Subgroups — #27164396

Order 12736.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₉₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#15691876	C₂	1	2	yes	maximal
3	#20263953	C₂	1	2	yes	maximal
4	C₈×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₁₆×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₈	C₂²	1	2	yes	—
7	Dic₇₉₆	—	2	2	no	—
8	#15691876	—	2	4	no	—
9	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
10	Dic₁₉₉⋊C₄	—	2	4	no	—
11	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
12	C₂×Q₈×C₁₉₉	C₂²	1	2	yes	—
13	Q₁₆×C₁₉₉	—	2	2	no	—
14	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
15	Dic₃₉₈	—	2	2	no	—
16	Dic₃₉₈	—	2	2	no	—
17	C₂×Dic₁₉₉	—	2	2	no	—
18	C₂×Dic₁₉₉	D₄	1	2	yes	—
19	C₁₉₉⋊C₈	D₄	1	2	yes	—
20	Q₈×C₁₉₉	—	2	2	no	—
21	Q₈×C₁₉₉	—	2	2	no	—
22	C₂×C₄×C₁₉₉	—	2	2	no	—
23	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
24	C₈×C₁₉₉	D₄	1	2	yes	—
25	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
26	C₄×C₁₉₉	—	4	2	no	—
27	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
28	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
29	Dic₁₉₉	—	4	2	no	—
30	Dic₁₉₉	Q₁₆	1	4	yes	complemented
31	C₂×C₁₉₉	#127	1	1	yes	characteristic
32	C₂×C₁₉₉	C₂×Q₁₆	1	2	yes	—
33	C₁₉₉	#494	1	1	yes	characteristic, complemented, Sylow
34	#494	—	199	1	no	maximal, Sylow
35	C₂×Q₁₆	—	199	2	no	—
36	C₂×Q₁₆	—	199	1	no	—
37	C₂×Q₁₆	D₁₉₉	1	1	yes	characteristic
38	C₄⋊Q₈	—	199	2	no	—
39	C₄×C₈	—	199	1	no	—
40	C₂×Q₈	—	199	2	no	—
41	C₂×Q₈	C₂×D₁₉₉	1	2	yes	—
42	Q₁₆	—	2	2	no	—
43	Q₁₆	—	398	4	no	—
44	Q₁₆	—	398	2	no	—
45	C₂×C₈	—	199	1	no	—
46	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
47	C₄⋊C₄	—	398	4	no	—
48	C₄²	—	199	1	no	—
49	Q₈	—	2	2	no	—
50	Q₈	—	2	2	no	—
51	Q₈	—	398	2	no	—
52	Q₈	—	398	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	398	2	no	—
55	C₂×C₄	—	199	2	no	—
56	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
57	C₈	—	199	2	no	—
58	C₈	C₂²⋊D₁₉₉	1	2	yes	—
59	C₂²	D₄×D₁₉₉	1	1	yes	characteristic
60	C₄	—	4	2	no	—
61	C₄	—	796	2	no	—
62	C₄	—	199	4	no	—
63	C₄	D₄×D₁₉₉	1	1	yes	characteristic
64	C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
65	C₂	#20263951	1	1	yes	characteristic
66	C₂	Q₁₆×D₁₉₉	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	C₂×Q₁₆×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂×D₄
hypercenter	C₂×Q₁₆	D₁₉₉
hypocenter	C₁₉₉	#494
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
30. Dic₁₉₉	Q₁₆	semidirect
33. C₁₉₉	#494	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₉₆	2
C₂×#15691876	2
#20263953	2
C₈×Dic₁₉₉	2
C₂×Q₁₆×C₁₉₉	2
#494	199