Subgroups — #33073672

Order 18688.

750 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620510	C₂	1	1	yes	maximal, characteristic
2	#22620491	C₂	1	2	yes	maximal, complemented
3	#22620256	C₂	1	2	yes	maximal, complemented
4	#22609544	C₂	1	2	yes	maximal, complemented
5	#22609540	C₂	1	1	yes	maximal, characteristic, complemented
6	#22609538	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609534	C₂	1	2	yes	maximal, complemented
8	#22609525	C₂	1	1	yes	maximal, characteristic, complemented
9	#22609524	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989137	C₂²	1	2	yes	—
13	#17988992	C₂²	1	1	yes	characteristic
14	#17988983	C₂²	1	2	yes	—
15	#17988964	C₂²	1	1	yes	characteristic
16	#17988961	C₂²	1	2	yes	—
17	#17988944	—	2	4	no	—
18	#17988939	C₂²	1	2	yes	—
19	#17988939	C₂²	1	2	yes	—
20	#17988937	C₂²	1	2	yes	—
21	#17988935	C₂²	1	2	yes	—
22	#17988935	C₂²	1	2	yes	—
23	#17988935	C₂²	1	2	yes	—
24	#17988934	C₂²	1	2	yes	—
25	#17988823	—	2	2	no	—
26	#17988818	—	2	4	no	—
27	C₄×#11924015	—	2	4	no	—
28	C₄×D₅₈₄	—	2	2	no	—
29	#17988581	—	2	4	no	—
30	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
31	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
32	C₈×#10596831	C₂²	1	1	yes	characteristic
33	C₈×#10596831	C₂²	1	1	yes	characteristic
34	C₈×#10596831	C₂²	1	1	yes	characteristic
35	C₈×#10596831	C₂²	1	1	yes	characteristic
36	#533×C₇₃	C₂²	1	2	yes	—
37	#500×C₇₃	C₂²	1	1	yes	characteristic
38	#494×C₇₃	C₂²	1	1	yes	characteristic
39	#487×C₇₃	C₂²	1	2	yes	—
40	#467×C₇₃	—	2	2	no	—
41	C₄×Q₁₆×C₇₃	—	2	2	no	—
42	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
43	C₂×#11924028	—	2	2	no	—
44	C₂×#11924015	—	2	4	no	—
45	C₂×D₅₈₄	—	2	2	no	—
46	C₂²×#10596831	D₄	1	2	yes	—
47	C₂²×#10596831	D₄	1	2	yes	—
48	#14556591	C₂³	1	2	yes	—
49	#14556588	C₂³	1	2	yes	—
50	#14556579	—	2	2	no	—
51	#14556565	—	2	2	no	—
52	#14556565	—	2	2	no	—
53	C₄×D₂₉₂	—	2	2	no	—
54	C₄×D₂₉₂	—	2	2	no	—
55	#14556525	—	2	2	no	—
56	#14556522	—	2	4	no	—
57	#14556515	—	2	4	no	—
58	#14556514	—	2	4	no	—
59	#14556514	—	2	4	no	—
60	#14556513	—	2	4	no	—
61	#14556511	—	2	4	no	—
62	#14556511	—	2	4	no	—
63	#14556511	—	2	4	no	—
64	C₄×#10596831	D₄	1	2	yes	—
65	C₄×#10596831	D₄	1	2	yes	—
66	C₄×#10596831	C₂³	1	1	yes	characteristic
67	C₄×#10596831	C₂³	1	1	yes	characteristic
68	C₄×#10596831	C₂³	1	1	yes	characteristic
69	C₄×#10596831	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₇₃	—	2	2	no	—
71	C₂²×C₈×C₇₃	D₄	1	2	yes	—
72	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
73	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
74	#125×C₇₃	—	2	2	no	—
75	C₂²⋊Q₈×C₇₃	—	2	2	no	—
76	C₂²⋊Q₈×C₇₃	—	2	2	no	—
77	C₄×Q₈×C₇₃	—	2	2	no	—
78	C₄×Q₈×C₇₃	—	2	2	no	—
79	#117×C₇₃	—	2	2	no	—
80	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
81	#107×C₇₃	—	2	2	no	—
82	#103×C₇₃	—	2	4	no	—
83	C₄×C₈×C₇₃	D₄	1	2	yes	—
84	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
86	#11924028	—	4	8	no	—
87	C₂×Dic₁₄₆	—	2	2	no	—
88	C₂×D₂₉₂	—	2	2	no	—
89	C₂×D₂₉₂	—	2	2	no	—
90	C₂×D₂₉₂	—	2	2	no	—
91	C₂×C₂²⋊D₇₃	—	4	2	no	—
92	C₂×C₄×D₇₃	—	4	2	no	—
93	#11924015	—	4	8	no	—
94	D₅₈₄	—	4	4	no	—
95	C₂×#10596831	—	2	4	no	—
96	C₂×#10596831	—	2	4	no	—
97	C₂×#10596831	C₂×D₄	1	2	yes	—
98	C₂×#10596831	C₂×D₄	1	2	yes	—
99	C₂×#10596831	C₂×D₄	1	2	yes	—
100	C₂×#10596831	C₂×D₄	1	2	yes	—
101	C₄⋊Dic₇₃	—	2	2	no	—
102	C₄⋊Dic₇₃	—	2	2	no	—
103	#11924005	—	4	2	no	—
104	#11924005	—	4	2	no	—
105	C₂×Q₈×C₇₃	—	2	2	no	—
106	C₂×Q₈×C₇₃	—	2	2	no	—
107	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
110	Q₁₆×C₇₃	—	4	4	no	—
111	C₂×C₈×C₇₃	—	2	4	no	—
112	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
114	C₄⋊C₄×C₇₃	—	2	2	no	—
115	C₄⋊C₄×C₇₃	—	2	2	no	—
116	C₄⋊C₄×C₇₃	—	2	2	no	—
117	C₄⋊C₄×C₇₃	—	2	2	no	—
118	C₄⋊C₄×C₇₃	—	4	2	no	—
119	C₄⋊C₄×C₇₃	—	4	2	no	—
120	C₂²⋊C₄×C₇₃	—	4	2	no	—
121	C₄²×C₇₃	—	4	2	no	—
122	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
126	C₂³×C₇₃	#127	1	1	yes	characteristic
127	C₂²×D₇₃	—	4	2	no	—
128	Q₈×C₇₃	—	4	4	no	—
129	Q₈×C₇₃	—	4	4	no	—
130	C₂×C₄×C₇₃	—	2	2	no	—
131	C₂×C₄×C₇₃	—	2	2	no	—
132	C₂×C₄×C₇₃	—	4	2	no	—
133	C₂×C₄×C₇₃	—	4	2	no	—
134	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
135	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
136	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
137	C₂×C₄×C₇₃	#135	1	4	yes	—
138	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
141	C₂²⋊D₇₃	—	8	4	no	—
142	C₂×Dic₇₃	—	4	2	no	—
143	D₂₉₂	—	4	4	no	—
144	D₂₉₂	—	4	4	no	—
145	D₂₉₂	—	4	4	no	—
146	C₄×D₇₃	—	8	4	no	—
147	Dic₁₄₆	—	4	4	no	—
148	C₈×C₇₃	—	2	2	no	—
149	C₈×C₇₃	#135	1	4	yes	—
150	#10596831	—	2	2	no	—
151	#10596831	—	2	2	no	—
152	#10596831	—	2	2	no	—
153	#10596831	#135	1	4	yes	—
154	C₂²×C₇₃	—	2	1	no	—
155	C₂²×C₇₃	—	2	1	no	—
156	C₂²×C₇₃	#495	1	2	yes	—
157	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
158	C₂×D₇₃	—	8	2	no	—
159	C₂×D₇₃	—	8	2	no	—
160	C₂×D₇₃	—	8	2	no	—
161	C₄×C₇₃	—	2	1	no	—
162	C₄×C₇₃	—	2	1	no	—
163	C₄×C₇₃	—	8	2	no	—
164	C₄×C₇₃	—	8	2	no	—
165	C₄×C₇₃	#495	1	2	yes	—
166	C₄×C₇₃	C₂×#135	1	2	yes	—
167	C₄×C₇₃	C₂×#135	1	2	yes	—
168	C₄×C₇₃	C₂×#135	1	2	yes	—
169	Dic₇₃	—	8	2	no	—
170	#19600	—	73	1	no	maximal, Sylow
171	C₂×C₇₃	—	2	2	no	—
172	C₂×C₇₃	C₂×#495	1	1	yes	characteristic
173	C₂×C₇₃	#3147	1	1	yes	characteristic
174	C₂×C₇₃	#3147	1	1	yes	characteristic
175	D₇₃	—	16	2	no	—
176	#3170	D₇₃	1	1	yes	characteristic, complemented
177	#3169	—	73	1	no	—
178	#3167	—	73	2	no	—
179	#3131	—	73	2	no	—
180	#1711	—	73	2	no	—
181	#1710	—	73	1	no	—
182	#1709	—	73	1	no	—
183	#1708	—	73	2	no	—
184	#1707	—	73	1	no	—
185	#1706	—	73	1	no	—
186	C₂×C₈²	—	73	1	no	—
187	C₇₃	#19600	1	1	yes	characteristic, complemented, Sylow
188	#533	C₂×D₇₃	1	2	yes	—
189	#532	—	73	2	no	—
190	#500	—	73	1	no	—
191	#500	C₂×D₇₃	1	1	yes	characteristic
192	#498	—	73	2	no	—
193	#494	C₂×D₇₃	1	1	yes	characteristic
194	#493	—	73	1	no	—
195	#492	—	73	2	no	—
196	#488	—	146	4	no	—
197	#487	—	73	2	no	—
198	#487	—	73	2	no	—
199	#487	—	73	2	no	—
200	#487	C₂×D₇₃	1	2	yes	—
201	#486	—	73	2	no	—
202	#486	—	73	2	no	—
203	#486	—	73	2	no	—
204	#486	—	73	2	no	—
205	#467	—	2	2	no	—
206	#466	—	146	2	no	—
207	#465	—	146	4	no	—
208	C₄×Q₁₆	—	2	2	no	—
209	C₄×SD₁₆	—	146	4	no	—
210	C₄×D₈	—	146	2	no	—
211	#416	—	146	4	no	—
212	C₂×C₄×C₈	—	73	1	no	—
213	C₂×C₄×C₈	—	73	1	no	—
214	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
215	C₈²	—	73	1	no	—
216	C₈²	—	73	1	no	—
217	C₈²	—	73	1	no	—
218	C₈²	—	73	1	no	—
219	C₂×#41	—	146	2	no	—
220	C₂×Q₁₆	—	2	2	no	—
221	C₂×SD₁₆	—	146	4	no	—
222	C₂×D₈	—	146	2	no	—
223	C₂²×C₈	—	73	2	no	—
224	C₂²×C₈	—	73	2	no	—
225	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
226	C₄⋊Q₈	—	73	2	no	—
227	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
228	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
229	#127	—	73	2	no	—
230	#125	—	2	2	no	—
231	#124	—	146	2	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	#121	—	146	2	no	—
235	#121	—	146	2	no	—
236	C₄×Q₈	—	2	2	no	—
237	C₄×Q₈	—	2	2	no	—
238	C₄×D₄	—	146	2	no	—
239	C₄×D₄	—	146	2	no	—
240	#117	—	2	2	no	—
241	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
242	#107	—	2	2	no	—
243	#107	—	146	2	no	—
244	#106	—	146	4	no	—
245	#103	—	2	4	no	—
246	#103	—	146	4	no	—
247	#103	—	146	4	no	—
248	#103	—	146	4	no	—
249	#102	—	146	4	no	—
250	#102	—	146	4	no	—
251	#102	—	146	4	no	—
252	#102	—	146	4	no	—
253	C₄×C₈	—	73	2	no	—
254	C₄×C₈	—	73	2	no	—
255	C₄×C₈	—	73	1	no	—
256	C₄×C₈	—	73	1	no	—
257	C₄×C₈	—	73	1	no	—
258	C₄×C₈	—	73	1	no	—
259	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
260	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
262	#41	—	292	8	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	146	2	no	—
266	C₂×D₄	—	292	2	no	—
267	C₂×D₄	—	146	2	no	—
268	C₂×D₄	—	146	2	no	—
269	C₂×D₄	—	146	2	no	—
270	C₂²×C₄	—	292	2	no	—
271	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
272	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
274	Q₁₆	—	4	4	no	—
275	SD₁₆	—	292	8	no	—
276	D₈	—	292	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	73	2	no	—
279	C₂×C₈	—	73	2	no	—
280	C₂×C₈	—	73	2	no	—
281	C₂×C₈	—	73	2	no	—
282	C₂×C₈	—	146	4	no	—
283	C₂×C₈	—	146	4	no	—
284	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
285	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	—	146	2	no	—
293	C₄⋊C₄	—	146	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	292	2	no	—
296	C₂²⋊C₄	—	292	2	no	—
297	C₄²	—	4	2	no	—
298	C₄²	D₄×D₇₃	1	1	yes	characteristic
299	C₄²	D₄×D₇₃	1	1	yes	characteristic
300	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
301	C₄²	C₂³×D₇₃	1	1	yes	characteristic
302	C₂³	—	292	2	no	—
303	C₂³	#14556587	1	1	yes	characteristic
304	Q₈	—	4	4	no	—
305	Q₈	—	4	4	no	—
306	Q₈	—	292	4	no	—
307	D₄	—	292	4	no	—
308	D₄	—	292	4	no	—
309	D₄	—	292	4	no	—
310	D₄	—	584	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	292	2	no	—
316	C₂×C₄	—	584	4	no	—
317	C₂×C₄	#14556587	1	1	yes	characteristic
318	C₂×C₄	#14556587	1	1	yes	characteristic
319	C₂×C₄	#14556587	1	1	yes	characteristic
320	C₂×C₄	#14556614	1	4	yes	—
321	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
322	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	73	4	no	—
326	C₈	—	146	2	no	—
327	C₈	—	146	2	no	—
328	C₈	—	146	2	no	—
329	C₈	#14556611	1	4	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	584	2	no	—
333	C₂²	—	584	2	no	—
334	C₂²	—	584	2	no	—
335	C₂²	#17988972	1	2	yes	—
336	C₂²	C₂×#14556587	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	8	2	no	—
340	C₄	—	8	2	no	—
341	C₄	—	584	2	no	—
342	C₄	#17988972	1	2	yes	—
343	C₄	#135×D₇₃	1	2	yes	—
344	C₄	C₂×#14556611	1	2	yes	—
345	C₄	C₂×#14556614	1	2	yes	—
346	C₂	—	2	2	no	—
347	C₂	—	1168	2	no	—
348	C₂	C₂×#17988972	1	1	yes	characteristic
349	C₂	#22620355	1	1	yes	characteristic
350	C₂	#22620357	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556587
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#3170	C₂
socle	C₂²×C₇₃	C₂×#127
prime radical	#22620510	C₂
hypercenter	#3170	D₇₃
hypocenter	C₇₃	#19600
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #22620491	C₂	semidirect
3. #22620256	C₂	semidirect
4. #22609544	C₂	semidirect
5. #22609540	C₂	semidirect
6. #22609538	C₂	semidirect
7. #22609534	C₂	semidirect
8. #22609525	C₂	semidirect
9. #22609524	C₂	semidirect
10. C₂×C₈×#10596831	C₂	semidirect
11. C₇₃×#3170	C₂	semidirect
176. #3170	D₇₃	semidirect
187. C₇₃	#19600	semidirect

Maximal subgroups

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

subgroup	index
#22620510	2
#22620491	2
#22620256	2
#22609544	2
#22609540	2
#22609538	2
#22609534	2
#22609525	2
#22609524	2
C₂×C₈×#10596831	2
C₇₃×#3170	2
#19600	73