Subgroups — #27163772

Order 12736.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₉₉	C₄	1	2	yes	—
5	C₂×C₁₉₉⋊C₈	—	2	1	no	—
6	C₂×C₁₉₉⋊C₈	—	2	1	no	—
7	C₂×C₁₉₉⋊C₈	C₄	1	2	yes	complemented
8	C₄×Dic₁₉₉	—	2	1	no	—
9	C₄×Dic₁₉₉	—	2	1	no	—
10	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₉₉	—	2	1	no	—
12	C₂×C₈×C₁₉₉	—	2	1	no	—
13	C₂×C₈×C₁₉₉	C₄	1	2	yes	complemented
14	C₂×Dic₁₉₉	—	2	2	no	—
15	C₂×Dic₁₉₉	—	2	2	no	—
16	C₂×Dic₁₉₉	C₈	1	4	yes	complemented
17	C₁₉₉⋊C₈	—	2	2	no	—
18	C₂³×C₁₉₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₉₉	Q₈	1	1	yes	characteristic
25	C₈×C₁₉₉	—	2	2	no	—
26	C₂²×C₁₉₉	C₄²	1	1	yes	characteristic
27	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₁₉₉	C₂×C₈	1	2	yes	—
30	C₂²×C₁₉₉	M₁₆	1	2	yes	—
31	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₄×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
34	C₄×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
35	Dic₁₉₉	—	2	4	no	—
36	C₂×C₁₉₉	#95	1	1	yes	characteristic
37	C₂×C₁₉₉	C₄×C₈	1	1	yes	characteristic
38	C₂×C₁₉₉	#97	1	1	yes	characteristic
39	C₂×C₁₉₉	C₂²⋊C₈	1	2	yes	—
40	C₂×C₁₉₉	C₄⋊C₈	1	2	yes	—
41	C₁₉₉	#336	1	1	yes	characteristic, complemented, Sylow
42	#336	—	199	1	no	maximal, Sylow
43	C₂²×C₈	—	199	1	no	—
44	C₂²×C₈	D₁₉₉	1	1	yes	characteristic
45	C₂×C₄²	—	199	1	no	—
46	C₂²×C₄	—	199	2	no	—
47	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
48	C₂×C₈	—	2	1	no	—
49	C₂×C₈	—	2	1	no	—
50	C₂×C₈	—	398	1	no	—
51	C₂×C₈	—	398	1	no	—
52	C₂×C₈	—	199	2	no	—
53	C₂×C₈	Dic₁₉₉	1	2	yes	complemented
54	C₄²	—	398	1	no	—
55	C₄²	—	398	1	no	—
56	C₂³	C₄×D₁₉₉	1	1	yes	characteristic
57	C₂×C₄	—	398	2	no	—
58	C₂×C₄	—	398	2	no	—
59	C₂×C₄	—	199	4	no	—
60	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
61	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
64	C₂×C₄	D₇₉₆	1	1	yes	characteristic
65	C₂×C₄	Dic₃₉₈	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₈	—	398	2	no	—
68	C₂²	C₄×Dic₁₉₉	1	1	yes	characteristic
69	C₂²	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
70	C₂²	Dic₁₉₉⋊C₄	1	1	yes	characteristic
71	C₂²	C₈×D₁₉₉	1	2	yes	—
72	C₂²	#15691869	1	2	yes	—
73	C₄	—	398	4	no	—
74	C₄	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
75	C₄	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
76	C₄	Dic₁₉₉⋊C₄	1	1	yes	characteristic
77	C₄	C₄⋊Dic₁₉₉	1	1	yes	characteristic
78	C₂	C₁₉₉⋊#95	1	1	yes	characteristic
79	C₂	C₈×Dic₁₉₉	1	1	yes	characteristic
80	C₂	#20263865	1	1	yes	characteristic
81	C₂	(C₂×D₁₉₉)⋊C₈	1	2	yes	—
82	C₂	Dic₁₉₉⋊C₈	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₁₉₉
commutator	C₂×C₁₉₉	C₄×C₈
Fitting	C₂²×C₈×C₁₉₉	C₂
socle, prime radical	C₂³×C₁₉₉	C₂×C₄
cosocle	C₂²×C₄×C₁₉₉	C₂²
hypercenter	C₂²×C₈	D₁₉₉
hypocenter	C₁₉₉	#336
LCS	G ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
7. C₂×C₁₉₉⋊C₈	C₄	semidirect
13. C₂×C₈×C₁₉₉	C₄	semidirect
16. C₂×Dic₁₉₉	C₈	semidirect
41. C₁₉₉	#336	semidirect
53. C₂×C₈	Dic₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₉₉⋊C₈	2
C₂×C₄×Dic₁₉₉	2
C₂²×C₈×C₁₉₉	2
#336	199