Subgroups — #441×C₃₆₇ #37553759

Order 23488.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×C₃₆₇	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×C₃₆₇	C₂	1	2	yes	maximal
3	#106×C₃₆₇	C₂	1	1	yes	maximal, characteristic
4	#103×C₃₆₇	C₂	1	2	yes	maximal
5	#97×C₃₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₃₆₇	C₂²	1	2	yes	—
7	Q₁₆×C₃₆₇	C₄	1	4	yes	complemented
8	C₂×C₈×C₃₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₃₆₇	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₃₆₇	—	2	2	no	—
11	C₄⋊C₄×C₃₆₇	C₂²	1	2	yes	—
12	C₄²×C₃₆₇	—	2	2	no	—
13	C₄²×C₃₆₇	C₂²	1	1	yes	characteristic
14	Q₈×C₃₆₇	—	2	2	no	—
15	Q₈×C₃₆₇	C₂×C₄	1	4	yes	—
16	C₂×C₄×C₃₆₇	—	2	2	no	—
17	C₂×C₄×C₃₆₇	—	2	2	no	—
18	C₂×C₄×C₃₆₇	D₄	1	2	yes	—
19	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
20	C₈×C₃₆₇	—	2	1	no	—
21	C₈×C₃₆₇	C₂×C₄	1	2	yes	—
22	C₂²×C₃₆₇	C₂×D₄	1	1	yes	characteristic
23	C₄×C₃₆₇	—	2	4	no	—
24	C₄×C₃₆₇	—	2	2	no	—
25	C₄×C₃₆₇	—	4	2	no	—
26	C₄×C₃₆₇	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₃₆₇	#41	1	1	yes	characteristic
28	C₂×C₃₆₇	C₄×D₄	1	1	yes	characteristic
29	C₂×C₃₆₇	#137	1	2	yes	—
30	C₃₆₇	#441	1	1	yes	characteristic, direct product factor, Sylow
31	#441	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂×Q₁₆	C₂×C₃₆₇	1	1	yes	characteristic
33	C₄×Q₈	C₂×C₃₆₇	1	2	yes	—
34	#106	C₂×C₃₆₇	1	1	yes	characteristic
35	#103	C₂×C₃₆₇	1	2	yes	—
36	#97	C₂×C₃₆₇	1	1	yes	characteristic
37	C₂×Q₈	C₂²×C₃₆₇	1	2	yes	—
38	Q₁₆	C₄×C₃₆₇	1	4	yes	complemented
39	C₂×C₈	C₂²×C₃₆₇	1	1	yes	characteristic
40	C₂×C₈	C₂²×C₃₆₇	1	1	yes	characteristic
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	C₂²×C₃₆₇	1	2	yes	—
43	C₄²	—	2	2	no	—
44	C₄²	C₂²×C₃₆₇	1	1	yes	characteristic
45	Q₈	—	2	2	no	—
46	Q₈	C₂×C₄×C₃₆₇	1	4	yes	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	D₄×C₃₆₇	1	2	yes	—
50	C₂×C₄	C₂³×C₃₆₇	1	1	yes	characteristic
51	C₈	—	2	1	no	—
52	C₈	C₂×C₄×C₃₆₇	1	2	yes	—
53	C₂²	C₂×D₄×C₃₆₇	1	1	yes	characteristic
54	C₄	—	2	4	no	—
55	C₄	—	2	2	no	—
56	C₄	—	4	2	no	—
57	C₄	C₂²×C₄×C₃₆₇	1	1	yes	characteristic
58	C₄	#41×C₃₆₇	1	1	yes	characteristic
59	C₂	C₄×D₄×C₃₆₇	1	1	yes	characteristic
60	C₂	#137×C₃₆₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₃₆₇	C₂×D₄
commutator	C₄	C₂²×C₄×C₃₆₇
Frattini, cosocle	C₂×C₄	C₂³×C₃₆₇
LCS	G ▷ 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
7. Q₁₆×C₃₆₇	C₄	semidirect
30. C₃₆₇	#441	direct (N × H)
31. #441	C₃₆₇	direct (N × H)
38. Q₁₆	C₄×C₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×C₃₆₇	2
C₄×Q₈×C₃₆₇	2
#106×C₃₆₇	2
#103×C₃₆₇	2
#97×C₃₆₇	2
#441	367

Subgroups — #441×C367 #37553759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #441×C₃₆₇ #37553759