Subgroups — #36351603

Order 22144.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011864	C₂	1	1	yes	maximal, characteristic
2	#26011312	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14583815	—	2	1	no	—
5	C₂×#14583815	—	2	1	no	—
6	#19149679	C₄	1	1	yes	characteristic
7	#19149676	C₄	1	1	yes	characteristic
8	#19149619	—	2	1	no	—
9	M₃₂×C₁₇₃	—	2	1	no	—
10	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₄₆	—	2	1	no	—
12	C₂×D₆₉₂	—	2	1	no	—
13	C₂×D₆₉₂	—	4	1	no	—
14	#14583815	—	4	1	no	—
15	#14583815	—	4	1	no	—
16	#14583815	—	4	1	no	—
17	#14583815	—	4	1	no	—
18	C₄⋊Dic₁₇₃	—	4	1	no	—
19	#14583801	—	4	1	no	—
20	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
22	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
23	C₁₆×C₁₇₃	—	4	1	no	—
24	C₂²×D₁₇₃	—	4	1	no	—
25	Dic₃₄₆	—	4	1	no	—
26	Dic₃₄₆	—	4	1	no	—
27	D₆₉₂	—	4	1	no	—
28	D₆₉₂	—	4	1	no	—
29	D₆₉₂	—	8	2	no	—
30	C₂×Dic₁₇₃	—	4	1	no	—
31	C₂×C₄×C₁₇₃	—	2	1	no	—
32	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₈×C₁₇₃	D₈	1	1	yes	characteristic
34	C₈×C₁₇₃	D₈	1	1	yes	characteristic
35	C₈×C₁₇₃	SD₁₆	1	1	yes	characteristic
36	C₈×C₁₇₃	SD₁₆	1	1	yes	characteristic
37	C₂²×C₁₇₃	#100	1	1	yes	characteristic
38	C₂×D₁₇₃	—	8	2	no	—
39	C₂×D₁₇₃	—	8	1	no	—
40	C₄×C₁₇₃	—	2	2	no	—
41	C₄×C₁₇₃	#102	1	1	yes	characteristic
42	C₄×C₁₇₃	#102	1	1	yes	characteristic
43	Dic₁₇₃	—	8	1	no	—
44	C₂×C₁₇₃	—	2	1	no	—
45	C₂×C₁₇₃	#331	1	1	yes	characteristic
46	D₁₇₃	—	16	1	no	—
47	C₁₇₃	#1359	1	1	yes	characteristic, complemented, Sylow
48	#1359	—	173	1	no	maximal, Sylow
49	#492	—	173	1	no	—
50	#364	—	173	1	no	—
51	#364	D₁₇₃	1	1	yes	characteristic, complemented
52	C₂×SD₁₆	—	346	1	no	—
53	C₂×SD₁₆	—	346	1	no	—
54	C₄⋊Q₈	—	173	1	no	—
55	#127	—	173	1	no	—
56	M₃₂	—	2	1	no	—
57	M₃₂	—	346	1	no	—
58	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
59	C₂×Q₈	—	346	1	no	—
60	C₂×D₄	—	346	1	no	—
61	C₂×D₄	—	692	1	no	—
62	SD₁₆	—	692	1	no	—
63	SD₁₆	—	692	1	no	—
64	SD₁₆	—	692	1	no	—
65	SD₁₆	—	692	1	no	—
66	C₂×C₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
67	C₂×C₈	D₆₉₂	1	1	yes	characteristic
68	C₄⋊C₄	—	692	1	no	—
69	C₄²	C₄×D₁₇₃	1	1	yes	characteristic
70	C₁₆	—	4	1	no	—
71	C₁₆	—	692	1	no	—
72	C₂³	—	692	1	no	—
73	Q₈	—	692	1	no	—
74	Q₈	—	692	1	no	—
75	D₄	—	1384	2	no	—
76	D₄	—	692	1	no	—
77	D₄	—	692	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	692	1	no	—
80	C₂×C₄	#14583803	1	1	yes	characteristic
81	C₈	#14583811	1	1	yes	characteristic
82	C₈	D₁₃₈₄	1	1	yes	characteristic
83	C₈	#14583813	1	1	yes	characteristic
84	C₈	#14583815	1	1	yes	characteristic
85	C₂²	—	1384	2	no	—
86	C₂²	—	1384	1	no	—
87	C₂²	#19149595	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	1384	1	no	—
90	C₄	#19149599	1	1	yes	characteristic
91	C₄	#19149601	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	2768	1	no	—
94	C₂	#26011234	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26011234
commutator	C₄²×C₁₇₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₇₃
Fitting	#364×C₁₇₃	C₂
socle	C₂×C₁₇₃	#331
prime radical	#19149676	C₄
cosocle	C₄×C₈×C₁₇₃	C₂²
hypercenter	#364	D₁₇₃
hypocenter	C₁₇₃	#1359
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. #26011312	C₂	semidirect
3. #364×C₁₇₃	C₂	semidirect
47. C₁₇₃	#1359	semidirect
51. #364	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26011864	2
#26011312	2
#364×C₁₇₃	2
#1359	173