Subgroups — #33077909

Order 18688.

624 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	#22622649	C₂	1	2	yes	maximal
2	#22622363	C₂	1	1	yes	maximal, characteristic, complemented
3	#22609525	C₂	1	1	yes	maximal, characteristic, complemented
4	#22609487	C₂	1	2	yes	maximal, complemented
5	#22609320	C₂	1	1	yes	maximal, characteristic, complemented
6	#22609293	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609199	C₂	1	2	yes	maximal, complemented
8	#22609194	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	D₄×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989310	C₂²	1	2	yes	—
13	#17988992	C₂²	1	1	yes	characteristic
14	#17988983	C₂²	1	2	yes	—
15	#17988949	—	2	2	no	—
16	#17988939	C₂²	1	2	yes	—
17	#17988927	—	2	2	no	—
18	#17988879	—	2	2	no	—
19	#17988872	C₂²	1	2	yes	—
20	#17988864	C₂²	1	2	yes	—
21	#17988825	—	2	1	no	—
22	#17988802	—	2	2	no	—
23	#17988792	C₂²	1	2	yes	—
24	#17988718	C₂²	1	2	yes	—
25	#17988687	—	2	2	no	—
26	C₄×#11924013	C₂²	1	1	yes	characteristic
27	C₈×D₂₉₂	C₂²	1	1	yes	characteristic
28	#17988624	—	2	1	no	—
29	#17988340	C₂²	1	1	yes	characteristic
30	#17988339	C₂²	1	1	yes	characteristic
31	#17988333	C₂²	1	2	yes	—
32	#17988332	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	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	—	2	2	no	—
46	C₂×#11924013	D₄	1	1	yes	characteristic
47	C₂×C₈×D₇₃	—	2	1	no	—
48	#14556591	C₂³	1	2	yes	—
49	#14556586	—	4	2	no	—
50	#14556579	—	2	2	no	—
51	#14556567	—	2	2	no	—
52	C₄×Dic₁₄₆	—	2	2	no	—
53	C₄×D₂₉₂	C₂³	1	1	yes	characteristic
54	#14556549	—	2	2	no	—
55	#14556525	—	2	1	no	—
56	#14556525	—	2	1	no	—
57	#14556524	—	2	2	no	—
58	#14556524	D₄	1	1	yes	characteristic
59	#14556522	—	2	2	no	—
60	#14556522	—	4	2	no	—
61	C₄⋊#10596831	—	2	2	no	—
62	C₄⋊#10596831	C₂³	1	2	yes	—
63	C₄⋊#10596831	C₂³	1	1	yes	characteristic
64	#14556515	—	2	2	no	—
65	#14556514	—	2	2	no	—
66	#14556514	—	2	2	no	—
67	#14556514	—	2	2	no	—
68	#14556512	D₄	1	1	yes	characteristic
69	#14556511	—	2	2	no	—
70	#14556511	D₄	1	1	yes	characteristic
71	#14556503	—	2	1	no	—
72	#14556502	—	2	2	no	—
73	C₄×#10596831	C₂³	1	1	yes	characteristic
74	C₄×#10596831	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
96	C₂×D₂₉₂	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₇₃	—	2	1	no	—
98	#11924016	—	4	4	no	—
99	#11924013	—	2	2	no	—
100	C₂×#10596831	—	2	1	no	—
101	C₂×#10596831	—	2	2	no	—
102	C₂×#10596831	—	2	1	no	—
103	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
104	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
105	C₈×D₇₃	—	4	2	no	—
106	C₄⋊Dic₇₃	—	2	2	no	—
107	C₄⋊Dic₇₃	—	2	2	no	—
108	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
109	#11924007	—	4	2	no	—
110	#11924005	—	2	1	no	—
111	#11924005	—	4	2	no	—
112	#11924005	—	4	2	no	—
113	C₄×Dic₇₃	—	4	2	no	—
114	C₂×Q₈×C₇₃	—	2	2	no	—
115	C₂×Q₈×C₇₃	—	4	2	no	—
116	C₂×Q₈×C₇₃	—	4	4	no	—
117	C₂×Q₈×C₇₃	—	8	2	no	—
118	C₂×Q₈×C₇₃	C₂×D₄	1	2	yes	—
119	C₂×D₄×C₇₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₇₃	—	2	1	no	—
121	C₂²×C₄×C₇₃	—	4	2	no	—
122	Q₁₆×C₇₃	—	2	2	no	—
123	Q₁₆×C₇₃	—	4	4	no	—
124	Q₁₆×C₇₃	—	4	2	no	—
125	Q₁₆×C₇₃	—	8	1	no	—
126	C₂×C₈×C₇₃	—	2	1	no	—
127	C₂×C₈×C₇₃	—	4	2	no	—
128	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄×C₇₃	—	2	2	no	—
131	C₄⋊C₄×C₇₃	—	2	2	no	—
132	C₄⋊C₄×C₇₃	—	2	2	no	—
133	C₄⋊C₄×C₇₃	—	4	2	no	—
134	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
136	C₂²⋊C₄×C₇₃	—	2	1	no	—
137	C₂²⋊C₄×C₇₃	—	4	2	no	—
138	C₄²×C₇₃	—	2	2	no	—
139	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
140	C₂³×C₇₃	—	2	1	no	—
141	C₂²×D₇₃	—	2	1	no	—
142	Q₈×C₇₃	—	2	2	no	—
143	Q₈×C₇₃	—	2	2	no	—
144	Q₈×C₇₃	—	4	2	no	—
145	Q₈×C₇₃	—	4	2	no	—
146	Q₈×C₇₃	—	8	4	no	—
147	D₄×C₇₃	—	2	1	no	—
148	D₄×C₇₃	#135	1	2	yes	—
149	C₂×C₄×C₇₃	—	2	2	no	—
150	C₂×C₄×C₇₃	—	2	2	no	—
151	C₂×C₄×C₇₃	—	2	1	no	—
152	C₂×C₄×C₇₃	—	4	1	no	—
153	C₂×C₄×C₇₃	—	4	1	no	—
154	C₂×C₄×C₇₃	—	4	2	no	—
155	C₂×C₄×C₇₃	—	8	4	no	—
156	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×Dic₇₃	—	2	1	no	—
160	C₂×Dic₇₃	—	4	2	no	—
161	D₂₉₂	—	2	1	no	—
162	D₂₉₂	—	2	1	no	—
163	C₄×D₇₃	—	4	1	no	—
164	C₄×D₇₃	—	4	1	no	—
165	Dic₁₄₆	—	4	2	no	—
166	Dic₁₄₆	—	4	2	no	—
167	C₈×C₇₃	—	2	1	no	—
168	C₈×C₇₃	—	4	1	no	—
169	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
170	#10596831	—	2	2	no	—
171	#10596831	—	2	1	no	—
172	#10596831	—	2	1	no	—
173	#10596831	—	4	2	no	—
174	#10596831	—	4	1	no	—
175	C₂²×C₇₃	—	2	2	no	—
176	C₂²×C₇₃	—	4	1	no	—
177	C₂²×C₇₃	—	4	1	no	—
178	C₂²×C₇₃	D₄²	1	1	yes	characteristic
179	C₂×D₇₃	—	4	1	no	—
180	C₂×D₇₃	—	4	1	no	—
181	C₂×D₇₃	—	4	1	no	—
182	C₄×C₇₃	—	2	1	no	—
183	C₄×C₇₃	—	4	1	no	—
184	C₄×C₇₃	—	4	2	no	—
185	C₄×C₇₃	—	4	2	no	—
186	C₄×C₇₃	—	8	2	no	—
187	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
188	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
189	C₄×C₇₃	#577	1	1	yes	characteristic
190	C₄×C₇₃	#578	1	1	yes	characteristic
191	Dic₇₃	—	4	1	no	—
192	Dic₇₃	—	8	2	no	—
193	#19933	—	73	1	no	maximal, Sylow
194	C₂×C₇₃	—	4	2	no	—
195	C₂×C₇₃	#3272	1	1	yes	characteristic
196	C₂×C₇₃	#3285	1	1	yes	characteristic
197	C₂×C₇₃	#3293	1	1	yes	characteristic
198	D₇₃	—	8	1	no	—
199	#3367	—	73	2	no	—
200	#3342	—	73	1	no	—
201	D₄×Q₁₆	D₇₃	1	1	yes	characteristic, complemented
202	#1707	—	73	1	no	—
203	#1700	—	73	2	no	—
204	#1673	—	73	1	no	—
205	#1669	—	73	1	no	—
206	#1652	—	73	2	no	—
207	#1651	—	73	2	no	—
208	#1591	—	73	1	no	—
209	C₈×D₈	—	73	1	no	—
210	C₇₃	#19933	1	1	yes	characteristic, complemented, Sylow
211	C₂²×Q₁₆	—	2	1	no	—
212	#556	—	73	2	no	—
213	D₄×Q₈	C₂×D₇₃	1	2	yes	—
214	#500	—	73	1	no	—
215	#498	—	73	2	no	—
216	#494	C₂×D₇₃	1	1	yes	characteristic
217	#489	—	146	2	no	—
218	#487	—	73	2	no	—
219	#485	—	146	2	no	—
220	#477	—	146	2	no	—
221	#475	—	73	2	no	—
222	#474	—	73	2	no	—
223	#467	—	2	1	no	—
224	#467	—	146	1	no	—
225	#463	—	146	2	no	—
226	#462	C₂×D₇₃	1	2	yes	—
227	#461	—	73	2	no	—
228	#451	—	2	2	no	—
229	#446	—	73	2	no	—
230	#440	—	146	2	no	—
231	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
232	C₄×D₈	—	73	1	no	—
233	C₈×D₄	—	73	1	no	—
234	C₈×D₄	C₂×D₇₃	1	1	yes	characteristic
235	#427	—	146	1	no	—
236	#335	—	73	1	no	—
237	#335	—	73	1	no	—
238	#332	—	73	2	no	—
239	#332	—	73	2	no	—
240	Q₈⋊C₈	—	73	2	no	—
241	D₄⋊C₈	—	73	1	no	—
242	D₄⋊C₈	—	73	1	no	—
243	C₈²	—	73	1	no	—
244	C₂²×Q₈	—	2	2	no	—
245	C₂×Q₁₆	—	2	2	no	—
246	C₂×Q₁₆	—	2	1	no	—
247	C₂×Q₁₆	—	2	1	no	—
248	C₂×Q₁₆	—	4	4	no	—
249	C₂×Q₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
250	C₂×SD₁₆	—	146	2	no	—
251	C₂×D₈	—	73	1	no	—
252	C₂²×C₈	—	2	1	no	—
253	C₂²×C₈	—	146	1	no	—
254	C₄⋊Q₈	—	2	2	no	—
255	C₄⋊Q₈	—	73	2	no	—
256	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
257	#126	—	292	2	no	—
258	#124	—	146	2	no	—
259	C₂²⋊Q₈	—	2	2	no	—
260	C₂²⋊Q₈	—	4	2	no	—
261	C₂²⋊Q₈	—	146	2	no	—
262	C₄×Q₈	—	146	2	no	—
263	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
264	C₄×D₄	—	2	2	no	—
265	C₄×D₄	—	73	1	no	—
266	C₄×D₄	C₂²×D₇₃	1	1	yes	characteristic
267	#117	—	146	2	no	—
268	#107	—	73	1	no	—
269	#107	—	146	2	no	—
270	#107	—	146	1	no	—
271	#107	—	146	1	no	—
272	#107	C₂²⋊D₇₃	1	1	yes	characteristic
273	#106	—	292	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	—	146	2	no	—
281	#103	—	146	2	no	—
282	#103	—	146	2	no	—
283	#103	—	146	2	no	—
284	#103	C₂²⋊D₇₃	1	2	yes	—
285	#102	—	73	1	no	—
286	#102	—	73	1	no	—
287	#102	—	146	2	no	—
288	C₂²⋊C₈	—	2	1	no	—
289	C₂²⋊C₈	—	146	1	no	—
290	#97	—	146	2	no	—
291	C₄×C₈	—	73	1	no	—
292	C₄×C₈	—	73	1	no	—
293	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	4	2	no	—
296	C₂×Q₈	—	4	4	no	—
297	C₂×Q₈	—	8	2	no	—
298	C₂×Q₈	—	146	2	no	—
299	C₂×Q₈	C₂×C₂²⋊D₇₃	1	2	yes	—
300	C₂×D₄	—	73	1	no	—
301	C₂×D₄	D₄×D₇₃	1	1	yes	characteristic
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	4	2	no	—
304	C₂²×C₄	—	146	1	no	—
305	Q₁₆	—	2	2	no	—
306	Q₁₆	—	4	4	no	—
307	Q₁₆	—	4	2	no	—
308	Q₁₆	—	8	1	no	—
309	SD₁₆	—	292	4	no	—
310	D₈	—	146	2	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	4	2	no	—
313	C₂×C₈	—	292	2	no	—
314	C₂×C₈	—	73	1	no	—
315	C₂×C₈	—	73	1	no	—
316	C₂×C₈	—	146	2	no	—
317	C₂×C₈	—	146	1	no	—
318	C₂×C₈	—	146	1	no	—
319	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
320	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	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	—	292	2	no	—
335	C₂²⋊C₄	—	146	1	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	292	2	no	—
338	C₄²	C₂³×D₇₃	1	1	yes	characteristic
339	C₂³	—	2	1	no	—
340	C₂³	—	146	1	no	—
341	Q₈	—	2	2	no	—
342	Q₈	—	2	2	no	—
343	Q₈	—	4	2	no	—
344	Q₈	—	4	2	no	—
345	Q₈	—	8	4	no	—
346	Q₈	—	292	2	no	—
347	Q₈	—	292	2	no	—
348	D₄	—	2	1	no	—
349	D₄	—	146	1	no	—
350	D₄	—	146	1	no	—
351	D₄	#14556614	1	2	yes	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	8	4	no	—
359	C₂×C₄	—	292	1	no	—
360	C₂×C₄	—	292	1	no	—
361	C₂×C₄	—	292	2	no	—
362	C₂×C₄	—	146	1	no	—
363	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
364	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
365	C₂×C₄	#14556636	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	4	1	no	—
368	C₈	—	292	2	no	—
369	C₈	—	292	1	no	—
370	C₈	—	146	2	no	—
371	C₈	—	146	1	no	—
372	C₈	—	146	1	no	—
373	C₈	#14556617	1	2	yes	—
374	C₂²	—	2	2	no	—
375	C₂²	—	4	1	no	—
376	C₂²	—	4	1	no	—
377	C₂²	—	292	1	no	—
378	C₂²	—	292	1	no	—
379	C₂²	—	292	1	no	—
380	C₂²	D₄×C₂²⋊D₇₃	1	1	yes	characteristic
381	C₄	—	2	1	no	—
382	C₄	—	4	2	no	—
383	C₄	—	4	1	no	—
384	C₄	—	4	2	no	—
385	C₄	—	8	2	no	—
386	C₄	—	292	1	no	—
387	C₄	—	584	2	no	—
388	C₄	C₂×#14556614	1	1	yes	characteristic
389	C₄	C₂×#14556617	1	1	yes	characteristic
390	C₄	#17989398	1	1	yes	characteristic
391	C₄	#17989400	1	1	yes	characteristic
392	C₂	—	4	2	no	—
393	C₂	—	584	1	no	—
394	C₂	#22621527	1	1	yes	characteristic
395	C₂	#22621672	1	1	yes	characteristic
396	C₂	#22621733	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	C₂×#11924013	D₄
hypercenter	D₄×Q₁₆	D₇₃
hypocenter	C₇₃	#19933
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
2. #22622363	C₂	semidirect
3. #22609525	C₂	semidirect
4. #22609487	C₂	semidirect
5. #22609320	C₂	semidirect
6. #22609293	C₂	semidirect
7. #22609199	C₂	semidirect
8. #22609194	C₂	semidirect
9. #22608828	C₂	semidirect
11. D₄×Q₁₆×C₇₃	C₂	semidirect
201. D₄×Q₁₆	D₇₃	semidirect
210. C₇₃	#19933	semidirect

Maximal subgroups

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

subgroup	index
#22622649	2
#22622363	2
#22609525	2
#22609487	2
#22609320	2
#22609293	2
#22609199	2
#22609194	2
#22608828	2
C₈×#11924013	2
D₄×Q₁₆×C₇₃	2
#19933	73