Subgroups — #32975802

Order 18688.

544 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	#22618561	C₂	1	1	yes	maximal, characteristic
2	#432×D₇₃	C₂	1	1	yes	maximal, characteristic
3	#431×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#22607794	C₂	1	1	yes	maximal, characteristic, complemented
5	#22607793	C₂	1	1	yes	maximal, characteristic, complemented
6	#22607792	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1287	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
9	#17988644	C₂²	1	1	yes	characteristic
10	#17988644	C₂²	1	1	yes	characteristic
11	#17988643	—	2	1	no	—
12	#17988642	—	2	1	no	—
13	#17988642	C₄	1	2	yes	—
14	#17988641	C₄	1	2	yes	—
15	#17988640	C₂²	1	1	yes	characteristic
16	#17988639	—	2	1	no	—
17	#17988639	—	2	1	no	—
18	#17988638	C₄	1	2	yes	—
19	#17988637	—	2	1	no	—
20	#17988637	—	2	1	no	—
21	#17988609	—	2	1	no	—
22	C₄⋊C₈×D₇₃	—	2	1	no	—
23	C₄⋊C₈×D₇₃	—	2	1	no	—
24	#17988549	—	2	1	no	—
25	C₂²⋊C₈×D₇₃	—	2	1	no	—
26	C₂²⋊C₈×D₇₃	C₄	1	2	yes	—
27	#17988539	C₄	1	2	yes	—
28	#97×D₇₃	—	2	1	no	—
29	C₄×C₈×D₇₃	C₄	1	2	yes	—
30	#432×C₇₃	C₂²	1	1	yes	characteristic
31	#432×C₇₃	C₂²	1	1	yes	characteristic
32	#431×C₇₃	C₂²	1	1	yes	characteristic
33	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×D₇₃	Q₈	1	1	yes	characteristic
36	C₂×#11924011	—	2	2	no	—
37	C₂×C₈×D₇₃	—	2	2	no	—
38	C₂×C₈×D₇₃	—	2	2	no	—
39	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
42	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
45	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×D₇₃	D₄	1	1	yes	characteristic
47	C₄²×D₇₃	D₄	1	1	yes	characteristic
48	C₄²×D₇₃	D₄	1	1	yes	characteristic
49	C₄²×D₇₃	D₄	1	1	yes	characteristic
50	C₄⋊#10596831	—	2	1	no	—
51	C₄⋊#10596831	—	2	1	no	—
52	C₄⋊#10596831	—	2	1	no	—
53	#14556520	—	2	2	no	—
54	#14556520	—	2	2	no	—
55	#14556520	—	2	2	no	—
56	C₂²⋊#10596831	—	2	1	no	—
57	C₂²⋊#10596831	—	2	1	no	—
58	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
59	#14556503	—	2	2	no	—
60	#14556503	—	2	2	no	—
61	#14556503	—	2	2	no	—
62	#14556502	—	2	1	no	—
63	#14556502	C₂×C₄	1	2	yes	—
64	#14556501	—	2	2	no	—
65	C₄×#10596831	C₂×C₄	1	2	yes	—
66	C₈×Dic₇₃	—	2	2	no	—
67	C₈×Dic₇₃	—	2	2	no	—
68	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₇₃	—	2	1	no	—
70	C₄⋊C₈×C₇₃	—	2	1	no	—
71	C₄⋊C₈×C₇₃	—	2	1	no	—
72	C₂²⋊C₈×C₇₃	—	2	1	no	—
73	C₂²⋊C₈×C₇₃	—	2	1	no	—
74	C₂²⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
75	#97×C₇₃	—	2	1	no	—
76	#97×C₇₃	C₂×C₄	1	2	yes	—
77	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
78	C₂³×D₇₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×D₇₃	—	2	4	no	—
83	C₂×C₄×D₇₃	—	4	2	no	—
84	C₂×C₄×D₇₃	—	4	2	no	—
85	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
86	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
87	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
97	#11924011	—	4	4	no	—
98	C₂×#10596831	—	2	2	no	—
99	C₂×#10596831	—	2	2	no	—
100	C₂×#10596831	—	2	2	no	—
101	C₈×D₇₃	—	4	4	no	—
102	C₈×D₇₃	—	4	4	no	—
103	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
104	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
105	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
106	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
107	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
108	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
109	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
110	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
111	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
112	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
113	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
114	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
118	C₂×C₈×C₇₃	—	2	2	no	—
119	C₂×C₈×C₇₃	—	2	2	no	—
120	C₂×C₈×C₇₃	—	2	2	no	—
121	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
125	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×D₇₃	—	2	2	no	—
127	C₂²×D₇₃	—	2	2	no	—
128	C₂²×D₇₃	—	4	1	no	—
129	C₂²×D₇₃	#95	1	1	yes	characteristic
130	C₂²×D₇₃	#95	1	1	yes	characteristic
131	C₂×C₄×C₇₃	—	2	2	no	—
132	C₂×C₄×C₇₃	—	4	1	no	—
133	C₂×C₄×C₇₃	—	4	1	no	—
134	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×Dic₇₃	—	2	2	no	—
141	C₂×Dic₇₃	—	2	2	no	—
142	C₂×Dic₇₃	—	4	1	no	—
143	C₂×Dic₇₃	#95	1	1	yes	characteristic
144	C₂×Dic₇₃	#95	1	1	yes	characteristic
145	C₂×Dic₇₃	#95	1	1	yes	characteristic
146	C₂×Dic₇₃	#95	1	1	yes	characteristic
147	C₂×Dic₇₃	#95	1	1	yes	characteristic
148	C₂×Dic₇₃	#95	1	1	yes	characteristic
149	C₄×D₇₃	—	2	2	no	—
150	C₄×D₇₃	—	2	2	no	—
151	C₄×D₇₃	—	2	2	no	—
152	C₄×D₇₃	—	2	2	no	—
153	C₄×D₇₃	—	2	2	no	—
154	C₄×D₇₃	—	2	2	no	—
155	C₄×D₇₃	—	2	2	no	—
156	C₄×D₇₃	—	4	1	no	—
157	C₄×D₇₃	—	4	1	no	—
158	C₄×D₇₃	—	4	1	no	—
159	C₄×D₇₃	—	4	1	no	—
160	C₄×D₇₃	#108	1	4	yes	—
161	C₈×C₇₃	—	4	2	no	—
162	C₈×C₇₃	—	4	2	no	—
163	C₈×C₇₃	—	4	2	no	—
164	#10596831	—	4	2	no	—
165	#10596831	—	4	2	no	—
166	#10596831	—	4	2	no	—
167	C₂²×C₇₃	—	2	1	no	—
168	C₂²×C₇₃	—	2	1	no	—
169	C₂²×C₇₃	—	2	1	no	—
170	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
171	C₂×D₇₃	—	2	1	no	—
172	C₂×D₇₃	—	2	1	no	—
173	C₂×D₇₃	—	2	1	no	—
174	C₂×D₇₃	—	2	1	no	—
175	C₂×D₇₃	—	4	4	no	—
176	C₂×D₇₃	H(3,ℤ/4)	1	2	yes	—
177	C₂×D₇₃	#341	1	2	yes	—
178	C₄×C₇₃	—	2	1	no	—
179	C₄×C₇₃	—	2	1	no	—
180	C₄×C₇₃	—	2	1	no	—
181	C₄×C₇₃	—	2	1	no	—
182	C₄×C₇₃	—	2	1	no	—
183	C₄×C₇₃	C₂×#108	1	2	yes	—
184	Dic₇₃	—	2	1	no	—
185	Dic₇₃	—	2	1	no	—
186	Dic₇₃	—	2	1	no	—
187	Dic₇₃	—	2	1	no	—
188	Dic₇₃	#341	1	2	yes	—
189	Dic₇₃	#344	1	2	yes	—
190	#7966	—	73	1	no	maximal, Sylow
191	C₂×C₇₃	—	4	1	no	—
192	C₂×C₇₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
193	C₂×C₇₃	C₂×#341	1	1	yes	characteristic
194	C₂×C₇₃	C₂×#344	1	1	yes	characteristic
195	D₇₃	—	2	2	no	—
196	D₇₃	—	2	2	no	—
197	#2920	—	73	1	no	—
198	C₂×#432	—	73	1	no	—
199	C₂×#431	—	73	1	no	—
200	#1287	—	73	1	no	—
201	#1287	—	73	1	no	—
202	#1287	—	73	1	no	—
203	#1287	D₇₃	1	1	yes	characteristic, complemented
204	C₇₃	#7966	1	1	yes	characteristic, complemented, Sylow
205	C₂²×C₄²	—	73	1	no	—
206	#432	—	73	1	no	—
207	#432	—	73	2	no	—
208	#432	—	73	1	no	—
209	#432	—	73	2	no	—
210	#432	—	146	1	no	—
211	#432	—	146	1	no	—
212	#432	C₂×D₇₃	1	1	yes	characteristic
213	#432	C₂×D₇₃	1	1	yes	characteristic
214	#431	—	73	1	no	—
215	#431	—	73	2	no	—
216	#431	—	146	1	no	—
217	#431	—	146	1	no	—
218	#431	—	146	1	no	—
219	#431	—	146	1	no	—
220	#431	C₂×D₇₃	1	1	yes	characteristic
221	#423	—	146	1	no	—
222	C₂×C₄⋊C₈	—	146	1	no	—
223	C₂×C₄⋊C₈	—	146	1	no	—
224	#407	—	146	1	no	—
225	C₂×C₂²⋊C₈	—	73	2	no	—
226	C₂×C₂²⋊C₈	—	146	1	no	—
227	C₄×M₁₆	—	73	2	no	—
228	C₂×#97	—	146	1	no	—
229	C₂×C₄×C₈	—	73	2	no	—
230	C₂³×C₄	—	73	1	no	—
231	C₂³×C₄	—	73	1	no	—
232	C₂³×C₄	—	73	1	no	—
233	C₂×M₁₆	—	146	2	no	—
234	C₂²×C₈	—	146	2	no	—
235	C₂²×C₈	—	146	2	no	—
236	C₂×C₄²	—	73	1	no	—
237	C₂×C₄²	—	73	1	no	—
238	C₂×C₄²	—	73	1	no	—
239	C₂×C₄²	—	73	1	no	—
240	C₂×C₄²	—	73	1	no	—
241	C₂×C₄²	—	73	1	no	—
242	C₂×C₄²	—	73	1	no	—
243	C₂×C₄²	—	73	1	no	—
244	C₂×C₄²	—	73	1	no	—
245	C₂×C₄²	—	73	1	no	—
246	C₂×C₄²	—	73	1	no	—
247	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	146	2	no	—
252	C₄⋊C₈	—	146	2	no	—
253	C₄⋊C₈	—	146	2	no	—
254	C₄⋊C₈	—	146	1	no	—
255	C₄⋊C₈	—	146	1	no	—
256	C₄⋊C₈	—	146	1	no	—
257	C₂²⋊C₈	—	2	1	no	—
258	C₂²⋊C₈	—	2	1	no	—
259	C₂²⋊C₈	—	73	2	no	—
260	C₂²⋊C₈	—	146	2	no	—
261	C₂²⋊C₈	—	146	2	no	—
262	C₂²⋊C₈	—	146	2	no	—
263	C₂²⋊C₈	—	146	1	no	—
264	C₂²⋊C₈	—	146	1	no	—
265	C₂²⋊C₈	C₄×D₇₃	1	2	yes	—
266	#97	—	2	1	no	—
267	#97	—	73	2	no	—
268	#97	—	146	2	no	—
269	#97	—	146	1	no	—
270	#97	C₄×D₇₃	1	2	yes	—
271	C₄×C₈	—	73	2	no	—
272	C₄×C₈	—	146	2	no	—
273	C₄×C₈	—	146	2	no	—
274	C₄×C₈	C₄×D₇₃	1	2	yes	—
275	C₂⁴	—	73	1	no	—
276	C₂²×C₄	—	292	2	no	—
277	C₂²×C₄	—	292	2	no	—
278	C₂²×C₄	—	73	1	no	—
279	C₂²×C₄	—	73	1	no	—
280	C₂²×C₄	—	73	1	no	—
281	C₂²×C₄	—	73	1	no	—
282	C₂²×C₄	—	73	1	no	—
283	C₂²×C₄	—	73	1	no	—
284	C₂²×C₄	—	73	1	no	—
285	C₂²×C₄	—	73	1	no	—
286	C₂²×C₄	—	73	1	no	—
287	C₂²×C₄	—	73	1	no	—
288	C₂²×C₄	—	73	1	no	—
289	C₂²×C₄	—	73	1	no	—
290	C₂²×C₄	—	73	1	no	—
291	C₂²×C₄	—	73	1	no	—
292	C₂²×C₄	—	73	1	no	—
293	C₂²×C₄	—	146	4	no	—
294	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
295	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
296	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
297	M₁₆	—	292	4	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	4	no	—
303	C₂×C₈	—	146	2	no	—
304	C₂×C₈	—	146	2	no	—
305	C₂×C₈	—	146	2	no	—
306	C₄²	—	73	1	no	—
307	C₄²	—	73	1	no	—
308	C₄²	—	73	1	no	—
309	C₄²	—	73	1	no	—
310	C₄²	—	73	1	no	—
311	C₄²	—	73	1	no	—
312	C₄²	—	73	1	no	—
313	C₄²	—	73	1	no	—
314	C₄²	—	73	1	no	—
315	C₄²	—	73	1	no	—
316	C₄²	—	73	1	no	—
317	C₄²	—	73	1	no	—
318	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
319	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
320	C₄²	D₄×D₇₃	1	1	yes	characteristic
321	C₄²	D₄×D₇₃	1	1	yes	characteristic
322	C₂³	—	292	1	no	—
323	C₂³	—	73	1	no	—
324	C₂³	—	73	1	no	—
325	C₂³	—	146	2	no	—
326	C₂³	—	146	2	no	—
327	C₂³	C₄⋊C₄×D₇₃	1	1	yes	characteristic
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	292	1	no	—
332	C₂×C₄	—	292	1	no	—
333	C₂×C₄	—	292	1	no	—
334	C₂×C₄	—	292	1	no	—
335	C₂×C₄	—	292	1	no	—
336	C₂×C₄	—	73	1	no	—
337	C₂×C₄	—	73	1	no	—
338	C₂×C₄	—	73	4	no	—
339	C₂×C₄	—	73	1	no	—
340	C₂×C₄	—	73	1	no	—
341	C₂×C₄	—	73	1	no	—
342	C₂×C₄	—	73	1	no	—
343	C₂×C₄	—	146	2	no	—
344	C₂×C₄	—	146	2	no	—
345	C₂×C₄	—	146	2	no	—
346	C₂×C₄	—	146	2	no	—
347	C₂×C₄	—	146	2	no	—
348	C₂×C₄	—	146	2	no	—
349	C₂×C₄	—	146	2	no	—
350	C₂×C₄	—	146	2	no	—
351	C₂×C₄	—	146	2	no	—
352	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
353	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
354	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
355	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
356	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
357	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
358	C₈	—	4	2	no	—
359	C₈	—	4	2	no	—
360	C₈	—	4	2	no	—
361	C₈	—	292	2	no	—
362	C₈	—	292	2	no	—
363	C₈	—	292	2	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	292	4	no	—
368	C₂²	—	73	2	no	—
369	C₂²	—	73	2	no	—
370	C₂²	—	146	1	no	—
371	C₂²	—	146	1	no	—
372	C₂²	—	146	1	no	—
373	C₂²	—	146	1	no	—
374	C₂²	#95×D₇₃	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	73	2	no	—
381	C₄	—	73	2	no	—
382	C₄	—	146	1	no	—
383	C₄	—	146	1	no	—
384	C₄	—	146	1	no	—
385	C₄	—	146	1	no	—
386	C₄	#108×D₇₃	1	2	yes	—
387	C₂	—	4	1	no	—
388	C₂	—	146	2	no	—
389	C₂	—	146	2	no	—
390	C₂	H(3,ℤ/4)×D₇₃	1	1	yes	characteristic
391	C₂	#341×D₇₃	1	1	yes	characteristic
392	C₂	#344×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#95×D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1287	C₂
socle	C₂²×C₇₃	C₂×#95
prime radical	C₂³×D₇₃	C₄⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1287	D₇₃
hypocenter	C₇₃	#7966
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1287

Complements

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

N	H ≅ G/N	type
4. #22607794	C₂	semidirect
5. #22607793	C₂	semidirect
6. #22607792	C₂	semidirect
7. C₇₃×#1287	C₂	semidirect
203. #1287	D₇₃	semidirect
204. C₇₃	#7966	semidirect

Maximal subgroups

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

subgroup	index
#22618561	2
#432×D₇₃	2
#431×D₇₃	2
#22607794	2
#22607793	2
#22607792	2
C₇₃×#1287	2
#7966	73