Subgroups — #557×C₃₆₇ #37553875

Order 23488.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₃₆₇	C₂	1	3	yes	maximal
2	#125×C₃₆₇	C₂	1	6	yes	maximal
3	C₄×Q₈×C₃₆₇	C₂	1	2	yes	maximal
4	C₄×Q₈×C₃₆₇	C₂	1	3	yes	maximal
5	C₄×Q₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₃₆₇	C₂²	1	3	yes	—
7	C₂×Q₈×C₃₆₇	C₂²	1	1	yes	characteristic
8	C₄⋊C₄×C₃₆₇	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₃₆₇	C₂²	1	6	yes	—
10	C₄⋊C₄×C₃₆₇	C₂²	1	3	yes	—
11	C₄⋊C₄×C₃₆₇	C₂²	1	6	yes	—
12	C₄⋊C₄×C₃₆₇	C₂²	1	6	yes	—
13	C₄²×C₃₆₇	C₂²	1	6	yes	—
14	C₄²×C₃₆₇	C₂²	1	3	yes	—
15	Q₈×C₃₆₇	—	2	6	no	—
16	Q₈×C₃₆₇	Q₈	1	4	yes	complemented
17	C₂×C₄×C₃₆₇	C₂³	1	2	yes	—
18	C₂×C₄×C₃₆₇	C₂³	1	3	yes	—
19	C₂×C₄×C₃₆₇	C₂³	1	6	yes	—
20	C₂×C₄×C₃₆₇	C₂³	1	3	yes	—
21	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
22	C₂²×C₃₆₇	C₂⁴	1	1	yes	characteristic
23	C₄×C₃₆₇	—	2	6	no	—
24	C₄×C₃₆₇	—	2	2	no	—
25	C₄×C₃₆₇	—	2	3	no	—
26	C₄×C₃₆₇	C₂×Q₈	1	6	yes	—
27	C₄×C₃₆₇	#41	1	2	yes	—
28	C₂×C₃₆₇	C₂²×Q₈	1	1	yes	characteristic
29	C₂×C₃₆₇	C₂×#41	1	1	yes	characteristic
30	C₂×C₃₆₇	2₋¹⁺⁴	1	1	yes	characteristic
31	C₃₆₇	#557	1	1	yes	characteristic, direct product factor, Sylow
32	#557	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₄⋊Q₈	C₂×C₃₆₇	1	3	yes	—
34	#125	C₂×C₃₆₇	1	6	yes	—
35	C₄×Q₈	C₂×C₃₆₇	1	2	yes	—
36	C₄×Q₈	C₂×C₃₆₇	1	3	yes	—
37	C₄×Q₈	C₂×C₃₆₇	1	1	yes	characteristic
38	C₂×Q₈	C₂²×C₃₆₇	1	3	yes	—
39	C₂×Q₈	C₂²×C₃₆₇	1	1	yes	characteristic
40	C₄⋊C₄	C₂²×C₃₆₇	1	1	yes	characteristic
41	C₄⋊C₄	C₂²×C₃₆₇	1	6	yes	—
42	C₄⋊C₄	C₂²×C₃₆₇	1	3	yes	—
43	C₄⋊C₄	C₂²×C₃₆₇	1	6	yes	—
44	C₄⋊C₄	C₂²×C₃₆₇	1	6	yes	—
45	C₄²	C₂²×C₃₆₇	1	6	yes	—
46	C₄²	C₂²×C₃₆₇	1	3	yes	—
47	Q₈	—	2	6	no	—
48	Q₈	Q₈×C₃₆₇	1	4	yes	complemented
49	C₂×C₄	C₂³×C₃₆₇	1	2	yes	—
50	C₂×C₄	C₂³×C₃₆₇	1	3	yes	—
51	C₂×C₄	C₂³×C₃₆₇	1	6	yes	—
52	C₂×C₄	C₂³×C₃₆₇	1	3	yes	—
53	C₂×C₄	C₂³×C₃₆₇	1	1	yes	characteristic
54	C₂²	C₂⁴×C₃₆₇	1	1	yes	characteristic
55	C₄	—	2	6	no	—
56	C₄	—	2	2	no	—
57	C₄	—	2	3	no	—
58	C₄	C₂×Q₈×C₃₆₇	1	6	yes	—
59	C₄	#41×C₃₆₇	1	2	yes	—
60	C₂	C₂²×Q₈×C₃₆₇	1	1	yes	characteristic
61	C₂	C₂×#41×C₃₆₇	1	1	yes	characteristic
62	C₂	2₋¹⁺⁴×C₃₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃₆₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃₆₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
16. Q₈×C₃₆₇	Q₈	semidirect
31. C₃₆₇	#557	direct (N × H)
32. #557	C₃₆₇	direct (N × H)
48. Q₈	Q₈×C₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×C₃₆₇	2
#125×C₃₆₇	2
C₄×Q₈×C₃₆₇	2
#557	367

Subgroups — #557×C367 #37553875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #557×C₃₆₇ #37553875