Subgroups — #33057414

Order 18688.

656 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618944	C₂	1	2	yes	maximal, complemented
2	C₄×#14556614	C₂	1	1	yes	maximal, characteristic
3	#22608828	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608770	C₂	1	2	yes	maximal, complemented
5	#22608765	C₂	1	2	yes	maximal, complemented
6	C₈×D₅₈₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608725	C₂	1	1	yes	maximal, characteristic, complemented
8	#22608693	C₂	1	1	yes	maximal, characteristic, complemented
9	#22608364	C₂	1	2	yes	maximal, complemented
10	C₈×#11924011	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556614	C₄	1	1	yes	characteristic
13	C₄×#11924029	C₂²	1	2	yes	—
14	#17988718	C₂²	1	2	yes	—
15	Q₁₆×Dic₇₃	C₄	1	1	yes	characteristic
16	C₄×#11924015	C₂²	1	2	yes	—
17	#17988672	C₄	1	2	yes	—
18	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
19	#17988669	C₄	1	1	yes	characteristic
20	#17988660	—	2	2	no	—
21	#17988657	—	2	2	no	—
22	C₈×D₂₉₂	C₂²	1	2	yes	—
23	#17988652	C₂²	1	2	yes	—
24	#17988643	—	2	2	no	—
25	#17988643	C₂²	1	2	yes	—
26	#17988638	C₂²	1	1	yes	characteristic
27	#17988624	C₄	1	1	yes	characteristic
28	#17988582	C₄	1	2	yes	—
29	C₄×#11924011	C₂²	1	1	yes	characteristic
30	#97×D₇₃	—	2	2	no	—
31	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
32	#17988340	C₂²	1	1	yes	characteristic
33	#17988337	C₂²	1	2	yes	—
34	Q₈⋊#10596831	C₂²	1	2	yes	—
35	#17988316	C₂²	1	2	yes	—
36	#17988314	C₂²	1	2	yes	—
37	#17988309	C₂²	1	1	yes	characteristic
38	#17988308	C₂²	1	1	yes	characteristic
39	C₈×#10596831	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×#11924029	C₂×C₄	1	2	yes	—
46	#14556614	—	2	4	no	—
47	C₂×#11924015	C₂×C₄	1	2	yes	—
48	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
49	C₂×#11924011	D₄	1	1	yes	characteristic
50	C₂×#11924011	D₄	1	1	yes	characteristic
51	C₂×C₈×D₇₃	—	2	2	no	—
52	C₂×C₈×D₇₃	—	2	2	no	—
53	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
55	Q₈×Dic₇₃	C₂×C₄	1	2	yes	—
56	C₄×D₂₉₂	—	2	2	no	—
57	C₄×D₂₉₂	C₂³	1	2	yes	—
58	#14556553	—	2	2	no	—
59	#14556553	C₂×C₄	1	2	yes	—
60	#14556549	—	2	2	no	—
61	#14556549	C₂×C₄	1	2	yes	—
62	C₄²×D₇₃	—	2	2	no	—
63	C₄²×D₇₃	C₂³	1	1	yes	characteristic
64	#14556525	C₂×C₄	1	1	yes	characteristic
65	#14556522	C₂×C₄	1	2	yes	—
66	C₄⋊#10596831	—	2	2	no	—
67	C₄⋊#10596831	C₂³	1	2	yes	—
68	#14556520	—	2	1	no	—
69	#14556515	C₂×C₄	1	2	yes	—
70	#14556513	C₂×C₄	1	2	yes	—
71	#14556511	C₂×C₄	1	2	yes	—
72	#14556503	—	2	2	no	—
73	#14556503	—	2	1	no	—
74	#14556503	—	2	2	no	—
75	#14556503	—	2	2	no	—
76	#14556502	—	2	2	no	—
77	#14556501	—	2	2	no	—
78	#14556501	D₄	1	1	yes	characteristic
79	#14556501	D₄	1	1	yes	characteristic
80	C₄×#10596831	C₂³	1	1	yes	characteristic
81	C₄×#10596831	C₂³	1	1	yes	characteristic
82	C₄×#10596831	C₂³	1	1	yes	characteristic
83	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
85	C₂×Q₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
87	#107×C₇₃	C₂×C₄	1	1	yes	characteristic
88	C₄⋊C₈×C₇₃	—	2	2	no	—
89	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
90	#103×C₇₃	C₂×C₄	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	#11924029	—	2	4	no	—
96	#11924029	—	4	2	no	—
97	C₂×D₂₉₂	—	2	2	no	—
98	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
99	C₂×C₄×D₇₃	—	2	2	no	—
100	C₂×C₄×D₇₃	—	2	2	no	—
101	C₂×C₄×D₇₃	—	2	2	no	—
102	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
105	#11924015	—	2	4	no	—
106	D₅₈₄	M₁₆	1	4	yes	complemented
107	#11924011	—	2	2	no	—
108	#11924011	—	2	2	no	—
109	C₂×#10596831	—	2	2	no	—
110	C₂×#10596831	—	2	1	no	—
111	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
112	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
113	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
114	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
115	C₈×D₇₃	—	2	2	no	—
116	C₈×D₇₃	—	4	1	no	—
117	C₈×D₇₃	—	4	4	no	—
118	C₈×D₇₃	—	4	4	no	—
119	C₄⋊Dic₇₃	—	2	2	no	—
120	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
121	#11924005	—	2	2	no	—
122	#11924005	—	2	2	no	—
123	#11924005	—	2	2	no	—
124	C₄×Dic₇₃	—	2	2	no	—
125	C₄×Dic₇₃	—	2	2	no	—
126	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
127	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
128	C₄×Dic₇₃	#41	1	1	yes	characteristic
129	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
130	Q₁₆×C₇₃	—	2	2	no	—
131	C₂×C₈×C₇₃	—	2	2	no	—
132	C₂×C₈×C₇₃	—	2	1	no	—
133	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₈×C₇₃	C₂²×C₄	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	C₂²×C₄	1	2	yes	—
139	C₄²×C₇₃	—	2	2	no	—
140	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
141	C₂²×D₇₃	—	2	2	no	—
142	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
143	Q₈×C₇₃	—	2	2	no	—
144	Q₈×C₇₃	—	2	2	no	—
145	C₂×C₄×C₇₃	—	2	2	no	—
146	C₂×C₄×C₇₃	—	2	2	no	—
147	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
150	C₂×Dic₇₃	—	2	2	no	—
151	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
152	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
153	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
154	D₂₉₂	—	2	2	no	—
155	D₂₉₂	—	4	4	no	—
156	D₂₉₂	C₂×M₁₆	1	4	yes	—
157	C₄×D₇₃	—	2	1	no	—
158	C₄×D₇₃	—	2	1	no	—
159	C₄×D₇₃	—	2	1	no	—
160	C₄×D₇₃	—	2	1	no	—
161	C₄×D₇₃	—	4	1	no	—
162	C₄×D₇₃	—	4	4	no	—
163	C₄×D₇₃	—	4	2	no	—
164	C₄×D₇₃	—	4	2	no	—
165	C₄×D₇₃	—	8	2	no	—
166	C₈×C₇₃	—	2	1	no	—
167	C₈×C₇₃	—	2	1	no	—
168	C₈×C₇₃	—	4	1	no	—
169	C₈×C₇₃	—	4	2	no	—
170	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
171	C₈×C₇₃	#135	1	2	yes	complemented
172	#10596831	—	2	1	no	—
173	#10596831	—	2	1	no	—
174	#10596831	—	2	1	no	—
175	#10596831	—	4	1	no	—
176	#10596831	—	4	2	no	—
177	#10596831	#135	1	2	yes	complemented
178	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
179	C₂×D₇₃	—	2	1	no	—
180	C₂×D₇₃	—	2	4	no	—
181	C₂×D₇₃	—	2	1	no	—
182	C₂×D₇₃	—	2	1	no	—
183	C₂×D₇₃	—	4	2	no	—
184	C₂×D₇₃	—	4	2	no	—
185	C₄×C₇₃	—	2	1	no	—
186	C₄×C₇₃	—	4	2	no	—
187	C₄×C₇₃	—	4	2	no	—
188	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
189	C₄×C₇₃	#568	1	1	yes	characteristic
190	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
191	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
192	Dic₇₃	—	2	1	no	—
193	Dic₇₃	—	2	1	no	—
194	Dic₇₃	—	2	1	no	—
195	Dic₇₃	—	4	2	no	—
196	#18035	—	73	1	no	maximal, Sylow
197	C₂×C₇₃	D₄×M₁₆	1	1	yes	characteristic
198	C₂×C₇₃	C₄×#135	1	1	yes	characteristic
199	C₂×C₇₃	#2957	1	1	yes	characteristic
200	D₇₃	—	4	1	no	—
201	D₇₃	—	4	4	no	—
202	#2970	—	73	2	no	—
203	C₄×#135	—	73	1	no	—
204	#1591	—	73	1	no	—
205	#1591	D₇₃	1	1	yes	characteristic, complemented
206	#1582	—	73	2	no	—
207	#1582	—	73	2	no	—
208	C₈×D₈	—	73	1	no	—
209	C₈×D₈	—	73	1	no	—
210	#1568	—	73	1	no	—
211	#1488	—	73	2	no	—
212	C₈×M₁₆	—	73	1	no	—
213	C₇₃	#18035	1	1	yes	characteristic, complemented, Sylow
214	C₂×#135	—	73	1	no	—
215	C₄×#41	—	73	2	no	—
216	#446	—	73	2	no	—
217	#446	C₂×D₇₃	1	2	yes	—
218	C₄×Q₁₆	—	73	1	no	—
219	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
220	C₄×SD₁₆	—	73	2	no	—
221	C₄×SD₁₆	—	73	2	no	—
222	C₄×D₈	—	73	1	no	—
223	C₄×D₈	—	73	1	no	—
224	#435	—	146	2	no	—
225	#435	—	146	2	no	—
226	C₈×D₄	—	73	2	no	—
227	C₈×D₄	—	73	2	no	—
228	#432	—	73	2	no	—
229	#432	—	146	2	no	—
230	#431	—	73	1	no	—
231	#427	—	73	1	no	—
232	#416	—	73	2	no	—
233	C₄×M₁₆	—	73	1	no	—
234	C₂×#97	—	146	2	no	—
235	C₂×C₄×C₈	—	73	1	no	—
236	#335	—	73	1	no	—
237	#335	C₂×D₇₃	1	1	yes	characteristic
238	#334	—	73	2	no	—
239	Q₈⋊C₈	—	73	2	no	—
240	Q₈⋊C₈	C₂×D₇₃	1	2	yes	—
241	D₄⋊C₈	—	73	2	no	—
242	D₄⋊C₈	—	73	2	no	—
243	#322	—	73	1	no	—
244	#322	—	73	1	no	—
245	C₈²	—	73	1	no	—
246	C₈²	C₂×D₇₃	1	1	yes	characteristic
247	C₂×#41	—	73	2	no	—
248	#135	—	146	4	no	—
249	C₂×Q₁₆	C₄×D₇₃	1	1	yes	characteristic
250	C₂×SD₁₆	—	73	2	no	—
251	C₂×D₈	—	73	1	no	—
252	C₂×M₁₆	—	73	1	no	—
253	C₂×M₁₆	—	73	1	no	—
254	C₂²×C₈	—	73	1	no	—
255	C₂²×C₈	—	73	1	no	—
256	C₂²×C₈	—	146	2	no	—
257	C₂²×C₈	—	146	2	no	—
258	C₄×Q₈	—	73	2	no	—
259	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
260	C₄×D₄	—	73	2	no	—
261	C₄×D₄	—	73	2	no	—
262	C₄×D₄	—	146	2	no	—
263	C₄×D₄	—	146	2	no	—
264	#117	—	73	2	no	—
265	#117	—	146	2	no	—
266	C₂×C₄²	—	73	1	no	—
267	C₂×C₄²	—	146	2	no	—
268	#107	—	73	1	no	—
269	#107	C₄×D₇₃	1	1	yes	characteristic
270	#106	—	73	2	no	—
271	C₄⋊C₈	—	2	2	no	—
272	C₄⋊C₈	—	73	2	no	—
273	C₄⋊C₈	—	146	2	no	—
274	C₄⋊C₈	—	146	1	no	—
275	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
276	#103	—	73	2	no	—
277	#103	C₄×D₇₃	1	2	yes	—
278	#102	—	73	2	no	—
279	#102	—	73	2	no	—
280	C₂²⋊C₈	—	146	2	no	—
281	C₂²⋊C₈	—	146	1	no	—
282	C₂²⋊C₈	—	146	2	no	—
283	C₂²⋊C₈	—	146	2	no	—
284	#97	—	2	2	no	—
285	#97	—	73	1	no	—
286	#97	—	73	1	no	—
287	#97	—	146	2	no	—
288	#97	—	146	2	no	—
289	C₄×C₈	—	73	1	no	—
290	C₄×C₈	—	73	1	no	—
291	C₄×C₈	—	73	1	no	—
292	C₄×C₈	—	73	1	no	—
293	C₄×C₈	—	73	1	no	—
294	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
295	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
296	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
297	#41	—	292	2	no	—
298	#41	—	146	4	no	—
299	C₂×Q₈	C₂×C₄×D₇₃	1	2	yes	—
300	C₂×D₄	—	73	2	no	—
301	C₂×D₄	—	146	2	no	—
302	C₂²×C₄	—	73	1	no	—
303	C₂²×C₄	—	73	1	no	—
304	C₂²×C₄	—	73	1	no	—
305	C₂²×C₄	—	146	2	no	—
306	C₂²×C₄	—	146	2	no	—
307	C₂²×C₄	—	146	2	no	—
308	Q₁₆	—	2	2	no	—
309	SD₁₆	—	146	4	no	—
310	D₈	—	73	4	no	—
311	M₁₆	—	146	2	no	—
312	M₁₆	—	146	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	292	1	no	—
316	C₂×C₈	—	292	4	no	—
317	C₂×C₈	—	292	4	no	—
318	C₂×C₈	—	73	1	no	—
319	C₂×C₈	—	73	1	no	—
320	C₂×C₈	—	73	1	no	—
321	C₂×C₈	—	73	1	no	—
322	C₂×C₈	—	146	2	no	—
323	C₂×C₈	—	146	1	no	—
324	C₂×C₈	—	146	2	no	—
325	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
326	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
327	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
328	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	73	2	no	—
331	C₄⋊C₄	—	146	2	no	—
332	C₄⋊C₄	C₂×C₄×D₇₃	1	2	yes	—
333	C₂²⋊C₄	—	146	2	no	—
334	C₂²⋊C₄	—	146	2	no	—
335	C₂²⋊C₄	—	146	2	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	73	1	no	—
338	C₄²	—	73	1	no	—
339	C₄²	—	73	1	no	—
340	C₄²	—	146	2	no	—
341	C₄²	—	146	2	no	—
342	C₄²	C₂³×D₇₃	1	1	yes	characteristic
343	C₂³	—	73	1	no	—
344	C₂³	—	146	2	no	—
345	Q₈	—	2	2	no	—
346	Q₈	—	2	2	no	—
347	D₄	—	292	4	no	—
348	D₄	—	73	4	no	—
349	D₄	—	146	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	292	1	no	—
353	C₂×C₄	—	292	4	no	—
354	C₂×C₄	—	292	2	no	—
355	C₂×C₄	—	292	2	no	—
356	C₂×C₄	—	584	2	no	—
357	C₂×C₄	—	73	1	no	—
358	C₂×C₄	—	73	1	no	—
359	C₂×C₄	—	73	1	no	—
360	C₂×C₄	—	146	2	no	—
361	C₂×C₄	—	146	1	no	—
362	C₂×C₄	—	146	1	no	—
363	C₂×C₄	—	146	1	no	—
364	C₂×C₄	—	146	1	no	—
365	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
366	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
367	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	—	4	1	no	—
371	C₈	—	4	2	no	—
372	C₈	—	292	1	no	—
373	C₈	—	292	2	no	—
374	C₈	—	73	2	no	—
375	C₈	—	146	1	no	—
376	C₈	—	146	1	no	—
377	C₈	—	146	1	no	—
378	C₈	M₁₆×D₇₃	1	2	yes	—
379	C₈	#14556614	1	2	yes	complemented
380	C₂²	—	292	2	no	—
381	C₂²	—	292	2	no	—
382	C₂²	—	146	4	no	—
383	C₂²	—	146	1	no	—
384	C₂²	—	146	1	no	—
385	C₂²	—	146	1	no	—
386	C₂²	C₄×D₄×D₇₃	1	1	yes	characteristic
387	C₄	—	2	1	no	—
388	C₄	—	4	2	no	—
389	C₄	—	4	2	no	—
390	C₄	—	292	2	no	—
391	C₄	—	146	1	no	—
392	C₄	—	146	1	no	—
393	C₄	—	146	1	no	—
394	C₄	C₂×M₁₆×D₇₃	1	1	yes	characteristic
395	C₄	#17989352	1	1	yes	characteristic
396	C₄	#135×D₇₃	1	1	yes	characteristic
397	C₄	C₂×#14556614	1	1	yes	characteristic
398	C₂	—	292	4	no	—
399	C₂	—	292	1	no	—
400	C₂	#22618667	1	1	yes	characteristic
401	C₂	C₄×#14556614	1	1	yes	characteristic
402	C₂	#22618843	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1591	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂×#14556614	C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1591	D₇₃
hypocenter	C₇₃	#18035
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
1. #22618944	C₂	semidirect
3. #22608828	C₂	semidirect
4. #22608770	C₂	semidirect
5. #22608765	C₂	semidirect
6. C₈×D₅₈₄	C₂	semidirect
7. #22608725	C₂	semidirect
8. #22608693	C₂	semidirect
9. #22608364	C₂	semidirect
10. C₈×#11924011	C₂	semidirect
11. C₇₃×#1591	C₂	semidirect
106. D₅₈₄	M₁₆	semidirect
171. C₈×C₇₃	#135	semidirect
177. #10596831	#135	semidirect
205. #1591	D₇₃	semidirect
213. C₇₃	#18035	semidirect
379. C₈	#14556614	semidirect

Maximal subgroups

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

subgroup	index
#22618944	2
C₄×#14556614	2
#22608828	2
#22608770	2
#22608765	2
C₈×D₅₈₄	2
#22608725	2
#22608693	2
#22608364	2
C₈×#11924011	2
C₇₃×#1591	2
#18035	73