Subgroups — #27163741

Order 12736.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×(C₂×D₁₉₉)⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₁₉₉	C₄	1	1	yes	characteristic
5	C₂²×Dic₁₉₉	C₄	1	1	yes	characteristic
6	C₂×C₁₉₉⋊C₈	—	2	1	no	—
7	(C₂×D₁₉₉)⋊C₄	—	2	1	no	—
8	(C₂×D₁₉₉)⋊C₄	—	2	1	no	—
9	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₁₉₉	—	2	1	no	—
11	C₂²×D₁₉₉	—	2	1	no	—
12	C₂²×D₁₉₉	—	2	1	no	—
13	C₂²×D₁₉₉	—	4	1	no	—
14	C₂²×D₁₉₉	—	4	1	no	—
15	C₂²×D₁₉₉	C₈	1	2	yes	complemented
16	C₂×Dic₁₉₉	—	2	1	no	—
17	C₂×Dic₁₉₉	—	4	1	no	—
18	C₁₉₉⋊C₈	—	4	1	no	—
19	C₂³×C₁₉₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₉	—	4	1	no	—
21	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
23	C₈×C₁₉₉	—	4	1	no	—
24	C₂²×C₁₉₉	—	2	1	no	—
25	C₂²×C₁₉₉	—	2	1	no	—
26	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₉₉	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₁₉₉	M₁₆	1	1	yes	characteristic
29	C₂×D₁₉₉	—	2	2	no	—
30	C₂×D₁₉₉	—	4	2	no	—
31	C₂×D₁₉₉	—	4	1	no	—
32	C₂×D₁₉₉	—	4	1	no	—
33	C₂×D₁₉₉	—	4	2	no	—
34	C₄×C₁₉₉	—	2	1	no	—
35	C₄×C₁₉₉	—	2	1	no	—
36	Dic₁₉₉	—	4	1	no	—
37	C₂×C₁₉₉	—	2	1	no	—
38	C₂×C₁₉₉	—	2	1	no	—
39	C₂×C₁₉₉	C₂²⋊C₈	1	1	yes	characteristic
40	C₂×C₁₉₉	#99	1	1	yes	characteristic
41	C₂×C₁₉₉	#100	1	1	yes	characteristic
42	D₁₉₉	—	4	2	no	—
43	C₁₉₉	#323	1	1	yes	characteristic, complemented, Sylow
44	#323	—	199	1	no	maximal, Sylow
45	C₂×C₂²⋊C₄	—	199	1	no	—
46	C₂²⋊C₈	—	199	1	no	—
47	C₂²⋊C₈	D₁₉₉	1	1	yes	characteristic, complemented
48	C₂⁴	—	199	1	no	—
49	C₂²×C₄	—	199	1	no	—
50	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	—	398	1	no	—
53	C₂²⋊C₄	—	398	1	no	—
54	C₂²⋊C₄	—	398	1	no	—
55	C₂³	—	796	1	no	—
56	C₂³	—	796	1	no	—
57	C₂³	—	398	1	no	—
58	C₂³	—	398	1	no	—
59	C₂³	—	199	2	no	—
60	C₂³	C₄×D₁₉₉	1	1	yes	characteristic
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	—	796	1	no	—
63	C₂×C₄	—	398	1	no	—
64	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
65	C₂×C₄	D₇₉₆	1	1	yes	characteristic
66	C₈	—	4	1	no	—
67	C₈	—	796	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	796	2	no	—
71	C₂²	—	796	2	no	—
72	C₂²	—	796	1	no	—
73	C₂²	—	796	1	no	—
74	C₂²	—	398	2	no	—
75	C₂²	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
76	C₂²	C₈×D₁₉₉	1	1	yes	characteristic
77	C₂²	#15691869	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	796	1	no	—
81	C₂	—	2	1	no	—
82	C₂	—	2	1	no	—
83	C₂	—	796	2	no	—
84	C₂	(C₂×D₁₉₉)⋊C₈	1	1	yes	characteristic
85	C₂	#20263869	1	1	yes	characteristic
86	C₂	#20263871	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₁₉₉)⋊C₄
commutator	C₂²×C₁₉₉	C₂×C₈
Frattini	C₂²×C₄	C₂×D₁₉₉
Fitting	C₂²⋊C₈×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂²⋊C₄
prime radical	C₂³×D₁₉₉	C₄
cosocle	C₂²×C₄×C₁₉₉	C₂²
hypercenter	C₂²⋊C₈	D₁₉₉
hypocenter	C₁₉₉	#323
LCS	G ▷ C₂²×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂²⋊(C₁₉₉⋊C₈)	C₂	semidirect
3. C₂²⋊C₈×C₁₉₉	C₂	semidirect
15. C₂²×D₁₉₉	C₈	semidirect
43. C₁₉₉	#323	semidirect
47. C₂²⋊C₈	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×(C₂×D₁₉₉)⋊C₄	2
C₂²⋊(C₁₉₉⋊C₈)	2
C₂²⋊C₈×C₁₉₉	2
#323	199