Subgroups — #491×C₃₆₇ #37553809

Order 23488.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic
2	#125×C₃₆₇	C₂	1	1	yes	maximal, characteristic
3	#103×C₃₆₇	C₂	1	4	yes	maximal
4	#97×C₃₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×C₃₆₇	C₂²	1	2	yes	—
6	C₂×C₈×C₃₆₇	C₂²	1	2	yes	—
7	C₄⋊C₄×C₃₆₇	—	2	2	no	—
8	C₄⋊C₄×C₃₆₇	—	2	2	no	—
9	C₄⋊C₄×C₃₆₇	C₂²	1	2	yes	—
10	C₄²×C₃₆₇	C₂²	1	1	yes	characteristic
11	Q₈×C₃₆₇	—	2	4	no	—
12	C₂×C₄×C₃₆₇	—	2	2	no	—
13	C₂×C₄×C₃₆₇	—	2	2	no	—
14	C₂×C₄×C₃₆₇	D₄	1	2	yes	—
15	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
16	C₈×C₃₆₇	—	2	2	no	—
17	C₂²×C₃₆₇	C₂×D₄	1	1	yes	characteristic
18	C₄×C₃₆₇	—	2	2	no	—
19	C₄×C₃₆₇	—	4	2	no	—
20	C₄×C₃₆₇	—	4	2	no	—
21	C₄×C₃₆₇	#41	1	2	yes	—
22	C₂×C₃₆₇	#124	1	1	yes	characteristic
23	C₂×C₃₆₇	#137	1	2	yes	—
24	C₃₆₇	#491	1	1	yes	characteristic, direct product factor, Sylow
25	#491	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₄⋊Q₈	C₂×C₃₆₇	1	1	yes	characteristic
27	#125	C₂×C₃₆₇	1	1	yes	characteristic
28	#103	C₂×C₃₆₇	1	4	yes	—
29	#97	C₂×C₃₆₇	1	1	yes	characteristic
30	C₂×Q₈	C₂²×C₃₆₇	1	2	yes	—
31	C₂×C₈	C₂²×C₃₆₇	1	2	yes	—
32	C₄⋊C₄	—	2	2	no	—
33	C₄⋊C₄	—	2	2	no	—
34	C₄⋊C₄	C₂²×C₃₆₇	1	2	yes	—
35	C₄²	C₂²×C₃₆₇	1	1	yes	characteristic
36	Q₈	—	2	4	no	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	—	2	2	no	—
39	C₂×C₄	D₄×C₃₆₇	1	2	yes	—
40	C₂×C₄	C₂³×C₃₆₇	1	1	yes	characteristic
41	C₈	—	2	2	no	—
42	C₂²	C₂×D₄×C₃₆₇	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	4	2	no	—
45	C₄	—	4	2	no	—
46	C₄	#41×C₃₆₇	1	2	yes	—
47	C₂	#124×C₃₆₇	1	1	yes	characteristic
48	C₂	#137×C₃₆₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₃₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₆₇ ◁ C₄²×C₃₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
24. C₃₆₇	#491	direct (N × H)
25. #491	C₃₆₇	direct (N × H)

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
#103×C₃₆₇	2
#97×C₃₆₇	2
#491	367

Subgroups — #491×C367 #37553809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #491×C₃₆₇ #37553809