Subgroups — #36355450

Order 22144.

144 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	#26011609	C₂	1	2	yes	maximal, complemented
3	#26011321	C₂	1	1	yes	maximal, characteristic, complemented
4	#26011299	C₂	1	2	yes	maximal, complemented
5	#346×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19149701	C₂²	1	2	yes	—
7	#19149685	—	2	2	no	—
8	Dic₁₃₈₄	C₄	1	1	yes	characteristic, complemented
9	#19149624	C₄	1	2	yes	complemented
10	D₂₇₆₈	C₄	1	1	yes	characteristic, complemented
11	#19149615	C₂²	1	2	yes	—
12	#19149607	—	2	2	no	—
13	C₂×C₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
16	#14583826	—	2	2	no	—
17	Dic₆₉₂	C₂×C₄	1	2	yes	—
18	#14583815	—	2	2	no	—
19	D₁₃₈₄	C₂×C₄	1	2	yes	—
20	#14583810	—	2	2	no	—
21	#14583809	—	4	2	no	—
22	C₈×D₁₇₃	—	4	2	no	—
23	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
25	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
26	C₁₆×C₁₇₃	—	2	1	no	—
27	C₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic
29	Dic₃₄₆	—	2	2	no	—
30	D₆₉₂	—	2	2	no	—
31	C₂²⋊D₁₇₃	—	4	2	no	—
32	C₄×D₁₇₃	—	4	2	no	—
33	#13063436	—	4	2	no	—
34	C₂×C₄×C₁₇₃	—	2	1	no	—
35	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
36	C₈×C₁₇₃	D₈	1	2	yes	—
37	C₈×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₇₃	#41	1	1	yes	characteristic
39	C₂²×C₁₇₃	#135	1	1	yes	characteristic
40	C₂×D₁₇₃	—	4	2	no	—
41	C₄×C₁₇₃	—	4	1	no	—
42	C₄×C₁₇₃	C₄×D₄	1	1	yes	characteristic
43	C₄×C₁₇₃	C₂×D₈	1	1	yes	characteristic
44	Dic₁₇₃	—	4	2	no	—
45	C₂×C₁₇₃	—	2	1	no	—
46	C₂×C₁₇₃	C₄×D₈	1	1	yes	characteristic
47	D₁₇₃	—	8	2	no	—
48	C₁₇₃	#2179	1	1	yes	characteristic, complemented, Sylow
49	#2179	—	173	1	no	maximal, Sylow
50	#508	—	173	1	no	—
51	#443	—	173	2	no	—
52	#368	—	173	1	no	—
53	#359	—	173	2	no	—
54	#346	D₁₇₃	1	1	yes	characteristic, complemented
55	#135	—	173	2	no	—
56	#131	—	346	2	no	—
57	Q₃₂	—	173	1	no	—
58	SD₃₂	—	173	2	no	—
59	D₁₆	—	173	1	no	—
60	C₂×C₁₆	C₂×D₁₇₃	1	1	yes	characteristic
61	C₂×C₁₆	C₂×D₁₇₃	1	1	yes	characteristic
62	#108	—	173	2	no	—
63	#104	—	346	2	no	—
64	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
65	#41	—	346	2	no	—
66	Q₁₆	—	173	2	no	—
67	SD₁₆	—	346	2	no	—
68	D₈	—	173	2	no	—
69	M₁₆	—	346	2	no	—
70	M₁₆	—	692	2	no	—
71	C₂×C₈	—	692	2	no	—
72	C₂×C₈	D₆₉₂	1	1	yes	characteristic
73	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
74	C₄²	D₆₉₂	1	1	yes	characteristic
75	C₁₆	—	2	1	no	—
76	C₁₆	C₄×D₁₇₃	1	1	yes	characteristic
77	C₁₆	C₄×D₁₇₃	1	1	yes	characteristic
78	Q₈	—	346	2	no	—
79	D₄	—	346	2	no	—
80	D₄	—	692	2	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	692	2	no	—
83	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
84	C₈	—	692	2	no	—
85	C₈	D₁₃₈₄	1	2	yes	—
86	C₈	C₂×C₄×D₁₇₃	1	1	yes	characteristic
87	C₈	#14583826	1	1	yes	characteristic
88	C₂²	—	692	2	no	—
89	C₂²	#19149701	1	1	yes	characteristic
90	C₄	—	4	1	no	—
91	C₄	—	692	2	no	—
92	C₄	C₄×D₆₉₂	1	1	yes	characteristic
93	C₄	C₂×D₁₃₈₄	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	1384	2	no	—
96	C₂	C₄×D₁₃₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₁₃₈₄
commutator	C₈×C₁₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₇₃
Fitting	#346×C₁₇₃	C₂
socle	C₂×C₁₇₃	C₄×D₈
prime radical	#26011923	C₂
cosocle	C₂×C₈×C₁₇₃	C₂³
hypercenter	#346	D₁₇₃
hypocenter	C₁₇₃	#2179
LCS	G ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. #26011609	C₂	semidirect
3. #26011321	C₂	semidirect
4. #26011299	C₂	semidirect
5. #346×C₁₇₃	C₂	semidirect
8. Dic₁₃₈₄	C₄	semidirect
9. #19149624	C₄	semidirect
10. D₂₇₆₈	C₄	semidirect
48. C₁₇₃	#2179	semidirect
54. #346	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
#26011609	2
#26011321	2
#26011299	2
#346×C₁₇₃	2
#2179	173