Subgroups — #33077935

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622941	C₂	1	1	yes	maximal, characteristic, complemented
2	#22622723	C₂	1	2	yes	maximal, complemented
3	#22609538	C₂	1	1	yes	maximal, characteristic, complemented
4	#22609501	C₂	1	2	yes	maximal, complemented
5	#22609298	C₂	1	1	yes	maximal, characteristic, complemented
6	#22609286	C₂	1	1	yes	maximal, characteristic
7	#22609115	C₂	1	2	yes	maximal, complemented
8	#22609034	C₂	1	2	yes	maximal, complemented
9	#22608828	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11924016	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989322	C₂²	1	2	yes	—
13	#17988972	—	2	1	no	—
14	#17988964	C₂²	1	1	yes	characteristic
15	#17988949	—	2	2	no	—
16	#17988935	C₂²	1	2	yes	—
17	#17988935	C₂²	1	2	yes	—
18	#17988874	—	2	2	no	—
19	#17988869	C₂²	1	2	yes	—
20	#17988805	—	2	2	no	—
21	#17988802	—	2	2	no	—
22	#17988787	C₂²	1	2	yes	—
23	#17988782	C₂²	1	2	yes	—
24	#17988718	C₂²	1	2	yes	—
25	C₈×Dic₁₄₆	C₂²	1	1	yes	characteristic
26	#17988697	—	2	2	no	—
27	C₄×#11924016	C₂²	1	1	yes	characteristic
28	#17988669	—	2	1	no	—
29	#17988340	C₂²	1	1	yes	characteristic
30	#17988337	C₂²	1	1	yes	characteristic
31	#17988333	C₂²	1	2	yes	—
32	#17988330	C₂²	1	2	yes	—
33	Q₈⋊#10596831	C₂²	1	2	yes	—
34	#17988317	C₂²	1	1	yes	characteristic
35	D₄⋊#10596831	C₂²	1	1	yes	characteristic
36	C₈×#10596831	C₂²	1	1	yes	characteristic
37	C₂²×Q₁₆×C₇₃	—	2	1	no	—
38	D₄×Q₈×C₇₃	C₂²	1	2	yes	—
39	#494×C₇₃	C₂²	1	1	yes	characteristic
40	#467×C₇₃	—	2	1	no	—
41	#462×C₇₃	C₂²	1	2	yes	—
42	#451×C₇₃	—	2	2	no	—
43	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
44	C₈×D₄×C₇₃	C₂²	1	1	yes	characteristic
45	C₂×#11924016	D₄	1	1	yes	characteristic
46	C₂×#11924015	—	2	2	no	—
47	C₂×D₅₈₄	—	2	1	no	—
48	C₂×D₅₈₄	—	2	1	no	—
49	C₂×#11924013	—	2	2	no	—
50	#14556588	C₂³	1	2	yes	—
51	#14556580	—	2	2	no	—
52	#14556579	—	2	2	no	—
53	#14556576	—	4	2	no	—
54	C₄×Dic₁₄₆	C₂³	1	1	yes	characteristic
55	C₄×D₂₉₂	—	2	2	no	—
56	#14556553	—	2	2	no	—
57	#14556522	—	2	2	no	—
58	#14556522	D₄	1	1	yes	characteristic
59	C₄⋊#10596831	—	2	2	no	—
60	C₄⋊#10596831	C₂³	1	2	yes	—
61	C₄⋊#10596831	C₂³	1	1	yes	characteristic
62	#14556520	—	2	1	no	—
63	#14556514	—	2	2	no	—
64	#14556514	D₄	1	1	yes	characteristic
65	#14556512	D₄	1	1	yes	characteristic
66	#14556511	—	2	2	no	—
67	#14556511	—	2	2	no	—
68	#14556511	—	2	2	no	—
69	#14556511	—	2	2	no	—
70	#14556511	—	4	2	no	—
71	#14556502	—	2	2	no	—
72	C₄×#10596831	C₂³	1	1	yes	characteristic
73	C₄×#10596831	C₂³	1	1	yes	characteristic
74	C₈×Dic₇₃	—	2	1	no	—
75	C₂²×Q₈×C₇₃	—	2	2	no	—
76	C₂×Q₁₆×C₇₃	—	2	2	no	—
77	C₂×Q₁₆×C₇₃	—	2	1	no	—
78	C₂×Q₁₆×C₇₃	—	2	1	no	—
79	C₂×Q₁₆×C₇₃	—	4	4	no	—
80	C₂×Q₁₆×C₇₃	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₇₃	—	2	1	no	—
82	C₄⋊Q₈×C₇₃	—	2	2	no	—
83	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
84	C₂²⋊Q₈×C₇₃	—	2	2	no	—
85	C₂²⋊Q₈×C₇₃	—	4	2	no	—
86	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
87	C₄×D₄×C₇₃	—	2	2	no	—
88	C₄×D₄×C₇₃	C₂³	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
96	C₂×D₂₉₂	—	2	2	no	—
97	C₂×D₂₉₂	—	2	2	no	—
98	C₂×D₂₉₂	—	2	2	no	—
99	C₂×C₄×D₇₃	—	4	2	no	—
100	#11924016	—	2	2	no	—
101	#11924015	—	8	2	no	—
102	D₅₈₄	—	4	2	no	—
103	D₅₈₄	—	8	1	no	—
104	#11924013	—	4	4	no	—
105	C₂×#10596831	—	2	1	no	—
106	C₂×#10596831	—	2	2	no	—
107	C₂×#10596831	—	2	1	no	—
108	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
109	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
110	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
111	#11924007	—	2	1	no	—
112	#11924005	—	4	2	no	—
113	#11924005	—	4	2	no	—
114	#11924005	—	4	2	no	—
115	C₄×Dic₇₃	—	2	1	no	—
116	C₂×Q₈×C₇₃	—	2	2	no	—
117	C₂×Q₈×C₇₃	—	4	2	no	—
118	C₂×Q₈×C₇₃	—	4	4	no	—
119	C₂×Q₈×C₇₃	—	8	2	no	—
120	C₂×Q₈×C₇₃	C₂×D₄	1	2	yes	—
121	C₂×D₄×C₇₃	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₇₃	—	2	1	no	—
123	C₂²×C₄×C₇₃	—	4	2	no	—
124	Q₁₆×C₇₃	—	2	2	no	—
125	Q₁₆×C₇₃	—	4	4	no	—
126	Q₁₆×C₇₃	—	4	2	no	—
127	Q₁₆×C₇₃	—	8	1	no	—
128	C₂×C₈×C₇₃	—	2	1	no	—
129	C₂×C₈×C₇₃	—	4	2	no	—
130	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₇₃	—	2	2	no	—
133	C₄⋊C₄×C₇₃	—	2	2	no	—
134	C₄⋊C₄×C₇₃	—	2	2	no	—
135	C₄⋊C₄×C₇₃	—	4	2	no	—
136	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
138	C₂²⋊C₄×C₇₃	—	2	1	no	—
139	C₂²⋊C₄×C₇₃	—	4	2	no	—
140	C₄²×C₇₃	—	2	2	no	—
141	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
142	C₂³×C₇₃	—	2	1	no	—
143	C₂²×D₇₃	—	4	2	no	—
144	Q₈×C₇₃	—	2	2	no	—
145	Q₈×C₇₃	—	2	2	no	—
146	Q₈×C₇₃	—	4	2	no	—
147	Q₈×C₇₃	—	4	2	no	—
148	Q₈×C₇₃	—	8	4	no	—
149	D₄×C₇₃	—	2	1	no	—
150	D₄×C₇₃	#135	1	2	yes	—
151	C₂×C₄×C₇₃	—	2	2	no	—
152	C₂×C₄×C₇₃	—	2	2	no	—
153	C₂×C₄×C₇₃	—	2	1	no	—
154	C₂×C₄×C₇₃	—	4	1	no	—
155	C₂×C₄×C₇₃	—	4	1	no	—
156	C₂×C₄×C₇₃	—	4	2	no	—
157	C₂×C₄×C₇₃	—	8	4	no	—
158	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂×Dic₇₃	—	2	1	no	—
162	C₂×Dic₇₃	—	2	1	no	—
163	D₂₉₂	—	4	2	no	—
164	D₂₉₂	—	4	2	no	—
165	D₂₉₂	—	4	2	no	—
166	D₂₉₂	—	4	2	no	—
167	D₂₉₂	—	8	2	no	—
168	C₄×D₇₃	—	16	2	no	—
169	Dic₁₄₆	—	2	1	no	—
170	Dic₁₄₆	—	2	1	no	—
171	C₈×C₇₃	—	2	1	no	—
172	C₈×C₇₃	—	4	1	no	—
173	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
174	#10596831	—	2	1	no	—
175	#10596831	—	2	1	no	—
176	#10596831	—	4	1	no	—
177	#10596831	—	4	2	no	—
178	#10596831	—	4	1	no	—
179	C₂²×C₇₃	—	2	2	no	—
180	C₂²×C₇₃	—	4	1	no	—
181	C₂²×C₇₃	—	4	1	no	—
182	C₂²×C₇₃	D₄²	1	1	yes	characteristic
183	C₂×D₇₃	—	8	2	no	—
184	C₂×D₇₃	—	8	2	no	—
185	C₂×D₇₃	—	8	2	no	—
186	C₄×C₇₃	—	2	1	no	—
187	C₄×C₇₃	—	4	1	no	—
188	C₄×C₇₃	—	4	2	no	—
189	C₄×C₇₃	—	4	2	no	—
190	C₄×C₇₃	—	8	2	no	—
191	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
192	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
193	C₄×C₇₃	#576	1	1	yes	characteristic
194	C₄×C₇₃	#578	1	1	yes	characteristic
195	Dic₇₃	—	4	1	no	—
196	Dic₇₃	—	4	1	no	—
197	#19935	—	73	1	no	maximal, Sylow
198	C₂×C₇₃	—	4	2	no	—
199	C₂×C₇₃	#3274	1	1	yes	characteristic
200	C₂×C₇₃	#3285	1	1	yes	characteristic
201	C₂×C₇₃	#3290	1	1	yes	characteristic
202	D₇₃	—	16	2	no	—
203	#3396	—	73	1	no	—
204	#3373	—	73	2	no	—
205	D₄×Q₁₆	D₇₃	1	1	yes	characteristic, complemented
206	#1709	—	73	1	no	—
207	#1702	—	73	2	no	—
208	#1670	—	73	1	no	—
209	#1667	—	73	1	no	—
210	#1640	—	73	2	no	—
211	#1627	—	73	2	no	—
212	#1591	—	73	1	no	—
213	C₈×SD₁₆	—	73	1	no	—
214	C₇₃	#19935	1	1	yes	characteristic, complemented, Sylow
215	C₂²×Q₁₆	—	2	1	no	—
216	#559	—	73	2	no	—
217	D₄×Q₈	C₂×D₇₃	1	2	yes	—
218	#495	—	146	1	no	—
219	#494	C₂×D₇₃	1	1	yes	characteristic
220	#493	—	73	1	no	—
221	#489	—	146	2	no	—
222	#486	—	73	2	no	—
223	#486	—	73	2	no	—
224	#476	—	146	2	no	—
225	#475	—	73	2	no	—
226	#467	—	2	1	no	—
227	#463	—	146	2	no	—
228	#463	—	146	2	no	—
229	#462	C₂×D₇₃	1	2	yes	—
230	#460	—	73	2	no	—
231	#459	—	73	2	no	—
232	#451	—	2	2	no	—
233	#446	—	73	2	no	—
234	C₈×Q₈	—	73	1	no	—
235	#442	—	146	2	no	—
236	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
237	C₄×SD₁₆	—	73	1	no	—
238	C₄×D₈	—	146	1	no	—
239	C₈×D₄	C₂×D₇₃	1	1	yes	characteristic
240	#335	—	73	1	no	—
241	#334	—	73	1	no	—
242	#332	—	73	2	no	—
243	#331	—	73	2	no	—
244	Q₈⋊C₈	—	73	2	no	—
245	Q₈⋊C₈	—	73	1	no	—
246	D₄⋊C₈	—	73	1	no	—
247	C₈²	—	73	1	no	—
248	C₂²×Q₈	—	2	2	no	—
249	C₂×Q₁₆	—	2	2	no	—
250	C₂×Q₁₆	—	2	1	no	—
251	C₂×Q₁₆	—	2	1	no	—
252	C₂×Q₁₆	—	4	4	no	—
253	C₂×Q₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
254	C₂×SD₁₆	—	73	1	no	—
255	C₂×SD₁₆	—	146	2	no	—
256	C₂×D₈	—	146	2	no	—
257	C₂×D₈	—	146	1	no	—
258	C₂×D₈	—	146	1	no	—
259	C₂²×C₈	—	2	1	no	—
260	C₄⋊Q₈	—	2	2	no	—
261	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
262	#127	—	73	2	no	—
263	#124	—	146	2	no	—
264	#124	—	146	2	no	—
265	#123	—	292	2	no	—
266	C₂²⋊Q₈	—	2	2	no	—
267	C₂²⋊Q₈	—	4	2	no	—
268	C₄×Q₈	—	73	1	no	—
269	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
270	C₄×D₄	—	2	2	no	—
271	C₄×D₄	—	146	2	no	—
272	C₄×D₄	—	146	2	no	—
273	C₄×D₄	C₂²×D₇₃	1	1	yes	characteristic
274	#107	C₂²⋊D₇₃	1	1	yes	characteristic
275	#106	—	73	1	no	—
276	#106	—	146	2	no	—
277	C₄⋊C₈	—	73	1	no	—
278	C₄⋊C₈	—	73	2	no	—
279	C₄⋊C₈	—	146	2	no	—
280	C₄⋊C₈	—	146	1	no	—
281	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
282	#103	—	2	2	no	—
283	#103	—	73	1	no	—
284	#103	—	146	2	no	—
285	#103	C₂²⋊D₇₃	1	2	yes	—
286	#102	—	292	2	no	—
287	#102	—	73	1	no	—
288	#102	—	146	2	no	—
289	#102	—	146	2	no	—
290	#102	—	146	2	no	—
291	#102	—	146	2	no	—
292	C₂²⋊C₈	—	2	1	no	—
293	#97	—	146	2	no	—
294	C₄×C₈	—	73	1	no	—
295	C₄×C₈	—	73	1	no	—
296	C₄×C₈	—	146	1	no	—
297	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	4	2	no	—
300	C₂×Q₈	—	4	4	no	—
301	C₂×Q₈	—	8	2	no	—
302	C₂×Q₈	—	73	1	no	—
303	C₂×Q₈	C₂×C₂²⋊D₇₃	1	2	yes	—
304	C₂×D₄	—	146	2	no	—
305	C₂×D₄	—	146	2	no	—
306	C₂×D₄	—	146	2	no	—
307	C₂×D₄	D₄×D₇₃	1	1	yes	characteristic
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	4	2	no	—
310	C₂²×C₄	—	292	2	no	—
311	Q₁₆	—	2	2	no	—
312	Q₁₆	—	4	4	no	—
313	Q₁₆	—	4	2	no	—
314	Q₁₆	—	8	1	no	—
315	SD₁₆	—	584	2	no	—
316	SD₁₆	—	146	2	no	—
317	D₈	—	292	4	no	—
318	D₈	—	292	2	no	—
319	D₈	—	584	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	4	2	no	—
322	C₂×C₈	—	73	1	no	—
323	C₂×C₈	—	73	1	no	—
324	C₂×C₈	—	146	2	no	—
325	C₂×C₈	—	146	1	no	—
326	C₂×C₈	—	146	1	no	—
327	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
328	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	4	2	no	—
333	C₄⋊C₄	—	73	1	no	—
334	C₄⋊C₄	—	146	1	no	—
335	C₄⋊C₄	D₄×D₇₃	1	1	yes	characteristic
336	C₄⋊C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
337	C₂²⋊C₄	—	2	1	no	—
338	C₂²⋊C₄	—	4	2	no	—
339	C₂²⋊C₄	—	292	2	no	—
340	C₂²⋊C₄	—	292	2	no	—
341	C₂²⋊C₄	—	292	2	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	146	1	no	—
344	C₄²	C₂³×D₇₃	1	1	yes	characteristic
345	C₂³	—	2	1	no	—
346	C₂³	—	292	2	no	—
347	Q₈	—	2	2	no	—
348	Q₈	—	2	2	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	4	2	no	—
351	Q₈	—	8	4	no	—
352	Q₈	—	146	1	no	—
353	Q₈	—	146	1	no	—
354	D₄	—	2	1	no	—
355	D₄	—	292	2	no	—
356	D₄	—	292	2	no	—
357	D₄	—	292	2	no	—
358	D₄	—	292	2	no	—
359	D₄	—	584	2	no	—
360	D₄	#14556614	1	2	yes	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	8	4	no	—
368	C₂×C₄	—	1168	2	no	—
369	C₂×C₄	—	146	1	no	—
370	C₂×C₄	—	146	1	no	—
371	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
372	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
373	C₂×C₄	#14556636	1	1	yes	characteristic
374	C₈	—	2	1	no	—
375	C₈	—	4	1	no	—
376	C₈	—	292	1	no	—
377	C₈	—	292	1	no	—
378	C₈	—	292	2	no	—
379	C₈	—	146	1	no	—
380	C₈	—	146	1	no	—
381	C₈	#14556617	1	2	yes	—
382	C₂²	—	2	2	no	—
383	C₂²	—	4	1	no	—
384	C₂²	—	4	1	no	—
385	C₂²	—	584	2	no	—
386	C₂²	—	584	2	no	—
387	C₂²	—	584	2	no	—
388	C₂²	D₄×C₂²⋊D₇₃	1	1	yes	characteristic
389	C₄	—	2	1	no	—
390	C₄	—	4	2	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	2	no	—
393	C₄	—	8	2	no	—
394	C₄	—	292	1	no	—
395	C₄	—	292	1	no	—
396	C₄	C₂×#14556614	1	1	yes	characteristic
397	C₄	C₂×#14556617	1	1	yes	characteristic
398	C₄	#17989390	1	1	yes	characteristic
399	C₄	#17989400	1	1	yes	characteristic
400	C₂	—	4	2	no	—
401	C₂	—	1168	2	no	—
402	C₂	#22621547	1	1	yes	characteristic
403	C₂	#22621672	1	1	yes	characteristic
404	C₂	#22621717	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₇₃
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	D₄×Q₁₆×C₇₃	C₂
socle	C₂²×C₇₃	D₄²
prime radical	#22609286	C₂
hypercenter	D₄×Q₁₆	D₇₃
hypocenter	C₇₃	#19935
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ D₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #22622941	C₂	semidirect
2. #22622723	C₂	semidirect
3. #22609538	C₂	semidirect
4. #22609501	C₂	semidirect
5. #22609298	C₂	semidirect
7. #22609115	C₂	semidirect
8. #22609034	C₂	semidirect
9. #22608828	C₂	semidirect
10. C₈×#11924016	C₂	semidirect
11. D₄×Q₁₆×C₇₃	C₂	semidirect
205. D₄×Q₁₆	D₇₃	semidirect
214. C₇₃	#19935	semidirect

Maximal subgroups

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

subgroup	index
#22622941	2
#22622723	2
#22609538	2
#22609501	2
#22609298	2
#22609286	2
#22609115	2
#22609034	2
#22608828	2
C₈×#11924016	2
D₄×Q₁₆×C₇₃	2
#19935	73