Subgroups — #36368623

Order 22144.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19149750	C₂	1	2	yes	maximal
2	C₂×C₁₆×D₁₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583836	—	2	2	no	—
4	#19149750	—	2	2	no	—
5	#19149750	C₄	1	4	yes	—
6	C₂×C₈×D₁₇₃	C₂²	1	1	yes	characteristic
7	C₂×#14583801	C₄	1	1	yes	characteristic
8	C₁₆×D₁₇₃	—	2	2	no	—
9	C₂×C₁₆×C₁₇₃	C₄	1	1	yes	characteristic
10	#14583836	—	2	4	no	—
11	#14583836	—	4	2	no	—
12	C₂×#13063443	—	4	2	no	—
13	C₂×C₄×D₁₇₃	D₄	1	1	yes	characteristic
14	C₂×#13063436	Q₈	1	1	yes	characteristic
15	C₈×D₁₇₃	C₂×C₄	1	2	yes	—
16	C₈×D₁₇₃	D₄	1	2	yes	—
17	#14583801	—	2	1	no	—
18	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₁₇₃	—	2	1	no	—
20	#13063443	—	4	4	no	—
21	C₂²×D₁₇₃	Q₁₆	1	1	yes	characteristic
22	C₂×Dic₁₇₃	D₈	1	1	yes	characteristic
23	C₄×D₁₇₃	C₂²⋊C₄	1	2	yes	—
24	C₄×D₁₇₃	SD₁₆	1	2	yes	—
25	#13063436	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
27	C₈×C₁₇₃	C₄²	1	1	yes	characteristic
28	C₈×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₁₇₃	#107	1	1	yes	characteristic
30	C₂×D₁₇₃	—	2	1	no	—
31	C₂×D₁₇₃	—	2	1	no	—
32	C₂×D₁₇₃	#103	1	2	yes	—
33	C₄×C₁₇₃	#95	1	1	yes	characteristic
34	C₄×C₁₇₃	#106	1	1	yes	characteristic
35	Dic₁₇₃	#102	1	2	yes	—
36	C₂×C₁₇₃	#340	1	1	yes	characteristic
37	C₂×C₁₇₃	#368	1	1	yes	characteristic
38	C₂×C₁₇₃	#368	1	1	yes	characteristic
39	D₁₇₃	—	2	2	no	—
40	C₁₇₃	#1381	1	1	yes	characteristic, complemented, Sylow
41	#1381	—	173	1	no	maximal, Sylow
42	C₂²×C₁₆	—	173	1	no	—
43	C₂×#108	—	173	2	no	—
44	C₂×M₁₆	—	346	2	no	—
45	C₂²×C₈	—	173	1	no	—
46	C₂×C₁₆	—	346	2	no	—
47	C₂×C₁₆	—	173	1	no	—
48	C₂×C₁₆	#10621924	1	1	yes	characteristic
49	#108	—	346	2	no	—
50	#108	—	173	4	no	—
51	C₂²×C₄	—	173	1	no	—
52	M₁₆	—	346	4	no	—
53	M₁₆	—	692	2	no	—
54	C₂×C₈	—	173	1	no	—
55	C₂×C₈	—	173	2	no	—
56	C₂×C₈	—	173	2	no	—
57	C₂×C₈	—	692	2	no	—
58	C₂×C₈	C₂×#10621924	1	1	yes	characteristic
59	C₁₆	—	2	1	no	—
60	C₁₆	—	346	1	no	—
61	C₂³	—	173	1	no	—
62	C₂×C₄	—	173	1	no	—
63	C₂×C₄	—	173	2	no	—
64	C₂×C₄	—	173	2	no	—
65	C₂×C₄	C₄⋊#10621924	1	1	yes	characteristic
66	C₈	—	173	2	no	—
67	C₈	—	692	4	no	—
68	C₈	C₄×#10621924	1	1	yes	characteristic
69	C₈	C₂²⋊#10621924	1	1	yes	characteristic
70	C₂²	—	346	1	no	—
71	C₂²	—	346	1	no	—
72	C₂²	—	173	2	no	—
73	C₂²	#19149748	1	1	yes	characteristic
74	C₄	—	173	2	no	—
75	C₄	(C₂×#10621924)⋊C₄	1	1	yes	characteristic
76	C₄	#19149747	1	1	yes	characteristic
77	C₂	—	346	2	no	—
78	C₂	#26012365	1	1	yes	characteristic
79	C₂	#26012419	1	1	yes	characteristic
80	C₂	#26012420	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19149748
commutator	C₈×C₁₇₃	C₄²
Frattini	C₂×C₈	C₂×#10621924
Fitting	C₂×C₁₆×C₁₇₃	C₄
socle	C₂²×C₁₇₃	#107
prime radical	C₂²×D₁₇₃	Q₁₆
cosocle	C₂×C₈×D₁₇₃	C₂²
hypercenter	C₂×C₁₆	#10621924
hypocenter	C₁₇₃	#1381
LCS	G ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
40. 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×#19149750	2
C₂×C₁₆×D₁₇₃	2
#1381	173