Subgroups — #28365273

Order 13952.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295428	C₂	1	1	yes	maximal, characteristic, complemented
2	#20294827	C₂	1	4	yes	maximal, complemented
3	#20294802	C₂	1	1	yes	maximal, characteristic, complemented
4	#493×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#16812691	—	2	1	no	—
6	C₂×#13182004	C₂²	1	2	yes	—
7	#16812633	C₂²	1	2	yes	—
8	#16812631	—	2	1	no	—
9	C₂×D₈×C₁₀₉	—	2	1	no	—
10	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
11	#127×C₁₀₉	—	2	1	no	—
12	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₀₉	—	2	2	no	—
14	#13182008	—	4	2	no	—
15	#13182004	—	2	2	no	—
16	C₂×D₄×C₁₀₉	—	2	2	no	—
17	C₂×D₄×C₁₀₉	—	4	2	no	—
18	D₈×C₁₀₉	—	2	4	no	—
19	D₈×C₁₀₉	—	4	2	no	—
20	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
23	C₂³×C₁₀₉	—	4	2	no	—
24	D₄×C₁₀₉	—	4	2	no	—
25	D₄×C₁₀₉	—	4	2	no	—
26	D₄×C₁₀₉	—	8	4	no	—
27	C₂×C₄×C₁₀₉	D₈	1	2	yes	—
28	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
29	C₂×Dic₁₀₉	—	4	2	no	—
30	C₈×C₁₀₉	SD₁₆	1	2	yes	—
31	C₈×C₁₀₉	#41	1	2	yes	—
32	C₂²×C₁₀₉	—	8	2	no	—
33	C₂²×C₁₀₉	—	8	4	no	—
34	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
35	C₄×C₁₀₉	—	2	2	no	—
36	C₄×C₁₀₉	#124	1	1	yes	characteristic
37	C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
38	Dic₁₀₉	—	8	2	no	—
39	C₂×C₁₀₉	—	16	2	no	—
40	C₂×C₁₀₉	#486	1	1	yes	characteristic
41	C₂×C₁₀₉	#509	1	2	yes	—
42	#2244	—	109	1	no	maximal, Sylow
43	C₁₀₉	#2244	1	1	yes	characteristic, complemented, Sylow
44	#499	—	109	1	no	—
45	#493	D₁₀₉	1	1	yes	characteristic
46	#357	—	109	4	no	—
47	#346	—	109	1	no	—
48	C₂×D₈	—	2	1	no	—
49	C₂×D₈	C₂×D₁₀₉	1	2	yes	—
50	#127	—	2	1	no	—
51	#125	—	218	1	no	—
52	C₂×C₁₆	—	109	2	no	—
53	#107	—	109	2	no	—
54	#106	—	218	1	no	—
55	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
56	C₂×D₄	—	2	2	no	—
57	C₂×D₄	—	4	2	no	—
58	D₈	—	2	4	no	—
59	D₈	—	4	2	no	—
60	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
62	C₄⋊C₄	—	436	2	no	—
63	C₄⋊C₄	—	218	2	no	—
64	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
65	C₁₆	—	218	2	no	—
66	C₂³	—	4	2	no	—
67	D₄	—	4	2	no	—
68	D₄	—	4	2	no	—
69	D₄	—	8	4	no	—
70	C₂×C₄	—	436	2	no	—
71	C₂×C₄	#13182014	1	2	yes	—
72	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
73	C₈	#13182017	1	2	yes	—
74	C₈	#13182028	1	2	yes	—
75	C₂²	—	8	4	no	—
76	C₂²	—	8	2	no	—
77	C₂²	C₂×#13182014	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	872	2	no	—
80	C₄	#16812686	1	1	yes	characteristic
81	C₄	C₂×#13182017	1	1	yes	characteristic
82	C₂	—	16	2	no	—
83	C₂	#20295372	1	1	yes	characteristic
84	C₂	#20295462	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13182014
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting, prime radical	#493×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₈
hypercenter	#493	D₁₀₉
hypocenter	C₁₀₉	#2244
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

Complements

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

N	H ≅ G/N	type
1. #20295428	C₂	semidirect
2. #20294827	C₂	semidirect
3. #20294802	C₂	semidirect
43. C₁₀₉	#2244	semidirect

Maximal subgroups

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

subgroup	index
#20295428	2
#20294827	2
#20294802	2
#493×C₁₀₉	2
#2244	109