Subgroups — #33241345

Order 18688.

598 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620444	C₂	1	1	yes	maximal, characteristic, complemented
2	#495×D₇₃	C₂	1	1	yes	maximal, characteristic
3	#22612169	C₂	1	2	yes	maximal, complemented
4	#22612166	C₂	1	2	yes	maximal, complemented
5	#22612165	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612163	C₂	1	1	yes	maximal, characteristic, complemented
7	#22612162	C₂	1	1	yes	maximal, characteristic, complemented
8	#22611649	C₂	1	2	yes	maximal, complemented
9	#22611645	C₂	1	2	yes	maximal
10	C₄×#14556530	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2242	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₃	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×D₇₃	—	2	1	no	—
14	C₂×D₈×D₇₃	C₂²	1	1	yes	characteristic
15	#17989143	—	2	1	no	—
16	#124×D₇₃	—	2	1	no	—
17	C₂×#14556530	C₂²	1	2	yes	—
18	#17988991	C₂²	1	2	yes	complemented
19	#17988989	C₂²	1	1	yes	characteristic, complemented
20	#17988986	—	2	1	no	—
21	#17988984	—	2	1	no	—
22	#17988972	C₂²	1	1	yes	characteristic
23	#17988971	C₂²	1	1	yes	characteristic
24	#17988970	C₂²	1	1	yes	characteristic
25	#17988969	—	2	1	no	—
26	#17988968	—	2	1	no	—
27	#17988624	C₂²	1	2	yes	—
28	#17988622	—	2	1	no	—
29	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
30	#17988392	C₂²	1	2	yes	complemented
31	#17988391	C₂²	1	2	yes	complemented
32	#17988390	C₂²	1	2	yes	complemented
33	#17988389	C₂²	1	2	yes	—
34	#17988388	C₂²	1	2	yes	—
35	#17988387	C₂²	1	2	yes	—
36	#17988362	C₂²	1	2	yes	complemented
37	C₄×#11924003	C₂²	1	1	yes	characteristic, complemented
38	#499×C₇₃	C₂²	1	1	yes	characteristic, complemented
39	#495×C₇₃	C₂²	1	1	yes	characteristic
40	#358×C₇₃	C₂²	1	2	yes	complemented
41	#357×C₇₃	C₂²	1	2	yes	—
42	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic, complemented
43	C₂×Q₈×D₇₃	—	2	1	no	—
44	C₂×D₄×D₇₃	—	2	1	no	—
45	C₂×Dic₂₉₂	C₂³	1	1	yes	characteristic
46	C₂×#11924018	—	2	1	no	—
47	Q₁₆×D₇₃	—	2	4	no	—
48	C₂×#11924017	—	2	1	no	—
49	C₂×#11924016	—	2	1	no	—
50	C₂×#11924015	—	2	1	no	—
51	SD₁₆×D₇₃	—	4	4	no	—
52	C₂×D₅₈₄	C₂³	1	1	yes	characteristic
53	C₂×#11924013	—	2	1	no	—
54	D₈×D₇₃	—	2	4	no	—
55	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
56	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
57	#14556590	—	2	2	no	—
58	#14556583	—	2	1	no	—
59	#14556580	—	2	1	no	—
60	#14556579	—	2	1	no	—
61	#14556578	—	2	1	no	—
62	#14556577	—	4	2	no	—
63	#14556569	—	4	2	no	—
64	#14556556	—	4	2	no	—
65	#14556549	—	2	2	no	—
66	C₂²⋊C₄×D₇₃	—	4	2	no	—
67	C₄²×D₇₃	D₄	1	1	yes	characteristic
68	#14556530	—	2	4	no	—
69	C₂×#11924003	C₂³	1	2	yes	—
70	#14556525	C₂³	1	2	yes	—
71	#14556524	—	2	2	no	—
72	#14556523	—	2	1	no	—
73	#14556522	—	2	2	no	—
74	C₄×#10596831	D₄	1	1	yes	characteristic
75	C₈×Dic₇₃	D₄	1	1	yes	characteristic
76	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
77	C₂×Q₁₆×C₇₃	C₂³	1	1	yes	characteristic
78	C₂×SD₁₆×C₇₃	—	2	1	no	—
79	C₂×D₈×C₇₃	C₂³	1	1	yes	characteristic
80	#125×C₇₃	—	2	1	no	—
81	#124×C₇₃	—	2	1	no	—
82	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
83	#107×C₇₃	C₂³	1	2	yes	—
84	#106×C₇₃	—	2	1	no	—
85	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
86	C₂³×D₇₃	—	4	1	no	—
87	C₂×Dic₁₄₆	—	2	1	no	—
88	C₂×Dic₁₄₆	—	4	2	no	—
89	C₂×Dic₁₄₆	—	4	1	no	—
90	Q₈×D₇₃	—	4	2	no	—
91	Q₈×D₇₃	—	4	2	no	—
92	C₂×D₂₉₂	—	2	1	no	—
93	C₂×C₂²⋊D₇₃	—	4	1	no	—
94	D₄×D₇₃	—	4	2	no	—
95	D₄×D₇₃	—	4	2	no	—
96	C₂×C₄×D₇₃	—	4	1	no	—
97	C₂×C₄×D₇₃	D₈	1	2	yes	—
98	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
99	Dic₂₉₂	—	2	2	no	—
100	#11924018	—	4	2	no	—
101	#11924017	—	4	2	no	—
102	#11924016	—	4	2	no	—
103	#11924015	—	4	2	no	—
104	D₅₈₄	—	2	2	no	—
105	#11924013	—	4	2	no	—
106	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
107	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
108	C₈×D₇₃	SD₁₆	1	4	yes	—
109	C₈×D₇₃	#41	1	4	yes	—
110	C₄⋊Dic₇₃	—	2	2	no	—
111	#11924007	—	4	2	no	—
112	#11924007	—	4	2	no	—
113	C₂²⋊Dic₇₃	—	4	2	no	—
114	#11924005	—	4	2	no	—
115	#11924005	—	4	2	no	—
116	#11924005	—	4	2	no	—
117	C₄×Dic₇₃	—	4	2	no	—
118	C₄×Dic₇₃	D₈	1	2	yes	—
119	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
120	#11924003	—	2	2	no	—
121	C₂×Q₈×C₇₃	—	2	1	no	—
122	C₂×D₄×C₇₃	—	2	1	no	—
123	Q₁₆×C₇₃	—	2	2	no	—
124	SD₁₆×C₇₃	—	4	2	no	—
125	D₈×C₇₃	—	2	2	no	—
126	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
128	C₄⋊C₄×C₇₃	—	2	2	no	—
129	C₄⋊C₄×C₇₃	—	4	2	no	—
130	C₂²⋊C₄×C₇₃	—	4	2	no	—
131	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
132	C₁₆×C₇₃	—	2	2	no	—
133	C₂³×C₇₃	—	4	1	no	—
134	C₂²×D₇₃	—	4	1	no	—
135	C₂²×D₇₃	—	8	4	no	—
136	C₂²×D₇₃	—	8	2	no	—
137	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
138	Q₈×C₇₃	—	4	1	no	—
139	Q₈×C₇₃	—	4	1	no	—
140	D₄×C₇₃	—	4	1	no	—
141	D₄×C₇₃	—	4	1	no	—
142	C₂×C₄×C₇₃	—	4	1	no	—
143	C₂×C₄×C₇₃	—	4	2	no	—
144	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₇₃	—	8	2	no	—
147	C₂×Dic₇₃	—	4	2	no	—
148	C₂×Dic₇₃	—	4	1	no	—
149	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
150	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
151	D₂₉₂	—	4	1	no	—
152	D₂₉₂	—	4	1	no	—
153	C₄×D₇₃	—	4	2	no	—
154	C₄×D₇₃	—	8	2	no	—
155	C₄×D₇₃	#124	1	2	yes	—
156	C₄×D₇₃	C₂×SD₁₆	1	2	yes	—
157	Dic₁₄₆	—	4	1	no	—
158	Dic₁₄₆	—	4	1	no	—
159	Dic₁₄₆	—	8	4	no	—
160	Dic₁₄₆	—	8	2	no	—
161	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
162	C₈×C₇₃	C₂×#41	1	2	yes	—
163	#10596831	#124	1	2	yes	—
164	#10596831	C₂×SD₁₆	1	2	yes	—
165	C₂²×C₇₃	—	8	1	no	—
166	C₂²×C₇₃	—	8	2	no	—
167	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
168	C₂×D₇₃	—	2	2	no	—
169	C₂×D₇₃	—	8	1	no	—
170	C₂×D₇₃	—	8	2	no	—
171	C₂×D₇₃	—	16	4	no	—
172	C₂×D₇₃	#486	1	2	yes	—
173	C₄×C₇₃	—	2	2	no	—
174	C₄×C₇₃	—	8	1	no	—
175	C₄×C₇₃	—	8	2	no	—
176	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
177	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
178	Dic₇₃	—	2	2	no	—
179	Dic₇₃	—	8	1	no	—
180	Dic₇₃	—	8	2	no	—
181	Dic₇₃	#486	1	2	yes	—
182	#33706	—	73	1	no	maximal, Sylow
183	C₂×C₇₃	—	16	1	no	—
184	C₂×C₇₃	C₂×#486	1	1	yes	characteristic
185	C₂×C₇₃	#3414	1	2	yes	—
186	D₇₃	—	2	2	no	—
187	D₇₃	—	16	1	no	—
188	#3160	—	73	1	no	—
189	C₂×#495	—	73	1	no	—
190	#2243	—	73	2	no	—
191	#2243	—	73	2	no	—
192	#2242	—	73	1	no	—
193	#2242	—	73	1	no	—
194	#2242	—	73	1	no	—
195	#2242	D₇₃	1	1	yes	characteristic, complemented
196	#2141	—	73	2	no	—
197	#2140	—	73	2	no	—
198	C₄×M₃₂	—	73	1	no	—
199	C₇₃	#33706	1	1	yes	characteristic, complemented, Sylow
200	C₂²×Q₁₆	—	73	1	no	—
201	C₂²×SD₁₆	—	146	1	no	—
202	C₂²×D₈	—	73	1	no	—
203	#533	—	146	1	no	—
204	C₂×#124	—	146	1	no	—
205	C₂×M₃₂	—	73	2	no	—
206	#500	—	73	2	no	—
207	#499	—	73	1	no	—
208	#499	—	146	1	no	—
209	#499	C₂×D₇₃	1	1	yes	characteristic, complemented
210	#498	—	146	1	no	—
211	#495	—	73	1	no	—
212	#495	—	73	1	no	—
213	#495	—	73	1	no	—
214	#495	—	146	1	no	—
215	#495	—	146	1	no	—
216	#495	C₂×D₇₃	1	1	yes	characteristic
217	#427	—	73	2	no	—
218	#427	—	146	1	no	—
219	C₂×C₄×C₈	—	73	1	no	—
220	#358	—	73	2	no	—
221	#358	—	73	2	no	—
222	#358	—	73	2	no	—
223	#358	C₂×D₇₃	1	2	yes	complemented
224	#357	—	73	2	no	—
225	#357	—	73	2	no	—
226	#357	—	73	2	no	—
227	#357	C₂×D₇₃	1	2	yes	—
228	#346	—	73	2	no	—
229	C₄×C₁₆	—	73	1	no	—
230	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic, complemented
231	C₂²×Q₈	—	146	1	no	—
232	C₂²×D₄	—	146	1	no	—
233	C₂×Q₁₆	—	73	1	no	—
234	C₂×Q₁₆	—	146	4	no	—
235	C₂×Q₁₆	—	146	1	no	—
236	C₂×Q₁₆	C₂²×D₇₃	1	1	yes	characteristic
237	C₂×SD₁₆	—	2	1	no	—
238	C₂×SD₁₆	—	292	4	no	—
239	C₂×SD₁₆	—	146	1	no	—
240	C₂×SD₁₆	—	146	1	no	—
241	C₂×SD₁₆	—	146	1	no	—
242	C₂×D₈	—	73	1	no	—
243	C₂×D₈	—	146	4	no	—
244	C₂×D₈	—	146	1	no	—
245	C₂×D₈	C₂²×D₇₃	1	1	yes	characteristic
246	C₂²×C₈	—	73	1	no	—
247	C₂²×C₈	—	73	1	no	—
248	C₄⋊Q₈	—	146	2	no	—
249	#125	—	2	1	no	—
250	#125	—	146	1	no	—
251	#124	—	2	1	no	—
252	#124	—	292	2	no	—
253	#124	—	146	1	no	—
254	#124	—	146	1	no	—
255	#124	—	146	1	no	—
256	C₂²⋊Q₈	—	292	2	no	—
257	C₄×Q₈	—	292	2	no	—
258	#117	—	146	2	no	—
259	C₂×C₂²⋊C₄	—	292	2	no	—
260	C₂×C₄²	—	73	1	no	—
261	M₃₂	—	146	4	no	—
262	C₂×C₁₆	—	73	2	no	—
263	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
264	#107	—	73	2	no	—
265	#107	—	146	2	no	—
266	#107	C₂²×D₇₃	1	2	yes	—
267	#106	—	2	1	no	—
268	#106	—	146	2	no	—
269	#106	—	146	1	no	—
270	C₄×C₈	—	73	1	no	—
271	C₄×C₈	—	73	1	no	—
272	C₄×C₈	—	73	1	no	—
273	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
274	C₂⁴	—	292	1	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	292	2	no	—
277	C₂×Q₈	—	292	1	no	—
278	C₂×Q₈	—	292	2	no	—
279	C₂×Q₈	—	292	2	no	—
280	C₂×Q₈	—	146	1	no	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	292	2	no	—
283	C₂×D₄	—	292	2	no	—
284	C₂×D₄	—	292	1	no	—
285	C₂×D₄	—	146	1	no	—
286	C₂²×C₄	—	292	1	no	—
287	C₂²×C₄	—	73	2	no	—
288	C₂²×C₄	—	73	1	no	—
289	Q₁₆	—	2	2	no	—
290	Q₁₆	—	292	2	no	—
291	Q₁₆	—	146	2	no	—
292	SD₁₆	—	4	2	no	—
293	SD₁₆	—	292	2	no	—
294	SD₁₆	—	292	2	no	—
295	SD₁₆	—	292	2	no	—
296	D₈	—	2	2	no	—
297	D₈	—	292	2	no	—
298	D₈	—	146	2	no	—
299	C₂×C₈	—	73	1	no	—
300	C₂×C₈	—	73	1	no	—
301	C₂×C₈	—	73	4	no	—
302	C₂×C₈	—	73	4	no	—
303	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
304	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	—	292	2	no	—
308	C₄⋊C₄	—	292	2	no	—
309	C₄⋊C₄	—	146	2	no	—
310	C₂²⋊C₄	—	4	2	no	—
311	C₂²⋊C₄	—	292	2	no	—
312	C₂²⋊C₄	—	292	2	no	—
313	C₂²⋊C₄	—	292	2	no	—
314	C₂²⋊C₄	—	292	2	no	—
315	C₄²	—	292	2	no	—
316	C₄²	—	73	2	no	—
317	C₄²	—	73	1	no	—
318	C₄²	D₄×D₇₃	1	1	yes	characteristic
319	C₁₆	—	2	2	no	—
320	C₁₆	—	146	2	no	—
321	C₂³	—	4	1	no	—
322	C₂³	—	292	1	no	—
323	C₂³	—	584	2	no	—
324	C₂³	—	584	4	no	—
325	C₂³	—	73	1	no	—
326	Q₈	—	4	1	no	—
327	Q₈	—	4	1	no	—
328	Q₈	—	292	1	no	—
329	Q₈	—	292	1	no	—
330	Q₈	—	584	2	no	—
331	Q₈	—	584	4	no	—
332	D₄	—	4	1	no	—
333	D₄	—	4	1	no	—
334	D₄	—	292	1	no	—
335	D₄	—	292	1	no	—
336	D₄	—	584	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	292	2	no	—
340	C₂×C₄	—	292	2	no	—
341	C₂×C₄	—	292	1	no	—
342	C₂×C₄	—	584	2	no	—
343	C₂×C₄	—	73	1	no	—
344	C₂×C₄	—	73	2	no	—
345	C₂×C₄	—	73	2	no	—
346	C₂×C₄	—	73	2	no	—
347	C₂×C₄	D₈×D₇₃	1	2	yes	—
348	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
349	C₈	—	73	2	no	—
350	C₈	—	73	2	no	—
351	C₈	SD₁₆×D₇₃	1	2	yes	—
352	C₈	#41×D₇₃	1	2	yes	—
353	C₂²	—	8	1	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	584	1	no	—
356	C₂²	—	584	2	no	—
357	C₂²	—	73	2	no	—
358	C₂²	—	1168	4	no	—
359	C₂²	—	146	2	no	—
360	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
361	C₄	—	2	2	no	—
362	C₄	—	8	1	no	—
363	C₄	—	8	2	no	—
364	C₄	—	584	1	no	—
365	C₄	—	584	2	no	—
366	C₄	—	73	2	no	—
367	C₄	—	146	2	no	—
368	C₄	#124×D₇₃	1	1	yes	characteristic
369	C₄	C₂×SD₁₆×D₇₃	1	1	yes	characteristic
370	C₂	—	16	1	no	—
371	C₂	—	1168	1	no	—
372	C₂	—	146	2	no	—
373	C₂	#486×D₇₃	1	1	yes	characteristic
374	C₂	#22623064	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2242	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×D₈×D₇₃	C₂²
hypercenter	#2242	D₇₃
hypocenter	C₇₃	#33706
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2242

Complements

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

N	H ≅ G/N	type
1. #22620444	C₂	semidirect
3. #22612169	C₂	semidirect
4. #22612166	C₂	semidirect
5. #22612165	C₂	semidirect
6. #22612163	C₂	semidirect
7. #22612162	C₂	semidirect
8. #22611649	C₂	semidirect
10. C₄×#14556530	C₂	semidirect
11. C₇₃×#2242	C₂	semidirect
18. #17988991	C₂²	semidirect
19. #17988989	C₂²	semidirect
30. #17988392	C₂²	semidirect
31. #17988391	C₂²	semidirect
32. #17988390	C₂²	semidirect
36. #17988362	C₂²	semidirect
37. C₄×#11924003	C₂²	semidirect
38. #499×C₇₃	C₂²	semidirect
40. #358×C₇₃	C₂²	semidirect
42. C₄×C₁₆×C₇₃	C₂²	semidirect
195. #2242	D₇₃	semidirect
199. C₇₃	#33706	semidirect
209. #499	C₂×D₇₃	semidirect
223. #358	C₂×D₇₃	semidirect
230. C₄×C₁₆	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22620444	2
#495×D₇₃	2
#22612169	2
#22612166	2
#22612165	2
#22612163	2
#22612162	2
#22611649	2
#22611645	2
C₄×#14556530	2
C₇₃×#2242	2
#33706	73