Subgroups — #33056532

Order 18688.

666 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620491	C₂	1	1	yes	maximal, characteristic
2	#22618644	C₂	1	1	yes	maximal, characteristic
3	#22608831	C₂	1	4	yes	maximal, complemented
4	#22608741	C₂	1	4	yes	maximal, complemented
5	#22608689	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608386	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608381	C₂	1	1	yes	maximal, characteristic
8	C₂×#17988309	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14556597	—	2	1	no	—
11	#17989137	C₂²	1	1	yes	characteristic
12	#17988983	C₄	1	1	yes	characteristic
13	#17988961	C₄	1	1	yes	characteristic
14	#17988937	C₄	1	1	yes	characteristic
15	#17988935	C₄	1	1	yes	characteristic
16	#17988818	C₄	1	4	yes	—
17	C₄×#11924015	C₂²	1	4	yes	—
18	#17988666	—	2	1	no	—
19	#17988666	C₂²	1	2	yes	—
20	C₈×D₂₉₂	—	2	1	no	—
21	C₈×D₂₉₂	C₂²	1	2	yes	—
22	#17988611	C₂²	1	1	yes	characteristic
23	#17988553	—	2	1	no	—
24	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
25	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
26	#17988339	C₂²	1	4	yes	—
27	Q₈⋊#10596831	C₂²	1	4	yes	—
28	#17988314	C₂²	1	4	yes	—
29	#17988309	C₂²	1	2	yes	—
30	#17988309	C₂²	1	2	yes	—
31	#533×C₇₃	C₂²	1	1	yes	characteristic
32	#446×C₇₃	—	2	1	no	—
33	#446×C₇₃	C₂²	1	2	yes	—
34	C₈×Q₈×C₇₃	—	2	1	no	—
35	C₈×Q₈×C₇₃	C₂²	1	2	yes	—
36	#424×C₇₃	—	2	1	no	—
37	#423×C₇₃	C₂²	1	1	yes	characteristic
38	#405×C₇₃	—	2	1	no	—
39	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
40	C₂×#11924028	—	2	1	no	—
41	C₂×#11924015	C₂×C₄	1	4	yes	—
42	#14556597	—	4	4	no	—
43	C₂×#11924012	—	2	1	no	—
44	C₂×#11924011	—	2	2	no	—
45	C₂²×#10596831	—	2	1	no	—
46	C₂²×#10596831	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₇₃	—	2	2	no	—
48	#14556591	C₂×C₄	1	1	yes	characteristic
49	#14556588	C₂×C₄	1	1	yes	characteristic
50	#14556579	—	2	1	no	—
51	#14556565	—	2	1	no	—
52	#14556565	C₂×C₄	1	2	yes	—
53	C₄×D₂₉₂	—	2	1	no	—
54	C₄×D₂₉₂	C₂³	1	2	yes	—
55	#14556522	C₂×C₄	1	4	yes	—
56	C₄⋊#10596831	—	2	1	no	—
57	C₄⋊#10596831	C₂³	1	2	yes	—
58	#14556515	C₂×C₄	1	4	yes	—
59	#14556511	C₂×C₄	1	4	yes	—
60	#14556503	—	2	2	no	—
61	#14556503	—	2	2	no	—
62	C₄×#10596831	—	2	1	no	—
63	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
64	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
65	C₄×#10596831	C₂³	1	2	yes	—
66	C₄×#10596831	C₂³	1	2	yes	—
67	C₂×M₁₆×C₇₃	—	2	1	no	—
68	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
69	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
70	C₄⋊Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
71	C₄⋊Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
72	#125×C₇₃	—	2	1	no	—
73	C₂²⋊Q₈×C₇₃	—	2	1	no	—
74	C₂²⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
75	C₄×Q₈×C₇₃	—	2	1	no	—
76	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
77	#117×C₇₃	—	2	1	no	—
78	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₇₃	—	2	2	no	—
80	C₄⋊C₈×C₇₃	—	2	2	no	—
81	C₄⋊C₈×C₇₃	—	2	1	no	—
82	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
83	#97×C₇₃	—	2	2	no	—
84	C₄×C₈×C₇₃	—	2	2	no	—
85	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
86	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
87	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
89	#11924028	—	4	2	no	—
90	#11924028	—	4	2	no	—
91	C₂×Dic₁₄₆	—	2	1	no	—
92	C₂×D₂₉₂	—	2	1	no	—
93	C₂×D₂₉₂	—	2	1	no	—
94	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
95	C₂×C₂²⋊D₇₃	—	2	2	no	—
96	C₂×C₄×D₇₃	—	2	2	no	—
97	#11924015	—	2	8	no	—
98	#11924012	—	4	2	no	—
99	#11924011	—	4	4	no	—
100	C₂×#10596831	—	2	4	no	—
101	C₂×#10596831	—	2	2	no	—
102	C₂×#10596831	—	2	2	no	—
103	C₂×#10596831	—	4	2	no	—
104	C₂×#10596831	C₂²×C₄	1	2	yes	—
105	C₂×#10596831	C₂²×C₄	1	2	yes	—
106	C₈×D₇₃	—	4	4	no	—
107	C₄⋊Dic₇₃	—	2	1	no	—
108	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
109	#11924005	—	2	2	no	—
110	#11924005	—	2	2	no	—
111	C₂×Q₈×C₇₃	—	2	1	no	—
112	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
116	M₁₆×C₇₃	—	4	2	no	—
117	C₂×C₈×C₇₃	—	2	2	no	—
118	C₂×C₈×C₇₃	—	2	2	no	—
119	C₂×C₈×C₇₃	—	2	2	no	—
120	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
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	Q₈×C₇₃	—	2	4	no	—
139	Q₈×C₇₃	—	4	2	no	—
140	C₂×C₄×C₇₃	—	2	2	no	—
141	C₂×C₄×C₇₃	—	2	2	no	—
142	C₂×C₄×C₇₃	—	2	2	no	—
143	C₂×C₄×C₇₃	—	2	2	no	—
144	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₇₃	#137	1	2	yes	—
149	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
152	C₂²⋊D₇₃	—	4	4	no	—
153	C₂×Dic₇₃	—	2	2	no	—
154	D₂₉₂	—	2	4	no	—
155	D₂₉₂	—	4	2	no	—
156	D₂₉₂	—	4	2	no	—
157	C₄×D₇₃	—	4	2	no	—
158	C₄×D₇₃	—	4	2	no	—
159	Dic₁₄₆	—	4	2	no	—
160	C₈×C₇₃	—	2	1	no	—
161	C₈×C₇₃	—	2	1	no	—
162	C₈×C₇₃	—	2	1	no	—
163	C₈×C₇₃	—	2	1	no	—
164	C₈×C₇₃	—	4	2	no	—
165	#10596831	—	2	2	no	—
166	#10596831	—	4	2	no	—
167	#10596831	—	4	2	no	—
168	#10596831	#131	1	4	yes	complemented
169	C₂²×C₇₃	—	2	1	no	—
170	C₂²×C₇₃	—	2	1	no	—
171	C₂²×C₇₃	#440	1	2	yes	—
172	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×D₇₃	—	4	2	no	—
174	C₂×D₇₃	—	4	2	no	—
175	C₂×D₇₃	—	4	2	no	—
176	C₄×C₇₃	—	2	1	no	—
177	C₄×C₇₃	—	2	1	no	—
178	C₄×C₇₃	—	4	2	no	—
179	C₄×C₇₃	—	4	2	no	—
180	C₄×C₇₃	#440	1	2	yes	—
181	C₄×C₇₃	#444	1	2	yes	—
182	C₄×C₇₃	C₂×#131	1	2	yes	—
183	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
185	Dic₇₃	—	4	2	no	—
186	#17960	—	73	1	no	maximal, Sylow
187	C₂×C₇₃	—	2	2	no	—
188	C₂×C₇₃	#2931	1	1	yes	characteristic
189	C₂×C₇₃	C₂×#440	1	1	yes	characteristic
190	C₂×C₇₃	C₂×#444	1	1	yes	characteristic
191	D₇₃	—	8	2	no	—
192	#3167	—	73	1	no	—
193	#2960	D₇₃	1	1	yes	characteristic, complemented
194	#2931	—	73	1	no	—
195	#1592	—	73	4	no	—
196	SD₁₆⋊C₈	—	73	4	no	—
197	#1567	—	73	1	no	—
198	#1493	—	73	1	no	—
199	#1492	—	73	1	no	—
200	C₂×#322	—	73	1	no	—
201	C₇₃	#17960	1	1	yes	characteristic, complemented, Sylow
202	C₂×#131	—	146	1	no	—
203	#533	C₂×D₇₃	1	1	yes	characteristic
204	#532	—	73	1	no	—
205	#498	—	73	1	no	—
206	#492	—	73	1	no	—
207	#487	—	73	1	no	—
208	#486	—	73	1	no	—
209	#465	—	73	4	no	—
210	#446	—	2	1	no	—
211	#446	C₂×D₇₃	1	2	yes	—
212	C₈×Q₈	—	2	1	no	—
213	C₈×Q₈	C₂×D₇₃	1	2	yes	—
214	C₄×SD₁₆	—	73	4	no	—
215	#436	—	73	2	no	—
216	#436	—	146	1	no	—
217	C₈×D₄	—	73	2	no	—
218	C₈×D₄	—	146	1	no	—
219	#424	—	2	1	no	—
220	#423	—	73	1	no	—
221	#423	C₂×D₇₃	1	1	yes	characteristic
222	#408	—	146	1	no	—
223	#405	—	2	1	no	—
224	C₂×C₄×C₈	—	73	1	no	—
225	C₂×C₄×C₈	—	73	1	no	—
226	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
227	#335	—	73	4	no	—
228	Q₈⋊C₈	—	73	4	no	—
229	D₄⋊C₈	—	73	4	no	—
230	#322	—	73	2	no	—
231	#322	—	73	2	no	—
232	C₂×#41	—	146	1	no	—
233	C₂×SD₁₆	—	73	4	no	—
234	#131	—	292	4	no	—
235	C₂×M₁₆	—	2	1	no	—
236	C₂×M₁₆	—	146	1	no	—
237	C₂×M₁₆	—	146	2	no	—
238	C₂²×C₈	—	73	2	no	—
239	C₂²×C₈	—	146	1	no	—
240	C₂²×C₈	—	146	2	no	—
241	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
242	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
243	C₄⋊Q₈	—	73	1	no	—
244	C₄⋊Q₈	C₄×D₇₃	1	1	yes	characteristic
245	C₄⋊Q₈	C₄×D₇₃	1	1	yes	characteristic
246	#127	—	73	1	no	—
247	#125	—	2	1	no	—
248	#124	—	146	1	no	—
249	C₂²⋊Q₈	—	2	1	no	—
250	C₂²⋊Q₈	C₄×D₇₃	1	2	yes	—
251	#121	—	73	2	no	—
252	#121	—	146	1	no	—
253	C₄×Q₈	—	2	1	no	—
254	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
255	C₄×D₄	—	73	2	no	—
256	C₄×D₄	—	146	1	no	—
257	#117	—	2	1	no	—
258	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
259	#106	—	73	4	no	—
260	C₄⋊C₈	—	2	2	no	—
261	C₄⋊C₈	—	2	2	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	73	2	no	—
264	C₄⋊C₈	—	146	1	no	—
265	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
266	#103	—	73	4	no	—
267	#102	—	73	4	no	—
268	C₂²⋊C₈	—	146	2	no	—
269	C₂²⋊C₈	—	146	2	no	—
270	#97	—	2	2	no	—
271	C₄×C₈	—	2	2	no	—
272	C₄×C₈	—	73	1	no	—
273	C₄×C₈	—	73	1	no	—
274	C₄×C₈	—	73	2	no	—
275	C₄×C₈	—	73	2	no	—
276	C₄×C₈	—	146	1	no	—
277	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
278	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
281	#41	—	292	2	no	—
282	#41	—	292	2	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×Q₈	—	146	1	no	—
285	C₂×Q₈	C₂×C₄×D₇₃	1	2	yes	—
286	C₂×D₄	—	73	2	no	—
287	C₂×D₄	—	146	2	no	—
288	C₂×D₄	—	146	1	no	—
289	C₂×D₄	—	146	1	no	—
290	C₂²×C₄	—	146	2	no	—
291	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
292	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
293	C₂²×C₄	#11924028	1	1	yes	characteristic
294	SD₁₆	—	146	8	no	—
295	M₁₆	—	4	2	no	—
296	M₁₆	—	292	4	no	—
297	M₁₆	—	292	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	292	4	no	—
302	C₂×C₈	—	292	2	no	—
303	C₂×C₈	—	73	2	no	—
304	C₂×C₈	—	73	2	no	—
305	C₂×C₈	—	146	2	no	—
306	C₂×C₈	—	146	2	no	—
307	C₂×C₈	—	146	4	no	—
308	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
309	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
310	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
311	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	73	2	no	—
318	C₄⋊C₄	—	146	1	no	—
319	C₄⋊C₄	C₂×C₄×D₇₃	1	2	yes	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₂²⋊C₄	—	146	2	no	—
322	C₂²⋊C₄	—	146	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
325	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
326	C₄²	#11924028	1	1	yes	characteristic
327	C₄²	C₂³×D₇₃	1	1	yes	characteristic
328	C₂³	—	146	2	no	—
329	C₂³	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
330	Q₈	—	2	4	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	292	2	no	—
333	D₄	—	292	4	no	—
334	D₄	—	292	2	no	—
335	D₄	—	292	2	no	—
336	D₄	—	146	4	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	292	2	no	—
342	C₂×C₄	—	292	2	no	—
343	C₂×C₄	—	146	2	no	—
344	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
345	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
346	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
347	C₂×C₄	#14556616	1	2	yes	—
348	C₂×C₄	#14556622	1	2	yes	—
349	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
350	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
351	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	4	2	no	—
357	C₈	—	292	2	no	—
358	C₈	—	292	2	no	—
359	C₈	—	73	4	no	—
360	C₈	—	146	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	292	2	no	—
364	C₂²	—	292	2	no	—
365	C₂²	—	292	2	no	—
366	C₂²	#17988685	1	2	yes	—
367	C₂²	C₂×C₄×C₂²⋊D₇₃	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	292	2	no	—
373	C₄	#17988685	1	2	yes	—
374	C₄	#17988707	1	2	yes	—
375	C₄	#131×D₇₃	1	2	yes	—
376	C₄	C₂×#14556616	1	1	yes	characteristic
377	C₄	C₂×#14556622	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	584	2	no	—
380	C₂	#22618641	1	1	yes	characteristic
381	C₂	C₂×#17988685	1	1	yes	characteristic
382	C₂	C₂×#17988707	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×#2960	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#17989137	C₂²
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#2960	D₇₃
hypocenter	C₇₃	#17960
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
3. #22608831	C₂	semidirect
4. #22608741	C₂	semidirect
5. #22608689	C₂	semidirect
6. #22608386	C₂	semidirect
8. C₂×#17988309	C₂	semidirect
9. C₇₃×#2960	C₂	semidirect
168. #10596831	#131	semidirect
193. #2960	D₇₃	semidirect
201. C₇₃	#17960	semidirect

Maximal subgroups

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

subgroup	index
#22620491	2
#22618644	2
#22608831	2
#22608741	2
#22608689	2
#22608386	2
#22608381	2
C₂×#17988309	2
C₇₃×#2960	2
#17960	73