Subgroups — #1746×D₇₃ #33083451

Order 18688.

1134 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#97×D₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
3	#376×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#22609684	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609683	C₂	1	1	yes	maximal, characteristic, complemented
6	#22609682	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609681	C₂	1	4	yes	maximal, complemented
8	#336×D₇₃	C₂	1	4	yes	maximal
9	C₇₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₇₃	C₂²	1	2	yes	—
11	C₂²×C₈×D₇₃	C₂²	1	2	yes	—
12	C₂×C₄²×D₇₃	C₂²	1	2	yes	—
13	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
14	C₂×#14556502	C₂²	1	1	yes	characteristic
15	C₂×#14556501	C₂²	1	2	yes	—
16	#97×D₇₃	C₄	1	8	yes	complemented
17	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
19	C₄×C₈×D₇₃	C₄	1	8	yes	complemented
20	#17988433	C₂²	1	2	yes	—
21	#17988433	C₂²	1	1	yes	characteristic
22	#17988432	C₄	1	4	yes	—
23	#95×D₇₃	C₄	1	4	yes	—
24	#17988342	C₂²	1	4	yes	—
25	#17988341	C₂²	1	4	yes	—
26	#17988341	C₂²	1	4	yes	—
27	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
29	#376×C₇₃	C₂²	1	1	yes	characteristic
30	#336×C₇₃	C₂²	1	4	yes	—
31	C₂²×C₄×D₇₃	C₂×C₄	1	4	yes	—
32	C₂²×C₄×D₇₃	C₂×C₄	1	2	yes	—
33	C₂²×C₄×D₇₃	C₂³	1	1	yes	characteristic
34	C₂²×#10596831	C₂³	1	2	yes	—
35	C₂²×#10596831	C₂³	1	2	yes	—
36	C₂×C₈×D₇₃	—	2	8	no	—
37	C₂×C₈×D₇₃	—	2	8	no	—
38	C₂×C₈×D₇₃	C₂×C₄	1	8	yes	—
39	C₂×C₈×D₇₃	C₂×C₄	1	8	yes	—
40	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
41	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
42	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
43	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
44	C₄²×D₇₃	—	2	8	no	—
45	C₄²×D₇₃	C₂×C₄	1	4	yes	—
46	C₄²×D₇₃	C₂×C₄	1	4	yes	—
47	#14556502	C₂×C₄	1	4	yes	complemented
48	#14556501	C₂×C₄	1	8	yes	complemented
49	C₄×#10596831	C₂×C₄	1	4	yes	complemented
50	C₈×Dic₇₃	C₂×C₄	1	8	yes	complemented
51	#14556498	C₂×C₄	1	8	yes	—
52	#14556498	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
57	#97×C₇₃	C₂×C₄	1	4	yes	complemented
58	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	complemented
59	#95×C₇₃	C₂×C₄	1	4	yes	—
60	C₂³×D₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×Dic₇₃	C₂²×C₄	1	4	yes	—
62	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
63	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×D₇₃	—	2	8	no	—
65	C₂×C₄×D₇₃	—	2	8	no	—
66	C₂×C₄×D₇₃	—	2	16	no	—
67	C₂×C₄×D₇₃	C₄²	1	8	yes	—
68	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
69	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
70	C₂×C₄×D₇₃	#41	1	8	yes	—
71	C₂×#10596831	—	2	4	no	—
72	C₂×#10596831	—	2	4	no	—
73	C₂×#10596831	C₂²×C₄	1	4	yes	—
74	C₂×#10596831	C₂²×C₄	1	4	yes	—
75	C₈×D₇₃	—	2	16	no	—
76	C₈×D₇₃	—	2	16	no	—
77	C₄×Dic₇₃	—	2	4	no	—
78	C₄×Dic₇₃	—	2	8	no	—
79	C₄×Dic₇₃	C₄²	1	8	yes	—
80	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
81	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₇₃	—	2	4	no	—
86	C₂×C₈×C₇₃	—	2	4	no	—
87	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
89	C₄²×C₇₃	—	2	4	no	—
90	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
91	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
92	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
93	C₂²×D₇₃	C₂×C₄²	1	2	yes	—
94	C₂²×D₇₃	#117	1	4	yes	—
95	C₂²×D₇₃	#131	1	8	yes	—
96	C₂×C₄×C₇₃	—	2	4	no	—
97	C₂×C₄×C₇₃	—	2	4	no	—
98	C₂×C₄×C₇₃	—	2	8	no	—
99	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
100	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂×#41	1	4	yes	—
103	C₂×Dic₇₃	—	2	4	no	—
104	C₂×Dic₇₃	—	2	4	no	—
105	C₂×Dic₇₃	—	2	8	no	—
106	C₂×Dic₇₃	C₂×C₄²	1	4	yes	—
107	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
108	C₂×Dic₇₃	#117	1	4	yes	—
109	C₄×D₇₃	—	2	16	no	—
110	C₄×D₇₃	—	4	16	no	—
111	C₄×D₇₃	#117	1	8	yes	—
112	C₄×D₇₃	#117	1	8	yes	—
113	C₈×C₇₃	—	2	4	no	—
114	C₈×C₇₃	—	2	4	no	—
115	#10596831	—	2	4	no	—
116	#10596831	—	2	4	no	—
117	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
118	C₂²×C₇₃	C₂×#117	1	2	yes	—
119	C₂²×C₇₃	C₂×#131	1	4	yes	—
120	C₂×D₇₃	#376	1	4	yes	—
121	C₂×D₇₃	#405	1	8	yes	—
122	C₂×D₇₃	#433	1	16	yes	—
123	C₄×C₇₃	—	2	4	no	—
124	C₄×C₇₃	—	4	4	no	—
125	C₄×C₇₃	C₂×#117	1	2	yes	—
126	C₄×C₇₃	C₂×#117	1	2	yes	—
127	Dic₇₃	—	2	4	no	—
128	Dic₇₃	—	4	4	no	—
129	Dic₇₃	#376	1	4	yes	—
130	C₂×#1746	—	73	1	no	maximal, Sylow
131	C₂×C₇₃	C₂×#376	1	1	yes	characteristic
132	C₂×C₇₃	C₂×#405	1	2	yes	—
133	C₂×C₇₃	C₂×#433	1	4	yes	—
134	D₇₃	#1746	1	8	yes	direct product factor
135	C₂²×#97	—	73	1	no	—
136	C₂²×C₄×C₈	—	73	1	no	—
137	C₂×#376	—	73	1	no	—
138	#1746	—	73	1	no	—
139	#1746	—	73	1	no	—
140	#1746	—	73	1	no	—
141	#1746	—	73	4	no	—
142	#1746	D₇₃	1	1	yes	characteristic, direct product factor
143	C₂×#336	—	73	4	no	—
144	C₇₃	C₂×#1746	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₈	—	73	2	no	—
146	C₂³×C₈	—	73	2	no	—
147	C₂²×C₄²	—	73	1	no	—
148	C₂²×C₄²	—	73	2	no	—
149	C₂×#97	—	73	2	no	—
150	C₂×#97	—	73	1	no	—
151	C₂×#97	—	73	8	no	—
152	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
153	C₂×C₄×C₈	—	73	2	no	—
154	C₂×C₄×C₈	—	73	1	no	—
155	C₂×C₄×C₈	—	73	8	no	—
156	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
157	#376	—	73	2	no	—
158	#376	—	73	1	no	—
159	#376	—	73	4	no	—
160	#376	C₂×D₇₃	1	1	yes	characteristic
161	C₂×#95	—	73	4	no	—
162	#336	—	73	4	no	—
163	#336	—	73	4	no	—
164	#336	—	73	4	no	—
165	#336	C₂×D₇₃	1	4	yes	—
166	C₂³×C₄	—	73	1	no	—
167	C₂³×C₄	—	73	2	no	—
168	C₂³×C₄	—	73	4	no	—
169	C₂²×C₈	—	73	2	no	—
170	C₂²×C₈	—	73	2	no	—
171	C₂²×C₈	—	73	8	no	—
172	C₂²×C₈	—	73	8	no	—
173	C₂²×C₈	—	146	8	no	—
174	C₂²×C₈	—	146	8	no	—
175	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
176	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
177	C₂×C₄²	—	73	2	no	—
178	C₂×C₄²	—	73	1	no	—
179	C₂×C₄²	—	73	4	no	—
180	C₂×C₄²	—	73	4	no	—
181	C₂×C₄²	—	73	4	no	—
182	C₂×C₄²	—	73	2	no	—
183	C₂×C₄²	—	146	8	no	—
184	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
185	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
186	#97	—	73	8	no	—
187	#97	—	73	4	no	—
188	#97	C₄×D₇₃	1	4	yes	complemented
189	C₄×C₈	—	73	8	no	—
190	C₄×C₈	—	73	4	no	—
191	C₄×C₈	C₄×D₇₃	1	4	yes	complemented
192	#95	—	73	8	no	—
193	#95	—	73	4	no	—
194	#95	C₄×D₇₃	1	4	yes	—
195	C₂⁴	—	73	1	no	—
196	C₂²×C₄	—	73	1	no	—
197	C₂²×C₄	—	73	2	no	—
198	C₂²×C₄	—	73	2	no	—
199	C₂²×C₄	—	73	8	no	—
200	C₂²×C₄	—	73	2	no	—
201	C₂²×C₄	—	73	8	no	—
202	C₂²×C₄	—	73	4	no	—
203	C₂²×C₄	—	146	8	no	—
204	C₂²×C₄	—	146	16	no	—
205	C₂²×C₄	—	146	8	no	—
206	C₂²×C₄	C₂×C₄×D₇₃	1	4	yes	—
207	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
208	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	73	4	no	—
212	C₂×C₈	—	73	4	no	—
213	C₂×C₈	—	146	16	no	—
214	C₂×C₈	—	146	16	no	—
215	C₂×C₈	—	146	4	no	—
216	C₂×C₈	—	146	4	no	—
217	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
218	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
219	C₄²	—	2	4	no	—
220	C₄²	—	73	8	no	—
221	C₄²	—	73	2	no	—
222	C₄²	—	73	2	no	—
223	C₄²	—	146	8	no	—
224	C₄²	—	146	4	no	—
225	C₄²	C₂×C₄×D₇₃	1	2	yes	—
226	C₄²	C₂×C₄×D₇₃	1	2	yes	—
227	C₂³	—	73	2	no	—
228	C₂³	—	73	4	no	—
229	C₂³	—	73	8	no	—
230	C₂³	C₂²×C₄×D₇₃	1	1	yes	characteristic
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	292	16	no	—
235	C₂×C₄	—	73	2	no	—
236	C₂×C₄	—	73	4	no	—
237	C₂×C₄	—	73	8	no	—
238	C₂×C₄	—	73	8	no	—
239	C₂×C₄	—	73	4	no	—
240	C₂×C₄	—	146	16	no	—
241	C₂×C₄	—	146	4	no	—
242	C₂×C₄	—	146	8	no	—
243	C₂×C₄	—	146	4	no	—
244	C₂×C₄	C₄²×D₇₃	1	4	yes	—
245	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
247	C₂×C₄	#41×D₇₃	1	4	yes	—
248	C₈	—	2	4	no	—
249	C₈	—	2	4	no	—
250	C₈	—	146	4	no	—
251	C₈	—	146	4	no	—
252	C₂²	—	73	4	no	—
253	C₂²	—	73	16	no	—
254	C₂²	—	73	8	no	—
255	C₂²	C₂×C₄²×D₇₃	1	1	yes	characteristic
256	C₂²	#117×D₇₃	1	2	yes	—
257	C₂²	#131×D₇₃	1	4	yes	—
258	C₄	—	2	4	no	—
259	C₄	—	4	4	no	—
260	C₄	—	292	4	no	—
261	C₄	—	73	4	no	—
262	C₄	—	146	4	no	—
263	C₄	#117×D₇₃	1	2	yes	—
264	C₄	#117×D₇₃	1	2	yes	—
265	C₂	—	73	8	no	—
266	C₂	#376×D₇₃	1	1	yes	characteristic
267	C₂	#405×D₇₃	1	2	yes	—
268	C₂	#433×D₇₃	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₃
commutator	C₂²×C₇₃	C₂²×C₄²
Fitting	C₇₃×#1746	C₂
socle	C₂³×C₇₃	C₂³×C₄
prime radical	C₂³×D₇₃	C₂²×C₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1746	D₇₃
hypocenter	C₇₃	C₂×#1746
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1746

Complements

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

N	H ≅ G/N	type
4. #22609684	C₂	semidirect
5. #22609683	C₂	semidirect
6. #22609682	C₂	semidirect
7. #22609681	C₂	semidirect
9. C₇₃×#1746	C₂	semidirect
16. #97×D₇₃	C₄	semidirect
19. C₄×C₈×D₇₃	C₄	semidirect
47. #14556502	C₂×C₄	semidirect
48. #14556501	C₂×C₄	semidirect
49. C₄×#10596831	C₂×C₄	semidirect
50. C₈×Dic₇₃	C₂×C₄	semidirect
57. #97×C₇₃	C₂×C₄	semidirect
58. C₄×C₈×C₇₃	C₂×C₄	semidirect
134. D₇₃	#1746	direct (N × H)
142. #1746	D₇₃	direct (N × H)
144. C₇₃	C₂×#1746	semidirect
188. #97	C₄×D₇₃	semidirect
191. C₄×C₈	C₄×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#97×D₇₃	2
C₂×C₄×C₈×D₇₃	2
#376×D₇₃	2
#22609684	2
#22609683	2
#22609682	2
#22609681	2
#336×D₇₃	2
C₇₃×#1746	2
C₂×#1746	73

Subgroups — #1746×D73 #33083451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1746×D₇₃ #33083451