Subgroups — #36351546

Order 22144.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011621	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#14583801	C₂	1	1	yes	maximal, characteristic
3	#346×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₃₄₆	C₄	1	1	yes	characteristic
5	C₂×#14583801	—	2	1	no	—
6	C₄⋊#13063436	C₄	1	1	yes	characteristic
7	#19149608	—	2	1	no	—
8	#19149589	—	2	1	no	—
9	C₂×C₁₆×C₁₇₃	—	2	1	no	—
10	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₄₆	C₈	1	1	yes	characteristic
12	C₂×#13063436	—	2	1	no	—
13	C₄⋊Dic₁₇₃	C₈	1	1	yes	characteristic
14	#14583805	—	2	1	no	—
15	C₄×Dic₁₇₃	—	2	1	no	—
16	#14583801	—	4	1	no	—
17	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
19	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₇₃	—	2	2	no	—
21	Dic₃₄₆	—	4	1	no	—
22	C₂×Dic₁₇₃	—	2	1	no	—
23	C₂×Dic₁₇₃	—	2	1	no	—
24	#13063436	—	4	1	no	—
25	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₇₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₇₃	M₁₆	1	1	yes	characteristic
28	C₈×C₁₇₃	—	2	1	no	—
29	C₈×C₁₇₃	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₇₃	Q₁₆	1	1	yes	characteristic
31	C₂²×C₁₇₃	C₂²⋊C₈	1	1	yes	characteristic
32	C₄×C₁₇₃	—	2	1	no	—
33	C₄×C₁₇₃	—	2	1	no	—
34	C₄×C₁₇₃	#103	1	1	yes	characteristic
35	C₄×C₁₇₃	#104	1	1	yes	characteristic
36	Dic₁₇₃	—	4	1	no	—
37	Dic₁₇₃	—	4	1	no	—
38	C₂×C₁₇₃	Q₈⋊C₈	1	1	yes	characteristic
39	C₂×C₁₇₃	#349	1	1	yes	characteristic
40	C₂×C₁₇₃	#350	1	1	yes	characteristic
41	C₁₇₃	#1338	1	1	yes	characteristic, complemented, Sylow
42	#1338	—	173	1	no	maximal, Sylow
43	#446	—	173	1	no	—
44	C₄⋊C₁₆	—	173	1	no	—
45	#346	D₁₇₃	1	1	yes	characteristic
46	C₄×Q₈	—	173	1	no	—
47	C₂×C₁₆	—	2	1	no	—
48	C₂×C₁₆	—	346	1	no	—
49	C₄⋊C₈	—	346	1	no	—
50	C₄⋊C₈	—	173	1	no	—
51	#97	—	346	1	no	—
52	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
53	C₂×Q₈	—	173	1	no	—
54	C₂×C₈	—	346	1	no	—
55	C₂×C₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
56	C₂×C₈	D₆₉₂	1	1	yes	characteristic
57	C₄⋊C₄	—	346	1	no	—
58	C₄⋊C₄	—	173	1	no	—
59	C₄²	—	346	1	no	—
60	C₄²	C₄×D₁₇₃	1	1	yes	characteristic
61	C₁₆	—	2	2	no	—
62	C₁₆	—	692	1	no	—
63	Q₈	—	692	1	no	—
64	C₂×C₄	—	346	1	no	—
65	C₂×C₄	—	346	1	no	—
66	C₂×C₄	#14583803	1	1	yes	characteristic
67	C₂×C₄	C₈×D₁₇₃	1	1	yes	characteristic
68	C₂×C₄	#14583809	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	692	1	no	—
71	C₈	#14583815	1	1	yes	characteristic
72	C₈	Dic₆₉₂	1	1	yes	characteristic
73	C₂²	#19149591	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	692	1	no	—
77	C₄	—	692	1	no	—
78	C₄	#19149604	1	1	yes	characteristic
79	C₄	#19149607	1	1	yes	characteristic
80	C₂	#26011223	1	1	yes	characteristic
81	C₂	#26011272	1	1	yes	characteristic
82	C₂	#26011276	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14583803
commutator	C₂×C₄×C₁₇₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₇₃
Fitting	#346×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	C₂²⋊C₈
cosocle	C₄×C₈×C₁₇₃	C₂²
hypercenter	#346	D₁₇₃
hypocenter	C₁₇₃	#1338
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
41. C₁₇₃	#1338	semidirect

Maximal subgroups

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

subgroup	index
#26011621	2
C₄⋊#14583801	2
#346×C₁₇₃	2
#1338	173