Subgroups — #36351610

Order 22144.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011868	C₂	1	1	yes	maximal, characteristic
2	#26011244	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₁₆×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₃₈₄	—	2	1	no	—
5	C₂×D₁₃₈₄	C₄	1	2	yes	—
6	#19149676	—	2	1	no	—
7	C₄⋊#13063436	—	2	1	no	—
8	C₂×C₁₆×C₁₇₃	—	2	1	no	—
9	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
10	C₂×D₆₉₂	—	2	2	no	—
11	C₂×D₆₉₂	—	4	1	no	—
12	C₂×D₆₉₂	—	4	1	no	—
13	D₁₃₈₄	—	4	2	no	—
14	D₁₃₈₄	—	4	1	no	—
15	D₁₃₈₄	—	4	1	no	—
16	C₂×#13063436	—	4	1	no	—
17	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
19	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
20	C₁₆×C₁₇₃	—	4	1	no	—
21	C₂²×D₁₇₃	—	4	2	no	—
22	D₆₉₂	—	4	2	no	—
23	D₆₉₂	—	4	2	no	—
24	D₆₉₂	—	8	2	no	—
25	D₆₉₂	—	8	1	no	—
26	D₆₉₂	—	8	1	no	—
27	#13063436	—	8	1	no	—
28	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
30	C₂×C₄×C₁₇₃	SD₁₆	1	1	yes	characteristic
31	C₈×C₁₇₃	—	2	1	no	—
32	C₈×C₁₇₃	D₈	1	1	yes	characteristic
33	C₈×C₁₇₃	SD₁₆	1	1	yes	characteristic
34	C₂²×C₁₇₃	#102	1	1	yes	characteristic
35	C₂×D₁₇₃	—	8	2	no	—
36	C₂×D₁₇₃	—	8	2	no	—
37	C₂×D₁₇₃	—	8	2	no	—
38	C₄×C₁₇₃	—	2	1	no	—
39	C₄×C₁₇₃	#100	1	1	yes	characteristic
40	C₄×C₁₇₃	#102	1	1	yes	characteristic
41	C₄×C₁₇₃	D₁₆	1	1	yes	characteristic
42	C₄×C₁₇₃	SD₃₂	1	1	yes	characteristic
43	C₂×C₁₇₃	#331	1	1	yes	characteristic
44	C₂×C₁₇₃	#357	1	1	yes	characteristic
45	C₂×C₁₇₃	#361	1	1	yes	characteristic
46	D₁₇₃	—	16	2	no	—
47	C₁₇₃	#1361	1	1	yes	characteristic, complemented, Sylow
48	#1361	—	173	1	no	maximal, Sylow
49	#493	—	173	1	no	—
50	C₄⋊C₁₆	D₁₇₃	1	1	yes	characteristic, complemented
51	#335	—	173	1	no	—
52	C₂×D₈	—	346	1	no	—
53	C₂×D₈	—	173	2	no	—
54	#127	—	346	1	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₄⋊C₈	—	346	1	no	—
57	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
58	C₂×D₄	—	346	2	no	—
59	C₂×D₄	—	692	1	no	—
60	C₂×D₄	—	692	1	no	—
61	D₈	—	692	2	no	—
62	D₈	—	692	1	no	—
63	D₈	—	692	1	no	—
64	C₂×C₈	—	692	1	no	—
65	C₂×C₈	C₄×D₁₇₃	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
67	C₄²	D₆₉₂	1	1	yes	characteristic
68	C₁₆	—	4	1	no	—
69	C₂³	—	692	2	no	—
70	D₄	—	1384	2	no	—
71	D₄	—	1384	1	no	—
72	D₄	—	1384	1	no	—
73	D₄	—	692	2	no	—
74	D₄	—	692	2	no	—
75	C₂×C₄	#14583803	1	1	yes	characteristic
76	C₂×C₄	D₁₃₈₄	1	1	yes	characteristic
77	C₂×C₄	#14583815	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	1384	1	no	—
80	C₈	#14583811	1	1	yes	characteristic
81	C₈	#14583813	1	1	yes	characteristic
82	C₂²	—	1384	2	no	—
83	C₂²	—	1384	2	no	—
84	C₂²	—	1384	2	no	—
85	C₂²	#19149601	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	#19149595	1	1	yes	characteristic
88	C₄	#19149599	1	1	yes	characteristic
89	C₄	D₂₇₆₈	1	1	yes	characteristic
90	C₄	#19149624	1	1	yes	characteristic
91	C₂	—	2768	2	no	—
92	C₂	#26011234	1	1	yes	characteristic
93	C₂	#26011293	1	1	yes	characteristic
94	C₂	#26011305	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19149601
commutator	C₂×C₈×C₁₇₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₇₃
Fitting	C₄⋊C₁₆×C₁₇₃	C₂
socle	C₂²×C₁₇₃	#102
prime radical	#26011868	C₂
cosocle	C₄×C₈×C₁₇₃	C₂²
hypercenter	C₄⋊C₁₆	D₁₇₃
hypocenter	C₁₇₃	#1361
LCS	G ▷ C₂×C₈×C₁₇₃ ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #26011244	C₂	semidirect
3. C₄⋊C₁₆×C₁₇₃	C₂	semidirect
47. C₁₇₃	#1361	semidirect
50. C₄⋊C₁₆	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26011868	2
#26011244	2
C₄⋊C₁₆×C₁₇₃	2
#1361	173