Group #22577774 — subgroups

Subgroups — #22577774

Order 8768.

142 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 47 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×#14555499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14555498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14555497	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984086	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984084	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₇₄	C₂²	1	1	yes	characteristic
9	C₂×D₅₄₈	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₁₃₇	—	2	2	no	—
11	#14555499	—	2	2	no	—
12	#14555498	—	2	2	no	—
13	#14555497	—	2	2	no	—
14	C₂×#11764040	C₂²	1	1	yes	characteristic
15	#14555489	—	2	2	no	—
16	C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
19	SD₁₆×C₁₃₇	—	2	2	no	—
20	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂²×D₁₃₇	—	2	1	no	—
22	Dic₂₇₄	—	2	1	no	—
23	Dic₂₇₄	—	2	1	no	—
24	D₅₄₈	—	2	1	no	—
25	D₅₄₈	—	2	1	no	—
26	C₂²⋊D₁₃₇	—	4	4	no	—
27	C₂×Dic₁₃₇	—	2	1	no	—
28	C₂×Dic₁₃₇	D₄	1	2	yes	—
29	#11764040	D₄	1	2	yes	complemented
30	C₂³×C₁₃₇	—	2	1	no	—
31	Q₈×C₁₃₇	—	2	1	no	—
32	Q₈×C₁₃₇	—	2	1	no	—
33	D₄×C₁₃₇	—	2	1	no	—
34	D₄×C₁₃₇	—	2	1	no	—
35	C₂×C₄×C₁₃₇	—	2	1	no	—
36	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₈×C₁₃₇	D₄	1	2	yes	complemented
38	C₂²×C₁₃₇	—	4	1	no	—
39	C₂²×C₁₃₇	—	4	2	no	—
40	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
41	C₂×D₁₃₇	—	4	1	no	—
42	C₂×D₁₃₇	—	4	2	no	—
43	C₄×C₁₃₇	—	4	1	no	—
44	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
45	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
46	Dic₁₃₇	—	4	1	no	—
47	Dic₁₃₇	SD₁₆	1	4	yes	complemented
48	C₂×C₁₃₇	—	8	1	no	—
49	C₂×C₁₃₇	#127	1	1	yes	characteristic
50	C₂×C₁₃₇	C₂×SD₁₆	1	2	yes	—
51	D₁₃₇	—	8	1	no	—
52	C₁₃₇	#492	1	1	yes	characteristic, complemented, Sylow
53	#492	—	137	1	no	maximal, Sylow
54	C₂×SD₁₆	—	137	1	no	—
55	C₂×SD₁₆	—	137	1	no	—
56	C₂×SD₁₆	—	137	1	no	—
57	C₂×SD₁₆	D₁₃₇	1	1	yes	characteristic, complemented
58	C₄⋊Q₈	—	137	1	no	—
59	#127	—	137	1	no	—
60	C₄×C₈	—	137	1	no	—
61	C₂×Q₈	—	137	1	no	—
62	C₂×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
63	C₂×D₄	—	274	2	no	—
64	C₂×D₄	—	137	1	no	—
65	C₂×D₄	C₂×D₁₃₇	1	1	yes	characteristic
66	SD₁₆	—	2	2	no	—
67	SD₁₆	—	274	2	no	—
68	SD₁₆	—	274	2	no	—
69	SD₁₆	—	274	2	no	—
70	C₂×C₈	—	137	1	no	—
71	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
72	C₄⋊C₄	—	274	2	no	—
73	C₄²	—	137	1	no	—
74	C₂³	—	2	1	no	—
75	C₂³	—	274	1	no	—
76	Q₈	—	2	1	no	—
77	Q₈	—	2	1	no	—
78	Q₈	—	274	1	no	—
79	Q₈	—	274	1	no	—
80	D₄	—	2	1	no	—
81	D₄	—	2	1	no	—
82	D₄	—	274	1	no	—
83	D₄	—	274	1	no	—
84	D₄	—	548	4	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	274	1	no	—
87	C₂×C₄	—	137	2	no	—
88	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
89	C₈	—	137	2	no	—
90	C₈	C₂²⋊D₁₃₇	1	2	yes	complemented
91	C₂²	—	4	1	no	—
92	C₂²	—	4	2	no	—
93	C₂²	—	548	1	no	—
94	C₂²	—	548	2	no	—
95	C₂²	D₄×D₁₃₇	1	1	yes	characteristic
96	C₄	—	4	1	no	—
97	C₄	—	548	1	no	—
98	C₄	—	137	4	no	—
99	C₄	D₄×D₁₃₇	1	1	yes	characteristic
100	C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
101	C₂	—	8	1	no	—
102	C₂	—	1096	1	no	—
103	C₂	#17984084	1	1	yes	characteristic
104	C₂	SD₁₆×D₁₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555499

C₂

yes

maximal, characteristic, complemented

C₂×#14555498

C₂

yes

maximal, characteristic, complemented

C₂×#14555497

C₂

yes

maximal, characteristic, complemented

#17984086

C₂

yes

maximal, characteristic, complemented

#17984084

C₂

yes

maximal, characteristic

C₈×Dic₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₇₄

C₂²

yes

characteristic

C₂×D₅₄₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₃₇

—

—

—

—

C₂²

yes

characteristic

#14555489

—

C₄×Dic₁₃₇

C₂²

yes

characteristic

C₂×Q₈×C₁₃₇

C₂²

yes

characteristic

C₂×D₄×C₁₃₇

C₂²

yes

characteristic

SD₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂²

yes

characteristic

C₂²×D₁₃₇

—

Dic₂₇₄

—

Dic₂₇₄

—

D₅₄₈

—

D₅₄₈

—

C₂²⋊D₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

D₄

yes

—

#11764040

D₄

yes

complemented

C₂³×C₁₃₇

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₈×C₁₃₇

D₄

yes

complemented

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×D₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₄×C₁₃₇

C₂×D₄

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

SD₁₆

yes

complemented

C₂×C₁₃₇

—

C₂×C₁₃₇

#127

yes

characteristic

C₂×C₁₃₇

C₂×SD₁₆

yes

—

D₁₃₇

—

C₁₃₇

#492

yes

characteristic, complemented, Sylow

#492

—

137

maximal, Sylow

C₂×SD₁₆

—

137

—

C₂×SD₁₆

—

137

—

C₂×SD₁₆

—

137

—

C₂×SD₁₆

D₁₃₇

yes

characteristic, complemented

C₄⋊Q₈

—

137

—

#127

—

137

—

C₄×C₈

—

137

—

C₂×Q₈

—

137

—

C₂×Q₈

C₂×D₁₃₇

yes

characteristic

C₂×D₄

—

274

—

C₂×D₄

—

137

—

C₂×D₄

C₂×D₁₃₇

yes

characteristic

SD₁₆

—

SD₁₆

—

274

—

SD₁₆

—

274

—

SD₁₆

—

274

—

C₂×C₈

—

137

—

C₂×C₈

C₂×D₁₃₇

yes

characteristic

C₄⋊C₄

—

274

—

C₄²

—

137

—

C₂³

—

C₂³

—

274

—

Q₈

—

Q₈

—

Q₈

—

274

—

Q₈

—

274

—

D₄

—

D₄

—

D₄

—

274

—

D₄

—

274

—

D₄

—

548

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₈

—

137

—

C₈

C₂²⋊D₁₃₇

yes

complemented

C₂²

—

C₂²

—

C₂²

—

548

—

C₂²

—

548

—

C₂²

D₄×D₁₃₇

yes

characteristic

C₄

—

C₄

—

548

—

C₄

—

137

—

C₄

D₄×D₁₃₇

yes

characteristic

100

C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

101

C₂

—

102

C₂

—

1096

—

103

C₂

#17984084

yes

characteristic

104

C₂

SD₁₆×D₁₃₇

yes

—

	subgroup	G/H
center	C₂²	D₄×D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	C₂×SD₁₆×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	C₂×SD₁₆	D₁₃₇
hypocenter	C₁₃₇	#492
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×D₁₃₇

commutator

C₂×C₄×C₁₃₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₃₇

Fitting

C₂×SD₁₆×C₁₃₇

C₂

socle

C₂²×C₁₃₇

C₂×D₄

hypercenter

C₂×SD₁₆

D₁₃₇

hypocenter

C₁₃₇

#492

LCS

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
1. C₂×#14555499	C₂	semidirect
2. C₂×#14555498	C₂	semidirect
3. C₂×#14555497	C₂	semidirect
4. #17984086	C₂	semidirect
6. C₈×Dic₁₃₇	C₂	semidirect
7. C₂×SD₁₆×C₁₃₇	C₂	semidirect
29. #11764040	D₄	semidirect
37. C₈×C₁₃₇	D₄	semidirect
47. Dic₁₃₇	SD₁₆	semidirect
52. C₁₃₇	#492	semidirect
57. C₂×SD₁₆	D₁₃₇	semidirect
90. C₈	C₂²⋊D₁₃₇	semidirect

H ≅ G/N

type

1. C₂×#14555499

C₂

semidirect

2. C₂×#14555498

C₂

semidirect

3. C₂×#14555497

C₂

semidirect

4. #17984086

C₂

semidirect

6. C₈×Dic₁₃₇

C₂

semidirect

7. C₂×SD₁₆×C₁₃₇

C₂

semidirect

29. #11764040

D₄

semidirect

37. C₈×C₁₃₇

D₄

semidirect

47. Dic₁₃₇

SD₁₆

semidirect

52. C₁₃₇

#492

semidirect

57. C₂×SD₁₆

D₁₃₇

semidirect

90. C₈

C₂²⋊D₁₃₇

semidirect

Subgroups — #22577774

Conjugacy classes of subgroups

Complements