Subgroups — #33077836

Order 18688.

626 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622830	C₂	1	2	yes	maximal
2	Q₁₆×D₂₉₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#22609529	C₂	1	1	yes	maximal, characteristic, complemented
4	#22609439	C₂	1	2	yes	maximal, complemented
5	#22609327	C₂	1	1	yes	maximal, characteristic, complemented
6	#22609296	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609145	C₂	1	2	yes	maximal, complemented
8	#22609094	C₂	1	2	yes	maximal, complemented
9	#22608828	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11924013	C₂	1	1	yes	maximal, characteristic
11	C₇₃×#3338	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₃	—	2	1	no	—
13	Q₈×D₂₉₂	C₂²	1	2	yes	—
14	#17988997	—	2	2	no	—
15	#17988983	C₂²	1	2	yes	—
16	#17988967	C₂²	1	1	yes	characteristic
17	#17988937	C₂²	1	2	yes	—
18	#17988884	—	2	2	no	—
19	#17988877	—	2	2	no	—
20	#17988874	C₂²	1	2	yes	—
21	#17988867	C₂²	1	2	yes	—
22	#17988829	—	2	1	no	—
23	#17988797	C₂²	1	2	yes	—
24	#17988744	—	2	2	no	—
25	#17988718	C₂²	1	2	yes	—
26	#17988697	—	2	2	no	—
27	C₄×#11924013	C₂²	1	1	yes	characteristic
28	C₈×D₂₉₂	C₂²	1	1	yes	characteristic
29	#17988340	C₂²	1	1	yes	characteristic
30	#17988339	C₂²	1	1	yes	characteristic
31	#17988336	C₂²	1	2	yes	—
32	#17988334	C₂²	1	2	yes	—
33	Q₈⋊#10596831	C₂²	1	2	yes	—
34	D₄⋊#10596831	C₂²	1	1	yes	characteristic
35	#17988314	C₂²	1	1	yes	characteristic
36	C₈×#10596831	C₂²	1	1	yes	characteristic
37	#554×C₇₃	C₂²	1	2	yes	—
38	#500×C₇₃	C₂²	1	1	yes	characteristic
39	#484×C₇₃	—	2	2	no	—
40	#477×C₇₃	C₂²	1	2	yes	—
41	#467×C₇₃	—	2	1	no	—
42	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
43	C₈×D₄×C₇₃	C₂²	1	1	yes	characteristic
44	C₂×#107×C₇₃	—	2	1	no	—
45	C₂×Q₈×D₇₃	—	2	2	no	—
46	C₂×Dic₂₉₂	—	2	1	no	—
47	C₂×Dic₂₉₂	—	2	1	no	—
48	C₂×#11924018	—	2	2	no	—
49	Q₁₆×D₇₃	—	4	4	no	—
50	C₂×#11924013	D₄	1	1	yes	characteristic
51	C₂×C₈×D₇₃	—	2	1	no	—
52	#14556591	—	2	2	no	—
53	#14556591	C₂³	1	2	yes	—
54	#14556571	—	2	2	no	—
55	#14556571	—	4	2	no	—
56	C₄×D₂₉₂	—	2	2	no	—
57	C₄×D₂₉₂	C₂³	1	1	yes	characteristic
58	#14556524	—	2	2	no	—
59	#14556524	D₄	1	1	yes	characteristic
60	#14556522	—	2	2	no	—
61	#14556522	—	2	2	no	—
62	#14556522	—	2	2	no	—
63	C₄⋊#10596831	—	2	2	no	—
64	C₄⋊#10596831	C₂³	1	2	yes	—
65	C₄⋊#10596831	C₂³	1	1	yes	characteristic
66	#14556515	—	4	2	no	—
67	#14556514	—	2	2	no	—
68	#14556512	—	2	2	no	—
69	#14556512	D₄	1	1	yes	characteristic
70	#14556511	—	2	2	no	—
71	#14556511	D₄	1	1	yes	characteristic
72	#14556503	—	2	1	no	—
73	#14556502	—	2	2	no	—
74	C₄×#10596831	C₂³	1	1	yes	characteristic
75	C₄×#10596831	C₂³	1	1	yes	characteristic
76	C₂×Q₁₆×C₇₃	D₄	1	1	yes	characteristic
77	C₂²×C₈×C₇₃	—	2	1	no	—
78	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
79	#125×C₇₃	—	2	2	no	—
80	C₂²⋊Q₈×C₇₃	—	2	2	no	—
81	C₂²⋊Q₈×C₇₃	—	4	2	no	—
82	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
83	C₄×D₄×C₇₃	—	2	2	no	—
84	C₄×D₄×C₇₃	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
86	#107×C₇₃	—	2	1	no	—
87	#107×C₇₃	—	2	1	no	—
88	#107×C₇₃	—	2	2	no	—
89	#107×C₇₃	D₄	1	1	yes	characteristic
90	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
91	#103×C₇₃	—	2	2	no	—
92	#103×C₇₃	D₄	1	2	yes	—
93	C₂²⋊C₈×C₇₃	—	2	1	no	—
94	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
95	C₂×Dic₁₄₆	—	2	2	no	—
96	C₂×Dic₁₄₆	—	4	2	no	—
97	Q₈×D₇₃	—	4	4	no	—
98	Q₈×D₇₃	—	8	2	no	—
99	C₂×D₂₉₂	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×D₇₃	—	2	1	no	—
101	C₂×C₄×D₇₃	—	4	2	no	—
102	Dic₂₉₂	—	4	2	no	—
103	Dic₂₉₂	—	8	1	no	—
104	#11924018	—	4	4	no	—
105	#11924013	Q₁₆	1	4	yes	complemented
106	C₂×#10596831	—	2	2	no	—
107	C₂×#10596831	—	2	1	no	—
108	C₂×#10596831	—	2	1	no	—
109	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
110	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
111	C₈×D₇₃	—	4	2	no	—
112	C₄⋊Dic₇₃	—	2	2	no	—
113	C₄⋊Dic₇₃	—	2	2	no	—
114	C₄⋊Dic₇₃	—	4	2	no	—
115	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
116	#11924005	—	2	1	no	—
117	#11924005	—	4	2	no	—
118	C₂×Q₈×C₇₃	—	2	2	no	—
119	C₂×Q₈×C₇₃	C₂×D₄	1	2	yes	—
120	C₂×D₄×C₇₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₇₃	—	2	1	no	—
122	C₂²×C₄×C₇₃	—	4	2	no	—
123	Q₁₆×C₇₃	—	2	2	no	—
124	C₂×C₈×C₇₃	—	2	1	no	—
125	C₂×C₈×C₇₃	—	4	2	no	—
126	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₄⋊C₄×C₇₃	—	2	2	no	—
129	C₄⋊C₄×C₇₃	—	2	2	no	—
130	C₄⋊C₄×C₇₃	—	2	2	no	—
131	C₄⋊C₄×C₇₃	—	4	2	no	—
132	C₄⋊C₄×C₇₃	—	4	2	no	—
133	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
134	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
135	C₂²⋊C₄×C₇₃	—	2	1	no	—
136	C₂²⋊C₄×C₇₃	—	4	2	no	—
137	C₄²×C₇₃	—	2	2	no	—
138	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
139	C₂³×C₇₃	—	2	1	no	—
140	C₂²×D₇₃	—	2	1	no	—
141	Q₈×C₇₃	—	2	2	no	—
142	Q₈×C₇₃	—	2	2	no	—
143	Q₈×C₇₃	—	8	2	no	—
144	D₄×C₇₃	—	2	1	no	—
145	D₄×C₇₃	C₂×Q₁₆	1	2	yes	complemented
146	C₂×C₄×C₇₃	—	2	2	no	—
147	C₂×C₄×C₇₃	—	2	2	no	—
148	C₂×C₄×C₇₃	—	2	1	no	—
149	C₂×C₄×C₇₃	—	4	1	no	—
150	C₂×C₄×C₇₃	—	4	1	no	—
151	C₂×C₄×C₇₃	—	4	2	no	—
152	C₂×C₄×C₇₃	—	16	2	no	—
153	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂×Dic₇₃	—	2	1	no	—
157	C₂×Dic₇₃	—	4	2	no	—
158	D₂₉₂	—	2	1	no	—
159	D₂₉₂	C₂×Q₁₆	1	2	yes	—
160	C₄×D₇₃	—	4	1	no	—
161	C₄×D₇₃	—	4	1	no	—
162	C₄×D₇₃	—	8	4	no	—
163	Dic₁₄₆	—	4	2	no	—
164	Dic₁₄₆	—	4	2	no	—
165	Dic₁₄₆	—	8	4	no	—
166	C₈×C₇₃	—	2	1	no	—
167	C₈×C₇₃	—	4	1	no	—
168	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
169	#10596831	—	2	1	no	—
170	#10596831	—	4	1	no	—
171	#10596831	—	4	1	no	—
172	#10596831	—	4	2	no	—
173	#10596831	C₂×Q₁₆	1	2	yes	complemented
174	C₂²×C₇₃	—	2	2	no	—
175	C₂²×C₇₃	—	4	1	no	—
176	C₂²×C₇₃	—	4	1	no	—
177	C₂²×C₇₃	D₄²	1	1	yes	characteristic
178	C₂×D₇₃	—	2	2	no	—
179	C₂×D₇₃	—	4	1	no	—
180	C₂×D₇₃	—	4	1	no	—
181	C₄×C₇₃	—	2	1	no	—
182	C₄×C₇₃	—	4	2	no	—
183	C₄×C₇₃	—	4	1	no	—
184	C₄×C₇₃	—	4	2	no	—
185	C₄×C₇₃	—	8	2	no	—
186	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
187	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₄×C₇₃	#577	1	1	yes	characteristic
189	C₄×C₇₃	#578	1	1	yes	characteristic
190	Dic₇₃	—	4	1	no	—
191	Dic₇₃	—	8	2	no	—
192	#19926	—	73	1	no	maximal, Sylow
193	C₂×C₇₃	—	4	2	no	—
194	C₂×C₇₃	#3272	1	1	yes	characteristic
195	C₂×C₇₃	D₄×Q₁₆	1	1	yes	characteristic
196	C₂×C₇₃	#3290	1	1	yes	characteristic
197	D₇₃	—	4	2	no	—
198	#3384	—	73	2	no	—
199	#3338	D₇₃	1	1	yes	characteristic, complemented
200	D₄×Q₁₆	—	73	1	no	—
201	#1707	—	73	1	no	—
202	#1692	—	73	2	no	—
203	#1675	—	73	1	no	—
204	#1669	—	73	1	no	—
205	#1644	—	73	2	no	—
206	#1637	—	73	2	no	—
207	#1591	—	73	1	no	—
208	C₈×D₈	—	73	1	no	—
209	C₇₃	#19926	1	1	yes	characteristic, complemented, Sylow
210	C₂²×Q₁₆	—	146	1	no	—
211	#554	C₂×D₇₃	1	2	yes	—
212	D₄×Q₈	—	73	2	no	—
213	#501	—	146	2	no	—
214	#500	C₂×D₇₃	1	1	yes	characteristic
215	#498	—	73	2	no	—
216	#494	—	73	1	no	—
217	#487	—	73	2	no	—
218	#484	—	2	2	no	—
219	#478	—	146	2	no	—
220	#477	C₂×D₇₃	1	2	yes	—
221	#476	—	73	2	no	—
222	#476	—	146	2	no	—
223	#474	—	73	2	no	—
224	#467	—	2	1	no	—
225	#467	—	146	1	no	—
226	#462	—	73	2	no	—
227	#451	—	146	2	no	—
228	#446	—	73	2	no	—
229	#442	—	146	2	no	—
230	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
231	C₄×D₈	—	73	1	no	—
232	C₈×D₄	—	73	1	no	—
233	C₈×D₄	C₂×D₇₃	1	1	yes	characteristic
234	C₂×#107	—	2	1	no	—
235	#335	—	73	1	no	—
236	#335	—	73	1	no	—
237	#333	—	73	2	no	—
238	#332	—	73	2	no	—
239	Q₈⋊C₈	—	73	2	no	—
240	D₄⋊C₈	—	73	1	no	—
241	D₄⋊C₈	—	73	1	no	—
242	C₈²	—	73	1	no	—
243	C₂²×Q₈	—	146	2	no	—
244	C₂×Q₁₆	—	292	4	no	—
245	C₂×Q₁₆	—	146	2	no	—
246	C₂×Q₁₆	—	146	1	no	—
247	C₂×Q₁₆	—	146	1	no	—
248	C₂×Q₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
249	C₂×D₈	—	73	1	no	—
250	C₂²×C₈	—	2	1	no	—
251	C₂²×C₈	—	146	1	no	—
252	C₄⋊Q₈	—	73	2	no	—
253	C₄⋊Q₈	—	146	2	no	—
254	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
255	#125	—	2	2	no	—
256	C₂²⋊Q₈	—	2	2	no	—
257	C₂²⋊Q₈	—	4	2	no	—
258	C₂²⋊Q₈	—	292	2	no	—
259	C₂²⋊Q₈	—	146	2	no	—
260	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
261	C₄×D₄	—	2	2	no	—
262	C₄×D₄	—	73	1	no	—
263	C₄×D₄	—	146	2	no	—
264	C₄×D₄	C₂²×D₇₃	1	1	yes	characteristic
265	C₂×C₄⋊C₄	—	2	2	no	—
266	#107	—	2	1	no	—
267	#107	—	2	1	no	—
268	#107	—	2	2	no	—
269	#107	—	73	1	no	—
270	#107	—	146	2	no	—
271	#107	C₂²⋊D₇₃	1	1	yes	characteristic
272	#106	—	146	2	no	—
273	#106	—	146	2	no	—
274	#106	—	146	2	no	—
275	C₄⋊C₈	—	73	1	no	—
276	C₄⋊C₈	—	73	2	no	—
277	C₄⋊C₈	—	146	2	no	—
278	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
279	#103	—	2	2	no	—
280	#103	—	292	2	no	—
281	#103	—	146	2	no	—
282	#103	C₂²⋊D₇₃	1	2	yes	—
283	#102	—	73	1	no	—
284	#102	—	73	1	no	—
285	#102	—	146	2	no	—
286	#102	—	146	2	no	—
287	C₂²⋊C₈	—	2	1	no	—
288	C₂²⋊C₈	—	146	1	no	—
289	#97	—	146	2	no	—
290	C₄×C₈	—	73	1	no	—
291	C₄×C₈	—	73	1	no	—
292	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	292	2	no	—
295	C₂×Q₈	—	292	4	no	—
296	C₂×Q₈	—	584	2	no	—
297	C₂×Q₈	—	146	2	no	—
298	C₂×Q₈	C₂×C₂²⋊D₇₃	1	2	yes	—
299	C₂×D₄	—	73	1	no	—
300	C₂×D₄	D₄×D₇₃	1	1	yes	characteristic
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	4	2	no	—
303	C₂²×C₄	—	292	2	no	—
304	C₂²×C₄	—	146	1	no	—
305	Q₁₆	—	2	2	no	—
306	Q₁₆	—	292	4	no	—
307	Q₁₆	—	292	2	no	—
308	Q₁₆	—	584	1	no	—
309	D₈	—	73	4	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	4	2	no	—
312	C₂×C₈	—	292	2	no	—
313	C₂×C₈	—	73	1	no	—
314	C₂×C₈	—	73	1	no	—
315	C₂×C₈	—	146	2	no	—
316	C₂×C₈	—	146	1	no	—
317	C₂×C₈	—	146	1	no	—
318	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
319	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	4	2	no	—
324	C₄⋊C₄	—	4	2	no	—
325	C₄⋊C₄	—	292	2	no	—
326	C₄⋊C₄	—	73	1	no	—
327	C₄⋊C₄	—	146	2	no	—
328	C₄⋊C₄	—	146	2	no	—
329	C₄⋊C₄	D₄×D₇₃	1	1	yes	characteristic
330	C₄⋊C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
331	C₂²⋊C₄	—	2	1	no	—
332	C₂²⋊C₄	—	4	2	no	—
333	C₂²⋊C₄	—	292	2	no	—
334	C₂²⋊C₄	—	146	1	no	—
335	C₄²	—	2	2	no	—
336	C₄²	C₂³×D₇₃	1	1	yes	characteristic
337	C₂³	—	2	1	no	—
338	C₂³	—	146	1	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	2	2	no	—
341	Q₈	—	8	2	no	—
342	Q₈	—	292	2	no	—
343	Q₈	—	292	2	no	—
344	Q₈	—	584	4	no	—
345	D₄	—	2	1	no	—
346	D₄	—	73	2	no	—
347	D₄	—	146	1	no	—
348	D₄	Q₁₆×D₇₃	1	2	yes	complemented
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	16	2	no	—
356	C₂×C₄	—	292	1	no	—
357	C₂×C₄	—	292	1	no	—
358	C₂×C₄	—	292	2	no	—
359	C₂×C₄	—	584	4	no	—
360	C₂×C₄	—	146	1	no	—
361	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
362	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
363	C₂×C₄	#14556636	1	1	yes	characteristic
364	C₈	—	2	1	no	—
365	C₈	—	4	1	no	—
366	C₈	—	292	1	no	—
367	C₈	—	292	2	no	—
368	C₈	—	292	1	no	—
369	C₈	—	73	2	no	—
370	C₈	—	146	1	no	—
371	C₈	#14556616	1	2	yes	—
372	C₂²	—	2	2	no	—
373	C₂²	—	4	1	no	—
374	C₂²	—	4	1	no	—
375	C₂²	—	292	1	no	—
376	C₂²	—	292	1	no	—
377	C₂²	—	146	2	no	—
378	C₂²	D₄×C₂²⋊D₇₃	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	4	2	no	—
381	C₄	—	4	1	no	—
382	C₄	—	4	2	no	—
383	C₄	—	8	2	no	—
384	C₄	—	292	1	no	—
385	C₄	—	584	2	no	—
386	C₄	C₂×Q₁₆×D₇₃	1	1	yes	characteristic
387	C₄	C₂×#14556616	1	1	yes	characteristic
388	C₄	#17989395	1	1	yes	characteristic
389	C₄	#17989400	1	1	yes	characteristic
390	C₂	—	4	2	no	—
391	C₂	—	292	2	no	—
392	C₂	#22621531	1	1	yes	characteristic
393	C₂	Q₁₆×C₂²⋊D₇₃	1	1	yes	characteristic
394	C₂	#22621713	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₇₃
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#3338	C₂
socle	C₂²×C₇₃	D₄²
prime radical	C₂×#11924013	D₄
hypercenter	#3338	D₇₃
hypocenter	C₇₃	#19926
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #3338

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₂₉₂	C₂	semidirect
3. #22609529	C₂	semidirect
4. #22609439	C₂	semidirect
5. #22609327	C₂	semidirect
6. #22609296	C₂	semidirect
7. #22609145	C₂	semidirect
8. #22609094	C₂	semidirect
9. #22608828	C₂	semidirect
11. C₇₃×#3338	C₂	semidirect
105. #11924013	Q₁₆	semidirect
145. D₄×C₇₃	C₂×Q₁₆	semidirect
173. #10596831	C₂×Q₁₆	semidirect
199. #3338	D₇₃	semidirect
209. C₇₃	#19926	semidirect
348. D₄	Q₁₆×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22622830	2
Q₁₆×D₂₉₂	2
#22609529	2
#22609439	2
#22609327	2
#22609296	2
#22609145	2
#22609094	2
#22608828	2
C₈×#11924013	2
C₇₃×#3338	2
#19926	73