Subgroups — #27164260

Order 12736.

114 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, complemented
2	C₂×C₈×D₁₉₉	C₂	1	1	yes	maximal, characteristic
3	#20263935	C₂	1	2	yes	maximal
4	#20263878	C₂	1	2	yes	maximal, complemented
5	#107×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₉₈	C₂²	1	2	yes	—
7	C₂×C₄×D₁₉₉	C₂²	1	1	yes	characteristic
8	Dic₇₉₆	—	2	2	no	—
9	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₈×D₁₉₉	—	2	2	no	—
11	C₄⋊Dic₁₉₉	—	2	2	no	—
12	(C₂×D₁₉₉)⋊C₄	—	2	2	no	—
13	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
15	C₂²×D₁₉₉	D₄	1	1	yes	characteristic
16	Dic₃₉₈	—	2	2	no	—
17	Dic₃₉₈	—	2	2	no	—
18	C₂×Dic₁₉₉	—	2	2	no	—
19	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
20	C₄×D₁₉₉	—	2	1	no	—
21	C₄×D₁₉₉	—	2	1	no	—
22	C₁₉₉⋊C₈	—	2	1	no	—
23	C₂×C₄×C₁₉₉	—	2	2	no	—
24	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
25	C₈×C₁₉₉	D₄	1	2	yes	—
26	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₉₉	—	2	1	no	—
28	C₂×D₁₉₉	—	2	1	no	—
29	C₂×D₁₉₉	Q₁₆	1	2	yes	complemented
30	C₄×C₁₉₉	—	4	2	no	—
31	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
32	C₄×C₁₉₉	#41	1	1	yes	characteristic
33	Dic₁₉₉	—	2	1	no	—
34	Dic₁₉₉	—	4	2	no	—
35	C₂×C₁₉₉	#121	1	1	yes	characteristic
36	C₂×C₁₉₉	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₁₉₉	#135	1	1	yes	characteristic
38	D₁₉₉	—	2	2	no	—
39	C₁₉₉	#467	1	1	yes	characteristic, complemented, Sylow
40	#467	—	199	1	no	maximal, Sylow
41	C₂×Q₁₆	—	199	1	no	—
42	C₂²×C₈	—	199	1	no	—
43	C₂²⋊Q₈	—	199	2	no	—
44	#107	D₁₉₉	1	1	yes	characteristic, complemented
45	#103	—	199	2	no	—
46	C₂×Q₈	—	199	2	no	—
47	C₂²×C₄	—	199	1	no	—
48	Q₁₆	—	398	2	no	—
49	C₂×C₈	—	398	2	no	—
50	C₂×C₈	—	199	1	no	—
51	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
52	C₄⋊C₄	—	398	2	no	—
53	C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
54	C₂²⋊C₄	—	398	2	no	—
55	C₂³	—	199	1	no	—
56	Q₈	—	398	2	no	—
57	Q₈	—	398	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	398	2	no	—
60	C₂×C₄	—	398	1	no	—
61	C₂×C₄	—	398	1	no	—
62	C₂×C₄	—	199	1	no	—
63	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
64	C₈	—	398	1	no	—
65	C₈	D₇₉₆	1	2	yes	—
66	C₂²	—	398	1	no	—
67	C₂²	—	398	1	no	—
68	C₂²	—	199	2	no	—
69	C₂²	D₄×D₁₉₉	1	1	yes	characteristic
70	C₄	—	4	2	no	—
71	C₄	—	796	2	no	—
72	C₄	—	398	1	no	—
73	C₄	C₂×D₇₉₆	1	1	yes	characteristic
74	C₄	#15691887	1	1	yes	characteristic
75	C₂	—	398	2	no	—
76	C₂	#20263930	1	1	yes	characteristic
77	C₂	Q₁₆×D₁₉₉	1	1	yes	characteristic
78	C₂	#20263974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#107×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
prime radical	C₂²×D₁₉₉	D₄
hypercenter	#107	D₁₉₉
hypocenter	C₁₉₉	#467
LCS	G ▷ C₂×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
1. C₂×Dic₇₉₆	C₂	semidirect
4. #20263878	C₂	semidirect
5. #107×C₁₉₉	C₂	semidirect
29. C₂×D₁₉₉	Q₁₆	semidirect
39. C₁₉₉	#467	semidirect
44. #107	D₁₉₉	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×C₈×D₁₉₉	2
#20263935	2
#20263878	2
#107×C₁₉₉	2
#467	199