Subgroups — #32975341

Order 18688.

574 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22609677	C₂	1	1	yes	maximal, characteristic
2	C₄²×#10596831	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988309	C₂	1	2	yes	maximal
4	C₂×C₈×#10596831	C₂	1	2	yes	maximal
5	C₇₃×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#10596831	C₄	1	4	yes	—
7	C₂×C₄×#10596831	C₂²	1	2	yes	—
8	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	#17988342	C₄	1	4	yes	—
11	#17988309	C₄	1	8	yes	complemented
12	C₈×#10596831	C₄	1	8	yes	complemented
13	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
14	C₄³×C₇₃	C₂²	1	1	yes	characteristic
15	#336×C₇₃	C₄	1	4	yes	—
16	C₂²×#10596831	C₂×C₄	1	4	yes	—
17	C₂²×#10596831	C₂×C₄	1	2	yes	—
18	C₂²×#10596831	C₂×C₄	1	2	yes	—
19	C₄×#10596831	C₈	1	16	yes	complemented
20	C₄×#10596831	C₂×C₄	1	4	yes	—
21	C₄×#10596831	C₂×C₄	1	4	yes	—
22	C₄×#10596831	C₂×C₄	1	4	yes	—
23	C₄×#10596831	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
31	C₂×#10596831	—	2	8	no	—
32	C₂×#10596831	C₄²	1	8	yes	—
33	C₂×#10596831	C₄²	1	4	yes	—
34	C₂×#10596831	C₄²	1	4	yes	—
35	C₂×#10596831	C₂×C₈	1	8	yes	—
36	C₂×#10596831	C₂×C₈	1	8	yes	—
37	C₂²×C₄×C₇₃	C₄²	1	4	yes	—
38	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	—	2	8	no	—
42	C₂×C₈×C₇₃	C₄²	1	8	yes	—
43	C₄²×C₇₃	—	2	4	no	—
44	C₄²×C₇₃	—	2	4	no	—
45	C₄²×C₇₃	C₂×C₈	1	8	yes	—
46	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
47	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
48	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₇₃	—	2	4	no	—
50	C₂×C₄×C₇₃	—	2	4	no	—
51	C₂×C₄×C₇₃	C₄×C₈	1	8	yes	—
52	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
55	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
56	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
57	C₂×C₄×C₇₃	#131	1	4	yes	—
58	C₂×C₄×C₇₃	#131	1	4	yes	—
59	C₈×C₇₃	—	2	8	no	—
60	#10596831	—	2	8	no	—
61	#10596831	C₄×C₈	1	8	yes	—
62	#10596831	C₄×C₈	1	8	yes	—
63	C₂²×C₇₃	C₄³	1	1	yes	characteristic
64	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
65	C₂²×C₇₃	#405	1	2	yes	—
66	C₂²×C₇₃	#405	1	2	yes	—
67	C₄×C₇₃	—	2	8	no	—
68	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
69	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
70	C₄×C₇₃	#405	1	4	yes	—
71	C₄×C₇₃	#405	1	4	yes	—
72	#7841	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	#1450	1	4	yes	—
74	C₂×C₇₃	C₄²×C₈	1	1	yes	characteristic
75	C₂×C₇₃	#1726	1	1	yes	characteristic
76	C₂×C₇₃	#1726	1	1	yes	characteristic
77	#1744	—	73	1	no	—
78	#1744	D₇₃	1	1	yes	characteristic
79	C₄²×C₈	—	73	1	no	—
80	C₂×#322	—	73	2	no	—
81	C₂×C₈²	—	73	2	no	—
82	C₇₃	#7841	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄×C₈	—	73	4	no	—
84	C₂×C₄×C₈	—	73	1	no	—
85	C₂×C₄×C₈	—	73	2	no	—
86	C₂×C₄×C₈	—	73	1	no	—
87	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
88	C₄³	C₂×D₇₃	1	1	yes	characteristic
89	#336	—	73	4	no	—
90	#336	Dic₇₃	1	4	yes	—
91	#322	—	73	8	no	—
92	C₈²	—	73	8	no	—
93	C₂²×C₈	—	73	2	no	—
94	C₂²×C₈	—	73	2	no	—
95	C₂²×C₈	—	73	4	no	—
96	C₂²×C₈	C₂×Dic₇₃	1	4	yes	—
97	C₂×C₄²	C₄×D₇₃	1	2	yes	—
98	C₂×C₄²	C₄×D₇₃	1	2	yes	—
99	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
100	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
101	C₄×C₈	—	73	16	no	—
102	C₄×C₈	—	73	4	no	—
103	C₄×C₈	—	73	4	no	—
104	C₄×C₈	—	73	4	no	—
105	C₄×C₈	—	73	4	no	—
106	C₄×C₈	C₄×D₇₃	1	4	yes	—
107	C₄×C₈	C₄×D₇₃	1	4	yes	—
108	C₂²×C₄	C₄×Dic₇₃	1	4	yes	—
109	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
110	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
111	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
112	C₂×C₈	—	2	8	no	—
113	C₂×C₈	—	73	4	no	—
114	C₂×C₈	—	73	4	no	—
115	C₂×C₈	—	73	8	no	—
116	C₂×C₈	—	73	8	no	—
117	C₂×C₈	—	73	8	no	—
118	C₂×C₈	—	146	8	no	—
119	C₂×C₈	C₄×Dic₇₃	1	8	yes	—
120	C₄²	—	2	4	no	—
121	C₄²	—	2	4	no	—
122	C₄²	C₈×D₇₃	1	8	yes	—
123	C₄²	C₂×C₄×D₇₃	1	2	yes	—
124	C₄²	C₂×C₄×D₇₃	1	2	yes	—
125	C₂³	C₂×C₄×Dic₇₃	1	1	yes	characteristic
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	C₈×Dic₇₃	1	8	yes	—
129	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
130	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
131	C₂×C₄	C₄²×D₇₃	1	2	yes	—
132	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
133	C₂×C₄	C₂×C₈×D₇₃	1	4	yes	—
134	C₂×C₄	#14556598	1	4	yes	—
135	C₂×C₄	#14556599	1	4	yes	—
136	C₈	—	2	8	no	—
137	C₈	—	73	8	no	—
138	C₈	—	73	8	no	—
139	C₈	—	146	8	no	—
140	C₂²	C₄²×Dic₇₃	1	1	yes	characteristic
141	C₂²	C₂×C₈×Dic₇₃	1	2	yes	—
142	C₂²	#17988543	1	2	yes	—
143	C₂²	#17988545	1	2	yes	—
144	C₄	—	2	8	no	—
145	C₄	C₄×C₈×D₇₃	1	2	yes	—
146	C₄	C₄×C₈×D₇₃	1	2	yes	—
147	C₄	#17988543	1	4	yes	—
148	C₄	#17988544	1	4	yes	—
149	C₂	#22608215	1	4	yes	—
150	C₂	C₄×C₈×Dic₇₃	1	1	yes	characteristic
151	C₂	#22609618	1	1	yes	characteristic
152	C₂	#22609619	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₃
commutator	C₂×C₇₃	C₄²×C₈
Fitting	C₇₃×#1744	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₄²
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1744	D₇₃
hypocenter	C₇₃	#7841
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄² ◁ #1744

Complements

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

N	H ≅ G/N	type
11. #17988309	C₄	semidirect
12. C₈×#10596831	C₄	semidirect
19. C₄×#10596831	C₈	semidirect
82. C₇₃	#7841	semidirect

Maximal subgroups

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

subgroup	index
#22609677	2
C₄²×#10596831	2
C₂×#17988309	2
C₂×C₈×#10596831	2
C₇₃×#1744	2
#7841	73