Subgroups — #33077292

Order 18688.

690 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	Q₈×D₅₈₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#22622278	C₂	1	1	yes	maximal, characteristic
3	#22621572	C₂	1	2	yes	maximal, complemented
4	#22609548	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609419	C₂	1	2	yes	maximal, complemented
6	#22609321	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609178	C₂	1	2	yes	maximal, complemented
8	#22609167	C₂	1	2	yes	maximal, complemented
9	#22608815	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11924015	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#1674	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556623	—	2	2	no	—
13	Q₈×D₂₉₂	C₂²	1	2	yes	—
14	#17989240	C₂²	1	2	yes	—
15	#17988992	—	2	1	no	—
16	#17988992	C₂²	1	1	yes	characteristic
17	#17988990	C₂²	1	2	yes	—
18	#17988966	C₂²	1	2	yes	—
19	#17988902	—	2	2	no	—
20	#17988868	—	2	2	no	—
21	#17988868	C₂²	1	2	yes	—
22	#17988864	C₂²	1	2	yes	—
23	#17988844	—	2	2	no	—
24	#17988824	—	2	1	no	—
25	#17988793	C₂²	1	2	yes	—
26	#17988752	—	2	2	no	—
27	#17988741	—	2	2	no	—
28	C₄×#11924015	C₂²	1	1	yes	characteristic
29	C₄×D₅₈₄	—	2	1	no	—
30	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
31	#17988664	C₂²	1	2	yes	—
32	C₈×D₂₉₂	C₂²	1	1	yes	characteristic
33	#107×D₇₃	—	2	1	no	—
34	#17988337	C₂²	1	1	yes	characteristic
35	#17988334	C₂²	1	2	yes	—
36	Q₈⋊#10596831	C₂²	1	1	yes	characteristic
37	#17988314	C₂²	1	1	yes	characteristic
38	#17988314	C₂²	1	2	yes	—
39	C₈×#10596831	C₂²	1	1	yes	characteristic
40	#500×C₇₃	C₂²	1	1	yes	characteristic
41	#462×C₇₃	C₂²	1	2	yes	—
42	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
43	#335×C₇₃	C₂²	1	1	yes	characteristic
44	#332×C₇₃	C₂²	1	2	yes	—
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×#11924018	—	2	2	no	—
50	C₂×#11924017	—	2	2	no	—
51	C₂×#11924015	D₄	1	1	yes	characteristic
52	C₂×D₅₈₄	D₄	1	1	yes	characteristic
53	C₂×#11924011	—	2	2	no	—
54	C₂×C₈×D₇₃	—	2	1	no	—
55	#14556591	—	2	2	no	—
56	#14556591	C₂³	1	2	yes	—
57	#14556576	—	4	2	no	—
58	#14556571	—	2	2	no	—
59	#14556571	—	4	2	no	—
60	#14556567	—	2	2	no	—
61	#14556566	—	2	2	no	—
62	C₄×D₂₉₂	—	2	2	no	—
63	C₄×D₂₉₂	C₂³	1	2	yes	—
64	C₄×D₂₉₂	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×D₇₃	—	2	2	no	—
66	#14556525	—	2	2	no	—
67	#14556525	—	2	1	no	—
68	#14556525	—	2	1	no	—
69	#14556525	D₄	1	1	yes	characteristic
70	#14556524	—	2	2	no	—
71	#14556524	—	4	2	no	—
72	#14556522	D₄	1	1	yes	characteristic
73	C₄⋊#10596831	C₂³	1	1	yes	characteristic
74	#14556516	—	2	2	no	—
75	#14556515	D₄	1	1	yes	characteristic
76	#14556513	—	2	2	no	—
77	#14556513	D₄	1	2	yes	—
78	#14556511	—	2	2	no	—
79	#14556511	D₄	1	1	yes	characteristic
80	#14556503	—	2	2	no	—
81	#14556503	—	2	1	no	—
82	C₄×#10596831	C₂³	1	1	yes	characteristic
83	C₄×#10596831	C₂³	1	1	yes	characteristic
84	C₂×Q₁₆×C₇₃	—	2	2	no	—
85	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
86	C₄×Q₈×C₇₃	C₂³	1	1	yes	characteristic
87	#107×C₇₃	—	2	1	no	—
88	#107×C₇₃	—	2	1	no	—
89	C₄⋊C₈×C₇₃	—	2	1	no	—
90	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
91	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
92	#103×C₇₃	—	2	2	no	—
93	C₄×C₈×C₇₃	—	2	1	no	—
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	—	4	2	no	—
102	C₂×D₂₉₂	C₂×D₄	1	2	yes	—
103	C₂×D₂₉₂	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×D₇₃	—	2	1	no	—
105	C₂×C₄×D₇₃	—	2	2	no	—
106	C₂×C₄×D₇₃	—	4	2	no	—
107	C₂×C₄×D₇₃	—	4	2	no	—
108	#11924018	—	4	4	no	—
109	#11924018	—	8	2	no	—
110	#11924017	—	4	4	no	—
111	#11924015	D₈	1	4	yes	complemented
112	D₅₈₄	—	2	2	no	—
113	#11924011	—	4	4	no	—
114	C₂×#10596831	—	2	1	no	—
115	C₂×#10596831	—	2	1	no	—
116	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
117	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
118	C₈×D₇₃	—	4	2	no	—
119	C₄⋊Dic₇₃	—	2	2	no	—
120	C₄⋊Dic₇₃	—	2	2	no	—
121	C₄⋊Dic₇₃	—	4	2	no	—
122	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
123	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
124	#11924007	—	4	2	no	—
125	#11924005	—	2	2	no	—
126	#11924005	—	2	1	no	—
127	#11924005	—	4	2	no	—
128	#11924005	—	4	2	no	—
129	C₂×Q₈×C₇₃	—	2	2	no	—
130	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
131	Q₁₆×C₇₃	—	4	4	no	—
132	C₂×C₈×C₇₃	—	2	1	no	—
133	C₂×C₈×C₇₃	—	2	2	no	—
134	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
136	C₄⋊C₄×C₇₃	—	2	1	no	—
137	C₄⋊C₄×C₇₃	—	2	2	no	—
138	C₄⋊C₄×C₇₃	—	2	2	no	—
139	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₇₃	—	2	1	no	—
141	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
142	C₂²×D₇₃	—	2	1	no	—
143	C₂²×D₇₃	—	2	2	no	—
144	Q₈×C₇₃	—	2	1	no	—
145	Q₈×C₇₃	—	4	2	no	—
146	Q₈×C₇₃	—	4	2	no	—
147	Q₈×C₇₃	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₇₃	—	2	1	no	—
149	C₂×C₄×C₇₃	—	2	1	no	—
150	C₂×C₄×C₇₃	—	4	2	no	—
151	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂×Dic₇₃	—	2	1	no	—
155	C₂×Dic₇₃	—	2	2	no	—
156	C₂×Dic₇₃	—	4	2	no	—
157	D₂₉₂	—	2	2	no	—
158	D₂₉₂	—	2	1	no	—
159	D₂₉₂	—	2	2	no	—
160	D₂₉₂	—	8	4	no	—
161	D₂₉₂	C₂×D₈	1	2	yes	—
162	C₄×D₇₃	—	4	2	no	—
163	C₄×D₇₃	—	4	1	no	—
164	C₄×D₇₃	—	4	2	no	—
165	C₄×D₇₃	—	4	1	no	—
166	C₄×D₇₃	—	8	4	no	—
167	C₄×D₇₃	—	8	4	no	—
168	Dic₁₄₆	—	4	2	no	—
169	Dic₁₄₆	—	4	2	no	—
170	Dic₁₄₆	—	8	4	no	—
171	C₈×C₇₃	—	2	1	no	—
172	C₈×C₇₃	—	4	2	no	—
173	C₈×C₇₃	—	4	1	no	—
174	C₈×C₇₃	#137	1	2	yes	complemented
175	#10596831	—	2	2	no	—
176	#10596831	—	2	1	no	—
177	#10596831	—	4	1	no	—
178	#10596831	C₂×D₈	1	2	yes	—
179	C₂²×C₇₃	D₄²	1	1	yes	characteristic
180	C₂×D₇₃	—	2	2	no	—
181	C₂×D₇₃	—	4	1	no	—
182	C₂×D₇₃	—	4	1	no	—
183	C₂×D₇₃	—	4	2	no	—
184	C₂×D₇₃	—	4	2	no	—
185	C₂×D₇₃	—	4	2	no	—
186	C₄×C₇₃	—	2	2	no	—
187	C₄×C₇₃	—	2	1	no	—
188	C₄×C₇₃	—	4	1	no	—
189	C₄×C₇₃	—	8	2	no	—
190	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
191	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
192	C₄×C₇₃	#576	1	1	yes	characteristic
193	C₄×C₇₃	#578	1	1	yes	characteristic
194	Dic₇₃	—	4	1	no	—
195	Dic₇₃	—	4	2	no	—
196	Dic₇₃	—	8	2	no	—
197	#19887	—	73	1	no	maximal, Sylow
198	C₂×C₇₃	#3276	1	1	yes	characteristic
199	C₂×C₇₃	D₄×D₈	1	1	yes	characteristic
200	C₂×C₇₃	#3293	1	1	yes	characteristic
201	D₇₃	—	4	2	no	—
202	D₇₃	—	8	2	no	—
203	Q₈×D₈	—	73	1	no	—
204	#3334	—	73	1	no	—
205	#3276	—	73	2	no	—
206	#1713	—	73	1	no	—
207	#1689	—	73	2	no	—
208	#1674	D₇₃	1	1	yes	characteristic, complemented
209	#1673	—	73	1	no	—
210	#1649	—	73	2	no	—
211	#1647	—	73	2	no	—
212	#1589	—	73	1	no	—
213	C₈×SD₁₆	—	73	1	no	—
214	C₇₃	#19887	1	1	yes	characteristic, complemented, Sylow
215	C₂×#137	—	146	2	no	—
216	D₄×Q₈	—	73	2	no	—
217	#547	—	73	2	no	—
218	#500	—	73	2	no	—
219	#500	—	73	1	no	—
220	#500	—	146	1	no	—
221	#500	C₂×D₇₃	1	1	yes	characteristic
222	#494	—	73	2	no	—
223	#480	—	146	2	no	—
224	#474	—	73	2	no	—
225	#474	—	73	2	no	—
226	#474	—	146	2	no	—
227	#470	—	146	2	no	—
228	#466	—	146	1	no	—
229	#462	C₂×D₇₃	1	2	yes	—
230	#461	—	73	2	no	—
231	#452	—	146	2	no	—
232	#451	—	146	2	no	—
233	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
234	C₄×SD₁₆	—	73	1	no	—
235	C₄×D₈	—	73	1	no	—
236	C₄×D₈	—	146	1	no	—
237	#436	—	73	2	no	—
238	C₈×D₄	—	73	1	no	—
239	C₂×#107	—	146	1	no	—
240	#335	C₂×D₇₃	1	1	yes	characteristic
241	#334	—	73	1	no	—
242	#332	—	73	2	no	—
243	#332	C₂×D₇₃	1	2	yes	—
244	Q₈⋊C₈	—	73	1	no	—
245	D₄⋊C₈	—	73	1	no	—
246	D₄⋊C₈	—	73	2	no	—
247	C₈²	—	73	1	no	—
248	C₂×#41	—	146	2	no	—
249	C₂²×Q₈	—	146	2	no	—
250	#137	—	292	8	no	—
251	C₂×Q₁₆	—	2	2	no	—
252	C₂×Q₁₆	—	146	2	no	—
253	C₂×Q₁₆	—	146	2	no	—
254	C₂×SD₁₆	—	73	1	no	—
255	C₂×SD₁₆	—	146	2	no	—
256	C₂×D₈	—	73	1	no	—
257	C₂×M₁₆	—	146	2	no	—
258	C₂²×C₈	—	146	1	no	—
259	C₄⋊Q₈	—	73	2	no	—
260	C₄⋊Q₈	—	146	2	no	—
261	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
262	#123	—	292	2	no	—
263	C₂²⋊Q₈	—	292	2	no	—
264	C₂²⋊Q₈	—	146	2	no	—
265	C₂²⋊Q₈	—	146	2	no	—
266	#121	—	146	2	no	—
267	C₄×Q₈	C₂²×D₇₃	1	1	yes	characteristic
268	C₄×D₄	—	73	1	no	—
269	C₄×D₄	—	73	2	no	—
270	C₄×D₄	—	146	2	no	—
271	C₂×C₄⋊C₄	—	146	2	no	—
272	#107	—	2	1	no	—
273	#107	—	2	1	no	—
274	#107	—	292	2	no	—
275	#107	—	73	1	no	—
276	#107	—	146	2	no	—
277	#107	—	146	2	no	—
278	#107	—	146	1	no	—
279	#107	—	146	1	no	—
280	#106	—	73	1	no	—
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	73	1	no	—
283	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
284	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
285	#103	—	2	2	no	—
286	#103	—	73	1	no	—
287	#103	—	146	2	no	—
288	#102	—	73	2	no	—
289	#102	—	73	1	no	—
290	#102	—	146	2	no	—
291	#102	—	146	2	no	—
292	C₂²⋊C₈	—	146	2	no	—
293	C₂²⋊C₈	—	146	1	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	—	73	1	no	—
296	C₄×C₈	—	73	1	no	—
297	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
298	#41	—	292	4	no	—
299	#41	—	584	2	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	292	2	no	—
302	C₂×Q₈	—	292	4	no	—
303	C₂×Q₈	—	584	2	no	—
304	C₂×Q₈	—	146	2	no	—
305	C₂×Q₈	D₄×D₇₃	1	1	yes	characteristic
306	C₂×D₄	—	292	2	no	—
307	C₂×D₄	—	73	1	no	—
308	C₂×D₄	—	73	2	no	—
309	C₂²×C₄	—	292	2	no	—
310	C₂²×C₄	—	292	2	no	—
311	C₂²×C₄	—	146	1	no	—
312	C₂²×C₄	—	146	2	no	—
313	Q₁₆	—	4	4	no	—
314	Q₁₆	—	292	4	no	—
315	Q₁₆	—	584	2	no	—
316	SD₁₆	—	292	4	no	—
317	SD₁₆	—	73	4	no	—
318	D₈	—	146	2	no	—
319	M₁₆	—	292	4	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	292	2	no	—
323	C₂×C₈	—	73	1	no	—
324	C₂×C₈	—	73	1	no	—
325	C₂×C₈	—	146	1	no	—
326	C₂×C₈	—	146	1	no	—
327	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
328	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	292	2	no	—
333	C₄⋊C₄	—	292	2	no	—
334	C₄⋊C₄	—	73	2	no	—
335	C₄⋊C₄	—	73	1	no	—
336	C₄⋊C₄	—	146	2	no	—
337	C₄⋊C₄	—	146	2	no	—
338	C₄⋊C₄	D₄×D₇₃	1	1	yes	characteristic
339	C₂²⋊C₄	—	292	2	no	—
340	C₂²⋊C₄	—	292	2	no	—
341	C₂²⋊C₄	—	146	2	no	—
342	C₂²⋊C₄	—	146	1	no	—
343	C₄²	—	2	1	no	—
344	C₄²	C₂³×D₇₃	1	1	yes	characteristic
345	C₂³	—	146	2	no	—
346	C₂³	—	146	1	no	—
347	Q₈	—	2	1	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	292	2	no	—
351	Q₈	—	292	2	no	—
352	Q₈	—	584	4	no	—
353	Q₈	D₈×D₇₃	1	2	yes	—
354	D₄	—	584	4	no	—
355	D₄	—	73	2	no	—
356	D₄	—	146	2	no	—
357	D₄	—	146	1	no	—
358	D₄	—	146	2	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	292	1	no	—
363	C₂×C₄	—	292	2	no	—
364	C₂×C₄	—	292	1	no	—
365	C₂×C₄	—	292	2	no	—
366	C₂×C₄	—	292	2	no	—
367	C₂×C₄	—	584	4	no	—
368	C₂×C₄	—	584	4	no	—
369	C₂×C₄	—	146	1	no	—
370	C₂×C₄	—	146	2	no	—
371	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
372	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
373	C₂×C₄	#14556637	1	1	yes	characteristic
374	C₈	—	2	1	no	—
375	C₈	—	4	1	no	—
376	C₈	—	4	2	no	—
377	C₈	—	292	1	no	—
378	C₈	—	73	2	no	—
379	C₈	—	146	2	no	—
380	C₈	—	146	1	no	—
381	C₈	#14556623	1	2	yes	complemented
382	C₂²	—	292	1	no	—
383	C₂²	—	292	1	no	—
384	C₂²	—	292	2	no	—
385	C₂²	—	292	2	no	—
386	C₂²	—	292	2	no	—
387	C₂²	—	146	2	no	—
388	C₂²	#17989221	1	1	yes	characteristic
389	C₄	—	2	1	no	—
390	C₄	—	2	2	no	—
391	C₄	—	4	1	no	—
392	C₄	—	8	2	no	—
393	C₄	—	292	1	no	—
394	C₄	—	292	2	no	—
395	C₄	—	584	2	no	—
396	C₄	C₂×D₈×D₇₃	1	1	yes	characteristic
397	C₄	C₂×#14556623	1	1	yes	characteristic
398	C₄	#17989389	1	1	yes	characteristic
399	C₄	#17989403	1	1	yes	characteristic
400	C₂	—	292	2	no	—
401	C₂	—	584	2	no	—
402	C₂	#22621572	1	1	yes	characteristic
403	C₂	#22621605	1	1	yes	characteristic
404	C₂	#22621734	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17989221
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1674	C₂
socle	C₂²×C₇₃	D₄²
prime radical	#22622278	C₂
hypercenter	#1674	D₇₃
hypocenter	C₇₃	#19887
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1674

Complements

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

N	H ≅ G/N	type
1. Q₈×D₅₈₄	C₂	semidirect
3. #22621572	C₂	semidirect
4. #22609548	C₂	semidirect
5. #22609419	C₂	semidirect
6. #22609321	C₂	semidirect
7. #22609178	C₂	semidirect
8. #22609167	C₂	semidirect
9. #22608815	C₂	semidirect
10. C₈×#11924015	C₂	semidirect
11. C₇₃×#1674	C₂	semidirect
111. #11924015	D₈	semidirect
174. C₈×C₇₃	#137	semidirect
208. #1674	D₇₃	semidirect
214. C₇₃	#19887	semidirect
381. C₈	#14556623	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₅₈₄	2
#22622278	2
#22621572	2
#22609548	2
#22609419	2
#22609321	2
#22609178	2
#22609167	2
#22608815	2
C₈×#11924015	2
C₇₃×#1674	2
#19887	73