Subgroups — #36353702

Order 22144.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011514	C₂	1	2	yes	maximal
2	C₂×#19149589	C₂	1	1	yes	maximal, characteristic
3	#26011256	C₂	1	1	yes	maximal, characteristic
4	#26011255	C₂	1	2	yes	maximal
5	#341×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14583810	C₂²	1	2	yes	—
7	C₂²×#13063436	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
9	#19149608	—	2	4	no	—
10	#19149589	C₄	1	4	yes	—
11	C₂×M₁₆×C₁₇₃	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₇₃	Q₈	1	2	yes	—
14	#14583810	—	2	2	no	—
15	#14583810	—	2	2	no	—
16	C₂×#13063436	—	2	2	no	—
17	C₂×#13063436	—	2	2	no	—
18	C₂×#13063436	C₂×C₄	1	2	yes	—
19	C₄×Dic₁₇₃	C₂×C₄	1	2	yes	—
20	C₄×Dic₁₇₃	D₄	1	2	yes	—
21	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₇₃	—	2	2	no	—
23	M₁₆×C₁₇₃	—	2	2	no	—
24	C₂×C₈×C₁₇₃	—	2	2	no	—
25	C₂×C₈×C₁₇₃	—	2	2	no	—
26	C₂×C₈×C₁₇₃	C₂×C₄	1	2	yes	—
27	C₂×Dic₁₇₃	—	4	2	no	—
28	C₂×Dic₁₇₃	C₄⋊C₄	1	4	yes	—
29	#13063436	—	2	2	no	—
30	#13063436	—	4	2	no	—
31	C₂³×C₁₇₃	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₇₃	#41	1	2	yes	—
35	C₂×C₄×C₁₇₃	#41	1	2	yes	—
36	C₈×C₁₇₃	—	2	2	no	—
37	C₈×C₁₇₃	—	4	2	no	—
38	C₂²×C₁₇₃	—	2	2	no	—
39	C₂²×C₁₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₇₃	#125	1	2	yes	—
41	C₄×C₁₇₃	#117	1	1	yes	characteristic
42	C₄×C₁₇₃	#117	1	1	yes	characteristic
43	C₄×C₁₇₃	#124	1	2	yes	—
44	Dic₁₇₃	—	2	4	no	—
45	C₂×C₁₇₃	—	2	2	no	—
46	C₂×C₁₇₃	#382	1	1	yes	characteristic
47	C₂×C₁₇₃	#430	1	2	yes	—
48	C₁₇₃	#1838	1	1	yes	characteristic, complemented, Sylow
49	#1838	—	173	1	no	maximal, Sylow
50	#423	—	173	2	no	—
51	C₂×#97	—	173	1	no	—
52	#341	—	173	2	no	—
53	#341	—	173	1	no	—
54	#341	D₁₇₃	1	1	yes	characteristic
55	C₂×M₁₆	—	173	2	no	—
56	C₂×M₁₆	C₂×D₁₇₃	1	2	yes	—
57	C₂²×C₈	—	173	1	no	—
58	C₂²×C₈	C₂×D₁₇₃	1	1	yes	characteristic
59	C₂×C₄²	—	173	1	no	—
60	C₄⋊C₈	—	346	4	no	—
61	#97	—	173	4	no	—
62	C₂²×C₄	—	173	2	no	—
63	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
64	M₁₆	—	2	2	no	—
65	M₁₆	—	2	2	no	—
66	M₁₆	—	346	2	no	—
67	M₁₆	—	346	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	346	2	no	—
71	C₂×C₈	—	346	2	no	—
72	C₂×C₈	—	173	2	no	—
73	C₂×C₈	C₄×D₁₇₃	1	2	yes	—
74	C₄²	—	173	2	no	—
75	C₄²	—	173	2	no	—
76	C₂³	Q₈×D₁₇₃	1	1	yes	characteristic
77	C₂×C₄	—	173	4	no	—
78	C₂×C₄	—	692	2	no	—
79	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
80	C₂×C₄	D₄×D₁₇₃	1	1	yes	characteristic
81	C₂×C₄	#14583825	1	2	yes	—
82	C₂×C₄	#14583826	1	2	yes	—
83	C₈	—	2	2	no	—
84	C₈	—	4	2	no	—
85	C₈	—	346	2	no	—
86	C₈	—	692	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	C₄⋊C₄×D₁₇₃	1	1	yes	characteristic
89	C₂²	#19149669	1	2	yes	—
90	C₄	—	346	4	no	—
91	C₄	#19149635	1	1	yes	characteristic
92	C₄	#19149636	1	1	yes	characteristic
93	C₄	#19149665	1	2	yes	—
94	C₂	—	2	2	no	—
95	C₂	#26011355	1	1	yes	characteristic
96	C₂	#26011538	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇₃
commutator	C₂×C₄×C₁₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	#341×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂×C₄⋊C₄
prime radical	C₂³×C₁₇₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	#341	D₁₇₃
hypocenter	C₁₇₃	#1838
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

Complements

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

N	H ≅ G/N	type
48. C₁₇₃	#1838	semidirect

Maximal subgroups

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

subgroup	index
#26011514	2
C₂×#19149589	2
#26011256	2
#26011255	2
#341×C₁₇₃	2
#1838	173