Subgroups — #36351657

Order 22144.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19149615	C₂	1	2	yes	maximal
2	C₂²×C₁₆×C₁₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583810	—	2	2	no	—
4	#19149615	—	2	2	no	—
5	#19149615	C₄	1	4	yes	—
6	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₆×C₁₇₃	—	2	2	no	—
8	C₂×C₁₆×C₁₇₃	C₄	1	2	yes	—
9	#14583810	—	2	4	no	—
10	#14583810	—	4	2	no	—
11	C₂×#13063436	—	4	2	no	—
12	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₇₃	C₂×C₄	1	2	yes	—
15	C₂×C₈×C₁₇₃	D₄	1	2	yes	—
16	C₂×C₈×C₁₇₃	Q₈	1	1	yes	characteristic
17	C₁₆×C₁₇₃	—	2	2	no	—
18	#13063436	—	4	4	no	—
19	C₂³×C₁₇₃	Q₁₆	1	1	yes	characteristic
20	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
21	C₂×C₄×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
23	C₂×C₄×C₁₇₃	SD₁₆	1	2	yes	—
24	C₈×C₁₇₃	C₄²	1	1	yes	characteristic
25	C₈×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₇₃	C₄⋊C₄	1	2	yes	—
27	C₂²×C₁₇₃	—	2	2	no	—
28	C₂²×C₁₇₃	#103	1	2	yes	—
29	C₂²×C₁₇₃	#107	1	1	yes	characteristic
30	C₄×C₁₇₃	#95	1	1	yes	characteristic
31	C₄×C₁₇₃	#102	1	2	yes	—
32	C₄×C₁₇₃	#106	1	1	yes	characteristic
33	C₂×C₁₇₃	—	2	2	no	—
34	C₂×C₁₇₃	#340	1	1	yes	characteristic
35	C₂×C₁₇₃	#368	1	2	yes	—
36	C₁₇₃	#1381	1	1	yes	characteristic, complemented, Sylow
37	#1381	—	173	1	no	maximal, Sylow
38	C₂²×C₁₆	D₁₇₃	1	1	yes	characteristic
39	C₂×#108	—	173	2	no	—
40	C₂×M₁₆	—	346	2	no	—
41	C₂²×C₈	C₂×D₁₇₃	1	1	yes	characteristic
42	C₂×C₁₆	—	2	2	no	—
43	C₂×C₁₆	Dic₁₇₃	1	2	yes	—
44	#108	—	346	2	no	—
45	#108	—	173	4	no	—
46	C₂²×C₄	D₆₉₂	1	1	yes	characteristic
47	M₁₆	—	346	4	no	—
48	M₁₆	—	692	2	no	—
49	C₂×C₈	—	692	2	no	—
50	C₂×C₈	C₄×D₁₇₃	1	2	yes	—
51	C₂×C₈	C₂×Dic₁₇₃	1	1	yes	characteristic
52	C₂×C₈	C₂²⋊D₁₇₃	1	2	yes	—
53	C₂×C₈	Dic₃₄₆	1	1	yes	characteristic
54	C₁₆	—	2	2	no	—
55	C₂³	Dic₆₉₂	1	1	yes	characteristic
56	C₂×C₄	#14583803	1	2	yes	—
57	C₂×C₄	C₄⋊Dic₁₇₃	1	1	yes	characteristic
58	C₂×C₄	D₁₃₈₄	1	1	yes	characteristic
59	C₂×C₄	#14583815	1	2	yes	—
60	C₈	—	692	4	no	—
61	C₈	C₄×Dic₁₇₃	1	1	yes	characteristic
62	C₈	C₂²⋊Dic₁₇₃	1	1	yes	characteristic
63	C₈	#14583805	1	2	yes	—
64	C₂²	—	2	2	no	—
65	C₂²	#19149604	1	2	yes	—
66	C₂²	#19149613	1	1	yes	characteristic
67	C₄	#19149586	1	1	yes	characteristic
68	C₄	#19149601	1	2	yes	—
69	C₄	#19149611	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	#26011254	1	1	yes	characteristic
72	C₂	#26011321	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₁₃₈₄
commutator	C₈×C₁₇₃	C₄²
Frattini	C₂²×C₈	C₂×D₁₇₃
Fitting	C₂²×C₁₆×C₁₇₃	C₂
socle	C₂²×C₁₇₃	#107
prime radical	C₂³×C₁₇₃	Q₁₆
cosocle	C₂²×C₈×C₁₇₃	C₂²
hypercenter	C₂²×C₁₆	D₁₇₃
hypocenter	C₁₇₃	#1381
LCS	G ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
36. C₁₇₃	#1381	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19149615	2
C₂²×C₁₆×C₁₇₃	2
#1381	173