Subgroups — #36355883

Order 22144.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011923	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₃₈₄	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₂₇₆₈	C₂	1	2	yes	maximal, complemented
4	#26011330	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19149701	—	2	1	no	—
7	C₂×Dic₆₉₂	—	2	1	no	—
8	Dic₁₃₈₄	—	2	1	no	—
9	Dic₁₃₈₄	C₂²	1	2	yes	—
10	Dic₁₃₈₄	C₂²	1	1	yes	characteristic
11	#19149624	—	2	1	no	—
12	D₂₇₆₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
14	C₃₂×C₁₇₃	C₂²	1	2	yes	—
15	#14583826	—	4	1	no	—
16	C₂×Dic₃₄₆	—	4	1	no	—
17	Dic₆₉₂	—	2	1	no	—
18	Dic₆₉₂	—	2	2	no	—
19	Dic₆₉₂	—	4	1	no	—
20	#14583815	—	4	1	no	—
21	D₁₃₈₄	—	2	1	no	—
22	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
23	C₁₆×C₁₇₃	D₄	1	1	yes	characteristic
24	C₁₆×C₁₇₃	C₂³	1	1	yes	characteristic
25	Dic₃₄₆	—	4	1	no	—
26	Dic₃₄₆	—	4	2	no	—
27	Dic₃₄₆	—	8	1	no	—
28	D₆₉₂	—	4	1	no	—
29	C₂²⋊D₁₇₃	—	8	1	no	—
30	C₂×Dic₁₇₃	—	8	1	no	—
31	C₄×D₁₇₃	—	8	1	no	—
32	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
33	C₈×C₁₇₃	D₈	1	1	yes	characteristic
34	C₈×C₁₇₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₇₃	D₁₆	1	1	yes	characteristic
36	C₂×D₁₇₃	—	8	1	no	—
37	C₄×C₁₇₃	D₁₆	1	1	yes	characteristic
38	C₄×C₁₇₃	C₂×D₈	1	1	yes	characteristic
39	Dic₁₇₃	—	8	1	no	—
40	Dic₁₇₃	—	8	2	no	—
41	C₂×C₁₇₃	—	2	1	no	—
42	C₂×C₁₇₃	C₂×D₁₆	1	1	yes	characteristic
43	D₁₇₃	—	16	1	no	—
44	C₁₇₃	#2264	1	1	yes	characteristic, complemented, Sylow
45	#2264	—	173	1	no	maximal, Sylow
46	#508	—	173	1	no	—
47	C₂×Q₃₂	—	173	1	no	—
48	Q₆₄	—	173	2	no	—
49	SD₆₄	—	173	2	no	—
50	M₆₄	D₁₇₃	1	1	yes	characteristic, complemented
51	#135	—	346	1	no	—
52	C₂×Q₁₆	—	346	1	no	—
53	Q₃₂	—	346	1	no	—
54	Q₃₂	—	173	2	no	—
55	Q₃₂	—	173	1	no	—
56	SD₃₂	—	346	1	no	—
57	D₁₆	—	173	1	no	—
58	C₂×C₁₆	C₂×D₁₇₃	1	1	yes	characteristic
59	C₃₂	C₂×D₁₇₃	1	2	yes	—
60	#41	—	692	1	no	—
61	C₂×Q₈	—	692	1	no	—
62	Q₁₆	—	346	1	no	—
63	Q₁₆	—	346	2	no	—
64	Q₁₆	—	692	1	no	—
65	SD₁₆	—	692	1	no	—
66	D₈	—	346	1	no	—
67	C₂×C₈	D₆₉₂	1	1	yes	characteristic
68	C₁₆	D₆₉₂	1	1	yes	characteristic
69	C₁₆	C₂²×D₁₇₃	1	1	yes	characteristic
70	Q₈	—	1384	1	no	—
71	Q₈	—	692	1	no	—
72	Q₈	—	692	2	no	—
73	D₄	—	1384	1	no	—
74	D₄	—	692	1	no	—
75	C₂×C₄	—	1384	1	no	—
76	C₂×C₄	—	1384	1	no	—
77	C₂×C₄	D₁₃₈₄	1	1	yes	characteristic
78	C₈	D₁₃₈₄	1	1	yes	characteristic
79	C₈	C₂×D₆₉₂	1	1	yes	characteristic
80	C₂²	—	1384	1	no	—
81	C₂²	D₂₇₆₈	1	1	yes	characteristic
82	C₄	—	1384	1	no	—
83	C₄	—	1384	2	no	—
84	C₄	D₂₇₆₈	1	1	yes	characteristic
85	C₄	C₂×D₁₃₈₄	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2768	1	no	—
88	C₂	C₂×D₂₇₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₇₆₈
commutator, cosocle	C₁₆×C₁₇₃	C₂³
Frattini	C₁₆	C₂²×D₁₇₃
Fitting	M₆₄×C₁₇₃	C₂
socle	C₂×C₁₇₃	C₂×D₁₆
prime radical	#26011923	C₂
hypercenter	M₆₄	D₁₇₃
hypocenter	C₁₇₃	#2264
LCS	G ▷ C₁₆×C₁₇₃ ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
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×Dic₁₃₈₄	C₂	semidirect
3. Dic₂₇₆₈	C₂	semidirect
4. #26011330	C₂	semidirect
5. M₆₄×C₁₇₃	C₂	semidirect
44. C₁₇₃	#2264	semidirect
50. M₆₄	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26011923	2
C₂×Dic₁₃₈₄	2
Dic₂₇₆₈	2
#26011330	2
M₆₄×C₁₇₃	2
#2264	173