Group #22578123 — subgroups

Subgroups — #22578123

Order 8768.

212 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 99 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	#17984088	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984081	C₂	1	4	yes	maximal, complemented
3	#17984075	C₂	1	2	yes	maximal
4	#17984067	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₂₇₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984053	C₂	1	2	yes	maximal, complemented
7	#17984047	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₇₄	C₂²	1	2	yes	—
10	C₂×Dic₂₇₄	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
13	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
14	#14555489	C₂²	1	4	yes	—
15	#14555489	C₂²	1	2	yes	—
16	C₂²⋊Dic₁₃₇	C₂²	1	4	yes	—
17	C₂²⋊Dic₁₃₇	C₂²	1	4	yes	—
18	C₄×Dic₁₃₇	C₂²	1	4	yes	—
19	C₄×Dic₁₃₇	C₂²	1	2	yes	—
20	C₂×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₃₇	C₂²	1	2	yes	—
22	C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
24	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
25	Dic₂₇₄	—	2	2	no	—
26	Dic₂₇₄	—	2	4	no	—
27	C₂×Dic₁₃₇	C₂³	1	2	yes	—
28	C₂×Dic₁₃₇	C₂³	1	2	yes	—
29	C₂×Dic₁₃₇	C₂³	1	4	yes	—
30	C₂³×C₁₃₇	C₂³	1	2	yes	—
31	D₄×C₁₃₇	—	2	2	no	—
32	C₂×C₄×C₁₃₇	—	2	4	no	—
33	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₁₃₇	C₂³	1	2	yes	—
35	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₂²×C₁₃₇	—	2	2	no	—
38	C₂²×C₁₃₇	—	2	2	no	—
39	C₂²×C₁₃₇	—	2	2	no	—
40	C₂²×C₁₃₇	C₂⁴	1	1	yes	characteristic
41	C₄×C₁₃₇	—	2	2	no	—
42	C₄×C₁₃₇	—	2	1	no	—
43	C₄×C₁₃₇	#41	1	2	yes	—
44	C₄×C₁₃₇	#41	1	2	yes	—
45	Dic₁₃₇	—	2	2	no	—
46	Dic₁₃₇	—	2	4	no	—
47	Dic₁₃₇	—	2	2	no	—
48	C₂×C₁₃₇	—	4	2	no	—
49	C₂×C₁₃₇	C₂×#41	1	1	yes	characteristic
50	C₂×C₁₃₇	C₂×#41	1	1	yes	characteristic
51	C₂×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
52	C₁₃₇	#556	1	1	yes	characteristic, complemented, Sylow
53	#556	—	137	1	no	maximal, Sylow
54	C₄⋊Q₈	—	137	1	no	—
55	#126	—	137	4	no	—
56	#124	—	137	2	no	—
57	C₂²⋊Q₈	—	137	2	no	—
58	C₄×Q₈	—	137	2	no	—
59	C₄×Q₈	—	137	1	no	—
60	C₄×D₄	D₁₃₇	1	1	yes	characteristic
61	#117	—	137	2	no	—
62	C₂×Q₈	—	137	1	no	—
63	C₂×Q₈	—	137	2	no	—
64	C₂×D₄	C₂×D₁₃₇	1	1	yes	characteristic
65	C₂²×C₄	C₂×D₁₃₇	1	2	yes	—
66	C₄⋊C₄	—	137	4	no	—
67	C₄⋊C₄	—	137	2	no	—
68	C₄⋊C₄	—	137	1	no	—
69	C₄⋊C₄	—	137	2	no	—
70	C₄⋊C₄	—	137	2	no	—
71	C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
72	C₂²⋊C₄	—	137	4	no	—
73	C₂²⋊C₄	—	137	4	no	—
74	C₂²⋊C₄	C₂×D₁₃₇	1	2	yes	—
75	C₄²	—	137	4	no	—
76	C₄²	—	137	2	no	—
77	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
78	C₂³	C₂²×D₁₃₇	1	2	yes	—
79	Q₈	—	274	2	no	—
80	Q₈	—	274	4	no	—
81	D₄	—	2	2	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	137	2	no	—
84	C₂×C₄	—	137	2	no	—
85	C₂×C₄	—	137	4	no	—
86	C₂×C₄	C₂²×D₁₃₇	1	2	yes	—
87	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	—	2	2	no	—
92	C₂²	—	2	2	no	—
93	C₂²	C₂³×D₁₃₇	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	—	274	2	no	—
97	C₄	—	274	4	no	—
98	C₄	—	274	2	no	—
99	C₄	#14555509	1	2	yes	—
100	C₄	#14555510	1	2	yes	—
101	C₂	—	4	2	no	—
102	C₂	C₂×#14555509	1	1	yes	characteristic
103	C₂	C₂×#14555510	1	1	yes	characteristic
104	C₂	#17984135	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984088

C₂

yes

maximal, characteristic, complemented

#17984081

C₂

yes

maximal, complemented

#17984075

C₂

yes

maximal

#17984067

C₂

yes

maximal, complemented

C₄×Dic₂₇₄

C₂

yes

maximal, characteristic, complemented

#17984053

C₂

yes

maximal, complemented

#17984047

C₂

yes

maximal, complemented

C₄×D₄×C₁₃₇

C₂

yes

maximal, characteristic

C₂×Dic₂₇₄

C₂²

yes

—

C₂×Dic₂₇₄

C₂²

yes

characteristic

C₄⋊Dic₁₃₇

C₂²

yes

characteristic

C₄⋊Dic₁₃₇

C₂²

yes

—

C₄⋊Dic₁₃₇

C₂²

yes

—

#14555489

C₂²

yes

—

#14555489

C₂²

yes

—

C₂²⋊Dic₁₃₇

C₂²

yes

—

C₂²⋊Dic₁₃₇

C₂²

yes

—

C₄×Dic₁₃₇

C₂²

yes

—

C₄×Dic₁₃₇

C₂²

yes

—

C₂×D₄×C₁₃₇

C₂²

yes

characteristic

C₂²×C₄×C₁₃₇

C₂²

yes

—

C₄⋊C₄×C₁₃₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₃₇

C₂²

yes

—

C₄²×C₁₃₇

C₂²

yes

characteristic

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

C₂³

yes

—

C₂×Dic₁₃₇

C₂³

yes

—

C₂×Dic₁₃₇

C₂³

yes

—

C₂³×C₁₃₇

C₂³

yes

—

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₄×C₁₃₇

C₂³

yes

—

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂⁴

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#41

yes

—

C₄×C₁₃₇

#41

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×#41

yes

characteristic

C₂×C₁₃₇

C₂×#41

yes

characteristic

C₂×C₁₃₇

2₋¹⁺⁴

yes

characteristic

C₁₃₇

#556

yes

characteristic, complemented, Sylow

#556

—

137

maximal, Sylow

C₄⋊Q₈

—

137

—

#126

—

137

—

#124

—

137

—

C₂²⋊Q₈

—

137

—

C₄×Q₈

—

137

—

C₄×Q₈

—

137

—

C₄×D₄

D₁₃₇

yes

characteristic

#117

—

137

—

C₂×Q₈

—

137

—

C₂×Q₈

—

137

—

C₂×D₄

C₂×D₁₃₇

yes

characteristic

C₂²×C₄

C₂×D₁₃₇

yes

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

C₂×D₁₃₇

yes

characteristic

C₂²⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₂²⋊C₄

C₂×D₁₃₇

yes

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂×D₁₃₇

yes

characteristic

C₂³

C₂²×D₁₃₇

yes

—

Q₈

—

274

—

Q₈

—

274

—

D₄

—

C₂×C₄

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂²×D₁₃₇

yes

—

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂³×D₁₃₇

yes

characteristic

C₄

—

C₄

—

C₄

—

274

—

C₄

—

274

—

C₄

—

274

—

C₄

#14555509

yes

—

100

C₄

#14555510

yes

—

101

C₂

—

102

C₂

C₂×#14555509

yes

characteristic

103

C₂

C₂×#14555510

yes

characteristic

104

C₂

#17984135

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂⁴
Fitting	C₄×D₄×C₁₃₇	C₂
hypercenter	C₄×D₄	D₁₃₇
hypocenter	C₁₃₇	#556
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₇

commutator, socle

C₂²×C₁₃₇

C₂⁴

Fitting

C₄×D₄×C₁₃₇

C₂

hypercenter

C₄×D₄

D₁₃₇

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #17984088	C₂	semidirect
2. #17984081	C₂	semidirect
4. #17984067	C₂	semidirect
5. C₄×Dic₂₇₄	C₂	semidirect
6. #17984053	C₂	semidirect
7. #17984047	C₂	semidirect
52. C₁₃₇	#556	semidirect

H ≅ G/N

type

1. #17984088

C₂

semidirect

2. #17984081

C₂

semidirect

4. #17984067

C₂

semidirect

5. C₄×Dic₂₇₄

C₂

semidirect

6. #17984053

C₂

semidirect

7. #17984047

C₂

semidirect

52. C₁₃₇

#556

semidirect

Subgroups — #22578123

Conjugacy classes of subgroups

Complements