Group #31780922 — subgroups

Subgroups — #31780922

Order 17536.

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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×#17983999	C₂	1	1	yes	maximal, characteristic
2	#22577273	C₂	1	1	yes	maximal, characteristic
3	#22577165	C₂	1	4	yes	maximal
4	#382×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11764040	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₃₇	—	2	2	no	—
7	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
8	#17983999	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
11	#95×C₁₃₇	—	2	2	no	—
12	C₂²×Dic₁₃₇	—	2	2	no	—
13	C₂×#11764040	—	2	2	no	—
14	C₂×#11764040	—	2	2	no	—
15	C₂×#11764040	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₁₃₇	—	2	4	no	—
17	C₄⋊Dic₁₃₇	—	4	4	no	—
18	C₂²×C₄×C₁₃₇	—	2	2	no	—
19	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
20	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₁₃₇	—	2	4	no	—
22	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
23	C₄²×C₁₃₇	Q₈	1	2	yes	—
24	C₂×Dic₁₃₇	—	4	2	no	—
25	C₂×Dic₁₃₇	—	4	4	no	—
26	#11764040	—	2	4	no	—
27	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₁₃₇	—	2	2	no	—
29	C₂×C₄×C₁₃₇	—	2	2	no	—
30	C₂×C₄×C₁₃₇	—	4	2	no	—
31	C₂×C₄×C₁₃₇	—	4	2	no	—
32	C₂×C₄×C₁₃₇	—	4	2	no	—
33	C₂×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
34	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₃₇	#41	1	2	yes	—
37	C₂×C₄×C₁₃₇	#41	1	2	yes	—
38	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₃₇	#124	1	1	yes	characteristic
40	C₂²×C₁₃₇	#124	1	1	yes	characteristic
41	C₂²×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₁₃₇	#137	1	2	yes	—
43	C₄×C₁₃₇	—	2	4	no	—
44	C₄×C₁₃₇	—	8	2	no	—
45	C₄×C₁₃₇	#117	1	2	yes	—
46	C₄×C₁₃₇	#125	1	2	yes	—
47	Dic₁₃₇	—	8	2	no	—
48	C₂×C₁₃₇	#382	1	1	yes	characteristic
49	C₂×C₁₃₇	#428	1	2	yes	—
50	C₂×C₁₃₇	#489	1	2	yes	—
51	C₂×C₁₃₇	#490	1	1	yes	characteristic
52	C₂×C₁₃₇	#491	1	1	yes	characteristic
53	C₁₃₇	#1836	1	1	yes	characteristic, complemented, Sylow
54	#1836	—	137	1	no	maximal, Sylow
55	C₂×#97	—	137	1	no	—
56	#384	—	137	1	no	—
57	#382	D₁₃₇	1	1	yes	characteristic
58	#340	—	137	4	no	—
59	C₂²×C₈	—	137	2	no	—
60	C₂×C₄⋊C₄	—	274	2	no	—
61	C₂×C₄⋊C₄	—	137	2	no	—
62	C₂×C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
63	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
64	#97	—	137	4	no	—
65	#95	—	2	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	274	2	no	—
68	C₂²×C₄	C₂²⋊D₁₃₇	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
70	C₂×C₈	—	274	2	no	—
71	C₂×C₈	—	274	2	no	—
72	C₂×C₈	—	137	4	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	274	4	no	—
75	C₄⋊C₄	—	548	4	no	—
76	C₄²	C₄×D₁₃₇	1	2	yes	—
77	C₄²	Dic₂₇₄	1	2	yes	—
78	C₂³	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	548	2	no	—
85	C₂×C₄	—	548	4	no	—
86	C₂×C₄	#14555489	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₂₇₄	1	1	yes	characteristic
89	C₂×C₄	#14555509	1	2	yes	—
90	C₂×C₄	#14555511	1	2	yes	—
91	C₈	—	274	4	no	—
92	C₂²	C₂×#14555489	1	1	yes	characteristic
93	C₂²	#17984075	1	1	yes	characteristic
94	C₂²	#17984077	1	1	yes	characteristic
95	C₂²	#17984113	1	2	yes	—
96	C₂²	#17984118	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	1096	2	no	—
100	C₄	#17984046	1	2	yes	—
101	C₄	#17984079	1	2	yes	—
102	C₂	#22577268	1	1	yes	characteristic
103	C₂	#22577446	1	2	yes	—
104	C₂	#22577762	1	2	yes	—
105	C₂	#22577769	1	1	yes	characteristic
106	C₂	#22577773	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17983999

C₂

yes

maximal, characteristic

#22577273

C₂

yes

maximal, characteristic

#22577165

C₂

yes

maximal

#382×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×#11764040

C₂²

yes

—

C₂×C₄⋊Dic₁₃₇

—

C₂×C₄⋊Dic₁₃₇

C₂²

yes

—

#17983999

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

#95×C₁₃₇

—

C₂²×Dic₁₃₇

—

C₂×#11764040

—

C₂×#11764040

—

C₂×#11764040

C₂×C₄

yes

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

Q₈

yes

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

#41

yes

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

—

C₂²×C₁₃₇

#137

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#117

yes

—

C₄×C₁₃₇

#125

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

#382

yes

characteristic

C₂×C₁₃₇

#428

yes

—

C₂×C₁₃₇

#489

yes

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#1836

—

137

maximal, Sylow

C₂×#97

—

137

—

#384

—

137

—

#382

D₁₃₇

yes

characteristic

#340

—

137

—

C₂²×C₈

—

137

—

C₂×C₄⋊C₄

—

274

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄⋊C₄

C₂×D₁₃₇

yes

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

#97

—

137

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

274

—

C₂²×C₄

C₂²⋊D₁₃₇

yes

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₄⋊C₄

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

548

—

C₄²

C₄×D₁₃₇

yes

—

C₄²

Dic₂₇₄

yes

—

C₂³

C₂×C₂²⋊D₁₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

548

—

C₂×C₄

—

548

—

C₂×C₄

#14555489

yes

—

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

C₂×C₄

C₂×Dic₂₇₄

yes

characteristic

C₂×C₄

#14555509

yes

—

C₂×C₄

#14555511

yes

—

C₈

—

274

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#17984118

yes

—

C₄

—

C₄

—

C₄

—

1096

—

100

C₄

#17984046

yes

—

101

C₄

#17984079

yes

—

102

C₂

#22577268

yes

characteristic

103

C₂

#22577446

yes

—

104

C₂

#22577762

yes

—

105

C₂

#22577769

yes

characteristic

106

C₂

#22577773

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	#382×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	C₂×D₄
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	#382	D₁₃₇
hypocenter	C₁₃₇	#1836
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₇

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₇

Fitting

#382×C₁₃₇

C₂

socle, prime radical

C₂³×C₁₃₇

C₂×D₄

cosocle

C₂²×C₄×C₁₃₇

C₂³

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
8. #17983999	C₄	semidirect
53. C₁₃₇	#1836	semidirect

H ≅ G/N

type

8. #17983999

C₄

semidirect

53. C₁₃₇

#1836

semidirect

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

subgroup	index
C₂×#17983999	2
#22577273	2
#22577165	2
#382×C₁₃₇	2
#1836	137

subgroup

index

#382×C₁₃₇

137

Subgroups — #31780922

Conjugacy classes of subgroups

Complements

Maximal subgroups