Subgroups — #28378036

Order 13952.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295918	C₂	1	2	yes	maximal, complemented
2	#20295392	C₂	1	1	yes	maximal, characteristic
3	#16812773	—	2	2	no	—
4	#16812696	C₄	1	1	yes	characteristic
5	#16812619	—	2	2	no	—
6	#16812610	C₂²	1	1	yes	characteristic
7	#125×C₁₀₉	C₄	1	1	yes	characteristic
8	#13182032	—	2	4	no	—
9	C₂×D₄₃₆	—	2	1	no	—
10	C₂×D₄₃₆	—	4	1	no	—
11	C₂×#11724784	D₄	1	2	yes	—
12	C₄⋊C₄×C₁₀₉	—	2	1	no	—
13	C₄⋊C₄×C₁₀₉	—	4	1	no	—
14	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×D₁₀₉	—	4	1	no	—
16	C₂×C₄×C₁₀₉	—	2	1	no	—
17	C₂×C₄×C₁₀₉	—	4	1	no	—
18	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
19	D₄₃₆	—	4	2	no	—
20	D₄₃₆	—	8	2	no	—
21	#11724784	—	2	2	no	—
22	C₂²×C₁₀₉	#100	1	1	yes	characteristic
23	C₂×D₁₀₉	—	8	2	no	—
24	C₂×D₁₀₉	—	8	1	no	—
25	C₄×C₁₀₉	—	4	1	no	—
26	C₄×C₁₀₉	—	8	1	no	—
27	C₄×C₁₀₉	#104	1	2	yes	—
28	C₂×C₁₀₉	—	2	1	no	—
29	C₂×C₁₀₉	#329	1	1	yes	characteristic
30	D₁₀₉	—	16	1	no	—
31	#1355	—	109	1	no	maximal, Sylow
32	C₁₀₉	#1355	1	1	yes	characteristic, complemented, Sylow
33	#490	—	109	1	no	—
34	#347	—	109	2	no	—
35	#127	—	109	1	no	—
36	#125	#89137	1	1	yes	characteristic
37	M₃₂	—	218	2	no	—
38	#102	—	218	2	no	—
39	#97	—	109	1	no	—
40	C₂×D₄	—	436	1	no	—
41	C₂×D₄	—	218	1	no	—
42	C₂×C₈	—	109	2	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	—	4	1	no	—
45	C₄²	C₂×#89137	1	1	yes	characteristic
46	C₁₆	—	218	4	no	—
47	C₂³	—	436	1	no	—
48	D₄	—	872	2	no	—
49	D₄	—	436	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	4	1	no	—
52	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
53	C₈	—	218	2	no	—
54	C₂²	—	872	2	no	—
55	C₂²	—	872	1	no	—
56	C₂²	#16812759	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	8	1	no	—
59	C₄	#16812764	1	2	yes	—
60	C₂	—	2	1	no	—
61	C₂	—	1744	1	no	—
62	C₂	#20295882	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20295882
commutator	C₄²×C₁₀₉	C₂×C₄
Frattini	C₄²	C₂×#89137
Fitting	#125×C₁₀₉	C₄
socle	C₂×C₁₀₉	#329
prime radical	#16812696	C₄
cosocle	#16812610	C₂²
hypercenter	#125	#89137
hypocenter	C₁₀₉	#1355
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #20295918	C₂	semidirect
32. C₁₀₉	#1355	semidirect

Maximal subgroups

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

subgroup	index
#20295918	2
#20295392	2
#1355	109