Subgroups — #33077318

Order 18688.

710 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622906	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₅₈₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#22621598	C₂	1	2	yes	maximal, complemented
4	#22609523	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609397	C₂	1	2	yes	maximal, complemented
6	#22609325	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609291	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609126	C₂	1	2	yes	maximal, complemented
9	#22609036	C₂	1	2	yes	maximal, complemented
10	#22608817	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556623	—	2	2	no	—
13	#17989259	C₂²	1	2	yes	—
14	Q₈×D₂₉₂	C₂²	1	2	yes	—
15	#17988992	—	2	1	no	—
16	#17988992	C₂²	1	1	yes	characteristic
17	#17988964	—	2	1	no	—
18	#17988964	C₂²	1	1	yes	characteristic
19	#17988960	C₂²	1	2	yes	—
20	#17988934	C₂²	1	2	yes	—
21	#17988868	—	2	2	no	—
22	#17988868	C₂²	1	2	yes	—
23	#17988834	—	2	2	no	—
24	#17988797	C₂²	1	2	yes	—
25	#17988787	C₂²	1	2	yes	—
26	#17988783	—	2	2	no	—
27	#17988783	C₂²	1	2	yes	—
28	#17988752	—	2	2	no	—
29	#17988727	—	2	2	no	—
30	C₄×D₅₈₄	—	2	1	no	—
31	C₄×D₅₈₄	—	2	1	no	—
32	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
33	#17988666	C₂²	1	2	yes	—
34	#17988340	C₂²	1	1	yes	characteristic
35	#17988332	C₂²	1	2	yes	—
36	#17988330	C₂²	1	2	yes	—
37	#17988316	C₂²	1	2	yes	—
38	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
39	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
41	#335×C₇₃	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
44	C₈²×C₇₃	C₂²	1	1	yes	characteristic
45	C₂×#11924029	—	2	2	no	—
46	C₂×Q₈×D₇₃	—	2	2	no	—
47	#14556623	—	4	8	no	—
48	C₂×#11924018	—	2	2	no	—
49	C₂×#11924017	—	2	2	no	—
50	C₂×#11924017	—	2	2	no	—
51	C₂×#11924015	—	2	2	no	—
52	C₂×D₅₈₄	—	2	2	no	—
53	C₂×D₅₈₄	—	2	1	no	—
54	C₂×D₅₈₄	—	2	1	no	—
55	C₂×D₅₈₄	D₄	1	1	yes	characteristic
56	C₂×#11924011	—	2	2	no	—
57	#14556591	—	2	2	no	—
58	#14556591	C₂³	1	2	yes	—
59	#14556588	—	2	2	no	—
60	#14556588	C₂³	1	2	yes	—
61	#14556571	—	2	2	no	—
62	#14556571	—	4	2	no	—
63	#14556566	—	2	2	no	—
64	#14556566	—	4	2	no	—
65	C₄×D₂₉₂	—	2	2	no	—
66	C₄×D₂₉₂	—	2	2	no	—
67	C₄×D₂₉₂	C₂³	1	2	yes	—
68	#14556525	—	2	2	no	—
69	#14556525	—	2	1	no	—
70	#14556525	—	2	1	no	—
71	#14556525	D₄	1	1	yes	characteristic
72	C₄⋊#10596831	C₂³	1	2	yes	—
73	#14556514	—	2	2	no	—
74	#14556513	—	2	2	no	—
75	#14556513	—	2	2	no	—
76	#14556513	—	4	2	no	—
77	#14556513	D₄	1	2	yes	—
78	#14556511	—	2	2	no	—
79	#14556503	—	2	2	no	—
80	C₂×Q₁₆×C₇₃	D₄	1	1	yes	characteristic
81	C₄×Q₈×C₇₃	C₂³	1	1	yes	characteristic
82	C₄×Q₈×C₇₃	C₂³	1	1	yes	characteristic
83	#107×C₇₃	D₄	1	1	yes	characteristic
84	C₄⋊C₈×C₇₃	—	2	1	no	—
85	C₄⋊C₈×C₇₃	—	2	1	no	—
86	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
88	#103×C₇₃	D₄	1	1	yes	characteristic
89	#103×C₇₃	D₄	1	1	yes	characteristic
90	C₄×C₈×C₇₃	—	2	1	no	—
91	C₄×C₈×C₇₃	—	2	1	no	—
92	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
95	#11924029	—	4	4	no	—
96	#11924029	—	8	2	no	—
97	C₂×Dic₁₄₆	—	2	2	no	—
98	C₂×Dic₁₄₆	—	4	2	no	—
99	Q₈×D₇₃	—	4	4	no	—
100	Q₈×D₇₃	—	8	2	no	—
101	C₂×D₂₉₂	—	2	2	no	—
102	C₂×D₂₉₂	—	2	2	no	—
103	C₂×D₂₉₂	—	2	2	no	—
104	C₂×D₂₉₂	—	4	2	no	—
105	C₂×D₂₉₂	—	4	2	no	—
106	C₂×D₂₉₂	C₂×D₄	1	2	yes	—
107	C₂×C₄×D₇₃	—	2	2	no	—
108	C₂×C₄×D₇₃	—	4	2	no	—
109	C₂×C₄×D₇₃	—	4	2	no	—
110	#11924018	—	4	4	no	—
111	#11924017	—	4	4	no	—
112	#11924017	—	8	2	no	—
113	#11924015	—	4	4	no	—
114	D₅₈₄	—	2	2	no	—
115	D₅₈₄	—	4	2	no	—
116	D₅₈₄	—	8	1	no	—
117	D₅₈₄	—	8	2	no	—
118	#11924011	—	4	4	no	—
119	C₂×#10596831	—	2	2	no	—
120	C₄⋊Dic₇₃	—	2	2	no	—
121	C₄⋊Dic₇₃	—	2	2	no	—
122	C₄⋊Dic₇₃	—	4	2	no	—
123	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
124	#11924005	—	2	2	no	—
125	#11924005	—	4	2	no	—
126	#11924005	—	4	2	no	—
127	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
129	Q₁₆×C₇₃	D₈	1	4	yes	complemented
130	C₂×C₈×C₇₃	—	2	1	no	—
131	C₂×C₈×C₇₃	—	2	1	no	—
132	C₂×C₈×C₇₃	—	2	1	no	—
133	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₇₃	—	2	1	no	—
138	C₄⋊C₄×C₇₃	—	2	1	no	—
139	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
140	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₇₃	—	2	1	no	—
142	C₄²×C₇₃	—	2	1	no	—
143	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
144	C₂²×D₇₃	—	2	2	no	—
145	C₂²×D₇₃	—	4	2	no	—
146	Q₈×C₇₃	—	2	1	no	—
147	Q₈×C₇₃	—	2	1	no	—
148	Q₈×C₇₃	C₂×D₈	1	2	yes	—
149	Q₈×C₇₃	C₂×D₈	1	2	yes	—
150	C₂×C₄×C₇₃	—	2	1	no	—
151	C₂×C₄×C₇₃	—	2	1	no	—
152	C₂×C₄×C₇₃	—	2	1	no	—
153	C₂×C₄×C₇₃	—	2	1	no	—
154	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×Dic₇₃	—	2	2	no	—
158	C₂×Dic₇₃	—	4	2	no	—
159	D₂₉₂	—	2	2	no	—
160	D₂₉₂	—	2	2	no	—
161	D₂₉₂	—	4	2	no	—
162	D₂₉₂	—	4	2	no	—
163	D₂₉₂	—	4	2	no	—
164	D₂₉₂	—	4	2	no	—
165	D₂₉₂	—	8	4	no	—
166	D₂₉₂	—	8	2	no	—
167	D₂₉₂	—	16	2	no	—
168	C₄×D₇₃	—	4	2	no	—
169	C₄×D₇₃	—	4	2	no	—
170	C₄×D₇₃	—	8	4	no	—
171	C₄×D₇₃	—	8	4	no	—
172	Dic₁₄₆	—	4	2	no	—
173	Dic₁₄₆	—	4	2	no	—
174	Dic₁₄₆	—	8	4	no	—
175	C₈×C₇₃	—	2	1	no	—
176	C₈×C₇₃	—	2	1	no	—
177	C₈×C₇₃	—	4	1	no	—
178	C₈×C₇₃	—	4	1	no	—
179	C₈×C₇₃	—	4	1	no	—
180	C₈×C₇₃	C₂×D₈	1	2	yes	—
181	C₈×C₇₃	#137	1	2	yes	complemented
182	#10596831	—	4	2	no	—
183	C₂²×C₇₃	D₄²	1	1	yes	characteristic
184	C₂×D₇₃	—	4	2	no	—
185	C₂×D₇₃	—	4	2	no	—
186	C₂×D₇₃	—	4	2	no	—
187	C₂×D₇₃	—	8	2	no	—
188	C₂×D₇₃	—	8	2	no	—
189	C₂×D₇₃	—	8	2	no	—
190	C₄×C₇₃	—	2	2	no	—
191	C₄×C₇₃	—	2	2	no	—
192	C₄×C₇₃	—	2	1	no	—
193	C₄×C₇₃	—	4	1	no	—
194	C₄×C₇₃	—	4	1	no	—
195	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
196	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
197	C₄×C₇₃	#576	1	1	yes	characteristic
198	C₄×C₇₃	#577	1	1	yes	characteristic
199	Dic₇₃	—	4	2	no	—
200	Dic₇₃	—	8	2	no	—
201	#19889	—	73	1	no	maximal, Sylow
202	C₂×C₇₃	#3278	1	1	yes	characteristic
203	C₂×C₇₃	D₄×D₈	1	1	yes	characteristic
204	C₂×C₇₃	#3290	1	1	yes	characteristic
205	D₇₃	—	8	2	no	—
206	D₇₃	—	16	2	no	—
207	#3391	—	73	1	no	—
208	Q₈×D₈	—	73	1	no	—
209	#3278	—	73	2	no	—
210	#1706	—	73	1	no	—
211	#1685	—	73	2	no	—
212	#1674	—	73	1	no	—
213	#1668	—	73	1	no	—
214	#1641	—	73	2	no	—
215	#1627	—	73	2	no	—
216	#1589	—	73	1	no	—
217	C₈×Q₁₆	D₇₃	1	1	yes	characteristic, complemented
218	C₇₃	#19889	1	1	yes	characteristic, complemented, Sylow
219	C₂×#137	—	146	2	no	—
220	#550	—	73	2	no	—
221	D₄×Q₈	—	73	2	no	—
222	#500	—	73	1	no	—
223	#500	—	146	1	no	—
224	#493	—	73	1	no	—
225	#493	—	146	1	no	—
226	#492	—	73	2	no	—
227	#486	—	73	2	no	—
228	#474	—	73	2	no	—
229	#474	—	146	2	no	—
230	#468	—	146	2	no	—
231	#462	—	73	2	no	—
232	#460	—	73	2	no	—
233	#459	—	73	2	no	—
234	#459	—	146	2	no	—
235	#452	—	146	2	no	—
236	#448	—	146	2	no	—
237	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
238	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
239	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
240	C₄×D₈	—	73	1	no	—
241	C₄×D₈	—	146	1	no	—
242	C₄×D₈	—	146	1	no	—
243	#436	—	73	2	no	—
244	#335	—	73	1	no	—
245	#335	C₂×D₇₃	1	1	yes	characteristic
246	#332	—	73	2	no	—
247	#331	—	73	2	no	—
248	Q₈⋊C₈	C₂×D₇₃	1	1	yes	characteristic
249	Q₈⋊C₈	C₂×D₇₃	1	1	yes	characteristic
250	D₄⋊C₈	—	73	2	no	—
251	C₈²	C₂×D₇₃	1	1	yes	characteristic
252	C₂×#41	—	146	2	no	—
253	C₂²×Q₈	—	146	2	no	—
254	#137	—	292	8	no	—
255	C₂×Q₁₆	—	146	2	no	—
256	C₂×Q₁₆	D₂₉₂	1	1	yes	characteristic
257	C₂×SD₁₆	—	146	2	no	—
258	C₂×SD₁₆	—	146	2	no	—
259	C₂×SD₁₆	—	146	2	no	—
260	C₂×D₈	—	73	1	no	—
261	C₂×D₈	—	146	2	no	—
262	C₂×D₈	—	146	1	no	—
263	C₂×D₈	—	146	1	no	—
264	C₂×M₁₆	—	146	2	no	—
265	C₄⋊Q₈	—	73	2	no	—
266	C₄⋊Q₈	—	146	2	no	—
267	#127	—	73	2	no	—
268	#127	—	146	2	no	—
269	C₂²⋊Q₈	—	292	2	no	—
270	C₂²⋊Q₈	—	146	2	no	—
271	#121	—	292	2	no	—
272	#121	—	146	2	no	—
273	C₄×Q₈	C₂²×D₇₃	1	1	yes	characteristic
274	C₄×Q₈	C₂²×D₇₃	1	1	yes	characteristic
275	C₄×D₄	—	73	2	no	—
276	C₄×D₄	—	146	2	no	—
277	C₄×D₄	—	146	2	no	—
278	#107	—	73	1	no	—
279	#107	—	146	2	no	—
280	#107	—	146	1	no	—
281	#107	—	146	1	no	—
282	#107	D₂₉₂	1	1	yes	characteristic
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	—	2	1	no	—
285	C₄⋊C₈	—	73	2	no	—
286	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
287	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
288	#103	—	146	2	no	—
289	#103	D₂₉₂	1	1	yes	characteristic
290	#103	D₂₉₂	1	1	yes	characteristic
291	#102	—	292	2	no	—
292	#102	—	73	2	no	—
293	#102	—	146	2	no	—
294	#102	—	146	2	no	—
295	#102	—	146	2	no	—
296	C₂²⋊C₈	—	146	2	no	—
297	C₄×C₈	—	2	1	no	—
298	C₄×C₈	—	2	1	no	—
299	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
300	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
301	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
302	#41	—	292	4	no	—
303	#41	—	584	2	no	—
304	C₂×Q₈	—	292	2	no	—
305	C₂×Q₈	—	292	4	no	—
306	C₂×Q₈	—	584	2	no	—
307	C₂×Q₈	—	146	2	no	—
308	C₂×Q₈	C₂×D₂₉₂	1	1	yes	characteristic
309	C₂×Q₈	C₂×D₂₉₂	1	1	yes	characteristic
310	C₂×D₄	—	292	2	no	—
311	C₂×D₄	—	292	2	no	—
312	C₂×D₄	—	73	2	no	—
313	C₂×D₄	—	146	2	no	—
314	C₂×D₄	—	146	2	no	—
315	C₂×D₄	—	146	2	no	—
316	C₂²×C₄	—	292	2	no	—
317	C₂²×C₄	—	292	2	no	—
318	C₂²×C₄	—	146	2	no	—
319	Q₁₆	—	292	4	no	—
320	Q₁₆	D₅₈₄	1	4	yes	complemented
321	SD₁₆	—	292	4	no	—
322	SD₁₆	—	292	4	no	—
323	SD₁₆	—	584	2	no	—
324	D₈	—	292	2	no	—
325	D₈	—	584	1	no	—
326	D₈	—	584	2	no	—
327	D₈	—	146	2	no	—
328	M₁₆	—	292	4	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	146	2	no	—
333	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
334	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
335	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
336	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	292	2	no	—
340	C₄⋊C₄	—	73	2	no	—
341	C₄⋊C₄	—	146	2	no	—
342	C₄⋊C₄	—	146	2	no	—
343	C₄⋊C₄	C₂×D₂₉₂	1	1	yes	characteristic
344	C₄⋊C₄	C₂×D₂₉₂	1	1	yes	characteristic
345	C₂²⋊C₄	—	292	2	no	—
346	C₂²⋊C₄	—	292	2	no	—
347	C₂²⋊C₄	—	146	2	no	—
348	C₄²	—	2	1	no	—
349	C₄²	—	2	1	no	—
350	C₄²	C₂³×D₇₃	1	1	yes	characteristic
351	C₂³	—	292	2	no	—
352	C₂³	—	146	2	no	—
353	Q₈	—	2	1	no	—
354	Q₈	—	2	1	no	—
355	Q₈	—	292	2	no	—
356	Q₈	—	292	2	no	—
357	Q₈	—	584	4	no	—
358	Q₈	C₂×D₅₈₄	1	2	yes	—
359	Q₈	C₂×D₅₈₄	1	2	yes	—
360	D₄	—	292	2	no	—
361	D₄	—	292	2	no	—
362	D₄	—	292	2	no	—
363	D₄	—	292	2	no	—
364	D₄	—	584	4	no	—
365	D₄	—	584	2	no	—
366	D₄	—	1168	2	no	—
367	D₄	—	146	2	no	—
368	D₄	—	146	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	292	2	no	—
374	C₂×C₄	—	292	2	no	—
375	C₂×C₄	—	292	2	no	—
376	C₂×C₄	—	584	4	no	—
377	C₂×C₄	—	584	4	no	—
378	C₂×C₄	—	146	2	no	—
379	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
380	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
381	C₂×C₄	#14556637	1	1	yes	characteristic
382	C₈	—	2	1	no	—
383	C₈	—	2	1	no	—
384	C₈	—	4	1	no	—
385	C₈	—	4	1	no	—
386	C₈	—	4	1	no	—
387	C₈	—	292	2	no	—
388	C₈	C₂×D₅₈₄	1	2	yes	—
389	C₈	#14556623	1	2	yes	complemented
390	C₂²	—	292	2	no	—
391	C₂²	—	292	2	no	—
392	C₂²	—	292	2	no	—
393	C₂²	—	584	2	no	—
394	C₂²	—	584	2	no	—
395	C₂²	—	584	2	no	—
396	C₂²	D₄×D₂₉₂	1	1	yes	characteristic
397	C₄	—	2	1	no	—
398	C₄	—	2	2	no	—
399	C₄	—	2	2	no	—
400	C₄	—	4	1	no	—
401	C₄	—	4	1	no	—
402	C₄	—	292	2	no	—
403	C₄	—	584	2	no	—
404	C₄	C₂²×D₅₈₄	1	1	yes	characteristic
405	C₄	C₂×#14556623	1	1	yes	characteristic
406	C₄	#17989392	1	1	yes	characteristic
407	C₄	#17989394	1	1	yes	characteristic
408	C₂	—	584	2	no	—
409	C₂	—	1168	2	no	—
410	C₂	#22621598	1	1	yes	characteristic
411	C₂	D₄×D₅₈₄	1	1	yes	characteristic
412	C₂	#22621718	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₉₂
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₈×Q₁₆×C₇₃	C₂
socle	C₂²×C₇₃	D₄²
prime radical	#22622906	C₂
hypercenter	C₈×Q₁₆	D₇₃
hypocenter	C₇₃	#19889
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₅₈₄	C₂	semidirect
3. #22621598	C₂	semidirect
4. #22609523	C₂	semidirect
5. #22609397	C₂	semidirect
6. #22609325	C₂	semidirect
7. #22609291	C₂	semidirect
8. #22609126	C₂	semidirect
9. #22609036	C₂	semidirect
10. #22608817	C₂	semidirect
11. C₈×Q₁₆×C₇₃	C₂	semidirect
129. Q₁₆×C₇₃	D₈	semidirect
181. C₈×C₇₃	#137	semidirect
217. C₈×Q₁₆	D₇₃	semidirect
218. C₇₃	#19889	semidirect
320. Q₁₆	D₅₈₄	semidirect
389. C₈	#14556623	semidirect

Maximal subgroups

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

subgroup	index
#22622906	2
Q₈×D₅₈₄	2
#22621598	2
#22609523	2
#22609397	2
#22609325	2
#22609291	2
#22609126	2
#22609036	2
#22608817	2
C₈×Q₁₆×C₇₃	2
#19889	73