Group #22578356 — subgroups

Subgroups — (C₂×#10587739)⋊Q₈ #22578356

Order 8768.

186 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 64 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×C₄⋊#10587739	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10587739	C₂	1	1	yes	maximal, characteristic
3	(C₂×#10587739)⋊C₄	C₂	1	4	yes	maximal
4	C₂×Q₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10587739	C₂²	1	2	yes	—
6	C₂²×#10587739	C₂²	1	2	yes	—
7	C₄⋊#10587739	—	2	2	no	—
8	C₄×#10587739	—	2	2	no	—
9	C₂×Dic₂₇₄	C₄	1	2	yes	complemented
10	C₂×Dic₂₇₄	C₄	1	1	yes	characteristic, complemented
11	Q₈×D₁₃₇	—	2	2	no	—
12	Q₈×D₁₃₇	—	2	2	no	—
13	C₂×C₄×D₁₃₇	C₂²	1	2	yes	—
14	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₁₃₇	C₄	1	1	yes	characteristic, complemented
16	C₂×#10587739	—	2	2	no	—
17	C₂×#10587739	—	2	2	no	—
18	C₂×#10587739	—	2	2	no	—
19	C₂×#10587739	—	2	4	no	—
20	C₂×#10587739	Q₈	1	4	yes	complemented
21	C₂²×D₁₃₇	C₂³	1	1	yes	characteristic
22	Dic₂₇₄	—	2	2	no	—
23	Dic₂₇₄	—	2	4	no	—
24	C₂×Dic₁₃₇	C₂×C₄	1	2	yes	—
25	C₂×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₄×D₁₃₇	—	2	2	no	—
27	C₄×D₁₃₇	—	2	2	no	—
28	C₄×D₁₃₇	D₄	1	2	yes	—
29	C₄×D₁₃₇	D₄	1	2	yes	—
30	Q₈×C₁₃₇	—	2	2	no	—
31	C₂×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
32	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×D₁₃₇	C₂×D₄	1	1	yes	characteristic
35	C₂×D₁₃₇	C₂×D₄	1	1	yes	characteristic
36	C₂×D₁₃₇	C₂×Q₈	1	2	yes	—
37	C₂×D₁₃₇	#41	1	2	yes	—
38	#10587739	—	2	4	no	—
39	#10587739	—	4	2	no	—
40	C₄×C₁₃₇	—	2	2	no	—
41	C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
42	Dic₁₃₇	—	2	2	no	—
43	Dic₁₃₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₁₃₇	C₄×Q₈	1	2	yes	—
46	D₁₃₇	C₂²⋊Q₈	1	2	yes	—
47	D₁₃₇	#124	1	1	yes	characteristic
48	D₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
49	C₁₃₇	#391	1	1	yes	characteristic, complemented, Sylow
50	#391	—	137	1	no	maximal, Sylow
51	C₂²×Q₈	—	137	1	no	—
52	C₂×C₄⋊C₄	—	137	1	no	—
53	C₂×C₄²	—	137	1	no	—
54	#95	—	137	4	no	—
55	C₂×Q₈	—	274	2	no	—
56	C₂×Q₈	—	274	2	no	—
57	C₂×Q₈	—	137	1	no	—
58	C₂×Q₈	—	137	2	no	—
59	C₂×Q₈	#10587739	1	1	yes	characteristic, complemented
60	C₂²×C₄	—	137	2	no	—
61	C₂²×C₄	—	137	2	no	—
62	C₂²×C₄	—	137	2	no	—
63	C₂²×C₄	—	137	1	no	—
64	C₄⋊C₄	—	274	2	no	—
65	C₄²	—	274	2	no	—
66	C₂³	—	137	1	no	—
67	Q₈	—	2	2	no	—
68	Q₈	—	274	4	no	—
69	Q₈	—	274	2	no	—
70	C₂×C₄	—	274	2	no	—
71	C₂×C₄	—	274	2	no	—
72	C₂×C₄	—	274	2	no	—
73	C₂×C₄	—	274	2	no	—
74	C₂×C₄	—	274	2	no	—
75	C₂×C₄	—	274	4	no	—
76	C₂×C₄	—	137	4	no	—
77	C₂×C₄	—	137	2	no	—
78	C₂×C₄	—	137	1	no	—
79	C₂×C₄	—	137	2	no	—
80	C₂×C₄	—	137	2	no	—
81	C₂×C₄	C₂×#10587739	1	2	yes	—
82	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
83	C₂²	—	137	1	no	—
84	C₂²	—	137	1	no	—
85	C₂²	—	137	2	no	—
86	C₂²	—	137	2	no	—
87	C₂²	C₂²×#10587739	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	274	4	no	—
90	C₄	—	274	2	no	—
91	C₄	—	548	2	no	—
92	C₄	—	137	2	no	—
93	C₄	C₂²⋊#10587739	1	2	yes	—
94	C₂	—	137	2	no	—
95	C₂	—	137	1	no	—
96	C₂	—	137	1	no	—
97	C₂	C₂×C₂²⋊#10587739	1	1	yes	characteristic
98	C₂	Q₈×#10587739	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#10587739

C₂

yes

maximal, characteristic

C₂×C₄×#10587739

C₂

yes

maximal, characteristic

(C₂×#10587739)⋊C₄

C₂

yes

maximal

C₂×Q₈×D₁₃₇

C₂

yes

maximal, characteristic

C₂²×#10587739

C₂²

yes

—

C₂²×#10587739

C₂²

yes

—

C₄⋊#10587739

—

C₄×#10587739

—

C₂×Dic₂₇₄

C₄

yes

complemented

C₂×Dic₂₇₄

C₄

yes

characteristic, complemented

Q₈×D₁₃₇

—

Q₈×D₁₃₇

—

C₂×C₄×D₁₃₇

C₂²

yes

—

C₂×C₄×D₁₃₇

C₂²

yes

characteristic

C₂×Q₈×C₁₃₇

C₄

yes

characteristic, complemented

—

—

—

—

Q₈

yes

complemented

C₂²×D₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

C₂×C₄

yes

—

C₂×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

D₄

yes

—

C₄×D₁₃₇

D₄

yes

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×C₄

yes

—

C₂×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×D₁₃₇

C₂×D₄

yes

characteristic

C₂×D₁₃₇

C₂×D₄

yes

characteristic

C₂×D₁₃₇

C₂×Q₈

yes

—

C₂×D₁₃₇

#41

yes

—

#10587739

—

#10587739

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂²⋊C₄

yes

—

Dic₁₃₇

—

Dic₁₃₇

C₂²⋊C₄

yes

—

C₂×C₁₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₃₇

C₄×Q₈

yes

—

D₁₃₇

C₂²⋊Q₈

yes

—

D₁₃₇

#124

yes

characteristic

D₁₃₇

C₄⋊Q₈

yes

characteristic

C₁₃₇

#391

yes

characteristic, complemented, Sylow

#391

—

137

maximal, Sylow

C₂²×Q₈

—

137

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄²

—

137

—

#95

—

137

—

C₂×Q₈

—

274

—

C₂×Q₈

—

274

—

C₂×Q₈

—

137

—

C₂×Q₈

—

137

—

C₂×Q₈

#10587739

yes

characteristic, complemented

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₄⋊C₄

—

274

—

C₄²

—

274

—

C₂³

—

137

—

Q₈

—

Q₈

—

274

—

Q₈

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂×#10587739

yes

—

C₂×C₄

C₂×#10587739

yes

characteristic

C₂²

—

137

—

C₂²

—

137

—

C₂²

—

137

—

C₂²

—

137

—

C₂²

C₂²×#10587739

yes

characteristic

C₄

—

C₄

—

274

—

C₄

—

274

—

C₄

—

548

—

C₄

—

137

—

C₄

C₂²⋊#10587739

yes

—

C₂

—

137

—

C₂

—

137

—

C₂

—

137

—

C₂

C₂×C₂²⋊#10587739

yes

characteristic

C₂

Q₈×#10587739

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#10587739
commutator, socle	C₂²×C₁₃₇	C₂²×C₄
Fitting	C₂×Q₈×C₁₃₇	C₄
hypercenter	C₂×Q₈	#10587739
hypocenter	C₁₃₇	#391
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂²×#10587739

commutator, socle

C₂²×C₁₃₇

C₂²×C₄

Fitting

C₂×Q₈×C₁₃₇

C₄

C₂×Q₈

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂² ◁ C₂×Q₈

N	H ≅ G/N	type
9. C₂×Dic₂₇₄	C₄	semidirect
10. C₂×Dic₂₇₄	C₄	semidirect
15. C₂×Q₈×C₁₃₇	C₄	semidirect
20. C₂×#10587739	Q₈	semidirect
49. C₁₃₇	#391	semidirect
59. C₂×Q₈	#10587739	semidirect

H ≅ G/N

type

9. C₂×Dic₂₇₄

C₄

semidirect

10. C₂×Dic₂₇₄

C₄

semidirect

15. C₂×Q₈×C₁₃₇

C₄

semidirect

20. C₂×#10587739

Q₈

semidirect

49. C₁₃₇

#391

semidirect

59. C₂×Q₈

#10587739

semidirect

Subgroups — (C₂×#10587739)⋊Q₈ #22578356

Conjugacy classes of subgroups

Complements

Subgroups — (C2×#10587739)⋊Q8 #22578356

Conjugacy classes of subgroups

Complements

Subgroups — (C₂×#10587739)⋊Q₈ #22578356