Group #22578430 — subgroups

Subgroups — #22578430

Order 8768.

134 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 56 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, complemented
2	C₂²×#11764047	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984155	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984145	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊#11764047	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11764047	C₂	1	1	yes	maximal, characteristic, complemented
7	#17984048	C₂	1	1	yes	maximal, characteristic
8	#14555518	—	2	2	no	—
9	C₂×#11764047	—	2	2	no	—
10	C₂×#11764047	C₂²	1	1	yes	characteristic
11	C₂×#11764047	C₂²	1	1	yes	characteristic
12	C₂×#11764047	C₂²	1	1	yes	characteristic
13	C₂×#11764047	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₁₃₇	C₄	1	1	yes	characteristic
15	C₂²×Dic₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
17	#14555489	C₄	1	1	yes	characteristic
18	#14555487	C₄	1	1	yes	characteristic
19	C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
20	C₂²⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic
21	#11764047	—	2	1	no	—
22	#11764047	—	2	1	no	—
23	#11764047	—	2	1	no	—
24	#11764047	D₄	1	2	yes	complemented
25	C₂²×D₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂²⋊D₁₃₇	C₈	1	4	yes	complemented
27	C₂×Dic₁₃₇	—	2	1	no	—
28	C₂×Dic₁₃₇	—	2	1	no	—
29	C₂×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂×Dic₁₃₇	C₂³	1	1	yes	characteristic
32	C₄×D₁₃₇	—	2	1	no	—
33	C₄×D₁₃₇	—	2	1	no	—
34	C₂³×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₁₃₇	—	2	1	no	—
38	C₂²×C₁₃₇	—	2	1	no	—
39	C₂²×C₁₃₇	C₂×C₈	1	2	yes	complemented
40	C₂²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×D₁₃₇	—	2	1	no	—
42	C₂×D₁₃₇	—	2	1	no	—
43	C₂×D₁₃₇	C₂×C₈	1	2	yes	complemented
44	C₄×C₁₃₇	—	2	1	no	—
45	C₄×C₁₃₇	—	2	1	no	—
46	Dic₁₃₇	—	2	1	no	—
47	Dic₁₃₇	C₂×C₈	1	2	yes	complemented
48	Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
49	Dic₁₃₇	#41	1	1	yes	characteristic
50	C₂×C₁₃₇	—	2	2	no	—
51	C₂×C₁₃₇	C₄×D₄	1	1	yes	characteristic
52	C₂×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₁₃₇	#131	1	1	yes	characteristic
54	D₁₃₇	—	2	2	no	—
55	C₁₃₇	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
56	C₈×D₄	—	137	1	no	maximal, Sylow
57	C₂²×C₈	—	137	1	no	—
58	C₂²×C₈	—	137	1	no	—
59	C₄×D₄	—	137	1	no	—
60	C₄⋊C₈	—	137	1	no	—
61	C₂²⋊C₈	—	137	1	no	—
62	C₂²⋊C₈	—	137	1	no	—
63	C₄×C₈	—	137	1	no	—
64	C₂×D₄	—	137	1	no	—
65	C₂²×C₄	—	137	1	no	—
66	C₂²×C₄	—	137	1	no	—
67	C₂×C₈	—	274	2	no	—
68	C₂×C₈	—	274	2	no	—
69	C₂×C₈	—	137	1	no	—
70	C₂×C₈	—	137	1	no	—
71	C₂×C₈	—	137	1	no	—
72	C₂×C₈	—	137	1	no	—
73	C₄⋊C₄	—	137	1	no	—
74	C₂²⋊C₄	—	137	1	no	—
75	C₂²⋊C₄	#10587739	1	1	yes	characteristic
76	C₄²	—	137	1	no	—
77	C₂³	—	137	1	no	—
78	C₂³	C₂×#10587739	1	1	yes	characteristic
79	D₄	—	137	4	no	—
80	C₂×C₄	—	274	1	no	—
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	C₂×C₄	—	137	1	no	—
86	C₂×C₄	—	137	1	no	—
87	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
88	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
89	C₈	—	274	1	no	—
90	C₈	—	274	1	no	—
91	C₈	—	274	1	no	—
92	C₈	—	137	2	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	274	1	no	—
96	C₂²	—	274	1	no	—
97	C₂²	—	137	2	no	—
98	C₂²	#14555518	1	2	yes	complemented
99	C₂²	C₂²×#10587739	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	274	1	no	—
103	C₄	—	137	1	no	—
104	C₄	—	137	1	no	—
105	C₄	—	137	2	no	—
106	C₂	—	2	2	no	—
107	C₂	—	274	2	no	—
108	C₂	D₄×#10587739	1	1	yes	characteristic
109	C₂	C₂×#14555518	1	1	yes	characteristic
110	C₂	#17984175	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555518

C₂

yes

maximal, characteristic, complemented

C₂²×#11764047

C₂

yes

maximal, characteristic, complemented

#17984155

C₂

yes

maximal, characteristic, complemented

#17984145

C₂

yes

maximal, characteristic, complemented

C₂²⋊#11764047

C₂

yes

maximal, characteristic, complemented

C₄×#11764047

C₂

yes

maximal, characteristic, complemented

#17984048

C₂

yes

maximal, characteristic

#14555518

—

C₂×#11764047

—

C₂×#11764047

C₂²

yes

characteristic

C₂×#11764047

C₂²

yes

characteristic

C₂×#11764047

C₂²

yes

characteristic

C₂×#11764047

C₂²

yes

characteristic

C₂×C₂²⋊D₁₃₇

C₄

yes

characteristic

C₂²×Dic₁₃₇

C₂²

yes

characteristic

C₂×C₄×D₁₃₇

C₂²

yes

characteristic

#14555489

C₄

yes

characteristic

#14555487

C₄

yes

characteristic

C₄×Dic₁₃₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₃₇

C₄

yes

—

—

—

D₄

yes

complemented

C₂²×D₁₃₇

C₂×C₄

yes

characteristic

C₂²⋊D₁₃₇

C₈

yes

complemented

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂×C₄

yes

characteristic

C₂×Dic₁₃₇

C₂×C₄

yes

characteristic

C₂×Dic₁₃₇

C₂³

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₂³×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₈

yes

complemented

C₂²×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

C₂×C₈

yes

complemented

C₄×C₁₃₇

—

C₄×C₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

C₂×C₈

yes

complemented

Dic₁₃₇

C₂×D₄

yes

characteristic

Dic₁₃₇

#41

yes

characteristic

C₂×C₁₃₇

—

C₂×C₁₃₇

C₄×D₄

yes

characteristic

C₂×C₁₃₇

C₂²×C₈

yes

characteristic

C₂×C₁₃₇

#131

yes

characteristic

D₁₃₇

—

C₁₃₇

C₈×D₄

yes

characteristic, complemented, Sylow

C₈×D₄

—

137

maximal, Sylow

C₂²×C₈

—

137

—

C₂²×C₈

—

137

—

C₄×D₄

—

137

—

C₄⋊C₈

—

137

—

C₂²⋊C₈

—

137

—

C₂²⋊C₈

—

137

—

C₄×C₈

—

137

—

C₂×D₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₂²⋊C₄

#10587739

yes

characteristic

C₄²

—

137

—

C₂³

—

137

—

C₂³

C₂×#10587739

yes

characteristic

D₄

—

137

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

274

—

C₈

—

274

—

C₈

—

274

—

C₈

—

137

—

C₂²

—

C₂²

—

C₂²

—

274

—

C₂²

—

274

—

C₂²

—

137

—

C₂²

yes

C₂²

yes

100

C₄

—

101

C₄

—

102

C₄

—

274

—

103

C₄

—

137

—

104

C₄

—

137

—

105

C₄

—

137

—

106

C₂

—

107

C₂

—

274

—

108

C₂

D₄×#10587739

yes

characteristic

109

C₂

C₂×#14555518

yes

characteristic

110

C₂

#17984175

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#10587739
commutator	C₂×C₁₃₇	C₂²×C₈
Fitting	C₂²⋊C₄×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂²×C₄
hypercenter	C₂²⋊C₄	#10587739
hypocenter	C₁₃₇	C₈×D₄
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂²×#10587739

commutator

C₂×C₁₃₇

C₂²×C₈

Fitting

C₂²⋊C₄×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₂²×C₄

hypercenter

C₂²⋊C₄

#10587739

hypocenter

C₁₃₇

C₈×D₄

LCS

G ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₂²⋊C₄

N	H ≅ G/N	type
1. C₂×#14555518	C₂	semidirect
2. C₂²×#11764047	C₂	semidirect
3. #17984155	C₂	semidirect
4. #17984145	C₂	semidirect
5. C₂²⋊#11764047	C₂	semidirect
6. C₄×#11764047	C₂	semidirect
24. #11764047	D₄	semidirect
26. C₂²⋊D₁₃₇	C₈	semidirect
39. C₂²×C₁₃₇	C₂×C₈	semidirect
43. C₂×D₁₃₇	C₂×C₈	semidirect
47. Dic₁₃₇	C₂×C₈	semidirect
55. C₁₃₇	C₈×D₄	semidirect
98. C₂²	#14555518	semidirect

H ≅ G/N

type

1. C₂×#14555518

C₂

semidirect

2. C₂²×#11764047

C₂

semidirect

3. #17984155

C₂

semidirect

4. #17984145

C₂

semidirect

5. C₂²⋊#11764047

C₂

semidirect

6. C₄×#11764047

C₂

semidirect

24. #11764047

D₄

semidirect

26. C₂²⋊D₁₃₇

C₈

semidirect

39. C₂²×C₁₃₇

C₂×C₈

semidirect

43. C₂×D₁₃₇

C₂×C₈

semidirect

47. Dic₁₃₇

C₂×C₈

semidirect

55. C₁₃₇

C₈×D₄

semidirect

98. C₂²

#14555518

semidirect

Subgroups — #22578430

Conjugacy classes of subgroups

Complements