Subgroups — #32975539

Order 18688.

798 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#10596831	C₂	1	1	yes	maximal, characteristic
2	#22612338	C₂	1	1	yes	maximal, characteristic
3	#22607770	C₂	1	4	yes	maximal, complemented
4	C₂×#423×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#10596831	—	2	1	no	—
6	C₂×#14556550	—	2	1	no	—
7	C₂×C₄×#10596831	—	2	1	no	—
8	C₂×C₄×#10596831	—	2	2	no	—
9	C₂×C₄×#10596831	C₄	1	4	yes	complemented
10	C₂×C₄×#10596831	C₂²	1	2	yes	—
11	#17988457	C₄	1	2	yes	—
12	#17988453	—	2	1	no	—
13	#17988447	—	2	1	no	—
14	#17988447	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₇₃	—	2	1	no	—
16	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₇₃	—	2	1	no	—
18	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
19	#423×C₇₃	—	2	2	no	—
20	#423×C₇₃	C₄	1	4	yes	complemented
21	C₂×C₄⋊C₈×C₇₃	—	2	1	no	—
22	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
23	C₂²×#10596831	—	2	8	no	—
24	C₂²×#10596831	—	2	4	no	—
25	C₂²×#10596831	—	2	2	no	—
26	#14556550	—	4	2	no	—
27	#14556550	—	4	2	no	—
28	C₂×C₄⋊Dic₇₃	—	2	2	no	—
29	C₂×C₄⋊Dic₇₃	—	2	2	no	—
30	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
31	C₂×C₂²⋊Dic₇₃	—	2	2	no	—
32	C₂×C₄×Dic₇₃	—	2	2	no	—
33	C₄×#10596831	—	2	4	no	—
34	C₄×#10596831	—	2	2	no	—
35	C₄×#10596831	C₂×C₄	1	4	yes	—
36	#14556498	—	2	2	no	—
37	#14556498	—	2	2	no	—
38	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
41	C₂×M₁₆×C₇₃	—	2	4	no	—
42	C₂×M₁₆×C₇₃	—	4	2	no	—
43	C₂×M₁₆×C₇₃	—	4	2	no	—
44	C₂²×C₈×C₇₃	—	2	2	no	—
45	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₇₃	D₄	1	2	yes	—
51	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
52	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₇₃	—	2	4	no	—
54	C₄⋊C₈×C₇₃	—	2	2	no	—
55	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
56	C₂²×Dic₇₃	—	2	4	no	—
57	C₂×#10596831	—	2	8	no	—
58	C₂×#10596831	—	2	16	no	—
59	C₂×#10596831	—	2	4	no	—
60	C₄⋊Dic₇₃	—	2	8	no	—
61	C₄⋊Dic₇₃	—	4	4	no	—
62	C₄⋊Dic₇₃	—	4	4	no	—
63	C₂²⋊Dic₇₃	—	4	4	no	—
64	C₄×Dic₇₃	—	4	2	no	—
65	C₄×Dic₇₃	—	4	2	no	—
66	C₂⁴×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	—	2	4	no	—
68	C₂²×C₄×C₇₃	—	2	4	no	—
69	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
70	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₇₃	D₈	1	2	yes	—
79	C₂²×C₄×C₇₃	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₇₃	SD₁₆	1	2	yes	—
81	C₂²×C₄×C₇₃	Q₁₆	1	2	yes	—
82	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
85	M₁₆×C₇₃	—	4	8	no	—
86	C₂×C₈×C₇₃	—	2	4	no	—
87	C₂×C₈×C₇₃	—	4	2	no	—
88	C₂×C₈×C₇₃	—	4	2	no	—
89	C₄²×C₇₃	C₄²	1	2	yes	—
90	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
91	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
93	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
94	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
95	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
99	C₂³×C₇₃	—	2	2	no	—
100	C₂³×C₇₃	—	2	2	no	—
101	C₂³×C₇₃	#95	1	2	yes	—
102	C₂³×C₇₃	#102	1	2	yes	—
103	C₂³×C₇₃	#103	1	2	yes	—
104	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	—	2	2	no	—
106	C₂×C₄×C₇₃	—	2	2	no	—
107	C₂×C₄×C₇₃	—	2	4	no	—
108	C₂×C₄×C₇₃	—	2	4	no	—
109	C₂×C₄×C₇₃	—	2	4	no	—
110	C₂×C₄×C₇₃	—	2	4	no	—
111	C₂×C₄×C₇₃	#95	1	2	yes	—
112	C₂×C₄×C₇₃	#95	1	2	yes	—
113	C₂×C₄×C₇₃	#95	1	2	yes	—
114	C₂×C₄×C₇₃	#102	1	2	yes	—
115	C₂×C₄×C₇₃	#102	1	4	yes	—
116	C₂×C₄×C₇₃	#103	1	2	yes	—
117	C₂×C₄×C₇₃	#103	1	4	yes	—
118	C₂×C₄×C₇₃	#104	1	4	yes	—
119	C₂×C₄×C₇₃	#106	1	4	yes	—
120	C₂×C₄×C₇₃	#107	1	4	yes	—
121	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₇₃	#131	1	4	yes	—
128	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
132	C₂×Dic₇₃	—	4	4	no	—
133	C₂×Dic₇₃	—	4	4	no	—
134	C₂×Dic₇₃	—	4	4	no	—
135	C₈×C₇₃	—	4	4	no	—
136	#10596831	—	2	8	no	—
137	C₂²×C₇₃	—	2	2	no	—
138	C₂²×C₇₃	—	2	2	no	—
139	C₂²×C₇₃	—	2	4	no	—
140	C₂²×C₇₃	—	2	4	no	—
141	C₂²×C₇₃	#340	1	4	yes	—
142	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
143	C₂²×C₇₃	#408	1	2	yes	—
144	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
145	C₂²×C₇₃	C₂×#103	1	1	yes	characteristic
146	C₂²×C₇₃	C₂×#104	1	2	yes	—
147	C₄×C₇₃	—	2	2	no	—
148	C₄×C₇₃	—	2	1	no	—
149	C₄×C₇₃	—	2	1	no	—
150	C₄×C₇₃	#339	1	4	yes	—
151	C₄×C₇₃	#340	1	4	yes	—
152	C₄×C₇₃	#405	1	2	yes	—
153	C₄×C₇₃	C₂×#102	1	1	yes	characteristic
154	C₄×C₇₃	C₂×#103	1	1	yes	characteristic
155	C₄×C₇₃	#424	1	2	yes	—
156	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
157	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
158	Dic₇₃	—	8	4	no	—
159	#7897	—	73	1	no	maximal, Sylow
160	C₂×C₇₃	—	2	4	no	—
161	C₂×C₇₃	#1492	1	2	yes	—
162	C₂×C₇₃	#1493	1	2	yes	—
163	C₂×C₇₃	#1729	1	1	yes	characteristic
164	C₂×C₇₃	C₂×#339	1	1	yes	characteristic
165	C₂×C₇₃	C₂×#340	1	1	yes	characteristic
166	C₂×#423	D₇₃	1	1	yes	characteristic
167	C₂²×C₄×C₈	—	73	1	no	—
168	#2285	—	73	1	no	—
169	#1279	—	73	4	no	—
170	C₇₃	#7897	1	1	yes	characteristic, complemented, Sylow
171	C₂²×M₁₆	—	2	1	no	—
172	C₂³×C₈	—	146	1	no	—
173	C₂×#117	—	146	1	no	—
174	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
175	#423	—	2	2	no	—
176	#423	Dic₇₃	1	4	yes	complemented
177	C₂×C₄⋊C₈	—	2	1	no	—
178	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
179	C₂×C₄×C₈	—	73	4	no	—
180	C₂×C₄×C₈	—	73	2	no	—
181	C₂×C₄×C₈	—	146	1	no	—
182	C₂×C₄×C₈	—	146	2	no	—
183	#384	—	73	2	no	—
184	#382	—	146	1	no	—
185	#380	—	73	2	no	—
186	#380	—	146	1	no	—
187	C₄×C₄⋊C₄	—	73	2	no	—
188	C₄×C₄⋊C₄	—	146	1	no	—
189	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
190	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
191	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
192	C₂×M₁₆	—	2	4	no	—
193	C₂×M₁₆	—	4	2	no	—
194	C₂×M₁₆	—	4	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	146	8	no	—
197	C₂²×C₈	—	146	4	no	—
198	C₂²×C₈	—	146	2	no	—
199	#117	—	292	2	no	—
200	#117	—	292	2	no	—
201	C₂×C₄⋊C₄	—	73	4	no	—
202	C₂×C₄⋊C₄	—	146	2	no	—
203	C₂×C₄⋊C₄	—	146	2	no	—
204	C₂×C₂²⋊C₄	—	146	2	no	—
205	C₂×C₄²	—	146	2	no	—
206	C₂×C₄²	Dic₁₄₆	1	2	yes	—
207	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
208	C₂×C₄²	C₄×D₇₃	1	2	yes	—
209	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
210	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
211	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
212	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
213	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
214	C₄⋊C₈	—	2	4	no	—
215	C₄⋊C₈	—	2	2	no	—
216	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
217	C₄×C₈	—	73	4	no	—
218	C₄×C₈	—	146	4	no	—
219	C₄×C₈	—	146	2	no	—
220	#95	—	146	2	no	—
221	#95	—	146	2	no	—
222	C₂⁴	#11924005	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	146	4	no	—
226	C₂²×C₄	C₄×Dic₇₃	1	2	yes	—
227	C₂²×C₄	#11924005	1	1	yes	characteristic
228	C₂²×C₄	#11924005	1	2	yes	—
229	C₂²×C₄	#11924005	1	1	yes	characteristic
230	C₂²×C₄	#11924005	1	1	yes	characteristic
231	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
232	C₂²×C₄	#11924007	1	2	yes	—
233	C₂²×C₄	#11924007	1	2	yes	—
234	C₂²×C₄	C₄⋊Dic₇₃	1	2	yes	—
235	C₂²×C₄	#11924013	1	2	yes	—
236	C₂²×C₄	#11924015	1	2	yes	—
237	C₂²×C₄	#11924016	1	2	yes	—
238	C₂²×C₄	#11924018	1	2	yes	—
239	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
242	M₁₆	—	4	8	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	146	8	no	—
247	C₂×C₈	—	146	16	no	—
248	C₂×C₈	—	146	4	no	—
249	C₄⋊C₄	—	292	4	no	—
250	C₄⋊C₄	—	292	4	no	—
251	C₄⋊C₄	—	146	8	no	—
252	C₂²⋊C₄	—	292	4	no	—
253	C₄²	—	292	2	no	—
254	C₄²	—	292	2	no	—
255	C₄²	C₄×Dic₇₃	1	2	yes	—
256	C₄²	#11924005	1	2	yes	—
257	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
258	C₄²	#11924007	1	2	yes	—
259	C₄²	#11924007	1	2	yes	—
260	C₄²	C₄⋊Dic₇₃	1	2	yes	—
261	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
262	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
263	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
264	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	#14556498	1	2	yes	—
268	C₂³	#14556511	1	2	yes	—
269	C₂³	#14556514	1	2	yes	—
270	C₂³	C₂×#11924005	1	1	yes	characteristic
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	292	4	no	—
278	C₂×C₄	—	292	4	no	—
279	C₂×C₄	—	292	4	no	—
280	C₂×C₄	#14556498	1	2	yes	—
281	C₂×C₄	#14556498	1	2	yes	—
282	C₂×C₄	#14556498	1	2	yes	—
283	C₂×C₄	#14556511	1	2	yes	—
284	C₂×C₄	#14556512	1	4	yes	—
285	C₂×C₄	#14556514	1	2	yes	—
286	C₂×C₄	#14556515	1	4	yes	—
287	C₂×C₄	#14556517	1	4	yes	—
288	C₂×C₄	#14556522	1	4	yes	—
289	C₂×C₄	#14556524	1	4	yes	—
290	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
291	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
292	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
293	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
294	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
295	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
296	C₂×C₄	#14556599	1	4	yes	—
297	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
298	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
299	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
300	C₂×C₄	C₂×#11924018	1	1	yes	characteristic
301	C₈	—	4	4	no	—
302	C₈	—	146	8	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	4	no	—
307	C₂²	#17988349	1	4	yes	—
308	C₂²	C₂×#14556498	1	1	yes	characteristic
309	C₂²	#17988555	1	2	yes	—
310	C₂²	C₂×#14556511	1	1	yes	characteristic
311	C₂²	C₂×#14556514	1	1	yes	characteristic
312	C₂²	C₂×#14556517	1	2	yes	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	584	4	no	—
317	C₄	#17988348	1	4	yes	—
318	C₄	#17988349	1	4	yes	—
319	C₄	#17988545	1	2	yes	—
320	C₄	C₂×#14556512	1	1	yes	characteristic
321	C₄	C₂×#14556515	1	1	yes	characteristic
322	C₄	#17988615	1	2	yes	—
323	C₄	C₂×#14556522	1	1	yes	characteristic
324	C₄	C₂×#14556524	1	1	yes	characteristic
325	C₂	—	2	4	no	—
326	C₂	#22608381	1	2	yes	—
327	C₂	#22608385	1	2	yes	—
328	C₂	#22609629	1	1	yes	characteristic
329	C₂	C₂×#17988348	1	1	yes	characteristic
330	C₂	C₂×#17988349	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#423×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#423	D₇₃
hypocenter	C₇₃	#7897
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #22607770	C₂	semidirect
9. C₂×C₄×#10596831	C₄	semidirect
20. #423×C₇₃	C₄	semidirect
170. C₇₃	#7897	semidirect
176. #423	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×#10596831	2
#22612338	2
#22607770	2
C₂×#423×C₇₃	2
#7897	73