Subgroups — #38766824

Order 25472.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27164136	C₂	1	1	yes	maximal, characteristic
2	#27163839	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈²×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263977	C₄	1	1	yes	characteristic
5	#20263961	—	2	1	no	—
6	#20263895	—	2	1	no	—
7	#20263891	C₄	1	1	yes	characteristic
8	#20263883	—	2	1	no	—
9	C₄×C₈×C₁₉₉	—	2	1	no	—
10	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
11	#15691886	—	2	1	no	—
12	Dic₇₉₆	C₈	1	1	yes	characteristic, complemented
13	#15691875	—	2	1	no	—
14	D₁₅₉₂	C₈	1	1	yes	characteristic, complemented
15	#15691870	—	2	1	no	—
16	#15691869	—	4	1	no	—
17	C₈×D₁₉₉	—	4	1	no	—
18	C₁₉₉⋊C₁₆	—	4	1	no	—
19	C₂×C₈×C₁₉₉	—	2	2	no	—
20	C₂×C₈×C₁₉₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₉₉	D₄	1	1	yes	characteristic
22	C₄²×C₁₉₉	D₄	1	1	yes	characteristic
23	Dic₃₉₈	—	2	1	no	—
24	D₇₉₆	—	2	1	no	—
25	C₂²⋊D₁₉₉	—	4	1	no	—
26	C₄×D₁₉₉	—	4	1	no	—
27	C₁₉₉⋊C₈	—	4	1	no	—
28	C₂×C₄×C₁₉₉	—	2	1	no	—
29	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₈×C₁₉₉	—	2	4	no	—
31	C₈×C₁₉₉	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₉₉	M₁₆	1	1	yes	characteristic
33	C₈×C₁₉₉	D₈	1	1	yes	characteristic
34	C₈×C₁₉₉	SD₁₆	1	1	yes	characteristic
35	C₂²×C₁₉₉	#104	1	1	yes	characteristic
36	C₂×D₁₉₉	—	4	1	no	—
37	C₄×C₁₉₉	—	2	2	no	—
38	C₄×C₁₉₉	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₉₉	#102	1	1	yes	characteristic
40	Dic₁₉₉	—	4	1	no	—
41	C₂×C₁₉₉	—	2	1	no	—
42	C₂×C₁₉₉	D₄⋊C₈	1	1	yes	characteristic
43	D₁₉₉	—	8	1	no	—
44	C₁₉₉	#1335	1	1	yes	characteristic, complemented, Sylow
45	#1335	—	199	1	no	maximal, Sylow
46	#443	—	199	1	no	—
47	#364	—	199	1	no	—
48	C₈²	D₁₉₉	1	1	yes	characteristic, complemented
49	#135	—	199	1	no	—
50	#131	—	398	1	no	—
51	M₃₂	—	398	1	no	—
52	#108	—	199	1	no	—
53	#104	—	398	1	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
56	#41	—	398	1	no	—
57	Q₁₆	—	199	1	no	—
58	SD₁₆	—	398	1	no	—
59	D₈	—	199	1	no	—
60	M₁₆	—	796	1	no	—
61	M₁₆	—	398	1	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	796	1	no	—
64	C₂×C₈	C₄×D₁₉₉	1	1	yes	characteristic
65	C₂×C₈	D₇₉₆	1	1	yes	characteristic
66	C₄²	C₂²⋊D₁₉₉	1	1	yes	characteristic
67	C₁₆	—	796	1	no	—
68	Q₈	—	398	1	no	—
69	D₄	—	796	1	no	—
70	D₄	—	398	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	796	1	no	—
73	C₂×C₄	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
74	C₈	—	2	4	no	—
75	C₈	—	796	1	no	—
76	C₈	C₈×D₁₉₉	1	1	yes	characteristic
77	C₈	#15691869	1	1	yes	characteristic
78	C₈	D₁₅₉₂	1	1	yes	characteristic
79	C₈	#15691875	1	1	yes	characteristic
80	C₂²	—	796	1	no	—
81	C₂²	#20263883	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	796	1	no	—
84	C₄	(C₂×D₁₉₉)⋊C₈	1	1	yes	characteristic
85	C₄	#20263877	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	1592	1	no	—
88	C₂	#27163747	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₁₅₉₂
commutator	C₈×C₁₉₉	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₉₉
Fitting	C₈²×C₁₉₉	C₂
socle	C₂×C₁₉₉	D₄⋊C₈
prime radical	#20263977	C₄
cosocle	C₄×C₈×C₁₉₉	C₂²
hypercenter	C₈²	D₁₉₉
hypocenter	C₁₉₉	#1335
LCS	G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #27163839	C₂	semidirect
3. C₈²×C₁₉₉	C₂	semidirect
12. Dic₇₉₆	C₈	semidirect
14. D₁₅₉₂	C₈	semidirect
44. C₁₉₉	#1335	semidirect
48. C₈²	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#27164136	2
#27163839	2
C₈²×C₁₉₉	2
#1335	199