Subgroups — #27163748

Order 12736.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₃₉₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₉₈	C₄	1	1	yes	characteristic
5	C₂×C₁₉₉⋊C₈	—	2	1	no	—
6	C₄⋊Dic₁₉₉	C₄	1	1	yes	characteristic
7	Dic₁₉₉⋊C₄	—	2	1	no	—
8	C₄×Dic₁₉₉	—	2	1	no	—
9	C₂×C₈×C₁₉₉	—	2	1	no	—
10	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
11	Dic₃₉₈	—	2	1	no	—
12	Dic₃₉₈	C₈	1	2	yes	complemented
13	C₂×Dic₁₉₉	—	2	1	no	—
14	C₂×Dic₁₉₉	—	2	1	no	—
15	C₁₉₉⋊C₈	—	2	2	no	—
16	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
19	C₈×C₁₉₉	—	4	1	no	—
20	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₁₉₉	—	2	1	no	—
22	C₄×C₁₉₉	C₂×C₈	1	1	yes	characteristic
23	C₄×C₁₉₉	M₁₆	1	1	yes	characteristic
24	C₄×C₁₉₉	SD₁₆	1	1	yes	characteristic
25	C₄×C₁₉₉	Q₁₆	1	1	yes	characteristic
26	Dic₁₉₉	—	2	2	no	—
27	Dic₁₉₉	—	4	1	no	—
28	C₂×C₁₉₉	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×C₁₉₉	#103	1	1	yes	characteristic
30	C₂×C₁₉₉	#104	1	1	yes	characteristic
31	C₁₉₉	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
32	Q₈⋊C₈	—	199	1	no	maximal, Sylow
33	C₄×Q₈	—	199	1	no	—
34	C₄⋊C₈	D₁₉₉	1	1	yes	characteristic
35	C₄×C₈	—	199	1	no	—
36	C₂×Q₈	—	199	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	398	1	no	—
39	C₄⋊C₄	—	398	1	no	—
40	C₄⋊C₄	—	199	1	no	—
41	C₄²	—	398	1	no	—
42	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
43	Q₈	—	398	1	no	—
44	Q₈	—	199	2	no	—
45	C₂×C₄	—	398	1	no	—
46	C₂×C₄	—	398	1	no	—
47	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
49	C₂×C₄	D₇₉₆	1	1	yes	characteristic
50	C₈	—	4	1	no	—
51	C₈	—	398	2	no	—
52	C₂²	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	796	1	no	—
55	C₄	—	398	2	no	—
56	C₄	C₈×D₁₉₉	1	1	yes	characteristic
57	C₄	#15691869	1	1	yes	characteristic
58	C₄	#15691874	1	1	yes	characteristic
59	C₄	#15691876	1	1	yes	characteristic
60	C₂	(C₂×D₁₉₉)⋊C₈	1	1	yes	characteristic
61	C₂	#20263878	1	1	yes	characteristic
62	C₂	#20263881	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₁₉₉
commutator	C₄×C₁₉₉	C₂×C₈
Frattini	C₄²	C₂×D₁₉₉
Fitting	C₄⋊C₈×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂²⋊C₄
cosocle	C₄²×C₁₉₉	C₂²
hypercenter	C₄⋊C₈	D₁₉₉
hypocenter	C₁₉₉	Q₈⋊C₈
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
12. Dic₃₉₈	C₈	semidirect
31. C₁₉₉	Q₈⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₃₉₈	2
C₄×C₁₉₉⋊C₈	2
C₄⋊C₈×C₁₉₉	2
Q₈⋊C₈	199