Subgroups — #1778×D₇₃ #33084083

Order 18688.

1046 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108×D₇₃	C₂	1	2	yes	maximal
2	C₄×M₁₆×D₇₃	C₂	1	2	yes	maximal
3	C₂×#97×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#22609839	C₂	1	2	yes	maximal, complemented
5	#22609838	C₂	1	2	yes	maximal, complemented
6	#22609837	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609836	C₂	1	2	yes	maximal, complemented
8	#341×D₇₃	C₂	1	2	yes	maximal
9	C₇₃×#1778	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×D₇₃	C₂²	1	4	yes	—
11	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
14	C₂×#14556527	C₂²	1	2	yes	—
15	C₂×#14556526	C₂²	1	4	yes	—
16	#108×D₇₃	C₄	1	16	yes	complemented
17	C₄×#11924012	C₂²	1	2	yes	—
18	#17988539	—	2	2	no	—
19	M₁₆×Dic₇₃	C₂²	1	4	yes	—
20	C₄×#11924011	—	2	2	no	—
21	C₂×#14556502	C₂²	1	1	yes	characteristic
22	C₂×#14556501	C₂²	1	1	yes	characteristic
23	C₂×#14556501	C₂²	1	1	yes	characteristic
24	#97×D₇₃	—	2	2	no	—
25	#97×D₇₃	C₄	1	4	yes	—
26	#97×D₇₃	C₄	1	4	yes	—
27	C₄×C₈×D₇₃	—	2	2	no	—
28	#17988352	C₂²	1	2	yes	—
29	#17988351	C₂²	1	4	yes	—
30	C₂×#108×C₇₃	C₂²	1	2	yes	—
31	C₄×M₁₆×C₇₃	C₂²	1	2	yes	—
32	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
33	#341×C₇₃	C₂²	1	2	yes	—
34	C₂²×C₄×D₇₃	Q₈	1	2	yes	—
35	C₂²×C₄×D₇₃	C₂³	1	1	yes	characteristic
36	C₂×#11924012	C₂³	1	4	yes	—
37	C₂×#11924011	—	2	4	no	—
38	M₁₆×D₇₃	—	2	8	no	—
39	M₁₆×D₇₃	C₂×C₄	1	16	yes	—
40	C₂²×#10596831	C₂³	1	1	yes	characteristic
41	C₂²×#10596831	C₂³	1	1	yes	characteristic
42	C₂×C₈×D₇₃	—	2	4	no	—
43	C₂×C₈×D₇₃	—	2	4	no	—
44	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₇₃	C₂×C₄	1	8	yes	—
47	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
50	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
51	C₄²×D₇₃	C₂×C₄	1	2	yes	—
52	C₄²×D₇₃	C₂×C₄	1	2	yes	—
53	C₄²×D₇₃	D₄	1	2	yes	—
54	C₄²×D₇₃	D₄	1	2	yes	—
55	#14556527	C₂×C₄	1	8	yes	complemented
56	#14556526	C₂×C₄	1	16	yes	complemented
57	#14556502	—	2	2	no	—
58	#14556502	C₂×C₄	1	4	yes	—
59	#14556501	—	2	4	no	—
60	#14556501	C₂×C₄	1	4	yes	—
61	#14556501	C₂×C₄	1	4	yes	—
62	C₄×#10596831	—	2	2	no	—
63	C₈×Dic₇₃	—	2	4	no	—
64	C₂×M₁₆×C₇₃	C₂³	1	4	yes	—
65	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
68	#108×C₇₃	C₂×C₄	1	8	yes	complemented
69	#97×C₇₃	—	2	2	no	—
70	#97×C₇₃	C₂×C₄	1	4	yes	—
71	C₄×C₈×C₇₃	—	2	2	no	—
72	C₂³×D₇₃	C₂×Q₈	1	1	yes	characteristic
73	C₂²×Dic₇₃	C₂×Q₈	1	2	yes	—
74	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₄×D₇₃	—	4	4	no	—
76	C₂×C₄×D₇₃	C₄⋊C₄	1	4	yes	—
77	C₂×C₄×D₇₃	C₄⋊C₄	1	4	yes	—
78	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×D₇₃	C₂²×C₄	1	4	yes	—
81	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×D₇₃	#41	1	4	yes	—
84	#11924012	—	2	4	no	—
85	#11924012	C₂²×C₄	1	8	yes	—
86	#11924011	—	2	16	no	—
87	C₂×#10596831	—	2	4	no	—
88	C₂×#10596831	—	2	2	no	—
89	C₂×#10596831	C₂²×C₄	1	2	yes	—
90	C₂×#10596831	C₂²×C₄	1	2	yes	—
91	C₂×#10596831	C₂²×C₄	1	4	yes	—
92	C₈×D₇₃	—	2	4	no	—
93	C₈×D₇₃	—	2	4	no	—
94	C₈×D₇₃	—	2	16	no	—
95	C₈×D₇₃	C₄²	1	8	yes	—
96	C₈×D₇₃	C₄²	1	8	yes	—
97	C₄×Dic₇₃	C₄⋊C₄	1	4	yes	—
98	C₄×Dic₇₃	C₄⋊C₄	1	4	yes	—
99	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
100	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
102	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
103	M₁₆×C₇₃	—	2	4	no	—
104	M₁₆×C₇₃	C₂²×C₄	1	8	yes	—
105	C₂×C₈×C₇₃	—	2	4	no	—
106	C₂×C₈×C₇₃	—	2	2	no	—
107	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
110	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
111	C₄²×C₇₃	C₂×D₄	1	2	yes	—
112	C₂³×C₇₃	C₂²×Q₈	1	1	yes	characteristic
113	C₂²×D₇₃	—	2	4	no	—
114	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×D₇₃	C₄×Q₈	1	4	yes	—
117	C₂×C₄×C₇₃	—	4	2	no	—
118	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
121	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
123	C₂×Dic₇₃	—	4	2	no	—
124	C₂×Dic₇₃	C₂×C₄⋊C₄	1	4	yes	—
125	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×Dic₇₃	C₄×Q₈	1	4	yes	—
128	C₄×D₇₃	—	2	8	no	—
129	C₄×D₇₃	—	4	4	no	—
130	C₄×D₇₃	C₂×C₄²	1	2	yes	—
131	C₄×D₇₃	C₂×C₄²	1	2	yes	—
132	C₄×D₇₃	#117	1	2	yes	—
133	C₄×D₇₃	#117	1	2	yes	—
134	C₄×D₇₃	C₄×D₄	1	4	yes	—
135	C₄×D₇₃	C₄×D₄	1	4	yes	—
136	C₈×C₇₃	—	2	2	no	—
137	C₈×C₇₃	—	2	8	no	—
138	C₈×C₇₃	C₂×C₄²	1	4	yes	—
139	#10596831	—	2	2	no	—
140	#10596831	—	2	8	no	—
141	#10596831	C₂×C₄²	1	4	yes	—
142	C₂²×C₇₃	—	2	2	no	—
143	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₇₃	C₂×C₄×Q₈	1	2	yes	—
145	C₂×D₇₃	—	2	2	no	—
146	C₂×D₇₃	—	2	8	no	—
147	C₂×D₇₃	C₄×C₄⋊C₄	1	2	yes	—
148	C₂×D₇₃	C₄×C₄⋊C₄	1	2	yes	—
149	C₂×D₇₃	#430	1	4	yes	—
150	C₄×C₇₃	—	2	4	no	—
151	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
152	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
153	C₄×C₇₃	C₂×C₄×D₄	1	2	yes	—
154	Dic₇₃	—	2	4	no	—
155	Dic₇₃	C₄×C₄⋊C₄	1	2	yes	—
156	Dic₇₃	C₄×C₄⋊C₄	1	2	yes	—
157	C₂×#1778	—	73	1	no	maximal, Sylow
158	C₂×C₇₃	—	2	2	no	—
159	C₂×C₇₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₇₃	C₂×#430	1	2	yes	—
161	D₇₃	—	2	2	no	—
162	D₇₃	#1778	1	4	yes	direct product factor
163	C₂²×#108	—	73	2	no	—
164	C₂×C₄×M₁₆	—	73	2	no	—
165	C₂²×#97	—	73	1	no	—
166	#1778	—	73	2	no	—
167	#1778	—	73	2	no	—
168	#1778	—	73	1	no	—
169	#1778	—	73	2	no	—
170	#1778	D₇₃	1	1	yes	characteristic, direct product factor
171	C₂×#341	—	73	2	no	—
172	C₇₃	C₂×#1778	1	1	yes	characteristic, complemented, Sylow
173	C₂²×M₁₆	—	73	4	no	—
174	C₂³×C₈	—	73	1	no	—
175	C₂³×C₈	—	73	1	no	—
176	C₂²×C₄²	—	73	1	no	—
177	C₂×#108	—	73	4	no	—
178	C₂×#108	—	73	2	no	—
179	C₂×#108	—	73	16	no	—
180	C₂×#108	C₂×D₇₃	1	2	yes	—
181	C₄×M₁₆	—	73	4	no	—
182	C₄×M₁₆	—	73	2	no	—
183	C₄×M₁₆	—	146	2	no	—
184	C₄×M₁₆	—	146	2	no	—
185	C₄×M₁₆	C₂×D₇₃	1	2	yes	—
186	C₂×#97	—	73	1	no	—
187	C₂×#97	—	73	1	no	—
188	C₂×#97	—	73	1	no	—
189	C₂×#97	—	73	4	no	—
190	C₂×#97	—	73	4	no	—
191	C₂×#97	—	146	2	no	—
192	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
193	C₂×C₄×C₈	—	146	2	no	—
194	#341	—	73	4	no	—
195	#341	—	73	2	no	—
196	#341	C₂×D₇₃	1	2	yes	—
197	C₂³×C₄	—	73	2	no	—
198	C₂³×C₄	—	73	1	no	—
199	C₂×M₁₆	—	73	4	no	—
200	C₂×M₁₆	—	73	16	no	—
201	C₂×M₁₆	—	146	4	no	—
202	C₂×M₁₆	—	146	8	no	—
203	C₂×M₁₆	C₂²×D₇₃	1	4	yes	—
204	C₂²×C₈	—	73	1	no	—
205	C₂²×C₈	—	73	2	no	—
206	C₂²×C₈	—	73	2	no	—
207	C₂²×C₈	—	73	8	no	—
208	C₂²×C₈	—	73	1	no	—
209	C₂²×C₈	—	73	2	no	—
210	C₂²×C₈	—	73	2	no	—
211	C₂²×C₈	—	146	4	no	—
212	C₂²×C₈	—	146	4	no	—
213	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
215	C₂×C₄²	—	73	2	no	—
216	C₂×C₄²	—	73	1	no	—
217	C₂×C₄²	—	73	2	no	—
218	C₂×C₄²	—	73	2	no	—
219	C₂×C₄²	—	73	2	no	—
220	C₂×C₄²	—	73	2	no	—
221	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
222	#108	—	73	16	no	—
223	#108	—	73	8	no	—
224	#108	C₄×D₇₃	1	8	yes	complemented
225	#97	—	2	2	no	—
226	#97	—	73	4	no	—
227	#97	—	73	4	no	—
228	#97	—	73	4	no	—
229	#97	—	146	4	no	—
230	#97	—	146	2	no	—
231	#97	C₄×D₇₃	1	4	yes	—
232	C₄×C₈	—	2	2	no	—
233	C₄×C₈	—	146	4	no	—
234	C₄×C₈	—	146	2	no	—
235	C₂⁴	—	73	1	no	—
236	C₂²×C₄	—	292	4	no	—
237	C₂²×C₄	—	73	2	no	—
238	C₂²×C₄	—	73	4	no	—
239	C₂²×C₄	—	73	1	no	—
240	C₂²×C₄	—	73	1	no	—
241	C₂²×C₄	—	73	1	no	—
242	C₂²×C₄	—	73	4	no	—
243	C₂²×C₄	—	73	1	no	—
244	C₂²×C₄	—	73	1	no	—
245	C₂²×C₄	—	73	4	no	—
246	C₂²×C₄	—	73	4	no	—
247	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
248	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
249	M₁₆	—	2	4	no	—
250	M₁₆	—	73	8	no	—
251	M₁₆	—	146	16	no	—
252	M₁₆	—	146	4	no	—
253	M₁₆	C₂×C₄×D₇₃	1	8	yes	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	73	2	no	—
257	C₂×C₈	—	73	4	no	—
258	C₂×C₈	—	73	8	no	—
259	C₂×C₈	—	73	8	no	—
260	C₂×C₈	—	73	2	no	—
261	C₂×C₈	—	146	4	no	—
262	C₂×C₈	—	146	4	no	—
263	C₂×C₈	—	146	16	no	—
264	C₂×C₈	—	146	4	no	—
265	C₂×C₈	—	146	2	no	—
266	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
267	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
268	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
269	C₄²	—	73	4	no	—
270	C₄²	—	73	4	no	—
271	C₄²	—	73	2	no	—
272	C₄²	—	73	2	no	—
273	C₄²	C₂×C₄×D₇₃	1	2	yes	—
274	C₄²	D₄×D₇₃	1	2	yes	—
275	C₂³	—	73	1	no	—
276	C₂³	—	73	1	no	—
277	C₂³	—	73	4	no	—
278	C₂³	—	146	4	no	—
279	C₂³	C₂×Q₈×D₇₃	1	1	yes	characteristic
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	292	4	no	—
282	C₂×C₄	—	292	2	no	—
283	C₂×C₄	—	73	1	no	—
284	C₂×C₄	—	73	1	no	—
285	C₂×C₄	—	73	4	no	—
286	C₂×C₄	—	73	2	no	—
287	C₂×C₄	—	73	2	no	—
288	C₂×C₄	—	73	2	no	—
289	C₂×C₄	—	73	2	no	—
290	C₂×C₄	—	73	4	no	—
291	C₂×C₄	—	73	4	no	—
292	C₂×C₄	—	73	4	no	—
293	C₂×C₄	—	146	8	no	—
294	C₂×C₄	C₄⋊C₄×D₇₃	1	4	yes	—
295	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
296	C₂×C₄	C₂²×C₄×D₇₃	1	2	yes	—
297	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
298	C₂×C₄	#41×D₇₃	1	2	yes	—
299	C₈	—	2	2	no	—
300	C₈	—	2	8	no	—
301	C₈	—	73	4	no	—
302	C₈	—	146	2	no	—
303	C₈	—	146	8	no	—
304	C₈	C₄²×D₇₃	1	4	yes	—
305	C₂²	—	2	2	no	—
306	C₂²	—	73	2	no	—
307	C₂²	—	73	4	no	—
308	C₂²	—	73	2	no	—
309	C₂²	—	146	8	no	—
310	C₂²	—	146	2	no	—
311	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
312	C₂²	C₄×Q₈×D₇₃	1	2	yes	—
313	C₄	—	2	4	no	—
314	C₄	—	73	2	no	—
315	C₄	—	73	2	no	—
316	C₄	—	146	4	no	—
317	C₄	C₂×C₄²×D₇₃	1	1	yes	characteristic
318	C₄	#117×D₇₃	1	1	yes	characteristic
319	C₄	C₄×D₄×D₇₃	1	2	yes	—
320	C₂	—	2	2	no	—
321	C₂	—	73	4	no	—
322	C₂	—	146	2	no	—
323	C₂	C₄×C₄⋊C₄×D₇₃	1	1	yes	characteristic
324	C₂	#430×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₄×C₇₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1778	C₂
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₇₃	C₂×Q₈
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1778	D₇₃
hypocenter	C₇₃	C₂×#1778
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1778

Complements

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

N	H ≅ G/N	type
4. #22609839	C₂	semidirect
5. #22609838	C₂	semidirect
6. #22609837	C₂	semidirect
7. #22609836	C₂	semidirect
9. C₇₃×#1778	C₂	semidirect
16. #108×D₇₃	C₄	semidirect
55. #14556527	C₂×C₄	semidirect
56. #14556526	C₂×C₄	semidirect
68. #108×C₇₃	C₂×C₄	semidirect
162. D₇₃	#1778	direct (N × H)
170. #1778	D₇₃	direct (N × H)
172. C₇₃	C₂×#1778	semidirect
224. #108	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×#108×D₇₃	2
C₄×M₁₆×D₇₃	2
C₂×#97×D₇₃	2
#22609839	2
#22609838	2
#22609837	2
#22609836	2
#341×D₇₃	2
C₇₃×#1778	2
C₂×#1778	73

Subgroups — #1778×D73 #33084083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1778×D₇₃ #33084083