Group #22578068 — subgroups

Subgroups — #22578068

Order 8768.

280 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 115 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×Q₈×D₁₃₇	C₂	1	2	yes	maximal, complemented
2	#17984087	C₂	1	2	yes	maximal, complemented
3	#17984066	C₂	1	4	yes	maximal, complemented
4	#17984064	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₂₇₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₅₄₈	C₂	1	1	yes	maximal, characteristic
7	#17984050	C₂	1	2	yes	maximal
8	C₄⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₇₄	C₂²	1	4	yes	—
10	C₂×Dic₂₇₄	C₂²	1	1	yes	characteristic
11	Q₈×D₁₃₇	—	2	8	no	—
12	C₂×D₅₄₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₃₇	C₂²	1	4	yes	—
14	C₂×C₄×D₁₃₇	C₂²	1	2	yes	—
15	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
17	#14555489	C₂²	1	2	yes	—
18	#14555489	C₂²	1	4	yes	—
19	#14555487	C₂²	1	4	yes	—
20	#14555487	C₂²	1	2	yes	—
21	C₄×Dic₁₃₇	C₂²	1	2	yes	—
22	C₂×Q₈×C₁₃₇	C₂²	1	2	yes	—
23	C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
24	C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
25	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₂²×D₁₃₇	C₂³	1	2	yes	—
27	Dic₂₇₄	—	2	8	no	—
28	Dic₂₇₄	D₄	1	4	yes	complemented
29	D₅₄₈	Q₈	1	4	yes	complemented
30	C₂×Dic₁₃₇	C₂³	1	2	yes	—
31	C₂×Dic₁₃₇	C₂³	1	2	yes	—
32	C₂×Dic₁₃₇	C₂³	1	2	yes	—
33	C₄×D₁₃₇	—	2	4	no	—
34	C₄×D₁₃₇	—	2	8	no	—
35	Q₈×C₁₃₇	—	2	4	no	—
36	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₁₃₇	C₂³	1	4	yes	—
38	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
40	C₂²×C₁₃₇	C₂⁴	1	1	yes	characteristic
41	C₂×D₁₃₇	—	2	2	no	—
42	C₂×D₁₃₇	—	2	2	no	—
43	C₂×D₁₃₇	C₂×Q₈	1	4	yes	complemented
44	C₄×C₁₃₇	—	2	4	no	—
45	C₄×C₁₃₇	—	2	1	no	—
46	C₄×C₁₃₇	C₂×D₄	1	2	yes	—
47	C₄×C₁₃₇	C₂×Q₈	1	2	yes	complemented
48	Dic₁₃₇	—	2	2	no	—
49	Dic₁₃₇	—	2	2	no	—
50	Dic₁₃₇	C₂×D₄	1	4	yes	—
51	C₂×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
52	C₂×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
53	C₂×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
54	D₁₃₇	—	2	4	no	—
55	C₁₃₇	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
56	D₄×Q₈	—	137	1	no	maximal, Sylow
57	C₂²×Q₈	—	137	2	no	—
58	C₄⋊Q₈	—	137	2	no	—
59	C₄⋊Q₈	D₁₃₇	1	1	yes	characteristic, complemented
60	C₂²⋊Q₈	—	137	2	no	—
61	C₂²⋊Q₈	—	137	4	no	—
62	C₄×Q₈	—	137	1	no	—
63	C₄×D₄	—	137	2	no	—
64	C₄×D₄	—	137	1	no	—
65	C₂×Q₈	—	274	8	no	—
66	C₂×Q₈	—	137	4	no	—
67	C₂×Q₈	—	137	1	no	—
68	C₂×Q₈	C₂×D₁₃₇	1	2	yes	—
69	C₂×D₄	—	137	1	no	—
70	C₂²×C₄	—	137	4	no	—
71	C₂²×C₄	—	137	2	no	—
72	C₄⋊C₄	—	137	2	no	—
73	C₄⋊C₄	—	137	4	no	—
74	C₄⋊C₄	—	137	1	no	—
75	C₄⋊C₄	—	137	1	no	—
76	C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
77	C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
78	C₂²⋊C₄	—	137	4	no	—
79	C₂²⋊C₄	—	137	2	no	—
80	C₄²	—	137	2	no	—
81	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
82	C₂³	—	137	2	no	—
83	Q₈	—	2	4	no	—
84	Q₈	—	274	8	no	—
85	Q₈	—	137	4	no	—
86	D₄	—	137	4	no	—
87	C₂×C₄	—	274	8	no	—
88	C₂×C₄	—	274	4	no	—
89	C₂×C₄	—	137	2	no	—
90	C₂×C₄	—	137	2	no	—
91	C₂×C₄	—	137	2	no	—
92	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
93	C₂×C₄	C₂²×D₁₃₇	1	4	yes	—
94	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
96	C₂²	—	274	2	no	—
97	C₂²	—	274	2	no	—
98	C₂²	—	137	4	no	—
99	C₂²	C₂³×D₁₃₇	1	1	yes	characteristic
100	C₄	—	2	4	no	—
101	C₄	—	2	1	no	—
102	C₄	—	274	2	no	—
103	C₄	—	274	2	no	—
104	C₄	—	137	4	no	—
105	C₄	D₄×D₁₃₇	1	2	yes	—
106	C₄	Q₈×D₁₃₇	1	2	yes	complemented
107	C₂	—	274	4	no	—
108	C₂	C₂×D₄×D₁₃₇	1	1	yes	characteristic
109	C₂	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
110	C₂	#17984136	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₁₃₇

C₂

yes

maximal, complemented

#17984087

C₂

yes

maximal, complemented

#17984066

C₂

yes

maximal, complemented

#17984064

C₂

yes

maximal, complemented

C₄×Dic₂₇₄

C₂

yes

maximal, characteristic, complemented

C₄×D₅₄₈

C₂

yes

maximal, characteristic

#17984050

C₂

yes

maximal

C₄⋊Q₈×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₇₄

C₂²

yes

—

C₂×Dic₂₇₄

C₂²

yes

characteristic

Q₈×D₁₃₇

—

C₂×D₅₄₈

C₂²

yes

characteristic

C₂×C₄×D₁₃₇

C₂²

yes

—

C₂×C₄×D₁₃₇

C₂²

yes

—

C₄⋊Dic₁₃₇

C₂²

yes

characteristic

C₄⋊Dic₁₃₇

C₂²

yes

characteristic

#14555489

C₂²

yes

—

#14555489

C₂²

yes

—

#14555487

C₂²

yes

—

#14555487

C₂²

yes

—

C₄×Dic₁₃₇

C₂²

yes

—

C₂×Q₈×C₁₃₇

C₂²

yes

—

C₄⋊C₄×C₁₃₇

C₂²

yes

—

C₄⋊C₄×C₁₃₇

C₂²

yes

—

C₄²×C₁₃₇

C₂²

yes

characteristic

C₂²×D₁₃₇

C₂³

yes

—

Dic₂₇₄

—

Dic₂₇₄

D₄

yes

complemented

D₅₄₈

Q₈

yes

complemented

C₂×Dic₁₃₇

C₂³

yes

—

C₂×Dic₁₃₇

C₂³

yes

—

C₂×Dic₁₃₇

C₂³

yes

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

Q₈×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

C₂⁴

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

C₂×Q₈

yes

complemented

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×D₄

yes

—

C₄×C₁₃₇

C₂×Q₈

yes

complemented

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

C₂×D₄

yes

—

C₂×C₁₃₇

C₂²×D₄

yes

characteristic

C₂×C₁₃₇

C₂²×Q₈

yes

characteristic

C₂×C₁₃₇

2₋¹⁺⁴

yes

characteristic

D₁₃₇

—

C₁₃₇

D₄×Q₈

yes

characteristic, complemented, Sylow

D₄×Q₈

—

137

maximal, Sylow

C₂²×Q₈

—

137

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

D₁₃₇

yes

characteristic, complemented

C₂²⋊Q₈

—

137

—

C₂²⋊Q₈

—

137

—

C₄×Q₈

—

137

—

C₄×D₄

—

137

—

C₄×D₄

—

137

—

C₂×Q₈

—

274

—

C₂×Q₈

—

137

—

C₂×Q₈

—

137

—

C₂×Q₈

C₂×D₁₃₇

yes

—

C₂×D₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

C₂×D₁₃₇

yes

—

C₄⋊C₄

C₂×D₁₃₇

yes

—

C₂²⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₄²

—

137

—

C₄²

C₂×D₁₃₇

yes

characteristic

C₂³

—

137

—

Q₈

—

Q₈

—

274

—

Q₈

—

137

—

D₄

—

137

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₄

C₂²×D₁₃₇

yes

—

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂²

—

274

—

C₂²

—

274

—

C₂²

—

137

—

C₂²

C₂³×D₁₃₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

274

—

103

C₄

—

274

—

104

C₄

—

137

—

105

C₄

D₄×D₁₃₇

yes

—

106

C₄

Q₈×D₁₃₇

yes

complemented

107

C₂

—

274

—

108

C₂

C₂×D₄×D₁₃₇

yes

characteristic

109

C₂

C₂×Q₈×D₁₃₇

yes

characteristic

110

C₂

#17984136

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂⁴
Fitting	C₄⋊Q₈×C₁₃₇	C₂
hypercenter	C₄⋊Q₈	D₁₃₇
hypocenter	C₁₃₇	D₄×Q₈
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₇

commutator, socle

C₂²×C₁₃₇

C₂⁴

Fitting

C₄⋊Q₈×C₁₃₇

C₂

hypercenter

C₄⋊Q₈

D₁₃₇

hypocenter

C₁₃₇

D₄×Q₈

LCS

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×Q₈×D₁₃₇	C₂	semidirect
2. #17984087	C₂	semidirect
3. #17984066	C₂	semidirect
4. #17984064	C₂	semidirect
5. C₄×Dic₂₇₄	C₂	semidirect
8. C₄⋊Q₈×C₁₃₇	C₂	semidirect
28. Dic₂₇₄	D₄	semidirect
29. D₅₄₈	Q₈	semidirect
43. C₂×D₁₃₇	C₂×Q₈	semidirect
47. C₄×C₁₃₇	C₂×Q₈	semidirect
55. C₁₃₇	D₄×Q₈	semidirect
59. C₄⋊Q₈	D₁₃₇	semidirect
106. C₄	Q₈×D₁₃₇	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₁₃₇

C₂

semidirect

2. #17984087

C₂

semidirect

3. #17984066

C₂

semidirect

4. #17984064

C₂

semidirect

5. C₄×Dic₂₇₄

C₂

semidirect

8. C₄⋊Q₈×C₁₃₇

C₂

semidirect

28. Dic₂₇₄

D₄

semidirect

29. D₅₄₈

Q₈

semidirect

43. C₂×D₁₃₇

C₂×Q₈

semidirect

47. C₄×C₁₃₇

C₂×Q₈

semidirect

55. C₁₃₇

D₄×Q₈

semidirect

59. C₄⋊Q₈

D₁₃₇

semidirect

106. C₄

Q₈×D₁₃₇

semidirect

Subgroups — #22578068

Conjugacy classes of subgroups

Complements