Subgroups — #33073648

Order 18688.

814 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620493	C₂	1	2	yes	maximal, complemented
2	#22620492	C₂	1	2	yes	maximal, complemented
3	#22620442	C₂	1	1	yes	maximal, characteristic, complemented
4	#22620360	C₂	1	1	yes	maximal, characteristic
5	#22609534	C₂	1	2	yes	maximal, complemented
6	#22609525	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609524	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609523	C₂	1	1	yes	maximal, characteristic, complemented
9	#22609519	C₂	1	2	yes	maximal, complemented
10	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556614	—	2	2	no	—
13	#17989142	C₂²	1	2	yes	—
14	#17989138	C₂²	1	2	yes	—
15	#17988992	C₂²	1	1	yes	characteristic
16	#17988988	—	2	2	no	—
17	#17988983	C₂²	1	2	yes	—
18	#17988983	C₂²	1	2	yes	—
19	#17988972	—	2	2	no	—
20	#17988964	C₂²	1	1	yes	characteristic
21	#17988961	C₂²	1	2	yes	—
22	#17988961	C₂²	1	2	yes	—
23	#17988960	C₂²	1	2	yes	—
24	#17988939	C₂²	1	2	yes	—
25	#17988935	C₂²	1	2	yes	—
26	#17988934	C₂²	1	2	yes	—
27	#17988819	—	2	4	no	—
28	#17988817	—	2	4	no	—
29	#17988674	—	2	4	no	—
30	#17988672	—	2	4	no	—
31	#17988624	—	2	2	no	—
32	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
35	C₈×#10596831	C₂²	1	1	yes	characteristic
36	C₈×#10596831	C₂²	1	1	yes	characteristic
37	C₈×#10596831	C₂²	1	1	yes	characteristic
38	#500×C₇₃	C₂²	1	1	yes	characteristic
39	#498×C₇₃	C₂²	1	2	yes	—
40	#494×C₇₃	C₂²	1	1	yes	characteristic
41	#487×C₇₃	C₂²	1	2	yes	—
42	C₈²×C₇₃	C₂²	1	1	yes	characteristic
43	C₂×#11924029	—	2	2	no	—
44	#14556614	—	4	8	no	—
45	C₂×#11924017	—	2	4	no	—
46	C₂×#11924015	—	2	4	no	—
47	C₂×#11924015	—	2	4	no	—
48	C₂×D₅₈₄	—	2	2	no	—
49	C₂×C₈×D₇₃	D₄	1	2	yes	—
50	C₂×C₈×D₇₃	D₄	1	2	yes	—
51	C₂×C₈×D₇₃	D₄	1	2	yes	—
52	#14556591	C₂³	1	2	yes	—
53	#14556588	C₂³	1	2	yes	—
54	#14556582	—	2	2	no	—
55	#14556580	—	2	2	no	—
56	#14556571	—	2	2	no	—
57	#14556567	—	2	2	no	—
58	#14556566	—	2	2	no	—
59	#14556566	—	2	2	no	—
60	Q₈×Dic₇₃	—	2	2	no	—
61	#14556556	—	2	2	no	—
62	#14556553	—	2	2	no	—
63	#14556553	—	2	2	no	—
64	#14556549	—	2	2	no	—
65	C₄²×D₇₃	C₂³	1	1	yes	characteristic
66	#14556525	—	2	2	no	—
67	#14556522	—	2	4	no	—
68	#14556522	—	2	4	no	—
69	#14556514	—	2	4	no	—
70	#14556513	—	2	4	no	—
71	#14556511	—	2	4	no	—
72	C₄×#10596831	C₂³	1	1	yes	characteristic
73	C₄×#10596831	C₂³	1	1	yes	characteristic
74	C₄×#10596831	C₂³	1	1	yes	characteristic
75	C₈×Dic₇₃	D₄	1	2	yes	—
76	C₈×Dic₇₃	D₄	1	2	yes	—
77	C₈×Dic₇₃	D₄	1	2	yes	—
78	C₂×Q₁₆×C₇₃	—	2	2	no	—
79	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
80	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
81	#107×C₇₃	—	2	2	no	—
82	#106×C₇₃	—	2	4	no	—
83	#103×C₇₃	—	2	4	no	—
84	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
87	#11924029	—	4	8	no	—
88	C₂×Dic₁₄₆	—	2	2	no	—
89	C₂×D₂₉₂	—	2	2	no	—
90	C₂×D₂₉₂	—	2	2	no	—
91	C₂×D₂₉₂	—	2	2	no	—
92	C₂×D₂₉₂	—	4	2	no	—
93	C₂×C₄×D₇₃	—	4	2	no	—
94	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
97	#11924017	—	4	8	no	—
98	#11924015	—	4	8	no	—
99	#11924015	—	4	8	no	—
100	D₅₈₄	—	4	4	no	—
101	C₂×#10596831	C₂×D₄	1	2	yes	—
102	C₂×#10596831	C₂×D₄	1	2	yes	—
103	C₂×#10596831	C₂×D₄	1	2	yes	—
104	C₈×D₇₃	—	2	4	no	—
105	C₈×D₇₃	—	2	4	no	—
106	C₈×D₇₃	SD₁₆	1	8	yes	complemented
107	C₄⋊Dic₇₃	—	2	2	no	—
108	C₄⋊Dic₇₃	—	2	2	no	—
109	C₄⋊Dic₇₃	—	4	2	no	—
110	#11924007	—	4	2	no	—
111	#11924005	—	4	2	no	—
112	#11924005	—	4	2	no	—
113	#11924005	—	4	2	no	—
114	C₄×Dic₇₃	—	4	2	no	—
115	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
116	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
117	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
118	C₂×Q₈×C₇₃	—	2	2	no	—
119	C₂×Q₈×C₇₃	—	2	2	no	—
120	Q₁₆×C₇₃	—	4	4	no	—
121	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
123	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
124	C₄⋊C₄×C₇₃	—	2	2	no	—
125	C₄⋊C₄×C₇₃	—	2	2	no	—
126	C₄⋊C₄×C₇₃	—	2	2	no	—
127	C₄⋊C₄×C₇₃	—	2	2	no	—
128	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
129	C₂²×D₇₃	—	4	2	no	—
130	C₂²×D₇₃	#127	1	1	yes	characteristic
131	Q₈×C₇₃	—	4	4	no	—
132	Q₈×C₇₃	—	4	4	no	—
133	C₂×C₄×C₇₃	—	4	2	no	—
134	C₂×C₄×C₇₃	—	4	2	no	—
135	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₇₃	—	4	2	no	—
139	C₂×Dic₇₃	#127	1	1	yes	characteristic
140	C₂×Dic₇₃	#127	1	1	yes	characteristic
141	C₂×Dic₇₃	#127	1	1	yes	characteristic
142	D₂₉₂	—	4	4	no	—
143	D₂₉₂	—	4	4	no	—
144	D₂₉₂	—	4	4	no	—
145	D₂₉₂	—	8	4	no	—
146	C₄×D₇₃	—	2	2	no	—
147	C₄×D₇₃	—	2	2	no	—
148	C₄×D₇₃	—	8	4	no	—
149	C₄×D₇₃	C₂×SD₁₆	1	4	yes	—
150	Dic₁₄₆	—	4	4	no	—
151	C₈×C₇₃	—	2	2	no	—
152	C₈×C₇₃	C₂×SD₁₆	1	4	yes	—
153	C₈×C₇₃	#135	1	4	yes	complemented
154	#10596831	—	2	2	no	—
155	#10596831	C₂×SD₁₆	1	4	yes	—
156	#10596831	#135	1	4	yes	complemented
157	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
158	C₂×D₇₃	—	2	1	no	—
159	C₂×D₇₃	—	2	1	no	—
160	C₂×D₇₃	—	8	2	no	—
161	C₂×D₇₃	—	8	2	no	—
162	C₂×D₇₃	—	8	2	no	—
163	C₂×D₇₃	#492	1	2	yes	—
164	C₄×C₇₃	—	8	2	no	—
165	C₄×C₇₃	—	8	2	no	—
166	C₄×C₇₃	C₂²×SD₁₆	1	2	yes	—
167	C₄×C₇₃	C₂×#135	1	2	yes	—
168	C₄×C₇₃	C₂×#135	1	2	yes	—
169	Dic₇₃	—	2	1	no	—
170	Dic₇₃	—	2	1	no	—
171	Dic₇₃	—	8	2	no	—
172	Dic₇₃	#492	1	2	yes	—
173	#19596	—	73	1	no	maximal, Sylow
174	C₂×C₇₃	C₂×#492	1	1	yes	characteristic
175	C₂×C₇₃	#3147	1	1	yes	characteristic
176	C₂×C₇₃	#3147	1	1	yes	characteristic
177	D₇₃	—	2	2	no	—
178	D₇₃	—	16	2	no	—
179	#3167	—	73	2	no	—
180	#3167	—	73	2	no	—
181	#3160	—	73	1	no	—
182	#3147	—	73	1	no	—
183	#1708	—	73	2	no	—
184	#1707	—	73	1	no	—
185	#1707	D₇₃	1	1	yes	characteristic, complemented
186	#1706	—	73	1	no	—
187	#1706	—	73	1	no	—
188	#1705	—	73	2	no	—
189	C₂×C₈²	—	73	1	no	—
190	C₇₃	#19596	1	1	yes	characteristic, complemented, Sylow
191	C₂×#135	—	146	2	no	—
192	#533	—	73	2	no	—
193	#532	—	73	2	no	—
194	#500	—	73	1	no	—
195	#500	C₂×D₇₃	1	1	yes	characteristic
196	#499	—	146	2	no	—
197	#498	—	73	2	no	—
198	#498	—	73	2	no	—
199	#498	C₂×D₇₃	1	2	yes	—
200	#495	—	146	2	no	—
201	#494	C₂×D₇₃	1	1	yes	characteristic
202	#493	—	73	1	no	—
203	#492	—	73	2	no	—
204	#492	—	73	2	no	—
205	#492	—	73	2	no	—
206	#487	—	73	2	no	—
207	#487	C₂×D₇₃	1	2	yes	—
208	#486	—	73	2	no	—
209	#486	—	73	2	no	—
210	#465	—	146	4	no	—
211	#465	—	146	4	no	—
212	C₄×SD₁₆	—	146	4	no	—
213	C₄×SD₁₆	—	146	4	no	—
214	#427	—	146	2	no	—
215	C₂×C₄×C₈	—	73	1	no	—
216	C₂×C₄×C₈	—	73	1	no	—
217	C₂×C₄×C₈	—	73	1	no	—
218	C₈²	—	73	1	no	—
219	C₈²	—	73	1	no	—
220	C₈²	—	73	1	no	—
221	C₈²	C₂×D₇₃	1	1	yes	characteristic
222	C₂×#41	—	146	2	no	—
223	#135	—	292	8	no	—
224	C₂×Q₁₆	—	2	2	no	—
225	C₂×SD₁₆	—	146	4	no	—
226	C₂×SD₁₆	—	146	4	no	—
227	C₂×SD₁₆	—	146	4	no	—
228	C₂×D₈	—	146	2	no	—
229	C₂²×C₈	—	73	2	no	—
230	C₂²×C₈	—	73	2	no	—
231	C₂²×C₈	—	73	2	no	—
232	C₄⋊Q₈	—	73	2	no	—
233	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
234	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
235	#127	—	73	2	no	—
236	#125	—	146	2	no	—
237	#124	—	146	2	no	—
238	C₂²⋊Q₈	—	146	2	no	—
239	C₂²⋊Q₈	—	146	2	no	—
240	#121	—	146	2	no	—
241	#121	—	146	2	no	—
242	C₄×Q₈	—	146	2	no	—
243	C₄×Q₈	—	146	2	no	—
244	C₄×D₄	—	146	2	no	—
245	C₄×D₄	—	146	2	no	—
246	#117	—	146	2	no	—
247	C₂×C₄²	—	73	1	no	—
248	#107	—	2	2	no	—
249	#107	—	146	2	no	—
250	#106	—	2	4	no	—
251	#106	—	146	4	no	—
252	#106	—	146	4	no	—
253	#103	—	2	4	no	—
254	#103	—	146	4	no	—
255	#102	—	146	4	no	—
256	#102	—	146	4	no	—
257	C₄×C₈	—	73	2	no	—
258	C₄×C₈	—	73	2	no	—
259	C₄×C₈	—	73	2	no	—
260	C₄×C₈	—	73	1	no	—
261	C₄×C₈	—	73	1	no	—
262	C₄×C₈	—	73	1	no	—
263	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
266	#41	—	292	8	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	146	2	no	—
270	C₂×D₄	—	292	2	no	—
271	C₂×D₄	—	146	2	no	—
272	C₂×D₄	—	146	2	no	—
273	C₂×D₄	—	146	2	no	—
274	C₂²×C₄	—	292	2	no	—
275	C₂²×C₄	—	73	1	no	—
276	C₂²×C₄	—	73	1	no	—
277	C₂²×C₄	—	73	1	no	—
278	Q₁₆	—	4	4	no	—
279	SD₁₆	—	292	8	no	—
280	SD₁₆	—	292	8	no	—
281	SD₁₆	—	292	8	no	—
282	D₈	—	292	4	no	—
283	C₂×C₈	—	73	2	no	—
284	C₂×C₈	—	73	2	no	—
285	C₂×C₈	—	73	8	no	—
286	C₂×C₈	—	73	2	no	—
287	C₂×C₈	—	146	4	no	—
288	C₂×C₈	—	146	4	no	—
289	C₂×C₈	D₄×D₇₃	1	2	yes	—
290	C₂×C₈	D₄×D₇₃	1	2	yes	—
291	C₂×C₈	D₄×D₇₃	1	2	yes	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	292	2	no	—
297	C₄⋊C₄	—	292	2	no	—
298	C₄⋊C₄	—	146	2	no	—
299	C₄⋊C₄	—	146	2	no	—
300	C₂²⋊C₄	—	292	2	no	—
301	C₂²⋊C₄	—	292	2	no	—
302	C₂²⋊C₄	—	292	2	no	—
303	C₄²	—	292	2	no	—
304	C₄²	—	73	1	no	—
305	C₄²	—	73	1	no	—
306	C₄²	—	73	1	no	—
307	C₄²	C₂³×D₇₃	1	1	yes	characteristic
308	C₂³	—	292	2	no	—
309	C₂³	—	73	1	no	—
310	Q₈	—	4	4	no	—
311	Q₈	—	4	4	no	—
312	Q₈	—	292	4	no	—
313	D₄	—	292	4	no	—
314	D₄	—	292	4	no	—
315	D₄	—	292	4	no	—
316	D₄	—	584	4	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	292	2	no	—
320	C₂×C₄	—	584	4	no	—
321	C₂×C₄	—	73	1	no	—
322	C₂×C₄	—	73	4	no	—
323	C₂×C₄	—	73	1	no	—
324	C₂×C₄	—	73	1	no	—
325	C₂×C₄	—	146	2	no	—
326	C₂×C₄	—	146	2	no	—
327	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	73	4	no	—
332	C₈	—	73	4	no	—
333	C₈	—	146	2	no	—
334	C₈	SD₁₆×D₇₃	1	4	yes	—
335	C₈	#14556614	1	4	yes	complemented
336	C₂²	—	584	2	no	—
337	C₂²	—	584	2	no	—
338	C₂²	—	584	2	no	—
339	C₂²	—	73	2	no	—
340	C₂²	—	146	1	no	—
341	C₂²	—	146	1	no	—
342	C₂²	#127×D₇₃	1	1	yes	characteristic
343	C₄	—	8	2	no	—
344	C₄	—	8	2	no	—
345	C₄	—	584	2	no	—
346	C₄	—	73	2	no	—
347	C₄	—	146	1	no	—
348	C₄	—	146	1	no	—
349	C₄	C₂×SD₁₆×D₇₃	1	2	yes	—
350	C₄	#135×D₇₃	1	2	yes	—
351	C₄	C₂×#14556614	1	2	yes	—
352	C₂	—	1168	2	no	—
353	C₂	—	146	2	no	—
354	C₂	#492×D₇₃	1	1	yes	characteristic
355	C₂	#22620356	1	1	yes	characteristic
356	C₂	#22620360	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₇₃
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1707	C₂
socle	C₂²×C₇₃	C₂×#127
prime radical	#22620360	C₂
hypercenter	#1707	D₇₃
hypocenter	C₇₃	#19596
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
1. #22620493	C₂	semidirect
2. #22620492	C₂	semidirect
3. #22620442	C₂	semidirect
5. #22609534	C₂	semidirect
6. #22609525	C₂	semidirect
7. #22609524	C₂	semidirect
8. #22609523	C₂	semidirect
9. #22609519	C₂	semidirect
10. C₈²×D₇₃	C₂	semidirect
11. C₇₃×#1707	C₂	semidirect
106. C₈×D₇₃	SD₁₆	semidirect
153. C₈×C₇₃	#135	semidirect
156. #10596831	#135	semidirect
185. #1707	D₇₃	semidirect
190. C₇₃	#19596	semidirect
335. C₈	#14556614	semidirect

Maximal subgroups

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

subgroup	index
#22620493	2
#22620492	2
#22620442	2
#22620360	2
#22609534	2
#22609525	2
#22609524	2
#22609523	2
#22609519	2
C₈²×D₇₃	2
C₇₃×#1707	2
#19596	73