Subgroups — #33239230

Order 18688.

642 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22623032	C₂	1	2	yes	maximal, complemented
2	#22618829	C₂	1	1	yes	maximal, characteristic
3	#22611774	C₂	1	1	yes	maximal, characteristic, complemented
4	#22611770	C₂	1	2	yes	maximal, complemented
5	#22611768	C₂	1	1	yes	maximal, characteristic, complemented
6	#22611767	C₂	1	1	yes	maximal, characteristic, complemented
7	#22611766	C₂	1	2	yes	maximal, complemented
8	#22611695	C₂	1	1	yes	maximal, characteristic, complemented
9	#350×D₇₃	C₂	1	2	yes	maximal, complemented
10	#22611529	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2171	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989384	C₄	1	1	yes	characteristic
13	#131×D₇₃	C₂²	1	2	yes	—
14	#17989005	—	2	2	no	—
15	#17989005	C₂²	1	2	yes	complemented
16	#17989004	C₂²	1	2	yes	complemented
17	#17989003	—	2	2	no	—
18	#17989003	C₂²	1	2	yes	complemented
19	C₂×#14556530	—	2	2	no	—
20	C₂×#14556530	C₂²	1	1	yes	characteristic
21	M₃₂×D₇₃	—	2	2	no	—
22	M₃₂×D₇₃	C₂²	1	2	yes	—
23	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
24	#17988709	C₄	1	1	yes	characteristic
25	#17988708	C₄	1	1	yes	characteristic
26	#17988706	C₄	1	2	yes	—
27	#17988705	C₂²	1	1	yes	characteristic
28	#17988702	C₂²	1	2	yes	—
29	#17988634	C₄	1	1	yes	characteristic
30	#17988602	C₄	1	2	yes	—
31	#17988541	C₂²	1	1	yes	characteristic
32	#17988408	C₂²	1	1	yes	characteristic, complemented
33	#17988407	C₂²	1	2	yes	complemented
34	#17988372	C₂²	1	2	yes	complemented
35	#17988371	C₂²	1	2	yes	complemented
36	#17988370	C₂²	1	2	yes	complemented
37	#17988362	C₂²	1	1	yes	characteristic, complemented
38	C₄×#11924003	C₂²	1	1	yes	characteristic, complemented
39	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic, complemented
40	#504×C₇₃	C₂²	1	2	yes	complemented
41	#443×C₇₃	C₂²	1	1	yes	characteristic
42	#364×C₇₃	C₂²	1	1	yes	characteristic, complemented
43	#350×C₇₃	C₂²	1	2	yes	complemented
44	#346×C₇₃	C₂²	1	1	yes	characteristic, complemented
45	#41×D₇₃	C₂×C₄	1	2	yes	—
46	#14556623	C₈	1	2	yes	—
47	#14556620	C₈	1	2	yes	—
48	#14556618	C₈	1	2	yes	—
49	#14556615	C₈	1	2	yes	—
50	#14556613	C₂×C₄	1	1	yes	characteristic
51	#14556611	C₂×C₄	1	2	yes	—
52	#14556599	—	2	2	no	—
53	#14556599	C₂×C₄	1	2	yes	—
54	#14556598	C₂×C₄	1	2	yes	—
55	#14556597	—	2	2	no	—
56	#14556597	C₂³	1	2	yes	—
57	C₂×#11924011	C₂×C₄	1	1	yes	characteristic
58	M₁₆×D₇₃	—	2	2	no	—
59	M₁₆×D₇₃	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₇₃	—	2	2	no	—
61	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
62	#14556548	C₂×C₄	1	1	yes	characteristic
63	#14556531	—	2	2	no	—
64	#14556531	C₂³	1	2	yes	—
65	#14556530	—	2	2	no	—
66	#14556530	—	2	2	no	—
67	#14556530	—	2	2	no	—
68	#14556530	D₄	1	2	yes	—
69	#14556530	D₄	1	2	yes	—
70	C₂×#11924003	—	2	2	no	—
71	C₂×#11924003	C₂³	1	1	yes	characteristic
72	C₂×#11924003	C₂³	1	1	yes	characteristic
73	C₁₆×D₇₃	—	2	2	no	—
74	C₁₆×D₇₃	—	2	2	no	—
75	C₁₆×D₇₃	—	2	2	no	—
76	C₁₆×D₇₃	—	2	2	no	—
77	#14556527	C₂×C₄	1	1	yes	characteristic
78	#14556526	C₂×C₄	1	2	yes	—
79	#14556519	C₂×C₄	1	2	yes	—
80	#14556518	C₂×C₄	1	2	yes	—
81	#14556517	C₂×C₄	1	2	yes	—
82	#14556502	C₂×C₄	1	1	yes	characteristic
83	#14556501	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
85	#135×C₇₃	C₂×C₄	1	1	yes	characteristic
86	#131×C₇₃	C₂³	1	2	yes	—
87	M₃₂×C₇₃	—	2	2	no	—
88	M₃₂×C₇₃	C₂³	1	2	yes	—
89	C₂×C₁₆×C₇₃	—	2	2	no	—
90	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
91	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
92	#108×C₇₃	C₂×C₄	1	1	yes	characteristic
93	#104×C₇₃	C₂×C₄	1	2	yes	—
94	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
95	#11924029	C₂×C₈	1	2	yes	—
96	#11924028	—	2	2	no	—
97	#11924028	C₂²×C₄	1	2	yes	—
98	#11924027	—	2	2	no	—
99	#11924027	C₂×C₈	1	2	yes	—
100	Q₈×D₇₃	C₂×C₈	1	2	yes	—
101	D₄×D₇₃	—	2	2	no	—
102	D₄×D₇₃	C₂×C₈	1	2	yes	—
103	C₂×C₄×D₇₃	—	2	2	no	—
104	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
105	Dic₂₉₂	C₂×C₈	1	1	yes	characteristic
106	#11924018	C₂×C₈	1	2	yes	—
107	#11924017	C₂×C₈	1	2	yes	—
108	#11924016	C₂×C₈	1	2	yes	—
109	#11924015	C₂×C₈	1	2	yes	—
110	D₅₈₄	C₂×C₈	1	1	yes	characteristic
111	#11924013	C₂×C₈	1	2	yes	—
112	#11924012	—	2	2	no	—
113	#11924012	C₂²×C₄	1	2	yes	—
114	#11924011	—	2	2	no	—
115	#11924011	—	2	2	no	—
116	#11924011	—	2	2	no	—
117	#11924011	C₂×C₈	1	2	yes	—
118	#11924011	C₂×C₈	1	2	yes	—
119	C₂×#10596831	—	2	2	no	—
120	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
121	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
122	C₈×D₇₃	—	2	2	no	—
123	C₈×D₇₃	—	2	2	no	—
124	C₈×D₇₃	—	2	2	no	—
125	C₈×D₇₃	C₂×D₄	1	2	yes	—
126	C₈×D₇₃	#41	1	2	yes	—
127	#11924007	—	2	1	no	—
128	#11924005	—	2	1	no	—
129	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
130	#11924003	—	2	2	no	—
131	#11924003	—	2	1	no	—
132	#11924003	C₂×D₄	1	1	yes	characteristic
133	#11924003	C₂×D₄	1	1	yes	characteristic
134	#41×C₇₃	C₂²×C₄	1	2	yes	—
135	Q₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
136	SD₁₆×C₇₃	C₂×C₈	1	2	yes	—
137	D₈×C₇₃	C₂×C₈	1	1	yes	characteristic
138	M₁₆×C₇₃	—	2	2	no	—
139	M₁₆×C₇₃	C₂²×C₄	1	2	yes	—
140	C₂×C₈×C₇₃	—	2	2	no	—
141	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
143	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
144	C₁₆×C₇₃	—	2	2	no	—
145	C₁₆×C₇₃	—	2	1	no	—
146	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
147	C₂²×D₇₃	—	2	2	no	—
148	C₂²×D₇₃	#131	1	1	yes	characteristic
149	Q₈×C₇₃	C₂²×C₈	1	2	yes	—
150	D₄×C₇₃	—	2	2	no	—
151	D₄×C₇₃	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₇₃	—	2	2	no	—
153	C₂×C₄×C₇₃	—	2	1	no	—
154	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
155	C₂²⋊D₇₃	—	2	2	no	—
156	C₂²⋊D₇₃	—	2	2	no	—
157	C₂²⋊D₇₃	—	2	2	no	—
158	C₂×Dic₇₃	—	2	2	no	—
159	C₂×Dic₇₃	—	2	1	no	—
160	C₂×Dic₇₃	#131	1	1	yes	characteristic
161	D₂₉₂	—	2	2	no	—
162	D₂₉₂	C₂²×C₈	1	2	yes	—
163	C₄×D₇₃	—	2	2	no	—
164	C₄×D₇₃	—	2	2	no	—
165	C₄×D₇₃	—	2	2	no	—
166	C₄×D₇₃	C₄×D₄	1	2	yes	—
167	C₄×D₇₃	C₂²×C₈	1	2	yes	—
168	Dic₁₄₆	—	2	2	no	—
169	Dic₁₄₆	C₂²×C₈	1	2	yes	—
170	C₈×C₇₃	—	2	2	no	—
171	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
172	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
173	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
174	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
175	#10596831	—	2	2	no	—
176	#10596831	C₄×D₄	1	2	yes	—
177	#10596831	C₂²×C₈	1	2	yes	—
178	C₂²×C₇₃	—	2	2	no	—
179	C₂²×C₇₃	C₂×#131	1	1	yes	characteristic
180	C₂×D₇₃	—	2	2	no	—
181	C₂×D₇₃	—	4	1	no	—
182	C₂×D₇₃	—	8	2	no	—
183	C₂×D₇₃	C₈×D₄	1	2	yes	—
184	C₄×C₇₃	—	2	2	no	—
185	C₄×C₇₃	—	4	1	no	—
186	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
187	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
188	Dic₇₃	—	2	2	no	—
189	Dic₇₃	—	4	1	no	—
190	Dic₇₃	C₈×D₄	1	2	yes	—
191	#33442	—	73	1	no	maximal, Sylow
192	C₂×C₇₃	—	2	1	no	—
193	C₂×C₇₃	—	4	2	no	—
194	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
195	D₇₃	—	2	2	no	—
196	D₇₃	—	4	2	no	—
197	#3407	—	73	2	no	—
198	#2955	—	73	1	no	—
199	#2171	—	73	2	no	—
200	#2171	—	73	1	no	—
201	#2171	D₇₃	1	1	yes	characteristic, complemented
202	#2170	—	73	2	no	—
203	#2170	—	73	1	no	—
204	#2170	—	73	1	no	—
205	#2153	—	73	1	no	—
206	C₂×#350	—	73	2	no	—
207	#2108	—	73	1	no	—
208	C₇₃	#33442	1	1	yes	characteristic, complemented, Sylow
209	#575	—	73	1	no	—
210	C₂×#131	—	73	2	no	—
211	#504	—	73	2	no	—
212	#504	—	73	2	no	—
213	#504	—	73	2	no	—
214	#504	—	146	2	no	—
215	#504	—	146	2	no	—
216	#504	C₂×D₇₃	1	2	yes	complemented
217	C₂×M₃₂	—	73	1	no	—
218	C₂×M₃₂	—	73	2	no	—
219	C₂×M₃₂	—	146	2	no	—
220	C₂×M₃₂	—	146	2	no	—
221	C₂²×C₁₆	—	73	1	no	—
222	#444	—	73	1	no	—
223	#444	—	73	1	no	—
224	#444	—	73	2	no	—
225	#443	—	73	2	no	—
226	#443	—	73	1	no	—
227	#443	C₂×D₇₃	1	1	yes	characteristic
228	#430	—	73	1	no	—
229	#421	—	73	2	no	—
230	#405	—	73	1	no	—
231	#364	—	73	2	no	—
232	#364	—	73	1	no	—
233	#364	C₂×D₇₃	1	1	yes	characteristic, complemented
234	#350	—	73	2	no	—
235	#350	—	73	2	no	—
236	#350	—	73	2	no	—
237	#350	C₂×D₇₃	1	2	yes	complemented
238	#346	—	73	1	no	—
239	#346	C₂×D₇₃	1	1	yes	characteristic, complemented
240	C₄×C₁₆	—	73	1	no	—
241	C₄×C₁₆	—	73	1	no	—
242	C₂×#41	—	73	2	no	—
243	#137	—	73	2	no	—
244	#137	—	73	2	no	—
245	AGL(1,ℤ/8)	—	73	2	no	—
246	AGL(1,ℤ/8)	—	73	2	no	—
247	#135	—	73	2	no	—
248	#135	—	73	1	no	—
249	#135	C₄×D₇₃	1	1	yes	characteristic
250	#131	—	73	2	no	—
251	#131	—	73	2	no	—
252	#131	—	73	2	no	—
253	#131	—	146	2	no	—
254	#131	—	146	2	no	—
255	#131	C₂²×D₇₃	1	2	yes	—
256	C₂×M₁₆	—	73	1	no	—
257	C₂×M₁₆	—	73	2	no	—
258	C₂×M₁₆	—	146	2	no	—
259	C₂²×C₈	—	73	1	no	—
260	C₂²×C₈	—	146	2	no	—
261	#117	—	73	1	no	—
262	M₃₂	—	2	2	no	—
263	M₃₂	—	73	2	no	—
264	M₃₂	—	73	2	no	—
265	M₃₂	—	73	2	no	—
266	M₃₂	—	146	2	no	—
267	M₃₂	—	146	2	no	—
268	M₃₂	—	146	2	no	—
269	M₃₂	—	146	2	no	—
270	M₃₂	C₂²×D₇₃	1	2	yes	—
271	C₂×C₁₆	—	2	2	no	—
272	C₂×C₁₆	—	73	1	no	—
273	C₂×C₁₆	—	73	1	no	—
274	C₂×C₁₆	—	146	2	no	—
275	C₂×C₁₆	—	146	2	no	—
276	C₂×C₁₆	—	146	2	no	—
277	C₂×C₁₆	—	146	2	no	—
278	C₂×C₁₆	—	146	2	no	—
279	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
280	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
281	#108	—	73	2	no	—
282	#108	—	73	1	no	—
283	#108	C₄×D₇₃	1	1	yes	characteristic
284	#104	—	73	2	no	—
285	#104	—	73	2	no	—
286	#104	—	73	2	no	—
287	#104	C₄×D₇₃	1	2	yes	—
288	#97	—	73	1	no	—
289	#97	—	73	1	no	—
290	C₄×C₈	—	73	1	no	—
291	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
292	#41	—	73	2	no	—
293	#41	—	73	2	no	—
294	#41	—	73	2	no	—
295	#41	—	146	2	no	—
296	#41	—	146	2	no	—
297	#41	C₂×C₄×D₇₃	1	2	yes	—
298	C₂×Q₈	—	73	2	no	—
299	C₂×D₄	—	73	2	no	—
300	C₂×D₄	—	146	2	no	—
301	C₂²×C₄	—	73	1	no	—
302	C₂²×C₄	—	146	2	no	—
303	Q₁₆	—	73	2	no	—
304	Q₁₆	—	73	1	no	—
305	Q₁₆	C₈×D₇₃	1	1	yes	characteristic
306	SD₁₆	—	73	2	no	—
307	SD₁₆	—	73	2	no	—
308	SD₁₆	—	73	2	no	—
309	SD₁₆	C₈×D₇₃	1	2	yes	—
310	D₈	—	73	2	no	—
311	D₈	—	73	1	no	—
312	D₈	C₈×D₇₃	1	1	yes	characteristic
313	M₁₆	—	2	2	no	—
314	M₁₆	—	73	2	no	—
315	M₁₆	—	73	2	no	—
316	M₁₆	—	73	2	no	—
317	M₁₆	—	146	2	no	—
318	M₁₆	—	146	2	no	—
319	M₁₆	—	146	2	no	—
320	M₁₆	—	146	2	no	—
321	M₁₆	C₂×C₄×D₇₃	1	2	yes	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	73	1	no	—
324	C₂×C₈	—	73	1	no	—
325	C₂×C₈	—	73	2	no	—
326	C₂×C₈	—	73	2	no	—
327	C₂×C₈	—	146	2	no	—
328	C₂×C₈	—	146	2	no	—
329	C₂×C₈	—	146	2	no	—
330	C₂×C₈	—	146	2	no	—
331	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
332	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
333	C₄⋊C₄	—	146	1	no	—
334	C₂²⋊C₄	—	146	1	no	—
335	C₄²	—	73	1	no	—
336	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
337	C₁₆	—	2	2	no	—
338	C₁₆	—	2	1	no	—
339	C₁₆	—	73	1	no	—
340	C₁₆	—	73	1	no	—
341	C₁₆	—	146	2	no	—
342	C₁₆	—	146	1	no	—
343	C₁₆	D₄×D₇₃	1	2	yes	—
344	C₂³	—	73	1	no	—
345	C₂³	—	146	2	no	—
346	Q₈	—	73	2	no	—
347	Q₈	—	146	2	no	—
348	Q₈	C₂×C₈×D₇₃	1	2	yes	—
349	D₄	—	2	2	no	—
350	D₄	—	73	2	no	—
351	D₄	—	146	2	no	—
352	D₄	—	146	2	no	—
353	D₄	—	146	2	no	—
354	D₄	—	146	2	no	—
355	D₄	C₂×C₈×D₇₃	1	2	yes	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	73	1	no	—
359	C₂×C₄	—	73	2	no	—
360	C₂×C₄	—	73	2	no	—
361	C₂×C₄	—	146	2	no	—
362	C₂×C₄	—	146	2	no	—
363	C₂×C₄	—	146	1	no	—
364	C₂×C₄	—	146	2	no	—
365	C₂×C₄	—	146	2	no	—
366	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	73	2	no	—
369	C₈	—	73	2	no	—
370	C₈	—	146	2	no	—
371	C₈	C₂×C₈×D₇₃	1	1	yes	characteristic
372	C₈	C₂×C₈×D₇₃	1	1	yes	characteristic
373	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
374	C₈	#41×D₇₃	1	1	yes	characteristic
375	C₂²	—	2	2	no	—
376	C₂²	—	292	1	no	—
377	C₂²	—	584	2	no	—
378	C₂²	—	73	2	no	—
379	C₂²	—	146	2	no	—
380	C₂²	#131×D₇₃	1	1	yes	characteristic
381	C₄	—	2	2	no	—
382	C₄	—	4	1	no	—
383	C₄	—	292	1	no	—
384	C₄	—	73	2	no	—
385	C₄	—	146	2	no	—
386	C₄	C₄×D₄×D₇₃	1	1	yes	characteristic
387	C₄	C₂²×C₈×D₇₃	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	4	2	no	—
390	C₂	—	292	2	no	—
391	C₂	—	146	2	no	—
392	C₂	C₈×D₄×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₇₃
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2171	C₂
socle	C₂×C₇₃	C₂×C₈×D₄
prime radical	#17989384	C₄
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#2171	D₇₃
hypocenter	C₇₃	#33442
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #2171

Complements

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

N	H ≅ G/N	type
1. #22623032	C₂	semidirect
3. #22611774	C₂	semidirect
4. #22611770	C₂	semidirect
5. #22611768	C₂	semidirect
6. #22611767	C₂	semidirect
7. #22611766	C₂	semidirect
8. #22611695	C₂	semidirect
9. #350×D₇₃	C₂	semidirect
10. #22611529	C₂	semidirect
11. C₇₃×#2171	C₂	semidirect
15. #17989005	C₂²	semidirect
16. #17989004	C₂²	semidirect
18. #17989003	C₂²	semidirect
32. #17988408	C₂²	semidirect
33. #17988407	C₂²	semidirect
34. #17988372	C₂²	semidirect
35. #17988371	C₂²	semidirect
36. #17988370	C₂²	semidirect
37. #17988362	C₂²	semidirect
38. C₄×#11924003	C₂²	semidirect
39. C₁₆×Dic₇₃	C₂²	semidirect
40. #504×C₇₃	C₂²	semidirect
42. #364×C₇₃	C₂²	semidirect
43. #350×C₇₃	C₂²	semidirect
44. #346×C₇₃	C₂²	semidirect
201. #2171	D₇₃	semidirect
208. C₇₃	#33442	semidirect
216. #504	C₂×D₇₃	semidirect
233. #364	C₂×D₇₃	semidirect
237. #350	C₂×D₇₃	semidirect
239. #346	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22623032	2
#22618829	2
#22611774	2
#22611770	2
#22611768	2
#22611767	2
#22611766	2
#22611695	2
#350×D₇₃	2
#22611529	2
C₇₃×#2171	2
#33442	73