Subgroups — #27169164

Order 12928.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265542	C₂	1	1	yes	maximal, characteristic
2	#20264955	C₂	1	1	yes	maximal, characteristic, complemented
3	#347×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#15692252	C₄	1	1	yes	characteristic
5	#15692247	C₄	1	1	yes	characteristic
6	#15692195	—	2	1	no	—
7	#15692177	—	2	1	no	—
8	#15692177	—	2	1	no	—
9	M₃₂×C₁₀₁	—	2	1	no	—
10	#97×C₁₀₁	C₂²	1	1	yes	characteristic
11	C₂×D₄₀₄	—	2	1	no	—
12	C₂×D₄₀₄	—	4	1	no	—
13	C₄⋊Dic₁₀₁	—	2	1	no	—
14	#13108053	—	4	1	no	—
15	#13108049	—	2	2	no	—
16	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
17	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
18	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₁₀₁	—	2	2	no	—
20	C₂²×D₁₀₁	—	4	1	no	—
21	C₂×C₄×C₁₀₁	—	2	1	no	—
22	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×Dic₁₀₁	—	4	1	no	—
24	D₄₀₄	—	4	1	no	—
25	D₄₀₄	—	4	1	no	—
26	D₄₀₄	—	8	2	no	—
27	C₈×C₁₀₁	—	2	1	no	—
28	C₈×C₁₀₁	—	2	1	no	—
29	C₂²×C₁₀₁	#100	1	1	yes	characteristic
30	C₂×D₁₀₁	—	8	2	no	—
31	C₂×D₁₀₁	—	8	1	no	—
32	C₄×C₁₀₁	—	4	1	no	—
33	C₄×C₁₀₁	#104	1	1	yes	characteristic
34	C₄×C₁₀₁	#104	1	1	yes	characteristic
35	Dic₁₀₁	—	8	1	no	—
36	C₂×C₁₀₁	—	2	1	no	—
37	C₂×C₁₀₁	#329	1	1	yes	characteristic
38	D₁₀₁	—	16	1	no	—
39	#1355	—	101	1	no	maximal, Sylow
40	C₁₀₁	#1355	1	1	yes	characteristic, complemented, Sylow
41	#490	—	101	1	no	—
42	#347	—	101	1	no	—
43	#347	D₁₀₁	1	1	yes	characteristic, complemented
44	#127	—	101	1	no	—
45	#125	—	101	1	no	—
46	M₃₂	—	2	1	no	—
47	M₃₂	—	202	1	no	—
48	#102	—	202	1	no	—
49	#102	—	202	1	no	—
50	#97	C₂×D₁₀₁	1	1	yes	characteristic
51	C₂×D₄	—	404	1	no	—
52	C₂×D₄	—	202	1	no	—
53	C₂×C₈	D₄₀₄	1	1	yes	characteristic
54	C₂×C₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
55	C₄⋊C₄	—	404	1	no	—
56	C₄⋊C₄	—	202	1	no	—
57	C₄²	C₄×D₁₀₁	1	1	yes	characteristic
58	C₁₆	—	2	2	no	—
59	C₁₆	—	202	2	no	—
60	C₂³	—	404	1	no	—
61	D₄	—	808	2	no	—
62	D₄	—	404	1	no	—
63	D₄	—	404	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	404	1	no	—
66	C₂×C₄	#13108051	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	2	1	no	—
69	C₂²	—	808	2	no	—
70	C₂²	—	808	1	no	—
71	C₂²	#15692171	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	808	1	no	—
74	C₄	#15692181	1	1	yes	characteristic
75	C₄	#15692183	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	1616	1	no	—
78	C₂	#20264916	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20264916
commutator	C₄²×C₁₀₁	C₂×C₄
Frattini	#97	C₂×D₁₀₁
Fitting	#347×C₁₀₁	C₂
socle	C₂×C₁₀₁	#329
prime radical	#15692252	C₄
cosocle	#97×C₁₀₁	C₂²
hypercenter	#347	D₁₀₁
hypocenter	C₁₀₁	#1355
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
2. #20264955	C₂	semidirect
3. #347×C₁₀₁	C₂	semidirect
40. C₁₀₁	#1355	semidirect
43. #347	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265542	2
#20264955	2
#347×C₁₀₁	2
#1355	101