Subgroups — #38766814

Order 25472.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₉₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#27163771	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₁₆×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉₉⋊C₁₆	—	2	1	no	—
5	C₄⋊(C₁₉₉⋊C₈)	—	2	1	no	—
6	C₂×Q₁₆×C₁₉₉	C₄	1	1	yes	characteristic
7	C₄×Q₈×C₁₉₉	—	2	1	no	—
8	#107×C₁₉₉	C₄	1	1	yes	characteristic
9	#103×C₁₉₉	—	2	1	no	—
10	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₉₉⋊C₈	—	4	1	no	—
12	C₁₉₉⋊C₁₆	—	2	2	no	—
13	C₂×Q₈×C₁₉₉	—	2	1	no	—
14	Q₁₆×C₁₉₉	—	2	1	no	—
15	Q₁₆×C₁₉₉	C₈	1	2	yes	complemented
16	C₂×C₈×C₁₉₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₁₉₉	D₄	1	1	yes	characteristic
18	C₄⋊C₄×C₁₉₉	—	2	1	no	—
19	C₄⋊C₄×C₁₉₉	—	4	1	no	—
20	C₄²×C₁₉₉	—	4	1	no	—
21	C₄²×C₁₉₉	D₄	1	1	yes	characteristic
22	C₁₉₉⋊C₈	—	8	1	no	—
23	Q₈×C₁₉₉	—	2	2	no	—
24	Q₈×C₁₉₉	—	4	1	no	—
25	C₂×C₄×C₁₉₉	—	4	1	no	—
26	C₂×C₄×C₁₉₉	—	4	1	no	—
27	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₉₉	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₉₉	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₉₉	—	2	1	no	—
31	C₈×C₁₉₉	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₉₉	M₁₆	1	1	yes	characteristic
33	C₂²×C₁₉₉	#102	1	1	yes	characteristic
34	C₄×C₁₉₉	—	2	1	no	—
35	C₄×C₁₉₉	—	4	2	no	—
36	C₄×C₁₉₉	—	8	1	no	—
37	C₄×C₁₉₉	C₂²⋊C₈	1	1	yes	characteristic
38	C₄×C₁₉₉	#104	1	1	yes	characteristic
39	C₄×C₁₉₉	SD₃₂	1	1	yes	characteristic
40	C₄×C₁₉₉	Q₃₂	1	1	yes	characteristic
41	C₂×C₁₉₉	D₄⋊C₈	1	1	yes	characteristic
42	C₂×C₁₉₉	#358	1	1	yes	characteristic
43	C₂×C₁₉₉	#359	1	1	yes	characteristic
44	C₁₉₉	#1332	1	1	yes	characteristic, complemented, Sylow
45	#1332	—	199	1	no	maximal, Sylow
46	C₄×Q₁₆	D₁₉₉	1	1	yes	characteristic
47	C₄×C₁₆	—	199	1	no	—
48	#335	—	199	1	no	—
49	C₂×Q₁₆	Dic₁₉₉	1	1	yes	characteristic
50	C₄×Q₈	—	2	1	no	—
51	C₂×C₁₆	—	398	1	no	—
52	#107	Dic₁₉₉	1	1	yes	characteristic
53	C₄⋊C₈	—	398	1	no	—
54	#103	—	2	1	no	—
55	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
56	C₂×Q₈	—	2	1	no	—
57	Q₁₆	—	2	1	no	—
58	Q₁₆	C₁₉₉⋊C₈	1	2	yes	complemented
59	C₂×C₈	—	796	1	no	—
60	C₂×C₈	C₂×Dic₁₉₉	1	1	yes	characteristic
61	C₂×C₈	C₂²⋊D₁₉₉	1	1	yes	characteristic
62	C₄⋊C₄	—	2	1	no	—
63	C₄⋊C₄	—	4	1	no	—
64	C₄²	—	4	1	no	—
65	C₄²	C₂²⋊D₁₉₉	1	1	yes	characteristic
66	C₁₆	—	398	2	no	—
67	Q₈	—	2	2	no	—
68	Q₈	—	4	1	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	—	4	1	no	—
71	C₂×C₄	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
72	C₂×C₄	#15691871	1	1	yes	characteristic
73	C₂×C₄	#15691874	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	—	1592	1	no	—
76	C₈	C₂×C₁₉₉⋊C₈	1	1	yes	characteristic
77	C₈	#15691870	1	1	yes	characteristic
78	C₂²	#20263876	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	4	2	no	—
81	C₄	—	8	1	no	—
82	C₄	C₂²⋊(C₁₉₉⋊C₈)	1	1	yes	characteristic
83	C₄	#20263882	1	1	yes	characteristic
84	C₄	#20263898	1	1	yes	characteristic
85	C₄	#20263901	1	1	yes	characteristic
86	C₂	D₄⋊(C₁₉₉⋊C₈)	1	1	yes	characteristic
87	C₂	#27163822	1	1	yes	characteristic
88	C₂	#27163825	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15691871
commutator	C₈×C₁₉₉	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₉₉
Fitting	C₄×Q₁₆×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	#102
cosocle	C₄×C₈×C₁₉₉	C₂²
hypercenter	C₄×Q₁₆	D₁₉₉
hypocenter	C₁₉₉	#1332
LCS	G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
15. Q₁₆×C₁₉₉	C₈	semidirect
44. C₁₉₉	#1332	semidirect
58. Q₁₆	C₁₉₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₁₉₉⋊C₁₆	2
#27163771	2
C₄×Q₁₆×C₁₉₉	2
#1332	199