Subgroups — #32985723

Order 18688.

642 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988961	C₂	1	1	yes	maximal, characteristic
2	#22610893	C₂	1	1	yes	maximal, characteristic, complemented
3	#22610737	C₂	1	1	yes	maximal, characteristic
4	#22610736	C₂	1	1	yes	maximal, characteristic
5	#22610087	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11924015	—	2	2	no	—
9	C₂×#14556588	C₂²	1	1	yes	characteristic
10	#17988961	—	2	2	no	—
11	#17988961	C₄	1	4	yes	complemented
12	C₂×#14556522	—	2	2	no	—
13	C₂×#14556513	—	2	2	no	—
14	C₂×#14556511	—	2	2	no	—
15	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
16	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
17	#17988457	C₂²	1	1	yes	characteristic
18	#17988349	—	2	2	no	—
19	C₈×#10596831	—	2	2	no	—
20	C₂×C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	#384×C₇₃	C₂²	1	1	yes	characteristic
23	#340×C₇₃	—	2	2	no	—
24	C₂²×D₂₉₂	—	2	1	no	—
25	C₂²×D₂₉₂	—	2	1	no	—
26	C₂²×D₂₉₂	—	2	1	no	—
27	C₂×#11924015	—	2	8	no	—
28	C₂×#11924015	—	4	4	no	—
29	C₂×#11924015	—	4	4	no	—
30	C₂²×#10596831	D₄	1	2	yes	—
31	C₂²×#10596831	D₄	1	2	yes	—
32	#14556588	—	2	1	no	—
33	#14556588	—	2	1	no	—
34	#14556588	—	2	1	no	—
35	#14556588	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₇₃	—	2	1	no	—
37	C₂×C₄⋊Dic₇₃	—	2	1	no	—
38	C₂×C₄⋊Dic₇₃	—	2	1	no	—
39	#14556522	—	4	4	no	—
40	#14556513	—	4	4	no	—
41	#14556511	—	4	4	no	—
42	C₄×#10596831	—	2	1	no	—
43	C₄×#10596831	—	2	1	no	—
44	C₄×#10596831	C₂×C₄	1	2	yes	—
45	C₄×#10596831	D₄	1	2	yes	—
46	C₂²×Q₈×C₇₃	—	2	1	no	—
47	C₂²×C₈×C₇₃	D₄	1	2	yes	—
48	C₄⋊Q₈×C₇₃	—	2	1	no	—
49	C₄⋊Q₈×C₇₃	—	2	1	no	—
50	C₄⋊Q₈×C₇₃	—	2	1	no	—
51	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
57	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₇₃	—	2	1	no	—
59	C₄×C₈×C₇₃	—	2	1	no	—
60	C₂³×D₇₃	—	4	1	no	—
61	C₂×D₂₉₂	—	2	2	no	—
62	C₂×D₂₉₂	—	2	2	no	—
63	C₂×D₂₉₂	—	2	2	no	—
64	C₂×D₂₉₂	—	4	2	no	—
65	C₂×D₂₉₂	—	4	2	no	—
66	C₂×D₂₉₂	—	4	2	no	—
67	C₂×D₂₉₂	—	4	2	no	—
68	C₂×D₂₉₂	—	4	2	no	—
69	C₂×D₂₉₂	—	4	2	no	—
70	C₂×D₂₉₂	—	4	1	no	—
71	C₂×D₂₉₂	—	4	1	no	—
72	C₂×D₂₉₂	—	4	1	no	—
73	C₂²×Dic₇₃	—	4	1	no	—
74	#11924015	—	4	16	no	—
75	C₂×#10596831	—	2	2	no	—
76	C₂×#10596831	—	2	2	no	—
77	C₂×#10596831	C₂²⋊C₄	1	4	yes	—
78	C₂×#10596831	D₈	1	4	yes	—
79	C₂×#10596831	SD₁₆	1	4	yes	—
80	C₂×#10596831	SD₁₆	1	4	yes	complemented
81	C₄⋊Dic₇₃	—	4	2	no	—
82	C₄⋊Dic₇₃	—	4	2	no	—
83	C₄⋊Dic₇₃	—	4	2	no	—
84	C₂×Q₈×C₇₃	—	2	2	no	—
85	C₂×Q₈×C₇₃	—	4	1	no	—
86	C₂×Q₈×C₇₃	—	4	2	no	—
87	C₂×Q₈×C₇₃	—	4	2	no	—
88	C₂²×C₄×C₇₃	—	4	1	no	—
89	C₂²×C₄×C₇₃	—	4	1	no	—
90	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₇₃	—	2	2	no	—
94	C₂×C₈×C₇₃	—	2	2	no	—
95	C₂×C₈×C₇₃	SD₁₆	1	4	yes	complemented
96	C₄⋊C₄×C₇₃	—	2	2	no	—
97	C₄⋊C₄×C₇₃	—	2	2	no	—
98	C₄⋊C₄×C₇₃	—	4	1	no	—
99	C₄⋊C₄×C₇₃	—	4	1	no	—
100	C₄⋊C₄×C₇₃	—	4	2	no	—
101	C₄⋊C₄×C₇₃	—	4	2	no	—
102	C₄⋊C₄×C₇₃	—	4	2	no	—
103	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇₃	#41	1	1	yes	characteristic
106	C₄²×C₇₃	#41	1	1	yes	characteristic
107	C₂³×C₇₃	#127	1	1	yes	characteristic
108	C₂²×D₇₃	—	4	2	no	—
109	C₂²×D₇₃	—	8	1	no	—
110	C₂²×D₇₃	—	8	1	no	—
111	C₂²×D₇₃	—	8	1	no	—
112	C₂²×D₇₃	—	8	1	no	—
113	C₂²×D₇₃	—	8	1	no	—
114	C₂²×D₇₃	—	8	1	no	—
115	Q₈×C₇₃	—	4	4	no	—
116	Q₈×C₇₃	—	8	2	no	—
117	C₂×C₄×C₇₃	—	4	2	no	—
118	C₂×C₄×C₇₃	—	8	1	no	—
119	C₂×C₄×C₇₃	—	8	1	no	—
120	C₂×C₄×C₇₃	—	8	1	no	—
121	C₂×C₄×C₇₃	—	8	1	no	—
122	C₂×C₄×C₇₃	—	8	1	no	—
123	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
127	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
131	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
132	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
133	C₂×C₄×C₇₃	#135	1	2	yes	—
134	C₂×C₄×C₇₃	#135	1	2	yes	—
135	C₂×Dic₇₃	—	8	1	no	—
136	C₂×Dic₇₃	—	8	1	no	—
137	C₂×Dic₇₃	—	8	1	no	—
138	D₂₉₂	—	4	4	no	—
139	D₂₉₂	—	4	4	no	—
140	D₂₉₂	—	4	4	no	—
141	D₂₉₂	—	8	2	no	—
142	D₂₉₂	—	8	2	no	—
143	D₂₉₂	—	8	2	no	—
144	C₈×C₇₃	—	2	4	no	—
145	#10596831	—	2	4	no	—
146	#10596831	#102	1	8	yes	complemented
147	C₂²×C₇₃	#390	1	1	yes	characteristic
148	C₂²×C₇₃	#492	1	1	yes	characteristic
149	C₂²×C₇₃	#492	1	1	yes	characteristic
150	C₂²×C₇₃	#492	1	1	yes	characteristic
151	C₂²×C₇₃	#493	1	1	yes	characteristic
152	C₂²×C₇₃	#495	1	1	yes	characteristic
153	C₂²×C₇₃	#495	1	1	yes	characteristic
154	C₂×D₇₃	—	8	2	no	—
155	C₂×D₇₃	—	8	2	no	—
156	C₂×D₇₃	—	8	2	no	—
157	C₂×D₇₃	—	16	1	no	—
158	C₂×D₇₃	—	16	1	no	—
159	C₂×D₇₃	—	16	1	no	—
160	C₂×D₇₃	—	16	1	no	—
161	C₄×C₇₃	—	8	2	no	—
162	C₄×C₇₃	—	16	1	no	—
163	C₄×C₇₃	C₂×#102	1	2	yes	—
164	C₄×C₇₃	C₄×SD₁₆	1	2	yes	—
165	C₄×C₇₃	C₄×SD₁₆	1	2	yes	—
166	C₄×C₇₃	#465	1	2	yes	—
167	C₄×C₇₃	#465	1	2	yes	—
168	C₄×C₇₃	#486	1	2	yes	—
169	Dic₇₃	—	16	1	no	—
170	#9820	—	73	1	no	maximal, Sylow
171	C₂×C₇₃	#1705	1	1	yes	characteristic
172	C₂×C₇₃	#1706	1	1	yes	characteristic
173	C₂×C₇₃	#1706	1	1	yes	characteristic
174	C₂×C₇₃	#1708	1	1	yes	characteristic
175	C₂×C₇₃	#1957	1	1	yes	characteristic
176	C₂×C₇₃	#1968	1	1	yes	characteristic
177	C₂×C₇₃	#1968	1	1	yes	characteristic
178	D₇₃	—	16	2	no	—
179	C₂×#492	—	73	1	no	—
180	#1990	—	73	1	no	—
181	#1990	D₇₃	1	1	yes	characteristic, complemented
182	#1957	—	73	1	no	—
183	#1957	—	73	1	no	—
184	#1844	—	73	1	no	—
185	C₂×C₈²	—	73	1	no	—
186	C₇₃	#9820	1	1	yes	characteristic, complemented, Sylow
187	C₂²×SD₁₆	—	146	2	no	—
188	C₂×C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
189	C₂×#127	—	73	1	no	—
190	#492	—	73	4	no	—
191	#492	—	146	2	no	—
192	C₂×#106	—	146	2	no	—
193	C₂×#102	—	146	2	no	—
194	C₂×#102	—	146	2	no	—
195	C₂×C₄×C₈	—	73	1	no	—
196	C₂×C₄×C₈	—	73	1	no	—
197	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
198	#384	—	73	1	no	—
199	#384	C₂×D₇₃	1	1	yes	characteristic
200	#340	—	2	2	no	—
201	#340	—	146	2	no	—
202	C₈²	—	146	2	no	—
203	C₂²×Q₈	—	2	1	no	—
204	C₂²×D₄	—	146	1	no	—
205	C₂²×D₄	—	146	1	no	—
206	C₂²×D₄	—	146	1	no	—
207	C₂×SD₁₆	—	292	4	no	—
208	C₂×SD₁₆	—	292	4	no	—
209	C₂×SD₁₆	—	146	8	no	—
210	C₂²×C₈	—	73	2	no	—
211	C₂²×C₈	—	73	2	no	—
212	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	C₄×D₇₃	1	2	yes	—
217	#127	—	73	2	no	—
218	#127	—	146	1	no	—
219	#127	—	146	1	no	—
220	#127	—	146	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	146	1	no	—
227	C₂×C₄⋊C₄	—	146	1	no	—
228	C₂×C₄⋊C₄	—	146	1	no	—
229	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
230	#106	—	292	4	no	—
231	#102	—	292	4	no	—
232	#102	—	292	4	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	—	73	2	no	—
236	C₄×C₈	—	73	2	no	—
237	C₄×C₈	—	146	1	no	—
238	C₄×C₈	—	146	1	no	—
239	C₂⁴	—	292	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×D₄	—	292	1	no	—
245	C₂×D₄	—	292	1	no	—
246	C₂×D₄	—	292	1	no	—
247	C₂×D₄	—	292	2	no	—
248	C₂×D₄	—	292	2	no	—
249	C₂×D₄	—	292	2	no	—
250	C₂×D₄	—	292	2	no	—
251	C₂×D₄	—	292	2	no	—
252	C₂×D₄	—	292	2	no	—
253	C₂×D₄	—	146	2	no	—
254	C₂×D₄	—	146	2	no	—
255	C₂×D₄	—	146	2	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	292	1	no	—
259	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
260	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
262	SD₁₆	—	292	16	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	73	4	no	—
266	C₂×C₈	—	73	4	no	—
267	C₂×C₈	—	73	4	no	—
268	C₂×C₈	—	73	4	no	—
269	C₂×C₈	—	146	2	no	—
270	C₂×C₈	—	146	2	no	—
271	C₂×C₈	#11924015	1	4	yes	complemented
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	292	2	no	—
280	C₄⋊C₄	—	292	2	no	—
281	C₄⋊C₄	—	292	2	no	—
282	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
283	C₄²	D₄×D₇₃	1	1	yes	characteristic
284	C₄²	#11924027	1	1	yes	characteristic
285	C₄²	#11924028	1	1	yes	characteristic
286	C₂³	—	292	2	no	—
287	C₂³	—	584	1	no	—
288	C₂³	—	584	1	no	—
289	C₂³	—	584	1	no	—
290	C₂³	—	584	1	no	—
291	C₂³	—	584	1	no	—
292	C₂³	—	584	1	no	—
293	C₂³	#14556587	1	1	yes	characteristic
294	Q₈	—	4	4	no	—
295	Q₈	—	8	2	no	—
296	D₄	—	292	4	no	—
297	D₄	—	292	4	no	—
298	D₄	—	292	4	no	—
299	D₄	—	584	2	no	—
300	D₄	—	584	2	no	—
301	D₄	—	584	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	8	1	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	—	584	1	no	—
309	C₂×C₄	—	584	1	no	—
310	C₂×C₄	—	584	1	no	—
311	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
312	C₂×C₄	#14556551	1	1	yes	characteristic
313	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
314	C₂×C₄	#14556563	1	1	yes	characteristic
315	C₂×C₄	#14556564	1	1	yes	characteristic
316	C₂×C₄	#14556577	1	1	yes	characteristic
317	C₂×C₄	D₈×D₇₃	1	2	yes	—
318	C₂×C₄	SD₁₆×D₇₃	1	2	yes	—
319	C₂×C₄	SD₁₆×D₇₃	1	2	yes	—
320	C₂×C₄	C₂×#11924015	1	2	yes	—
321	C₂×C₄	#14556611	1	2	yes	—
322	C₂×C₄	#14556614	1	2	yes	—
323	C₈	—	2	4	no	—
324	C₈	—	73	8	no	—
325	C₈	—	146	4	no	—
326	C₂²	—	584	2	no	—
327	C₂²	—	584	2	no	—
328	C₂²	—	584	2	no	—
329	C₂²	—	1168	1	no	—
330	C₂²	—	1168	1	no	—
331	C₂²	—	1168	1	no	—
332	C₂²	—	1168	1	no	—
333	C₂²	#17988487	1	1	yes	characteristic
334	C₂²	#17988958	1	1	yes	characteristic
335	C₂²	#17988958	1	1	yes	characteristic
336	C₂²	#17988961	1	1	yes	characteristic
337	C₂²	#17988962	1	1	yes	characteristic
338	C₂²	#17988969	1	1	yes	characteristic
339	C₂²	#17988972	1	1	yes	characteristic
340	C₄	—	8	2	no	—
341	C₄	—	16	1	no	—
342	C₄	—	1168	1	no	—
343	C₄	#102×D₇₃	1	2	yes	—
344	C₄	#17988672	1	2	yes	—
345	C₄	C₄×#11924015	1	2	yes	—
346	C₄	#17988818	1	2	yes	—
347	C₄	#17988819	1	2	yes	—
348	C₄	#17988932	1	2	yes	—
349	C₂	—	1168	2	no	—
350	C₂	#22609519	1	1	yes	characteristic
351	C₂	#22609522	1	1	yes	characteristic
352	C₂	#22609524	1	1	yes	characteristic
353	C₂	#22609534	1	1	yes	characteristic
354	C₂	#22610735	1	1	yes	characteristic
355	C₂	#22610782	1	1	yes	characteristic
356	C₂	#22610788	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556587
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1990	C₂
socle	C₂³×C₇₃	#127
prime radical	C₂×#14556588	C₂²
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1990	D₇₃
hypocenter	C₇₃	#9820
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
2. #22610893	C₂	semidirect
5. #22610087	C₂	semidirect
6. C₂×C₈×#10596831	C₂	semidirect
7. C₇₃×#1990	C₂	semidirect
11. #17988961	C₄	semidirect
80. C₂×#10596831	SD₁₆	semidirect
95. C₂×C₈×C₇₃	SD₁₆	semidirect
146. #10596831	#102	semidirect
181. #1990	D₇₃	semidirect
186. C₇₃	#9820	semidirect
271. C₂×C₈	#11924015	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988961	2
#22610893	2
#22610737	2
#22610736	2
#22610087	2
C₂×C₈×#10596831	2
C₇₃×#1990	2
#9820	73