Subgroups — #36351597

Order 22144.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011860	C₂	1	1	yes	maximal, characteristic
2	#26011269	C₂	1	1	yes	maximal, characteristic
3	#347×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	#19149679	C₄	1	1	yes	characteristic
5	#19149671	C₄	1	1	yes	characteristic
6	#19149619	—	2	1	no	—
7	#19149604	—	2	1	no	—
8	#19149604	—	2	1	no	—
9	M₃₂×C₁₇₃	—	2	1	no	—
10	#97×C₁₇₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₄₆	—	2	1	no	—
12	C₄⋊Dic₁₇₃	—	2	1	no	—
13	C₄⋊Dic₁₇₃	—	4	1	no	—
14	#14583805	—	4	1	no	—
15	#14583801	—	2	2	no	—
16	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
17	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
18	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₁₇₃	—	2	2	no	—
20	Dic₃₄₆	—	4	1	no	—
21	Dic₃₄₆	—	4	1	no	—
22	C₂×Dic₁₇₃	—	4	1	no	—
23	C₂×Dic₁₇₃	—	4	1	no	—
24	C₂×C₄×C₁₇₃	—	2	1	no	—
25	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₇₃	—	2	1	no	—
27	C₈×C₁₇₃	—	2	1	no	—
28	C₂²×C₁₇₃	#100	1	1	yes	characteristic
29	C₄×C₁₇₃	—	4	1	no	—
30	C₄×C₁₇₃	#104	1	1	yes	characteristic
31	C₄×C₁₇₃	#104	1	1	yes	characteristic
32	Dic₁₇₃	—	8	1	no	—
33	Dic₁₇₃	—	8	1	no	—
34	C₂×C₁₇₃	—	2	1	no	—
35	C₂×C₁₇₃	#329	1	1	yes	characteristic
36	C₁₇₃	#1356	1	1	yes	characteristic, complemented, Sylow
37	#1356	—	173	1	no	maximal, Sylow
38	#491	—	173	1	no	—
39	#347	—	173	1	no	—
40	#347	D₁₇₃	1	1	yes	characteristic
41	C₄⋊Q₈	—	173	1	no	—
42	#125	—	173	1	no	—
43	M₃₂	—	2	1	no	—
44	M₃₂	—	346	1	no	—
45	#103	—	346	1	no	—
46	#103	—	346	1	no	—
47	#97	C₂×D₁₇₃	1	1	yes	characteristic
48	C₂×Q₈	—	346	1	no	—
49	C₂×C₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
50	C₂×C₈	D₆₉₂	1	1	yes	characteristic
51	C₄⋊C₄	—	346	1	no	—
52	C₄⋊C₄	—	692	1	no	—
53	C₄⋊C₄	—	692	1	no	—
54	C₄²	C₄×D₁₇₃	1	1	yes	characteristic
55	C₁₆	—	2	2	no	—
56	C₁₆	—	346	2	no	—
57	Q₈	—	692	1	no	—
58	Q₈	—	692	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	692	1	no	—
61	C₂×C₄	—	692	1	no	—
62	C₂×C₄	#14583803	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	2	1	no	—
65	C₂²	#19149595	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	—	1384	1	no	—
68	C₄	—	1384	1	no	—
69	C₄	#19149605	1	1	yes	characteristic
70	C₄	#19149607	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#26011230	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26011230
commutator	C₄²×C₁₇₃	C₂×C₄
Frattini	#97	C₂×D₁₇₃
Fitting	#347×C₁₇₃	C₂
socle	C₂×C₁₇₃	#329
prime radical	C₂²×C₁₇₃	#100
cosocle	#97×C₁₇₃	C₂²
hypercenter	#347	D₁₇₃
hypocenter	C₁₇₃	#1356
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
36. C₁₇₃	#1356	semidirect

Maximal subgroups

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

subgroup	index
#26011860	2
#26011269	2
#347×C₁₇₃	2
#1356	173