Subgroups — #36351737

Order 22144.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011845	C₂	1	1	yes	maximal, characteristic
2	#26011288	C₂	1	1	yes	maximal, characteristic
3	#354×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	#19149671	C₄	1	1	yes	characteristic, complemented
5	#19149603	—	2	1	no	—
6	#19149600	—	2	1	no	—
7	#19149598	—	2	1	no	—
8	C₄×#13063436	C₄	1	1	yes	characteristic, complemented
9	#124×C₁₇₃	C₂²	1	1	yes	characteristic
10	#99×C₁₇₃	—	2	1	no	—
11	#14583810	—	4	1	no	—
12	C₂×#13063436	—	2	1	no	—
13	C₄⋊Dic₁₇₃	—	2	1	no	—
14	#14583805	—	4	1	no	—
15	C₂×Q₈×C₁₇₃	D₄	1	1	yes	characteristic
16	C₂×D₄×C₁₇₃	D₄	1	1	yes	characteristic
17	C₂²⋊C₄×C₁₇₃	—	4	1	no	—
18	C₂²⋊C₄×C₁₇₃	—	4	1	no	—
19	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₇₃	—	4	1	no	—
21	#13063436	—	4	1	no	—
22	#13063436	—	8	1	no	—
23	C₂³×C₁₇₃	—	2	1	no	—
24	Q₈×C₁₇₃	—	4	1	no	—
25	D₄×C₁₇₃	—	4	1	no	—
26	C₂×C₄×C₁₇₃	—	2	1	no	—
27	C₂×C₄×C₁₇₃	—	2	1	no	—
28	C₂×C₄×C₁₇₃	—	8	1	no	—
29	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₁₇₃	—	4	1	no	—
31	C₂²×C₁₇₃	—	8	1	no	—
32	C₂²×C₁₇₃	#99	1	1	yes	characteristic
33	C₄×C₁₇₃	—	2	1	no	—
34	C₄×C₁₇₃	—	4	1	no	—
35	C₄×C₁₇₃	—	4	1	no	—
36	C₄×C₁₇₃	—	16	1	no	—
37	Dic₁₇₃	—	8	1	no	—
38	C₂×C₁₇₃	—	2	1	no	—
39	C₂×C₁₇₃	—	8	1	no	—
40	C₂×C₁₇₃	#354	1	1	yes	characteristic
41	C₁₇₃	#1414	1	1	yes	characteristic, complemented, Sylow
42	#1414	—	173	1	no	maximal, Sylow
43	#488	—	173	1	no	—
44	#355	—	173	1	no	—
45	#354	D₁₇₃	1	1	yes	characteristic
46	#125	—	173	1	no	—
47	#124	C₂×D₁₇₃	1	1	yes	characteristic
48	#103	—	346	1	no	—
49	#102	—	346	1	no	—
50	#101	—	346	1	no	—
51	#99	—	2	1	no	—
52	C₄×C₈	—	173	1	no	—
53	C₂×Q₈	D₆₉₂	1	1	yes	characteristic
54	C₂×D₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
55	M₁₆	—	692	1	no	—
56	C₂×C₈	—	346	1	no	—
57	C₄⋊C₄	—	346	1	no	—
58	C₄⋊C₄	—	692	1	no	—
59	C₂²⋊C₄	—	4	1	no	—
60	C₂²⋊C₄	—	4	1	no	—
61	C₄²	C₄×D₁₇₃	1	1	yes	characteristic
62	C₂³	—	2	1	no	—
63	Q₈	—	4	1	no	—
64	D₄	—	4	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	8	1	no	—
68	C₂×C₄	—	692	1	no	—
69	C₂×C₄	#14583803	1	1	yes	characteristic
70	C₈	—	1384	1	no	—
71	C₈	—	692	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	8	1	no	—
74	C₂²	#19149593	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	16	1	no	—
79	C₄	—	1384	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	#26011285	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26011285
commutator	C₄²×C₁₇₃	C₂×C₄
Frattini	#124	C₂×D₁₇₃
Fitting	#354×C₁₇₃	C₂
socle	C₂×C₁₇₃	#354
prime radical	C₂×D₄×C₁₇₃	D₄
cosocle	#124×C₁₇₃	C₂²
hypercenter	#354	D₁₇₃
hypocenter	C₁₇₃	#1414
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂×C₄×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #354

Complements

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

N	H ≅ G/N	type
4. #19149671	C₄	semidirect
8. C₄×#13063436	C₄	semidirect
41. C₁₇₃	#1414	semidirect

Maximal subgroups

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

subgroup	index
#26011845	2
#26011288	2
#354×C₁₇₃	2
#1414	173