Subgroups — #27164112

Order 12736.

110 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	#20263920	C₂	1	2	yes	maximal
3	#20263878	C₂	1	2	yes	maximal
4	C₈×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
5	#107×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₈	C₂²	1	2	yes	—
7	Dic₇₉₆	C₄	1	4	yes	complemented
8	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
9	Dic₁₉₉⋊C₄	—	2	2	no	—
10	C₄×Dic₁₉₉	—	2	2	no	—
11	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
14	Dic₃₉₈	—	2	2	no	—
15	Dic₃₉₈	C₂×C₄	1	4	yes	—
16	C₂×Dic₁₉₉	—	2	2	no	—
17	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
18	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
19	C₁₉₉⋊C₈	—	2	1	no	—
20	C₂×C₄×C₁₉₉	—	2	2	no	—
21	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
22	C₈×C₁₉₉	C₂×C₄	1	2	yes	—
23	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
24	C₄×C₁₉₉	—	4	2	no	—
25	C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
26	C₄×C₁₉₉	#41	1	1	yes	characteristic
27	Dic₁₉₉	—	2	4	no	—
28	Dic₁₉₉	—	2	1	no	—
29	Dic₁₉₉	Q₁₆	1	2	yes	complemented
30	C₂×C₁₉₉	C₄×D₄	1	1	yes	characteristic
31	C₂×C₁₉₉	C₂×Q₁₆	1	1	yes	characteristic
32	C₂×C₁₉₉	#135	1	1	yes	characteristic
33	C₁₉₉	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×Q₁₆	—	199	1	no	maximal, Sylow
35	C₂×Q₁₆	—	199	1	no	—
36	C₄×Q₈	—	199	2	no	—
37	#107	D₁₉₉	1	1	yes	characteristic
38	#103	—	199	2	no	—
39	C₄×C₈	—	199	1	no	—
40	C₂×Q₈	—	199	2	no	—
41	Q₁₆	—	199	4	no	—
42	C₂×C₈	—	199	1	no	—
43	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
44	C₄⋊C₄	—	398	2	no	—
45	C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
46	C₄²	—	398	2	no	—
47	C₄²	—	199	1	no	—
48	Q₈	—	398	2	no	—
49	Q₈	—	199	4	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	398	2	no	—
52	C₂×C₄	—	199	1	no	—
53	C₂×C₄	—	199	1	no	—
54	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
55	C₈	—	398	1	no	—
56	C₈	C₄×D₁₉₉	1	2	yes	—
57	C₂²	D₄×D₁₉₉	1	1	yes	characteristic
58	C₄	—	4	2	no	—
59	C₄	—	398	1	no	—
60	C₄	—	398	4	no	—
61	C₄	—	199	2	no	—
62	C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
63	C₄	#15691887	1	1	yes	characteristic
64	C₂	#20263917	1	1	yes	characteristic
65	C₂	Q₁₆×D₁₉₉	1	1	yes	characteristic
66	C₂	#20263974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₉
commutator	C₄×C₁₉₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#107×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂×D₄
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	#107	D₁₉₉
hypocenter	C₁₉₉	C₄×Q₁₆
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
7. Dic₇₉₆	C₄	semidirect
29. Dic₁₉₉	Q₁₆	semidirect
33. C₁₉₉	C₄×Q₁₆	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
#20263920	2
#20263878	2
C₈×Dic₁₉₉	2
#107×C₁₉₉	2
C₄×Q₁₆	199