Subgroups — #27163999

Order 12736.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691885	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₁₉₉	C₂	1	2	yes	maximal, complemented
3	#20263873	C₂	1	2	yes	maximal, complemented
4	#20263872	C₂	1	1	yes	maximal, characteristic, complemented
5	#100×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15691885	—	2	2	no	—
7	#15691885	—	2	2	no	—
8	C₂×Dic₃₉₈	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊D₁₉₉	C₄	1	2	yes	—
10	C₂²×Dic₁₉₉	C₄	1	2	yes	—
11	C₂×C₄×D₁₉₉	C₂²	1	1	yes	characteristic
12	#15691870	C₂²	1	2	yes	—
13	#15691869	—	2	2	no	—
14	C₈×D₁₉₉	—	2	2	no	—
15	C₂×D₄×C₁₉₉	C₂²	1	1	yes	characteristic
16	M₁₆×C₁₉₉	C₂²	1	2	yes	—
17	C₂²×D₁₉₉	C₂×C₄	1	1	yes	characteristic
18	Dic₃₉₈	—	2	2	no	—
19	C₂²⋊D₁₉₉	—	2	4	no	—
20	C₂×Dic₁₉₉	—	2	2	no	—
21	C₂×Dic₁₉₉	—	2	2	no	—
22	C₂×Dic₁₉₉	C₂×C₄	1	2	yes	—
23	C₂×Dic₁₉₉	C₂×C₄	1	1	yes	characteristic
24	C₄×D₁₉₉	D₄	1	2	yes	—
25	C₄×D₁₉₉	D₄	1	2	yes	—
26	C₁₉₉⋊C₈	—	2	2	no	—
27	C₂³×C₁₉₉	C₂×C₄	1	2	yes	—
28	D₄×C₁₉₉	—	2	2	no	—
29	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
30	C₈×C₁₉₉	—	2	2	no	—
31	C₂²×C₁₉₉	—	2	2	no	—
32	C₂²×C₁₉₉	—	4	2	no	—
33	C₂²×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×D₁₉₉	—	4	1	no	—
35	C₂×D₁₉₉	C₂²⋊C₄	1	2	yes	—
36	C₄×C₁₉₉	C₂×D₄	1	2	yes	—
37	Dic₁₉₉	—	2	2	no	—
38	Dic₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
39	Dic₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₁₉₉	—	2	1	no	—
41	C₂×C₁₉₉	—	4	2	no	—
42	C₂×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	D₁₉₉	—	2	2	no	—
44	C₁₉₉	#413	1	1	yes	characteristic, complemented, Sylow
45	#413	—	199	1	no	maximal, Sylow
46	C₂×#41	—	199	1	no	—
47	C₂×M₁₆	—	199	2	no	—
48	#101	—	199	2	no	—
49	#100	—	199	1	no	—
50	#100	D₁₉₉	1	1	yes	characteristic, complemented
51	#41	—	398	2	no	—
52	#41	—	398	2	no	—
53	C₂×Q₈	—	199	1	no	—
54	C₂×D₄	—	199	2	no	—
55	C₂×D₄	C₂×D₁₉₉	1	1	yes	characteristic
56	C₂²×C₄	—	199	2	no	—
57	C₂²×C₄	—	199	1	no	—
58	M₁₆	—	398	2	no	—
59	M₁₆	—	199	2	no	—
60	M₁₆	C₂×D₁₉₉	1	2	yes	—
61	C₂×C₈	—	398	2	no	—
62	C₂³	—	199	1	no	—
63	C₂³	C₄×D₁₉₉	1	2	yes	—
64	Q₈	—	398	2	no	—
65	D₄	—	2	2	no	—
66	D₄	—	398	4	no	—
67	C₂×C₄	—	398	2	no	—
68	C₂×C₄	—	398	2	no	—
69	C₂×C₄	—	199	2	no	—
70	C₂×C₄	—	199	1	no	—
71	C₂×C₄	—	199	2	no	—
72	C₂×C₄	—	199	2	no	—
73	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₈	—	398	2	no	—
76	C₂²	—	2	2	no	—
77	C₂²	—	4	2	no	—
78	C₂²	—	796	1	no	—
79	C₂²	—	199	2	no	—
80	C₂²	C₂×C₄×D₁₉₉	1	1	yes	characteristic
81	C₄	—	398	2	no	—
82	C₄	—	199	1	no	—
83	C₄	—	199	1	no	—
84	C₄	D₄×D₁₉₉	1	2	yes	—
85	C₂	—	2	1	no	—
86	C₂	—	4	2	no	—
87	C₂	—	398	2	no	—
88	C₂	C₂²⋊C₄×D₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊C₄×D₁₉₉
commutator	C₂²×C₁₉₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#100×C₁₉₉	C₂
socle	C₂×C₁₉₉	C₂×C₂²⋊C₄
prime radical	C₂×#15691885	C₂
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	#100	D₁₉₉
hypocenter	C₁₉₉	#413
LCS	G ▷ C₂²×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

Complements

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

N	H ≅ G/N	type
2. M₁₆×D₁₉₉	C₂	semidirect
3. #20263873	C₂	semidirect
4. #20263872	C₂	semidirect
5. #100×C₁₉₉	C₂	semidirect
44. C₁₉₉	#413	semidirect
50. #100	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691885	2
M₁₆×D₁₉₉	2
#20263873	2
#20263872	2
#100×C₁₉₉	2
#413	199