Subgroups — #27164285

Order 12736.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263981	C₂	1	2	yes	maximal, complemented
2	#20263961	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263891	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263872	C₂	1	2	yes	maximal, complemented
5	C₂×D₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15691886	C₂²	1	1	yes	characteristic
7	#15691874	—	2	2	no	—
8	#15691871	—	2	2	no	—
9	#15691870	C₂²	1	1	yes	characteristic
10	#15691870	C₂²	1	2	yes	—
11	#15691869	—	2	1	no	—
12	C₈×D₁₉₉	—	2	1	no	—
13	C₂×D₄×C₁₉₉	C₂²	1	2	yes	—
14	D₈×C₁₉₉	—	2	1	no	—
15	D₈×C₁₉₉	—	2	1	no	—
16	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
17	Dic₃₉₈	D₄	1	1	yes	characteristic
18	D₇₉₆	D₄	1	1	yes	characteristic
19	C₂²⋊D₁₉₉	—	2	1	no	—
20	C₄×D₁₉₉	—	2	1	no	—
21	C₁₉₉⋊C₈	—	2	1	no	—
22	C₁₉₉⋊C₈	—	2	2	no	—
23	C₂³×C₁₉₉	—	2	2	no	—
24	D₄×C₁₉₉	—	2	2	no	—
25	D₄×C₁₉₉	—	2	2	no	—
26	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
27	C₈×C₁₉₉	D₄	1	1	yes	characteristic
28	C₈×C₁₉₉	D₄	1	1	yes	characteristic
29	C₂²×C₁₉₉	—	4	2	no	—
30	C₂²×C₁₉₉	—	8	2	no	—
31	C₂²×C₁₉₉	#41	1	1	yes	characteristic
32	C₂×D₁₉₉	—	2	1	no	—
33	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
34	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
35	Dic₁₉₉	—	2	1	no	—
36	C₂×C₁₉₉	—	2	1	no	—
37	C₂×C₁₉₉	—	8	2	no	—
38	C₂×C₁₉₉	#121	1	1	yes	characteristic
39	D₁₉₉	—	4	1	no	—
40	C₁₉₉	#472	1	1	yes	characteristic, complemented, Sylow
41	#472	—	199	1	no	maximal, Sylow
42	AGL(1,ℤ/8)	—	199	2	no	—
43	C₂×D₈	D₁₉₉	1	1	yes	characteristic, complemented
44	#131	—	199	1	no	—
45	#108	—	199	1	no	—
46	#100	—	199	2	no	—
47	#41	—	199	1	no	—
48	C₂×D₄	C₂×D₁₉₉	1	2	yes	—
49	SD₁₆	—	398	2	no	—
50	D₈	—	2	1	no	—
51	D₈	—	2	1	no	—
52	D₈	—	398	2	no	—
53	M₁₆	—	398	1	no	—
54	M₁₆	—	199	1	no	—
55	M₁₆	—	199	2	no	—
56	C₂×C₈	—	398	1	no	—
57	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
58	C₂³	—	2	2	no	—
59	Q₈	—	199	1	no	—
60	D₄	—	2	2	no	—
61	D₄	—	2	2	no	—
62	D₄	—	398	1	no	—
63	D₄	—	199	1	no	—
64	C₂×C₄	—	398	1	no	—
65	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
66	C₈	—	398	2	no	—
67	C₈	—	398	1	no	—
68	C₈	C₂²⋊D₁₉₉	1	1	yes	characteristic
69	C₈	C₂²⋊D₁₉₉	1	1	yes	characteristic
70	C₂²	—	4	2	no	—
71	C₂²	—	8	2	no	—
72	C₂²	—	398	1	no	—
73	C₂²	#15691885	1	1	yes	characteristic
74	C₄	—	398	1	no	—
75	C₄	D₄×D₁₉₉	1	1	yes	characteristic
76	C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	8	2	no	—
79	C₂	—	796	1	no	—
80	C₂	#20263926	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20263926
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	C₂×D₈×C₁₉₉	C₂
socle	C₂×C₁₉₉	#121
hypercenter	C₂×D₈	D₁₉₉
hypocenter	C₁₉₉	#472
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #20263981	C₂	semidirect
2. #20263961	C₂	semidirect
3. #20263891	C₂	semidirect
4. #20263872	C₂	semidirect
5. C₂×D₈×C₁₉₉	C₂	semidirect
40. C₁₉₉	#472	semidirect
43. C₂×D₈	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263981	2
#20263961	2
#20263891	2
#20263872	2
C₂×D₈×C₁₉₉	2
#472	199