Subgroups — #28371421

Order 13952.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20294917	C₂	1	7	yes	maximal
2	#20294910	C₂	1	7	yes	maximal, complemented
3	C₄³×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13182008	C₂²	1	7	yes	—
5	C₂×#13182006	C₂²	1	7	yes	—
6	#16812606	C₂²	1	14	yes	—
7	C₂×C₄²×C₁₀₉	C₂²	1	7	yes	—
8	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
9	C₂²×Dic₁₀₉	C₂³	1	7	yes	—
10	#13182008	—	2	14	no	—
11	#13182006	—	2	14	no	—
12	C₂²×C₄×C₁₀₉	C₂³	1	7	yes	—
13	C₄²×C₁₀₉	—	2	14	no	—
14	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
15	C₂²×D₁₀₉	—	2	7	no	—
16	C₂×C₄×C₁₀₉	—	2	14	no	—
17	C₂×C₄×C₁₀₉	#41	1	14	yes	—
18	C₂×Dic₁₀₉	—	2	7	no	—
19	C₂×Dic₁₀₉	—	2	14	no	—
20	C₂²×C₁₀₉	C₂×#41	1	7	yes	—
21	C₂×D₁₀₉	—	4	7	no	—
22	C₄×C₁₀₉	—	2	14	no	—
23	Dic₁₀₉	—	4	7	no	—
24	C₂×C₁₀₉	#529	1	7	yes	—
25	D₁₀₉	—	8	1	no	—
26	#2865	—	109	1	no	maximal, Sylow
27	C₁₀₉	#2865	1	1	yes	characteristic, complemented, Sylow
28	#388	—	109	7	no	—
29	#387	—	109	7	no	—
30	C₄³	D₁₀₉	1	1	yes	characteristic, complemented
31	C₂×C₄⋊C₄	—	109	7	no	—
32	C₂×C₂²⋊C₄	—	109	7	no	—
33	C₂×C₄²	C₂×D₁₀₉	1	7	yes	—
34	#95	—	109	14	no	—
35	C₂⁴	—	109	1	no	—
36	C₂²×C₄	—	109	7	no	—
37	C₂²×C₄	C₂²×D₁₀₉	1	7	yes	—
38	C₄⋊C₄	—	218	14	no	—
39	C₂²⋊C₄	—	218	14	no	—
40	C₄²	—	2	14	no	—
41	C₂³	—	218	7	no	—
42	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
43	C₂×C₄	—	2	14	no	—
44	C₂×C₄	—	218	7	no	—
45	C₂×C₄	—	218	14	no	—
46	C₂×C₄	#13182029	1	14	yes	—
47	C₂²	—	436	7	no	—
48	C₂²	C₂×#13182029	1	7	yes	—
49	C₄	—	2	14	no	—
50	C₄	—	436	7	no	—
51	C₂	—	872	1	no	—
52	C₂	#20295560	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₉
commutator, socle, cosocle	C₂³×C₁₀₉	C₂⁴
Fitting	C₄³×C₁₀₉	C₂
prime radical	C₂³×D₁₀₉	C₂³
hypercenter	C₄³	D₁₀₉
hypocenter	C₁₀₉	#2865
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
2. #20294910	C₂	semidirect
3. C₄³×C₁₀₉	C₂	semidirect
27. C₁₀₉	#2865	semidirect
30. C₄³	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20294917	2
#20294910	2
C₄³×C₁₀₉	2
#2865	109