Subgroups — #27164376

Order 12736.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263954	C₂	1	1	yes	maximal, characteristic, complemented
2	#20263942	C₂	1	1	yes	maximal, characteristic
3	#20263880	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263878	C₂	1	1	yes	maximal, characteristic, complemented
5	#20263876	C₂	1	1	yes	maximal, characteristic
6	#20263865	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₁₉₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₉₈	—	2	1	no	—
9	C₂×Dic₃₉₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
12	Dic₁₉₉⋊C₄	—	2	1	no	—
13	C₂²⋊Dic₁₉₉	—	2	1	no	—
14	C₂²⋊Dic₁₉₉	—	2	1	no	—
15	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₁₉₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
19	Dic₃₉₈	—	2	1	no	—
20	Dic₃₉₈	—	2	1	no	—
21	Dic₃₉₈	—	4	2	no	—
22	C₂×Dic₁₉₉	—	2	1	no	—
23	C₂×Dic₁₉₉	—	2	1	no	—
24	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
25	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
26	C₁₉₉⋊C₈	—	2	1	no	—
27	C₂³×C₁₉₉	—	2	1	no	—
28	D₄×C₁₉₉	—	2	1	no	—
29	D₄×C₁₉₉	—	2	1	no	—
30	C₂×C₄×C₁₉₉	—	2	1	no	—
31	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
32	C₈×C₁₉₉	—	2	1	no	—
33	C₂²×C₁₉₉	—	4	1	no	—
34	C₂²×C₁₉₉	—	4	1	no	—
35	C₂²×C₁₉₉	—	4	1	no	—
36	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
37	C₄×C₁₉₉	—	4	1	no	—
38	C₄×C₁₉₉	#41	1	1	yes	characteristic
39	C₄×C₁₉₉	#41	1	1	yes	characteristic
40	Dic₁₉₉	—	2	1	no	—
41	Dic₁₉₉	—	2	1	no	—
42	Dic₁₉₉	—	4	1	no	—
43	Dic₁₉₉	—	4	1	no	—
44	C₂×C₁₉₉	—	8	1	no	—
45	C₂×C₁₉₉	#124	1	1	yes	characteristic
46	C₂×C₁₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic
47	C₂×C₁₉₉	#137	1	1	yes	characteristic
48	C₁₉₉	#489	1	1	yes	characteristic, complemented, Sylow
49	#489	—	199	1	no	maximal, Sylow
50	C₄⋊Q₈	—	199	1	no	—
51	#124	—	199	1	no	—
52	#103	—	199	1	no	—
53	#103	—	199	1	no	—
54	#102	—	199	1	no	—
55	#102	D₁₉₉	1	1	yes	characteristic
56	#97	—	199	1	no	—
57	C₂×Q₈	—	398	1	no	—
58	C₂×Q₈	—	199	1	no	—
59	C₂×D₄	C₂×D₁₉₉	1	1	yes	characteristic
60	C₂×C₈	—	199	1	no	—
61	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
62	C₄⋊C₄	—	398	1	no	—
63	C₄⋊C₄	—	199	1	no	—
64	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
65	C₂²⋊C₄	—	398	1	no	—
66	C₂²⋊C₄	—	398	1	no	—
67	C₄²	—	199	1	no	—
68	C₂³	—	2	1	no	—
69	Q₈	—	796	2	no	—
70	Q₈	—	398	1	no	—
71	Q₈	—	398	1	no	—
72	D₄	—	2	1	no	—
73	D₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	398	1	no	—
76	C₂×C₄	—	398	1	no	—
77	C₂×C₄	—	199	1	no	—
78	C₂×C₄	—	199	1	no	—
79	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	398	1	no	—
82	C₂²	—	4	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	4	1	no	—
85	C₂²	D₄×D₁₉₉	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	796	1	no	—
88	C₄	—	796	1	no	—
89	C₄	—	398	1	no	—
90	C₄	—	398	1	no	—
91	C₄	#15691885	1	1	yes	characteristic
92	C₄	#15691886	1	1	yes	characteristic
93	C₂	—	8	1	no	—
94	C₂	#20263941	1	1	yes	characteristic
95	C₂	#20263979	1	1	yes	characteristic
96	C₂	#20263984	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#102×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
prime radical	C₂×D₄×C₁₉₉	C₂²
hypercenter	#102	D₁₉₉
hypocenter	C₁₉₉	#489
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
1. #20263954	C₂	semidirect
3. #20263880	C₂	semidirect
4. #20263878	C₂	semidirect
6. #20263865	C₂	semidirect
48. C₁₉₉	#489	semidirect

Maximal subgroups

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

subgroup	index
#20263954	2
#20263942	2
#20263880	2
#20263878	2
#20263876	2
#20263865	2
#102×C₁₉₉	2
#489	199