Subgroups — #33242709

Order 18688.

664 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₇₃	C₂	1	2	yes	maximal, complemented
2	Q₁₆×D₂₉₂	C₂	1	1	yes	maximal, characteristic
3	#22621681	C₂	1	1	yes	maximal, characteristic
4	#22612183	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612021	C₂	1	2	yes	maximal, complemented
6	#22611982	C₂	1	1	yes	maximal, characteristic, complemented
7	#22611866	C₂	1	2	yes	maximal, complemented
8	#22611863	C₂	1	2	yes	maximal, complemented
9	C₄×#14556537	C₂	1	1	yes	maximal, characteristic, complemented
10	#22611750	C₂	1	1	yes	maximal, characteristic
11	C₇₃×#2209	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₃	C₂²	1	2	yes	—
13	C₂×Q₁₆×D₇₃	C₂²	1	2	yes	—
14	Q₈×D₂₉₂	—	2	1	no	—
15	#17989252	—	2	1	no	—
16	C₂×Dic₅₈₄	C₂²	1	2	yes	—
17	C₂×#14556537	C₂²	1	2	yes	—
18	C₂×#14556537	C₂²	1	2	yes	—
19	C₂×#14556537	C₂²	1	1	yes	characteristic
20	Q₃₂×D₇₃	—	2	8	no	—
21	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
22	#17988967	C₂²	1	1	yes	characteristic
23	#17988829	C₂²	1	2	yes	—
24	#17988825	C₂²	1	2	yes	—
25	#17988797	—	2	1	no	—
26	#17988794	—	2	1	no	—
27	#17988744	—	2	2	no	—
28	#17988741	—	2	2	no	—
29	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
30	C₈×D₂₉₂	C₂²	1	1	yes	characteristic
31	#17988412	C₂²	1	1	yes	characteristic
32	#17988392	C₂²	1	2	yes	—
33	#17988391	C₂²	1	1	yes	characteristic
34	#17988390	C₂²	1	1	yes	characteristic
35	#17988366	C₂²	1	2	yes	—
36	C₄×#11924003	C₂²	1	1	yes	characteristic
37	C₂×Q₃₂×C₇₃	C₂²	1	2	yes	—
38	#494×C₇₃	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
40	C₄⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
41	#358×C₇₃	C₂²	1	2	yes	—
42	C₂×Q₈×D₇₃	—	2	2	no	—
43	C₂×Q₈×D₇₃	—	2	2	no	—
44	C₂×Dic₂₉₂	—	2	1	no	—
45	C₂×Dic₂₉₂	C₂³	1	2	yes	—
46	C₂×Dic₂₉₂	C₂³	1	1	yes	characteristic
47	C₂×#11924018	—	2	2	no	—
48	C₂×#11924018	—	2	2	no	—
49	Q₁₆×D₇₃	—	2	4	no	—
50	Q₁₆×D₇₃	—	2	4	no	—
51	Q₁₆×D₇₃	—	4	2	no	—
52	Q₁₆×D₇₃	—	4	2	no	—
53	C₂×C₈×D₇₃	C₂³	1	2	yes	—
54	#14556591	—	2	1	no	—
55	#14556591	—	4	1	no	—
56	#14556590	—	4	1	no	—
57	#14556571	—	2	2	no	—
58	#14556571	—	4	2	no	—
59	#14556569	—	4	2	no	—
60	#14556567	—	2	2	no	—
61	C₄×Dic₁₄₆	—	2	1	no	—
62	C₄×D₂₉₂	—	4	1	no	—
63	C₄×D₂₉₂	D₄	1	1	yes	characteristic
64	#14556553	—	4	1	no	—
65	Dic₅₈₄	—	2	4	no	—
66	#14556537	—	2	4	no	—
67	#14556537	—	4	2	no	—
68	#14556537	D₄	1	4	yes	complemented
69	C₂×#11924003	C₂³	1	1	yes	characteristic
70	C₂×#11924003	C₂³	1	1	yes	characteristic
71	C₁₆×D₇₃	—	2	4	no	—
72	#14556525	C₂³	1	1	yes	characteristic
73	C₄⋊#10596831	D₄	1	1	yes	characteristic
74	#14556516	—	2	1	no	—
75	#14556515	—	2	2	no	—
76	#14556514	—	2	2	no	—
77	#14556503	D₄	1	2	yes	—
78	C₂×Q₁₆×C₇₃	—	2	1	no	—
79	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
80	C₂×Q₁₆×C₇₃	C₂³	1	1	yes	characteristic
81	C₄⋊Q₈×C₇₃	—	2	1	no	—
82	C₄×Q₈×C₇₃	—	2	1	no	—
83	Q₃₂×C₇₃	—	2	4	no	—
84	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
85	#107×C₇₃	C₂³	1	1	yes	characteristic
86	#103×C₇₃	—	2	1	no	—
87	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
88	C₂×Dic₁₄₆	—	2	1	no	—
89	C₂×Dic₁₄₆	—	2	2	no	—
90	C₂×Dic₁₄₆	—	4	2	no	—
91	C₂×Dic₁₄₆	—	4	2	no	—
92	Q₈×D₇₃	—	4	4	no	—
93	Q₈×D₇₃	—	4	4	no	—
94	Q₈×D₇₃	—	8	2	no	—
95	Q₈×D₇₃	—	8	2	no	—
96	C₂×D₂₉₂	D₈	1	1	yes	characteristic
97	C₂×C₄×D₇₃	—	4	2	no	—
98	C₂×C₄×D₇₃	—	4	2	no	—
99	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
100	Dic₂₉₂	—	2	2	no	—
101	Dic₂₉₂	—	2	1	no	—
102	Dic₂₉₂	—	2	2	no	—
103	Dic₂₉₂	—	8	1	no	—
104	Dic₂₉₂	C₂×D₄	1	2	yes	—
105	#11924018	—	4	4	no	—
106	#11924018	—	4	4	no	—
107	C₂×#10596831	C₂×D₄	1	2	yes	—
108	C₈×D₇₃	—	2	2	no	—
109	C₈×D₇₃	—	2	2	no	—
110	C₄⋊Dic₇₃	—	2	1	no	—
111	C₄⋊Dic₇₃	—	4	1	no	—
112	C₄⋊Dic₇₃	—	4	1	no	—
113	C₄⋊Dic₇₃	—	4	2	no	—
114	C₄⋊Dic₇₃	D₈	1	1	yes	characteristic
115	#11924007	—	4	1	no	—
116	#11924007	—	4	2	no	—
117	#11924005	—	4	2	no	—
118	#11924005	—	4	2	no	—
119	#11924005	D₈	1	2	yes	—
120	C₄×Dic₇₃	—	4	1	no	—
121	#11924003	—	2	1	no	—
122	#11924003	C₂×D₄	1	2	yes	—
123	C₂×Q₈×C₇₃	—	2	1	no	—
124	C₂×Q₈×C₇₃	—	2	2	no	—
125	Q₁₆×C₇₃	—	2	2	no	—
126	Q₁₆×C₇₃	—	2	1	no	—
127	Q₁₆×C₇₃	—	2	2	no	—
128	Q₁₆×C₇₃	—	8	1	no	—
129	Q₁₆×C₇₃	C₂×D₄	1	2	yes	—
130	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
132	C₄⋊C₄×C₇₃	—	2	1	no	—
133	C₄⋊C₄×C₇₃	—	4	1	no	—
134	C₄⋊C₄×C₇₃	—	4	1	no	—
135	C₄⋊C₄×C₇₃	—	4	1	no	—
136	C₄²×C₇₃	—	4	1	no	—
137	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
138	C₁₆×C₇₃	—	2	2	no	—
139	C₂²×D₇₃	C₂×D₈	1	2	yes	—
140	Q₈×C₇₃	—	2	2	no	—
141	Q₈×C₇₃	—	4	1	no	—
142	Q₈×C₇₃	—	4	2	no	—
143	Q₈×C₇₃	—	4	2	no	—
144	C₂×C₄×C₇₃	—	4	1	no	—
145	C₂×C₄×C₇₃	—	4	1	no	—
146	C₂×C₄×C₇₃	—	4	2	no	—
147	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₇₃	—	4	1	no	—
151	C₂×Dic₇₃	—	4	1	no	—
152	C₂×Dic₇₃	—	4	2	no	—
153	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
154	D₂₉₂	Q₃₂	1	4	yes	complemented
155	C₄×D₇₃	—	2	2	no	—
156	C₄×D₇₃	—	2	2	no	—
157	C₄×D₇₃	—	8	4	no	—
158	C₄×D₇₃	—	8	4	no	—
159	Dic₁₄₆	—	2	2	no	—
160	Dic₁₄₆	—	4	1	no	—
161	Dic₁₄₆	—	4	2	no	—
162	Dic₁₄₆	—	4	2	no	—
163	Dic₁₄₆	—	8	4	no	—
164	Dic₁₄₆	—	8	4	no	—
165	C₈×C₇₃	—	2	1	no	—
166	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
167	C₈×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
168	#10596831	—	2	2	no	—
169	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
170	C₂×D₇₃	—	2	2	no	—
171	C₂×D₇₃	—	2	2	no	—
172	C₂×D₇₃	C₂×Q₃₂	1	4	yes	complemented
173	C₄×C₇₃	—	2	1	no	—
174	C₄×C₇₃	—	4	2	no	—
175	C₄×C₇₃	—	8	1	no	—
176	C₄×C₇₃	—	8	2	no	—
177	C₄×C₇₃	C₂×Q₃₂	1	2	yes	complemented
178	C₄×C₇₃	D₄²	1	1	yes	characteristic
179	C₄×C₇₃	#576	1	1	yes	characteristic
180	Dic₇₃	—	2	2	no	—
181	Dic₇₃	—	4	2	no	—
182	Dic₇₃	—	8	1	no	—
183	Dic₇₃	—	8	2	no	—
184	D₄×Q₃₂	—	73	1	no	maximal, Sylow
185	C₂×C₇₃	D₄×D₈	1	1	yes	characteristic
186	C₂×C₇₃	C₂²×Q₃₂	1	1	yes	characteristic
187	C₂×C₇₃	#3417	1	1	yes	characteristic
188	D₇₃	—	2	4	no	—
189	C₂²×Q₃₂	—	73	2	no	—
190	D₄×Q₁₆	—	73	1	no	—
191	D₄×Q₁₆	—	73	1	no	—
192	#2247	—	73	1	no	—
193	#2216	—	73	2	no	—
194	#2209	—	73	1	no	—
195	#2209	D₇₃	1	1	yes	characteristic, complemented
196	#2188	—	73	2	no	—
197	#2188	—	73	2	no	—
198	C₄×Q₃₂	—	73	1	no	—
199	D₄×C₁₆	—	73	1	no	—
200	C₇₃	D₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
201	C₂²×Q₁₆	—	73	2	no	—
202	C₂²×Q₁₆	—	73	2	no	—
203	D₄×Q₈	—	146	1	no	—
204	D₄×Q₈	—	146	1	no	—
205	C₂×Q₃₂	—	73	2	no	—
206	C₂×Q₃₂	—	73	1	no	—
207	C₂×Q₃₂	—	73	2	no	—
208	C₂×Q₃₂	—	73	2	no	—
209	C₂×Q₃₂	—	146	8	no	—
210	C₂×Q₃₂	C₂×D₇₃	1	2	yes	—
211	C₂²×C₁₆	—	73	2	no	—
212	#494	—	73	1	no	—
213	#494	C₂×D₇₃	1	1	yes	characteristic
214	#467	—	73	2	no	—
215	#467	—	73	2	no	—
216	#462	—	146	1	no	—
217	#462	—	146	1	no	—
218	#451	—	146	2	no	—
219	#451	—	146	2	no	—
220	C₄×Q₁₆	—	73	1	no	—
221	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
222	C₈×D₄	—	73	1	no	—
223	#366	—	73	1	no	—
224	C₄⋊C₁₆	C₂×D₇₃	1	1	yes	characteristic
225	#358	—	73	1	no	—
226	#358	—	73	2	no	—
227	#358	—	73	1	no	—
228	#358	C₂×D₇₃	1	2	yes	—
229	C₂²⋊C₁₆	—	73	2	no	—
230	C₄×C₁₆	—	73	1	no	—
231	C₂²×Q₈	—	146	2	no	—
232	C₂²×Q₈	—	146	2	no	—
233	C₂×Q₁₆	—	2	1	no	—
234	C₂×Q₁₆	—	292	2	no	—
235	C₂×Q₁₆	—	292	2	no	—
236	C₂×Q₁₆	—	73	1	no	—
237	C₂×Q₁₆	—	73	2	no	—
238	C₂×Q₁₆	—	146	2	no	—
239	C₂×Q₁₆	—	146	2	no	—
240	C₂×Q₁₆	—	146	4	no	—
241	C₂×Q₁₆	—	146	4	no	—
242	C₂×Q₁₆	—	146	1	no	—
243	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
244	C₂×Q₁₆	C₂²×D₇₃	1	1	yes	characteristic
245	C₂²×C₈	—	73	2	no	—
246	C₄⋊Q₈	—	2	1	no	—
247	C₄⋊Q₈	—	292	1	no	—
248	C₄⋊Q₈	—	292	1	no	—
249	C₄⋊Q₈	—	146	1	no	—
250	C₂²⋊Q₈	—	292	2	no	—
251	C₂²⋊Q₈	—	292	2	no	—
252	C₂²⋊Q₈	—	146	2	no	—
253	C₂²⋊Q₈	—	146	2	no	—
254	C₄×Q₈	—	2	1	no	—
255	C₄×Q₈	—	146	1	no	—
256	C₄×D₄	—	292	1	no	—
257	C₄×D₄	—	292	1	no	—
258	C₄×D₄	—	73	1	no	—
259	Q₃₂	—	2	4	no	—
260	Q₃₂	—	292	2	no	—
261	Q₃₂	—	73	4	no	—
262	Q₃₂	—	146	4	no	—
263	Q₃₂	—	146	4	no	—
264	C₂×C₁₆	—	73	1	no	—
265	C₂×C₁₆	—	73	1	no	—
266	C₂×C₁₆	—	146	4	no	—
267	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
268	#107	—	73	1	no	—
269	#107	C₂²×D₇₃	1	1	yes	characteristic
270	C₄⋊C₈	—	73	1	no	—
271	#103	—	2	1	no	—
272	#103	—	146	2	no	—
273	#103	—	146	2	no	—
274	#103	—	146	1	no	—
275	C₂²⋊C₈	—	73	2	no	—
276	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	—	292	2	no	—
280	C₂×Q₈	—	292	2	no	—
281	C₂×Q₈	—	292	4	no	—
282	C₂×Q₈	—	292	4	no	—
283	C₂×Q₈	—	584	2	no	—
284	C₂×Q₈	—	584	2	no	—
285	C₂×Q₈	—	146	2	no	—
286	C₂×Q₈	—	146	1	no	—
287	C₂×D₄	—	73	1	no	—
288	C₂²×C₄	—	292	2	no	—
289	C₂²×C₄	—	292	2	no	—
290	C₂²×C₄	—	73	2	no	—
291	Q₁₆	—	2	2	no	—
292	Q₁₆	—	2	1	no	—
293	Q₁₆	—	2	2	no	—
294	Q₁₆	—	8	1	no	—
295	Q₁₆	—	292	4	no	—
296	Q₁₆	—	292	4	no	—
297	Q₁₆	—	584	1	no	—
298	Q₁₆	—	73	2	no	—
299	Q₁₆	—	146	2	no	—
300	Q₁₆	—	146	2	no	—
301	Q₁₆	—	146	1	no	—
302	Q₁₆	D₄×D₇₃	1	2	yes	—
303	C₂×C₈	—	73	2	no	—
304	C₂×C₈	—	146	2	no	—
305	C₂×C₈	—	146	2	no	—
306	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
307	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	292	2	no	—
313	C₄⋊C₄	—	292	2	no	—
314	C₄⋊C₄	—	292	1	no	—
315	C₄⋊C₄	—	292	1	no	—
316	C₄⋊C₄	—	292	1	no	—
317	C₄⋊C₄	—	73	1	no	—
318	C₄⋊C₄	—	146	1	no	—
319	C₂²⋊C₄	—	292	2	no	—
320	C₂²⋊C₄	—	292	2	no	—
321	C₂²⋊C₄	—	73	2	no	—
322	C₄²	—	4	1	no	—
323	C₄²	—	292	1	no	—
324	C₄²	D₄×D₇₃	1	1	yes	characteristic
325	C₁₆	—	2	2	no	—
326	C₁₆	—	73	2	no	—
327	C₁₆	—	146	1	no	—
328	C₂³	—	73	2	no	—
329	Q₈	—	2	2	no	—
330	Q₈	—	4	1	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	4	2	no	—
333	Q₈	—	292	1	no	—
334	Q₈	—	292	2	no	—
335	Q₈	—	292	2	no	—
336	Q₈	—	584	4	no	—
337	Q₈	—	584	4	no	—
338	Q₈	—	146	2	no	—
339	D₄	—	73	4	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	292	1	no	—
344	C₂×C₄	—	292	1	no	—
345	C₂×C₄	—	292	2	no	—
346	C₂×C₄	—	584	4	no	—
347	C₂×C₄	—	584	4	no	—
348	C₂×C₄	—	73	2	no	—
349	C₂×C₄	—	146	2	no	—
350	C₂×C₄	—	146	2	no	—
351	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
352	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
354	C₈	—	2	1	no	—
355	C₈	—	146	2	no	—
356	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
357	C₈	#14556637	1	1	yes	characteristic
358	C₂²	—	73	4	no	—
359	C₂²	—	146	2	no	—
360	C₂²	—	146	2	no	—
361	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	4	2	no	—
364	C₄	—	8	1	no	—
365	C₄	—	8	2	no	—
366	C₄	—	292	2	no	—
367	C₄	—	584	1	no	—
368	C₄	—	584	2	no	—
369	C₄	—	146	2	no	—
370	C₄	Q₃₂×D₇₃	1	2	yes	complemented
371	C₄	#17989221	1	1	yes	characteristic
372	C₄	#17989391	1	1	yes	characteristic
373	C₂	—	146	4	no	—
374	C₂	#22621603	1	1	yes	characteristic
375	C₂	C₂×Q₃₂×D₇₃	1	1	yes	characteristic
376	C₂	#22623083	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×#2209	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×D₂₉₂	D₈
hypercenter	#2209	D₇₃
hypocenter	C₇₃	D₄×Q₃₂
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2209

Complements

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

N	H ≅ G/N	type
1. C₂×Q₃₂×D₇₃	C₂	semidirect
4. #22612183	C₂	semidirect
5. #22612021	C₂	semidirect
6. #22611982	C₂	semidirect
7. #22611866	C₂	semidirect
8. #22611863	C₂	semidirect
9. C₄×#14556537	C₂	semidirect
11. C₇₃×#2209	C₂	semidirect
68. #14556537	D₄	semidirect
154. D₂₉₂	Q₃₂	semidirect
172. C₂×D₇₃	C₂×Q₃₂	semidirect
177. C₄×C₇₃	C₂×Q₃₂	semidirect
195. #2209	D₇₃	semidirect
200. C₇₃	D₄×Q₃₂	semidirect
370. C₄	Q₃₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₇₃	2
Q₁₆×D₂₉₂	2
#22621681	2
#22612183	2
#22612021	2
#22611982	2
#22611866	2
#22611863	2
C₄×#14556537	2
#22611750	2
C₇₃×#2209	2
D₄×Q₃₂	73