Subgroups — #33056524

Order 18688.

630 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620504	C₂	1	1	yes	maximal, characteristic
2	#22618905	C₂	1	2	yes	maximal, complemented
3	#22608821	C₂	1	2	yes	maximal, complemented
4	#22608817	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₁₆×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608728	C₂	1	2	yes	maximal, complemented
7	#22608725	C₂	1	1	yes	maximal, characteristic, complemented
8	#22608693	C₂	1	1	yes	maximal, characteristic, complemented
9	#22608390	C₂	1	2	yes	maximal, complemented
10	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989124	C₂²	1	2	yes	—
13	#17988988	C₄	1	1	yes	characteristic
14	#17988972	C₄	1	1	yes	characteristic
15	#17988943	C₄	1	2	yes	—
16	#17988825	C₄	1	1	yes	characteristic
17	#17988824	C₄	1	1	yes	characteristic
18	#17988819	C₄	1	2	yes	—
19	Q₈×#10596831	C₂²	1	2	yes	—
20	C₄×#11924015	C₂²	1	2	yes	—
21	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
22	#17988666	C₂²	1	2	yes	—
23	#17988660	—	2	2	no	—
24	#17988652	—	2	2	no	—
25	#17988652	C₂²	1	2	yes	—
26	#17988647	—	2	2	no	—
27	#17988643	C₂²	1	2	yes	—
28	#17988544	—	2	2	no	—
29	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
32	#17988340	C₂²	1	1	yes	characteristic
33	#17988337	C₂²	1	2	yes	—
34	Q₈⋊#10596831	C₂²	1	2	yes	—
35	#17988316	C₂²	1	2	yes	—
36	#17988314	C₂²	1	2	yes	—
37	C₈×#10596831	C₂²	1	1	yes	characteristic
38	C₈×#10596831	C₂²	1	1	yes	characteristic
39	C₈×#10596831	C₂²	1	1	yes	characteristic
40	#446×C₇₃	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
42	#335×C₇₃	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₇₃	C₂²	1	2	yes	—
44	C₈²×C₇₃	C₂²	1	1	yes	characteristic
45	C₂×#11924015	C₂×C₄	1	2	yes	—
46	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
47	C₂×#11924011	—	2	2	no	—
48	C₂×C₈×D₇₃	—	2	1	no	—
49	C₂×C₈×D₇₃	—	2	2	no	—
50	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
53	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
54	#14556586	—	2	2	no	—
55	#14556582	C₂×C₄	1	2	yes	—
56	#14556580	C₂×C₄	1	2	yes	—
57	#14556576	—	2	2	no	—
58	#14556567	C₂×C₄	1	2	yes	—
59	#14556566	C₂×C₄	1	2	yes	—
60	C₄×D₂₉₂	—	2	2	no	—
61	C₄×D₂₉₂	C₂³	1	2	yes	—
62	#14556548	—	2	2	no	—
63	C₄²×D₇₃	C₂³	1	1	yes	characteristic
64	#14556525	C₂×C₄	1	1	yes	characteristic
65	#14556522	C₂×C₄	1	2	yes	—
66	C₄⋊#10596831	—	2	2	no	—
67	C₄⋊#10596831	C₂³	1	2	yes	—
68	#14556515	C₂×C₄	1	2	yes	—
69	#14556513	C₂×C₄	1	2	yes	—
70	#14556511	C₂×C₄	1	2	yes	—
71	#14556503	—	2	2	no	—
72	#14556503	—	2	2	no	—
73	#14556503	—	2	2	no	—
74	#14556501	—	2	2	no	—
75	C₄×#10596831	—	2	2	no	—
76	C₄×#10596831	C₂³	1	1	yes	characteristic
77	C₄×#10596831	C₂³	1	1	yes	characteristic
78	C₄×#10596831	C₂³	1	1	yes	characteristic
79	C₈×Dic₇₃	—	2	1	no	—
80	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₇₃	D₄	1	1	yes	characteristic
83	C₈×Dic₇₃	D₄	1	1	yes	characteristic
84	C₂×Q₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
86	#107×C₇₃	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₇₃	—	2	2	no	—
88	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
89	#103×C₇₃	C₂×C₄	1	2	yes	—
90	#97×C₇₃	—	2	2	no	—
91	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
94	C₂×D₂₉₂	—	2	2	no	—
95	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₇₃	—	2	2	no	—
97	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
100	#11924015	—	2	4	no	—
101	D₅₈₄	—	2	2	no	—
102	#11924011	—	4	4	no	—
103	C₂×#10596831	—	2	1	no	—
104	C₂×#10596831	—	2	2	no	—
105	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
106	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
107	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
108	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
109	C₈×D₇₃	—	2	4	no	—
110	C₈×D₇₃	—	2	2	no	—
111	C₈×D₇₃	—	2	2	no	—
112	C₈×D₇₃	—	2	2	no	—
113	C₈×D₇₃	—	2	2	no	—
114	C₈×D₇₃	—	4	4	no	—
115	C₄⋊Dic₇₃	—	2	2	no	—
116	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
117	#11924007	—	2	2	no	—
118	#11924007	—	2	2	no	—
119	#11924005	—	2	2	no	—
120	#11924005	—	2	2	no	—
121	#11924005	—	2	2	no	—
122	#11924005	—	2	2	no	—
123	#11924005	—	2	2	no	—
124	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
125	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
126	C₄×Dic₇₃	#41	1	1	yes	characteristic
127	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
128	Q₁₆×C₇₃	—	2	2	no	—
129	C₂×C₈×C₇₃	—	2	1	no	—
130	C₂×C₈×C₇₃	—	2	2	no	—
131	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₇₃	—	2	2	no	—
136	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
137	C₄²×C₇₃	—	2	2	no	—
138	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
139	C₂²×D₇₃	—	2	2	no	—
140	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
141	Q₈×C₇₃	—	2	2	no	—
142	Q₈×C₇₃	—	2	2	no	—
143	C₂×C₄×C₇₃	—	2	2	no	—
144	C₂×C₄×C₇₃	—	2	2	no	—
145	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₇₃	—	2	2	no	—
149	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
150	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
151	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
152	D₂₉₂	—	2	2	no	—
153	D₂₉₂	—	2	2	no	—
154	D₂₉₂	—	4	4	no	—
155	C₄×D₇₃	—	2	2	no	—
156	C₄×D₇₃	—	2	1	no	—
157	C₄×D₇₃	—	2	1	no	—
158	C₄×D₇₃	—	2	1	no	—
159	C₄×D₇₃	—	2	1	no	—
160	C₄×D₇₃	—	4	2	no	—
161	C₄×D₇₃	—	4	2	no	—
162	C₈×C₇₃	—	2	2	no	—
163	C₈×C₇₃	—	2	1	no	—
164	C₈×C₇₃	—	2	1	no	—
165	C₈×C₇₃	—	4	2	no	—
166	C₈×C₇₃	#131	1	2	yes	—
167	C₈×C₇₃	#135	1	2	yes	—
168	#10596831	—	2	2	no	—
169	#10596831	—	2	1	no	—
170	#10596831	—	2	1	no	—
171	#10596831	—	4	2	no	—
172	#10596831	#131	1	2	yes	—
173	#10596831	#135	1	2	yes	—
174	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂×D₇₃	—	2	1	no	—
176	C₂×D₇₃	—	2	1	no	—
177	C₂×D₇₃	—	4	2	no	—
178	C₂×D₇₃	—	4	2	no	—
179	C₂×D₇₃	—	4	2	no	—
180	C₂×D₇₃	#443	1	2	yes	—
181	C₄×C₇₃	—	2	1	no	—
182	C₄×C₇₃	—	4	2	no	—
183	C₄×C₇₃	—	4	2	no	—
184	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
185	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
186	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
187	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
188	Dic₇₃	—	2	1	no	—
189	Dic₇₃	—	2	1	no	—
190	Dic₇₃	—	2	1	no	—
191	Dic₇₃	—	4	2	no	—
192	#17959	—	73	1	no	maximal, Sylow
193	C₂×C₇₃	#2931	1	1	yes	characteristic
194	C₂×C₇₃	C₄×#135	1	1	yes	characteristic
195	C₂×C₇₃	C₂×#443	1	1	yes	characteristic
196	D₇₃	—	2	2	no	—
197	D₇₃	—	8	2	no	—
198	#3168	—	73	1	no	—
199	#2966	—	73	2	no	—
200	#1591	D₇₃	1	1	yes	characteristic, complemented
201	#1590	—	73	2	no	—
202	#1589	—	73	1	no	—
203	C₈×Q₁₆	—	73	1	no	—
204	C₈×SD₁₆	—	73	2	no	—
205	C₈×D₈	—	73	1	no	—
206	#1568	—	73	1	no	—
207	#1494	—	73	2	no	—
208	C₂×C₈²	—	73	1	no	—
209	C₇₃	#17959	1	1	yes	characteristic, complemented, Sylow
210	#529	—	73	2	no	—
211	#499	—	73	1	no	—
212	#495	—	73	1	no	—
213	#488	—	73	2	no	—
214	#467	—	73	1	no	—
215	#466	—	73	1	no	—
216	#465	—	73	2	no	—
217	#446	C₂×D₇₃	1	2	yes	—
218	C₈×Q₈	—	73	2	no	—
219	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
220	C₄×SD₁₆	—	73	2	no	—
221	C₄×D₈	—	73	1	no	—
222	#436	—	73	2	no	—
223	#435	—	146	2	no	—
224	C₈×D₄	—	73	2	no	—
225	C₈×D₄	—	146	2	no	—
226	#433	—	146	2	no	—
227	#432	—	73	2	no	—
228	#405	—	146	2	no	—
229	C₂×C₄×C₈	—	73	1	no	—
230	C₂×C₄×C₈	—	73	1	no	—
231	C₂×C₄×C₈	—	73	1	no	—
232	#335	—	73	1	no	—
233	#335	C₂×D₇₃	1	1	yes	characteristic
234	#334	—	73	2	no	—
235	Q₈⋊C₈	—	73	2	no	—
236	Q₈⋊C₈	C₂×D₇₃	1	2	yes	—
237	D₄⋊C₈	—	73	2	no	—
238	D₄⋊C₈	—	73	2	no	—
239	C₈²	—	73	1	no	—
240	C₈²	—	73	1	no	—
241	C₈²	—	73	1	no	—
242	C₈²	C₂×D₇₃	1	1	yes	characteristic
243	C₂×Q₁₆	C₄×D₇₃	1	1	yes	characteristic
244	C₂×SD₁₆	—	73	2	no	—
245	C₂×D₈	—	73	1	no	—
246	C₂×M₁₆	—	146	2	no	—
247	C₂²×C₈	—	73	1	no	—
248	C₂²×C₈	—	73	1	no	—
249	C₂²×C₈	—	73	1	no	—
250	C₂²×C₈	—	73	1	no	—
251	C₂²×C₈	—	146	2	no	—
252	C₂²×C₈	—	146	1	no	—
253	#126	—	146	2	no	—
254	#125	—	73	2	no	—
255	#124	—	73	2	no	—
256	#123	—	146	2	no	—
257	C₂²⋊Q₈	—	73	2	no	—
258	#121	—	73	2	no	—
259	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
260	C₄×D₄	—	73	2	no	—
261	C₄×D₄	—	146	2	no	—
262	#117	—	146	2	no	—
263	C₂×C₄²	—	73	1	no	—
264	#107	—	73	1	no	—
265	#107	C₄×D₇₃	1	1	yes	characteristic
266	#106	—	73	2	no	—
267	C₄⋊C₈	—	2	2	no	—
268	C₄⋊C₈	—	73	2	no	—
269	C₄⋊C₈	—	146	2	no	—
270	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
271	#103	—	73	2	no	—
272	#103	C₄×D₇₃	1	2	yes	—
273	#102	—	73	2	no	—
274	#102	—	73	2	no	—
275	C₂²⋊C₈	—	146	2	no	—
276	C₂²⋊C₈	—	146	2	no	—
277	C₂²⋊C₈	—	146	2	no	—
278	#97	—	2	2	no	—
279	#97	—	146	2	no	—
280	C₄×C₈	—	73	1	no	—
281	C₄×C₈	—	73	1	no	—
282	C₄×C₈	—	73	1	no	—
283	C₄×C₈	—	73	1	no	—
284	C₄×C₈	—	73	1	no	—
285	C₄×C₈	—	73	1	no	—
286	C₄×C₈	—	73	1	no	—
287	C₄×C₈	—	146	2	no	—
288	C₄×C₈	—	146	1	no	—
289	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
291	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
292	C₂×Q₈	C₂×C₄×D₇₃	1	2	yes	—
293	C₂×D₄	—	73	2	no	—
294	C₂×D₄	—	146	2	no	—
295	C₂²×C₄	—	73	1	no	—
296	C₂²×C₄	—	73	1	no	—
297	C₂²×C₄	—	73	1	no	—
298	C₂²×C₄	—	146	2	no	—
299	Q₁₆	—	2	2	no	—
300	SD₁₆	—	146	4	no	—
301	D₈	—	146	2	no	—
302	M₁₆	—	292	4	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	292	4	no	—
306	C₂×C₈	—	73	1	no	—
307	C₂×C₈	—	73	1	no	—
308	C₂×C₈	—	73	1	no	—
309	C₂×C₈	—	73	1	no	—
310	C₂×C₈	—	146	2	no	—
311	C₂×C₈	—	146	2	no	—
312	C₂×C₈	—	146	4	no	—
313	C₂×C₈	—	146	2	no	—
314	C₂×C₈	—	146	1	no	—
315	C₂×C₈	—	146	2	no	—
316	C₂×C₈	—	146	2	no	—
317	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
318	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
319	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
320	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	73	2	no	—
323	C₄⋊C₄	—	146	2	no	—
324	C₄⋊C₄	—	146	2	no	—
325	C₄⋊C₄	—	146	2	no	—
326	C₄⋊C₄	C₂×C₄×D₇₃	1	2	yes	—
327	C₂²⋊C₄	—	146	2	no	—
328	C₂²⋊C₄	—	146	2	no	—
329	C₂²⋊C₄	—	146	2	no	—
330	C₂²⋊C₄	—	146	2	no	—
331	C₂²⋊C₄	—	146	2	no	—
332	C₄²	—	2	2	no	—
333	C₄²	—	73	1	no	—
334	C₄²	—	73	1	no	—
335	C₄²	—	73	1	no	—
336	C₄²	C₂³×D₇₃	1	1	yes	characteristic
337	C₂³	—	73	1	no	—
338	C₂³	—	146	2	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	2	2	no	—
341	D₄	—	292	4	no	—
342	D₄	—	146	2	no	—
343	D₄	—	146	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	292	2	no	—
347	C₂×C₄	—	292	2	no	—
348	C₂×C₄	—	73	1	no	—
349	C₂×C₄	—	73	1	no	—
350	C₂×C₄	—	73	1	no	—
351	C₂×C₄	—	146	2	no	—
352	C₂×C₄	—	146	2	no	—
353	C₂×C₄	—	146	1	no	—
354	C₂×C₄	—	146	1	no	—
355	C₂×C₄	—	146	1	no	—
356	C₂×C₄	—	146	1	no	—
357	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
358	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
359	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
360	C₈	—	2	2	no	—
361	C₈	—	2	1	no	—
362	C₈	—	2	1	no	—
363	C₈	—	4	2	no	—
364	C₈	—	292	2	no	—
365	C₈	—	73	2	no	—
366	C₈	—	73	2	no	—
367	C₈	—	146	2	no	—
368	C₈	—	146	1	no	—
369	C₈	—	146	1	no	—
370	C₈	#14556599	1	2	yes	—
371	C₈	#14556614	1	2	yes	—
372	C₂²	—	292	2	no	—
373	C₂²	—	292	2	no	—
374	C₂²	—	292	2	no	—
375	C₂²	—	73	2	no	—
376	C₂²	—	146	1	no	—
377	C₂²	—	146	1	no	—
378	C₂²	C₄×D₄×D₇₃	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	4	2	no	—
381	C₄	—	4	2	no	—
382	C₄	—	292	2	no	—
383	C₄	—	146	1	no	—
384	C₄	—	146	1	no	—
385	C₄	—	146	1	no	—
386	C₄	#131×D₇₃	1	1	yes	characteristic
387	C₄	C₂×#14556599	1	1	yes	characteristic
388	C₄	#135×D₇₃	1	1	yes	characteristic
389	C₄	C₂×#14556614	1	1	yes	characteristic
390	C₂	—	584	2	no	—
391	C₂	—	146	2	no	—
392	C₂	#22618640	1	1	yes	characteristic
393	C₂	C₄×#14556614	1	1	yes	characteristic
394	C₂	#443×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1591	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#17988824	C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1591	D₇₃
hypocenter	C₇₃	#17959
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
2. #22618905	C₂	semidirect
3. #22608821	C₂	semidirect
4. #22608817	C₂	semidirect
5. Q₁₆×#10596831	C₂	semidirect
6. #22608728	C₂	semidirect
7. #22608725	C₂	semidirect
8. #22608693	C₂	semidirect
9. #22608390	C₂	semidirect
10. C₈²×D₇₃	C₂	semidirect
11. C₇₃×#1591	C₂	semidirect
200. #1591	D₇₃	semidirect
209. C₇₃	#17959	semidirect

Maximal subgroups

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

subgroup	index
#22620504	2
#22618905	2
#22608821	2
#22608817	2
Q₁₆×#10596831	2
#22608728	2
#22608725	2
#22608693	2
#22608390	2
C₈²×D₇₃	2
C₇₃×#1591	2
#17959	73