Subgroups — #33000879

Order 18688.

570 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556530	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₇₃	C₂	1	1	yes	maximal, characteristic
3	#427×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#22608028	C₂	1	1	yes	maximal, characteristic, complemented
5	#22608027	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608026	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₇₃	—	2	1	no	—
9	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	#117×D₇₃	—	2	1	no	—
11	C₂×#14556530	—	2	1	no	—
12	C₂×#14556530	—	2	1	no	—
13	C₂×#14556530	—	2	2	no	—
14	C₂×#14556530	C₄	1	4	yes	complemented
15	C₂²×#11924003	C₂²	1	1	yes	characteristic
16	C₂×#14556527	C₂²	1	1	yes	characteristic
17	C₂×#14556526	—	2	1	no	—
18	#108×D₇₃	—	2	2	no	—
19	#108×D₇₃	C₄	1	4	yes	complemented
20	#17988625	C₂²	1	1	yes	characteristic
21	#17988624	C₄	1	2	yes	—
22	#17988623	—	2	1	no	—
23	#17988622	—	2	1	no	—
24	#107×D₇₃	C₄	1	2	yes	—
25	#106×D₇₃	—	2	1	no	—
26	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
27	C₂×#108×C₇₃	C₂²	1	1	yes	characteristic
28	#427×C₇₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
30	C₂×#11924012	—	2	1	no	—
31	C₂×#11924011	—	4	1	no	—
32	M₁₆×D₇₃	—	2	4	no	—
33	M₁₆×D₇₃	—	4	2	no	—
34	C₂²×#10596831	D₄	1	1	yes	characteristic
35	C₂×C₈×D₇₃	—	4	1	no	—
36	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
41	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
42	C₂×C₈×D₇₃	D₄	1	2	yes	—
43	C₂×C₈×D₇₃	Q₈	1	2	yes	—
44	#14556550	—	2	1	no	—
45	#14556549	—	2	2	no	—
46	#14556548	—	4	1	no	—
47	#14556547	—	4	1	no	—
48	C₄⋊C₄×D₇₃	—	2	2	no	—
49	C₂²⋊C₄×D₇₃	—	4	1	no	—
50	C₄²×D₇₃	—	4	1	no	—
51	#14556530	—	2	4	no	—
52	#14556530	—	2	4	no	—
53	C₂×#11924003	—	2	1	no	—
54	C₂×#11924003	—	2	1	no	—
55	C₂×#11924003	C₂×C₄	1	2	yes	complemented
56	#14556527	—	2	1	no	—
57	#14556527	C₂×C₄	1	2	yes	complemented
58	#14556526	—	2	4	no	—
59	#14556525	C₂×C₄	1	2	yes	—
60	#14556524	—	2	2	no	—
61	#14556523	—	2	1	no	—
62	#14556522	—	2	2	no	—
63	C₂×M₁₆×C₇₃	—	2	1	no	—
64	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
65	#117×C₇₃	—	2	1	no	—
66	C₂×C₁₆×C₇₃	—	2	1	no	—
67	C₂×C₁₆×C₇₃	—	2	1	no	—
68	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	complemented
69	#108×C₇₃	—	2	1	no	—
70	#108×C₇₃	C₂×C₄	1	2	yes	complemented
71	#107×C₇₃	C₂×C₄	1	2	yes	—
72	#106×C₇₃	—	2	1	no	—
73	C₂³×D₇₃	SD₁₆	1	1	yes	characteristic
74	C₂²×Dic₇₃	SD₁₆	1	1	yes	characteristic
75	C₂×C₄×D₇₃	—	4	2	no	—
76	C₂×C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
80	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
81	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
82	C₂×C₄×D₇₃	SD₁₆	1	2	yes	—
83	C₂×C₄×D₇₃	Q₁₆	1	2	yes	—
84	#11924012	—	2	2	no	—
85	#11924012	—	4	1	no	—
86	#11924011	—	4	4	no	—
87	C₂×#10596831	—	4	1	no	—
88	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
91	C₂×#10596831	C₄⋊C₄	1	2	yes	—
92	C₈×D₇₃	—	4	4	no	—
93	C₈×D₇₃	C₄²	1	2	yes	—
94	C₈×D₇₃	C₄²	1	2	yes	—
95	C₈×D₇₃	C₂²⋊C₄	1	2	yes	—
96	C₈×D₇₃	C₂²⋊C₄	1	2	yes	—
97	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
98	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
99	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
100	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
101	C₄⋊Dic₇₃	—	2	2	no	—
102	#11924007	—	4	2	no	—
103	C₂²⋊Dic₇₃	—	4	1	no	—
104	#11924005	—	4	2	no	—
105	C₄×Dic₇₃	—	4	1	no	—
106	C₄×Dic₇₃	—	4	2	no	—
107	#11924003	—	2	2	no	—
108	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
109	M₁₆×C₇₃	—	2	2	no	—
110	M₁₆×C₇₃	—	4	1	no	—
111	C₂×C₈×C₇₃	—	4	1	no	—
112	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
118	C₄⋊C₄×C₇₃	—	2	2	no	—
119	C₂²⋊C₄×C₇₃	—	4	1	no	—
120	C₄²×C₇₃	—	4	1	no	—
121	C₁₆×C₇₃	—	2	2	no	—
122	C₂³×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
123	C₂²×D₇₃	—	4	2	no	—
124	C₂²×D₇₃	#102	1	1	yes	characteristic
125	C₂²×D₇₃	#102	1	1	yes	characteristic
126	C₂²×D₇₃	#103	1	2	yes	—
127	C₂²×D₇₃	#106	1	2	yes	—
128	C₂×C₄×C₇₃	—	4	2	no	—
129	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
135	C₂×Dic₇₃	—	4	2	no	—
136	C₂×Dic₇₃	#102	1	1	yes	characteristic
137	C₂×Dic₇₃	#102	1	1	yes	characteristic
138	C₂×Dic₇₃	#103	1	2	yes	—
139	C₂×Dic₇₃	#106	1	2	yes	—
140	C₄×D₇₃	—	8	4	no	—
141	C₄×D₇₃	#95	1	2	yes	—
142	C₄×D₇₃	#95	1	2	yes	—
143	C₄×D₇₃	#102	1	2	yes	—
144	C₄×D₇₃	#102	1	2	yes	—
145	C₄×D₇₃	#103	1	2	yes	—
146	C₄×D₇₃	#103	1	2	yes	—
147	C₄×D₇₃	#107	1	2	yes	—
148	C₄×D₇₃	#107	1	2	yes	—
149	C₈×C₇₃	—	4	2	no	—
150	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
151	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
154	#10596831	—	4	2	no	—
155	#10596831	#95	1	2	yes	—
156	#10596831	#95	1	2	yes	—
157	C₂²×C₇₃	—	2	2	no	—
158	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
159	C₂²×C₇₃	C₂×#103	1	1	yes	characteristic
160	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
161	C₂×D₇₃	—	2	2	no	—
162	C₂×D₇₃	—	2	2	no	—
163	C₂×D₇₃	—	4	2	no	—
164	C₂×D₇₃	—	4	2	no	—
165	C₂×D₇₃	#340	1	2	yes	—
166	C₂×D₇₃	#340	1	2	yes	—
167	C₄×C₇₃	—	8	2	no	—
168	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
169	C₄×C₇₃	C₂×#102	1	1	yes	characteristic
170	C₄×C₇₃	C₂×#103	1	1	yes	characteristic
171	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
172	Dic₇₃	—	8	2	no	—
173	Dic₇₃	#340	1	2	yes	—
174	Dic₇₃	#340	1	2	yes	—
175	#12561	—	73	1	no	maximal, Sylow
176	C₂×C₇₃	—	2	1	no	—
177	C₂×C₇₃	—	2	1	no	—
178	C₂×C₇₃	C₂×#340	1	1	yes	characteristic
179	C₂×C₇₃	#2143	1	2	yes	—
180	D₇₃	—	2	2	no	—
181	D₇₃	—	2	2	no	—
182	C₂²×M₃₂	—	73	1	no	—
183	C₂²×#108	—	73	1	no	—
184	C₂×#427	—	73	1	no	—
185	#1382	—	73	1	no	—
186	#1382	—	73	1	no	—
187	#1382	—	73	1	no	—
188	#1382	D₇₃	1	1	yes	characteristic, complemented
189	C₇₃	#12561	1	1	yes	characteristic, complemented, Sylow
190	C₂²×M₁₆	—	146	1	no	—
191	C₂³×C₈	—	73	1	no	—
192	C₂×#117	—	146	1	no	—
193	C₂×M₃₂	—	73	4	no	—
194	C₂×M₃₂	—	146	1	no	—
195	C₂×M₃₂	—	146	1	no	—
196	C₂×M₃₂	—	146	2	no	—
197	C₂²×C₁₆	—	73	1	no	—
198	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
199	C₂×#108	—	73	1	no	—
200	C₂×#108	—	73	4	no	—
201	C₂×#108	—	146	2	no	—
202	C₂×#108	—	146	1	no	—
203	C₂×#108	C₂×D₇₃	1	1	yes	characteristic
204	#427	—	73	1	no	—
205	#427	—	73	2	no	—
206	#427	—	146	1	no	—
207	#427	—	146	1	no	—
208	#427	C₂×D₇₃	1	1	yes	characteristic
209	C₂×#107	—	73	2	no	—
210	C₂×#106	—	146	1	no	—
211	C₂³×C₄	—	73	1	no	—
212	C₂×M₁₆	—	2	1	no	—
213	C₂×M₁₆	—	292	1	no	—
214	C₂×M₁₆	—	292	2	no	—
215	C₂×M₁₆	—	146	4	no	—
216	C₂×M₁₆	—	146	1	no	—
217	C₂²×C₈	—	292	1	no	—
218	C₂²×C₈	—	73	1	no	—
219	C₂²×C₈	—	73	1	no	—
220	C₂²×C₈	—	73	1	no	—
221	C₂²×C₈	—	73	2	no	—
222	C₂²×C₈	—	73	2	no	—
223	C₂²×C₈	—	73	1	no	—
224	C₂²×C₈	—	73	1	no	—
225	C₂²×C₈	—	73	2	no	—
226	C₂²×C₈	—	73	2	no	—
227	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
228	#117	—	2	1	no	—
229	#117	—	292	1	no	—
230	#117	—	292	1	no	—
231	#117	—	146	2	no	—
232	#117	—	146	1	no	—
233	C₂×C₄⋊C₄	—	146	2	no	—
234	C₂×C₂²⋊C₄	—	292	1	no	—
235	C₂×C₄²	—	292	1	no	—
236	M₃₂	—	146	4	no	—
237	M₃₂	—	146	4	no	—
238	C₂×C₁₆	—	2	1	no	—
239	C₂×C₁₆	—	2	1	no	—
240	C₂×C₁₆	—	73	2	no	—
241	C₂×C₁₆	—	146	1	no	—
242	C₂×C₁₆	—	146	1	no	—
243	C₂×C₁₆	C₄×D₇₃	1	2	yes	complemented
244	#108	—	2	1	no	—
245	#108	—	73	2	no	—
246	#108	—	146	4	no	—
247	#108	—	146	1	no	—
248	#108	C₄×D₇₃	1	2	yes	complemented
249	#107	—	73	2	no	—
250	#107	—	146	2	no	—
251	#107	C₄×D₇₃	1	2	yes	—
252	#106	—	2	1	no	—
253	#106	—	146	2	no	—
254	#106	—	146	1	no	—
255	C₂⁴	—	73	1	no	—
256	C₂²×C₄	—	292	2	no	—
257	C₂²×C₄	—	73	1	no	—
258	C₂²×C₄	—	73	1	no	—
259	C₂²×C₄	—	73	1	no	—
260	C₂²×C₄	—	73	2	no	—
261	C₂²×C₄	—	73	2	no	—
262	C₂²×C₄	—	73	1	no	—
263	C₂²×C₄	—	73	1	no	—
264	C₂²×C₄	—	73	2	no	—
265	C₂²×C₄	—	73	2	no	—
266	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
267	M₁₆	—	2	2	no	—
268	M₁₆	—	4	1	no	—
269	M₁₆	—	292	4	no	—
270	M₁₆	—	292	1	no	—
271	M₁₆	—	146	2	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	292	4	no	—
274	C₂×C₈	—	292	1	no	—
275	C₂×C₈	—	73	1	no	—
276	C₂×C₈	—	73	1	no	—
277	C₂×C₈	—	73	2	no	—
278	C₂×C₈	—	73	2	no	—
279	C₂×C₈	—	73	2	no	—
280	C₂×C₈	—	73	2	no	—
281	C₂×C₈	—	73	2	no	—
282	C₂×C₈	—	73	2	no	—
283	C₂×C₈	—	73	2	no	—
284	C₂×C₈	—	73	2	no	—
285	C₂×C₈	—	73	2	no	—
286	C₂×C₈	—	73	2	no	—
287	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
288	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
290	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
291	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
292	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	292	2	no	—
295	C₄⋊C₄	—	146	2	no	—
296	C₂²⋊C₄	—	4	1	no	—
297	C₂²⋊C₄	—	292	2	no	—
298	C₂²⋊C₄	—	292	1	no	—
299	C₄²	—	4	1	no	—
300	C₄²	—	292	2	no	—
301	C₄²	—	292	1	no	—
302	C₁₆	—	2	2	no	—
303	C₁₆	—	146	2	no	—
304	C₂³	—	292	2	no	—
305	C₂³	—	73	2	no	—
306	C₂³	—	73	2	no	—
307	C₂³	—	73	1	no	—
308	C₂³	—	73	1	no	—
309	C₂³	SD₁₆×D₇₃	1	1	yes	characteristic
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	292	2	no	—
312	C₂×C₄	—	584	4	no	—
313	C₂×C₄	—	73	2	no	—
314	C₂×C₄	—	73	1	no	—
315	C₂×C₄	—	73	1	no	—
316	C₂×C₄	—	73	2	no	—
317	C₂×C₄	—	73	2	no	—
318	C₂×C₄	—	73	2	no	—
319	C₂×C₄	—	73	2	no	—
320	C₂×C₄	—	73	2	no	—
321	C₂×C₄	—	73	2	no	—
322	C₂×C₄	—	73	2	no	—
323	C₂×C₄	—	73	2	no	—
324	C₂×C₄	—	73	2	no	—
325	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
326	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
327	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
328	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
329	C₂×C₄	SD₁₆×D₇₃	1	1	yes	characteristic
330	C₂×C₄	Q₁₆×D₇₃	1	1	yes	characteristic
331	C₈	—	4	2	no	—
332	C₈	—	292	2	no	—
333	C₈	—	73	2	no	—
334	C₈	—	73	2	no	—
335	C₈	C₄²×D₇₃	1	1	yes	characteristic
336	C₈	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
337	C₈	C₄⋊C₄×D₇₃	1	1	yes	characteristic
338	C₈	C₄⋊C₄×D₇₃	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	292	2	no	—
341	C₂²	—	292	2	no	—
342	C₂²	—	73	2	no	—
343	C₂²	—	73	2	no	—
344	C₂²	—	146	2	no	—
345	C₂²	—	146	2	no	—
346	C₂²	#102×D₇₃	1	1	yes	characteristic
347	C₂²	#103×D₇₃	1	1	yes	characteristic
348	C₂²	#106×D₇₃	1	1	yes	characteristic
349	C₄	—	8	2	no	—
350	C₄	—	584	2	no	—
351	C₄	—	73	2	no	—
352	C₄	—	73	2	no	—
353	C₄	#95×D₇₃	1	1	yes	characteristic
354	C₄	#102×D₇₃	1	1	yes	characteristic
355	C₄	#103×D₇₃	1	1	yes	characteristic
356	C₄	#107×D₇₃	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	146	2	no	—
360	C₂	—	146	2	no	—
361	C₂	#340×D₇₃	1	1	yes	characteristic
362	C₂	#22611657	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₇₃
commutator	C₈×C₇₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1382	C₂
socle	C₂²×C₇₃	C₂×#102
prime radical	C₂³×D₇₃	SD₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1382	D₇₃
hypocenter	C₇₃	#12561
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1382

Complements

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

N	H ≅ G/N	type
4. #22608028	C₂	semidirect
5. #22608027	C₂	semidirect
6. #22608026	C₂	semidirect
7. C₇₃×#1382	C₂	semidirect
14. C₂×#14556530	C₄	semidirect
19. #108×D₇₃	C₄	semidirect
55. C₂×#11924003	C₂×C₄	semidirect
57. #14556527	C₂×C₄	semidirect
68. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
70. #108×C₇₃	C₂×C₄	semidirect
188. #1382	D₇₃	semidirect
189. C₇₃	#12561	semidirect
243. C₂×C₁₆	C₄×D₇₃	semidirect
248. #108	C₄×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556530	2
C₂×#108×D₇₃	2
#427×D₇₃	2
#22608028	2
#22608027	2
#22608026	2
C₇₃×#1382	2
#12561	73