Subgroups — #36355379

Order 22144.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011312	C₂	1	2	yes	maximal, complemented
2	#26011273	C₂	1	2	yes	maximal
3	#26011269	C₂	1	2	yes	maximal, complemented
4	#405×C₁₇₃	C₂	1	1	yes	maximal, characteristic
5	#19149619	C₂²	1	4	yes	—
6	C₂×M₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₇₃	C₄	1	1	yes	characteristic
8	#117×C₁₇₃	C₄	1	1	yes	characteristic
9	#97×C₁₇₃	C₄	1	1	yes	characteristic
10	#97×C₁₇₃	C₂²	1	1	yes	characteristic
11	C₄×C₈×C₁₇₃	C₄	1	1	yes	characteristic
12	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
13	#14583801	—	2	4	no	—
14	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
15	M₁₆×C₁₇₃	—	2	2	no	—
16	C₂×C₈×C₁₇₃	—	2	2	no	—
17	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₁₇₃	—	2	1	no	—
22	C₂²⋊C₄×C₁₇₃	—	2	1	no	—
23	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₁₇₃	M₁₆	1	1	yes	characteristic
26	C₂×C₄×C₁₇₃	—	2	1	no	—
27	C₂×C₄×C₁₇₃	—	2	1	no	—
28	C₂×C₄×C₁₇₃	—	2	1	no	—
29	C₂×C₄×C₁₇₃	—	2	1	no	—
30	C₂×C₄×C₁₇₃	M₁₆	1	1	yes	characteristic
31	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₁₇₃	—	2	1	no	—
33	C₈×C₁₇₃	—	2	1	no	—
34	C₈×C₁₇₃	M₁₆	1	2	yes	—
35	C₈×C₁₇₃	#41	1	2	yes	—
36	C₂²×C₁₇₃	—	2	1	no	—
37	C₂²×C₁₇₃	—	4	1	no	—
38	C₂²×C₁₇₃	C₂×M₁₆	1	1	yes	characteristic
39	C₄×C₁₇₃	—	2	1	no	—
40	C₄×C₁₇₃	—	4	1	no	—
41	C₄×C₁₇₃	—	4	1	no	—
42	C₄×C₁₇₃	#117	1	1	yes	characteristic
43	C₄×C₁₇₃	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₁₇₃	—	2	1	no	—
45	C₂×C₁₇₃	—	4	1	no	—
46	C₂×C₁₇₃	#432	1	1	yes	characteristic
47	C₁₇₃	#2166	1	1	yes	characteristic, complemented, Sylow
48	#2166	—	173	1	no	maximal, Sylow
49	#405	D₁₇₃	1	1	yes	characteristic
50	#364	—	173	2	no	—
51	#349	—	173	2	no	—
52	#347	—	173	2	no	—
53	C₂×M₁₆	C₂×D₁₇₃	1	1	yes	characteristic
54	C₂²×C₈	Dic₁₇₃	1	1	yes	characteristic
55	#117	Dic₁₇₃	1	1	yes	characteristic
56	M₃₂	—	173	4	no	—
57	#97	Dic₁₇₃	1	1	yes	characteristic
58	#97	C₂×D₁₇₃	1	1	yes	characteristic
59	C₄×C₈	Dic₁₇₃	1	1	yes	characteristic
60	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
61	C₂²×C₄	C₂×Dic₁₇₃	1	1	yes	characteristic
62	M₁₆	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	C₂×Dic₁₇₃	1	1	yes	characteristic
65	C₂×C₈	C₂×Dic₁₇₃	1	1	yes	characteristic
66	C₂×C₈	C₂×Dic₁₇₃	1	1	yes	characteristic
67	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
68	C₄⋊C₄	—	2	1	no	—
69	C₂²⋊C₄	—	2	1	no	—
70	C₄²	C₂×Dic₁₇₃	1	1	yes	characteristic
71	C₄²	C₂×Dic₁₇₃	1	1	yes	characteristic
72	C₁₆	—	346	4	no	—
73	C₂³	#14583810	1	1	yes	characteristic
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	#14583810	1	1	yes	characteristic
79	C₂×C₄	C₂²×Dic₁₇₃	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	2	1	no	—
82	C₈	#14583810	1	2	yes	—
83	C₈	#14583826	1	2	yes	—
84	C₂²	—	2	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	C₂×#14583810	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	4	1	no	—
89	C₄	—	4	1	no	—
90	C₄	#19149638	1	1	yes	characteristic
91	C₄	C₂×#14583810	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	4	1	no	—
94	C₂	#26011548	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19149638
commutator	C₂×C₄×C₁₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₇₃
Fitting	#405×C₁₇₃	C₂
socle	C₂×C₁₇₃	#432
prime radical	C₂³×C₁₇₃	M₁₆
cosocle	C₂×C₈×C₁₇₃	C₂³
hypercenter	#405	D₁₇₃
hypocenter	C₁₇₃	#2166
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄ ◁ #97 ◁ #405

Complements

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

N	H ≅ G/N	type
1. #26011312	C₂	semidirect
3. #26011269	C₂	semidirect
47. C₁₇₃	#2166	semidirect

Maximal subgroups

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

subgroup	index
#26011312	2
#26011273	2
#26011269	2
#405×C₁₇₃	2
#2166	173