Group #22578446 — subgroups

Subgroups — #22578446

Order 8768.

106 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 106 conjugacy classes of subgroups — too many to draw. Only its 40 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	#17984176	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984175	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984158	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984153	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984152	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#10587739	C₂	1	1	yes	maximal, characteristic, complemented
7	#17984109	C₂	1	1	yes	maximal, characteristic
8	#14555520	—	2	1	no	—
9	#14555520	C₂²	1	1	yes	characteristic
10	#14555520	C₂²	1	1	yes	characteristic
11	#14555519	—	2	1	no	—
12	#14555518	—	2	1	no	—
13	#14555518	C₂²	1	1	yes	characteristic
14	C₂×#11764047	—	2	1	no	—
15	C₄×#10587739	C₂²	1	1	yes	characteristic
16	#14555511	C₂²	1	1	yes	characteristic
17	#14555509	C₂²	1	1	yes	characteristic
18	#14555500	C₄	1	1	yes	characteristic, complemented
19	#14555499	C₄	1	1	yes	characteristic, complemented
20	#14555495	C₄	1	1	yes	characteristic, complemented
21	C₈×D₁₃₇	C₂²	1	1	yes	characteristic
22	SD₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
23	C₂×#10587739	—	2	1	no	—
24	#11764047	—	2	1	no	—
25	#11764047	—	2	1	no	—
26	#11764047	D₄	1	1	yes	characteristic
27	#11764047	D₄	1	1	yes	characteristic
28	Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
29	D₅₄₈	—	2	1	no	—
30	D₅₄₈	C₂×C₄	1	1	yes	characteristic
31	C₂²⋊D₁₃₇	—	2	1	no	—
32	C₂×Dic₁₃₇	—	2	1	no	—
33	C₄×D₁₃₇	—	2	1	no	—
34	C₄×D₁₃₇	C₂³	1	1	yes	characteristic
35	#11764040	C₂×C₄	1	1	yes	characteristic
36	Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
38	C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₁₃₇	—	2	1	no	—
40	C₂×D₁₃₇	—	2	1	no	—
41	C₂×D₁₃₇	#41	1	1	yes	characteristic
42	#10587739	—	2	2	no	—
43	C₄×C₁₃₇	—	2	1	no	—
44	C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
45	Dic₁₃₇	—	2	1	no	—
46	Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₁₃₇	—	4	1	no	—
48	C₂×C₁₃₇	C₄×D₄	1	1	yes	characteristic
49	D₁₃₇	—	2	1	no	—
50	D₁₃₇	—	4	1	no	—
51	C₁₃₇	#443	1	1	yes	characteristic, complemented, Sylow
52	#443	—	137	1	no	maximal, Sylow
53	#135	—	137	1	no	—
54	#131	—	137	1	no	—
55	#131	—	137	1	no	—
56	#108	—	137	1	no	—
57	#104	—	137	1	no	—
58	#104	—	137	1	no	—
59	C₄×C₈	—	137	1	no	—
60	#41	—	137	1	no	—
61	#41	—	137	1	no	—
62	Q₁₆	—	137	1	no	—
63	SD₁₆	—	137	1	no	—
64	SD₁₆	#10587739	1	1	yes	characteristic, complemented
65	D₈	—	137	1	no	—
66	M₁₆	—	274	1	no	—
67	M₁₆	—	274	1	no	—
68	M₁₆	—	137	1	no	—
69	M₁₆	—	137	1	no	—
70	C₂×C₈	—	274	1	no	—
71	C₂×C₈	—	274	1	no	—
72	C₂×C₈	—	137	1	no	—
73	C₂×C₈	—	137	1	no	—
74	C₄²	—	137	1	no	—
75	Q₈	—	137	1	no	—
76	Q₈	C₂×#10587739	1	1	yes	characteristic
77	D₄	—	274	1	no	—
78	D₄	—	274	1	no	—
79	D₄	—	137	1	no	—
80	D₄	C₂×#10587739	1	1	yes	characteristic
81	C₂×C₄	—	274	1	no	—
82	C₂×C₄	—	274	1	no	—
83	C₂×C₄	—	274	1	no	—
84	C₂×C₄	—	137	1	no	—
85	C₈	—	274	1	no	—
86	C₈	—	274	1	no	—
87	C₈	—	137	1	no	—
88	C₈	—	137	1	no	—
89	C₈	—	137	1	no	—
90	C₈	C₂×#10587739	1	1	yes	characteristic
91	C₂²	—	2	1	no	—
92	C₂²	—	274	1	no	—
93	C₂²	—	137	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	274	2	no	—
96	C₄	—	274	1	no	—
97	C₄	—	137	1	no	—
98	C₄	C₂²×#10587739	1	1	yes	characteristic
99	C₂	—	4	1	no	—
100	C₂	—	274	1	no	—
101	C₂	—	548	1	no	—
102	C₂	D₄×#10587739	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984176

C₂

yes

maximal, characteristic, complemented

#17984175

C₂

yes

maximal, characteristic, complemented

#17984158

C₂

yes

maximal, characteristic, complemented

#17984153

C₂

yes

maximal, characteristic, complemented

#17984152

C₂

yes

maximal, characteristic, complemented

C₈×#10587739

C₂

yes

maximal, characteristic, complemented

#17984109

C₂

yes

maximal, characteristic

#14555520

—

#14555520

C₂²

yes

characteristic

#14555520

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×#11764047

—

C₄×#10587739

C₂²

yes

characteristic

#14555511

C₂²

yes

characteristic

#14555509

C₂²

yes

characteristic

#14555500

C₄

yes

characteristic, complemented

#14555499

C₄

yes

characteristic, complemented

#14555495

C₄

yes

characteristic, complemented

C₈×D₁₃₇

C₂²

yes

characteristic

SD₁₆×C₁₃₇

C₄

yes

characteristic, complemented

—

—

—

D₄

yes

characteristic

#11764047

D₄

yes

characteristic

Dic₂₇₄

C₂×C₄

yes

characteristic

D₅₄₈

—

D₅₄₈

C₂×C₄

yes

characteristic

C₂²⋊D₁₃₇

—

C₂×Dic₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

C₂³

yes

characteristic

#11764040

C₂×C₄

yes

characteristic

Q₈×C₁₃₇

C₂×C₄

yes

characteristic

D₄×C₁₃₇

C₂×C₄

yes

characteristic

C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

#41

yes

characteristic

#10587739

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂²×C₄

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

C₂×D₄

yes

characteristic

C₂×C₁₃₇

—

C₂×C₁₃₇

C₄×D₄

yes

characteristic

D₁₃₇

—

D₁₃₇

—

C₁₃₇

#443

yes

characteristic, complemented, Sylow

#443

—

137

maximal, Sylow

#135

—

137

—

#131

—

137

—

#131

—

137

—

#108

—

137

—

#104

—

137

—

#104

—

137

—

C₄×C₈

—

137

—

#41

—

137

—

#41

—

137

—

Q₁₆

—

137

—

SD₁₆

—

137

—

SD₁₆

#10587739

yes

characteristic, complemented

D₈

—

137

—

M₁₆

—

274

—

M₁₆

—

274

—

M₁₆

—

137

—

M₁₆

—

137

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₄²

—

137

—

Q₈

—

137

—

Q₈

C₂×#10587739

yes

characteristic

D₄

—

274

—

D₄

—

274

—

D₄

—

137

—

D₄

C₂×#10587739

yes

characteristic

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₈

—

274

—

C₈

—

274

—

C₈

—

137

—

C₈

—

137

—

C₈

—

137

—

C₈

C₂×#10587739

yes

characteristic

C₂²

—

C₂²

—

274

—

C₂²

—

137

—

C₄

—

C₄

—

274

—

C₄

—

274

—

C₄

—

137

—

C₄

C₂²×#10587739

yes

characteristic

C₂

—

100

C₂

—

274

—

101

C₂

—

548

—

102

C₂

D₄×#10587739

yes

characteristic

	subgroup	G/H
center	C₂	D₄×#10587739
commutator	C₄×C₁₃₇	C₂²×C₄
Frattini	C₄	C₂²×#10587739
Fitting	SD₁₆×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₄×D₄
hypercenter	SD₁₆	#10587739
hypocenter	C₁₃₇	#443
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

subgroup

G/H

center

C₂

D₄×#10587739

commutator

C₄×C₁₃₇

C₂²×C₄

Frattini

C₄

C₂²×#10587739

Fitting

SD₁₆×C₁₃₇

C₄

socle

C₂×C₁₃₇

C₄×D₄

SD₁₆

C₁₃₇

G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

N	H ≅ G/N	type
1. #17984176	C₂	semidirect
2. #17984175	C₂	semidirect
3. #17984158	C₂	semidirect
4. #17984153	C₂	semidirect
5. #17984152	C₂	semidirect
6. C₈×#10587739	C₂	semidirect
18. #14555500	C₄	semidirect
19. #14555499	C₄	semidirect
20. #14555495	C₄	semidirect
22. SD₁₆×C₁₃₇	C₄	semidirect
51. C₁₃₇	#443	semidirect
64. SD₁₆	#10587739	semidirect

H ≅ G/N

type

1. #17984176

C₂

semidirect

2. #17984175

C₂

semidirect

3. #17984158

C₂

semidirect

4. #17984153

C₂

semidirect

5. #17984152

C₂

semidirect

6. C₈×#10587739

C₂

semidirect

18. #14555500

C₄

semidirect

19. #14555499

C₄

semidirect

20. #14555495

C₄

semidirect

22. SD₁₆×C₁₃₇

C₄

semidirect

51. C₁₃₇

#443

semidirect

64. SD₁₆

#10587739

semidirect

Subgroups — #22578446

Conjugacy classes of subgroups

Complements