Subgroups — #27179521

Order 12928.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265068	C₂	1	7	yes	maximal
2	#20265061	C₂	1	7	yes	maximal, complemented
3	C₄³×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13108053	C₂²	1	7	yes	—
5	C₂×#13108051	C₂²	1	7	yes	—
6	#15692162	C₂²	1	14	yes	—
7	C₂×C₄²×C₁₀₁	C₂²	1	7	yes	—
8	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
9	C₂²×Dic₁₀₁	C₂³	1	7	yes	—
10	#13108053	—	2	14	no	—
11	#13108051	—	2	14	no	—
12	C₂²×C₄×C₁₀₁	C₂³	1	7	yes	—
13	C₄²×C₁₀₁	—	2	14	no	—
14	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
15	C₂²×D₁₀₁	—	2	7	no	—
16	C₂×C₄×C₁₀₁	—	2	14	no	—
17	C₂×C₄×C₁₀₁	#41	1	14	yes	—
18	C₂×Dic₁₀₁	—	2	7	no	—
19	C₂×Dic₁₀₁	—	2	14	no	—
20	C₂²×C₁₀₁	C₂×#41	1	7	yes	—
21	C₂×D₁₀₁	—	4	7	no	—
22	C₄×C₁₀₁	—	2	14	no	—
23	Dic₁₀₁	—	4	7	no	—
24	C₂×C₁₀₁	#529	1	7	yes	—
25	D₁₀₁	—	8	1	no	—
26	#2865	—	101	1	no	maximal, Sylow
27	C₁₀₁	#2865	1	1	yes	characteristic, complemented, Sylow
28	#388	—	101	7	no	—
29	#387	—	101	7	no	—
30	C₄³	D₁₀₁	1	1	yes	characteristic, complemented
31	C₂×C₄⋊C₄	—	101	7	no	—
32	C₂×C₂²⋊C₄	—	101	7	no	—
33	C₂×C₄²	C₂×D₁₀₁	1	7	yes	—
34	#95	—	101	14	no	—
35	C₂⁴	—	101	1	no	—
36	C₂²×C₄	—	101	7	no	—
37	C₂²×C₄	C₂²×D₁₀₁	1	7	yes	—
38	C₄⋊C₄	—	202	14	no	—
39	C₂²⋊C₄	—	202	14	no	—
40	C₄²	—	2	14	no	—
41	C₂³	—	202	7	no	—
42	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
43	C₂×C₄	—	2	14	no	—
44	C₂×C₄	—	202	7	no	—
45	C₂×C₄	—	202	14	no	—
46	C₂×C₄	#13108074	1	14	yes	—
47	C₂²	—	404	7	no	—
48	C₂²	C₂×#13108074	1	7	yes	—
49	C₄	—	2	14	no	—
50	C₄	—	404	7	no	—
51	C₂	—	808	1	no	—
52	C₂	#20265711	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator, socle, cosocle	C₂³×C₁₀₁	C₂⁴
Fitting	C₄³×C₁₀₁	C₂
prime radical	C₂³×D₁₀₁	C₂³
hypercenter	C₄³	D₁₀₁
hypocenter	C₁₀₁	#2865
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
2. #20265061	C₂	semidirect
3. C₄³×C₁₀₁	C₂	semidirect
27. C₁₀₁	#2865	semidirect
30. C₄³	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265068	2
#20265061	2
C₄³×C₁₀₁	2
#2865	101