Subgroups — #108×D₁₉₉ #27164062

Order 12736.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×D₁₉₉	C₂	1	2	yes	maximal
2	C₂×C₈×D₁₉₉	C₂	1	1	yes	maximal, characteristic
3	#20263891	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263890	C₂	1	2	yes	maximal, complemented
5	#108×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₁₉₉	C₂²	1	1	yes	characteristic
7	#15691870	C₂²	1	2	yes	—
8	#15691869	—	2	2	no	—
9	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₈×D₁₉₉	—	2	2	no	—
11	C₈×D₁₉₉	C₄	1	2	yes	—
12	C₈×D₁₉₉	C₄	1	2	yes	—
13	M₁₆×C₁₉₉	C₂²	1	2	yes	—
14	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₂²×D₁₉₉	Q₈	1	1	yes	characteristic
16	C₂×Dic₁₉₉	Q₈	1	1	yes	characteristic
17	C₄×D₁₉₉	C₂×C₄	1	1	yes	characteristic
18	C₄×D₁₉₉	C₂×C₄	1	1	yes	characteristic
19	C₄×D₁₉₉	D₄	1	1	yes	characteristic
20	C₄×D₁₉₉	D₄	1	1	yes	characteristic
21	C₁₉₉⋊C₈	—	2	2	no	—
22	C₁₉₉⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
24	C₈×C₁₉₉	—	2	2	no	—
25	C₈×C₁₉₉	C₂×C₄	1	2	yes	—
26	C₂²×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
27	C₂×D₁₉₉	—	2	2	no	—
28	C₂×D₁₉₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂×D₁₉₉	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
32	Dic₁₉₉	C₄⋊C₄	1	1	yes	characteristic
33	Dic₁₉₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₁₉₉	—	2	1	no	—
35	C₂×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
36	D₁₉₉	—	2	1	no	—
37	D₁₉₉	#108	1	2	yes	direct product factor
38	C₁₉₉	C₂×#108	1	1	yes	characteristic, complemented, Sylow
39	C₂×#108	—	199	1	no	maximal, Sylow
40	C₂×M₁₆	—	199	2	no	—
41	C₂²×C₈	—	199	1	no	—
42	#108	—	199	2	no	—
43	#108	—	199	1	no	—
44	#108	D₁₉₉	1	1	yes	characteristic, direct product factor
45	C₂²×C₄	—	199	1	no	—
46	M₁₆	—	398	2	no	—
47	M₁₆	—	199	2	no	—
48	M₁₆	C₂×D₁₉₉	1	2	yes	—
49	C₂×C₈	—	398	2	no	—
50	C₂×C₈	—	199	1	no	—
51	C₂×C₈	—	199	2	no	—
52	C₂×C₈	—	199	2	no	—
53	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
54	C₂³	—	199	1	no	—
55	C₂×C₄	—	199	1	no	—
56	C₂×C₄	—	199	1	no	—
57	C₂×C₄	—	199	1	no	—
58	C₂×C₄	—	199	1	no	—
59	C₂×C₄	—	199	1	no	—
60	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
61	C₈	—	2	2	no	—
62	C₈	—	398	2	no	—
63	C₈	—	199	2	no	—
64	C₈	C₄×D₁₉₉	1	2	yes	—
65	C₂²	—	398	2	no	—
66	C₂²	—	199	1	no	—
67	C₂²	—	199	1	no	—
68	C₂²	Q₈×D₁₉₉	1	1	yes	characteristic
69	C₄	—	199	1	no	—
70	C₄	—	199	1	no	—
71	C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
72	C₄	D₄×D₁₉₉	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	—	398	1	no	—
75	C₂	—	199	2	no	—
76	C₂	C₄⋊C₄×D₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₁₉₉
commutator	C₄×C₁₉₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#108×C₁₉₉	C₂
socle	C₂×C₁₉₉	C₂×C₄⋊C₄
prime radical	C₂²×D₁₉₉	Q₈
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	#108	D₁₉₉
hypocenter	C₁₉₉	C₂×#108
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
3. #20263891	C₂	semidirect
4. #20263890	C₂	semidirect
5. #108×C₁₉₉	C₂	semidirect
37. D₁₉₉	#108	direct (N × H)
38. C₁₉₉	C₂×#108	semidirect
44. #108	D₁₉₉	direct (N × H)

Maximal subgroups

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

subgroup	index
M₁₆×D₁₉₉	2
C₂×C₈×D₁₉₉	2
#20263891	2
#20263890	2
#108×C₁₉₉	2
C₂×#108	199

Subgroups — #108×D199 #27164062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #108×D₁₉₉ #27164062