Subgroups — #28362703

Order 13952.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295431	C₂	1	2	yes	maximal, complemented
2	#20295406	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295399	C₂	1	1	yes	maximal, characteristic
4	#20295373	C₂	1	2	yes	maximal
5	#322×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₃₆	—	2	2	no	—
7	C₂×#13182018	—	2	2	no	—
8	#16812699	C₂²	1	1	yes	characteristic
9	#16812699	C₂²	1	2	yes	—
10	#16812696	C₂²	1	1	yes	characteristic
11	#16812633	—	2	4	no	—
12	#16812621	—	2	4	no	—
13	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
14	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₁₈	—	2	1	no	—
16	C₂×Dic₂₁₈	—	2	2	no	—
17	C₂×D₄₃₆	—	2	2	no	—
18	C₂×D₄₃₆	—	2	1	no	—
19	Dic₄₃₆	—	4	4	no	—
20	#13182018	—	4	4	no	—
21	C₄⋊Dic₁₀₉	—	2	2	no	—
22	C₄⋊Dic₁₀₉	—	2	2	no	—
23	C₄⋊Dic₁₀₉	—	2	2	no	—
24	C₂×C₈×C₁₀₉	D₄	1	4	yes	—
25	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
26	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂²×D₁₀₉	—	4	1	no	—
28	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
30	C₂×Dic₁₀₉	—	4	2	no	—
31	C₂×Dic₁₀₉	—	4	1	no	—
32	D₄₃₆	—	4	2	no	—
33	D₄₃₆	—	4	4	no	—
34	Dic₂₁₈	—	4	2	no	—
35	Dic₂₁₈	—	4	4	no	—
36	C₈×C₁₀₉	—	2	4	no	—
37	C₈×C₁₀₉	D₈	1	4	yes	—
38	C₂²×C₁₀₉	#127	1	1	yes	characteristic
39	C₂×D₁₀₉	—	8	1	no	—
40	C₂×D₁₀₉	—	8	2	no	—
41	C₄×C₁₀₉	C₂×D₈	1	2	yes	—
42	C₄×C₁₀₉	#137	1	4	yes	—
43	Dic₁₀₉	—	8	1	no	—
44	Dic₁₀₉	—	8	2	no	—
45	C₂×C₁₀₉	#493	1	1	yes	characteristic
46	C₂×C₁₀₉	#497	1	2	yes	—
47	D₁₀₉	—	16	1	no	—
48	#1722	—	109	1	no	maximal, Sylow
49	C₁₀₉	#1722	1	1	yes	characteristic, complemented, Sylow
50	#500	—	109	2	no	—
51	#494	—	109	1	no	—
52	#492	—	109	1	no	—
53	#486	—	109	2	no	—
54	#322	D₁₀₉	1	1	yes	characteristic, complemented
55	C₂×Q₁₆	—	218	2	no	—
56	C₂×SD₁₆	—	218	2	no	—
57	C₄⋊Q₈	—	109	2	no	—
58	C₄⋊Q₈	—	109	1	no	—
59	#127	—	109	1	no	—
60	#107	—	218	4	no	—
61	#102	—	218	4	no	—
62	C₄×C₈	C₂×D₁₀₉	1	2	yes	—
63	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
64	C₂×Q₈	—	218	1	no	—
65	C₂×Q₈	—	218	2	no	—
66	C₂×D₄	—	218	2	no	—
67	C₂×D₄	—	218	1	no	—
68	Q₁₆	—	436	4	no	—
69	SD₁₆	—	436	4	no	—
70	C₂×C₈	D₄₃₆	1	4	yes	—
71	C₂×C₈	D₄₃₆	1	2	yes	—
72	C₄⋊C₄	—	218	2	no	—
73	C₄⋊C₄	—	218	2	no	—
74	C₄⋊C₄	—	218	2	no	—
75	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
76	C₂³	—	436	1	no	—
77	Q₈	—	436	2	no	—
78	Q₈	—	436	4	no	—
79	D₄	—	436	2	no	—
80	D₄	—	436	4	no	—
81	C₂×C₄	—	436	2	no	—
82	C₂×C₄	—	436	1	no	—
83	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
84	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
85	C₈	—	2	4	no	—
86	C₈	D₈₇₂	1	4	yes	—
87	C₂²	—	872	1	no	—
88	C₂²	—	872	2	no	—
89	C₂²	#16812696	1	1	yes	characteristic
90	C₄	—	872	1	no	—
91	C₄	—	872	2	no	—
92	C₄	C₂×D₈₇₂	1	2	yes	—
93	C₄	#16812732	1	4	yes	—
94	C₂	—	1744	1	no	—
95	C₂	#20295403	1	1	yes	characteristic
96	C₂	#20295420	1	2	yes	—

	subgroup	G/H
center	C₂²	#16812696
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#322×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#127
prime radical	#16812696	C₂²
hypercenter	#322	D₁₀₉
hypocenter	C₁₀₉	#1722
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
1. #20295431	C₂	semidirect
2. #20295406	C₂	semidirect
5. #322×C₁₀₉	C₂	semidirect
49. C₁₀₉	#1722	semidirect
54. #322	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295431	2
#20295406	2
#20295399	2
#20295373	2
#322×C₁₀₉	2
#1722	109