Subgroups — #28361085

Order 13952.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295428	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#13182004	C₂	1	1	yes	maximal, characteristic
3	#334×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812691	—	2	1	no	—
5	C₂×#13182004	—	2	1	no	—
6	#16812633	C₄	1	1	yes	characteristic
7	#16812633	C₄	1	1	yes	characteristic
8	#16812631	—	2	1	no	—
9	C₄⋊C₈×C₁₀₉	—	2	1	no	—
10	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₀₉	—	2	1	no	—
12	C₄⋊Dic₁₀₉	—	2	1	no	—
13	#13182008	—	4	1	no	—
14	#13182008	—	4	1	no	—
15	#13182004	—	4	1	no	—
16	C₂×C₈×C₁₀₉	—	4	1	no	—
17	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
19	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
22	C₂×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
23	C₂×Dic₁₀₉	—	4	1	no	—
24	C₂×Dic₁₀₉	—	4	1	no	—
25	C₈×C₁₀₉	—	2	1	no	—
26	C₈×C₁₀₉	—	8	1	no	—
27	C₈×C₁₀₉	SD₁₆	1	1	yes	characteristic
28	C₈×C₁₀₉	Q₁₆	1	1	yes	characteristic
29	C₂²×C₁₀₉	#102	1	1	yes	characteristic
30	C₄×C₁₀₉	—	2	1	no	—
31	C₄×C₁₀₉	—	2	1	no	—
32	C₄×C₁₀₉	#101	1	1	yes	characteristic
33	C₄×C₁₀₉	#103	1	1	yes	characteristic
34	Dic₁₀₉	—	8	1	no	—
35	Dic₁₀₉	—	8	1	no	—
36	C₂×C₁₀₉	#332	1	1	yes	characteristic
37	C₂×C₁₀₉	#359	1	1	yes	characteristic
38	C₂×C₁₀₉	#360	1	1	yes	characteristic
39	#1363	—	109	1	no	maximal, Sylow
40	C₁₀₉	#1363	1	1	yes	characteristic, complemented, Sylow
41	#499	—	109	1	no	—
42	C₄⋊C₁₆	—	109	1	no	—
43	#334	D₁₀₉	1	1	yes	characteristic
44	#125	—	218	1	no	—
45	C₂×C₁₆	—	218	1	no	—
46	#107	—	109	1	no	—
47	#107	—	109	1	no	—
48	#106	—	218	1	no	—
49	C₄⋊C₈	—	2	1	no	—
50	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
51	C₂×C₈	—	4	1	no	—
52	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
53	C₂×C₈	D₄₃₆	1	1	yes	characteristic
54	C₄⋊C₄	—	436	1	no	—
55	C₄⋊C₄	—	436	1	no	—
56	C₄⋊C₄	—	218	1	no	—
57	C₄⋊C₄	—	218	1	no	—
58	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
59	C₁₆	—	436	1	no	—
60	C₂×C₄	—	436	1	no	—
61	C₂×C₄	—	436	1	no	—
62	C₂×C₄	#13182006	1	1	yes	characteristic
63	C₂×C₄	#13182014	1	1	yes	characteristic
64	C₂×C₄	#13182016	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	8	1	no	—
67	C₈	#13182018	1	1	yes	characteristic
68	C₈	Dic₄₃₆	1	1	yes	characteristic
69	C₂²	#16812619	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	2	1	no	—
72	C₄	—	872	1	no	—
73	C₄	—	872	1	no	—
74	C₄	#16812617	1	1	yes	characteristic
75	C₄	#16812624	1	1	yes	characteristic
76	C₂	#20294773	1	1	yes	characteristic
77	C₂	#20294832	1	1	yes	characteristic
78	C₂	#20294835	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16812619
commutator	C₂×C₈×C₁₀₉	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₀₉
Fitting	#334×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	#102
cosocle	C₄×C₈×C₁₀₉	C₂²
hypercenter	#334	D₁₀₉
hypocenter	C₁₀₉	#1363
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

Complements

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

N	H ≅ G/N	type
40. C₁₀₉	#1363	semidirect

Maximal subgroups

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

subgroup	index
#20295428	2
C₄⋊#13182004	2
#334×C₁₀₉	2
#1363	109