Group #20295979 — subgroups

Subgroups — (C₂×#89137)⋊Q₈ #20295979

Order 6976.

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⋊#89137	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#89137	C₂	1	1	yes	maximal, characteristic
3	(C₂×#89137)⋊C₄	C₂	1	4	yes	maximal
4	C₂×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#89137	C₂²	1	2	yes	—
6	C₂²×#89137	C₂²	1	2	yes	—
7	C₄⋊#89137	—	2	2	no	—
8	C₄×#89137	—	2	2	no	—
9	C₂×Dic₂₁₈	C₄	1	1	yes	characteristic, complemented
10	C₂×Dic₂₁₈	C₄	1	2	yes	complemented
11	Q₈×D₁₀₉	—	2	2	no	—
12	Q₈×D₁₀₉	—	2	2	no	—
13	C₂×C₄×D₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
16	C₂²×D₁₀₉	C₂³	1	1	yes	characteristic
17	C₂×#89137	—	2	2	no	—
18	C₂×#89137	—	2	2	no	—
19	C₂×#89137	—	2	2	no	—
20	C₂×#89137	—	2	4	no	—
21	C₂×#89137	Q₈	1	4	yes	complemented
22	Q₈×C₁₀₉	—	2	2	no	—
23	C₂×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₁₀₉	C₂×C₄	1	2	yes	—
26	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄×D₁₀₉	—	2	2	no	—
28	C₄×D₁₀₉	—	2	2	no	—
29	C₄×D₁₀₉	D₄	1	2	yes	—
30	C₄×D₁₀₉	D₄	1	2	yes	—
31	Dic₂₁₈	—	2	2	no	—
32	Dic₂₁₈	—	2	4	no	—
33	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×D₁₀₉	C₂×D₄	1	1	yes	characteristic
35	C₂×D₁₀₉	C₂×D₄	1	1	yes	characteristic
36	C₂×D₁₀₉	C₂×Q₈	1	2	yes	—
37	C₂×D₁₀₉	#41	1	2	yes	—
38	#89137	—	2	4	no	—
39	#89137	—	4	2	no	—
40	C₄×C₁₀₉	—	2	2	no	—
41	C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
42	Dic₁₀₉	—	2	2	no	—
43	Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
44	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₁₀₉	C₄×Q₈	1	2	yes	—
46	D₁₀₉	C₂²⋊Q₈	1	2	yes	—
47	D₁₀₉	#124	1	1	yes	characteristic
48	D₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
49	C₁₀₉	#391	1	1	yes	characteristic, complemented, Sylow
50	#391	—	109	1	no	maximal, Sylow
51	C₂²×Q₈	—	109	1	no	—
52	C₂×C₄⋊C₄	—	109	1	no	—
53	C₂×C₄²	—	109	1	no	—
54	#95	—	109	4	no	—
55	C₂×Q₈	—	109	2	no	—
56	C₂×Q₈	—	109	1	no	—
57	C₂×Q₈	—	218	2	no	—
58	C₂×Q₈	—	218	2	no	—
59	C₂×Q₈	#89137	1	1	yes	characteristic, complemented
60	C₂²×C₄	—	109	1	no	—
61	C₂²×C₄	—	109	2	no	—
62	C₂²×C₄	—	109	2	no	—
63	C₂²×C₄	—	109	2	no	—
64	C₄⋊C₄	—	218	2	no	—
65	C₄²	—	218	2	no	—
66	C₂³	—	109	1	no	—
67	Q₈	—	2	2	no	—
68	Q₈	—	218	4	no	—
69	Q₈	—	218	2	no	—
70	C₂×C₄	—	109	1	no	—
71	C₂×C₄	—	109	2	no	—
72	C₂×C₄	—	109	2	no	—
73	C₂×C₄	—	109	4	no	—
74	C₂×C₄	—	109	2	no	—
75	C₂×C₄	—	218	2	no	—
76	C₂×C₄	—	218	2	no	—
77	C₂×C₄	—	218	2	no	—
78	C₂×C₄	—	218	4	no	—
79	C₂×C₄	—	218	2	no	—
80	C₂×C₄	—	218	2	no	—
81	C₂×C₄	C₂×#89137	1	2	yes	—
82	C₂×C₄	C₂×#89137	1	1	yes	characteristic
83	C₂²	—	109	1	no	—
84	C₂²	—	109	1	no	—
85	C₂²	—	109	2	no	—
86	C₂²	—	109	2	no	—
87	C₂²	C₂²×#89137	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	109	2	no	—
90	C₄	—	436	2	no	—
91	C₄	—	218	4	no	—
92	C₄	—	218	2	no	—
93	C₄	C₂²⋊#89137	1	2	yes	—
94	C₂	—	109	1	no	—
95	C₂	—	109	2	no	—
96	C₂	—	109	1	no	—
97	C₂	C₂×C₂²⋊#89137	1	1	yes	characteristic
98	C₂	Q₈×#89137	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#89137

C₂

yes

maximal, characteristic

C₂×C₄×#89137

C₂

yes

maximal, characteristic

(C₂×#89137)⋊C₄

C₂

yes

maximal

C₂×Q₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂²×#89137

C₂²

yes

—

C₂²×#89137

C₂²

yes

—

C₄⋊#89137

—

C₄×#89137

—

C₂×Dic₂₁₈

C₄

yes

characteristic, complemented

C₂×Dic₂₁₈

C₄

yes

complemented

Q₈×D₁₀₉

—

Q₈×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂²

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

C₂×Q₈×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×D₁₀₉

C₂³

yes

—

—

—

—

Q₈

yes

complemented

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₄

yes

—

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

D₄

yes

—

C₄×D₁₀₉

D₄

yes

—

Dic₂₁₈

—

Dic₂₁₈

—

C₂²×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×D₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

C₂×Q₈

yes

—

C₂×D₁₀₉

#41

yes

—

#89137

—

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊C₄

yes

—

Dic₁₀₉

—

Dic₁₀₉

C₂²⋊C₄

yes

—

C₂×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₀₉

C₄×Q₈

yes

—

D₁₀₉

C₂²⋊Q₈

yes

—

D₁₀₉

#124

yes

characteristic

D₁₀₉

C₄⋊Q₈

yes

characteristic

C₁₀₉

#391

yes

characteristic, complemented, Sylow

#391

—

109

maximal, Sylow

C₂²×Q₈

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

—

109

—

#95

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

#89137

yes

characteristic, complemented

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₄⋊C₄

—

218

—

C₄²

—

218

—

C₂³

—

109

—

Q₈

—

Q₈

—

218

—

Q₈

—

218

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂×#89137

yes

—

C₂×C₄

C₂×#89137

yes

characteristic

C₂²

—

109

—

C₂²

—

109

—

C₂²

—

109

—

C₂²

—

109

—

C₂²

C₂²×#89137

yes

characteristic

C₄

—

C₄

—

109

—

C₄

—

436

—

C₄

—

218

—

C₄

—

218

—

C₄

C₂²⋊#89137

yes

—

C₂

—

109

—

C₂

—

109

—

C₂

—

109

—

C₂

C₂×C₂²⋊#89137

yes

characteristic

C₂

Q₈×#89137

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#89137
commutator, socle	C₂²×C₁₀₉	C₂²×C₄
Fitting	C₂×Q₈×C₁₀₉	C₄
hypercenter	C₂×Q₈	#89137
hypocenter	C₁₀₉	#391
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂²×#89137

commutator, socle

C₂²×C₁₀₉

C₂²×C₄

Fitting

C₂×Q₈×C₁₀₉

C₄

C₂×Q₈

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ C₂×Q₈

N	H ≅ G/N	type
9. C₂×Dic₂₁₈	C₄	semidirect
10. C₂×Dic₂₁₈	C₄	semidirect
15. C₂×Q₈×C₁₀₉	C₄	semidirect
21. C₂×#89137	Q₈	semidirect
49. C₁₀₉	#391	semidirect
59. C₂×Q₈	#89137	semidirect

H ≅ G/N

type

9. C₂×Dic₂₁₈

C₄

semidirect

10. C₂×Dic₂₁₈

C₄

semidirect

15. C₂×Q₈×C₁₀₉

C₄

semidirect

21. C₂×#89137

Q₈

semidirect

49. C₁₀₉

#391

semidirect

59. C₂×Q₈

#89137

semidirect

Subgroups — (C₂×#89137)⋊Q₈ #20295979

Conjugacy classes of subgroups

Complements

Subgroups — (C2×#89137)⋊Q8 #20295979

Conjugacy classes of subgroups

Complements

Subgroups — (C₂×#89137)⋊Q₈ #20295979