Subgroups — #36368469

Order 22144.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012512	C₂	1	2	yes	maximal
2	#26011514	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13063443	—	2	2	no	—
4	C₂²⋊#13063443	C₄	1	4	yes	—
5	#19149734	—	2	2	no	—
6	C₂×#14583810	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
8	#19149608	—	2	2	no	—
9	C₂×M₁₆×C₁₇₃	C₄	1	1	yes	characteristic
10	C₂×#13063443	—	2	4	no	—
11	C₂²×Dic₁₇₃	C₂×C₄	1	2	yes	—
12	#14583810	—	4	1	no	—
13	C₂×#13063436	—	2	1	no	—
14	C₄×Dic₁₇₃	D₄	1	2	yes	—
15	C₄×Dic₁₇₃	D₄	1	1	yes	characteristic
16	C₄×Dic₁₇₃	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
18	M₁₆×C₁₇₃	—	4	1	no	—
19	C₂×C₈×C₁₇₃	—	2	1	no	—
20	#13063443	—	4	4	no	—
21	C₂×Dic₁₇₃	—	4	2	no	—
22	C₂×Dic₁₇₃	C₂²⋊C₄	1	2	yes	—
23	C₂×Dic₁₇₃	C₄⋊C₄	1	2	yes	—
24	#13063436	—	4	1	no	—
25	C₂³×C₁₇₃	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₁₇₃	—	2	1	no	—
27	C₂×C₄×C₁₇₃	—	2	1	no	—
28	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
30	C₈×C₁₇₃	—	4	1	no	—
31	C₂²×C₁₇₃	—	2	1	no	—
32	C₂²×C₁₇₃	—	2	2	no	—
33	C₂²×C₁₇₃	#95	1	1	yes	characteristic
34	C₄×C₁₇₃	—	2	1	no	—
35	C₄×C₁₇₃	#100	1	1	yes	characteristic
36	C₄×C₁₇₃	#101	1	1	yes	characteristic
37	Dic₁₇₃	—	2	2	no	—
38	Dic₁₇₃	—	2	2	no	—
39	C₂×C₁₇₃	—	4	1	no	—
40	C₂×C₁₇₃	#343	1	1	yes	characteristic
41	C₂×C₁₇₃	#344	1	2	yes	—
42	C₁₇₃	#1307	1	1	yes	characteristic, complemented, Sylow
43	#1307	—	173	1	no	maximal, Sylow
44	#432	—	173	2	no	—
45	#423	—	173	1	no	—
46	C₂×M₁₆	—	173	1	no	—
47	C₂×M₁₆	#10621924	1	1	yes	characteristic
48	C₂×C₄²	—	173	1	no	—
49	C₄⋊C₈	—	346	2	no	—
50	C₄⋊C₈	—	346	2	no	—
51	C₂²⋊C₈	—	173	4	no	—
52	#97	—	346	2	no	—
53	C₂²×C₄	—	173	2	no	—
54	C₂²×C₄	C₂×#10621924	1	1	yes	characteristic
55	M₁₆	—	4	1	no	—
56	M₁₆	—	692	1	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	346	4	no	—
59	C₂×C₈	—	346	1	no	—
60	C₄²	—	173	2	no	—
61	C₄²	—	173	1	no	—
62	C₄²	—	173	1	no	—
63	C₂³	C₄×#10621924	1	1	yes	characteristic
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	173	2	no	—
67	C₂×C₄	—	173	2	no	—
68	C₂×C₄	—	692	2	no	—
69	C₂×C₄	C₂²⋊#10621924	1	1	yes	characteristic
70	C₂×C₄	C₄⋊#10621924	1	1	yes	characteristic
71	C₈	—	4	1	no	—
72	C₈	—	692	4	no	—
73	C₈	—	692	1	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	2	no	—
76	C₂²	(C₂×#10621924)⋊C₄	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	346	2	no	—
79	C₄	—	346	2	no	—
80	C₄	#19149739	1	1	yes	characteristic
81	C₄	#19149740	1	1	yes	characteristic
82	C₂	—	4	1	no	—
83	C₂	#26012371	1	1	yes	characteristic
84	C₂	#26012374	1	2	yes	—

	subgroup	G/H
center	C₂²	(C₂×#10621924)⋊C₄
commutator, prime radical	C₂³×C₁₇₃	C₄²
Frattini	C₂²×C₄	C₂×#10621924
Fitting	C₂×M₁₆×C₁₇₃	C₄
socle	C₂²×C₁₇₃	#95
cosocle	C₂×C₄×Dic₁₇₃	C₂²
hypercenter	C₂×M₁₆	#10621924
hypocenter	C₁₇₃	#1307
LCS	G ▷ C₂³×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
42. C₁₇₃	#1307	semidirect

Maximal subgroups

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

subgroup	index
#26012512	2
#26011514	2
#1307	173