Subgroups — #28365352

Order 13952.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295458	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₈₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₁₇₄₄	C₂	1	2	yes	maximal, complemented
4	#20294865	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812721	—	2	1	no	—
7	C₂×Dic₄₃₆	—	2	1	no	—
8	Dic₈₇₂	—	2	1	no	—
9	Dic₈₇₂	C₂²	1	2	yes	—
10	Dic₈₇₂	C₂²	1	1	yes	characteristic
11	#16812644	—	2	1	no	—
12	D₁₇₄₄	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₃₂×C₁₀₉	C₂²	1	2	yes	—
15	#13182029	—	4	1	no	—
16	C₂×Dic₂₁₈	—	4	1	no	—
17	Dic₄₃₆	—	2	1	no	—
18	Dic₄₃₆	—	2	2	no	—
19	Dic₄₃₆	—	4	1	no	—
20	#13182018	—	4	1	no	—
21	D₈₇₂	—	2	1	no	—
22	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
23	C₁₆×C₁₀₉	D₄	1	1	yes	characteristic
24	C₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
26	C₂²⋊D₁₀₉	—	8	1	no	—
27	C₂×Dic₁₀₉	—	8	1	no	—
28	D₄₃₆	—	4	1	no	—
29	C₄×D₁₀₉	—	8	1	no	—
30	Dic₂₁₈	—	4	1	no	—
31	Dic₂₁₈	—	4	2	no	—
32	Dic₂₁₈	—	8	1	no	—
33	C₈×C₁₀₉	D₈	1	1	yes	characteristic
34	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₀₉	D₁₆	1	1	yes	characteristic
36	C₂×D₁₀₉	—	8	1	no	—
37	C₄×C₁₀₉	D₁₆	1	1	yes	characteristic
38	C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
39	Dic₁₀₉	—	8	2	no	—
40	Dic₁₀₉	—	8	1	no	—
41	C₂×C₁₀₉	—	2	1	no	—
42	C₂×C₁₀₉	C₂×D₁₆	1	1	yes	characteristic
43	D₁₀₉	—	16	1	no	—
44	#2264	—	109	1	no	maximal, Sylow
45	C₁₀₉	#2264	1	1	yes	characteristic, complemented, Sylow
46	#508	—	109	1	no	—
47	C₂×Q₃₂	—	109	1	no	—
48	Q₆₄	—	109	2	no	—
49	SD₆₄	—	109	2	no	—
50	M₆₄	D₁₀₉	1	1	yes	characteristic, complemented
51	#135	—	218	1	no	—
52	C₂×Q₁₆	—	218	1	no	—
53	Q₃₂	—	109	2	no	—
54	Q₃₂	—	109	1	no	—
55	Q₃₂	—	218	1	no	—
56	SD₃₂	—	218	1	no	—
57	D₁₆	—	109	1	no	—
58	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
59	C₃₂	C₂×D₁₀₉	1	2	yes	—
60	#41	—	436	1	no	—
61	C₂×Q₈	—	436	1	no	—
62	Q₁₆	—	436	1	no	—
63	Q₁₆	—	218	1	no	—
64	Q₁₆	—	218	2	no	—
65	SD₁₆	—	436	1	no	—
66	D₈	—	218	1	no	—
67	C₂×C₈	D₄₃₆	1	1	yes	characteristic
68	C₁₆	D₄₃₆	1	1	yes	characteristic
69	C₁₆	C₂²×D₁₀₉	1	1	yes	characteristic
70	Q₈	—	872	1	no	—
71	Q₈	—	436	1	no	—
72	Q₈	—	436	2	no	—
73	D₄	—	872	1	no	—
74	D₄	—	436	1	no	—
75	C₂×C₄	—	872	1	no	—
76	C₂×C₄	—	872	1	no	—
77	C₂×C₄	D₈₇₂	1	1	yes	characteristic
78	C₈	D₈₇₂	1	1	yes	characteristic
79	C₈	C₂×D₄₃₆	1	1	yes	characteristic
80	C₂²	—	872	1	no	—
81	C₂²	D₁₇₄₄	1	1	yes	characteristic
82	C₄	—	872	1	no	—
83	C₄	—	872	2	no	—
84	C₄	D₁₇₄₄	1	1	yes	characteristic
85	C₄	C₂×D₈₇₂	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	1744	1	no	—
88	C₂	C₂×D₁₇₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₇₄₄
commutator, cosocle	C₁₆×C₁₀₉	C₂³
Frattini	C₁₆	C₂²×D₁₀₉
Fitting	M₆₄×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×D₁₆
prime radical	#20295458	C₂
hypercenter	M₆₄	D₁₀₉
hypocenter	C₁₀₉	#2264
LCS	G ▷ C₁₆×C₁₀₉ ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₈₇₂	C₂	semidirect
3. Dic₁₇₄₄	C₂	semidirect
4. #20294865	C₂	semidirect
5. M₆₄×C₁₀₉	C₂	semidirect
45. C₁₀₉	#2264	semidirect
50. M₆₄	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295458	2
C₂×Dic₈₇₂	2
Dic₁₇₄₄	2
#20294865	2
M₆₄×C₁₀₉	2
#2264	109