Subgroups — #33056122

Order 18688.

638 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22619935	C₂	1	1	yes	maximal, characteristic
2	#22618954	C₂	1	1	yes	maximal, characteristic
3	#22608786	C₂	1	2	yes	maximal, complemented
4	#22608765	C₂	1	2	yes	maximal, complemented
5	#22608741	C₂	1	2	yes	maximal, complemented
6	C₈×#11924015	C₂	1	2	yes	maximal, complemented
7	#22608712	C₂	1	1	yes	maximal, characteristic, complemented
8	#22608373	C₂	1	1	yes	maximal, characteristic
9	#22608352	C₂	1	1	yes	maximal, characteristic, complemented
10	#22608226	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2985	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
13	#17988915	C₄	1	2	yes	—
14	#17988874	C₄	1	2	yes	—
15	#17988789	C₄	1	2	yes	—
16	#17988737	C₄	1	2	yes	—
17	C₄×#11924015	C₂²	1	2	yes	—
18	C₄×#11924015	C₂²	1	2	yes	—
19	C₈×D₂₉₂	C₂²	1	2	yes	—
20	C₈×D₂₉₂	C₂²	1	2	yes	—
21	#17988652	—	2	2	no	—
22	#17988640	C₂²	1	1	yes	characteristic
23	#17988640	C₂²	1	1	yes	characteristic
24	C₂²⋊C₈×D₇₃	—	2	2	no	—
25	C₄×#11924012	C₂²	1	1	yes	characteristic
26	#17988339	C₂²	1	2	yes	complemented
27	#17988337	C₂²	1	2	yes	complemented
28	Q₈⋊#10596831	C₂²	1	2	yes	complemented
29	Q₈⋊#10596831	C₂²	1	2	yes	complemented
30	#17988314	C₂²	1	2	yes	—
31	#17988314	C₂²	1	2	yes	—
32	#17988309	C₂²	1	1	yes	characteristic, complemented
33	#17988309	C₂²	1	1	yes	characteristic, complemented
34	#17988308	C₂²	1	1	yes	characteristic, complemented
35	C₈×#10596831	C₂²	1	1	yes	characteristic, complemented
36	#533×C₇₃	C₂²	1	1	yes	characteristic
37	#446×C₇₃	—	2	2	no	—
38	#446×C₇₃	C₂²	1	2	yes	complemented
39	C₈×Q₈×C₇₃	C₂²	1	2	yes	complemented
40	#433×C₇₃	—	2	2	no	—
41	#432×C₇₃	C₂²	1	1	yes	characteristic
42	#431×C₇₃	C₂²	1	1	yes	characteristic
43	C₂²×D₂₉₂	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×D₇₃	—	2	1	no	—
45	C₂×#11924015	C₂×C₄	1	2	yes	—
46	C₂×#11924015	C₂×C₄	1	2	yes	—
47	C₂×#11924012	—	2	1	no	—
48	C₂×C₈×D₇₃	—	2	2	no	—
49	C₂×C₈×D₇₃	—	2	2	no	—
50	C₄×D₂₉₂	—	2	1	no	—
51	C₄×D₂₉₂	—	2	1	no	—
52	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
53	C₄×D₂₉₂	C₂³	1	2	yes	—
54	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
55	C₂×#11924005	—	2	1	no	—
56	#14556522	C₂×C₄	1	2	yes	—
57	#14556522	C₂×C₄	1	2	yes	—
58	C₄⋊#10596831	C₂×C₄	1	2	yes	—
59	C₄⋊#10596831	C₂³	1	2	yes	—
60	#14556515	C₂×C₄	1	2	yes	—
61	#14556515	C₂×C₄	1	2	yes	—
62	#14556511	C₂×C₄	1	2	yes	—
63	#14556511	C₂×C₄	1	2	yes	—
64	C₂²⋊#10596831	—	2	1	no	—
65	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
66	#14556503	—	2	2	no	—
67	#14556503	—	2	2	no	—
68	#14556502	—	2	1	no	—
69	C₄×#10596831	—	2	1	no	—
70	C₄×#10596831	C₂³	1	1	yes	characteristic
71	C₄×#10596831	C₂³	1	1	yes	characteristic
72	C₄×#10596831	C₂³	1	1	yes	characteristic
73	C₄×#10596831	C₂³	1	1	yes	characteristic
74	C₄⋊Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
75	C₄⋊Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
76	#125×C₇₃	—	2	1	no	—
77	C₂²⋊Q₈×C₇₃	—	2	1	no	—
78	C₂²⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
79	C₄×Q₈×C₇₃	—	2	1	no	—
80	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
81	#117×C₇₃	—	2	1	no	—
82	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₇₃	—	2	2	no	—
84	C₄⋊C₈×C₇₃	—	2	2	no	—
85	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
86	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
87	C₂²⋊C₈×C₇₃	—	2	1	no	—
88	C₂²⋊C₈×C₇₃	D₄	1	2	yes	—
89	#97×C₇₃	—	2	2	no	—
90	#97×C₇₃	—	2	1	no	—
91	C₄×C₈×C₇₃	—	2	2	no	—
92	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
94	C₂³×D₇₃	—	2	1	no	—
95	C₂×D₂₉₂	—	2	1	no	—
96	C₂×D₂₉₂	—	2	4	no	—
97	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
98	C₂²×Dic₇₃	—	2	1	no	—
99	C₂×C₄×D₇₃	—	2	2	no	—
100	C₂×C₄×D₇₃	—	2	2	no	—
101	C₂×C₄×D₇₃	—	8	2	no	—
102	#11924015	—	2	4	no	—
103	#11924015	—	2	4	no	—
104	#11924012	—	4	2	no	—
105	C₂×#10596831	—	2	1	no	—
106	C₂×#10596831	—	2	2	no	—
107	C₂×#10596831	—	2	1	no	—
108	C₂×#10596831	C₂²×C₄	1	2	yes	—
109	C₂×#10596831	C₂²×C₄	1	2	yes	—
110	C₈×D₇₃	—	8	2	no	—
111	C₈×D₇₃	—	8	2	no	—
112	C₄⋊Dic₇₃	—	2	1	no	—
113	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
114	#11924005	—	2	2	no	—
115	#11924005	—	2	2	no	—
116	C₂×Q₈×C₇₃	—	2	1	no	—
117	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
121	C₂×C₈×C₇₃	—	2	2	no	—
122	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
123	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
124	C₄⋊C₄×C₇₃	—	2	1	no	—
125	C₄⋊C₄×C₇₃	—	2	1	no	—
126	C₄⋊C₄×C₇₃	—	2	1	no	—
127	C₄⋊C₄×C₇₃	—	2	2	no	—
128	C₄⋊C₄×C₇₃	—	2	2	no	—
129	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
130	C₂²⋊C₄×C₇₃	—	2	2	no	—
131	C₄²×C₇₃	—	2	2	no	—
132	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₇₃	#41	1	1	yes	characteristic
135	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
136	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₇₃	—	2	2	no	—
138	C₂²×D₇₃	—	2	4	no	—
139	C₂²×D₇₃	—	8	1	no	—
140	C₂²×D₇₃	—	8	1	no	—
141	Q₈×C₇₃	—	2	2	no	—
142	Q₈×C₇₃	—	2	2	no	—
143	Q₈×C₇₃	—	8	1	no	—
144	C₂×C₄×C₇₃	—	2	2	no	—
145	C₂×C₄×C₇₃	—	2	2	no	—
146	C₂×C₄×C₇₃	—	2	2	no	—
147	C₂×C₄×C₇₃	—	2	2	no	—
148	C₂×C₄×C₇₃	—	4	1	no	—
149	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
155	C₂×Dic₇₃	—	2	2	no	—
156	C₂×Dic₇₃	—	8	1	no	—
157	D₂₉₂	—	2	4	no	—
158	D₂₉₂	—	2	2	no	—
159	D₂₉₂	#131	1	4	yes	—
160	C₄×D₇₃	—	4	2	no	—
161	C₄×D₇₃	—	4	2	no	—
162	C₄×D₇₃	—	8	2	no	—
163	C₈×C₇₃	—	2	2	no	—
164	C₈×C₇₃	—	2	2	no	—
165	C₈×C₇₃	—	4	2	no	—
166	#10596831	—	2	2	no	—
167	#10596831	—	2	2	no	—
168	#10596831	—	2	2	no	—
169	#10596831	—	4	1	no	—
170	#10596831	—	4	2	no	—
171	C₂²×C₇₃	—	2	1	no	—
172	C₂²×C₇₃	—	2	1	no	—
173	C₂²×C₇₃	—	2	1	no	—
174	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂×D₇₃	—	2	4	no	—
176	C₂×D₇₃	—	4	2	no	—
177	C₂×D₇₃	—	4	2	no	—
178	C₂×D₇₃	—	8	4	no	—
179	C₄×C₇₃	—	2	1	no	—
180	C₄×C₇₃	—	2	1	no	—
181	C₄×C₇₃	—	2	1	no	—
182	C₄×C₇₃	—	2	1	no	—
183	C₄×C₇₃	—	4	2	no	—
184	C₄×C₇₃	—	4	2	no	—
185	C₄×C₇₃	C₂×#131	1	2	yes	—
186	C₄×C₇₃	#577	1	2	yes	—
187	Dic₇₃	—	4	2	no	—
188	#17924	—	73	1	no	maximal, Sylow
189	C₂×C₇₃	—	4	1	no	—
190	C₂×C₇₃	#2929	1	1	yes	characteristic
191	C₂×C₇₃	#2949	1	1	yes	characteristic
192	C₂×C₇₃	#2956	1	1	yes	characteristic
193	D₇₃	—	4	4	no	—
194	#3101	—	73	1	no	—
195	#2985	D₇₃	1	1	yes	characteristic, complemented
196	#2972	—	73	1	no	—
197	#1585	—	73	2	no	—
198	#1582	—	73	2	no	—
199	SD₁₆⋊C₈	—	73	2	no	—
200	C₈×SD₁₆	—	73	2	no	—
201	#1572	—	73	1	no	—
202	#1490	—	73	1	no	—
203	#1485	—	73	1	no	—
204	#1453	—	73	1	no	—
205	C₇₃	#17924	1	1	yes	characteristic, complemented, Sylow
206	#533	C₂×D₇₃	1	1	yes	characteristic
207	C₂×C₄×D₄	—	73	1	no	—
208	#483	—	73	2	no	—
209	#476	—	73	2	no	—
210	#461	—	73	2	no	—
211	#450	—	73	2	no	—
212	#446	—	2	2	no	—
213	#446	C₂×D₇₃	1	2	yes	complemented
214	C₈×Q₈	C₂×D₇₃	1	2	yes	complemented
215	C₄×SD₁₆	—	73	2	no	—
216	C₄×SD₁₆	—	73	2	no	—
217	C₈×D₄	—	73	2	no	—
218	C₈×D₄	—	73	2	no	—
219	C₈×D₄	—	146	2	no	—
220	#433	—	2	2	no	—
221	#432	C₂×D₇₃	1	1	yes	characteristic
222	#431	—	73	1	no	—
223	#431	—	73	1	no	—
224	#431	C₂×D₇₃	1	1	yes	characteristic
225	C₂×C₂²⋊C₈	—	146	2	no	—
226	C₄×M₁₆	—	73	1	no	—
227	#335	—	73	2	no	—
228	#334	—	73	2	no	—
229	Q₈⋊C₈	—	73	2	no	—
230	Q₈⋊C₈	—	73	2	no	—
231	D₄⋊C₈	—	73	2	no	—
232	D₄⋊C₈	—	73	2	no	—
233	#322	—	73	1	no	—
234	#322	—	73	1	no	—
235	#322	—	73	1	no	—
236	C₈²	—	73	1	no	—
237	C₂²×D₄	—	73	1	no	—
238	C₂³×C₄	—	146	1	no	—
239	C₂×SD₁₆	—	73	2	no	—
240	C₂×SD₁₆	—	73	2	no	—
241	C₂×M₁₆	—	146	1	no	—
242	C₂²×C₈	—	146	2	no	—
243	C₂²×C₈	—	146	2	no	—
244	C₄⋊Q₈	C₄×D₇₃	1	1	yes	characteristic
245	C₄⋊Q₈	C₄×D₇₃	1	1	yes	characteristic
246	#125	—	2	1	no	—
247	C₂²⋊Q₈	—	2	1	no	—
248	C₂²⋊Q₈	C₄×D₇₃	1	2	yes	—
249	C₄×Q₈	—	2	1	no	—
250	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
251	C₄×D₄	—	73	2	no	—
252	C₄×D₄	—	73	2	no	—
253	C₄×D₄	—	146	1	no	—
254	C₄×D₄	—	146	1	no	—
255	#117	—	2	1	no	—
256	C₂×C₄⋊C₄	—	73	1	no	—
257	C₂×C₂²⋊C₄	—	146	1	no	—
258	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
259	#106	—	73	2	no	—
260	#106	—	73	2	no	—
261	C₄⋊C₈	—	2	2	no	—
262	C₄⋊C₈	—	2	2	no	—
263	C₄⋊C₈	—	73	2	no	—
264	C₄⋊C₈	—	73	2	no	—
265	C₄⋊C₈	C₂²⋊D₇₃	1	2	yes	—
266	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
267	#103	—	73	2	no	—
268	#103	—	73	2	no	—
269	#102	—	73	2	no	—
270	#102	—	73	2	no	—
271	C₂²⋊C₈	—	2	1	no	—
272	C₂²⋊C₈	—	73	2	no	—
273	C₂²⋊C₈	—	146	1	no	—
274	C₂²⋊C₈	—	146	2	no	—
275	C₂²⋊C₈	—	146	2	no	—
276	C₂²⋊C₈	C₂²⋊D₇₃	1	2	yes	—
277	#97	—	2	2	no	—
278	#97	—	2	1	no	—
279	#97	—	146	1	no	—
280	C₄×C₈	—	2	2	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	—	146	1	no	—
286	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
288	C₂⁴	—	146	1	no	—
289	C₂×Q₈	—	2	1	no	—
290	C₂×Q₈	C₂×C₄×D₇₃	1	2	yes	—
291	C₂×D₄	—	73	2	no	—
292	C₂×D₄	—	146	4	no	—
293	C₂×D₄	—	146	1	no	—
294	C₂²×C₄	—	584	2	no	—
295	C₂²×C₄	—	146	2	no	—
296	C₂²×C₄	—	146	1	no	—
297	C₂²×C₄	—	146	2	no	—
298	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
299	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
300	C₂²×C₄	#11924028	1	1	yes	characteristic
301	SD₁₆	—	146	4	no	—
302	SD₁₆	—	146	4	no	—
303	M₁₆	—	292	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	584	2	no	—
306	C₂×C₈	—	584	2	no	—
307	C₂×C₈	—	73	2	no	—
308	C₂×C₈	—	73	2	no	—
309	C₂×C₈	—	146	2	no	—
310	C₂×C₈	—	146	1	no	—
311	C₂×C₈	—	146	1	no	—
312	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
313	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	1	no	—
316	C₄⋊C₄	—	2	1	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	73	2	no	—
320	C₄⋊C₄	—	146	1	no	—
321	C₄⋊C₄	C₂×C₄×D₇₃	1	2	yes	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₂²⋊C₄	—	146	2	no	—
324	C₂²⋊C₄	—	146	2	no	—
325	C₄²	—	2	2	no	—
326	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
327	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
328	C₄²	#11924028	1	1	yes	characteristic
329	C₄²	C₂³×D₇₃	1	1	yes	characteristic
330	C₂³	—	584	1	no	—
331	C₂³	—	584	1	no	—
332	C₂³	—	146	4	no	—
333	C₂³	—	146	2	no	—
334	C₂³	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
335	Q₈	—	2	2	no	—
336	Q₈	—	2	2	no	—
337	Q₈	—	8	1	no	—
338	D₄	—	73	4	no	—
339	D₄	—	146	4	no	—
340	D₄	—	146	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	292	2	no	—
347	C₂×C₄	—	292	2	no	—
348	C₂×C₄	—	584	2	no	—
349	C₂×C₄	—	584	1	no	—
350	C₂×C₄	—	146	2	no	—
351	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
352	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
353	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
354	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
356	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
357	C₈	—	2	2	no	—
358	C₈	—	2	2	no	—
359	C₈	—	4	2	no	—
360	C₈	—	292	1	no	—
361	C₈	—	292	2	no	—
362	C₈	—	146	2	no	—
363	C₈	—	146	2	no	—
364	C₈	—	146	2	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	292	2	no	—
369	C₂²	—	292	2	no	—
370	C₂²	—	584	4	no	—
371	C₂²	—	146	4	no	—
372	C₂²	C₂×C₄×C₂²⋊D₇₃	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	2	no	—
379	C₄	—	292	2	no	—
380	C₄	#131×D₇₃	1	2	yes	—
381	C₄	#17989395	1	2	yes	—
382	C₂	—	4	1	no	—
383	C₂	—	292	4	no	—
384	C₂	#22618623	1	1	yes	characteristic
385	C₂	#22618790	1	1	yes	characteristic
386	C₂	#22618838	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#2985	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂²×D₂₉₂	C₂×C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#2985	D₇₃
hypocenter	C₇₃	#17924
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #2985

Complements

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

N	H ≅ G/N	type
3. #22608786	C₂	semidirect
4. #22608765	C₂	semidirect
5. #22608741	C₂	semidirect
6. C₈×#11924015	C₂	semidirect
7. #22608712	C₂	semidirect
9. #22608352	C₂	semidirect
10. #22608226	C₂	semidirect
11. C₇₃×#2985	C₂	semidirect
26. #17988339	C₂²	semidirect
27. #17988337	C₂²	semidirect
28. Q₈⋊#10596831	C₂²	semidirect
29. Q₈⋊#10596831	C₂²	semidirect
32. #17988309	C₂²	semidirect
33. #17988309	C₂²	semidirect
34. #17988308	C₂²	semidirect
35. C₈×#10596831	C₂²	semidirect
38. #446×C₇₃	C₂²	semidirect
39. C₈×Q₈×C₇₃	C₂²	semidirect
195. #2985	D₇₃	semidirect
205. C₇₃	#17924	semidirect
213. #446	C₂×D₇₃	semidirect
214. C₈×Q₈	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22619935	2
#22618954	2
#22608786	2
#22608765	2
#22608741	2
C₈×#11924015	2
#22608712	2
#22608373	2
#22608352	2
#22608226	2
C₇₃×#2985	2
#17924	73