Subgroups — #33056537

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	#22618866	C₂	1	1	yes	maximal, characteristic, complemented
3	#22618642	C₂	1	1	yes	maximal, characteristic
4	#22608836	C₂	1	4	yes	maximal, complemented
5	#22608746	C₂	1	4	yes	maximal, complemented
6	#22608689	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608383	C₂	1	1	yes	maximal, characteristic
8	C₂×#17988309	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14556599	—	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	#17988718	—	2	1	no	—
18	#17988718	C₂²	1	2	yes	—
19	Q₈×#10596831	—	2	1	no	—
20	Q₈×#10596831	C₂²	1	2	yes	—
21	C₄×#11924015	C₂²	1	4	yes	—
22	#17988664	—	2	1	no	—
23	#17988664	C₂²	1	2	yes	—
24	#17988652	—	2	1	no	—
25	#17988652	C₂²	1	2	yes	—
26	#17988614	—	2	1	no	—
27	#17988611	C₂²	1	1	yes	characteristic
28	#17988555	—	2	1	no	—
29	#17988543	—	2	1	no	—
30	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
31	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
32	#17988339	C₂²	1	4	yes	—
33	#17988316	C₂²	1	4	yes	—
34	#17988309	C₂²	1	2	yes	—
35	#17988309	C₂²	1	2	yes	—
36	#533×C₇₃	C₂²	1	1	yes	characteristic
37	#423×C₇₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
39	Q₈⋊C₈×C₇₃	C₂²	1	4	yes	—
40	C₂×#11924028	—	2	1	no	—
41	C₂×#11924015	C₂×C₄	1	4	yes	—
42	#14556599	—	4	4	no	—
43	C₂×#11924012	—	2	1	no	—
44	C₂×#11924011	—	2	2	no	—
45	C₂²×#10596831	C₂×C₄	1	2	yes	—
46	C₂²×#10596831	D₄	1	1	yes	characteristic
47	C₂²×#10596831	D₄	1	1	yes	characteristic
48	C₂×C₈×D₇₃	—	2	2	no	—
49	#14556591	C₂×C₄	1	1	yes	characteristic
50	#14556588	C₂×C₄	1	1	yes	characteristic
51	#14556579	—	2	1	no	—
52	#14556565	—	2	1	no	—
53	#14556565	C₂×C₄	1	2	yes	—
54	C₄×D₂₉₂	—	2	1	no	—
55	C₄×D₂₉₂	C₂³	1	2	yes	—
56	#14556522	C₂×C₄	1	4	yes	—
57	C₄⋊#10596831	—	2	2	no	—
58	C₄⋊#10596831	—	2	2	no	—
59	C₄⋊#10596831	—	2	1	no	—
60	C₄⋊#10596831	C₂³	1	2	yes	—
61	#14556515	C₂×C₄	1	4	yes	—
62	#14556511	C₂×C₄	1	4	yes	—
63	#14556503	—	2	2	no	—
64	#14556503	—	2	2	no	—
65	#14556502	—	2	2	no	—
66	C₄×#10596831	—	2	2	no	—
67	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
68	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
69	C₄×#10596831	D₄	1	1	yes	characteristic
70	C₄×#10596831	D₄	1	1	yes	characteristic
71	C₄×#10596831	C₂³	1	2	yes	—
72	C₄×#10596831	C₂³	1	1	yes	characteristic
73	C₄×#10596831	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₇₃	—	2	1	no	—
75	C₂²×C₈×C₇₃	—	2	1	no	—
76	C₄⋊Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
77	C₄⋊Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
78	#125×C₇₃	—	2	1	no	—
79	C₂²⋊Q₈×C₇₃	—	2	1	no	—
80	C₂²⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
81	C₄×Q₈×C₇₃	—	2	1	no	—
82	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
83	#117×C₇₃	—	2	1	no	—
84	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
85	C₄⋊C₈×C₇₃	—	2	1	no	—
86	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
87	C₄×C₈×C₇₃	—	2	1	no	—
88	C₄×C₈×C₇₃	C₂³	1	2	yes	—
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	2	no	—
101	C₂×#10596831	—	2	2	no	—
102	C₂×#10596831	—	2	4	no	—
103	C₂×#10596831	—	2	2	no	—
104	C₂×#10596831	C₂²×C₄	1	2	yes	—
105	C₂×#10596831	C₂²×C₄	1	2	yes	—
106	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
107	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
108	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
109	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
110	C₈×D₇₃	—	4	4	no	—
111	C₄⋊Dic₇₃	—	2	1	no	—
112	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
113	#11924005	—	2	2	no	—
114	#11924005	—	2	2	no	—
115	C₂×Q₈×C₇₃	—	2	1	no	—
116	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
120	M₁₆×C₇₃	—	4	2	no	—
121	C₂×C₈×C₇₃	—	2	2	no	—
122	C₂×C₈×C₇₃	—	2	2	no	—
123	C₂×C₈×C₇₃	—	4	2	no	—
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	—	4	2	no	—
161	C₈×C₇₃	—	4	2	no	—
162	#10596831	—	2	2	no	—
163	#10596831	—	2	1	no	—
164	#10596831	—	2	1	no	—
165	#10596831	—	2	1	no	—
166	#10596831	—	2	1	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	—	73	1	no	—
194	#2931	—	73	1	no	—
195	#1592	—	73	4	no	—
196	SD₁₆⋊C₈	—	73	4	no	—
197	#1567	—	73	1	no	—
198	#1493	D₇₃	1	1	yes	characteristic, complemented
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	—	73	2	no	—
211	#446	—	146	1	no	—
212	C₈×Q₈	—	73	2	no	—
213	C₈×Q₈	—	146	1	no	—
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	—	146	1	no	—
220	#423	—	73	1	no	—
221	#423	C₂×D₇₃	1	1	yes	characteristic
222	#408	—	146	1	no	—
223	#405	—	146	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	C₂×D₇₃	1	4	yes	—
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	2	no	—
237	C₂×M₁₆	—	146	1	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₂²×C₈	—	73	2	no	—
240	C₂²×C₈	—	73	1	no	—
241	C₂²×C₈	—	73	1	no	—
242	C₂²×C₈	—	146	2	no	—
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	1	no	—
261	C₄⋊C₈	—	73	2	no	—
262	C₄⋊C₈	—	146	2	no	—
263	C₄⋊C₈	—	146	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	—	146	2	no	—
271	C₄×C₈	—	2	1	no	—
272	C₄×C₈	—	73	1	no	—
273	C₄×C₈	—	73	1	no	—
274	C₄×C₈	—	73	1	no	—
275	C₄×C₈	—	73	1	no	—
276	C₄×C₈	—	73	2	no	—
277	C₄×C₈	—	73	1	no	—
278	C₄×C₈	—	73	1	no	—
279	C₄×C₈	—	146	2	no	—
280	C₄×C₈	C₂²×D₇₃	1	2	yes	—
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	D₄×D₇₃	1	1	yes	characteristic
293	C₂²×C₄	#11924027	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	—	4	2	no	—
301	C₂×C₈	—	292	4	no	—
302	C₂×C₈	—	73	2	no	—
303	C₂×C₈	—	73	2	no	—
304	C₂×C₈	—	73	1	no	—
305	C₂×C₈	—	73	1	no	—
306	C₂×C₈	—	73	1	no	—
307	C₂×C₈	—	73	1	no	—
308	C₂×C₈	—	146	2	no	—
309	C₂×C₈	—	146	4	no	—
310	C₂×C₈	—	146	2	no	—
311	C₂×C₈	—	146	2	no	—
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	D₄×D₇₃	1	1	yes	characteristic
326	C₄²	#11924027	1	1	yes	characteristic
327	C₄²	C₂³×D₇₃	1	1	yes	characteristic
328	C₂³	—	146	2	no	—
329	C₂³	#14556551	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	#14556551	1	1	yes	characteristic
345	C₂×C₄	#14556551	1	1	yes	characteristic
346	C₂×C₄	#14556551	1	1	yes	characteristic
347	C₂×C₄	#14556618	1	2	yes	—
348	C₂×C₄	#14556623	1	2	yes	—
349	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
350	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
351	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
352	C₈	—	4	2	no	—
353	C₈	—	4	2	no	—
354	C₈	—	292	2	no	—
355	C₈	—	73	4	no	—
356	C₈	—	146	1	no	—
357	C₈	—	146	1	no	—
358	C₈	—	146	2	no	—
359	C₈	—	146	1	no	—
360	C₈	—	146	1	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	#17988690	1	2	yes	—
367	C₂²	C₂×#14556551	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	#17988690	1	2	yes	—
374	C₄	#17988706	1	2	yes	—
375	C₄	#131×D₇₃	1	2	yes	—
376	C₄	C₂×#14556618	1	1	yes	characteristic
377	C₄	C₂×#14556623	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	584	2	no	—
380	C₂	#22618639	1	1	yes	characteristic
381	C₂	C₂×#17988690	1	1	yes	characteristic
382	C₂	C₂×#17988706	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1493	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#17989137	C₂²
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1493	D₇₃
hypocenter	C₇₃	#17960
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1493

Complements

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

N	H ≅ G/N	type
2. #22618866	C₂	semidirect
4. #22608836	C₂	semidirect
5. #22608746	C₂	semidirect
6. #22608689	C₂	semidirect
8. C₂×#17988309	C₂	semidirect
9. C₇₃×#1493	C₂	semidirect
168. #10596831	#131	semidirect
198. #1493	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
#22618866	2
#22618642	2
#22608836	2
#22608746	2
#22608689	2
#22608383	2
C₂×#17988309	2
C₇₃×#1493	2
#17960	73