Subgroups — #36368471

Order 22144.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012494	C₂	1	2	yes	maximal
2	#26011514	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583835	C₄	1	4	yes	—
4	#19149746	—	2	2	no	—
5	C₄⋊#13063443	—	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	#14583835	—	4	4	no	—
11	C₂×#13063443	—	2	4	no	—
12	C₂²×Dic₁₇₃	C₂×C₄	1	2	yes	—
13	#14583810	—	4	1	no	—
14	C₂×#13063436	—	2	1	no	—
15	C₄×Dic₁₇₃	D₄	1	2	yes	—
16	C₄×Dic₁₇₃	D₄	1	1	yes	characteristic
17	C₄×Dic₁₇₃	Q₈	1	1	yes	characteristic
18	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
19	M₁₆×C₁₇₃	—	4	1	no	—
20	C₂×C₈×C₁₇₃	—	2	1	no	—
21	#13063443	—	4	4	no	—
22	C₂×Dic₁₇₃	—	2	2	no	—
23	C₂×Dic₁₇₃	—	2	2	no	—
24	C₂×Dic₁₇₃	C₂²⋊C₄	1	2	yes	—
25	C₂×Dic₁₇₃	C₄⋊C₄	1	2	yes	—
26	#13063436	—	4	1	no	—
27	C₂³×C₁₇₃	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₁₇₃	—	2	1	no	—
29	C₂×C₄×C₁₇₃	—	2	1	no	—
30	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
32	C₈×C₁₇₃	—	4	1	no	—
33	C₂²×C₁₇₃	—	2	1	no	—
34	C₂²×C₁₇₃	—	2	2	no	—
35	C₂²×C₁₇₃	#95	1	1	yes	characteristic
36	C₄×C₁₇₃	—	2	1	no	—
37	C₄×C₁₇₃	#100	1	1	yes	characteristic
38	C₄×C₁₇₃	#101	1	1	yes	characteristic
39	Dic₁₇₃	—	2	2	no	—
40	Dic₁₇₃	#100	1	2	yes	—
41	Dic₁₇₃	#101	1	2	yes	—
42	C₂×C₁₇₃	—	4	1	no	—
43	C₂×C₁₇₃	#343	1	2	yes	—
44	C₂×C₁₇₃	#343	1	1	yes	characteristic
45	C₁₇₃	#1309	1	1	yes	characteristic, complemented, Sylow
46	#1309	—	173	1	no	maximal, Sylow
47	#423	—	173	2	no	—
48	#423	—	173	1	no	—
49	C₂×M₁₆	—	173	4	no	—
50	C₂×M₁₆	—	173	1	no	—
51	C₂×M₁₆	#10621924	1	1	yes	characteristic
52	C₂×C₄²	—	173	1	no	—
53	C₄⋊C₈	—	346	2	no	—
54	C₄⋊C₈	—	346	2	no	—
55	C₄⋊C₈	—	346	2	no	—
56	C₂²×C₄	—	173	2	no	—
57	C₂²×C₄	C₂×#10621924	1	1	yes	characteristic
58	M₁₆	—	4	1	no	—
59	M₁₆	—	692	1	no	—
60	M₁₆	—	692	4	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	346	4	no	—
63	C₂×C₈	—	346	1	no	—
64	C₄²	—	173	2	no	—
65	C₄²	—	173	1	no	—
66	C₄²	—	173	1	no	—
67	C₂³	C₄×#10621924	1	1	yes	characteristic
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	346	2	no	—
71	C₂×C₄	—	346	2	no	—
72	C₂×C₄	—	173	2	no	—
73	C₂×C₄	—	173	2	no	—
74	C₂×C₄	C₂²⋊#10621924	1	1	yes	characteristic
75	C₂×C₄	C₄⋊#10621924	1	1	yes	characteristic
76	C₈	—	4	1	no	—
77	C₈	—	692	4	no	—
78	C₈	—	692	1	no	—
79	C₂²	—	2	2	no	—
80	C₂²	—	2	1	no	—
81	C₂²	(C₂×#10621924)⋊C₄	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	346	2	no	—
84	C₄	—	173	2	no	—
85	C₄	—	173	2	no	—
86	C₄	#19149739	1	1	yes	characteristic
87	C₄	#19149740	1	1	yes	characteristic
88	C₂	—	4	1	no	—
89	C₂	#26012370	1	2	yes	—
90	C₂	#26012371	1	1	yes	characteristic

	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	#1309
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
45. C₁₇₃	#1309	semidirect

Maximal subgroups

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

subgroup	index
#26012494	2
#26011514	2
#1309	173