Subgroups — #33070004

Order 18688.

658 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22619724	C₂	1	2	yes	maximal
2	C₂×#17988666	C₂	1	1	yes	maximal, characteristic
3	#22609503	C₂	1	4	yes	maximal, complemented
4	#22609451	C₂	1	2	yes	maximal, complemented
5	#22609417	C₂	1	2	yes	maximal, complemented
6	#22608687	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608621	C₂	1	2	yes	maximal, complemented
8	C₇₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11924011	—	2	1	no	—
10	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
11	#17988884	—	2	4	no	—
12	#17988874	C₂²	1	4	yes	—
13	#17988864	C₂²	1	4	yes	—
14	#17988666	—	2	1	no	—
15	#17988666	—	2	1	no	—
16	#17988666	C₂²	1	2	yes	—
17	#17988666	C₂²	1	2	yes	—
18	#17988630	—	2	2	no	—
19	#17988611	C₂²	1	2	yes	—
20	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
21	#17988591	—	2	2	no	—
22	C₂×#14556503	—	2	1	no	—
23	#17988338	C₂²	1	2	yes	complemented
24	#17988338	C₂²	1	2	yes	complemented
25	#17988336	C₂²	1	4	yes	complemented
26	#17988333	C₂²	1	4	yes	complemented
27	#533×C₇₃	C₂²	1	2	yes	—
28	#500×C₇₃	C₂²	1	1	yes	characteristic, complemented
29	#494×C₇₃	C₂²	1	1	yes	characteristic, complemented
30	#487×C₇₃	C₂²	1	2	yes	complemented
31	#467×C₇₃	—	2	1	no	—
32	#467×C₇₃	—	2	1	no	—
33	C₄×Q₁₆×C₇₃	—	2	1	no	—
34	C₄×Q₁₆×C₇₃	—	2	1	no	—
35	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
36	C₂²×D₂₉₂	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₇₃	—	2	1	no	—
38	C₂×#11924012	—	2	2	no	—
39	C₂×#11924011	—	2	2	no	—
40	C₂×#11924011	—	2	2	no	—
41	C₂×#11924011	—	4	4	no	—
42	C₂²×#10596831	—	2	1	no	—
43	C₄×D₂₉₂	—	2	1	no	—
44	C₄×D₂₉₂	—	2	1	no	—
45	C₄×D₂₉₂	D₄	1	2	yes	—
46	C₄×D₂₉₂	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
48	C₂×#11924005	—	2	1	no	—
49	#14556524	—	2	4	no	—
50	#14556522	—	2	4	no	—
51	C₄⋊#10596831	—	2	1	no	—
52	C₄⋊#10596831	—	2	2	no	—
53	C₄⋊#10596831	C₂³	1	2	yes	—
54	C₄⋊#10596831	C₂³	1	4	yes	—
55	#14556511	—	2	4	no	—
56	#14556511	—	2	4	no	—
57	#14556503	—	2	2	no	—
58	#14556503	—	2	2	no	—
59	C₂×Q₁₆×C₇₃	—	2	1	no	—
60	C₂×Q₁₆×C₇₃	—	2	1	no	—
61	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
65	#125×C₇₃	—	2	2	no	—
66	C₂²⋊Q₈×C₇₃	—	2	2	no	—
67	C₂²⋊Q₈×C₇₃	—	2	2	no	—
68	C₄×Q₈×C₇₃	—	2	2	no	—
69	C₄×Q₈×C₇₃	—	2	2	no	—
70	#117×C₇₃	—	2	2	no	—
71	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
72	#107×C₇₃	—	2	1	no	—
73	#107×C₇₃	—	2	1	no	—
74	#103×C₇₃	—	2	2	no	—
75	#103×C₇₃	—	2	2	no	—
76	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
78	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
80	C₂³×D₇₃	—	2	1	no	—
81	C₂×D₂₉₂	—	2	4	no	—
82	C₂×D₂₉₂	—	2	1	no	—
83	C₂×D₂₉₂	C₂×D₄	1	2	yes	—
84	C₂²×Dic₇₃	—	2	1	no	—
85	C₂×C₄×D₇₃	—	2	2	no	—
86	C₂×C₄×D₇₃	—	2	2	no	—
87	C₂×C₄×D₇₃	—	4	2	no	—
88	C₂×C₄×D₇₃	—	4	2	no	—
89	#11924012	—	4	4	no	—
90	#11924011	—	4	4	no	—
91	#11924011	—	4	4	no	—
92	C₂×#10596831	—	2	2	no	—
93	C₂×#10596831	—	2	4	no	—
94	C₂×#10596831	—	4	2	no	—
95	C₄⋊Dic₇₃	—	2	1	no	—
96	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
97	#11924005	—	2	2	no	—
98	#11924005	—	2	2	no	—
99	C₂×Q₈×C₇₃	—	2	2	no	—
100	C₂×Q₈×C₇₃	—	2	2	no	—
101	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
104	Q₁₆×C₇₃	—	4	2	no	—
105	Q₁₆×C₇₃	—	4	2	no	—
106	C₂×C₈×C₇₃	—	2	2	no	—
107	C₂×C₈×C₇₃	—	4	1	no	—
108	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₄⋊C₄×C₇₃	—	2	2	no	—
113	C₄⋊C₄×C₇₃	—	2	2	no	—
114	C₄⋊C₄×C₇₃	—	2	2	no	—
115	C₄⋊C₄×C₇₃	—	2	2	no	—
116	C₄⋊C₄×C₇₃	—	4	2	no	—
117	C₄⋊C₄×C₇₃	—	4	2	no	—
118	C₂²⋊C₄×C₇₃	—	4	2	no	—
119	C₄²×C₇₃	—	4	2	no	—
120	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₇₃	#41	1	1	yes	characteristic
123	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
124	C₂³×C₇₃	#121	1	1	yes	characteristic
125	C₂²×D₇₃	—	2	2	no	—
126	C₂²×D₇₃	—	4	2	no	—
127	C₂²×D₇₃	—	4	2	no	—
128	C₂²×D₇₃	—	4	2	no	—
129	Q₈×C₇₃	—	4	2	no	—
130	Q₈×C₇₃	—	4	2	no	—
131	Q₈×C₇₃	—	8	2	no	—
132	C₂×C₄×C₇₃	—	2	1	no	—
133	C₂×C₄×C₇₃	—	2	1	no	—
134	C₂×C₄×C₇₃	—	4	1	no	—
135	C₂×C₄×C₇₃	—	4	2	no	—
136	C₂×C₄×C₇₃	—	4	2	no	—
137	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
138	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₇₃	#137	1	4	yes	—
141	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
144	C₂×Dic₇₃	—	2	2	no	—
145	C₂×Dic₇₃	—	4	2	no	—
146	D₂₉₂	—	2	4	no	—
147	D₂₉₂	—	4	2	no	—
148	C₄×D₇₃	—	4	4	no	—
149	C₄×D₇₃	—	4	2	no	—
150	C₄×D₇₃	—	4	2	no	—
151	C₈×C₇₃	—	2	1	no	—
152	C₈×C₇₃	—	2	1	no	—
153	C₈×C₇₃	—	2	1	no	—
154	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
155	#10596831	—	4	2	no	—
156	#10596831	—	4	4	no	—
157	C₂²×C₇₃	—	2	1	no	—
158	C₂²×C₇₃	—	2	1	no	—
159	C₂²×C₇₃	#470	1	2	yes	—
160	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
161	C₂×D₇₃	—	4	2	no	—
162	C₂×D₇₃	—	4	2	no	—
163	C₂×D₇₃	—	4	2	no	—
164	C₂×D₇₃	—	8	4	no	—
165	C₄×C₇₃	—	2	1	no	—
166	C₄×C₇₃	—	2	1	no	—
167	C₄×C₇₃	—	2	1	no	—
168	C₄×C₇₃	—	8	2	no	—
169	C₄×C₇₃	—	8	2	no	—
170	C₄×C₇₃	#470	1	2	yes	—
171	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₄×C₇₃	C₂×#137	1	2	yes	—
173	C₄×C₇₃	#575	1	1	yes	characteristic
174	Dic₇₃	—	4	2	no	—
175	#19281	—	73	1	no	maximal, Sylow
176	C₂×C₇₃	—	2	2	no	—
177	C₂×C₇₃	#3036	1	1	yes	characteristic
178	C₂×C₇₃	C₂×#470	1	1	yes	characteristic
179	C₂×C₇₃	#3067	1	1	yes	characteristic
180	D₇₃	—	8	2	no	—
181	#3170	D₇₃	1	1	yes	characteristic, complemented
182	#3077	—	73	2	no	—
183	C₂×#436	—	73	1	no	—
184	#1703	—	73	4	no	—
185	#1695	—	73	2	no	—
186	#1689	—	73	2	no	—
187	#1566	—	73	1	no	—
188	#1550	—	73	2	no	—
189	C₇₃	#19281	1	1	yes	characteristic, complemented, Sylow
190	C₂²×M₁₆	—	146	1	no	—
191	#533	C₂×D₇₃	1	2	yes	—
192	C₂×C₄×D₄	—	73	1	no	—
193	#500	C₂×D₇₃	1	1	yes	characteristic, complemented
194	#494	C₂×D₇₃	1	1	yes	characteristic, complemented
195	#487	C₂×D₇₃	1	2	yes	complemented
196	#478	—	146	4	no	—
197	#476	—	73	4	no	—
198	#474	—	73	4	no	—
199	#467	—	2	1	no	—
200	#467	—	2	1	no	—
201	C₄×Q₁₆	—	2	1	no	—
202	C₄×Q₁₆	—	2	1	no	—
203	#436	—	73	2	no	—
204	#436	—	73	2	no	—
205	#436	—	146	1	no	—
206	#436	—	146	1	no	—
207	#428	—	146	2	no	—
208	#423	—	73	2	no	—
209	C₂×C₄⋊C₈	—	73	1	no	—
210	#418	—	146	2	no	—
211	C₂×C₂²⋊C₈	—	146	1	no	—
212	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
213	#334	—	73	2	no	—
214	#334	—	73	2	no	—
215	#333	—	73	4	no	—
216	#332	—	73	4	no	—
217	C₂²×D₄	—	73	1	no	—
218	C₂³×C₄	—	146	1	no	—
219	C₂×Q₁₆	—	2	1	no	—
220	C₂×Q₁₆	—	2	1	no	—
221	C₂×M₁₆	—	292	4	no	—
222	C₂×M₁₆	—	146	2	no	—
223	C₂×M₁₆	—	146	2	no	—
224	C₂×M₁₆	—	146	2	no	—
225	C₂²×C₈	—	146	1	no	—
226	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
227	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
228	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
229	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
230	#125	—	2	2	no	—
231	C₂²⋊Q₈	—	2	2	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	C₄×Q₈	—	2	2	no	—
234	C₄×Q₈	—	2	2	no	—
235	C₄×D₄	—	73	2	no	—
236	C₄×D₄	—	73	2	no	—
237	C₄×D₄	—	146	1	no	—
238	C₄×D₄	—	146	1	no	—
239	#117	—	2	2	no	—
240	C₂×C₄⋊C₄	—	73	1	no	—
241	C₂×C₂²⋊C₄	—	146	1	no	—
242	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
243	#107	—	2	1	no	—
244	#107	—	2	1	no	—
245	#107	—	146	4	no	—
246	#106	—	146	4	no	—
247	C₄⋊C₈	—	73	2	no	—
248	C₄⋊C₈	—	73	4	no	—
249	C₄⋊C₈	—	146	2	no	—
250	C₄⋊C₈	—	146	1	no	—
251	#103	—	2	2	no	—
252	#103	—	2	2	no	—
253	#102	—	146	4	no	—
254	#102	—	146	4	no	—
255	C₂²⋊C₈	—	146	2	no	—
256	C₂²⋊C₈	—	146	2	no	—
257	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
258	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
259	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
261	C₂⁴	—	146	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×D₄	—	73	2	no	—
265	C₂×D₄	—	146	1	no	—
266	C₂×D₄	—	146	4	no	—
267	C₂²×C₄	—	292	2	no	—
268	C₂²×C₄	—	292	2	no	—
269	C₂²×C₄	—	146	1	no	—
270	C₂²×C₄	—	146	2	no	—
271	C₂²×C₄	—	146	2	no	—
272	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
274	C₂²×C₄	#11924027	1	1	yes	characteristic
275	Q₁₆	—	4	2	no	—
276	Q₁₆	—	4	2	no	—
277	M₁₆	—	292	4	no	—
278	M₁₆	—	292	4	no	—
279	M₁₆	—	292	4	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	292	2	no	—
283	C₂×C₈	—	146	4	no	—
284	C₂×C₈	—	146	2	no	—
285	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
286	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
287	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	—	73	2	no	—
296	C₄⋊C₄	—	146	1	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	146	2	no	—
299	C₂²⋊C₄	—	146	2	no	—
300	C₄²	—	4	2	no	—
301	C₄²	D₄×D₇₃	1	1	yes	characteristic
302	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
303	C₄²	#11924027	1	1	yes	characteristic
304	C₄²	C₂³×D₇₃	1	1	yes	characteristic
305	C₂³	—	292	2	no	—
306	C₂³	—	292	2	no	—
307	C₂³	—	292	2	no	—
308	C₂³	—	146	2	no	—
309	C₂³	#14556562	1	1	yes	characteristic
310	Q₈	—	4	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	8	2	no	—
313	D₄	—	292	2	no	—
314	D₄	—	146	4	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	292	4	no	—
321	C₂×C₄	—	292	2	no	—
322	C₂×C₄	—	292	2	no	—
323	C₂×C₄	—	292	2	no	—
324	C₂×C₄	—	146	2	no	—
325	C₂×C₄	#14556562	1	1	yes	characteristic
326	C₂×C₄	#14556562	1	1	yes	characteristic
327	C₂×C₄	#14556562	1	1	yes	characteristic
328	C₂×C₄	#14556623	1	4	yes	—
329	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
331	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	292	4	no	—
336	C₈	—	292	2	no	—
337	C₈	#14556616	1	2	yes	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	292	2	no	—
341	C₂²	—	292	2	no	—
342	C₂²	—	292	2	no	—
343	C₂²	—	584	4	no	—
344	C₂²	#17988842	1	2	yes	—
345	C₂²	C₂×#14556562	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	8	2	no	—
350	C₄	—	8	2	no	—
351	C₄	—	292	2	no	—
352	C₄	#17988842	1	2	yes	—
353	C₄	C₂×#14556616	1	1	yes	characteristic
354	C₄	C₂×#14556623	1	2	yes	—
355	C₄	#17989385	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	584	2	no	—
358	C₂	#22619391	1	1	yes	characteristic
359	C₂	C₂×#17988842	1	1	yes	characteristic
360	C₂	#22619647	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556562
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#3170	C₂
socle	C₂²×C₇₃	C₂×#121
prime radical	C₂²×D₂₉₂	D₄
hypercenter	#3170	D₇₃
hypocenter	C₇₃	#19281
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #22609503	C₂	semidirect
4. #22609451	C₂	semidirect
5. #22609417	C₂	semidirect
6. #22608687	C₂	semidirect
7. #22608621	C₂	semidirect
8. C₇₃×#3170	C₂	semidirect
23. #17988338	C₂²	semidirect
24. #17988338	C₂²	semidirect
25. #17988336	C₂²	semidirect
26. #17988333	C₂²	semidirect
28. #500×C₇₃	C₂²	semidirect
29. #494×C₇₃	C₂²	semidirect
30. #487×C₇₃	C₂²	semidirect
181. #3170	D₇₃	semidirect
189. C₇₃	#19281	semidirect
193. #500	C₂×D₇₃	semidirect
194. #494	C₂×D₇₃	semidirect
195. #487	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22619724	2
C₂×#17988666	2
#22609503	2
#22609451	2
#22609417	2
#22608687	2
#22608621	2
C₇₃×#3170	2
#19281	73