Subgroups — #27164367

Order 12736.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263954	C₂	1	1	yes	maximal, characteristic
2	#20263953	C₂	1	1	yes	maximal, characteristic
3	#20263880	C₂	1	1	yes	maximal, characteristic
4	#20263879	C₂	1	1	yes	maximal, characteristic
5	#20263878	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
7	#103×C₁₉₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₉₈	—	2	1	no	—
9	C₂×Dic₃₉₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
12	Dic₁₉₉⋊C₄	—	2	1	no	—
13	Dic₁₉₉⋊C₄	—	2	1	no	—
14	Dic₁₉₉⋊C₄	—	2	1	no	—
15	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
19	Dic₃₉₈	—	2	1	no	—
20	Dic₃₉₈	—	2	1	no	—
21	Dic₃₉₈	—	4	2	no	—
22	C₂×Dic₁₉₉	—	2	1	no	—
23	C₂×Dic₁₉₉	—	2	1	no	—
24	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
25	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
26	C₁₉₉⋊C₈	—	2	1	no	—
27	Q₈×C₁₉₉	—	2	1	no	—
28	Q₈×C₁₉₉	—	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	C₄×C₁₉₉	—	4	1	no	—
35	C₄×C₁₉₉	—	4	1	no	—
36	C₄×C₁₉₉	#41	1	1	yes	characteristic
37	C₄×C₁₉₉	#41	1	1	yes	characteristic
38	Dic₁₉₉	—	4	1	no	—
39	Dic₁₉₉	—	4	1	no	—
40	Dic₁₉₉	SD₁₆	1	2	yes	—
41	Dic₁₉₉	Q₁₆	1	2	yes	—
42	C₂×C₁₉₉	#124	1	1	yes	characteristic
43	C₂×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
44	C₂×C₁₉₉	C₂×Q₁₆	1	1	yes	characteristic
45	C₁₉₉	#487	1	1	yes	characteristic, complemented, Sylow
46	#487	—	199	1	no	maximal, Sylow
47	C₄⋊Q₈	—	199	1	no	—
48	C₄⋊Q₈	—	199	1	no	—
49	#103	—	199	1	no	—
50	#103	—	199	1	no	—
51	#103	—	199	1	no	—
52	#103	D₁₉₉	1	1	yes	characteristic
53	C₄×C₈	—	199	1	no	—
54	C₂×Q₈	—	398	1	no	—
55	C₂×Q₈	—	199	1	no	—
56	C₂×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
57	C₂×C₈	—	199	1	no	—
58	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
59	C₄⋊C₄	—	398	1	no	—
60	C₄⋊C₄	—	398	1	no	—
61	C₄⋊C₄	—	398	1	no	—
62	C₄⋊C₄	—	199	1	no	—
63	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
64	C₄²	—	199	1	no	—
65	Q₈	—	2	1	no	—
66	Q₈	—	2	1	no	—
67	Q₈	—	796	2	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	—	199	1	no	—
75	C₂×C₄	—	199	1	no	—
76	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	398	1	no	—
79	C₂²	D₄×D₁₉₉	1	1	yes	characteristic
80	C₄	—	4	1	no	—
81	C₄	—	4	1	no	—
82	C₄	—	796	1	no	—
83	C₄	—	796	1	no	—
84	C₄	—	199	2	no	—
85	C₄	—	199	2	no	—
86	C₄	#15691885	1	1	yes	characteristic
87	C₄	#15691886	1	1	yes	characteristic
88	C₂	#20263941	1	1	yes	characteristic
89	C₂	SD₁₆×D₁₉₉	1	1	yes	characteristic
90	C₂	Q₁₆×D₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#103×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂×D₄
hypercenter	#103	D₁₉₉
hypocenter	C₁₉₉	#487
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
45. C₁₉₉	#487	semidirect

Maximal subgroups

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

subgroup	index
#20263954	2
#20263953	2
#20263880	2
#20263879	2
#20263878	2
C₈×Dic₁₉₉	2
#103×C₁₉₉	2
#487	199