Subgroups — #33056686

Order 18688.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620501	C₂	1	1	yes	maximal, characteristic
2	#22619082	C₂	1	2	yes	maximal, complemented
3	#22608846	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608841	C₂	1	1	yes	maximal, characteristic, complemented
5	#22608823	C₂	1	2	yes	maximal, complemented
6	#22608756	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608751	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#11924017	C₂	1	2	yes	maximal, complemented
9	#22608687	C₂	1	1	yes	maximal, characteristic, complemented
10	#22608397	C₂	1	2	yes	maximal, complemented
11	C₇₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989121	C₂²	1	2	yes	—
13	#17988989	C₄	1	1	yes	characteristic
14	#17988971	C₄	1	1	yes	characteristic
15	#17988942	C₄	1	2	yes	—
16	#17988828	C₄	1	1	yes	characteristic
17	#17988823	C₄	1	1	yes	characteristic
18	#17988816	C₄	1	2	yes	—
19	#17988717	C₂²	1	2	yes	—
20	C₈×Dic₁₄₆	C₂²	1	2	yes	—
21	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
22	C₄×#11924017	C₂²	1	2	yes	—
23	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
24	#17988666	C₂²	1	2	yes	—
25	#17988662	—	2	2	no	—
26	#17988661	—	2	2	no	—
27	C₈×D₂₉₂	C₂²	1	2	yes	—
28	#17988647	—	2	2	no	—
29	#17988612	—	2	2	no	—
30	#17988611	C₂²	1	2	yes	—
31	#17988338	C₂²	1	2	yes	—
32	#17988338	C₂²	1	1	yes	characteristic
33	#17988338	C₂²	1	1	yes	characteristic
34	#17988319	C₂²	1	2	yes	—
35	#17988319	C₂²	1	2	yes	—
36	#17988316	C₂²	1	2	yes	—
37	#17988316	C₂²	1	2	yes	—
38	#431×C₇₃	C₂²	1	1	yes	characteristic
39	#431×C₇₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
41	#322×C₇₃	C₂²	1	1	yes	characteristic
42	#322×C₇₃	C₂²	1	1	yes	characteristic
43	C₈²×C₇₃	C₂²	1	1	yes	characteristic
44	C₈²×C₇₃	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#11924017	C₂×C₄	1	2	yes	—
47	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
48	C₂×#11924012	—	2	2	no	—
49	C₂×#11924011	—	2	2	no	—
50	C₂×C₈×D₇₃	—	2	2	no	—
51	#14556585	—	2	2	no	—
52	#14556583	C₂×C₄	1	2	yes	—
53	#14556579	C₂×C₄	1	2	yes	—
54	#14556574	—	2	2	no	—
55	#14556570	C₂×C₄	1	2	yes	—
56	#14556565	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₄₆	C₂³	1	2	yes	—
58	C₄×D₂₉₂	C₂³	1	2	yes	—
59	C₄×C₂²⋊D₇₃	—	2	2	no	—
60	#14556548	—	2	2	no	—
61	#14556525	C₂×C₄	1	1	yes	characteristic
62	#14556525	C₂×C₄	1	1	yes	characteristic
63	#14556523	C₂×C₄	1	2	yes	—
64	C₄⋊#10596831	—	2	2	no	—
65	C₄⋊#10596831	C₂³	1	2	yes	—
66	C₄⋊#10596831	C₂³	1	2	yes	—
67	#14556520	—	2	2	no	—
68	#14556520	—	2	2	no	—
69	#14556516	C₂×C₄	1	2	yes	—
70	#14556516	C₂×C₄	1	2	yes	—
71	#14556513	C₂×C₄	1	2	yes	—
72	#14556513	C₂×C₄	1	2	yes	—
73	#14556503	—	2	2	no	—
74	#14556503	—	2	2	no	—
75	#14556501	—	2	2	no	—
76	C₈×Dic₇₃	—	2	2	no	—
77	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₇₃	—	2	1	no	—
81	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
82	C₂²⋊C₈×C₇₃	—	2	1	no	—
83	C₂²⋊C₈×C₇₃	D₄	1	2	yes	—
84	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
86	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
92	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
93	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
94	C₂×C₂²⋊D₇₃	—	2	2	no	—
95	C₂×C₄×D₇₃	—	2	2	no	—
96	Dic₂₉₂	—	2	2	no	—
97	#11924017	—	2	4	no	—
98	D₅₈₄	—	2	2	no	—
99	#11924012	—	8	2	no	—
100	#11924011	—	8	2	no	—
101	C₂×#10596831	—	2	2	no	—
102	C₂×#10596831	—	2	2	no	—
103	C₈×D₇₃	—	8	2	no	—
104	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
105	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
106	#11924007	—	2	2	no	—
107	#11924007	—	2	2	no	—
108	#11924007	—	2	2	no	—
109	#11924007	—	2	2	no	—
110	C₂²⋊Dic₇₃	—	2	2	no	—
111	#11924005	—	2	2	no	—
112	#11924005	—	2	2	no	—
113	#11924005	—	2	2	no	—
114	#11924005	—	2	2	no	—
115	C₄×Dic₇₃	—	2	2	no	—
116	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
119	C₂×C₈×C₇₃	—	2	2	no	—
120	C₂×C₈×C₇₃	—	2	2	no	—
121	C₂×C₈×C₇₃	—	2	1	no	—
122	C₂×C₈×C₇₃	—	2	1	no	—
123	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
129	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₇₃	#41	1	1	yes	characteristic
132	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
133	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
134	C₂²×D₇₃	—	2	2	no	—
135	C₂×C₄×C₇₃	—	2	1	no	—
136	C₂×C₄×C₇₃	—	2	1	no	—
137	C₂×C₄×C₇₃	—	2	2	no	—
138	C₂×C₄×C₇₃	—	4	1	no	—
139	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
145	C₂²⋊D₇₃	—	8	2	no	—
146	C₂×Dic₇₃	—	2	2	no	—
147	C₂×Dic₇₃	—	2	2	no	—
148	C₂×Dic₇₃	—	2	2	no	—
149	D₂₉₂	—	2	2	no	—
150	D₂₉₂	—	2	2	no	—
151	C₄×D₇₃	—	4	2	no	—
152	C₄×D₇₃	—	4	2	no	—
153	Dic₁₄₆	—	2	2	no	—
154	Dic₁₄₆	—	2	2	no	—
155	C₈×C₇₃	—	2	1	no	—
156	C₈×C₇₃	—	2	1	no	—
157	C₈×C₇₃	—	2	2	no	—
158	C₈×C₇₃	—	2	2	no	—
159	C₈×C₇₃	—	4	1	no	—
160	C₈×C₇₃	—	4	1	no	—
161	C₈×C₇₃	#131	1	2	yes	—
162	C₈×C₇₃	#131	1	2	yes	—
163	#10596831	—	4	2	no	—
164	#10596831	—	4	2	no	—
165	C₂²×C₇₃	—	2	1	no	—
166	C₂²×C₇₃	—	2	1	no	—
167	C₂²×C₇₃	—	2	1	no	—
168	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂×D₇₃	—	4	2	no	—
170	C₂×D₇₃	—	4	2	no	—
171	C₂×D₇₃	—	4	2	no	—
172	C₄×C₇₃	—	2	1	no	—
173	C₄×C₇₃	—	2	1	no	—
174	C₄×C₇₃	—	2	1	no	—
175	C₄×C₇₃	—	2	1	no	—
176	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
177	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
178	C₄×C₇₃	#577	1	2	yes	—
179	Dic₇₃	—	4	2	no	—
180	Dic₇₃	—	4	2	no	—
181	Dic₇₃	—	4	2	no	—
182	#17974	—	73	1	no	maximal, Sylow
183	C₂×C₇₃	—	4	1	no	—
184	C₂×C₇₃	#2931	1	1	yes	characteristic
185	C₂×C₇₃	#2952	1	1	yes	characteristic
186	C₂×C₇₃	#2957	1	1	yes	characteristic
187	D₇₃	—	8	2	no	—
188	#3168	—	73	1	no	—
189	#2993	—	73	2	no	—
190	#1594	—	73	1	no	—
191	#1593	—	73	1	no	—
192	#1590	—	73	2	no	—
193	#1580	—	73	1	no	—
194	#1579	—	73	1	no	—
195	C₈×SD₁₆	—	73	2	no	—
196	#1566	—	73	1	no	—
197	#1495	—	73	2	no	—
198	#1450	D₇₃	1	1	yes	characteristic, complemented
199	C₇₃	#17974	1	1	yes	characteristic, complemented, Sylow
200	#529	—	73	2	no	—
201	#499	—	73	1	no	—
202	#495	—	73	1	no	—
203	#488	—	73	2	no	—
204	#467	—	73	1	no	—
205	#466	—	73	1	no	—
206	#465	—	73	2	no	—
207	#446	—	73	2	no	—
208	C₈×Q₈	—	73	2	no	—
209	C₄×Q₁₆	—	73	1	no	—
210	C₄×SD₁₆	—	73	2	no	—
211	C₄×D₈	—	73	1	no	—
212	#436	—	73	2	no	—
213	#436	—	146	2	no	—
214	#435	—	146	2	no	—
215	C₈×D₄	—	73	2	no	—
216	#433	—	146	2	no	—
217	#431	C₂×D₇₃	1	1	yes	characteristic
218	#431	C₂×D₇₃	1	1	yes	characteristic
219	#424	—	146	2	no	—
220	#423	—	73	2	no	—
221	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
222	#334	—	73	2	no	—
223	#334	—	73	1	no	—
224	#334	—	73	1	no	—
225	Q₈⋊C₈	—	73	2	no	—
226	Q₈⋊C₈	—	73	2	no	—
227	D₄⋊C₈	—	73	2	no	—
228	D₄⋊C₈	—	73	2	no	—
229	#322	C₂×D₇₃	1	1	yes	characteristic
230	#322	C₂×D₇₃	1	1	yes	characteristic
231	C₈²	C₂×D₇₃	1	1	yes	characteristic
232	C₈²	C₂×D₇₃	1	1	yes	characteristic
233	C₂×Q₁₆	—	73	1	no	—
234	C₂×SD₁₆	—	73	2	no	—
235	C₂×D₈	—	73	1	no	—
236	C₂×M₁₆	—	146	2	no	—
237	C₂×M₁₆	—	146	2	no	—
238	C₂²×C₈	—	146	2	no	—
239	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
240	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
241	#126	—	146	2	no	—
242	#125	—	73	2	no	—
243	#124	—	73	2	no	—
244	#123	—	146	2	no	—
245	C₂²⋊Q₈	—	73	2	no	—
246	#121	—	73	2	no	—
247	C₄×Q₈	—	73	2	no	—
248	C₄×D₄	—	73	2	no	—
249	C₄×D₄	—	146	2	no	—
250	#117	—	146	2	no	—
251	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
252	#107	—	73	1	no	—
253	#107	—	73	1	no	—
254	#106	—	73	2	no	—
255	C₄⋊C₈	—	2	1	no	—
256	C₄⋊C₈	—	73	2	no	—
257	C₄⋊C₈	—	73	2	no	—
258	C₄⋊C₈	—	146	2	no	—
259	C₄⋊C₈	—	146	2	no	—
260	C₄⋊C₈	—	146	2	no	—
261	C₄⋊C₈	D₂₉₂	1	2	yes	—
262	#103	—	73	2	no	—
263	#103	—	73	2	no	—
264	#102	—	73	2	no	—
265	#102	—	73	2	no	—
266	C₂²⋊C₈	—	2	1	no	—
267	C₂²⋊C₈	—	146	2	no	—
268	C₂²⋊C₈	—	146	2	no	—
269	C₂²⋊C₈	D₂₉₂	1	2	yes	—
270	#97	—	146	2	no	—
271	C₄×C₈	—	146	2	no	—
272	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
273	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
280	C₂×Q₈	—	73	2	no	—
281	C₂×D₄	—	73	2	no	—
282	C₂×D₄	—	146	2	no	—
283	C₂²×C₄	—	146	2	no	—
284	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
286	C₂²×C₄	#11924028	1	1	yes	characteristic
287	Q₁₆	—	146	2	no	—
288	SD₁₆	—	146	4	no	—
289	D₈	—	146	2	no	—
290	M₁₆	—	584	2	no	—
291	M₁₆	—	584	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	584	2	no	—
297	C₂×C₈	—	146	2	no	—
298	C₂×C₈	—	146	2	no	—
299	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
300	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
301	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
303	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
304	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
305	C₄⋊C₄	—	73	2	no	—
306	C₄⋊C₄	—	73	2	no	—
307	C₄⋊C₄	—	146	2	no	—
308	C₄⋊C₄	—	146	2	no	—
309	C₄⋊C₄	—	146	2	no	—
310	C₄⋊C₄	—	146	2	no	—
311	C₂²⋊C₄	—	146	2	no	—
312	C₂²⋊C₄	—	146	2	no	—
313	C₂²⋊C₄	—	146	2	no	—
314	C₂²⋊C₄	—	146	2	no	—
315	C₂²⋊C₄	—	146	2	no	—
316	C₄²	—	146	2	no	—
317	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
318	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
319	C₄²	#11924028	1	1	yes	characteristic
320	C₄²	C₂³×D₇₃	1	1	yes	characteristic
321	C₂³	—	146	2	no	—
322	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
323	Q₈	—	146	2	no	—
324	Q₈	—	146	2	no	—
325	D₄	—	584	2	no	—
326	D₄	—	146	2	no	—
327	D₄	—	146	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	292	2	no	—
333	C₂×C₄	—	292	2	no	—
334	C₂×C₄	—	146	2	no	—
335	C₂×C₄	—	146	2	no	—
336	C₂×C₄	—	146	2	no	—
337	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
338	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
339	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
340	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
341	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
342	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
343	C₈	—	2	2	no	—
344	C₈	—	2	2	no	—
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	4	1	no	—
348	C₈	—	4	1	no	—
349	C₈	—	292	2	no	—
350	C₈	—	292	2	no	—
351	C₈	#14556597	1	2	yes	—
352	C₈	#14556597	1	2	yes	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	292	2	no	—
357	C₂²	—	292	2	no	—
358	C₂²	—	292	2	no	—
359	C₂²	C₂×C₄×D₂₉₂	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	292	2	no	—
365	C₄	—	292	2	no	—
366	C₄	—	292	2	no	—
367	C₄	C₂×#14556597	1	1	yes	characteristic
368	C₄	C₂×#14556597	1	1	yes	characteristic
369	C₄	#17989399	1	2	yes	—
370	C₂	—	4	1	no	—
371	C₂	—	584	2	no	—
372	C₂	#22618644	1	1	yes	characteristic
373	C₂	#22618816	1	1	yes	characteristic
374	C₂	#22618849	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1450	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#17988823	C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1450	D₇₃
hypocenter	C₇₃	#17974
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
2. #22619082	C₂	semidirect
3. #22608846	C₂	semidirect
4. #22608841	C₂	semidirect
5. #22608823	C₂	semidirect
6. #22608756	C₂	semidirect
7. #22608751	C₂	semidirect
8. C₈×#11924017	C₂	semidirect
9. #22608687	C₂	semidirect
10. #22608397	C₂	semidirect
11. C₇₃×#1450	C₂	semidirect
198. #1450	D₇₃	semidirect
199. C₇₃	#17974	semidirect

Maximal subgroups

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

subgroup	index
#22620501	2
#22619082	2
#22608846	2
#22608841	2
#22608823	2
#22608756	2
#22608751	2
C₈×#11924017	2
#22608687	2
#22608397	2
C₇₃×#1450	2
#17974	73