Group #22578067 — subgroups

Subgroups — Q₈×C₂²⋊D₁₃₇ #22578067

Order 8768.

280 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 123 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	1	yes	maximal, characteristic, complemented
2	#17984086	C₂	1	3	yes	maximal, complemented
3	#17984067	C₂	1	3	yes	maximal, complemented
4	#17984064	C₂	1	3	yes	maximal, complemented
5	Q₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₂²⋊D₁₃₇	C₂	1	3	yes	maximal
7	C₂²×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₇₄	C₂²	1	3	yes	—
9	Q₈×D₁₃₇	—	2	4	no	—
10	C₂×C₂²⋊D₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₃₇	C₂²	1	3	yes	—
12	C₄⋊Dic₁₃₇	C₂²	1	3	yes	—
13	#14555489	C₂²	1	3	yes	—
14	#14555489	C₂²	1	6	yes	—
15	C₂²⋊Dic₁₃₇	C₂²	1	3	yes	—
16	#14555487	C₂²	1	3	yes	—
17	C₄×Dic₁₃₇	C₂²	1	3	yes	—
18	C₂×Q₈×C₁₃₇	—	2	4	no	—
19	C₂×Q₈×C₁₃₇	C₂²	1	3	yes	—
20	C₂×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₃₇	C₂²	1	3	yes	—
22	C₂²×D₁₃₇	C₂³	1	1	yes	characteristic
23	Dic₂₇₄	—	2	6	no	—
24	C₂²⋊D₁₃₇	Q₈	1	4	yes	direct product factor
25	C₂×Dic₁₃₇	C₂³	1	3	yes	—
26	C₂×Dic₁₃₇	C₂³	1	3	yes	—
27	C₂×Dic₁₃₇	C₂³	1	1	yes	characteristic
28	C₄×D₁₃₇	—	2	6	no	—
29	C₂³×C₁₃₇	C₂³	1	1	yes	characteristic
30	Q₈×C₁₃₇	—	2	6	no	—
31	Q₈×C₁₃₇	D₄	1	4	yes	complemented
32	C₂×C₄×C₁₃₇	—	2	6	no	—
33	C₂×C₄×C₁₃₇	C₂³	1	3	yes	—
34	C₂×C₄×C₁₃₇	C₂³	1	3	yes	—
35	C₂²×C₁₃₇	—	2	1	no	—
36	C₂²×C₁₃₇	—	2	1	no	—
37	C₂²×C₁₃₇	C₂×Q₈	1	2	yes	complemented
38	C₂²×C₁₃₇	C₂⁴	1	1	yes	characteristic
39	C₂×D₁₃₇	—	2	1	no	—
40	C₂×D₁₃₇	—	2	1	no	—
41	C₂×D₁₃₇	C₂×Q₈	1	2	yes	complemented
42	C₄×C₁₃₇	—	2	3	no	—
43	C₄×C₁₃₇	C₂×D₄	1	6	yes	—
44	Dic₁₃₇	—	2	3	no	—
45	Dic₁₃₇	—	2	3	no	—
46	Dic₁₃₇	C₂×Q₈	1	2	yes	complemented
47	C₂×C₁₃₇	—	2	2	no	—
48	C₂×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
49	C₂×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
50	C₂×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
51	D₁₃₇	—	2	2	no	—
52	C₁₃₇	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
53	D₄×Q₈	—	137	1	no	maximal, Sylow
54	C₂²×Q₈	—	137	1	no	—
55	C₂²×Q₈	D₁₃₇	1	1	yes	characteristic, complemented
56	C₄⋊Q₈	—	137	3	no	—
57	C₂²⋊Q₈	—	137	3	no	—
58	C₂²⋊Q₈	—	137	3	no	—
59	C₄×Q₈	—	137	1	no	—
60	C₄×D₄	—	137	3	no	—
61	C₂×Q₈	—	2	4	no	—
62	C₂×Q₈	—	274	4	no	—
63	C₂×Q₈	—	137	3	no	—
64	C₂×Q₈	C₂×D₁₃₇	1	3	yes	—
65	C₂×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
66	C₂×D₄	—	137	1	no	—
67	C₂²×C₄	—	137	3	no	—
68	C₂²×C₄	C₂×D₁₃₇	1	3	yes	—
69	C₄⋊C₄	—	137	3	no	—
70	C₄⋊C₄	—	137	6	no	—
71	C₄⋊C₄	—	137	3	no	—
72	C₂²⋊C₄	—	137	3	no	—
73	C₂²⋊C₄	—	137	3	no	—
74	C₄²	—	137	3	no	—
75	C₂³	—	137	1	no	—
76	C₂³	C₂²×D₁₃₇	1	1	yes	characteristic
77	Q₈	—	2	6	no	—
78	Q₈	—	274	6	no	—
79	Q₈	C₂²⋊D₁₃₇	1	4	yes	direct product factor
80	D₄	—	137	4	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	274	6	no	—
83	C₂×C₄	—	137	3	no	—
84	C₂×C₄	—	137	3	no	—
85	C₂×C₄	—	137	1	no	—
86	C₂×C₄	C₂²×D₁₃₇	1	3	yes	—
87	C₂×C₄	C₂²×D₁₃₇	1	3	yes	—
88	C₂²	—	2	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	274	1	no	—
91	C₂²	—	274	1	no	—
92	C₂²	—	137	2	no	—
93	C₂²	Q₈×D₁₃₇	1	2	yes	complemented
94	C₂²	C₂³×D₁₃₇	1	1	yes	characteristic
95	C₄	—	2	3	no	—
96	C₄	—	274	3	no	—
97	C₄	—	274	3	no	—
98	C₄	—	137	2	no	—
99	C₄	C₂×C₂²⋊D₁₃₇	1	6	yes	—
100	C₂	—	2	2	no	—
101	C₂	—	274	2	no	—
102	C₂	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
103	C₂	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
104	C₂	#17984136	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₁₃₇

C₂

yes

maximal, characteristic, complemented

#17984086

C₂

yes

maximal, complemented

#17984067

C₂

yes

maximal, complemented

#17984064

C₂

yes

maximal, complemented

Q₈×Dic₁₃₇

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₃₇

C₂

yes

maximal

C₂²×Q₈×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₇₄

C₂²

yes

—

Q₈×D₁₃₇

—

C₂×C₂²⋊D₁₃₇

C₂²

yes

characteristic

C₂×C₄×D₁₃₇

C₂²

yes

—

C₄⋊Dic₁₃₇

C₂²

yes

—

#14555489

C₂²

yes

—

#14555489

C₂²

yes

—

C₂²⋊Dic₁₃₇

C₂²

yes

—

#14555487

C₂²

yes

—

C₄×Dic₁₃₇

C₂²

yes

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

C₂²

yes

—

C₂×Q₈×C₁₃₇

C₂²

yes

characteristic

C₂²×C₄×C₁₃₇

C₂²

yes

—

C₂²×D₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

—

C₂²⋊D₁₃₇

Q₈

yes

direct product factor

C₂×Dic₁₃₇

C₂³

yes

—

C₂×Dic₁₃₇

C₂³

yes

—

C₂×Dic₁₃₇

C₂³

yes

characteristic

C₄×D₁₃₇

—

C₂³×C₁₃₇

C₂³

yes

characteristic

Q₈×C₁₃₇

—

Q₈×C₁₃₇

D₄

yes

complemented

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂³

yes

—

C₂×C₄×C₁₃₇

C₂³

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×Q₈

yes

complemented

C₂²×C₁₃₇

C₂⁴

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

C₂×Q₈

yes

complemented

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×D₄

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

C₂×Q₈

yes

complemented

C₂×C₁₃₇

—

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

—

C₂²×Q₈

D₁₃₇

yes

characteristic, complemented

C₄⋊Q₈

—

137

—

C₂²⋊Q₈

—

137

—

C₂²⋊Q₈

—

137

—

C₄×Q₈

—

137

—

C₄×D₄

—

137

—

C₂×Q₈

—

C₂×Q₈

—

274

—

C₂×Q₈

—

137

—

C₂×Q₈

C₂×D₁₃₇

yes

—

C₂×Q₈

C₂×D₁₃₇

yes

characteristic

C₂×D₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂×D₁₃₇

yes

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₄²

—

137

—

C₂³

—

137

—

C₂³

C₂²×D₁₃₇

yes

characteristic

Q₈

—

Q₈

—

274

—

Q₈

C₂²⋊D₁₃₇

yes

direct product factor

D₄

—

137

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂²×D₁₃₇

yes

—

C₂×C₄

C₂²×D₁₃₇

yes

—

C₂²

—

C₂²

—

C₂²

—

274

—

C₂²

—

274

—

C₂²

—

137

—

C₂²

Q₈×D₁₃₇

yes

complemented

C₂²

C₂³×D₁₃₇

yes

characteristic

C₄

—

C₄

—

274

—

C₄

—

274

—

C₄

—

137

—

C₄

C₂×C₂²⋊D₁₃₇

yes

—

100

C₂

—

101

C₂

—

274

—

102

C₂

C₂²×C₂²⋊D₁₃₇

yes

characteristic

103

C₂

C₂×Q₈×D₁₃₇

yes

characteristic

104

C₂

#17984136

yes

characteristic

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

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₇

commutator, socle

C₂²×C₁₃₇

C₂⁴

Fitting

C₂²×Q₈×C₁₃₇

C₂

hypercenter

C₂²×Q₈

D₁₃₇

hypocenter

C₁₃₇

D₄×Q₈

LCS

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
1. C₂×Q₈×D₁₃₇	C₂	semidirect
2. #17984086	C₂	semidirect
3. #17984067	C₂	semidirect
4. #17984064	C₂	semidirect
5. Q₈×Dic₁₃₇	C₂	semidirect
7. C₂²×Q₈×C₁₃₇	C₂	semidirect
24. C₂²⋊D₁₃₇	Q₈	direct (N × H)
31. Q₈×C₁₃₇	D₄	semidirect
37. C₂²×C₁₃₇	C₂×Q₈	semidirect
41. C₂×D₁₃₇	C₂×Q₈	semidirect
46. Dic₁₃₇	C₂×Q₈	semidirect
52. C₁₃₇	D₄×Q₈	semidirect
55. C₂²×Q₈	D₁₃₇	semidirect
79. Q₈	C₂²⋊D₁₃₇	direct (N × H)
93. C₂²	Q₈×D₁₃₇	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₁₃₇

C₂

semidirect

2. #17984086

C₂

semidirect

3. #17984067

C₂

semidirect

4. #17984064

C₂

semidirect

5. Q₈×Dic₁₃₇

C₂

semidirect

7. C₂²×Q₈×C₁₃₇

C₂

semidirect

24. C₂²⋊D₁₃₇

Q₈

direct (N × H)

31. Q₈×C₁₃₇

D₄

semidirect

37. C₂²×C₁₃₇

C₂×Q₈

semidirect

41. C₂×D₁₃₇

C₂×Q₈

semidirect

46. Dic₁₃₇

C₂×Q₈

semidirect

52. C₁₃₇

D₄×Q₈

semidirect

55. C₂²×Q₈

D₁₃₇

semidirect

79. Q₈

C₂²⋊D₁₃₇

direct (N × H)

93. C₂²

Q₈×D₁₃₇

semidirect

Subgroups — Q₈×C₂²⋊D₁₃₇ #22578067

Conjugacy classes of subgroups

Complements

Subgroups — Q8×C22⋊D137 #22578067

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×C₂²⋊D₁₃₇ #22578067