Subgroups — #32976258

Order 18688.

618 conjugacy classes of subgroups.

The subgroup lattice has 402 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	#22618559	C₂	1	1	yes	maximal, characteristic
3	C₂×#97×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#22607831	C₂	1	1	yes	maximal, characteristic, complemented
5	#22607830	C₂	1	1	yes	maximal, characteristic, complemented
6	#22607829	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1302	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₇₃	—	2	1	no	—
9	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
10	#17988644	C₂²	1	1	yes	characteristic
11	#17988643	—	2	1	no	—
12	#17988642	—	2	1	no	—
13	#17988641	C₄	1	2	yes	—
14	#17988640	C₂²	1	1	yes	characteristic
15	#17988639	C₄	1	2	yes	—
16	#17988638	—	2	1	no	—
17	#17988637	—	2	1	no	—
18	#17988609	—	2	1	no	—
19	#17988609	C₄	1	2	yes	—
20	#17988549	—	2	1	no	—
21	#17988549	C₄	1	2	yes	—
22	#17988539	—	2	1	no	—
23	C₄×#11924011	—	2	1	no	—
24	C₂×#14556502	C₂²	1	1	yes	characteristic
25	C₂×#14556501	—	2	1	no	—
26	#97×D₇₃	—	2	1	no	—
27	#97×D₇₃	—	2	1	no	—
28	#97×D₇₃	C₄	1	2	yes	—
29	#97×D₇₃	C₄	1	2	yes	—
30	#432×C₇₃	C₂²	1	1	yes	characteristic
31	#431×C₇₃	C₂²	1	1	yes	characteristic
32	C₂×#97×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×#11924011	—	2	2	no	—
38	C₂²×#10596831	—	2	1	no	—
39	C₂×C₈×D₇₃	—	2	8	no	—
40	C₂×C₈×D₇₃	—	2	2	no	—
41	C₂×C₈×D₇₃	—	2	2	no	—
42	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
44	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
45	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
49	C₄²×D₇₃	D₄	1	1	yes	characteristic
50	C₄²×D₇₃	D₄	1	1	yes	characteristic
51	C₄²×D₇₃	D₄	1	1	yes	characteristic
52	C₄²×D₇₃	D₄	1	1	yes	characteristic
53	C₄⋊#10596831	—	2	1	no	—
54	C₄⋊#10596831	C₂×C₄	1	2	yes	—
55	#14556520	—	2	2	no	—
56	#14556520	—	2	2	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	#14556502	—	2	1	no	—
62	#14556502	—	2	1	no	—
63	#14556502	C₂×C₄	1	2	yes	—
64	#14556501	—	2	2	no	—
65	#14556501	—	2	2	no	—
66	#14556501	—	2	2	no	—
67	C₄×#10596831	—	2	1	no	—
68	C₈×Dic₇₃	—	2	2	no	—
69	C₂²×C₈×C₇₃	—	2	1	no	—
70	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₇₃	—	2	1	no	—
72	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
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	—	2	1	no	—
77	#97×C₇₃	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₇₃	—	2	1	no	—
79	C₂³×D₇₃	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	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×D₇₃	—	2	4	no	—
84	C₂×C₄×D₇₃	—	4	2	no	—
85	C₂×C₄×D₇₃	—	4	2	no	—
86	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
87	C₂×C₄×D₇₃	C₄²	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	C₂²⋊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	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
98	#11924011	—	4	4	no	—
99	#11924011	—	4	4	no	—
100	C₂×#10596831	—	2	4	no	—
101	C₂×#10596831	—	2	2	no	—
102	C₂×#10596831	—	2	2	no	—
103	C₂×#10596831	—	2	2	no	—
104	C₈×D₇₃	—	2	8	no	—
105	C₈×D₇₃	—	2	8	no	—
106	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
107	C₄×Dic₇₃	C₄²	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	C₂²⋊C₄	1	1	yes	characteristic
112	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
113	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
114	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
115	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
116	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
117	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
121	C₂×C₈×C₇₃	—	2	4	no	—
122	C₂×C₈×C₇₃	—	2	2	no	—
123	C₂×C₈×C₇₃	—	2	2	no	—
124	C₂×C₈×C₇₃	—	2	2	no	—
125	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
129	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×D₇₃	—	2	2	no	—
131	C₂²×D₇₃	—	2	2	no	—
132	C₂²×D₇₃	—	2	2	no	—
133	C₂²×D₇₃	#95	1	1	yes	characteristic
134	C₂²×D₇₃	#95	1	1	yes	characteristic
135	C₂×C₄×C₇₃	—	2	2	no	—
136	C₂×C₄×C₇₃	—	2	2	no	—
137	C₂×C₄×C₇₃	—	4	1	no	—
138	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×Dic₇₃	—	2	2	no	—
145	C₂×Dic₇₃	—	2	2	no	—
146	C₂×Dic₇₃	—	4	1	no	—
147	C₂×Dic₇₃	#95	1	1	yes	characteristic
148	C₂×Dic₇₃	#95	1	1	yes	characteristic
149	C₂×Dic₇₃	#95	1	1	yes	characteristic
150	C₂×Dic₇₃	#95	1	1	yes	characteristic
151	C₂×Dic₇₃	#95	1	1	yes	characteristic
152	C₂×Dic₇₃	#95	1	1	yes	characteristic
153	C₄×D₇₃	—	2	2	no	—
154	C₄×D₇₃	—	2	2	no	—
155	C₄×D₇₃	—	2	2	no	—
156	C₄×D₇₃	—	2	2	no	—
157	C₄×D₇₃	—	2	2	no	—
158	C₄×D₇₃	—	4	1	no	—
159	C₄×D₇₃	—	4	1	no	—
160	C₄×D₇₃	—	4	1	no	—
161	C₄×D₇₃	—	4	1	no	—
162	C₄×D₇₃	—	4	1	no	—
163	C₄×D₇₃	—	4	1	no	—
164	C₄×D₇₃	#104	1	4	yes	—
165	C₈×C₇₃	—	2	4	no	—
166	C₈×C₇₃	—	4	2	no	—
167	C₈×C₇₃	—	4	2	no	—
168	#10596831	—	2	4	no	—
169	#10596831	—	4	2	no	—
170	#10596831	—	4	2	no	—
171	C₂²×C₇₃	—	2	1	no	—
172	C₂²×C₇₃	—	2	1	no	—
173	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
174	C₂²×C₇₃	#430	1	2	yes	—
175	C₂×D₇₃	—	2	8	no	—
176	C₂×D₇₃	—	2	1	no	—
177	C₂×D₇₃	—	2	1	no	—
178	C₂×D₇₃	—	2	1	no	—
179	C₂×D₇₃	—	2	1	no	—
180	C₂×D₇₃	#339	1	2	yes	—
181	C₂×D₇₃	#344	1	2	yes	—
182	C₄×C₇₃	—	2	1	no	—
183	C₄×C₇₃	—	2	1	no	—
184	C₄×C₇₃	—	2	1	no	—
185	C₄×C₇₃	—	2	1	no	—
186	C₄×C₇₃	—	2	1	no	—
187	C₄×C₇₃	C₂×#104	1	2	yes	—
188	Dic₇₃	—	2	1	no	—
189	Dic₇₃	—	2	1	no	—
190	Dic₇₃	—	2	1	no	—
191	Dic₇₃	—	2	1	no	—
192	Dic₇₃	—	2	1	no	—
193	Dic₇₃	#339	1	2	yes	—
194	#8088	—	73	1	no	maximal, Sylow
195	C₂×C₇₃	—	2	2	no	—
196	C₂×C₇₃	C₂×#339	1	1	yes	characteristic
197	C₂×C₇₃	#1738	1	1	yes	characteristic
198	C₂×C₇₃	C₂×#344	1	1	yes	characteristic
199	D₇₃	—	2	2	no	—
200	D₇₃	—	2	2	no	—
201	#2920	—	73	1	no	—
202	#2920	—	73	1	no	—
203	C₂²×#97	—	73	1	no	—
204	#1302	—	73	1	no	—
205	#1302	—	73	1	no	—
206	#1302	—	73	1	no	—
207	#1302	D₇₃	1	1	yes	characteristic, complemented
208	C₇₃	#8088	1	1	yes	characteristic, complemented, Sylow
209	C₂³×C₈	—	146	1	no	—
210	C₂²×C₄²	—	73	1	no	—
211	#432	—	73	2	no	—
212	#432	—	73	1	no	—
213	#432	—	146	1	no	—
214	#432	—	146	1	no	—
215	#432	C₂×D₇₃	1	1	yes	characteristic
216	#431	—	73	1	no	—
217	#431	—	73	2	no	—
218	#431	—	146	1	no	—
219	#431	—	146	1	no	—
220	#431	C₂×D₇₃	1	1	yes	characteristic
221	#423	—	73	2	no	—
222	#423	—	146	1	no	—
223	#407	—	73	2	no	—
224	#407	—	146	1	no	—
225	C₄×M₁₆	—	146	1	no	—
226	C₄×M₁₆	—	146	1	no	—
227	C₂×#97	—	73	1	no	—
228	C₂×#97	—	73	2	no	—
229	C₂×#97	—	73	2	no	—
230	C₂×#97	—	146	1	no	—
231	C₂×#97	—	146	1	no	—
232	C₂×#97	—	146	1	no	—
233	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
234	C₂³×C₄	—	73	1	no	—
235	C₂³×C₄	—	73	1	no	—
236	C₂³×C₄	—	73	1	no	—
237	C₂×M₁₆	—	146	2	no	—
238	C₂×M₁₆	—	146	2	no	—
239	C₂²×C₈	—	2	1	no	—
240	C₂²×C₈	—	146	2	no	—
241	C₂²×C₈	—	146	2	no	—
242	C₂²×C₈	—	146	8	no	—
243	C₂²×C₈	—	146	1	no	—
244	C₂×C₄²	—	73	1	no	—
245	C₂×C₄²	—	73	1	no	—
246	C₂×C₄²	—	73	1	no	—
247	C₂×C₄²	—	73	1	no	—
248	C₂×C₄²	—	73	1	no	—
249	C₂×C₄²	—	73	1	no	—
250	C₂×C₄²	—	73	1	no	—
251	C₂×C₄²	—	73	1	no	—
252	C₂×C₄²	—	73	1	no	—
253	C₂×C₄²	—	73	1	no	—
254	C₂×C₄²	—	73	1	no	—
255	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
256	C₄⋊C₈	—	2	1	no	—
257	C₄⋊C₈	—	73	2	no	—
258	C₄⋊C₈	—	146	2	no	—
259	C₄⋊C₈	—	146	2	no	—
260	C₄⋊C₈	—	146	1	no	—
261	C₄⋊C₈	C₄×D₇₃	1	2	yes	—
262	C₂²⋊C₈	—	2	1	no	—
263	C₂²⋊C₈	—	73	2	no	—
264	C₂²⋊C₈	—	146	2	no	—
265	C₂²⋊C₈	—	146	2	no	—
266	C₂²⋊C₈	—	146	1	no	—
267	C₂²⋊C₈	C₄×D₇₃	1	2	yes	—
268	#97	—	2	1	no	—
269	#97	—	2	1	no	—
270	#97	—	73	2	no	—
271	#97	—	146	2	no	—
272	#97	—	146	2	no	—
273	#97	—	146	2	no	—
274	#97	—	146	1	no	—
275	#97	—	146	1	no	—
276	#97	C₄×D₇₃	1	2	yes	—
277	C₄×C₈	—	2	1	no	—
278	C₄×C₈	—	146	2	no	—
279	C₄×C₈	—	146	1	no	—
280	C₂⁴	—	73	1	no	—
281	C₂²×C₄	—	292	2	no	—
282	C₂²×C₄	—	292	2	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	—	73	1	no	—
294	C₂²×C₄	—	73	1	no	—
295	C₂²×C₄	—	73	1	no	—
296	C₂²×C₄	—	73	1	no	—
297	C₂²×C₄	—	73	1	no	—
298	C₂²×C₄	—	146	4	no	—
299	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
300	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
301	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
302	M₁₆	—	292	4	no	—
303	M₁₆	—	292	4	no	—
304	C₂×C₈	—	2	4	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	146	8	no	—
309	C₂×C₈	—	146	8	no	—
310	C₂×C₈	—	146	2	no	—
311	C₂×C₈	—	146	2	no	—
312	C₂×C₈	—	146	4	no	—
313	C₂×C₈	—	146	2	no	—
314	C₄²	—	73	1	no	—
315	C₄²	—	73	1	no	—
316	C₄²	—	73	1	no	—
317	C₄²	—	73	1	no	—
318	C₄²	—	73	1	no	—
319	C₄²	—	73	1	no	—
320	C₄²	—	73	1	no	—
321	C₄²	—	73	1	no	—
322	C₄²	—	73	1	no	—
323	C₄²	—	73	1	no	—
324	C₄²	—	73	1	no	—
325	C₄²	—	73	1	no	—
326	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
327	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
328	C₄²	D₄×D₇₃	1	1	yes	characteristic
329	C₄²	D₄×D₇₃	1	1	yes	characteristic
330	C₂³	—	73	1	no	—
331	C₂³	—	73	1	no	—
332	C₂³	—	146	2	no	—
333	C₂³	—	146	2	no	—
334	C₂³	—	146	2	no	—
335	C₂³	C₄⋊C₄×D₇₃	1	1	yes	characteristic
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	292	1	no	—
340	C₂×C₄	—	292	1	no	—
341	C₂×C₄	—	292	1	no	—
342	C₂×C₄	—	292	1	no	—
343	C₂×C₄	—	292	1	no	—
344	C₂×C₄	—	292	1	no	—
345	C₂×C₄	—	292	1	no	—
346	C₂×C₄	—	73	1	no	—
347	C₂×C₄	—	73	4	no	—
348	C₂×C₄	—	73	1	no	—
349	C₂×C₄	—	73	1	no	—
350	C₂×C₄	—	73	1	no	—
351	C₂×C₄	—	73	1	no	—
352	C₂×C₄	—	73	1	no	—
353	C₂×C₄	—	146	2	no	—
354	C₂×C₄	—	146	2	no	—
355	C₂×C₄	—	146	2	no	—
356	C₂×C₄	—	146	2	no	—
357	C₂×C₄	—	146	2	no	—
358	C₂×C₄	—	146	2	no	—
359	C₂×C₄	—	146	2	no	—
360	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
361	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
362	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
363	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
364	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
365	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
366	C₈	—	2	4	no	—
367	C₈	—	4	2	no	—
368	C₈	—	4	2	no	—
369	C₈	—	292	2	no	—
370	C₈	—	292	2	no	—
371	C₈	—	146	4	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	2	1	no	—
374	C₂²	—	73	2	no	—
375	C₂²	—	73	2	no	—
376	C₂²	—	146	8	no	—
377	C₂²	—	146	1	no	—
378	C₂²	—	146	1	no	—
379	C₂²	—	146	1	no	—
380	C₂²	—	146	1	no	—
381	C₂²	#95×D₇₃	1	1	yes	characteristic
382	C₂²	#17988634	1	2	yes	—
383	C₄	—	2	1	no	—
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	73	2	no	—
389	C₄	—	146	1	no	—
390	C₄	—	146	1	no	—
391	C₄	—	146	1	no	—
392	C₄	—	146	1	no	—
393	C₄	—	146	1	no	—
394	C₄	#104×D₇₃	1	2	yes	—
395	C₂	—	2	2	no	—
396	C₂	—	146	2	no	—
397	C₂	—	146	2	no	—
398	C₂	#339×D₇₃	1	1	yes	characteristic
399	C₂	#22609656	1	1	yes	characteristic
400	C₂	#344×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1302	C₂
socle	C₂²×C₇₃	C₂×#95
prime radical	C₂³×D₇₃	C₄⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1302	D₇₃
hypocenter	C₇₃	#8088
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1302

Complements

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

N	H ≅ G/N	type
4. #22607831	C₂	semidirect
5. #22607830	C₂	semidirect
6. #22607829	C₂	semidirect
7. C₇₃×#1302	C₂	semidirect
207. #1302	D₇₃	semidirect
208. C₇₃	#8088	semidirect

Maximal subgroups

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

subgroup	index
#22618561	2
#22618559	2
C₂×#97×D₇₃	2
#22607831	2
#22607830	2
#22607829	2
C₇₃×#1302	2
#8088	73