Subgroups — #33077858

Order 18688.

622 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622936	C₂	1	1	yes	maximal, characteristic, complemented
2	#22622854	C₂	1	2	yes	maximal
3	Q₁₆×D₂₉₂	C₂	1	1	yes	maximal, characteristic
4	#22609549	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609449	C₂	1	2	yes	maximal, complemented
6	#22609302	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609297	C₂	1	1	yes	maximal, characteristic
8	#22609182	C₂	1	2	yes	maximal
9	#22609175	C₂	1	2	yes	maximal, complemented
10	C₈×#11924017	C₂	1	1	yes	maximal, characteristic
11	C₇₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₃	—	2	1	no	—
13	#17989315	C₂²	1	2	yes	—
14	Q₈×D₂₉₂	C₂²	1	2	yes	—
15	#17988996	—	2	2	no	—
16	#17988992	C₂²	1	2	yes	—
17	#17988992	C₂²	1	2	yes	—
18	#17988967	C₂²	1	1	yes	characteristic
19	#17988967	C₂²	1	1	yes	characteristic
20	#17988965	—	2	1	no	—
21	#17988888	—	2	2	no	—
22	#17988883	—	2	2	no	—
23	#17988873	C₂²	1	2	yes	—
24	#17988868	C₂²	1	2	yes	—
25	#17988829	—	2	1	no	—
26	#17988797	C₂²	1	2	yes	—
27	#17988797	C₂²	1	2	yes	—
28	#17988796	—	2	2	no	—
29	#17988744	—	2	2	no	—
30	C₈×Dic₁₄₆	C₂²	1	1	yes	characteristic
31	#17988689	—	2	2	no	—
32	#17988681	—	2	1	no	—
33	C₄×#11924017	C₂²	1	1	yes	characteristic
34	C₈×D₂₉₂	C₂²	1	1	yes	characteristic
35	#17988338	C₂²	1	1	yes	characteristic
36	#17988332	C₂²	1	2	yes	—
37	#17988332	C₂²	1	2	yes	—
38	#17988319	C₂²	1	1	yes	characteristic
39	#17988316	C₂²	1	1	yes	characteristic
40	#446×C₇₃	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
42	#335×C₇₃	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₇₃	C₂²	1	2	yes	—
44	C₈²×C₇₃	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₇₃	—	2	2	no	—
46	C₂×Dic₂₉₂	—	2	1	no	—
47	C₂×Dic₂₉₂	—	2	1	no	—
48	C₂×Dic₂₉₂	—	2	1	no	—
49	C₂×Dic₂₉₂	—	2	1	no	—
50	C₂×#11924018	—	2	2	no	—
51	C₂×#11924018	—	2	2	no	—
52	Q₁₆×D₇₃	—	4	4	no	—
53	C₂×#11924017	D₄	1	1	yes	characteristic
54	C₂×C₈×D₇₃	—	2	1	no	—
55	#14556591	—	2	2	no	—
56	#14556591	C₂³	1	2	yes	—
57	#14556591	C₂³	1	2	yes	—
58	#14556589	—	2	2	no	—
59	#14556583	—	2	2	no	—
60	#14556581	—	4	2	no	—
61	#14556571	—	2	2	no	—
62	#14556571	—	4	2	no	—
63	C₄×Dic₁₄₆	—	2	2	no	—
64	C₄×Dic₁₄₆	C₂³	1	1	yes	characteristic
65	#14556556	—	2	2	no	—
66	C₄×D₂₉₂	—	2	2	no	—
67	C₄×D₂₉₂	C₂³	1	1	yes	characteristic
68	#14556525	—	2	2	no	—
69	#14556525	—	2	2	no	—
70	#14556525	—	2	2	no	—
71	#14556525	—	4	2	no	—
72	#14556523	—	2	2	no	—
73	#14556523	D₄	1	1	yes	characteristic
74	C₄⋊#10596831	C₂³	1	1	yes	characteristic
75	C₄⋊#10596831	C₂³	1	1	yes	characteristic
76	#14556520	—	2	1	no	—
77	#14556516	—	2	2	no	—
78	#14556516	D₄	1	1	yes	characteristic
79	#14556514	—	2	2	no	—
80	#14556514	—	2	2	no	—
81	#14556513	—	2	2	no	—
82	#14556513	D₄	1	1	yes	characteristic
83	#14556503	—	2	1	no	—
84	C₈×Dic₇₃	—	2	1	no	—
85	C₂×Q₁₆×C₇₃	D₄	1	1	yes	characteristic
86	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
87	#107×C₇₃	D₄	1	1	yes	characteristic
88	C₄⋊C₈×C₇₃	—	2	2	no	—
89	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
90	#103×C₇₃	D₄	1	2	yes	—
91	#97×C₇₃	—	2	2	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	C₂×Dic₁₄₆	—	2	2	no	—
96	C₂×Dic₁₄₆	—	2	2	no	—
97	C₂×Dic₁₄₆	—	4	2	no	—
98	C₂×Dic₁₄₆	C₂×D₄	1	1	yes	characteristic
99	Q₈×D₇₃	—	4	4	no	—
100	Q₈×D₇₃	—	8	2	no	—
101	C₂×D₂₉₂	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×D₇₃	—	2	1	no	—
103	C₂×C₄×D₇₃	—	4	2	no	—
104	Dic₂₉₂	—	4	2	no	—
105	Dic₂₉₂	—	4	2	no	—
106	Dic₂₉₂	—	8	1	no	—
107	Dic₂₉₂	—	8	1	no	—
108	#11924018	—	4	4	no	—
109	#11924018	—	8	2	no	—
110	#11924017	Q₁₆	1	4	yes	complemented
111	C₂×#10596831	—	2	1	no	—
112	C₂×#10596831	—	2	1	no	—
113	C₈×D₇₃	—	4	2	no	—
114	C₄⋊Dic₇₃	—	2	2	no	—
115	C₄⋊Dic₇₃	—	2	2	no	—
116	C₄⋊Dic₇₃	—	2	2	no	—
117	C₄⋊Dic₇₃	—	2	2	no	—
118	C₄⋊Dic₇₃	—	4	2	no	—
119	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
120	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
121	#11924007	—	2	1	no	—
122	#11924007	—	4	2	no	—
123	#11924007	—	4	2	no	—
124	#11924007	—	4	2	no	—
125	#11924005	—	2	1	no	—
126	#11924005	—	4	2	no	—
127	C₄×Dic₇₃	—	2	1	no	—
128	C₄×Dic₇₃	—	4	2	no	—
129	C₂×Q₈×C₇₃	C₂×D₄	1	2	yes	—
130	Q₁₆×C₇₃	—	2	2	no	—
131	C₂×C₈×C₇₃	—	2	1	no	—
132	C₂×C₈×C₇₃	—	2	2	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	2	no	—
138	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
139	C₄²×C₇₃	—	2	2	no	—
140	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
141	C₂²×D₇₃	—	2	1	no	—
142	Q₈×C₇₃	—	2	2	no	—
143	Q₈×C₇₃	—	2	2	no	—
144	C₂×C₄×C₇₃	—	2	2	no	—
145	C₂×C₄×C₇₃	—	2	2	no	—
146	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂×Dic₇₃	—	2	1	no	—
150	C₂×Dic₇₃	—	2	1	no	—
151	C₂×Dic₇₃	—	2	1	no	—
152	C₂×Dic₇₃	—	4	2	no	—
153	C₂×Dic₇₃	—	4	2	no	—
154	D₂₉₂	—	2	1	no	—
155	D₂₉₂	C₂×Q₁₆	1	2	yes	—
156	C₄×D₇₃	—	4	1	no	—
157	C₄×D₇₃	—	4	1	no	—
158	C₄×D₇₃	—	8	4	no	—
159	Dic₁₄₆	—	2	1	no	—
160	Dic₁₄₆	—	4	2	no	—
161	Dic₁₄₆	—	4	2	no	—
162	Dic₁₄₆	—	4	2	no	—
163	Dic₁₄₆	—	4	2	no	—
164	Dic₁₄₆	—	8	4	no	—
165	Dic₁₄₆	C₂×Q₁₆	1	2	yes	complemented
166	C₈×C₇₃	—	2	2	no	—
167	C₈×C₇₃	—	2	1	no	—
168	C₈×C₇₃	—	2	1	no	—
169	C₈×C₇₃	—	4	2	no	—
170	C₈×C₇₃	C₂×Q₁₆	1	2	yes	complemented
171	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
172	#10596831	—	4	1	no	—
173	#10596831	—	4	1	no	—
174	C₂²×C₇₃	D₄²	1	1	yes	characteristic
175	C₂×D₇₃	—	2	2	no	—
176	C₂×D₇₃	—	4	1	no	—
177	C₂×D₇₃	—	4	1	no	—
178	C₄×C₇₃	—	2	1	no	—
179	C₄×C₇₃	—	4	2	no	—
180	C₄×C₇₃	—	4	2	no	—
181	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
182	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₄×C₇₃	#576	1	1	yes	characteristic
184	C₄×C₇₃	#578	1	1	yes	characteristic
185	Dic₇₃	—	2	2	no	—
186	Dic₇₃	—	4	1	no	—
187	Dic₇₃	—	4	1	no	—
188	Dic₇₃	—	8	2	no	—
189	Dic₇₃	—	8	2	no	—
190	#19928	—	73	1	no	maximal, Sylow
191	C₂×C₇₃	#3274	1	1	yes	characteristic
192	C₂×C₇₃	D₄×Q₁₆	1	1	yes	characteristic
193	C₂×C₇₃	#3293	1	1	yes	characteristic
194	D₇₃	—	4	2	no	—
195	#3395	—	73	1	no	—
196	#3386	—	73	2	no	—
197	D₄×Q₁₆	—	73	1	no	—
198	#1713	—	73	1	no	—
199	#1694	—	73	2	no	—
200	#1670	—	73	1	no	—
201	#1669	—	73	1	no	—
202	#1649	—	73	2	no	—
203	#1648	—	73	2	no	—
204	#1591	D₇₃	1	1	yes	characteristic, complemented
205	C₈×SD₁₆	—	73	1	no	—
206	C₇₃	#19928	1	1	yes	characteristic, complemented, Sylow
207	C₂²×Q₁₆	—	146	1	no	—
208	#557	—	73	2	no	—
209	D₄×Q₈	—	73	2	no	—
210	#501	—	146	2	no	—
211	#500	—	73	2	no	—
212	#500	—	73	2	no	—
213	#494	—	73	1	no	—
214	#494	—	73	1	no	—
215	#494	—	146	1	no	—
216	#478	—	146	2	no	—
217	#477	—	146	2	no	—
218	#475	—	73	2	no	—
219	#474	—	73	2	no	—
220	#467	—	146	1	no	—
221	#462	—	73	2	no	—
222	#462	—	73	2	no	—
223	#462	—	146	2	no	—
224	#451	—	146	2	no	—
225	#446	C₂×D₇₃	1	2	yes	—
226	C₈×Q₈	—	73	1	no	—
227	#440	—	146	2	no	—
228	C₄×Q₁₆	—	146	1	no	—
229	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
230	C₄×SD₁₆	—	73	1	no	—
231	C₈×D₄	—	73	1	no	—
232	#335	C₂×D₇₃	1	1	yes	characteristic
233	#334	—	73	1	no	—
234	#332	—	73	2	no	—
235	#332	—	73	2	no	—
236	Q₈⋊C₈	—	73	1	no	—
237	Q₈⋊C₈	C₂×D₇₃	1	2	yes	—
238	D₄⋊C₈	—	73	1	no	—
239	C₈²	C₂×D₇₃	1	1	yes	characteristic
240	C₂²×Q₈	—	146	2	no	—
241	C₂×Q₁₆	—	292	4	no	—
242	C₂×Q₁₆	—	146	2	no	—
243	C₂×Q₁₆	—	146	2	no	—
244	C₂×Q₁₆	—	146	1	no	—
245	C₂×Q₁₆	—	146	1	no	—
246	C₂×Q₁₆	—	146	1	no	—
247	C₂×Q₁₆	—	146	1	no	—
248	C₂×Q₁₆	D₂₉₂	1	1	yes	characteristic
249	C₂×SD₁₆	—	73	1	no	—
250	C₂²×C₈	—	146	1	no	—
251	C₄⋊Q₈	—	73	2	no	—
252	C₄⋊Q₈	—	73	2	no	—
253	C₄⋊Q₈	—	146	2	no	—
254	C₄⋊Q₈	—	146	2	no	—
255	#125	—	292	2	no	—
256	#125	—	146	2	no	—
257	C₂²⋊Q₈	—	292	2	no	—
258	C₂²⋊Q₈	—	146	2	no	—
259	C₄×Q₈	—	73	1	no	—
260	C₄×Q₈	—	146	2	no	—
261	C₄×Q₈	—	146	2	no	—
262	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
263	C₄×D₄	—	73	1	no	—
264	C₄×D₄	—	146	2	no	—
265	#107	—	292	2	no	—
266	#107	—	146	2	no	—
267	#107	—	146	2	no	—
268	#107	—	146	2	no	—
269	#107	D₂₉₂	1	1	yes	characteristic
270	#106	—	73	1	no	—
271	#106	—	146	2	no	—
272	C₄⋊C₈	—	2	2	no	—
273	C₄⋊C₈	—	73	1	no	—
274	C₄⋊C₈	—	73	1	no	—
275	C₄⋊C₈	—	146	1	no	—
276	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
277	#103	—	73	1	no	—
278	#103	—	146	2	no	—
279	#103	—	146	2	no	—
280	#103	—	146	2	no	—
281	#103	D₂₉₂	1	2	yes	—
282	#102	—	73	1	no	—
283	#102	—	146	2	no	—
284	C₂²⋊C₈	—	146	1	no	—
285	#97	—	2	2	no	—
286	C₄×C₈	—	146	1	no	—
287	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
290	C₂×Q₈	—	292	2	no	—
291	C₂×Q₈	—	292	4	no	—
292	C₂×Q₈	—	584	2	no	—
293	C₂×Q₈	—	73	1	no	—
294	C₂×Q₈	—	146	2	no	—
295	C₂×Q₈	—	146	2	no	—
296	C₂×Q₈	C₂×D₂₉₂	1	2	yes	—
297	C₂×D₄	—	73	1	no	—
298	C₂²×C₄	—	292	2	no	—
299	C₂²×C₄	—	146	1	no	—
300	Q₁₆	—	2	2	no	—
301	Q₁₆	—	292	4	no	—
302	Q₁₆	—	292	2	no	—
303	Q₁₆	—	292	2	no	—
304	Q₁₆	—	584	1	no	—
305	Q₁₆	—	584	1	no	—
306	Q₁₆	—	584	2	no	—
307	SD₁₆	—	73	4	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	292	2	no	—
311	C₂×C₈	—	146	1	no	—
312	C₂×C₈	—	146	1	no	—
313	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
314	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
315	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
316	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	292	2	no	—
319	C₄⋊C₄	—	292	2	no	—
320	C₄⋊C₄	—	292	2	no	—
321	C₄⋊C₄	—	292	2	no	—
322	C₄⋊C₄	—	73	1	no	—
323	C₄⋊C₄	—	73	1	no	—
324	C₄⋊C₄	—	146	2	no	—
325	C₄⋊C₄	—	146	2	no	—
326	C₄⋊C₄	—	146	2	no	—
327	C₄⋊C₄	—	146	2	no	—
328	C₄⋊C₄	—	146	1	no	—
329	C₄⋊C₄	C₂×D₂₉₂	1	2	yes	—
330	C₂²⋊C₄	—	292	2	no	—
331	C₂²⋊C₄	—	146	1	no	—
332	C₄²	—	2	2	no	—
333	C₄²	—	292	2	no	—
334	C₄²	—	146	1	no	—
335	C₄²	C₂³×D₇₃	1	1	yes	characteristic
336	C₂³	—	146	1	no	—
337	Q₈	—	2	2	no	—
338	Q₈	—	2	2	no	—
339	Q₈	—	292	2	no	—
340	Q₈	—	292	2	no	—
341	Q₈	—	292	2	no	—
342	Q₈	—	292	2	no	—
343	Q₈	—	584	4	no	—
344	Q₈	—	73	2	no	—
345	Q₈	—	146	1	no	—
346	D₄	—	73	2	no	—
347	D₄	—	146	1	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	292	1	no	—
351	C₂×C₄	—	292	1	no	—
352	C₂×C₄	—	292	2	no	—
353	C₂×C₄	—	292	2	no	—
354	C₂×C₄	—	584	4	no	—
355	C₂×C₄	—	146	1	no	—
356	C₂×C₄	—	146	1	no	—
357	C₂×C₄	—	146	1	no	—
358	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
359	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
360	C₂×C₄	#14556637	1	1	yes	characteristic
361	C₈	—	2	2	no	—
362	C₈	—	2	1	no	—
363	C₈	—	2	1	no	—
364	C₈	—	4	2	no	—
365	C₈	—	292	1	no	—
366	C₈	—	292	1	no	—
367	C₈	Q₁₆×D₇₃	1	2	yes	complemented
368	C₈	#14556619	1	2	yes	—
369	C₂²	—	292	1	no	—
370	C₂²	—	292	1	no	—
371	C₂²	—	146	2	no	—
372	C₂²	D₄×D₂₉₂	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	4	2	no	—
375	C₄	—	4	2	no	—
376	C₄	—	292	1	no	—
377	C₄	—	292	1	no	—
378	C₄	—	584	2	no	—
379	C₄	—	584	2	no	—
380	C₄	—	146	2	no	—
381	C₄	C₂×Q₁₆×D₇₃	1	1	yes	characteristic
382	C₄	C₂×#14556619	1	1	yes	characteristic
383	C₄	#17989392	1	1	yes	characteristic
384	C₄	#17989401	1	1	yes	characteristic
385	C₂	—	292	2	no	—
386	C₂	#22621553	1	1	yes	characteristic
387	C₂	Q₁₆×D₂₉₂	1	1	yes	characteristic
388	C₂	#22621736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₉₂
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1591	C₂
socle	C₂²×C₇₃	D₄²
prime radical	C₂×D₂₉₂	C₂×D₄
hypercenter	#1591	D₇₃
hypocenter	C₇₃	#19928
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1591

Complements

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

N	H ≅ G/N	type
1. #22622936	C₂	semidirect
4. #22609549	C₂	semidirect
5. #22609449	C₂	semidirect
6. #22609302	C₂	semidirect
9. #22609175	C₂	semidirect
11. C₇₃×#1591	C₂	semidirect
110. #11924017	Q₁₆	semidirect
165. Dic₁₄₆	C₂×Q₁₆	semidirect
170. C₈×C₇₃	C₂×Q₁₆	semidirect
204. #1591	D₇₃	semidirect
206. C₇₃	#19928	semidirect
367. C₈	Q₁₆×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22622936	2
#22622854	2
Q₁₆×D₂₉₂	2
#22609549	2
#22609449	2
#22609302	2
#22609297	2
#22609182	2
#22609175	2
C₈×#11924017	2
C₇₃×#1591	2
#19928	73