Subgroups — C₂×Q₈×Dic₃₆₇ #37554652

Order 23488.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₃₆₇	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊Dic₃₆₇	C₂	1	3	yes	maximal
3	C₂×C₄×Dic₃₆₇	C₂	1	3	yes	maximal
4	C₂²×Q₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₆₇	C₂²	1	3	yes	—
6	C₂²×Dic₃₆₇	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₃₆₇	C₂²	1	12	yes	—
8	C₄×Dic₃₆₇	C₂²	1	12	yes	—
9	C₂×Q₈×C₃₆₇	C₄	1	8	yes	complemented
10	C₂×Q₈×C₃₆₇	C₂²	1	4	yes	—
11	C₂²×C₄×C₃₆₇	C₂²	1	3	yes	—
12	C₂×Dic₃₆₇	—	2	6	no	—
13	C₂×Dic₃₆₇	Q₈	1	4	yes	direct product factor
14	C₂×Dic₃₆₇	C₂³	1	6	yes	—
15	C₂×Dic₃₆₇	C₂³	1	2	yes	—
16	C₂³×C₃₆₇	C₂³	1	1	yes	characteristic
17	Q₈×C₃₆₇	C₂×C₄	1	16	yes	complemented
18	C₂×C₄×C₃₆₇	C₂×C₄	1	12	yes	—
19	C₂×C₄×C₃₆₇	C₂³	1	6	yes	—
20	C₂²×C₃₆₇	C₂²×C₄	1	2	yes	—
21	C₂²×C₃₆₇	C₂×Q₈	1	2	yes	—
22	C₂²×C₃₆₇	#41	1	2	yes	—
23	C₂²×C₃₆₇	C₂⁴	1	1	yes	characteristic
24	C₄×C₃₆₇	C₂²×C₄	1	12	yes	—
25	Dic₃₆₇	—	2	6	no	—
26	Dic₃₆₇	C₂×Q₈	1	4	yes	direct product factor
27	C₂×C₃₆₇	C₄×Q₈	1	4	yes	complemented
28	C₂×C₃₆₇	C₂³×C₄	1	1	yes	characteristic
29	C₂×C₃₆₇	C₂²×Q₈	1	1	yes	characteristic
30	C₂×C₃₆₇	C₂×#41	1	1	yes	characteristic
31	C₃₆₇	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
32	C₂×C₄×Q₈	—	367	1	no	maximal, Sylow
33	C₂²×Q₈	D₃₆₇	1	1	yes	characteristic
34	C₄×Q₈	—	367	8	no	—
35	C₂×C₄⋊C₄	—	367	3	no	—
36	C₂×C₄²	—	367	3	no	—
37	C₂×Q₈	Dic₃₆₇	1	8	yes	direct product factor
38	C₂×Q₈	C₂×D₃₆₇	1	4	yes	—
39	C₂²×C₄	—	367	1	no	—
40	C₂²×C₄	—	367	3	no	—
41	C₂²×C₄	C₂×D₃₆₇	1	3	yes	—
42	C₄⋊C₄	—	367	12	no	—
43	C₄²	—	367	12	no	—
44	C₂³	C₂²×D₃₆₇	1	1	yes	characteristic
45	Q₈	C₂×Dic₃₆₇	1	16	yes	direct product factor
46	C₂×C₄	—	367	4	no	—
47	C₂×C₄	—	367	2	no	—
48	C₂×C₄	—	367	6	no	—
49	C₂×C₄	—	734	6	no	—
50	C₂×C₄	C₂×Dic₃₆₇	1	12	yes	—
51	C₂×C₄	C₂²×D₃₆₇	1	6	yes	—
52	C₂²	C₂²×Dic₃₆₇	1	2	yes	—
53	C₂²	Q₈×D₃₆₇	1	2	yes	—
54	C₂²	#19154578	1	2	yes	—
55	C₂²	C₂³×D₃₆₇	1	1	yes	characteristic
56	C₄	—	367	4	no	—
57	C₄	—	734	6	no	—
58	C₄	C₂²×Dic₃₆₇	1	12	yes	—
59	C₂	Q₈×Dic₃₆₇	1	4	yes	direct product factor
60	C₂	C₂³×Dic₃₆₇	1	1	yes	characteristic
61	C₂	C₂×Q₈×D₃₆₇	1	1	yes	characteristic
62	C₂	C₂×#19154578	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃₆₇
commutator	C₂×C₃₆₇	C₂³×C₄
Frattini	C₂²	C₂³×D₃₆₇
Fitting	C₂²×Q₈×C₃₆₇	C₂
socle, prime radical	C₂³×C₃₆₇	C₂³
cosocle	C₂²×C₃₆₇	C₂⁴
hypercenter	C₂²×Q₈	D₃₆₇
hypocenter	C₃₆₇	C₂×C₄×Q₈
LCS	G ▷ C₂×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. Q₈×Dic₃₆₇	C₂	direct (N × H)
9. C₂×Q₈×C₃₆₇	C₄	semidirect
13. C₂×Dic₃₆₇	Q₈	direct (N × H)
17. Q₈×C₃₆₇	C₂×C₄	semidirect
26. Dic₃₆₇	C₂×Q₈	direct (N × H)
27. C₂×C₃₆₇	C₄×Q₈	semidirect
31. C₃₆₇	C₂×C₄×Q₈	semidirect
37. C₂×Q₈	Dic₃₆₇	direct (N × H)
45. Q₈	C₂×Dic₃₆₇	direct (N × H)
59. C₂	Q₈×Dic₃₆₇	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈×Dic₃₆₇	2
C₂×C₄⋊Dic₃₆₇	2
C₂×C₄×Dic₃₆₇	2
C₂²×Q₈×C₃₆₇	2
C₂×C₄×Q₈	367

Subgroups — C2×Q8×Dic367 #37554652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×Dic₃₆₇ #37554652