Subgroups — #36351655

Order 22144.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19149613	C₂	1	2	yes	maximal
2	C₂²×C₁₆×C₁₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊Dic₁₇₃	—	2	2	no	—
4	#19149613	—	2	2	no	—
5	#19149613	C₄	1	4	yes	complemented
6	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₆×C₁₇₃	—	2	2	no	—
8	C₂×C₁₆×C₁₇₃	C₄	1	2	yes	complemented
9	C₂²×Dic₁₇₃	—	4	2	no	—
10	C₄⋊Dic₁₇₃	—	2	4	no	—
11	C₄⋊Dic₁₇₃	—	4	2	no	—
12	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₇₃	C₂×C₄	1	2	yes	—
15	C₂×C₈×C₁₇₃	D₄	1	2	yes	—
16	C₂×C₈×C₁₇₃	Q₈	1	1	yes	characteristic
17	C₁₆×C₁₇₃	—	2	2	no	—
18	C₂×Dic₁₇₃	—	4	4	no	—
19	C₂×Dic₁₇₃	—	8	2	no	—
20	C₂×Dic₁₇₃	—	8	2	no	—
21	C₂³×C₁₇₃	D₈	1	1	yes	characteristic
22	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
23	C₂×C₄×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₇₃	SD₁₆	1	2	yes	—
25	C₂×C₄×C₁₇₃	Q₁₆	1	1	yes	characteristic
26	C₈×C₁₇₃	C₄²	1	1	yes	characteristic
27	C₈×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₈×C₁₇₃	C₄⋊C₄	1	2	yes	—
29	C₂²×C₁₇₃	#102	1	2	yes	—
30	C₂²×C₁₇₃	#107	1	1	yes	characteristic
31	C₂²×C₁₇₃	D₁₆	1	1	yes	characteristic
32	C₂²×C₁₇₃	SD₃₂	1	2	yes	—
33	C₂²×C₁₇₃	Q₃₂	1	1	yes	characteristic
34	C₄×C₁₇₃	#95	1	1	yes	characteristic
35	C₄×C₁₇₃	#103	1	2	yes	—
36	C₄×C₁₇₃	#106	1	1	yes	characteristic
37	Dic₁₇₃	—	8	4	no	—
38	C₂×C₁₇₃	#340	1	1	yes	characteristic
39	C₂×C₁₇₃	#357	1	2	yes	—
40	C₂×C₁₇₃	#358	1	2	yes	—
41	C₂×C₁₇₃	#366	1	1	yes	characteristic
42	C₂×C₁₇₃	#367	1	1	yes	characteristic
43	C₁₇₃	#1380	1	1	yes	characteristic, complemented, Sylow
44	#1380	—	173	1	no	maximal, Sylow
45	C₂²×C₁₆	D₁₇₃	1	1	yes	characteristic
46	C₂×#107	—	173	2	no	—
47	C₂²×C₈	C₂×D₁₇₃	1	1	yes	characteristic
48	C₂×C₄⋊C₄	—	346	2	no	—
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	Dic₁₇₃	1	2	yes	complemented
51	#107	—	346	2	no	—
52	#107	—	173	4	no	—
53	C₂²×C₄	—	692	2	no	—
54	C₂²×C₄	D₆₉₂	1	1	yes	characteristic
55	C₂×C₈	C₄×D₁₇₃	1	2	yes	—
56	C₂×C₈	C₂×Dic₁₇₃	1	1	yes	characteristic
57	C₂×C₈	C₂²⋊D₁₇₃	1	2	yes	—
58	C₂×C₈	Dic₃₄₆	1	1	yes	characteristic
59	C₄⋊C₄	—	346	4	no	—
60	C₄⋊C₄	—	692	2	no	—
61	C₁₆	—	2	2	no	—
62	C₂³	D₁₃₈₄	1	1	yes	characteristic
63	C₂×C₄	—	1384	2	no	—
64	C₂×C₄	—	1384	2	no	—
65	C₂×C₄	—	692	4	no	—
66	C₂×C₄	#14583803	1	2	yes	—
67	C₂×C₄	C₄⋊Dic₁₇₃	1	1	yes	characteristic
68	C₂×C₄	#14583815	1	2	yes	—
69	C₂×C₄	Dic₆₉₂	1	1	yes	characteristic
70	C₈	C₄×Dic₁₇₃	1	1	yes	characteristic
71	C₈	C₂²⋊Dic₁₇₃	1	1	yes	characteristic
72	C₈	#14583805	1	2	yes	—
73	C₂²	#19149601	1	2	yes	—
74	C₂²	#19149613	1	1	yes	characteristic
75	C₂²	D₂₇₆₈	1	1	yes	characteristic
76	C₂²	#19149624	1	2	yes	—
77	C₂²	Dic₁₃₈₄	1	1	yes	characteristic
78	C₄	—	1384	4	no	—
79	C₄	#19149586	1	1	yes	characteristic
80	C₄	#19149604	1	2	yes	—
81	C₄	#19149611	1	1	yes	characteristic
82	C₂	#26011254	1	1	yes	characteristic
83	C₂	#26011293	1	2	yes	—
84	C₂	#26011296	1	2	yes	—
85	C₂	#26011317	1	1	yes	characteristic
86	C₂	#26011319	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₁₃₈₄
commutator	C₈×C₁₇₃	C₄²
Frattini	C₂²×C₈	C₂×D₁₇₃
Fitting	C₂²×C₁₆×C₁₇₃	C₂
socle, prime radical	C₂³×C₁₇₃	D₈
cosocle	C₂²×C₈×C₁₇₃	C₂²
hypercenter	C₂²×C₁₆	D₁₇₃
hypocenter	C₁₇₃	#1380
LCS	G ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
5. #19149613	C₄	semidirect
8. C₂×C₁₆×C₁₇₃	C₄	semidirect
43. C₁₇₃	#1380	semidirect
50. C₂×C₁₆	Dic₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19149613	2
C₂²×C₁₆×C₁₇₃	2
#1380	173