Subgroups — #36353316

Order 22144.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#13063436	C₂	1	1	yes	maximal, characteristic
2	#26011337	C₂	1	1	yes	maximal, characteristic
3	#26011245	C₂	1	4	yes	maximal
4	C₂×#97×C₁₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13063436	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₇₃	C₂²	1	2	yes	—
7	C₄×#13063436	C₄	1	4	yes	complemented
8	#19149586	C₄	1	4	yes	—
9	C₂²×C₈×C₁₇₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
11	#97×C₁₇₃	C₄	1	4	yes	complemented
12	C₂²×Dic₁₇₃	C₂×C₄	1	4	yes	—
13	C₂×#13063436	—	2	4	no	—
14	C₂×#13063436	C₂×C₄	1	4	yes	—
15	C₄×Dic₁₇₃	—	2	4	no	—
16	C₂²×C₄×C₁₇₃	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₁₇₃	—	2	4	no	—
19	C₂×C₈×C₁₇₃	C₂×C₄	1	4	yes	—
20	C₄²×C₁₇₃	C₂×C₄	1	2	yes	—
21	C₄²×C₁₇₃	C₂×C₄	1	2	yes	—
22	C₂×Dic₁₇₃	—	2	4	no	—
23	C₂×Dic₁₇₃	—	2	8	no	—
24	#13063436	—	2	4	no	—
25	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₇₃	—	2	4	no	—
27	C₂×C₄×C₁₇₃	C₄²	1	4	yes	—
28	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₇₃	#41	1	4	yes	—
31	C₈×C₁₇₃	—	2	4	no	—
32	C₂²×C₁₇₃	C₂×C₄²	1	1	yes	characteristic
33	C₂²×C₁₇₃	#117	1	2	yes	—
34	C₂²×C₁₇₃	#131	1	4	yes	—
35	C₄×C₁₇₃	—	2	4	no	—
36	C₄×C₁₇₃	#117	1	2	yes	—
37	C₄×C₁₇₃	#117	1	2	yes	—
38	Dic₁₇₃	—	4	4	no	—
39	C₂×C₁₇₃	#376	1	1	yes	characteristic
40	C₂×C₁₇₃	#405	1	2	yes	—
41	C₂×C₁₇₃	#433	1	4	yes	—
42	C₁₇₃	#1746	1	1	yes	characteristic, complemented, Sylow
43	#1746	—	173	1	no	maximal, Sylow
44	C₂×#97	D₁₇₃	1	1	yes	characteristic
45	C₂×C₄×C₈	—	173	1	no	—
46	#376	—	173	1	no	—
47	#336	—	173	4	no	—
48	C₂²×C₈	—	173	2	no	—
49	C₂²×C₈	C₂×D₁₇₃	1	2	yes	—
50	C₂×C₄²	—	173	2	no	—
51	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
52	#97	Dic₁₇₃	1	4	yes	complemented
53	C₄×C₈	—	173	4	no	—
54	#95	—	173	4	no	—
55	C₂²×C₄	—	173	4	no	—
56	C₂²×C₄	C₄×D₁₇₃	1	2	yes	—
57	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
58	C₂×C₈	—	2	4	no	—
59	C₂×C₈	—	346	4	no	—
60	C₂×C₈	—	173	4	no	—
61	C₂×C₈	C₂×Dic₁₇₃	1	4	yes	—
62	C₄²	—	346	4	no	—
63	C₄²	C₄×D₁₇₃	1	2	yes	—
64	C₄²	C₂×Dic₁₇₃	1	2	yes	—
65	C₂³	C₂×C₄×D₁₇₃	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	346	4	no	—
68	C₂×C₄	—	346	8	no	—
69	C₂×C₄	C₄×Dic₁₇₃	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
71	C₂×C₄	C₂²×Dic₁₇₃	1	1	yes	characteristic
72	C₂×C₄	#14583826	1	4	yes	—
73	C₈	—	2	4	no	—
74	C₈	—	346	4	no	—
75	C₂²	C₂×C₄×Dic₁₇₃	1	1	yes	characteristic
76	C₂²	#19149636	1	2	yes	—
77	C₂²	#19149687	1	4	yes	—
78	C₄	—	2	4	no	—
79	C₄	—	692	4	no	—
80	C₄	#19149636	1	2	yes	—
81	C₄	#19149638	1	2	yes	—
82	C₂	#26011337	1	1	yes	characteristic
83	C₂	#26011449	1	2	yes	—
84	C₂	#26011553	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₇₃
commutator	C₂²×C₁₇₃	C₂×C₄²
Fitting	C₂×#97×C₁₇₃	C₂
socle, prime radical	C₂³×C₁₇₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	C₂×#97	D₁₇₃
hypocenter	C₁₇₃	#1746
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
7. C₄×#13063436	C₄	semidirect
11. #97×C₁₇₃	C₄	semidirect
42. C₁₇₃	#1746	semidirect
52. #97	Dic₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#13063436	2
#26011337	2
#26011245	2
C₂×#97×C₁₇₃	2
#1746	173