Subgroups — #27164303

Order 12736.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263955	C₂	1	1	yes	maximal, characteristic
2	#20263886	C₂	1	2	yes	maximal
3	C₄⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#20263879	C₂	1	2	yes	maximal
5	C₄×Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₈	—	2	1	no	—
7	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₁₉₉	—	2	1	no	—
9	C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
10	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₁₉₉	—	2	1	no	—
12	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₄²×C₁₉₉	—	2	1	no	—
14	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
15	Dic₃₉₈	—	4	2	no	—
16	C₂×Dic₁₉₉	—	2	2	no	—
17	C₁₉₉⋊C₈	—	2	2	no	—
18	Q₈×C₁₉₉	—	2	1	no	—
19	Q₈×C₁₉₉	Q₈	1	2	yes	complemented
20	C₂×C₄×C₁₉₉	—	2	1	no	—
21	C₂×C₄×C₁₉₉	—	2	1	no	—
22	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
25	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
26	C₄×C₁₉₉	—	2	2	no	—
27	C₄×C₁₉₉	—	2	1	no	—
28	C₄×C₁₉₉	—	4	1	no	—
29	C₄×C₁₉₉	SD₁₆	1	2	yes	—
30	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
31	C₄×C₁₉₉	#41	1	1	yes	characteristic
32	Dic₁₉₉	—	4	2	no	—
33	C₂×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
35	C₂×C₁₉₉	#137	1	1	yes	characteristic
36	C₁₉₉	#475	1	1	yes	characteristic, complemented, Sylow
37	#475	—	199	1	no	maximal, Sylow
38	C₄⋊Q₈	—	199	1	no	—
39	C₄×Q₈	D₁₉₉	1	1	yes	characteristic
40	#106	—	199	2	no	—
41	C₄⋊C₈	—	199	1	no	—
42	#103	—	199	2	no	—
43	C₂×Q₈	—	398	1	no	—
44	C₂×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
45	C₂×C₈	—	199	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	398	1	no	—
48	C₄⋊C₄	—	199	2	no	—
49	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
50	C₄²	—	2	1	no	—
51	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
52	Q₈	—	2	1	no	—
53	Q₈	—	796	2	no	—
54	Q₈	Dic₃₉₈	1	2	yes	complemented
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	398	2	no	—
58	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
61	C₈	—	398	2	no	—
62	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	2	2	no	—
65	C₄	—	4	1	no	—
66	C₄	—	796	2	no	—
67	C₄	#15691873	1	2	yes	—
68	C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
69	C₄	#15691886	1	1	yes	characteristic
70	C₂	#20263934	1	1	yes	characteristic
71	C₂	C₂×#15691873	1	1	yes	characteristic
72	C₂	#20263985	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	C₄×Q₈×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂×D₄
hypercenter	C₄×Q₈	D₁₉₉
hypocenter	C₁₉₉	#475
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
19. Q₈×C₁₉₉	Q₈	semidirect
36. C₁₉₉	#475	semidirect
54. Q₈	Dic₃₉₈	semidirect

Maximal subgroups

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

subgroup	index
#20263955	2
#20263886	2
C₄⋊(C₁₉₉⋊C₈)	2
#20263879	2
C₄×Q₈×C₁₉₉	2
#475	199