Subgroups — #33069775

Order 18688.

798 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620510	C₂	1	1	yes	maximal, characteristic, complemented
2	#22619397	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×D₂₉₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#22609369	C₂	1	4	yes	maximal, complemented
5	#22609321	C₂	1	2	yes	maximal, complemented
6	#22609287	C₂	1	2	yes	maximal, complemented
7	#22608609	C₂	1	2	yes	maximal, complemented
8	C₇₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556619	—	2	2	no	—
10	C₂²×C₈×D₇₃	—	2	1	no	—
11	#17989137	C₂²	1	2	yes	—
12	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
13	#17988992	C₂²	1	1	yes	characteristic, complemented
14	#17988964	C₂²	1	1	yes	characteristic, complemented
15	#17988934	C₂²	1	2	yes	complemented
16	#17988823	—	2	1	no	—
17	#17988823	—	2	1	no	—
18	#17988803	—	2	4	no	—
19	#17988793	C₂²	1	4	yes	—
20	#17988783	C₂²	1	4	yes	—
21	C₄×D₅₈₄	—	2	1	no	—
22	C₄×D₅₈₄	—	2	1	no	—
23	C₈×D₂₉₂	—	2	1	no	—
24	C₈×D₂₉₂	—	2	1	no	—
25	C₈×D₂₉₂	C₂²	1	2	yes	—
26	C₈×D₂₉₂	C₂²	1	2	yes	—
27	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
28	#17988589	—	2	2	no	—
29	C₂×#14556503	—	2	1	no	—
30	#17988334	C₂²	1	4	yes	complemented
31	#17988330	C₂²	1	4	yes	complemented
32	#423×C₇₃	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
34	#335×C₇₃	C₂²	1	2	yes	complemented
35	#335×C₇₃	C₂²	1	2	yes	complemented
36	C₂×#11924028	—	2	2	no	—
37	C₂²×D₂₉₂	D₄	1	1	yes	characteristic
38	C₂²×C₄×D₇₃	—	2	1	no	—
39	#14556619	—	4	8	no	—
40	C₂×#11924017	—	2	4	no	—
41	C₂×D₅₈₄	—	2	4	no	—
42	C₂×D₅₈₄	—	2	1	no	—
43	C₂×D₅₈₄	—	2	1	no	—
44	C₂²×#10596831	—	2	1	no	—
45	C₂×C₈×D₇₃	—	2	2	no	—
46	C₂×C₈×D₇₃	—	2	2	no	—
47	C₂×C₈×D₇₃	—	4	4	no	—
48	#14556591	C₂³	1	2	yes	—
49	#14556588	C₂³	1	2	yes	—
50	#14556579	—	2	2	no	—
51	#14556565	—	2	2	no	—
52	#14556565	—	2	2	no	—
53	C₄×D₂₉₂	—	2	1	no	—
54	C₄×D₂₉₂	—	2	2	no	—
55	C₄×D₂₉₂	—	2	1	no	—
56	C₄×D₂₉₂	—	2	2	no	—
57	C₄×D₂₉₂	D₄	1	2	yes	—
58	C₄×D₂₉₂	C₂³	1	2	yes	—
59	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
60	C₂×#11924005	—	2	1	no	—
61	#14556525	—	2	1	no	—
62	#14556525	—	2	1	no	—
63	C₄⋊#10596831	—	2	1	no	—
64	C₄⋊#10596831	C₂³	1	2	yes	—
65	#14556516	—	2	4	no	—
66	#14556513	—	2	4	no	—
67	#14556513	—	2	2	no	—
68	#14556513	—	2	2	no	—
69	#14556503	—	2	2	no	—
70	#14556503	—	2	2	no	—
71	C₂×M₁₆×C₇₃	—	2	2	no	—
72	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
73	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₇₃	—	2	2	no	—
76	C₄⋊C₈×C₇₃	C₂³	1	4	yes	—
77	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
78	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
79	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
81	C₂³×D₇₃	—	2	1	no	—
82	#11924028	—	4	4	no	—
83	#11924028	—	4	4	no	—
84	C₂×Dic₁₄₆	—	2	2	no	—
85	C₂×D₂₉₂	—	2	1	no	—
86	C₂×D₂₉₂	—	2	2	no	—
87	C₂×D₂₉₂	—	2	2	no	—
88	C₂×D₂₉₂	—	2	2	no	—
89	C₂×D₂₉₂	—	2	2	no	—
90	C₂×D₂₉₂	D₈	1	4	yes	complemented
91	C₂×D₂₉₂	C₂×D₄	1	2	yes	—
92	C₂×C₂²⋊D₇₃	—	4	2	no	—
93	C₂²×Dic₇₃	—	2	1	no	—
94	C₂×C₄×D₇₃	—	2	2	no	—
95	C₂×C₄×D₇₃	—	2	2	no	—
96	C₂×C₄×D₇₃	—	4	2	no	—
97	C₂×C₄×D₇₃	—	4	2	no	—
98	C₂×C₄×D₇₃	—	4	2	no	—
99	#11924017	—	4	8	no	—
100	D₅₈₄	—	4	2	no	—
101	D₅₈₄	—	4	8	no	—
102	D₅₈₄	—	4	2	no	—
103	C₂×#10596831	—	2	2	no	—
104	C₂×#10596831	—	4	2	no	—
105	C₈×D₇₃	—	4	4	no	—
106	C₈×D₇₃	—	4	4	no	—
107	C₄⋊Dic₇₃	—	2	1	no	—
108	C₄⋊Dic₇₃	—	2	2	no	—
109	C₄⋊Dic₇₃	—	2	2	no	—
110	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
111	#11924005	—	2	2	no	—
112	#11924005	—	2	2	no	—
113	#11924005	—	4	2	no	—
114	#11924005	—	4	2	no	—
115	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
118	M₁₆×C₇₃	—	4	4	no	—
119	C₂×C₈×C₇₃	—	2	2	no	—
120	C₂×C₈×C₇₃	—	2	4	no	—
121	C₂×C₈×C₇₃	—	4	1	no	—
122	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₇₃	#41	1	1	yes	characteristic
129	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
130	C₂³×C₇₃	#121	1	1	yes	characteristic
131	C₂²×D₇₃	—	2	2	no	—
132	C₂²×D₇₃	—	2	4	no	—
133	C₂²×D₇₃	—	4	2	no	—
134	C₂²×D₇₃	—	4	2	no	—
135	C₂²×D₇₃	—	4	2	no	—
136	C₂×C₄×C₇₃	—	2	1	no	—
137	C₂×C₄×C₇₃	—	2	1	no	—
138	C₂×C₄×C₇₃	—	4	1	no	—
139	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
142	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₇₃	#135	1	2	yes	—
144	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
147	C₂²⋊D₇₃	—	8	4	no	—
148	C₂×Dic₇₃	—	2	2	no	—
149	C₂×Dic₇₃	—	4	2	no	—
150	C₂×Dic₇₃	—	4	2	no	—
151	D₂₉₂	—	2	2	no	—
152	D₂₉₂	—	2	4	no	—
153	D₂₉₂	—	4	2	no	—
154	D₂₉₂	—	4	2	no	—
155	D₂₉₂	—	4	2	no	—
156	D₂₉₂	—	4	2	no	—
157	D₂₉₂	—	8	2	no	—
158	D₂₉₂	C₂×D₈	1	4	yes	—
159	C₄×D₇₃	—	4	4	no	—
160	C₄×D₇₃	—	4	2	no	—
161	C₄×D₇₃	—	4	2	no	—
162	C₄×D₇₃	—	8	4	no	—
163	Dic₁₄₆	—	4	2	no	—
164	Dic₁₄₆	—	4	2	no	—
165	C₈×C₇₃	—	2	1	no	—
166	C₈×C₇₃	—	2	1	no	—
167	C₈×C₇₃	—	2	1	no	—
168	C₈×C₇₃	—	4	4	no	—
169	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	complemented
170	#10596831	—	4	2	no	—
171	C₂²×C₇₃	—	2	1	no	—
172	C₂²×C₇₃	—	2	1	no	—
173	C₂²×C₇₃	#466	1	2	yes	—
174	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
175	C₂×D₇₃	—	2	4	no	—
176	C₂×D₇₃	—	4	8	no	—
177	C₂×D₇₃	—	4	2	no	—
178	C₂×D₇₃	—	4	2	no	—
179	C₂×D₇₃	—	8	2	no	—
180	C₂×D₇₃	—	8	2	no	—
181	C₂×D₇₃	—	8	2	no	—
182	C₄×C₇₃	—	2	1	no	—
183	C₄×C₇₃	—	2	1	no	—
184	C₄×C₇₃	—	2	1	no	—
185	C₄×C₇₃	#466	1	2	yes	—
186	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
187	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
188	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
189	C₄×C₇₃	#575	1	1	yes	characteristic
190	Dic₇₃	—	4	2	no	—
191	Dic₇₃	—	8	2	no	—
192	#19255	—	73	1	no	maximal, Sylow
193	C₂×C₇₃	—	2	2	no	—
194	C₂×C₇₃	#3036	1	1	yes	characteristic
195	C₂×C₇₃	C₂×#466	1	1	yes	characteristic
196	C₂×C₇₃	#3067	1	1	yes	characteristic
197	D₇₃	—	4	4	no	—
198	D₇₃	—	16	2	no	—
199	#3169	—	73	1	no	—
200	#3036	—	73	2	no	—
201	C₂×C₈×D₄	—	73	1	no	—
202	#1681	—	73	4	no	—
203	#1673	—	73	2	no	—
204	#1667	—	73	2	no	—
205	#1567	D₇₃	1	1	yes	characteristic, complemented
206	#1547	—	73	2	no	—
207	C₇₃	#19255	1	1	yes	characteristic, complemented, Sylow
208	C₂×AGL(1,ℤ/8)	—	146	2	no	—
209	C₂³×C₈	—	146	1	no	—
210	#532	—	73	2	no	—
211	C₂×C₄×D₄	—	73	1	no	—
212	#500	—	73	1	no	—
213	#493	—	73	1	no	—
214	#486	—	73	2	no	—
215	#466	—	146	1	no	—
216	#466	—	146	1	no	—
217	#463	—	146	4	no	—
218	#461	—	73	4	no	—
219	#459	—	73	4	no	—
220	C₄×D₈	—	146	1	no	—
221	C₄×D₈	—	146	1	no	—
222	C₈×D₄	—	73	2	no	—
223	C₈×D₄	—	73	2	no	—
224	C₈×D₄	—	146	1	no	—
225	C₈×D₄	—	146	1	no	—
226	#423	C₂×D₇₃	1	2	yes	—
227	C₂×C₄⋊C₈	—	73	1	no	—
228	#417	—	146	2	no	—
229	C₂×C₂²⋊C₈	—	146	1	no	—
230	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
231	#335	C₂×D₇₃	1	2	yes	complemented
232	#335	C₂×D₇₃	1	2	yes	complemented
233	#332	—	73	4	no	—
234	#331	—	73	4	no	—
235	C₂×#41	—	146	2	no	—
236	C₂²×D₄	—	73	1	no	—
237	C₂³×C₄	—	146	1	no	—
238	AGL(1,ℤ/8)	—	292	8	no	—
239	C₂×SD₁₆	—	146	4	no	—
240	C₂×D₈	—	146	4	no	—
241	C₂×D₈	—	146	1	no	—
242	C₂×D₈	—	146	1	no	—
243	C₂×M₁₆	—	2	2	no	—
244	C₂²×C₈	—	292	4	no	—
245	C₂²×C₈	—	146	1	no	—
246	C₂²×C₈	—	146	2	no	—
247	C₂²×C₈	—	146	2	no	—
248	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
249	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
250	C₄⋊Q₈	—	73	2	no	—
251	#127	—	73	2	no	—
252	#124	—	146	2	no	—
253	#121	—	146	2	no	—
254	#121	—	146	2	no	—
255	C₄×D₄	—	73	2	no	—
256	C₄×D₄	—	73	2	no	—
257	C₄×D₄	—	146	2	no	—
258	C₄×D₄	—	146	1	no	—
259	C₄×D₄	—	146	2	no	—
260	C₄×D₄	—	146	1	no	—
261	C₂×C₄⋊C₄	—	73	1	no	—
262	C₂×C₂²⋊C₄	—	146	1	no	—
263	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
264	#107	—	146	1	no	—
265	#107	—	146	1	no	—
266	C₄⋊C₈	—	2	2	no	—
267	C₄⋊C₈	—	73	2	no	—
268	C₄⋊C₈	—	146	1	no	—
269	C₄⋊C₈	C₂²×D₇₃	1	4	yes	—
270	#103	—	146	4	no	—
271	#102	—	146	4	no	—
272	#102	—	146	2	no	—
273	#102	—	146	2	no	—
274	C₂²⋊C₈	—	146	2	no	—
275	C₂²⋊C₈	—	146	2	no	—
276	C₄×C₈	D₂₉₂	1	1	yes	characteristic
277	C₄×C₈	D₂₉₂	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
280	C₂⁴	—	146	1	no	—
281	#41	—	292	4	no	—
282	#41	—	292	4	no	—
283	C₂×Q₈	—	146	2	no	—
284	C₂×D₄	—	292	2	no	—
285	C₂×D₄	—	73	4	no	—
286	C₂×D₄	—	73	2	no	—
287	C₂×D₄	—	146	1	no	—
288	C₂×D₄	—	146	2	no	—
289	C₂×D₄	—	146	2	no	—
290	C₂×D₄	—	146	2	no	—
291	C₂×D₄	—	146	2	no	—
292	C₂²×C₄	—	292	2	no	—
293	C₂²×C₄	—	292	2	no	—
294	C₂²×C₄	—	292	2	no	—
295	C₂²×C₄	—	146	1	no	—
296	C₂²×C₄	—	146	2	no	—
297	C₂²×C₄	—	146	2	no	—
298	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
300	C₂²×C₄	#11924029	1	1	yes	characteristic
301	SD₁₆	—	292	8	no	—
302	D₈	—	292	2	no	—
303	D₈	—	292	8	no	—
304	D₈	—	292	2	no	—
305	M₁₆	—	4	4	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	4	no	—
308	C₂×C₈	—	4	1	no	—
309	C₂×C₈	—	292	4	no	—
310	C₂×C₈	—	292	2	no	—
311	C₂×C₈	—	292	4	no	—
312	C₂×C₈	—	146	2	no	—
313	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
314	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
315	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
316	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
317	C₄⋊C₄	—	73	2	no	—
318	C₄⋊C₄	—	146	2	no	—
319	C₄⋊C₄	—	146	2	no	—
320	C₄⋊C₄	—	146	1	no	—
321	C₂²⋊C₄	—	292	2	no	—
322	C₂²⋊C₄	—	292	2	no	—
323	C₂²⋊C₄	—	146	2	no	—
324	C₂²⋊C₄	—	146	2	no	—
325	C₄²	D₄×D₇₃	1	1	yes	characteristic
326	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
327	C₄²	#11924029	1	1	yes	characteristic
328	C₄²	C₂³×D₇₃	1	1	yes	characteristic
329	C₂³	—	292	2	no	—
330	C₂³	—	292	2	no	—
331	C₂³	—	292	2	no	—
332	C₂³	—	146	4	no	—
333	C₂³	—	146	2	no	—
334	C₂³	#14556566	1	1	yes	characteristic
335	Q₈	—	292	2	no	—
336	Q₈	—	292	2	no	—
337	D₄	—	292	2	no	—
338	D₄	—	292	2	no	—
339	D₄	—	292	2	no	—
340	D₄	—	292	2	no	—
341	D₄	—	584	2	no	—
342	D₄	—	584	4	no	—
343	D₄	—	73	4	no	—
344	D₄	—	146	4	no	—
345	D₄	—	146	2	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	292	4	no	—
350	C₂×C₄	—	292	2	no	—
351	C₂×C₄	—	292	2	no	—
352	C₂×C₄	—	292	2	no	—
353	C₂×C₄	—	292	2	no	—
354	C₂×C₄	—	584	4	no	—
355	C₂×C₄	—	146	2	no	—
356	C₂×C₄	#14556566	1	1	yes	characteristic
357	C₂×C₄	#14556566	1	1	yes	characteristic
358	C₂×C₄	#14556566	1	1	yes	characteristic
359	C₂×C₄	D₈×D₇₃	1	2	yes	—
360	C₂×C₄	#14556614	1	2	yes	—
361	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
362	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
363	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
364	C₈	—	2	1	no	—
365	C₈	—	2	1	no	—
366	C₈	—	2	1	no	—
367	C₈	—	4	4	no	—
368	C₈	—	292	2	no	—
369	C₈	#14556619	1	2	yes	complemented
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	292	8	no	—
373	C₂²	—	292	2	no	—
374	C₂²	—	292	2	no	—
375	C₂²	—	584	2	no	—
376	C₂²	—	584	2	no	—
377	C₂²	—	584	2	no	—
378	C₂²	—	146	4	no	—
379	C₂²	#17988824	1	2	yes	—
380	C₂²	C₂×#14556566	1	1	yes	characteristic
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	292	2	no	—
385	C₄	—	584	2	no	—
386	C₄	#17988824	1	2	yes	—
387	C₄	C₂×D₈×D₇₃	1	1	yes	characteristic
388	C₄	C₂×#14556614	1	1	yes	characteristic
389	C₄	C₂×#14556619	1	1	yes	characteristic
390	C₄	#17989387	1	1	yes	characteristic
391	C₂	—	2	2	no	—
392	C₂	—	292	4	no	—
393	C₂	—	1168	2	no	—
394	C₂	#22619397	1	1	yes	characteristic
395	C₂	C₂×#17988824	1	1	yes	characteristic
396	C₂	#22619652	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556566
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1567	C₂
socle	C₂²×C₇₃	C₂×#121
hypercenter	#1567	D₇₃
hypocenter	C₇₃	#19255
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
1. #22620510	C₂	semidirect
2. #22619397	C₂	semidirect
3. C₂×C₈×D₂₉₂	C₂	semidirect
4. #22609369	C₂	semidirect
5. #22609321	C₂	semidirect
6. #22609287	C₂	semidirect
7. #22608609	C₂	semidirect
8. C₇₃×#1567	C₂	semidirect
13. #17988992	C₂²	semidirect
14. #17988964	C₂²	semidirect
15. #17988934	C₂²	semidirect
30. #17988334	C₂²	semidirect
31. #17988330	C₂²	semidirect
34. #335×C₇₃	C₂²	semidirect
35. #335×C₇₃	C₂²	semidirect
90. C₂×D₂₉₂	D₈	semidirect
169. C₈×C₇₃	AGL(1,ℤ/8)	semidirect
205. #1567	D₇₃	semidirect
207. C₇₃	#19255	semidirect
231. #335	C₂×D₇₃	semidirect
232. #335	C₂×D₇₃	semidirect
369. C₈	#14556619	semidirect

Maximal subgroups

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

subgroup	index
#22620510	2
#22619397	2
C₂×C₈×D₂₉₂	2
#22609369	2
#22609321	2
#22609287	2
#22608609	2
C₇₃×#1567	2
#19255	73