Subgroups — #33240424

Order 18688.

710 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989020	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#22619514	C₂	1	2	yes	maximal, complemented
4	#22612030	C₂	1	2	yes	maximal, complemented
5	#22612029	C₂	1	2	yes	maximal, complemented
6	#22612019	C₂	1	1	yes	maximal, characteristic, complemented
7	#22612018	C₂	1	1	yes	maximal, characteristic, complemented
8	#22612017	C₂	1	1	yes	maximal, characteristic, complemented
9	#22611712	C₂	1	1	yes	maximal, characteristic, complemented
10	#22611637	C₂	1	2	yes	maximal, complemented
11	C₇₃×#2216	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556614	C₂²	1	2	yes	—
13	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
14	#17989100	—	2	2	no	—
15	#17989020	—	2	4	no	—
16	C₂×#14556534	C₂²	1	2	yes	complemented
17	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic, complemented
18	C₂²×#11924003	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×D₇₃	—	2	4	no	—
20	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
22	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
23	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
24	#17988825	C₂²	1	2	yes	—
25	#17988824	C₂²	1	2	yes	—
26	#17988823	C₂²	1	2	yes	—
27	#17988819	—	2	2	no	—
28	#17988818	—	2	2	no	—
29	#17988624	C₂²	1	2	yes	—
30	#17988582	—	2	2	no	—
31	#17988414	C₂²	1	2	yes	complemented
32	#17988413	C₂²	1	1	yes	characteristic, complemented
33	#17988391	C₂²	1	2	yes	complemented
34	#17988389	C₂²	1	2	yes	complemented
35	#17988387	C₂²	1	2	yes	complemented
36	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic, complemented
37	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
38	#467×C₇₃	C₂²	1	2	yes	—
39	#366×C₇₃	C₂²	1	1	yes	characteristic, complemented
40	#358×C₇₃	C₂²	1	2	yes	complemented
41	C₂×#11924029	—	2	2	no	—
42	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
43	#14556614	—	2	4	no	—
44	#14556614	—	2	4	no	—
45	C₂×#11924015	—	2	2	no	—
46	C₂×D₅₈₄	C₂³	1	2	yes	—
47	C₂²×#10596831	D₄	1	1	yes	characteristic
48	C₂×C₈×D₇₃	—	2	2	no	—
49	C₂×C₈×D₇₃	—	2	2	no	—
50	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
51	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
52	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
54	#14556576	—	4	2	no	—
55	#14556567	—	2	2	no	—
56	#14556566	—	2	2	no	—
57	#14556565	—	2	2	no	—
58	#14556561	—	4	2	no	—
59	#14556553	—	4	2	no	—
60	#14556551	—	4	2	no	—
61	#14556549	—	2	2	no	—
62	#14556534	—	2	4	no	—
63	D₁₁₆₈	—	2	2	no	—
64	C₂×#11924003	—	2	2	no	—
65	C₂×#11924003	C₂³	1	1	yes	characteristic
66	C₂×#11924003	C₂³	1	1	yes	characteristic
67	C₁₆×D₇₃	—	2	2	no	—
68	C₁₆×D₇₃	—	2	2	no	—
69	C₁₆×D₇₃	—	2	2	no	—
70	C₁₆×D₇₃	D₄	1	4	yes	complemented
71	#14556525	C₂³	1	2	yes	—
72	#14556522	—	2	2	no	—
73	#14556515	—	2	2	no	—
74	#14556513	—	2	2	no	—
75	#14556511	—	2	2	no	—
76	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
77	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
78	C₂²⋊Q₈×C₇₃	—	2	2	no	—
79	Q₃₂×C₇₃	—	2	2	no	—
80	C₂×C₁₆×C₇₃	—	2	2	no	—
81	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
83	#107×C₇₃	C₂³	1	2	yes	—
84	#103×C₇₃	—	2	2	no	—
85	C₂³×D₇₃	D₈	1	1	yes	characteristic
86	#11924029	—	4	4	no	—
87	#11924029	—	4	4	no	—
88	C₂×D₂₉₂	—	2	2	no	—
89	C₂×D₂₉₂	—	4	2	no	—
90	C₂×D₂₉₂	—	4	2	no	—
91	C₂×C₂²⋊D₇₃	—	4	2	no	—
92	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
93	C₂×C₄×D₇₃	—	2	2	no	—
94	C₂×C₄×D₇₃	—	2	2	no	—
95	C₂×C₄×D₇₃	—	4	2	no	—
96	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
97	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
98	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
100	#11924015	—	4	4	no	—
101	D₅₈₄	—	2	2	no	—
102	D₅₈₄	—	2	2	no	—
103	C₂×#10596831	—	2	2	no	—
104	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
105	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
106	C₈×D₇₃	—	2	2	no	—
107	C₈×D₇₃	—	2	1	no	—
108	C₈×D₇₃	—	2	1	no	—
109	C₈×D₇₃	—	2	2	no	—
110	C₈×D₇₃	C₂×D₄	1	2	yes	—
111	C₈×D₇₃	#41	1	2	yes	—
112	C₄⋊Dic₇₃	—	2	2	no	—
113	#11924007	—	4	2	no	—
114	#11924005	—	4	2	no	—
115	#11924005	—	4	2	no	—
116	#11924005	—	4	2	no	—
117	#11924005	—	4	2	no	—
118	C₄×Dic₇₃	—	4	2	no	—
119	#11924003	—	2	1	no	—
120	#11924003	C₂×D₄	1	2	yes	—
121	C₂×Q₈×C₇₃	—	2	2	no	—
122	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
123	Q₁₆×C₇₃	—	2	2	no	—
124	Q₁₆×C₇₃	—	2	2	no	—
125	C₂×C₈×C₇₃	—	2	1	no	—
126	C₂×C₈×C₇₃	—	2	1	no	—
127	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
129	C₄⋊C₄×C₇₃	—	2	2	no	—
130	C₄⋊C₄×C₇₃	—	4	2	no	—
131	C₂²⋊C₄×C₇₃	—	4	2	no	—
132	C₁₆×C₇₃	—	2	1	no	—
133	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
134	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₇₃	—	2	2	no	—
136	C₂²×D₇₃	—	2	2	no	—
137	C₂²×D₇₃	—	2	2	no	—
138	C₂²×D₇₃	—	4	2	no	—
139	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
140	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
141	Q₈×C₇₃	—	4	2	no	—
142	Q₈×C₇₃	—	4	2	no	—
143	C₂×C₄×C₇₃	—	2	1	no	—
144	C₂×C₄×C₇₃	—	2	1	no	—
145	C₂×C₄×C₇₃	—	4	2	no	—
146	C₂×C₄×C₇₃	—	4	2	no	—
147	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
149	C₂²⋊D₇₃	—	8	4	no	—
150	C₂×Dic₇₃	—	2	2	no	—
151	C₂×Dic₇₃	—	4	2	no	—
152	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
153	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
154	D₂₉₂	—	4	2	no	—
155	D₂₉₂	—	4	2	no	—
156	D₂₉₂	—	8	4	no	—
157	D₂₉₂	—	8	4	no	—
158	C₄×D₇₃	—	2	2	no	—
159	C₄×D₇₃	—	2	1	no	—
160	C₄×D₇₃	—	2	1	no	—
161	C₄×D₇₃	—	2	2	no	—
162	C₄×D₇₃	—	8	4	no	—
163	C₄×D₇₃	#121	1	2	yes	—
164	C₄×D₇₃	#135	1	2	yes	—
165	C₈×C₇₃	—	2	1	no	—
166	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
167	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
168	#10596831	—	2	1	no	—
169	#10596831	#121	1	2	yes	—
170	C₂²×C₇₃	—	2	1	no	—
171	C₂²×C₇₃	—	2	1	no	—
172	C₂²×C₇₃	#508	1	2	yes	complemented
173	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
174	C₂×D₇₃	—	2	8	no	—
175	C₂×D₇₃	—	2	1	no	—
176	C₂×D₇₃	—	2	1	no	—
177	C₂×D₇₃	—	2	1	no	—
178	C₂×D₇₃	—	2	1	no	—
179	C₂×D₇₃	—	8	2	no	—
180	C₂×D₇₃	—	8	2	no	—
181	C₂×D₇₃	—	8	2	no	—
182	C₂×D₇₃	#466	1	2	yes	—
183	C₂×D₇₃	#508	1	2	yes	complemented
184	C₄×C₇₃	—	2	1	no	—
185	C₄×C₇₃	—	8	2	no	—
186	C₄×C₇₃	—	8	2	no	—
187	C₄×C₇₃	C₂×#121	1	1	yes	characteristic
188	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
189	Dic₇₃	—	2	1	no	—
190	Dic₇₃	—	8	2	no	—
191	Dic₇₃	#466	1	2	yes	—
192	#33600	—	73	1	no	maximal, Sylow
193	C₂×C₇₃	—	2	2	no	—
194	C₂×C₇₃	C₂×#466	1	1	yes	characteristic
195	C₂×C₇₃	C₂×#508	1	1	yes	characteristic
196	C₂×C₇₃	C₂×#508	1	1	yes	characteristic
197	D₇₃	—	2	2	no	—
198	D₇₃	—	2	2	no	—
199	D₇₃	—	16	2	no	—
200	C₂×#508	—	73	1	no	—
201	C₂³×C₁₆	—	73	1	no	—
202	#3050	—	73	2	no	—
203	#2217	—	73	2	no	—
204	#2217	—	73	2	no	—
205	#2216	—	73	1	no	—
206	#2216	D₇₃	1	1	yes	characteristic, complemented
207	#2215	—	73	1	no	—
208	#2215	—	73	1	no	—
209	#2158	—	73	1	no	—
210	#2138	—	73	2	no	—
211	C₇₃	#33600	1	1	yes	characteristic, complemented, Sylow
212	C₂×#135	—	73	2	no	—
213	C₂³×C₈	—	73	1	no	—
214	#525	—	146	2	no	—
215	#508	—	146	4	no	—
216	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic, complemented
217	C₂×SD₃₂	—	73	2	no	—
218	C₂×D₁₆	—	73	1	no	—
219	C₂²×C₁₆	—	73	1	no	—
220	C₂²×C₁₆	—	73	1	no	—
221	C₂²×C₁₆	—	73	1	no	—
222	C₂²×C₁₆	—	73	1	no	—
223	C₂²×C₁₆	—	73	1	no	—
224	C₂²×C₁₆	—	146	4	no	—
225	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
226	#467	—	73	2	no	—
227	#467	C₂×D₇₃	1	2	yes	—
228	#466	—	73	2	no	—
229	#466	—	73	2	no	—
230	#465	—	146	2	no	—
231	#465	—	146	2	no	—
232	#427	—	73	2	no	—
233	#416	—	146	2	no	—
234	#367	—	73	2	no	—
235	#366	—	73	1	no	—
236	#366	C₂×D₇₃	1	1	yes	characteristic, complemented
237	#358	—	73	2	no	—
238	#358	C₂×D₇₃	1	2	yes	complemented
239	#357	—	73	2	no	—
240	#357	—	73	2	no	—
241	C₂×#41	—	146	2	no	—
242	C₂³×C₄	—	73	1	no	—
243	#135	—	146	4	no	—
244	#135	—	146	4	no	—
245	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
246	C₂×SD₁₆	—	146	2	no	—
247	C₂×D₈	—	73	2	no	—
248	C₂²×C₈	—	73	1	no	—
249	C₂²×C₈	—	73	1	no	—
250	C₂²×C₈	—	73	1	no	—
251	C₂²×C₈	—	73	1	no	—
252	C₂²×C₈	—	73	1	no	—
253	C₂²×C₈	—	146	2	no	—
254	C₂²×C₈	—	146	2	no	—
255	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
256	#123	—	292	2	no	—
257	C₂²⋊Q₈	—	2	2	no	—
258	C₂²⋊Q₈	—	146	2	no	—
259	#121	—	146	2	no	—
260	#121	—	146	2	no	—
261	#120	—	292	2	no	—
262	C₄×D₄	—	292	2	no	—
263	C₄×D₄	—	292	2	no	—
264	#117	—	146	2	no	—
265	Q₃₂	—	2	2	no	—
266	SD₃₂	—	146	4	no	—
267	D₁₆	—	146	2	no	—
268	C₂×C₁₆	—	2	2	no	—
269	C₂×C₁₆	—	73	1	no	—
270	C₂×C₁₆	—	73	4	no	—
271	C₂×C₁₆	—	73	1	no	—
272	C₂×C₁₆	—	146	2	no	—
273	C₂×C₁₆	—	146	2	no	—
274	C₂×C₁₆	—	146	2	no	—
275	C₂×C₁₆	—	146	2	no	—
276	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
277	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
278	#107	—	73	2	no	—
279	#107	C₂²×D₇₃	1	2	yes	—
280	#106	—	146	2	no	—
281	#103	—	2	2	no	—
282	#103	—	146	2	no	—
283	#102	—	146	2	no	—
284	#102	—	146	2	no	—
285	C₂⁴	—	73	1	no	—
286	#41	—	292	4	no	—
287	#41	—	292	4	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×D₄	—	292	2	no	—
290	C₂×D₄	—	292	2	no	—
291	C₂×D₄	—	292	2	no	—
292	C₂×D₄	—	146	2	no	—
293	C₂²×C₄	—	292	2	no	—
294	C₂²×C₄	—	73	1	no	—
295	C₂²×C₄	—	73	1	no	—
296	C₂²×C₄	—	73	1	no	—
297	C₂²×C₄	—	73	1	no	—
298	C₂²×C₄	—	73	1	no	—
299	C₂²×C₄	—	146	2	no	—
300	C₂²×C₄	—	146	2	no	—
301	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
302	Q₁₆	—	2	2	no	—
303	Q₁₆	—	2	2	no	—
304	SD₁₆	—	292	4	no	—
305	D₈	—	146	2	no	—
306	D₈	—	146	2	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	73	1	no	—
310	C₂×C₈	—	73	2	no	—
311	C₂×C₈	—	73	2	no	—
312	C₂×C₈	—	73	1	no	—
313	C₂×C₈	—	146	2	no	—
314	C₂×C₈	—	146	2	no	—
315	C₂×C₈	—	146	2	no	—
316	C₂×C₈	—	146	1	no	—
317	C₂×C₈	—	146	1	no	—
318	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
319	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	4	2	no	—
322	C₄⋊C₄	—	292	2	no	—
323	C₄⋊C₄	—	146	2	no	—
324	C₂²⋊C₄	—	4	2	no	—
325	C₂²⋊C₄	—	292	2	no	—
326	C₂²⋊C₄	—	292	2	no	—
327	C₂²⋊C₄	—	292	2	no	—
328	C₂²⋊C₄	—	292	2	no	—
329	C₄²	—	292	2	no	—
330	C₁₆	—	2	1	no	—
331	C₁₆	—	73	2	no	—
332	C₁₆	—	146	1	no	—
333	C₁₆	D₄×D₇₃	1	2	yes	—
334	C₂³	—	292	2	no	—
335	C₂³	—	73	1	no	—
336	C₂³	—	73	1	no	—
337	C₂³	—	146	2	no	—
338	C₂³	—	146	2	no	—
339	C₂³	—	146	2	no	—
340	C₂³	D₈×D₇₃	1	1	yes	characteristic
341	Q₈	—	4	2	no	—
342	Q₈	—	4	2	no	—
343	D₄	—	292	2	no	—
344	D₄	—	292	2	no	—
345	D₄	—	584	4	no	—
346	D₄	—	584	4	no	—
347	D₄	—	584	4	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	292	2	no	—
353	C₂×C₄	—	584	4	no	—
354	C₂×C₄	—	73	1	no	—
355	C₂×C₄	—	73	2	no	—
356	C₂×C₄	—	73	2	no	—
357	C₂×C₄	—	73	1	no	—
358	C₂×C₄	—	146	2	no	—
359	C₂×C₄	—	146	2	no	—
360	C₂×C₄	—	146	2	no	—
361	C₂×C₄	—	146	1	no	—
362	C₂×C₄	—	146	1	no	—
363	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
364	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
365	C₈	—	2	1	no	—
366	C₈	—	73	2	no	—
367	C₈	—	146	1	no	—
368	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
369	C₈	#41×D₇₃	1	1	yes	characteristic
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	584	2	no	—
373	C₂²	—	584	2	no	—
374	C₂²	—	584	2	no	—
375	C₂²	—	73	2	no	—
376	C₂²	—	73	2	no	—
377	C₂²	—	146	8	no	—
378	C₂²	—	146	1	no	—
379	C₂²	—	146	1	no	—
380	C₂²	—	146	1	no	—
381	C₂²	—	146	1	no	—
382	C₂²	#17989020	1	2	yes	complemented
383	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	8	2	no	—
386	C₄	—	8	2	no	—
387	C₄	—	584	2	no	—
388	C₄	—	73	2	no	—
389	C₄	—	146	1	no	—
390	C₄	#121×D₇₃	1	1	yes	characteristic
391	C₄	#135×D₇₃	1	1	yes	characteristic
392	C₂	—	2	2	no	—
393	C₂	—	1168	2	no	—
394	C₂	—	146	2	no	—
395	C₂	—	146	2	no	—
396	C₂	#466×D₇₃	1	1	yes	characteristic
397	C₂	#508×D₇₃	1	1	yes	characteristic
398	C₂	C₂×#17989020	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2216	C₂
socle	C₂²×C₇₃	C₂²×D₈
hypercenter	#2216	D₇₃
hypocenter	C₇₃	#33600
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

Complements

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

N	H ≅ G/N	type
1. C₂×#17989020	C₂	semidirect
2. C₂²×C₁₆×D₇₃	C₂	semidirect
3. #22619514	C₂	semidirect
4. #22612030	C₂	semidirect
5. #22612029	C₂	semidirect
6. #22612019	C₂	semidirect
7. #22612018	C₂	semidirect
8. #22612017	C₂	semidirect
9. #22611712	C₂	semidirect
10. #22611637	C₂	semidirect
11. C₇₃×#2216	C₂	semidirect
16. C₂×#14556534	C₂²	semidirect
17. C₂×D₁₁₆₈	C₂²	semidirect
31. #17988414	C₂²	semidirect
32. #17988413	C₂²	semidirect
33. #17988391	C₂²	semidirect
34. #17988389	C₂²	semidirect
35. #17988387	C₂²	semidirect
36. C₂×Q₃₂×C₇₃	C₂²	semidirect
39. #366×C₇₃	C₂²	semidirect
40. #358×C₇₃	C₂²	semidirect
70. C₁₆×D₇₃	D₄	semidirect
172. C₂²×C₇₃	#508	semidirect
183. C₂×D₇₃	#508	semidirect
206. #2216	D₇₃	semidirect
211. C₇₃	#33600	semidirect
216. C₂×Q₃₂	C₂×D₇₃	semidirect
236. #366	C₂×D₇₃	semidirect
238. #358	C₂×D₇₃	semidirect
382. C₂²	#17989020	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989020	2
C₂²×C₁₆×D₇₃	2
#22619514	2
#22612030	2
#22612029	2
#22612019	2
#22612018	2
#22612017	2
#22611712	2
#22611637	2
C₇₃×#2216	2
#33600	73