Group #31779927 — subgroups

Subgroups — #31779927

Order 17536.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22578133	C₂	1	1	yes	maximal, characteristic
2	#22577695	C₂	1	2	yes	maximal
3	#22577152	C₂	1	1	yes	maximal, characteristic
4	#22577135	C₂	1	2	yes	maximal
5	#494×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#17984087	—	2	2	no	—
7	#17984086	—	2	2	no	—
8	C₄×Dic₂₇₄	C₂²	1	2	yes	—
9	#17984053	—	2	2	no	—
10	#17984021	—	2	2	no	—
11	C₄⋊#11764040	C₂²	1	2	yes	—
12	#17984011	—	2	2	no	—
13	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
14	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₇₄	—	2	2	no	—
18	C₂×Dic₂₇₄	D₄	1	2	yes	—
19	C₂×#11764040	—	2	2	no	—
20	C₄⋊Dic₁₃₇	D₄	1	2	yes	—
21	#14555489	—	2	4	no	—
22	#14555489	—	2	2	no	—
23	C₄×Dic₁₃₇	—	2	2	no	—
24	C₄×Dic₁₃₇	—	2	2	no	—
25	C₂×Q₈×C₁₃₇	—	2	1	no	—
26	C₂×Q₈×C₁₃₇	D₄	1	2	yes	—
27	Q₁₆×C₁₃₇	—	4	4	no	—
28	C₂×C₈×C₁₃₇	—	2	1	no	—
29	C₄⋊C₄×C₁₃₇	—	2	2	no	—
30	C₄⋊C₄×C₁₃₇	—	2	2	no	—
31	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
32	Dic₂₇₄	—	2	2	no	—
33	Dic₂₇₄	—	8	2	no	—
34	Dic₂₇₄	Q₁₆	1	4	yes	complemented
35	C₂×Dic₁₃₇	—	2	2	no	—
36	C₂×Dic₁₃₇	—	2	2	no	—
37	#11764040	—	4	2	no	—
38	Q₈×C₁₃₇	—	4	2	no	—
39	Q₈×C₁₃₇	—	4	2	no	—
40	C₂×C₄×C₁₃₇	—	2	2	no	—
41	C₂×C₄×C₁₃₇	—	4	1	no	—
42	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₃₇	—	2	2	no	—
45	C₂²×C₁₃₇	#120	1	1	yes	characteristic
46	C₄×C₁₃₇	—	2	1	no	—
47	C₄×C₁₃₇	—	4	2	no	—
48	C₄×C₁₃₇	—	8	1	no	—
49	C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
50	C₄×C₁₃₇	#137	1	2	yes	—
51	Dic₁₃₇	—	2	4	no	—
52	Dic₁₃₇	—	4	2	no	—
53	C₂×C₁₃₇	#451	1	2	yes	—
54	C₂×C₁₃₇	#453	1	1	yes	characteristic
55	C₁₃₇	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	137	1	no	maximal, Sylow
57	Q₈²	—	137	1	no	—
58	#494	D₁₃₇	1	1	yes	characteristic
59	#477	—	137	2	no	—
60	#333	—	137	1	no	—
61	Q₈⋊C₈	—	137	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	274	2	no	—
64	C₄⋊Q₈	—	274	2	no	—
65	C₄⋊Q₈	C₂×D₁₃₇	1	1	yes	characteristic
66	C₄⋊Q₈	C₂×D₁₃₇	1	1	yes	characteristic
67	C₄×Q₈	—	274	2	no	—
68	C₄×Q₈	—	137	2	no	—
69	#107	—	274	2	no	—
70	C₄⋊C₈	—	137	2	no	—
71	#103	—	274	2	no	—
72	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	274	2	no	—
75	C₂×Q₈	—	137	2	no	—
76	C₂×Q₈	C₂²⋊D₁₃₇	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	274	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	274	4	no	—
83	C₄⋊C₄	—	274	2	no	—
84	C₄⋊C₄	—	137	2	no	—
85	C₄²	—	274	2	no	—
86	C₄²	—	274	2	no	—
87	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	274	2	no	—
91	Q₈	—	1096	2	no	—
92	Q₈	—	137	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	274	2	no	—
96	C₂×C₄	—	274	2	no	—
97	C₂×C₄	D₄×D₁₃₇	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	548	2	no	—
101	C₂²	#17984056	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	274	4	no	—
106	C₄	—	548	2	no	—
107	C₄	Q₁₆×D₁₃₇	1	2	yes	—
108	C₄	#17984120	1	2	yes	—
109	C₂	#22577557	1	2	yes	—
110	C₂	#22577570	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578133

C₂

yes

maximal, characteristic

#22577695

C₂

yes

maximal

#22577152

C₂

yes

maximal, characteristic

#22577135

C₂

yes

maximal

#494×C₁₃₇

C₂

yes

maximal, characteristic

#17984087

—

#17984086

—

C₄×Dic₂₇₄

C₂²

yes

—

#17984053

—

#17984021

—

C₄⋊#11764040

C₂²

yes

—

#17984011

—

C₂×Q₁₆×C₁₃₇

—

C₄⋊Q₈×C₁₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₇

C₂²

yes

characteristic

C₄×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

D₄

yes

—

C₂×#11764040

—

C₄⋊Dic₁₃₇

D₄

yes

—

#14555489

—

#14555489

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

D₄

yes

—

Q₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

—

Dic₂₇₄

—

Dic₂₇₄

Q₁₆

yes

complemented

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

#120

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×Q₁₆

yes

—

C₄×C₁₃₇

#137

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

#451

yes

—

C₂×C₁₃₇

#453

yes

characteristic

C₁₃₇

#1633

yes

characteristic, complemented, Sylow

#1633

—

137

maximal, Sylow

Q₈²

—

137

—

#494

D₁₃₇

yes

characteristic

#477

—

137

—

#333

—

137

—

Q₈⋊C₈

—

137

—

C₂×Q₁₆

—

C₄⋊Q₈

—

274

—

C₄⋊Q₈

—

274

—

C₄⋊Q₈

C₂×D₁₃₇

yes

characteristic

C₄⋊Q₈

C₂×D₁₃₇

yes

characteristic

C₄×Q₈

—

274

—

C₄×Q₈

—

137

—

#107

—

274

—

C₄⋊C₈

—

137

—

#103

—

274

—

C₄×C₈

C₂×D₁₃₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

274

—

C₂×Q₈

—

137

—

C₂×Q₈

C₂²⋊D₁₃₇

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

274

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

137

—

C₄²

—

274

—

C₄²

—

274

—

C₄²

C₂²×D₁₃₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

274

—

Q₈

—

1096

—

Q₈

—

137

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

D₄×D₁₃₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

C₈

—

100

C₈

—

548

—

101

C₂²

#17984056

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

274

—

106

C₄

—

548

—

107

C₄

Q₁₆×D₁₃₇

yes

—

108

C₄

#17984120

yes

—

109

C₂

#22577557

yes

—

110

C₂

#22577570

yes

characteristic

	subgroup	G/H
center	C₂²	#17984056
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting	#494×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	#120
hypercenter	#494	D₁₃₇
hypocenter	C₁₃₇	#1633
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#17984056

commutator, cosocle

C₄²×C₁₃₇

C₂³

Frattini

C₄²

C₂²×D₁₃₇

Fitting

#494×C₁₃₇

C₂

C₂²×C₁₃₇

D₁₃₇

C₁₃₇

G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
34. Dic₂₇₄	Q₁₆	semidirect
55. C₁₃₇	#1633	semidirect

H ≅ G/N

type

34. Dic₂₇₄

Q₁₆

semidirect

55. C₁₃₇

#1633

semidirect

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

subgroup	index
#22578133	2
#22577695	2
#22577152	2
#22577135	2
#494×C₁₃₇	2
#1633	137

subgroup

index

#494×C₁₃₇

137

Subgroups — #31779927

Conjugacy classes of subgroups

Complements

Maximal subgroups