Subgroups — #36355448

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	#26011607	C₂	1	2	yes	maximal, complemented
3	#26011297	C₂	1	2	yes	maximal, complemented
4	#26011266	C₂	1	1	yes	maximal, characteristic, complemented
5	#368×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19149701	C₂²	1	2	yes	—
7	#19149687	—	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	C₂×#14583801	C₂²	1	1	yes	characteristic
12	#19149605	—	2	2	no	—
13	C₈×Dic₁₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
15	#108×C₁₇₃	C₂²	1	2	yes	—
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	#14583809	—	4	2	no	—
21	C₂×#13063436	D₄	1	1	yes	characteristic
22	C₈×D₁₇₃	—	4	2	no	—
23	C₄×Dic₁₇₃	D₄	1	1	yes	characteristic
24	#14583801	—	2	1	no	—
25	M₁₆×C₁₇₃	—	2	2	no	—
26	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
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	C₂×Dic₁₇₃	—	2	1	no	—
33	C₄×D₁₇₃	—	4	2	no	—
34	#13063436	D₈	1	2	yes	—
35	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
36	C₈×C₁₇₃	—	4	2	no	—
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	C₄×D₄	1	1	yes	characteristic
42	C₄×C₁₇₃	C₂×D₈	1	1	yes	characteristic
43	Dic₁₇₃	—	4	1	no	—
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	D₁₇₃	1	1	yes	characteristic, complemented
53	#359	—	173	2	no	—
54	#346	—	173	1	no	—
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	—	173	1	no	—
61	C₂×C₁₆	C₂×D₁₇₃	1	1	yes	characteristic
62	#108	C₂×D₁₇₃	1	2	yes	—
63	#104	—	346	2	no	—
64	C₄×C₈	—	173	1	no	—
65	#41	—	346	2	no	—
66	Q₁₆	—	173	2	no	—
67	SD₁₆	—	346	2	no	—
68	D₈	—	173	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	692	2	no	—
71	C₂×C₈	—	173	1	no	—
72	C₂×C₈	—	692	2	no	—
73	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
74	C₄²	—	173	1	no	—
75	C₁₆	—	346	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	—	346	1	no	—
82	C₂×C₄	—	692	2	no	—
83	C₂×C₄	D₄×D₁₇₃	1	1	yes	characteristic
84	C₈	—	4	2	no	—
85	C₈	—	173	2	no	—
86	C₈	C₂×C₄×D₁₇₃	1	1	yes	characteristic
87	C₈	#14583827	1	1	yes	characteristic
88	C₂²	—	692	2	no	—
89	C₂²	#19149702	1	1	yes	characteristic
90	C₄	—	692	1	no	—
91	C₄	—	692	2	no	—
92	C₄	#19149641	1	1	yes	characteristic
93	C₄	D₈×D₁₇₃	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	1384	2	no	—
96	C₂	#26011573	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₁₇₃
commutator	C₈×C₁₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₇₃
Fitting	#368×C₁₇₃	C₂
socle	C₂×C₁₇₃	C₄×D₈
prime radical	#26011923	C₂
cosocle	C₂×C₈×C₁₇₃	C₂³
hypercenter	#368	D₁₇₃
hypocenter	C₁₇₃	#2179
LCS	G ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

Complements

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

N	H ≅ G/N	type
2. #26011607	C₂	semidirect
3. #26011297	C₂	semidirect
4. #26011266	C₂	semidirect
5. #368×C₁₇₃	C₂	semidirect
8. Dic₁₃₈₄	C₄	semidirect
9. #19149624	C₄	semidirect
10. D₂₇₆₈	C₄	semidirect
48. C₁₇₃	#2179	semidirect
52. #368	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
#26011607	2
#26011297	2
#26011266	2
#368×C₁₇₃	2
#2179	173