Group #31780908 — subgroups

Subgroups — #31780908

Order 17536.

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

normal subgroups only

This group has 216 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×C₄⋊#11764040	C₂	1	1	yes	maximal, characteristic
2	#22577249	C₂	1	1	yes	maximal, characteristic
3	#22577157	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11764040	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₃₇	C₂²	1	2	yes	—
7	C₄⋊#11764040	—	2	2	no	—
8	#17983995	C₄	1	2	yes	—
9	#17983995	C₄	1	2	yes	—
10	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₃₇	—	2	2	no	—
13	C₂²×Dic₁₃₇	C₂×C₄	1	4	yes	—
14	C₂×#11764040	—	2	2	no	—
15	C₂×#11764040	—	2	4	no	—
16	C₄×Dic₁₃₇	—	2	2	no	—
17	C₄×Dic₁₃₇	—	2	2	no	—
18	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₁₃₇	—	2	2	no	—
22	C₂×C₈×C₁₃₇	—	2	4	no	—
23	C₄²×C₁₃₇	D₄	1	2	yes	—
24	C₄²×C₁₃₇	Q₈	1	2	yes	—
25	C₂×Dic₁₃₇	—	2	4	no	—
26	C₂×Dic₁₃₇	—	2	4	no	—
27	C₂×Dic₁₃₇	—	2	4	no	—
28	#11764040	—	4	2	no	—
29	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₃₇	—	2	2	no	—
31	C₂×C₄×C₁₃₇	—	2	2	no	—
32	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
33	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₃₇	C₂×D₄	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	C₈×C₁₃₇	—	4	2	no	—
39	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₃₇	#117	1	2	yes	—
41	C₂²×C₁₃₇	#131	1	2	yes	—
42	C₂²×C₁₃₇	#131	1	2	yes	—
43	C₄×C₁₃₇	—	2	2	no	—
44	C₄×C₁₃₇	—	2	2	no	—
45	C₄×C₁₃₇	#124	1	1	yes	characteristic
46	C₄×C₁₃₇	#124	1	1	yes	characteristic
47	C₄×C₁₃₇	#125	1	2	yes	—
48	Dic₁₃₇	—	4	4	no	—
49	C₂×C₁₃₇	#382	1	1	yes	characteristic
50	C₂×C₁₃₇	#424	1	1	yes	characteristic
51	C₂×C₁₃₇	#424	1	1	yes	characteristic
52	C₂×C₁₃₇	#433	1	4	yes	—
53	C₁₃₇	#1832	1	1	yes	characteristic, complemented, Sylow
54	#1832	—	137	1	no	maximal, Sylow
55	C₂×C₄⋊C₈	—	137	1	no	—
56	C₂×C₄⋊C₈	D₁₃₇	1	1	yes	characteristic
57	#376	—	137	1	no	—
58	#336	—	137	4	no	—
59	C₂²×C₈	—	137	2	no	—
60	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
61	C₂×C₄²	—	137	2	no	—
62	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	274	2	no	—
65	#95	—	137	2	no	—
66	#95	—	137	2	no	—
67	C₂²×C₄	—	137	4	no	—
68	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
69	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	274	2	no	—
74	C₂×C₈	—	274	4	no	—
75	C₄²	—	274	2	no	—
76	C₄²	—	274	2	no	—
77	C₄²	C₂²⋊D₁₃₇	1	2	yes	—
78	C₄²	Dic₂₇₄	1	2	yes	—
79	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	274	4	no	—
83	C₂×C₄	—	274	4	no	—
84	C₂×C₄	—	274	4	no	—
85	C₂×C₄	#14555489	1	2	yes	—
86	C₂×C₄	#14555489	1	2	yes	—
87	C₂×C₄	C₂×C₂²⋊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	—	4	2	no	—
92	C₈	—	548	2	no	—
93	C₂²	C₂×#14555489	1	1	yes	characteristic
94	C₂²	#17984046	1	2	yes	—
95	C₂²	#17984094	1	2	yes	—
96	C₂²	#17984096	1	2	yes	—
97	C₄	—	2	2	no	—
98	C₄	—	2	2	no	—
99	C₄	—	548	4	no	—
100	C₄	#17984075	1	1	yes	characteristic
101	C₄	#17984077	1	1	yes	characteristic
102	C₄	#17984079	1	2	yes	—
103	C₂	#22577268	1	1	yes	characteristic
104	C₂	#22577429	1	1	yes	characteristic
105	C₂	#22577431	1	1	yes	characteristic
106	C₂	#22577463	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11764040

C₂

yes

maximal, characteristic

#22577249

C₂

yes

maximal, characteristic

#22577157

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×#11764040

C₂²

yes

—

C₂×C₄×Dic₁₃₇

C₂²

yes

—

C₄⋊#11764040

—

#17983995

C₄

yes

—

#17983995

C₄

yes

—

C₂²×C₈×C₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₇

—

C₂²×Dic₁₃₇

C₂×C₄

yes

—

C₂×#11764040

—

C₂×#11764040

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₄²×C₁₃₇

D₄

yes

—

C₄²×C₁₃₇

Q₈

yes

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

—

C₂³×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

#41

yes

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₇

#117

yes

—

C₂²×C₁₃₇

#131

yes

—

C₂²×C₁₃₇

#131

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

—

C₁₃₇

#1832

yes

characteristic, complemented, Sylow

#1832

—

137

maximal, Sylow

C₂×C₄⋊C₈

—

137

—

C₂×C₄⋊C₈

D₁₃₇

yes

characteristic

#376

—

137

—

#336

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

C₂×D₁₃₇

yes

—

C₂×C₄²

—

137

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

274

—

#95

—

137

—

#95

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₄×D₁₃₇

yes

characteristic

C₂²×C₄

C₄×D₁₃₇

yes

characteristic

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₄²

—

274

—

C₄²

—

274

—

C₄²

C₂²⋊D₁₃₇

yes

—

C₄²

Dic₂₇₄

yes

—

C₂³

C₂×C₄×D₁₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

#14555489

yes

—

C₂×C₄

#14555489

yes

—

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

C₂×C₄

C₂×Dic₂₇₄

yes

characteristic

C₂×C₄

#14555509

yes

—

C₂×C₄

#14555511

yes

—

C₈

—

C₈

—

548

—

C₂²

C₂×#14555489

yes

characteristic

C₂²

#17984046

yes

—

C₂²

#17984094

yes

—

C₂²

#17984096

yes

—

C₄

—

C₄

—

C₄

—

548

—

100

C₄

#17984075

yes

characteristic

101

C₄

#17984077

yes

characteristic

102

C₄

#17984079

yes

—

103

C₂

#22577268

yes

characteristic

104

C₂

#22577429

yes

characteristic

105

C₂

#22577431

yes

characteristic

106

C₂

#22577463

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₃₇
commutator, socle, prime radical	C₂³×C₁₃₇	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₃₇	C₂
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₃₇
hypocenter	C₁₃₇	#1832
LCS	G ▷ C₂³×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₃₇

commutator, socle, prime radical

C₂³×C₁₃₇

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₃₇

C₂

cosocle

C₂²×C₄×C₁₃₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₃₇

C₁₃₇

G ▷ C₂³×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
53. C₁₃₇	#1832	semidirect

H ≅ G/N

type

53. C₁₃₇

#1832

semidirect

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

subgroup	index
C₂×C₄⋊#11764040	2
#22577249	2
#22577157	2
C₂×C₄⋊C₈×C₁₃₇	2
#1832	137

subgroup

index

C₂×C₄⋊C₈×C₁₃₇

137

Subgroups — #31780908

Conjugacy classes of subgroups

Complements

Maximal subgroups