Group #22578276 — subgroups

Subgroups — #22578276

Order 8768.

114 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 114 conjugacy classes of subgroups — too many to draw. Only its 46 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×#14555518	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#10587739	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10587739	—	2	1	no	—
5	#14555518	—	2	1	no	—
6	#14555518	C₄	1	2	yes	complemented
7	C₂×#11764047	—	2	1	no	—
8	C₄⋊#10587739	—	2	1	no	—
9	C₄⋊#10587739	C₄	1	2	yes	complemented
10	C₂×C₄×D₁₃₇	—	2	1	no	—
11	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₃₇	C₄	1	1	yes	characteristic, complemented
13	#14555489	—	2	1	no	—
14	C₄⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic, complemented
15	C₂×#10587739	—	2	2	no	—
16	C₂×#10587739	—	4	1	no	—
17	C₂×#10587739	—	4	1	no	—
18	#11764047	—	2	2	no	—
19	C₂²×D₁₃₇	D₄	1	1	yes	characteristic
20	C₂×Dic₁₃₇	—	2	1	no	—
21	C₂×Dic₁₃₇	D₄	1	1	yes	characteristic
22	C₄×D₁₃₇	—	4	1	no	—
23	C₄×D₁₃₇	—	4	1	no	—
24	C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₄×D₁₃₇	D₄	1	1	yes	characteristic
27	C₄×D₁₃₇	Q₈	1	1	yes	characteristic
28	C₂×C₄×C₁₃₇	—	2	1	no	—
29	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×D₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×D₁₃₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×D₁₃₇	D₈	1	1	yes	characteristic
34	C₂×D₁₃₇	SD₁₆	1	1	yes	characteristic
35	C₂×D₁₃₇	SD₁₆	1	1	yes	characteristic
36	C₂×D₁₃₇	Q₁₆	1	1	yes	characteristic
37	#10587739	—	4	2	no	—
38	C₄×C₁₃₇	—	4	1	no	—
39	C₄×C₁₃₇	C₄²	1	1	yes	characteristic
40	C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
41	Dic₁₃₇	—	4	1	no	—
42	Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
43	Dic₁₃₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₁₃₇	#95	1	1	yes	characteristic
45	C₂×C₁₃₇	#102	1	1	yes	characteristic
46	C₂×C₁₃₇	#103	1	1	yes	characteristic
47	D₁₃₇	#102	1	1	yes	characteristic
48	D₁₃₇	#103	1	1	yes	characteristic
49	D₁₃₇	#106	1	1	yes	characteristic
50	D₁₃₇	#107	1	1	yes	characteristic
51	C₁₃₇	#340	1	1	yes	characteristic, complemented, Sylow
52	#340	—	137	1	no	maximal, Sylow
53	C₂²×C₈	—	137	1	no	—
54	C₂×C₄⋊C₄	—	137	1	no	—
55	C₂×C₄⋊C₄	—	137	1	no	—
56	C₂²×C₄	—	274	1	no	—
57	C₂²×C₄	—	274	1	no	—
58	C₂²×C₄	—	137	1	no	—
59	C₂×C₈	—	274	1	no	—
60	C₂×C₈	—	274	1	no	—
61	C₂×C₈	—	137	2	no	—
62	C₄⋊C₄	—	274	1	no	—
63	C₄⋊C₄	—	274	1	no	—
64	C₄⋊C₄	—	137	2	no	—
65	C₄⋊C₄	—	137	1	no	—
66	C₄⋊C₄	#10587739	1	1	yes	characteristic, complemented
67	C₂³	—	137	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	274	2	no	—
70	C₂×C₄	—	274	1	no	—
71	C₂×C₄	—	548	1	no	—
72	C₂×C₄	—	548	1	no	—
73	C₂×C₄	—	548	1	no	—
74	C₂×C₄	—	548	1	no	—
75	C₂×C₄	—	137	1	no	—
76	C₂×C₄	—	137	1	no	—
77	C₂×C₄	—	137	1	no	—
78	C₂×C₄	—	137	1	no	—
79	C₂×C₄	—	137	1	no	—
80	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
81	C₈	—	274	2	no	—
82	C₂²	—	137	1	no	—
83	C₂²	—	137	1	no	—
84	C₂²	—	137	1	no	—
85	C₂²	—	137	1	no	—
86	C₂²	—	137	1	no	—
87	C₂²	—	137	1	no	—
88	C₂²	C₂²⋊#10587739	1	1	yes	characteristic
89	C₄	—	4	1	no	—
90	C₄	—	548	2	no	—
91	C₄	—	548	1	no	—
92	C₄	—	137	1	no	—
93	C₄	—	137	1	no	—
94	C₄	C₄×#10587739	1	1	yes	characteristic
95	C₄	C₄⋊#10587739	1	1	yes	characteristic
96	C₂	—	137	1	no	—
97	C₂	—	137	1	no	—
98	C₂	—	137	1	no	—
99	C₂	—	137	1	no	—
100	C₂	(C₂×#10587739)⋊C₄	1	1	yes	characteristic
101	C₂	#17984150	1	1	yes	characteristic
102	C₂	#17984151	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555518

C₂

yes

maximal, characteristic

C₂×C₄⋊#10587739

C₂

yes

maximal, characteristic

C₄⋊C₄×D₁₃₇

C₂

yes

maximal, characteristic

C₂²×#10587739

—

#14555518

—

#14555518

C₄

yes

—

—

C₄

yes

complemented

C₂×C₄×D₁₃₇

—

C₂×C₄×D₁₃₇

C₂²

yes

characteristic

C₄⋊Dic₁₃₇

C₄

yes

characteristic, complemented

#14555489

—

C₄⋊C₄×C₁₃₇

C₄

yes

characteristic, complemented

—

—

—

—

C₂²×D₁₃₇

D₄

yes

characteristic

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

D₄

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

C₂×C₄

yes

characteristic

C₄×D₁₃₇

C₂×C₄

yes

characteristic

C₄×D₁₃₇

D₄

yes

characteristic

C₄×D₁₃₇

Q₈

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₁₃₇

C₂²⋊C₄

yes

characteristic

C₂×D₁₃₇

C₂²⋊C₄

yes

characteristic

C₂×D₁₃₇

C₄⋊C₄

yes

characteristic

C₂×D₁₃₇

D₈

yes

characteristic

C₂×D₁₃₇

SD₁₆

yes

characteristic

C₂×D₁₃₇

SD₁₆

yes

characteristic

C₂×D₁₃₇

Q₁₆

yes

characteristic

#10587739

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₄²

yes

characteristic

C₄×C₁₃₇

C₄⋊C₄

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

C₂²⋊C₄

yes

characteristic

Dic₁₃₇

C₄⋊C₄

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

D₁₃₇

yes

D₁₃₇

yes

D₁₃₇

yes

D₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#340

—

137

maximal, Sylow

C₂²×C₈

—

137

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄⋊C₄

—

137

—

C₂²×C₄

—

274

—

C₂²×C₄

—

274

—

C₂²×C₄

—

137

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

#10587739

yes

characteristic, complemented

C₂³

—

137

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

548

—

C₂×C₄

—

548

—

C₂×C₄

—

548

—

C₂×C₄

—

548

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂×#10587739

yes

characteristic

C₈

—

274

—

C₂²

—

137

—

C₂²

—

137

—

C₂²

—

137

—

C₂²

—

137

—

C₂²

—

137

—

C₂²

—

137

—

C₂²

C₂²⋊#10587739

yes

characteristic

C₄

—

C₄

—

548

—

C₄

—

548

—

C₄

—

137

—

C₄

—

137

—

C₄

yes

C₄

yes

C₂

—

137

—

C₂

—

137

—

C₂

—

137

—

C₂

—

137

—

100

C₂

(C₂×#10587739)⋊C₄

yes

characteristic

101

C₂

#17984150

yes

characteristic

102

C₂

#17984151

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#10587739
commutator	C₄×C₁₃₇	C₄²
Frattini	C₂×C₄	C₂×#10587739
Fitting	C₄⋊C₄×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂²⋊C₄
hypercenter	C₄⋊C₄	#10587739
hypocenter	C₁₃₇	#340
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

subgroup

G/H

center

C₂²

C₂²⋊#10587739

commutator

C₄×C₁₃₇

C₄²

Frattini

C₂×C₄

C₂×#10587739

Fitting

C₄⋊C₄×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₂²⋊C₄

C₄⋊C₄

C₁₃₇

G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

N	H ≅ G/N	type
6. #14555518	C₄	semidirect
9. C₄⋊#10587739	C₄	semidirect
12. C₄⋊Dic₁₃₇	C₄	semidirect
14. C₄⋊C₄×C₁₃₇	C₄	semidirect
51. C₁₃₇	#340	semidirect
66. C₄⋊C₄	#10587739	semidirect

H ≅ G/N

type

6. #14555518

C₄

semidirect

9. C₄⋊#10587739

C₄

semidirect

12. C₄⋊Dic₁₃₇

C₄

semidirect

14. C₄⋊C₄×C₁₃₇

C₄

semidirect

51. C₁₃₇

#340

semidirect

66. C₄⋊C₄

#10587739

semidirect

Subgroups — #22578276

Conjugacy classes of subgroups

Complements