Subgroups — #28365308

Order 13952.

116 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	#20294853	C₂	1	2	yes	maximal
3	#20294851	C₂	1	2	yes	maximal
4	C₄×#13182004	C₂	1	1	yes	maximal, characteristic
5	#499×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#16812691	—	2	1	no	—
7	C₂×#13182004	C₂²	1	1	yes	characteristic
8	C₂×#13182004	C₂²	1	1	yes	characteristic
9	#16812633	C₂²	1	2	yes	—
10	#16812631	—	2	1	no	—
11	#125×C₁₀₉	—	2	1	no	—
12	#107×C₁₀₉	C₂²	1	2	yes	—
13	#106×C₁₀₉	—	2	1	no	—
14	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₀₉	—	2	2	no	—
16	#13182008	—	4	2	no	—
17	#13182004	Q₈	1	2	yes	—
18	#13182004	Q₈	1	2	yes	—
19	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₁₀₉	—	2	2	no	—
22	C₄⋊C₄×C₁₀₉	—	4	2	no	—
23	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₀₉	—	4	2	no	—
25	C₂×C₄×C₁₀₉	D₈	1	2	yes	—
26	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₁₀₉	—	4	2	no	—
28	C₈×C₁₀₉	Q₁₆	1	2	yes	—
29	C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
30	C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
32	C₄×C₁₀₉	—	2	2	no	—
33	C₄×C₁₀₉	—	8	2	no	—
34	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
35	C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
36	Dic₁₀₉	—	8	2	no	—
37	C₂×C₁₀₉	#500	1	1	yes	characteristic
38	C₂×C₁₀₉	#508	1	2	yes	—
39	#2253	—	109	1	no	maximal, Sylow
40	C₁₀₉	#2253	1	1	yes	characteristic, complemented, Sylow
41	#499	—	109	1	no	—
42	#499	D₁₀₉	1	1	yes	characteristic
43	#367	—	109	2	no	—
44	#366	—	109	2	no	—
45	C₄×C₁₆	—	109	1	no	—
46	#125	—	2	1	no	—
47	#125	—	218	1	no	—
48	C₂×C₁₆	—	109	1	no	—
49	C₂×C₁₆	—	109	1	no	—
50	#107	—	109	2	no	—
51	#107	C₂×D₁₀₉	1	2	yes	—
52	#106	—	2	1	no	—
53	#106	—	218	1	no	—
54	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
55	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
56	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
57	C₄⋊C₄	—	2	2	no	—
58	C₄⋊C₄	—	4	2	no	—
59	C₄⋊C₄	—	436	2	no	—
60	C₄⋊C₄	—	218	2	no	—
61	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
62	C₁₆	—	109	2	no	—
63	C₁₆	—	109	2	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	436	2	no	—
66	C₂×C₄	#13182014	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
68	C₈	#13182019	1	2	yes	—
69	C₈	Q₈×D₁₀₉	1	1	yes	characteristic
70	C₈	Q₈×D₁₀₉	1	1	yes	characteristic
71	C₂²	C₂×#13182014	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	8	2	no	—
74	C₄	—	872	2	no	—
75	C₄	#16812697	1	1	yes	characteristic
76	C₄	C₂×#13182019	1	1	yes	characteristic
77	C₂	#20295429	1	1	yes	characteristic
78	C₂	#20295456	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13182014
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#499×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₂×D₈
hypercenter	#499	D₁₀₉
hypocenter	C₁₀₉	#2253
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
40. C₁₀₉	#2253	semidirect

Maximal subgroups

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

subgroup	index
#20295428	2
#20294853	2
#20294851	2
C₄×#13182004	2
#499×C₁₀₉	2
#2253	109