Subgroups — #33077538

Order 18688.

604 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₅₈₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#22622354	C₂	1	1	yes	maximal, characteristic
3	#22622227	C₂	1	2	yes	maximal, complemented
4	#22609541	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609507	C₂	1	2	yes	maximal, complemented
6	#22609335	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609320	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609161	C₂	1	2	yes	maximal, complemented
9	#22609098	C₂	1	2	yes	maximal, complemented
10	#22608817	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989302	C₂²	1	2	yes	—
13	Q₈×D₂₉₂	C₂²	1	2	yes	—
14	#17988992	—	2	1	no	—
15	#17988992	C₂²	1	1	yes	characteristic
16	#17988992	C₂²	1	1	yes	characteristic
17	#17988938	C₂²	1	2	yes	—
18	#17988938	C₂²	1	2	yes	—
19	#17988931	—	2	2	no	—
20	#17988912	—	2	2	no	—
21	#17988882	C₂²	1	2	yes	—
22	#17988878	C₂²	1	2	yes	—
23	#17988868	—	2	2	no	—
24	#17988868	C₂²	1	2	yes	—
25	#17988832	—	2	2	no	—
26	#17988823	—	2	1	no	—
27	#17988792	C₂²	1	2	yes	—
28	#17988724	—	2	2	no	—
29	C₄×D₅₈₄	—	2	1	no	—
30	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
31	#17988666	C₂²	1	2	yes	—
32	#17988625	—	2	1	no	—
33	#17988594	—	2	2	no	—
34	#17988340	C₂²	1	1	yes	characteristic
35	#17988335	C₂²	1	2	yes	—
36	#17988332	C₂²	1	2	yes	—
37	#17988316	C₂²	1	2	yes	—
38	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
39	C₄×SD₁₆×C₇₃	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₇₃	C₂²	1	1	yes	characteristic
41	#334×C₇₃	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
44	C₈²×C₇₃	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₇₃	—	2	2	no	—
46	C₂×#11924016	—	2	2	no	—
47	C₂×D₅₈₄	D₄	1	1	yes	characteristic
48	C₂×#11924011	—	2	2	no	—
49	#14556591	—	2	2	no	—
50	#14556591	C₂³	1	2	yes	—
51	#14556591	C₂³	1	2	yes	—
52	#14556577	—	4	2	no	—
53	#14556571	—	2	2	no	—
54	#14556571	—	4	2	no	—
55	#14556565	—	2	2	no	—
56	#14556562	—	2	2	no	—
57	C₄×D₂₉₂	—	2	2	no	—
58	C₄×D₂₉₂	C₂³	1	2	yes	—
59	#14556550	—	2	2	no	—
60	#14556549	—	2	2	no	—
61	#14556525	—	2	2	no	—
62	#14556525	—	2	1	no	—
63	#14556525	—	2	1	no	—
64	#14556525	—	2	1	no	—
65	#14556525	—	2	1	no	—
66	#14556525	D₄	1	1	yes	characteristic
67	#14556524	—	2	2	no	—
68	#14556523	—	2	2	no	—
69	C₄⋊#10596831	C₂³	1	2	yes	—
70	#14556516	—	2	2	no	—
71	#14556516	—	2	2	no	—
72	#14556516	—	4	2	no	—
73	#14556515	—	2	2	no	—
74	#14556514	—	2	2	no	—
75	#14556513	—	2	2	no	—
76	#14556513	—	2	2	no	—
77	#14556513	D₄	1	2	yes	—
78	#14556503	—	2	2	no	—
79	C₂×SD₁₆×C₇₃	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₇₃	—	2	1	no	—
81	C₄×Q₈×C₇₃	C₂³	1	1	yes	characteristic
82	C₄×D₄×C₇₃	C₂³	1	1	yes	characteristic
83	#106×C₇₃	D₄	1	1	yes	characteristic
84	C₄⋊C₈×C₇₃	—	2	1	no	—
85	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
87	#103×C₇₃	D₄	1	1	yes	characteristic
88	#102×C₇₃	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₇₃	—	2	1	no	—
90	C₄×C₈×C₇₃	—	2	1	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×Dic₁₄₆	—	2	2	no	—
95	C₂×Dic₁₄₆	—	2	2	no	—
96	C₂×Dic₁₄₆	—	4	2	no	—
97	Q₈×D₇₃	—	4	4	no	—
98	Q₈×D₇₃	—	8	2	no	—
99	C₂×D₂₉₂	C₂×D₄	1	2	yes	—
100	C₂×C₂²⋊D₇₃	—	4	2	no	—
101	C₂×C₄×D₇₃	—	2	2	no	—
102	C₂×C₄×D₇₃	—	4	2	no	—
103	#11924016	—	8	2	no	—
104	D₅₈₄	—	2	2	no	—
105	#11924011	—	8	2	no	—
106	C₂×#10596831	—	2	2	no	—
107	C₄⋊Dic₇₃	—	2	2	no	—
108	C₄⋊Dic₇₃	—	2	2	no	—
109	C₄⋊Dic₇₃	—	2	2	no	—
110	C₄⋊Dic₇₃	—	2	2	no	—
111	C₄⋊Dic₇₃	—	4	2	no	—
112	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
113	C₂²⋊Dic₇₃	—	4	2	no	—
114	#11924005	—	2	2	no	—
115	#11924005	—	4	2	no	—
116	#11924005	—	4	2	no	—
117	C₄×Dic₇₃	—	4	2	no	—
118	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
119	C₂×D₄×C₇₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₇₃	—	2	1	no	—
121	SD₁₆×C₇₃	—	2	2	no	—
122	C₂×C₈×C₇₃	—	2	1	no	—
123	C₂×C₈×C₇₃	—	2	1	no	—
124	C₂×C₈×C₇₃	—	2	1	no	—
125	C₂×C₈×C₇₃	—	4	2	no	—
126	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄×C₇₃	—	2	1	no	—
131	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
133	C₂²⋊C₄×C₇₃	—	2	1	no	—
134	C₄²×C₇₃	—	2	1	no	—
135	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
136	C₂³×C₇₃	—	2	1	no	—
137	C₂²×D₇₃	—	2	2	no	—
138	Q₈×C₇₃	—	2	1	no	—
139	Q₈×C₇₃	#135	1	2	yes	—
140	D₄×C₇₃	—	2	1	no	—
141	D₄×C₇₃	—	2	1	no	—
142	C₂×C₄×C₇₃	—	2	1	no	—
143	C₂×C₄×C₇₃	—	2	1	no	—
144	C₂×C₄×C₇₃	—	2	1	no	—
145	C₂×C₄×C₇₃	—	4	1	no	—
146	C₂×C₄×C₇₃	—	4	1	no	—
147	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂²⋊D₇₃	—	16	2	no	—
151	C₂×Dic₇₃	—	2	2	no	—
152	C₂×Dic₇₃	—	4	2	no	—
153	C₂×Dic₇₃	—	4	2	no	—
154	D₂₉₂	—	2	2	no	—
155	D₂₉₂	—	2	2	no	—
156	C₄×D₇₃	—	4	2	no	—
157	C₄×D₇₃	—	4	2	no	—
158	C₄×D₇₃	—	8	4	no	—
159	Dic₁₄₆	—	4	2	no	—
160	Dic₁₄₆	—	4	2	no	—
161	Dic₁₄₆	—	8	4	no	—
162	Dic₁₄₆	—	8	2	no	—
163	C₈×C₇₃	—	2	1	no	—
164	C₈×C₇₃	—	2	1	no	—
165	C₈×C₇₃	—	2	1	no	—
166	C₈×C₇₃	—	4	1	no	—
167	C₈×C₇₃	—	4	1	no	—
168	C₈×C₇₃	—	4	1	no	—
169	C₈×C₇₃	#137	1	2	yes	—
170	#10596831	—	4	2	no	—
171	C₂²×C₇₃	—	4	1	no	—
172	C₂²×C₇₃	—	4	1	no	—
173	C₂²×C₇₃	—	4	1	no	—
174	C₂²×C₇₃	D₄²	1	1	yes	characteristic
175	C₂×D₇₃	—	4	2	no	—
176	C₂×D₇₃	—	4	2	no	—
177	C₂×D₇₃	—	4	2	no	—
178	C₄×C₇₃	—	2	2	no	—
179	C₄×C₇₃	—	2	1	no	—
180	C₄×C₇₃	—	4	1	no	—
181	C₄×C₇₃	—	4	1	no	—
182	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
183	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
184	C₄×C₇₃	#576	1	1	yes	characteristic
185	C₄×C₇₃	#578	1	1	yes	characteristic
186	Dic₇₃	—	4	2	no	—
187	Dic₇₃	—	8	2	no	—
188	Dic₇₃	—	8	2	no	—
189	#19905	—	73	1	no	maximal, Sylow
190	C₂×C₇₃	—	8	1	no	—
191	C₂×C₇₃	#3276	1	1	yes	characteristic
192	C₂×C₇₃	#3280	1	1	yes	characteristic
193	C₂×C₇₃	#3290	1	1	yes	characteristic
194	D₇₃	—	8	2	no	—
195	Q₈×D₈	—	73	1	no	—
196	#3341	—	73	1	no	—
197	#3329	—	73	2	no	—
198	#1710	—	73	1	no	—
199	#1703	—	73	2	no	—
200	#1676	—	73	1	no	—
201	#1673	—	73	1	no	—
202	#1646	—	73	2	no	—
203	#1637	—	73	2	no	—
204	#1589	—	73	1	no	—
205	C₈×SD₁₆	D₇₃	1	1	yes	characteristic, complemented
206	C₇₃	#19905	1	1	yes	characteristic, complemented, Sylow
207	#555	—	73	2	no	—
208	D₄×Q₈	—	73	2	no	—
209	#500	—	73	1	no	—
210	#500	—	73	1	no	—
211	#500	—	146	1	no	—
212	#487	—	73	2	no	—
213	#487	—	73	2	no	—
214	#485	—	146	2	no	—
215	#482	—	146	2	no	—
216	#477	—	73	2	no	—
217	#476	—	73	2	no	—
218	#474	—	73	2	no	—
219	#474	—	146	2	no	—
220	#468	—	146	2	no	—
221	#466	—	146	1	no	—
222	#461	—	73	2	no	—
223	#448	—	146	2	no	—
224	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
225	C₄×SD₁₆	C₂×D₇₃	1	1	yes	characteristic
226	C₄×D₈	—	73	1	no	—
227	C₄×D₈	—	146	1	no	—
228	#436	—	73	2	no	—
229	C₈×D₄	C₂×D₇₃	1	1	yes	characteristic
230	#427	—	146	1	no	—
231	#419	—	146	2	no	—
232	#335	—	73	1	no	—
233	#334	C₂×D₇₃	1	1	yes	characteristic
234	#333	—	73	2	no	—
235	#332	—	73	2	no	—
236	Q₈⋊C₈	C₂×D₇₃	1	1	yes	characteristic
237	D₄⋊C₈	—	73	2	no	—
238	D₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
239	C₈²	C₂×D₇₃	1	1	yes	characteristic
240	C₂²×Q₈	—	146	2	no	—
241	C₂×SD₁₆	—	146	2	no	—
242	C₂×SD₁₆	D₂₉₂	1	1	yes	characteristic
243	C₂×D₈	—	73	1	no	—
244	C₂×M₁₆	—	146	2	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₄⋊Q₈	—	73	2	no	—
247	C₄⋊Q₈	—	73	2	no	—
248	C₄⋊Q₈	—	146	2	no	—
249	#124	—	292	2	no	—
250	C₂²⋊Q₈	—	292	2	no	—
251	C₂²⋊Q₈	—	146	2	no	—
252	#121	—	146	2	no	—
253	#121	—	146	2	no	—
254	C₄×Q₈	C₂²×D₇₃	1	1	yes	characteristic
255	C₄×D₄	—	73	2	no	—
256	C₄×D₄	—	146	2	no	—
257	C₄×D₄	C₂²×D₇₃	1	1	yes	characteristic
258	#117	—	146	2	no	—
259	#117	—	146	2	no	—
260	#107	—	73	1	no	—
261	#107	—	146	2	no	—
262	#107	—	146	1	no	—
263	#107	—	146	1	no	—
264	#107	—	146	1	no	—
265	#107	—	146	1	no	—
266	#107	—	146	2	no	—
267	#106	—	146	2	no	—
268	#106	D₂₉₂	1	1	yes	characteristic
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	73	2	no	—
271	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
272	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
273	#103	—	292	2	no	—
274	#103	—	146	2	no	—
275	#103	—	146	2	no	—
276	#103	—	146	2	no	—
277	#103	—	146	2	no	—
278	#103	D₂₉₂	1	1	yes	characteristic
279	#102	—	73	2	no	—
280	#102	—	146	2	no	—
281	#102	—	146	2	no	—
282	#102	D₂₉₂	1	1	yes	characteristic
283	C₂²⋊C₈	—	2	1	no	—
284	C₂²⋊C₈	—	146	2	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
289	C₂×Q₈	—	292	2	no	—
290	C₂×Q₈	—	292	4	no	—
291	C₂×Q₈	—	584	2	no	—
292	C₂×Q₈	—	146	2	no	—
293	C₂×Q₈	—	146	2	no	—
294	C₂×Q₈	C₂×D₂₉₂	1	1	yes	characteristic
295	C₂×D₄	—	292	2	no	—
296	C₂×D₄	—	73	2	no	—
297	C₂×D₄	C₂×D₂₉₂	1	1	yes	characteristic
298	C₂²×C₄	—	2	1	no	—
299	C₂²×C₄	—	292	2	no	—
300	C₂²×C₄	—	146	2	no	—
301	SD₁₆	—	2	2	no	—
302	SD₁₆	—	584	2	no	—
303	D₈	—	146	2	no	—
304	M₁₆	—	584	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	4	2	no	—
309	C₂×C₈	—	146	2	no	—
310	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
311	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
312	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
313	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	292	2	no	—
316	C₄⋊C₄	—	73	2	no	—
317	C₄⋊C₄	—	146	2	no	—
318	C₄⋊C₄	—	146	2	no	—
319	C₄⋊C₄	—	146	2	no	—
320	C₄⋊C₄	—	146	2	no	—
321	C₄⋊C₄	C₂×D₂₉₂	1	1	yes	characteristic
322	C₄⋊C₄	C₂×D₂₉₂	1	1	yes	characteristic
323	C₂²⋊C₄	—	2	1	no	—
324	C₂²⋊C₄	—	292	2	no	—
325	C₂²⋊C₄	—	292	2	no	—
326	C₂²⋊C₄	—	292	2	no	—
327	C₂²⋊C₄	—	146	2	no	—
328	C₄²	—	2	1	no	—
329	C₄²	—	292	2	no	—
330	C₄²	C₂³×D₇₃	1	1	yes	characteristic
331	C₂³	—	2	1	no	—
332	C₂³	—	146	2	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	292	2	no	—
335	Q₈	—	292	2	no	—
336	Q₈	—	584	4	no	—
337	Q₈	—	584	2	no	—
338	Q₈	#14556613	1	2	yes	—
339	D₄	—	2	1	no	—
340	D₄	—	2	1	no	—
341	D₄	—	1168	2	no	—
342	D₄	—	146	2	no	—
343	D₄	—	146	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	292	2	no	—
350	C₂×C₄	—	292	2	no	—
351	C₂×C₄	—	292	2	no	—
352	C₂×C₄	—	292	2	no	—
353	C₂×C₄	—	584	4	no	—
354	C₂×C₄	—	146	2	no	—
355	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
357	C₂×C₄	#14556637	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	4	1	no	—
362	C₈	—	4	1	no	—
363	C₈	—	4	1	no	—
364	C₈	—	292	2	no	—
365	C₈	#14556620	1	2	yes	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	292	2	no	—
370	C₂²	—	292	2	no	—
371	C₂²	—	292	2	no	—
372	C₂²	D₄×D₂₉₂	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	2	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	1	no	—
377	C₄	—	292	2	no	—
378	C₄	—	584	2	no	—
379	C₄	—	584	2	no	—
380	C₄	C₂×#14556613	1	1	yes	characteristic
381	C₄	C₂×#14556620	1	1	yes	characteristic
382	C₄	#17989392	1	1	yes	characteristic
383	C₄	#17989401	1	1	yes	characteristic
384	C₂	—	8	1	no	—
385	C₂	—	584	2	no	—
386	C₂	#22621578	1	1	yes	characteristic
387	C₂	#22621618	1	1	yes	characteristic
388	C₂	#22621718	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₉₂
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₈×SD₁₆×C₇₃	C₂
socle	C₂²×C₇₃	D₄²
prime radical	#22622354	C₂
hypercenter	C₈×SD₁₆	D₇₃
hypocenter	C₇₃	#19905
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆

Complements

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

N	H ≅ G/N	type
1. Q₈×D₅₈₄	C₂	semidirect
3. #22622227	C₂	semidirect
4. #22609541	C₂	semidirect
5. #22609507	C₂	semidirect
6. #22609335	C₂	semidirect
7. #22609320	C₂	semidirect
8. #22609161	C₂	semidirect
9. #22609098	C₂	semidirect
10. #22608817	C₂	semidirect
11. C₈×SD₁₆×C₇₃	C₂	semidirect
205. C₈×SD₁₆	D₇₃	semidirect
206. C₇₃	#19905	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₅₈₄	2
#22622354	2
#22622227	2
#22609541	2
#22609507	2
#22609335	2
#22609320	2
#22609161	2
#22609098	2
#22608817	2
C₈×SD₁₆×C₇₃	2
#19905	73