Subgroups — #28365347

Order 13952.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295458	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₇₄₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#20294865	C₂	1	2	yes	maximal, complemented
4	D₃₄₈₈	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812721	—	2	1	no	—
7	C₂×D₈₇₂	—	2	1	no	—
8	Dic₈₇₂	C₂²	1	1	yes	characteristic, complemented
9	#16812644	—	2	1	no	—
10	D₁₇₄₄	—	2	1	no	—
11	D₁₇₄₄	C₂²	1	2	yes	—
12	D₁₇₄₄	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₃₂×C₁₀₉	C₂²	1	2	yes	complemented
15	#13182029	—	4	1	no	—
16	C₂×D₄₃₆	—	4	1	no	—
17	Dic₄₃₆	—	2	1	no	—
18	#13182018	—	4	1	no	—
19	D₈₇₂	—	2	2	no	—
20	D₈₇₂	—	2	1	no	—
21	D₈₇₂	—	4	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	C₂²×D₁₀₉	—	8	1	no	—
26	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
27	C₂²⋊D₁₀₉	—	8	1	no	—
28	D₄₃₆	—	4	2	no	—
29	D₄₃₆	—	4	1	no	—
30	D₄₃₆	—	8	1	no	—
31	C₄×D₁₀₉	—	8	1	no	—
32	Dic₂₁₈	—	4	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	C₂×D₁₀₉	—	8	2	no	—
38	C₂×D₁₀₉	—	16	2	no	—
39	C₄×C₁₀₉	D₁₆	1	1	yes	characteristic
40	C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
41	Dic₁₀₉	—	8	1	no	—
42	C₂×C₁₀₉	—	2	1	no	—
43	C₂×C₁₀₉	C₂×D₁₆	1	1	yes	characteristic
44	D₁₀₉	—	16	1	no	—
45	D₁₀₉	—	16	2	no	—
46	#2263	—	109	1	no	maximal, Sylow
47	C₁₀₉	#2263	1	1	yes	characteristic, complemented, Sylow
48	#508	—	109	1	no	—
49	C₂×D₁₆	—	109	1	no	—
50	SD₆₄	—	109	2	no	—
51	D₃₂	—	109	2	no	—
52	M₆₄	D₁₀₉	1	1	yes	characteristic, complemented
53	#135	—	218	1	no	—
54	C₂×D₈	—	218	1	no	—
55	Q₃₂	—	109	1	no	—
56	SD₃₂	—	218	1	no	—
57	D₁₆	—	109	2	no	—
58	D₁₆	—	109	1	no	—
59	D₁₆	—	218	1	no	—
60	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
61	C₃₂	C₂×D₁₀₉	1	2	yes	complemented
62	#41	—	436	1	no	—
63	C₂×D₄	—	436	1	no	—
64	Q₁₆	—	218	1	no	—
65	SD₁₆	—	436	1	no	—
66	D₈	—	436	1	no	—
67	D₈	—	218	2	no	—
68	D₈	—	218	1	no	—
69	C₂×C₈	D₄₃₆	1	1	yes	characteristic
70	C₁₆	D₄₃₆	1	1	yes	characteristic
71	C₁₆	C₂²×D₁₀₉	1	1	yes	characteristic
72	C₂³	—	872	1	no	—
73	Q₈	—	436	1	no	—
74	D₄	—	872	1	no	—
75	D₄	—	872	1	no	—
76	D₄	—	436	2	no	—
77	D₄	—	436	1	no	—
78	C₂×C₄	—	872	1	no	—
79	C₂×C₄	D₈₇₂	1	1	yes	characteristic
80	C₈	D₈₇₂	1	1	yes	characteristic
81	C₈	C₂×D₄₃₆	1	1	yes	characteristic
82	C₂²	—	872	1	no	—
83	C₂²	—	872	2	no	—
84	C₂²	—	1744	2	no	—
85	C₂²	D₁₇₄₄	1	1	yes	characteristic
86	C₄	—	872	1	no	—
87	C₄	D₁₇₄₄	1	1	yes	characteristic
88	C₄	C₂×D₈₇₂	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	1744	2	no	—
91	C₂	—	1744	1	no	—
92	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
hypercenter	M₆₄	D₁₀₉
hypocenter	C₁₀₉	#2263
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
1. #20295458	C₂	semidirect
2. C₂×D₁₇₄₄	C₂	semidirect
3. #20294865	C₂	semidirect
4. D₃₄₈₈	C₂	semidirect
5. M₆₄×C₁₀₉	C₂	semidirect
8. Dic₈₇₂	C₂²	semidirect
12. D₁₇₄₄	C₂²	semidirect
14. C₃₂×C₁₀₉	C₂²	semidirect
47. C₁₀₉	#2263	semidirect
52. M₆₄	D₁₀₉	semidirect
61. C₃₂	C₂×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×D₁₇₄₄	2
#20294865	2
D₃₄₈₈	2
M₆₄×C₁₀₉	2
#2263	109