Subgroups — Q₈×Dic₇₃₄ #37554917

Order 23488.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26044261	C₂	1	3	yes	maximal
2	#26044260	C₂	1	6	yes	maximal
3	Q₈×Dic₃₆₇	C₂	1	2	yes	maximal
4	C₄×Dic₇₃₄	C₂	1	3	yes	maximal
5	C₄×Q₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₃₄	C₂²	1	6	yes	—
7	C₂×Dic₇₃₄	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₃₆₇	C₂²	1	3	yes	—
9	C₄⋊Dic₃₆₇	C₂²	1	6	yes	—
10	Dic₃₆₇⋊C₄	C₂²	1	6	yes	—
11	C₄×Dic₃₆₇	C₂²	1	6	yes	—
12	C₂×Q₈×C₃₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₃₆₇	C₂²	1	3	yes	—
14	C₄²×C₃₆₇	C₂²	1	3	yes	—
15	Dic₇₃₄	—	4	6	no	—
16	Dic₇₃₄	Q₈	1	4	yes	direct product factor
17	C₂×Dic₃₆₇	C₂³	1	6	yes	—
18	C₂×Dic₃₆₇	C₂³	1	2	yes	—
19	Q₈×C₃₆₇	Q₈	1	4	yes	complemented
20	C₂×C₄×C₃₆₇	C₂³	1	3	yes	—
21	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₃₆₇	C₂³	1	3	yes	—
23	C₂²×C₃₆₇	C₂⁴	1	1	yes	characteristic
24	C₄×C₃₆₇	—	2	3	no	—
25	C₄×C₃₆₇	C₂×Q₈	1	2	yes	—
26	C₄×C₃₆₇	C₂×Q₈	1	6	yes	—
27	Dic₃₆₇	—	2	6	no	—
28	Dic₃₆₇	C₂×Q₈	1	4	yes	—
29	C₂×C₃₆₇	C₂²×Q₈	1	1	yes	characteristic
30	C₂×C₃₆₇	C₂²×Q₈	1	1	yes	characteristic
31	C₂×C₃₆₇	H(5,𝔽₂)	1	1	yes	characteristic
32	C₃₆₇	Q₈²	1	1	yes	characteristic, complemented, Sylow
33	Q₈²	—	367	1	no	maximal, Sylow
34	C₄⋊Q₈	—	367	6	no	—
35	C₄⋊Q₈	—	367	3	no	—
36	C₄×Q₈	—	367	2	no	—
37	C₄×Q₈	—	367	3	no	—
38	C₄×Q₈	D₃₆₇	1	1	yes	characteristic
39	C₂×Q₈	—	367	1	no	—
40	C₂×Q₈	—	367	6	no	—
41	C₂×Q₈	C₂×D₃₆₇	1	1	yes	characteristic
42	C₄⋊C₄	—	367	6	no	—
43	C₄⋊C₄	—	367	3	no	—
44	C₄⋊C₄	—	367	6	no	—
45	C₄⋊C₄	C₂×D₃₆₇	1	3	yes	—
46	C₄²	—	367	6	no	—
47	C₄²	C₂×D₃₆₇	1	3	yes	—
48	Q₈	—	367	4	no	—
49	Q₈	—	1468	6	no	—
50	Q₈	Dic₇₃₄	1	4	yes	direct product factor
51	C₂×C₄	—	367	2	no	—
52	C₂×C₄	—	367	6	no	—
53	C₂×C₄	C₂²×D₃₆₇	1	3	yes	—
54	C₂×C₄	C₂²×D₃₆₇	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₃₆₇	1	3	yes	—
56	C₂²	C₂³×D₃₆₇	1	1	yes	characteristic
57	C₄	—	2	3	no	—
58	C₄	—	367	4	no	—
59	C₄	—	734	6	no	—
60	C₄	Q₈×D₃₆₇	1	2	yes	—
61	C₄	C₂×Dic₇₃₄	1	6	yes	—
62	C₂	C₂×Q₈×D₃₆₇	1	1	yes	characteristic
63	C₂	C₂²×Dic₇₃₄	1	1	yes	characteristic
64	C₂	#26044307	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₆₇
commutator, socle, prime radical, cosocle	C₂²×C₃₆₇	C₂⁴
Fitting	C₄×Q₈×C₃₆₇	C₂
hypercenter	C₄×Q₈	D₃₆₇
hypocenter	C₃₆₇	Q₈²
LCS	G ▷ C₂²×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
16. Dic₇₃₄	Q₈	direct (N × H)
19. Q₈×C₃₆₇	Q₈	semidirect
32. C₃₆₇	Q₈²	semidirect
50. Q₈	Dic₇₃₄	direct (N × H)

Maximal subgroups

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

subgroup	index
#26044261	2
#26044260	2
Q₈×Dic₃₆₇	2
C₄×Dic₇₃₄	2
C₄×Q₈×C₃₆₇	2
Q₈²	367

Subgroups — Q8×Dic734 #37554917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Dic₇₃₄ #37554917