Subgroups — #28378175

Order 13952.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295937	C₂	1	2	yes	maximal
2	#20295396	C₂	1	1	yes	maximal, characteristic
3	#16812760	—	2	2	no	—
4	#16812699	C₂²	1	1	yes	characteristic
5	#16812622	—	2	2	no	—
6	#16812610	C₄	1	1	yes	characteristic
7	#125×C₁₀₉	C₄	1	1	yes	characteristic
8	#13182038	—	4	2	no	—
9	C₂×Dic₂₁₈	D₄	1	2	yes	—
10	C₂×#11724784	—	2	1	no	—
11	C₄⋊Dic₁₀₉	—	4	1	no	—
12	C₄⋊C₄×C₁₀₉	—	2	1	no	—
13	C₄⋊C₄×C₁₀₉	—	4	1	no	—
14	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₀₉	—	2	1	no	—
16	C₂×C₄×C₁₀₉	—	4	1	no	—
17	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
18	C₂×Dic₁₀₉	—	2	2	no	—
19	Dic₂₁₈	—	4	2	no	—
20	#11724786	—	8	2	no	—
21	#11724784	—	4	1	no	—
22	C₂²×C₁₀₉	#99	1	1	yes	characteristic
23	C₄×C₁₀₉	—	2	1	no	—
24	C₄×C₁₀₉	—	4	1	no	—
25	C₄×C₁₀₉	—	8	1	no	—
26	Dic₁₀₉	—	4	2	no	—
27	C₂×C₁₀₉	—	2	1	no	—
28	C₂×C₁₀₉	#354	1	1	yes	characteristic
29	#1413	—	109	1	no	maximal, Sylow
30	C₁₀₉	#1413	1	1	yes	characteristic, complemented, Sylow
31	#491	—	109	1	no	—
32	#356	—	109	2	no	—
33	C₄⋊Q₈	—	109	1	no	—
34	#125	#89137	1	1	yes	characteristic
35	#103	—	218	2	no	—
36	#101	—	218	2	no	—
37	#97	—	109	1	no	—
38	C₂×Q₈	—	109	2	no	—
39	M₁₆	—	436	2	no	—
40	C₂×C₈	—	218	1	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	—	4	1	no	—
43	C₄⋊C₄	—	436	1	no	—
44	C₄²	C₂×#89137	1	1	yes	characteristic
45	Q₈	—	436	2	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	—	218	2	no	—
49	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
50	C₈	—	872	2	no	—
51	C₈	—	436	1	no	—
52	C₂²	#16812757	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	4	1	no	—
55	C₄	—	8	1	no	—
56	C₄	—	436	2	no	—
57	C₂	—	2	1	no	—
58	C₂	#20295932	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20295932
commutator	C₄²×C₁₀₉	C₂×C₄
Frattini	C₄²	C₂×#89137
Fitting	#125×C₁₀₉	C₄
socle	C₂×C₁₀₉	#354
prime radical	C₂²×C₁₀₉	#99
cosocle	#16812699	C₂²
hypercenter	#125	#89137
hypocenter	C₁₀₉	#1413
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
30. C₁₀₉	#1413	semidirect

Maximal subgroups

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

subgroup	index
#20295937	2
#20295396	2
#1413	109