Subgroups — #27164215

Order 12736.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691874	C₂	1	2	yes	maximal, complemented
2	C₄×Dic₃₉₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263876	C₂	1	2	yes	maximal, complemented
5	#127×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₈	C₂²	1	1	yes	characteristic
7	#15691874	—	2	4	no	—
8	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
10	Dic₁₉₉⋊C₄	—	2	1	no	—
11	C₄×Dic₁₉₉	—	2	1	no	—
12	C₂×D₄×C₁₉₉	—	2	1	no	—
13	C₂×D₄×C₁₉₉	—	2	1	no	—
14	C₂×D₄×C₁₉₉	C₂²	1	2	yes	—
15	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
16	Dic₃₉₈	—	2	1	no	—
17	Dic₃₉₈	D₄	1	2	yes	complemented
18	C₂×Dic₁₉₉	—	2	1	no	—
19	C₂×Dic₁₉₉	—	2	1	no	—
20	C₁₉₉⋊C₈	—	2	2	no	—
21	C₂³×C₁₉₉	—	2	2	no	—
22	D₄×C₁₉₉	—	2	2	no	—
23	D₄×C₁₉₉	—	2	2	no	—
24	D₄×C₁₉₉	—	4	2	no	—
25	D₄×C₁₉₉	—	4	2	no	—
26	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
28	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
29	C₂²×C₁₉₉	—	4	2	no	—
30	C₂²×C₁₉₉	—	4	2	no	—
31	C₂²×C₁₉₉	—	4	2	no	—
32	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
33	C₄×C₁₉₉	—	2	1	no	—
34	C₄×C₁₉₉	SD₁₆	1	2	yes	complemented
35	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
36	C₄×C₁₉₉	#41	1	1	yes	characteristic
37	Dic₁₉₉	—	2	2	no	—
38	Dic₁₉₉	—	4	1	no	—
39	C₂×C₁₉₉	—	8	2	no	—
40	C₂×C₁₉₉	#121	1	1	yes	characteristic
41	C₂×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₁₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic
43	C₁₉₉	#460	1	1	yes	characteristic, complemented, Sylow
44	#460	—	199	1	no	maximal, Sylow
45	C₂×SD₁₆	—	199	2	no	—
46	#127	D₁₉₉	1	1	yes	characteristic
47	C₄×Q₈	—	199	1	no	—
48	C₄⋊C₈	—	199	1	no	—
49	#102	—	199	2	no	—
50	C₂×Q₈	—	199	1	no	—
51	C₂×D₄	—	2	1	no	—
52	C₂×D₄	—	2	1	no	—
53	C₂×D₄	C₂×D₁₉₉	1	2	yes	—
54	SD₁₆	—	398	4	no	—
55	C₂×C₈	—	199	2	no	—
56	C₄⋊C₄	—	398	1	no	—
57	C₄⋊C₄	—	199	1	no	—
58	C₄²	—	398	1	no	—
59	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
60	C₂³	—	2	2	no	—
61	Q₈	—	398	1	no	—
62	Q₈	—	199	2	no	—
63	D₄	—	2	2	no	—
64	D₄	—	2	2	no	—
65	D₄	—	4	2	no	—
66	D₄	—	4	2	no	—
67	C₂×C₄	—	398	1	no	—
68	C₂×C₄	—	398	1	no	—
69	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
70	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
72	C₈	—	398	2	no	—
73	C₂²	—	4	2	no	—
74	C₂²	—	4	2	no	—
75	C₂²	—	4	2	no	—
76	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	796	1	no	—
79	C₄	—	398	2	no	—
80	C₄	#15691874	1	2	yes	complemented
81	C₄	D₄×D₁₉₉	1	1	yes	characteristic
82	C₄	#15691885	1	1	yes	characteristic
83	C₂	—	8	2	no	—
84	C₂	#20263926	1	1	yes	characteristic
85	C₂	C₂×#15691874	1	1	yes	characteristic
86	C₂	#20263981	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting, prime radical	#127×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
hypercenter	#127	D₁₉₉
hypocenter	C₁₉₉	#460
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×#15691874	C₂	semidirect
2. C₄×Dic₃₉₈	C₂	semidirect
3. C₄⋊(C₁₉₉⋊C₈)	C₂	semidirect
4. #20263876	C₂	semidirect
17. Dic₃₉₈	D₄	semidirect
34. C₄×C₁₉₉	SD₁₆	semidirect
43. C₁₉₉	#460	semidirect
80. C₄	#15691874	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691874	2
C₄×Dic₃₉₈	2
C₄⋊(C₁₉₉⋊C₈)	2
#20263876	2
#127×C₁₉₉	2
#460	199