Subgroups — #27164362

Order 12736.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263935	C₂	1	1	yes	maximal, characteristic
2	#20263913	C₂	1	1	yes	maximal, characteristic
3	#20263888	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263880	C₂	1	1	yes	maximal, characteristic, complemented
5	#20263878	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₁₉₉)⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#106×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₃₉₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₉₉	C₂²	1	1	yes	characteristic
10	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₉₉	—	2	1	no	—
12	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
13	(C₂×D₁₉₉)⋊C₄	—	2	1	no	—
14	(C₂×D₁₉₉)⋊C₄	—	2	1	no	—
15	C₄×Dic₁₉₉	—	2	1	no	—
16	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
19	C₂²×D₁₉₉	D₄	1	1	yes	characteristic
20	Dic₃₉₈	—	2	1	no	—
21	Dic₃₉₈	—	2	1	no	—
22	C₂×Dic₁₉₉	—	2	1	no	—
23	C₂×Dic₁₉₉	—	2	1	no	—
24	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
25	C₄×D₁₉₉	—	2	1	no	—
26	C₄×D₁₉₉	—	2	1	no	—
27	C₁₉₉⋊C₈	—	2	1	no	—
28	C₂×C₄×C₁₉₉	—	2	1	no	—
29	C₂×C₄×C₁₉₉	—	2	1	no	—
30	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
31	C₈×C₁₉₉	—	2	1	no	—
32	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
33	C₂×D₁₉₉	—	2	1	no	—
34	C₂×D₁₉₉	—	2	1	no	—
35	C₂×D₁₉₉	—	2	1	no	—
36	C₄×C₁₉₉	—	4	1	no	—
37	C₄×C₁₉₉	—	4	1	no	—
38	C₄×C₁₉₉	#41	1	1	yes	characteristic
39	C₄×C₁₉₉	#41	1	1	yes	characteristic
40	Dic₁₉₉	—	2	1	no	—
41	Dic₁₉₉	—	4	1	no	—
42	Dic₁₉₉	—	4	1	no	—
43	C₂×C₁₉₉	#123	1	1	yes	characteristic
44	C₂×C₁₉₉	#135	1	1	yes	characteristic
45	C₂×C₁₉₉	#137	1	1	yes	characteristic
46	D₁₉₉	—	4	1	no	—
47	C₁₉₉	#485	1	1	yes	characteristic, complemented, Sylow
48	#485	—	199	1	no	maximal, Sylow
49	C₂²⋊Q₈	—	199	1	no	—
50	#117	—	199	1	no	—
51	#107	—	199	1	no	—
52	#106	D₁₉₉	1	1	yes	characteristic, complemented
53	#103	—	199	1	no	—
54	#103	—	199	1	no	—
55	C₂²⋊C₈	—	199	1	no	—
56	C₂×Q₈	—	199	1	no	—
57	C₂²×C₄	—	199	1	no	—
58	C₂×C₈	—	199	1	no	—
59	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
60	C₄⋊C₄	—	398	1	no	—
61	C₄⋊C₄	—	199	1	no	—
62	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
63	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
64	C₂²⋊C₄	—	398	1	no	—
65	C₂²⋊C₄	—	398	1	no	—
66	C₄²	—	398	1	no	—
67	C₂³	—	199	1	no	—
68	Q₈	—	398	1	no	—
69	Q₈	—	398	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	398	1	no	—
73	C₂×C₄	—	398	1	no	—
74	C₂×C₄	—	398	1	no	—
75	C₂×C₄	—	398	1	no	—
76	C₂×C₄	—	199	1	no	—
77	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	398	1	no	—
80	C₂²	—	398	1	no	—
81	C₂²	—	398	1	no	—
82	C₂²	—	398	1	no	—
83	C₂²	D₄×D₁₉₉	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	796	1	no	—
87	C₄	—	796	1	no	—
88	C₄	—	398	1	no	—
89	C₄	#15691886	1	1	yes	characteristic
90	C₄	#15691887	1	1	yes	characteristic
91	C₂	—	796	1	no	—
92	C₂	#20263940	1	1	yes	characteristic
93	C₂	#20263976	1	1	yes	characteristic
94	C₂	#20263984	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#106×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
prime radical	C₂²×D₁₉₉	D₄
hypercenter	#106	D₁₉₉
hypocenter	C₁₉₉	#485
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
3. #20263888	C₂	semidirect
4. #20263880	C₂	semidirect
5. #20263878	C₂	semidirect
7. #106×C₁₉₉	C₂	semidirect
47. C₁₉₉	#485	semidirect
52. #106	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263935	2
#20263913	2
#20263888	2
#20263880	2
#20263878	2
(C₂×D₁₉₉)⋊C₈	2
#106×C₁₉₉	2
#485	199