆

—

137

—

C₂×M₁₆

C₂×D₁₃₇

yes

—

C₂²×C₈

—

137

—

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

C₂×C₄²

—

137

—

#108

—

137

—

#97

—

274

—

#97

—

137

—

C₄×C₈

—

274

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

M₁₆

—

M₁₆

—

274

—

M₁₆

—

137

—

M₁₆

C₄×D₁₃₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

C₄×D₁₃₇

yes

—

C₄²

—

137

—

C₄²

—

137

—

C₂³

Q₈×D₁₃₇

yes

characteristic

C₂×C₄

—

548

—

C₂×C₄

—

137

—

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

C₂×C₄

C₂×C₄×D₁₃₇

yes

—

C₂×C₄

D₄×D₁₃₇

yes

characteristic

C₂×C₄

#14555509

yes

—

C₈

—

C₈

—

C₈

—

274

—

C₈

—

137

—

C₂²

—

C₂²

C₄⋊C₄×D₁₃₇

yes

characteristic

C₂²

#17984053

yes

—

C₄

—

274

—

C₄

C₄²×D₁₃₇

yes

C₄

yes

C₄

yes

—

C₂

—

C₂

#22577254

yes

characteristic

100

C₂

#22577450

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₇
commutator	C₄×C₁₃₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	#341×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	#341	D₁₃₇
hypocenter	C₁₃₇	#1778
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₃₇

commutator

C₄×C₁₃₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₃₇

Fitting

#341×C₁₃₇

C₂

socle

C₂²×C₁₃₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₃₇

C₂×Q₈

cosocle

C₂²×C₄×C₁₃₇

C₂³

D₁₃₇

C₁₃₇

G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #17984023	C₄	semidirect
50. C₁₃₇	#1778	semidirect

H ≅ G/N

type

9. #17984023

C₄

semidirect

50. C₁₃₇

#1778

semidirect

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

subgroup	index
C₂×#17984023	2
M₁₆×Dic₁₃₇	2
C₂×#17983998	2
#22577168	2
#341×C₁₃₇	2
#1778	137

subgroup

index

M₁₆×Dic₁₃₇

#341×C₁₃₇

137

Subgroups — #31780694

Conjugacy classes of subgroups

Complements

Maximal subgroups