Subgroups — #27164350

Order 12736.

104 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	C₄⋊C₄×D₁₉₉	C₂	1	1	yes	maximal, characteristic
3	#20263886	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	—	2	1	no	—
10	C₂×C₄×D₁₉₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₉₉	—	2	1	no	—
13	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
14	Dic₁₉₉⋊C₄	—	2	1	no	—
15	(C₂×D₁₉₉)⋊C₄	—	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	C₄×D₁₉₉	—	4	2	no	—
28	C₁₉₉⋊C₈	—	2	1	no	—
29	C₂×C₄×C₁₉₉	—	2	1	no	—
30	C₂×C₄×C₁₉₉	—	2	1	no	—
31	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
32	C₈×C₁₉₉	—	2	1	no	—
33	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
34	C₂×D₁₉₉	—	2	1	no	—
35	C₂×D₁₉₉	—	2	1	no	—
36	C₂×D₁₉₉	SD₁₆	1	2	yes	—
37	C₄×C₁₉₉	—	4	1	no	—
38	C₄×C₁₉₉	—	4	1	no	—
39	C₄×C₁₉₉	#41	1	1	yes	characteristic
40	C₄×C₁₉₉	#41	1	1	yes	characteristic
41	Dic₁₉₉	—	2	1	no	—
42	Dic₁₉₉	—	4	1	no	—
43	Dic₁₉₉	—	4	1	no	—
44	C₂×C₁₉₉	#123	1	1	yes	characteristic
45	C₂×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
46	C₂×C₁₉₉	#137	1	1	yes	characteristic
47	D₁₉₉	—	2	2	no	—
48	C₁₉₉	#483	1	1	yes	characteristic, complemented, Sylow
49	#483	—	199	1	no	maximal, Sylow
50	C₂²⋊Q₈	—	199	1	no	—
51	C₂×C₄⋊C₄	—	199	1	no	—
52	#106	—	199	1	no	—
53	#106	D₁₉₉	1	1	yes	characteristic, complemented
54	#103	—	199	1	no	—
55	#103	—	199	1	no	—
56	C₂²⋊C₈	—	199	1	no	—
57	C₂×Q₈	—	199	1	no	—
58	C₂²×C₄	—	398	1	no	—
59	C₂²×C₄	—	199	1	no	—
60	C₂×C₈	—	199	1	no	—
61	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
62	C₄⋊C₄	—	398	1	no	—
63	C₄⋊C₄	—	398	1	no	—
64	C₄⋊C₄	—	199	1	no	—
65	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
66	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
67	C₂²⋊C₄	—	398	1	no	—
68	C₂³	—	199	1	no	—
69	Q₈	—	398	1	no	—
70	Q₈	—	398	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	796	2	no	—
74	C₂×C₄	—	398	1	no	—
75	C₂×C₄	—	398	1	no	—
76	C₂×C₄	—	398	1	no	—
77	C₂×C₄	—	398	1	no	—
78	C₂×C₄	—	199	1	no	—
79	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	398	1	no	—
82	C₂²	—	398	1	no	—
83	C₂²	—	398	1	no	—
84	C₂²	—	199	2	no	—
85	C₂²	D₄×D₁₉₉	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	796	1	no	—
89	C₄	—	796	1	no	—
90	C₄	—	398	1	no	—
91	C₄	#15691886	1	1	yes	characteristic
92	C₄	#15691887	1	1	yes	characteristic
93	C₂	—	398	2	no	—
94	C₂	#20263940	1	1	yes	characteristic
95	C₂	SD₁₆×D₁₉₉	1	1	yes	characteristic
96	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	#483
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. #20263886	C₂	semidirect
4. #20263880	C₂	semidirect
5. #20263878	C₂	semidirect
7. #106×C₁₉₉	C₂	semidirect
48. C₁₉₉	#483	semidirect
53. #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
C₄⋊C₄×D₁₉₉	2
#20263886	2
#20263880	2
#20263878	2
(C₂×D₁₉₉)⋊C₈	2
#106×C₁₉₉	2
#483	199