Subgroups — #33242711

Order 18688.

664 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₈₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×Q₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#22621680	C₂	1	2	yes	maximal
4	#22612182	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612023	C₂	1	1	yes	maximal, characteristic, complemented
6	#22611980	C₂	1	2	yes	maximal, complemented
7	#22611866	C₂	1	2	yes	maximal, complemented
8	#22611864	C₂	1	2	yes	maximal, complemented
9	#22611789	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₆×C₂²⋊D₇₃	C₂	1	1	yes	maximal, characteristic
11	C₇₃×#2216	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₂₉₂	C₂²	1	2	yes	—
13	C₂×Q₁₆×D₇₃	C₂²	1	2	yes	—
14	#17989250	—	2	2	no	—
15	C₂×Dic₅₈₄	—	2	4	no	—
16	C₂×Dic₅₈₄	C₂²	1	2	yes	—
17	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic
18	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic
19	C₂×#14556537	C₂²	1	2	yes	—
20	C₂×#14556537	C₂²	1	2	yes	—
21	Q₃₂×D₇₃	—	2	4	no	—
22	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
23	#17988965	C₂²	1	2	yes	—
24	#17988829	C₂²	1	2	yes	—
25	#17988796	—	2	2	no	—
26	#17988744	—	2	2	no	—
27	#17988742	—	2	2	no	—
28	#17988681	C₂²	1	2	yes	—
29	C₈×C₂²⋊D₇₃	C₂²	1	1	yes	characteristic
30	#17988405	C₂²	1	1	yes	characteristic
31	#17988390	C₂²	1	2	yes	—
32	#17988390	C₂²	1	2	yes	—
33	C₂²⋊#11924003	C₂²	1	1	yes	characteristic
34	#17988366	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
36	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
37	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
38	#467×C₇₃	C₂²	1	2	yes	—
39	#366×C₇₃	C₂²	1	1	yes	characteristic
40	#358×C₇₃	C₂²	1	2	yes	—
41	C₂²×Dic₁₄₆	—	2	2	no	—
42	C₂×Q₈×D₇₃	—	2	2	no	—
43	C₂×Dic₂₉₂	—	2	4	no	—
44	C₂×Dic₂₉₂	—	2	2	no	—
45	C₂×Dic₂₉₂	—	4	2	no	—
46	C₂×Dic₂₉₂	C₂³	1	2	yes	—
47	C₂×Dic₂₉₂	C₂³	1	2	yes	—
48	C₂×#11924018	—	2	2	no	—
49	C₂×#11924018	—	2	2	no	—
50	Q₁₆×D₇₃	—	2	4	no	—
51	Q₁₆×D₇₃	—	4	2	no	—
52	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
53	#14556590	—	4	2	no	—
54	#14556589	—	2	2	no	—
55	#14556571	—	2	2	no	—
56	#14556569	—	4	2	no	—
57	#14556568	—	4	2	no	—
58	#14556556	—	2	2	no	—
59	C₄×C₂²⋊D₇₃	D₄	1	1	yes	characteristic
60	#14556551	—	4	2	no	—
61	Dic₅₈₄	—	2	4	no	—
62	Dic₅₈₄	—	2	2	no	—
63	Dic₅₈₄	D₄	1	4	yes	complemented
64	#14556537	—	2	4	no	—
65	#14556537	—	4	2	no	—
66	C₂×#11924003	C₂³	1	1	yes	characteristic
67	C₂×#11924003	C₂³	1	1	yes	characteristic
68	C₁₆×D₇₃	—	2	2	no	—
69	#14556520	D₄	1	1	yes	characteristic
70	#14556514	—	2	2	no	—
71	#14556514	—	2	2	no	—
72	C₂²⋊#10596831	D₄	1	1	yes	characteristic
73	#14556503	D₄	1	1	yes	characteristic
74	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
75	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
76	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
77	C₂²⋊Q₈×C₇₃	—	2	2	no	—
78	Q₃₂×C₇₃	—	2	2	no	—
79	C₂×C₁₆×C₇₃	—	2	2	no	—
80	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
81	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
82	#107×C₇₃	C₂³	1	2	yes	—
83	#103×C₇₃	—	2	2	no	—
84	C₂×Dic₁₄₆	—	2	2	no	—
85	C₂×Dic₁₄₆	—	2	2	no	—
86	C₂×Dic₁₄₆	—	4	4	no	—
87	C₂×Dic₁₄₆	—	4	2	no	—
88	C₂×Dic₁₄₆	—	4	2	no	—
89	C₂×Dic₁₄₆	—	8	2	no	—
90	Q₈×D₇₃	—	4	4	no	—
91	Q₈×D₇₃	—	8	2	no	—
92	C₂×C₂²⋊D₇₃	D₈	1	1	yes	characteristic
93	C₂²×Dic₇₃	—	4	2	no	—
94	C₂×C₄×D₇₃	—	4	2	no	—
95	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
96	Dic₂₉₂	—	2	2	no	—
97	Dic₂₉₂	—	2	2	no	—
98	Dic₂₉₂	—	2	2	no	—
99	Dic₂₉₂	—	4	4	no	—
100	Dic₂₉₂	C₂×D₄	1	4	yes	—
101	#11924018	—	4	4	no	—
102	#11924018	—	8	2	no	—
103	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
104	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
105	C₈×D₇₃	—	2	1	no	—
106	C₈×D₇₃	—	2	1	no	—
107	C₄⋊Dic₇₃	—	4	2	no	—
108	#11924007	—	4	2	no	—
109	#11924007	—	4	2	no	—
110	#11924007	—	4	2	no	—
111	#11924007	D₈	1	1	yes	characteristic
112	C₂²⋊Dic₇₃	D₈	1	1	yes	characteristic
113	#11924005	—	4	2	no	—
114	#11924005	D₈	1	1	yes	characteristic
115	C₄×Dic₇₃	—	4	2	no	—
116	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
117	#11924003	—	2	1	no	—
118	#11924003	—	2	1	no	—
119	C₂×Q₈×C₇₃	—	2	2	no	—
120	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
121	Q₁₆×C₇₃	—	2	2	no	—
122	Q₁₆×C₇₃	—	2	2	no	—
123	C₂×C₈×C₇₃	—	2	1	no	—
124	C₂×C₈×C₇₃	—	2	1	no	—
125	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄×C₇₃	—	2	2	no	—
128	C₄⋊C₄×C₇₃	—	4	2	no	—
129	C₂²⋊C₄×C₇₃	—	4	2	no	—
130	C₁₆×C₇₃	—	2	1	no	—
131	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
132	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
133	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
134	Q₈×C₇₃	—	4	2	no	—
135	Q₈×C₇₃	—	4	2	no	—
136	C₂×C₄×C₇₃	—	2	1	no	—
137	C₂×C₄×C₇₃	—	2	1	no	—
138	C₂×C₄×C₇₃	—	4	2	no	—
139	C₂×C₄×C₇₃	—	4	2	no	—
140	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
142	C₂²⋊D₇₃	Q₃₂	1	4	yes	complemented
143	C₂×Dic₇₃	—	4	2	no	—
144	C₂×Dic₇₃	—	4	2	no	—
145	C₂×Dic₇₃	—	8	4	no	—
146	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
147	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
148	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
149	C₄×D₇₃	—	2	1	no	—
150	C₄×D₇₃	—	2	1	no	—
151	C₄×D₇₃	—	8	4	no	—
152	Dic₁₄₆	—	2	4	no	—
153	Dic₁₄₆	—	4	2	no	—
154	Dic₁₄₆	—	4	2	no	—
155	Dic₁₄₆	—	4	2	no	—
156	Dic₁₄₆	—	8	4	no	—
157	Dic₁₄₆	—	8	4	no	—
158	C₈×C₇₃	—	2	1	no	—
159	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
160	C₈×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
161	#10596831	—	2	1	no	—
162	#10596831	—	2	1	no	—
163	C₂²×C₇₃	—	2	1	no	—
164	C₂²×C₇₃	—	2	1	no	—
165	C₂²×C₇₃	C₂×Q₃₂	1	2	yes	complemented
166	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₇₃	—	2	1	no	—
168	C₂×D₇₃	—	2	1	no	—
169	C₂×D₇₃	C₂×Q₃₂	1	2	yes	complemented
170	C₄×C₇₃	—	2	1	no	—
171	C₄×C₇₃	—	8	2	no	—
172	C₄×C₇₃	—	8	2	no	—
173	C₄×C₇₃	D₄²	1	1	yes	characteristic
174	C₄×C₇₃	#576	1	1	yes	characteristic
175	Dic₇₃	—	2	1	no	—
176	Dic₇₃	—	2	1	no	—
177	Dic₇₃	—	4	4	no	—
178	Dic₇₃	—	8	2	no	—
179	Dic₇₃	C₂×Q₃₂	1	2	yes	complemented
180	D₄×Q₃₂	—	73	1	no	maximal, Sylow
181	C₂×C₇₃	—	2	2	no	—
182	C₂×C₇₃	D₄×D₈	1	1	yes	characteristic
183	C₂×C₇₃	C₂²×Q₃₂	1	1	yes	characteristic
184	C₂×C₇₃	#3417	1	1	yes	characteristic
185	D₇₃	—	2	2	no	—
186	C₂²×Q₃₂	—	73	1	no	—
187	C₂²×Q₃₂	—	73	1	no	—
188	D₄×Q₁₆	—	73	2	no	—
189	#2247	—	73	1	no	—
190	#2216	—	73	1	no	—
191	#2216	D₇₃	1	1	yes	characteristic, complemented
192	#2209	—	73	2	no	—
193	#2188	—	73	2	no	—
194	#2188	—	73	2	no	—
195	C₄×Q₃₂	—	73	1	no	—
196	D₄×C₁₆	—	73	1	no	—
197	C₇₃	D₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
198	C₂²×Q₁₆	—	73	2	no	—
199	C₂²×Q₁₆	—	73	2	no	—
200	D₄×Q₈	—	146	2	no	—
201	C₂×Q₃₂	—	73	2	no	—
202	C₂×Q₃₂	—	73	2	no	—
203	C₂×Q₃₂	—	73	1	no	—
204	C₂×Q₃₂	—	73	2	no	—
205	C₂×Q₃₂	—	73	1	no	—
206	C₂×Q₃₂	—	146	4	no	—
207	C₂×Q₃₂	—	146	4	no	—
208	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic
209	C₂²×C₁₆	—	73	1	no	—
210	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
211	#494	—	73	2	no	—
212	#467	—	73	2	no	—
213	#467	C₂×D₇₃	1	2	yes	—
214	#462	—	146	2	no	—
215	#451	—	146	2	no	—
216	#451	—	146	2	no	—
217	C₄×Q₁₆	—	73	2	no	—
218	C₈×D₄	—	73	1	no	—
219	#366	C₂×D₇₃	1	1	yes	characteristic
220	C₄⋊C₁₆	—	73	1	no	—
221	#358	—	73	2	no	—
222	#358	—	73	2	no	—
223	#358	C₂×D₇₃	1	2	yes	—
224	C₂²⋊C₁₆	—	73	1	no	—
225	C₂²⋊C₁₆	—	73	1	no	—
226	C₄×C₁₆	—	73	1	no	—
227	C₂²×Q₈	—	146	2	no	—
228	C₂²×Q₈	—	146	2	no	—
229	C₂×Q₁₆	—	292	2	no	—
230	C₂×Q₁₆	—	292	2	no	—
231	C₂×Q₁₆	—	73	2	no	—
232	C₂×Q₁₆	—	73	2	no	—
233	C₂×Q₁₆	—	146	4	no	—
234	C₂×Q₁₆	—	146	2	no	—
235	C₂×Q₁₆	—	146	2	no	—
236	C₂×Q₁₆	—	146	4	no	—
237	C₂×Q₁₆	—	146	2	no	—
238	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
239	C₂²×C₈	—	73	1	no	—
240	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
241	C₄⋊Q₈	—	292	2	no	—
242	C₄⋊Q₈	—	146	2	no	—
243	C₂²⋊Q₈	—	2	2	no	—
244	C₂²⋊Q₈	—	292	2	no	—
245	C₂²⋊Q₈	—	292	2	no	—
246	C₂²⋊Q₈	—	146	2	no	—
247	C₄×Q₈	—	146	2	no	—
248	C₄×D₄	—	292	2	no	—
249	C₄×D₄	—	73	1	no	—
250	Q₃₂	—	2	2	no	—
251	Q₃₂	—	292	2	no	—
252	Q₃₂	—	73	4	no	—
253	Q₃₂	—	146	4	no	—
254	Q₃₂	—	146	4	no	—
255	Q₃₂	—	146	2	no	—
256	C₂×C₁₆	—	2	2	no	—
257	C₂×C₁₆	—	73	1	no	—
258	C₂×C₁₆	—	73	1	no	—
259	C₂×C₁₆	—	146	2	no	—
260	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
261	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
262	#107	C₂²×D₇₃	1	2	yes	—
263	C₄⋊C₈	—	73	1	no	—
264	#103	—	2	2	no	—
265	#103	—	146	2	no	—
266	#103	—	146	2	no	—
267	C₂²⋊C₈	—	73	1	no	—
268	C₂²⋊C₈	—	73	1	no	—
269	C₄×C₈	—	73	1	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	292	4	no	—
272	C₂×Q₈	—	292	2	no	—
273	C₂×Q₈	—	292	2	no	—
274	C₂×Q₈	—	292	4	no	—
275	C₂×Q₈	—	584	2	no	—
276	C₂×Q₈	—	584	2	no	—
277	C₂×Q₈	—	146	2	no	—
278	C₂×Q₈	—	146	2	no	—
279	C₂×D₄	—	73	1	no	—
280	C₂²×C₄	—	292	2	no	—
281	C₂²×C₄	—	292	2	no	—
282	C₂²×C₄	—	73	1	no	—
283	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
284	Q₁₆	—	2	2	no	—
285	Q₁₆	—	2	2	no	—
286	Q₁₆	—	292	4	no	—
287	Q₁₆	—	292	4	no	—
288	Q₁₆	—	584	2	no	—
289	Q₁₆	—	73	4	no	—
290	Q₁₆	—	146	2	no	—
291	Q₁₆	—	146	2	no	—
292	Q₁₆	—	146	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	73	1	no	—
296	C₂×C₈	—	73	1	no	—
297	C₂×C₈	—	146	1	no	—
298	C₂×C₈	—	146	1	no	—
299	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
300	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	292	2	no	—
304	C₄⋊C₄	—	292	2	no	—
305	C₄⋊C₄	—	292	2	no	—
306	C₄⋊C₄	—	292	2	no	—
307	C₄⋊C₄	—	73	1	no	—
308	C₂²⋊C₄	—	4	2	no	—
309	C₂²⋊C₄	—	292	2	no	—
310	C₂²⋊C₄	—	73	1	no	—
311	C₂²⋊C₄	—	73	1	no	—
312	C₄²	—	292	2	no	—
313	C₄²	—	73	1	no	—
314	C₁₆	—	2	1	no	—
315	C₁₆	—	146	1	no	—
316	C₁₆	—	146	1	no	—
317	C₁₆	D₄×D₇₃	1	2	yes	—
318	C₂³	—	73	1	no	—
319	C₂³	D₈×D₇₃	1	1	yes	characteristic
320	Q₈	—	4	2	no	—
321	Q₈	—	4	2	no	—
322	Q₈	—	292	2	no	—
323	Q₈	—	292	2	no	—
324	Q₈	—	292	2	no	—
325	Q₈	—	584	4	no	—
326	Q₈	—	584	4	no	—
327	Q₈	—	146	4	no	—
328	D₄	—	73	4	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	292	2	no	—
334	C₂×C₄	—	292	2	no	—
335	C₂×C₄	—	584	4	no	—
336	C₂×C₄	—	584	4	no	—
337	C₂×C₄	—	73	1	no	—
338	C₂×C₄	—	73	1	no	—
339	C₂×C₄	—	73	1	no	—
340	C₂×C₄	—	146	1	no	—
341	C₂×C₄	—	146	1	no	—
342	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
344	C₈	—	2	1	no	—
345	C₈	—	146	1	no	—
346	C₈	—	146	1	no	—
347	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
348	C₈	#14556637	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	73	2	no	—
352	C₂²	—	146	1	no	—
353	C₂²	—	146	1	no	—
354	C₂²	Q₃₂×D₇₃	1	2	yes	complemented
355	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	8	2	no	—
358	C₄	—	8	2	no	—
359	C₄	—	292	4	no	—
360	C₄	—	584	2	no	—
361	C₄	—	73	2	no	—
362	C₄	—	146	1	no	—
363	C₄	—	146	1	no	—
364	C₄	#17989221	1	1	yes	characteristic
365	C₄	#17989389	1	1	yes	characteristic
366	C₂	—	2	2	no	—
367	C₂	—	146	2	no	—
368	C₂	#22621605	1	1	yes	characteristic
369	C₂	C₂×Q₃₂×D₇₃	1	1	yes	characteristic
370	C₂	#22623081	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2216	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×C₂²⋊D₇₃	D₈
hypercenter	#2216	D₇₃
hypocenter	C₇₃	D₄×Q₃₂
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₅₈₄	C₂	semidirect
2. C₂×Q₃₂×D₇₃	C₂	semidirect
4. #22612182	C₂	semidirect
5. #22612023	C₂	semidirect
6. #22611980	C₂	semidirect
7. #22611866	C₂	semidirect
8. #22611864	C₂	semidirect
9. #22611789	C₂	semidirect
11. C₇₃×#2216	C₂	semidirect
63. Dic₅₈₄	D₄	semidirect
142. C₂²⋊D₇₃	Q₃₂	semidirect
165. C₂²×C₇₃	C₂×Q₃₂	semidirect
169. C₂×D₇₃	C₂×Q₃₂	semidirect
179. Dic₇₃	C₂×Q₃₂	semidirect
191. #2216	D₇₃	semidirect
197. C₇₃	D₄×Q₃₂	semidirect
354. C₂²	Q₃₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₅₈₄	2
C₂×Q₃₂×D₇₃	2
#22621680	2
#22612182	2
#22612023	2
#22611980	2
#22611866	2
#22611864	2
#22611789	2
C₁₆×C₂²⋊D₇₃	2
C₇₃×#2216	2
D₄×Q₃₂	73