Subgroups — #36351528

Order 22144.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₆₉₂	C₂	1	1	yes	maximal, characteristic
2	#26011244	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₉₂	C₄	1	1	yes	characteristic
5	C₄×Dic₃₄₆	—	2	1	no	—
6	#19149613	C₄	1	1	yes	characteristic
7	C₄⋊#13063436	—	2	1	no	—
8	#19149604	—	2	1	no	—
9	C₂×C₁₆×C₁₇₃	—	2	1	no	—
10	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₄₆	—	2	1	no	—
12	Dic₆₉₂	—	2	1	no	—
13	Dic₆₉₂	C₈	1	2	yes	complemented
14	C₂×#13063436	—	4	1	no	—
15	C₄⋊Dic₁₇₃	—	2	1	no	—
16	#14583805	—	4	1	no	—
17	C₄×Dic₁₇₃	—	4	1	no	—
18	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
20	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
21	C₁₆×C₁₇₃	—	2	2	no	—
22	Dic₃₄₆	—	2	2	no	—
23	Dic₃₄₆	—	4	1	no	—
24	C₂×Dic₁₇₃	—	4	1	no	—
25	C₂×Dic₁₇₃	—	4	1	no	—
26	#13063436	—	8	1	no	—
27	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₇₃	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₇₃	—	2	1	no	—
31	C₈×C₁₇₃	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₇₃	M₁₆	1	1	yes	characteristic
33	C₂²×C₁₇₃	#102	1	1	yes	characteristic
34	C₄×C₁₇₃	—	2	1	no	—
35	C₄×C₁₇₃	C₂²⋊C₈	1	1	yes	characteristic
36	C₄×C₁₇₃	#104	1	1	yes	characteristic
37	C₄×C₁₇₃	SD₃₂	1	1	yes	characteristic
38	C₄×C₁₇₃	Q₃₂	1	1	yes	characteristic
39	Dic₁₇₃	—	4	2	no	—
40	Dic₁₇₃	—	8	1	no	—
41	C₂×C₁₇₃	D₄⋊C₈	1	1	yes	characteristic
42	C₂×C₁₇₃	#358	1	1	yes	characteristic
43	C₂×C₁₇₃	#359	1	1	yes	characteristic
44	C₁₇₃	#1332	1	1	yes	characteristic, complemented, Sylow
45	#1332	—	173	1	no	maximal, Sylow
46	C₄×Q₁₆	—	173	1	no	—
47	C₄×C₁₆	D₁₇₃	1	1	yes	characteristic
48	#335	—	173	1	no	—
49	C₂×Q₁₆	—	173	1	no	—
50	C₄×Q₈	—	346	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	#107	—	173	1	no	—
53	C₄⋊C₈	—	346	1	no	—
54	#103	—	346	1	no	—
55	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
56	C₂×Q₈	—	346	1	no	—
57	Q₁₆	—	346	1	no	—
58	Q₁₆	—	173	2	no	—
59	C₂×C₈	—	692	1	no	—
60	C₂×C₈	C₄×D₁₇₃	1	1	yes	characteristic
61	C₂×C₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
62	C₄⋊C₄	—	346	1	no	—
63	C₄⋊C₄	—	692	1	no	—
64	C₄²	—	692	1	no	—
65	C₄²	D₆₉₂	1	1	yes	characteristic
66	C₁₆	—	2	2	no	—
67	Q₈	—	346	2	no	—
68	Q₈	—	692	1	no	—
69	C₂×C₄	—	692	1	no	—
70	C₂×C₄	—	692	1	no	—
71	C₂×C₄	#14583803	1	1	yes	characteristic
72	C₂×C₄	D₁₃₈₄	1	1	yes	characteristic
73	C₂×C₄	#14583815	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	—	1384	1	no	—
76	C₈	C₈×D₁₇₃	1	1	yes	characteristic
77	C₈	#14583809	1	1	yes	characteristic
78	C₂²	#19149601	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	1384	1	no	—
81	C₄	—	692	2	no	—
82	C₄	#19149591	1	1	yes	characteristic
83	C₄	#19149607	1	1	yes	characteristic
84	C₄	#19149624	1	1	yes	characteristic
85	C₄	Dic₁₃₈₄	1	1	yes	characteristic
86	C₂	#26011220	1	1	yes	characteristic
87	C₂	#26011296	1	1	yes	characteristic
88	C₂	#26011299	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₃₈₄
commutator	C₈×C₁₇₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₇₃
Fitting	C₄×C₁₆×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	#102
cosocle	C₄×C₈×C₁₇₃	C₂²
hypercenter	C₄×C₁₆	D₁₇₃
hypocenter	C₁₇₃	#1332
LCS	G ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
13. Dic₆₉₂	C₈	semidirect
44. C₁₇₃	#1332	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₆₉₂	2
#26011244	2
C₄×C₁₆×C₁₇₃	2
#1332	173