Subgroups — #33055836

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22619047	C₂	1	2	yes	maximal, complemented
2	C₄×#14556614	C₂	1	1	yes	maximal, characteristic
3	#22608836	C₂	1	2	yes	maximal, complemented
4	#22608831	C₂	1	2	yes	maximal, complemented
5	#22608828	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608817	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608815	C₂	1	1	yes	maximal, characteristic, complemented
8	#335×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#22608317	C₂	1	2	yes	maximal, complemented
10	#22608213	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	#17988666	C₂²	1	2	yes	—
21	#17988664	C₂²	1	2	yes	—
22	#17988660	—	2	2	no	—
23	#17988657	—	2	2	no	—
24	#17988642	—	2	2	no	—
25	#17988638	C₂²	1	1	yes	characteristic
26	#17988638	C₂²	1	1	yes	characteristic
27	#17988624	C₄	1	1	yes	characteristic
28	#17988609	—	2	2	no	—
29	C₄⋊C₈×D₇₃	C₂²	1	2	yes	—
30	#17988582	C₄	1	2	yes	—
31	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
32	#17988340	C₂²	1	1	yes	characteristic
33	#17988339	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	#17988309	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	—	2	2	no	—
50	C₂×#11924011	—	2	2	no	—
51	C₂×C₈×D₇₃	—	2	2	no	—
52	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
54	Q₈×Dic₇₃	C₂×C₄	1	2	yes	—
55	C₄×D₂₉₂	—	2	2	no	—
56	C₄×D₂₉₂	C₂³	1	2	yes	—
57	#14556553	—	2	2	no	—
58	#14556553	C₂×C₄	1	2	yes	—
59	#14556549	—	2	2	no	—
60	#14556549	C₂×C₄	1	2	yes	—
61	C₄²×D₇₃	—	2	2	no	—
62	C₄²×D₇₃	C₂³	1	1	yes	characteristic
63	#14556525	C₂×C₄	1	1	yes	characteristic
64	#14556522	C₂×C₄	1	2	yes	—
65	C₄⋊#10596831	—	2	2	no	—
66	C₄⋊#10596831	C₂³	1	2	yes	—
67	#14556520	—	2	2	no	—
68	#14556520	—	2	1	no	—
69	#14556520	D₄	1	1	yes	characteristic
70	#14556520	D₄	1	1	yes	characteristic
71	#14556515	C₂×C₄	1	2	yes	—
72	#14556513	C₂×C₄	1	2	yes	—
73	#14556511	C₂×C₄	1	2	yes	—
74	#14556503	—	2	1	no	—
75	#14556503	—	2	2	no	—
76	#14556503	—	2	2	no	—
77	#14556503	D₄	1	1	yes	characteristic
78	#14556503	D₄	1	1	yes	characteristic
79	#14556502	—	2	2	no	—
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	—	2	4	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	—	2	2	no	—
107	#11924011	—	8	2	no	—
108	#11924011	—	8	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	—	8	2	no	—
117	C₈×D₇₃	M₁₆	1	4	yes	complemented
118	C₄⋊Dic₇₃	—	2	2	no	—
119	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
120	#11924005	—	2	2	no	—
121	#11924005	—	2	2	no	—
122	#11924005	—	2	2	no	—
123	C₄×Dic₇₃	—	2	2	no	—
124	C₄×Dic₇₃	—	2	2	no	—
125	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
126	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
127	C₄×Dic₇₃	#41	1	1	yes	characteristic
128	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
129	Q₁₆×C₇₃	—	2	2	no	—
130	C₂×C₈×C₇₃	—	2	2	no	—
131	C₂×C₈×C₇₃	—	2	1	no	—
132	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₇₃	C₂²×C₄	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	C₄⋊C₄×C₇₃	—	2	2	no	—
137	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
138	C₄²×C₇₃	—	2	2	no	—
139	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
140	C₂²×D₇₃	—	2	2	no	—
141	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
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	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
149	C₂×Dic₇₃	—	2	2	no	—
150	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
151	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
152	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
153	D₂₉₂	—	2	2	no	—
154	D₂₉₂	—	2	2	no	—
155	D₂₉₂	—	4	4	no	—
156	C₄×D₇₃	—	2	1	no	—
157	C₄×D₇₃	—	2	1	no	—
158	C₄×D₇₃	—	4	1	no	—
159	C₄×D₇₃	—	4	2	no	—
160	C₄×D₇₃	—	4	2	no	—
161	C₄×D₇₃	—	4	4	no	—
162	C₄×D₇₃	—	8	2	no	—
163	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
164	C₄×D₇₃	#131	1	2	yes	—
165	C₈×C₇₃	—	2	1	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	#10596831	—	2	1	no	—
172	#10596831	—	2	1	no	—
173	#10596831	—	2	1	no	—
174	#10596831	—	4	1	no	—
175	#10596831	—	4	2	no	—
176	#10596831	C₂×M₁₆	1	2	yes	—
177	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂×D₇₃	—	2	1	no	—
179	C₂×D₇₃	—	2	1	no	—
180	C₂×D₇₃	—	4	2	no	—
181	C₂×D₇₃	—	4	2	no	—
182	C₂×D₇₃	—	4	2	no	—
183	C₂×D₇₃	#436	1	2	yes	—
184	C₄×C₇₃	—	2	1	no	—
185	C₄×C₇₃	—	4	2	no	—
186	C₄×C₇₃	—	4	2	no	—
187	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
188	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
189	C₄×C₇₃	#576	1	1	yes	characteristic
190	C₄×C₇₃	#578	1	1	yes	characteristic
191	Dic₇₃	—	2	1	no	—
192	Dic₇₃	—	2	1	no	—
193	Dic₇₃	—	4	2	no	—
194	Dic₇₃	#436	1	2	yes	—
195	#17898	—	73	1	no	maximal, Sylow
196	C₂×C₇₃	C₂×#436	1	1	yes	characteristic
197	C₂×C₇₃	#2951	1	1	yes	characteristic
198	C₂×C₇₃	#2957	1	1	yes	characteristic
199	D₇₃	—	2	2	no	—
200	D₇₃	—	8	2	no	—
201	#2989	—	73	2	no	—
202	C₄×#135	—	73	1	no	—
203	#1592	—	73	2	no	—
204	#1592	—	73	2	no	—
205	#1591	—	73	1	no	—
206	#1591	D₇₃	1	1	yes	characteristic, complemented
207	#1589	—	73	1	no	—
208	#1589	—	73	1	no	—
209	C₂×#335	—	73	1	no	—
210	#1477	—	73	2	no	—
211	#1450	—	73	1	no	—
212	C₇₃	#17898	1	1	yes	characteristic, complemented, Sylow
213	C₂×#135	—	73	1	no	—
214	C₄×#41	—	73	2	no	—
215	#446	—	73	2	no	—
216	#446	C₂×D₇₃	1	2	yes	—
217	C₄×Q₁₆	—	73	1	no	—
218	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
219	C₄×SD₁₆	—	73	2	no	—
220	C₄×SD₁₆	—	73	2	no	—
221	C₄×D₈	—	73	1	no	—
222	C₄×D₈	—	73	1	no	—
223	#436	—	73	2	no	—
224	#436	—	73	2	no	—
225	#435	—	146	2	no	—
226	#435	—	146	2	no	—
227	#432	—	146	2	no	—
228	#431	—	73	1	no	—
229	#431	—	73	1	no	—
230	#427	—	73	1	no	—
231	#423	—	146	2	no	—
232	C₂×C₄⋊C₈	—	73	2	no	—
233	#416	—	73	2	no	—
234	C₂×C₄×C₈	—	73	1	no	—
235	#335	—	73	2	no	—
236	#335	—	73	1	no	—
237	#335	C₂×D₇₃	1	1	yes	characteristic
238	Q₈⋊C₈	—	73	2	no	—
239	Q₈⋊C₈	C₂×D₇₃	1	2	yes	—
240	D₄⋊C₈	—	73	2	no	—
241	D₄⋊C₈	—	73	2	no	—
242	#322	—	73	1	no	—
243	#322	—	73	1	no	—
244	C₈²	—	73	1	no	—
245	C₈²	C₂×D₇₃	1	1	yes	characteristic
246	C₂×#41	—	73	2	no	—
247	#135	—	146	4	no	—
248	C₂×Q₁₆	C₄×D₇₃	1	1	yes	characteristic
249	C₂×SD₁₆	—	73	2	no	—
250	C₂×D₈	—	73	1	no	—
251	C₂×M₁₆	—	146	2	no	—
252	C₂×M₁₆	—	146	2	no	—
253	C₂²×C₈	—	73	1	no	—
254	C₂²×C₈	—	73	1	no	—
255	C₂²×C₈	—	146	2	no	—
256	C₄×Q₈	—	73	2	no	—
257	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
258	C₄×D₄	—	73	2	no	—
259	C₄×D₄	—	73	2	no	—
260	C₄×D₄	—	146	2	no	—
261	C₄×D₄	—	146	2	no	—
262	#117	—	73	2	no	—
263	#117	—	146	2	no	—
264	C₂×C₄²	—	73	1	no	—
265	C₂×C₄²	—	146	2	no	—
266	#107	—	73	1	no	—
267	#107	C₄×D₇₃	1	1	yes	characteristic
268	#106	—	73	2	no	—
269	C₄⋊C₈	—	2	2	no	—
270	C₄⋊C₈	—	73	1	no	—
271	C₄⋊C₈	—	73	1	no	—
272	C₄⋊C₈	—	73	2	no	—
273	C₄⋊C₈	—	146	2	no	—
274	C₄⋊C₈	—	146	2	no	—
275	C₄⋊C₈	—	146	1	no	—
276	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
277	#103	—	73	2	no	—
278	#103	C₄×D₇₃	1	2	yes	—
279	#102	—	73	2	no	—
280	#102	—	73	2	no	—
281	C₂²⋊C₈	—	73	1	no	—
282	C₂²⋊C₈	—	73	1	no	—
283	C₂²⋊C₈	—	146	2	no	—
284	C₂²⋊C₈	—	146	1	no	—
285	C₂²⋊C₈	—	146	2	no	—
286	#97	—	2	2	no	—
287	#97	—	146	2	no	—
288	C₄×C₈	—	73	1	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	C₂²×D₇₃	1	1	yes	characteristic
294	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
295	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
296	#41	—	146	4	no	—
297	#41	—	146	4	no	—
298	C₂×Q₈	C₂×C₄×D₇₃	1	2	yes	—
299	C₂×D₄	—	73	2	no	—
300	C₂×D₄	—	146	2	no	—
301	C₂²×C₄	—	73	1	no	—
302	C₂²×C₄	—	73	1	no	—
303	C₂²×C₄	—	73	1	no	—
304	C₂²×C₄	—	146	2	no	—
305	C₂²×C₄	—	146	2	no	—
306	C₂²×C₄	—	146	2	no	—
307	Q₁₆	—	2	2	no	—
308	SD₁₆	—	146	4	no	—
309	D₈	—	146	2	no	—
310	M₁₆	—	584	2	no	—
311	M₁₆	—	584	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	584	2	no	—
315	C₂×C₈	—	73	1	no	—
316	C₂×C₈	—	73	4	no	—
317	C₂×C₈	—	73	1	no	—
318	C₂×C₈	—	73	1	no	—
319	C₂×C₈	—	73	1	no	—
320	C₂×C₈	—	146	2	no	—
321	C₂×C₈	—	146	2	no	—
322	C₂×C₈	—	146	1	no	—
323	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
324	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
325	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
326	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	73	2	no	—
329	C₄⋊C₄	—	146	2	no	—
330	C₄⋊C₄	C₂×C₄×D₇₃	1	2	yes	—
331	C₂²⋊C₄	—	146	2	no	—
332	C₂²⋊C₄	—	146	2	no	—
333	C₂²⋊C₄	—	146	2	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	73	1	no	—
336	C₄²	—	73	1	no	—
337	C₄²	—	73	1	no	—
338	C₄²	—	146	2	no	—
339	C₄²	—	146	2	no	—
340	C₄²	C₂³×D₇₃	1	1	yes	characteristic
341	C₂³	—	73	1	no	—
342	C₂³	—	146	2	no	—
343	Q₈	—	2	2	no	—
344	Q₈	—	2	2	no	—
345	D₄	—	292	4	no	—
346	D₄	—	146	2	no	—
347	D₄	—	146	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	292	1	no	—
351	C₂×C₄	—	292	4	no	—
352	C₂×C₄	—	292	2	no	—
353	C₂×C₄	—	292	2	no	—
354	C₂×C₄	—	584	2	no	—
355	C₂×C₄	—	73	1	no	—
356	C₂×C₄	—	73	2	no	—
357	C₂×C₄	—	73	2	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	C₂²×C₄×D₇₃	1	1	yes	characteristic
364	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
365	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	2	1	no	—
368	C₈	—	2	1	no	—
369	C₈	—	4	1	no	—
370	C₈	—	4	2	no	—
371	C₈	—	292	1	no	—
372	C₈	—	292	2	no	—
373	C₈	—	73	2	no	—
374	C₈	—	146	1	no	—
375	C₈	—	146	1	no	—
376	C₈	—	146	1	no	—
377	C₈	M₁₆×D₇₃	1	2	yes	—
378	C₂²	—	292	2	no	—
379	C₂²	—	292	2	no	—
380	C₂²	—	292	2	no	—
381	C₂²	—	73	2	no	—
382	C₂²	—	146	1	no	—
383	C₂²	—	146	1	no	—
384	C₂²	C₄×D₄×D₇₃	1	1	yes	characteristic
385	C₄	—	2	1	no	—
386	C₄	—	4	2	no	—
387	C₄	—	4	2	no	—
388	C₄	—	292	2	no	—
389	C₄	—	73	2	no	—
390	C₄	—	146	1	no	—
391	C₄	—	146	1	no	—
392	C₄	C₂×M₁₆×D₇₃	1	1	yes	characteristic
393	C₄	#131×D₇₃	1	1	yes	characteristic
394	C₄	#17989389	1	1	yes	characteristic
395	C₄	#17989400	1	1	yes	characteristic
396	C₂	—	584	2	no	—
397	C₂	—	146	2	no	—
398	C₂	#436×D₇₃	1	1	yes	characteristic
399	C₂	#22618806	1	1	yes	characteristic
400	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	#17898
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. #22619047	C₂	semidirect
3. #22608836	C₂	semidirect
4. #22608831	C₂	semidirect
5. #22608828	C₂	semidirect
6. #22608817	C₂	semidirect
7. #22608815	C₂	semidirect
8. #335×D₇₃	C₂	semidirect
9. #22608317	C₂	semidirect
10. #22608213	C₂	semidirect
11. C₇₃×#1591	C₂	semidirect
117. C₈×D₇₃	M₁₆	semidirect
206. #1591	D₇₃	semidirect
212. C₇₃	#17898	semidirect

Maximal subgroups

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

subgroup	index
#22619047	2
C₄×#14556614	2
#22608836	2
#22608831	2
#22608828	2
#22608817	2
#22608815	2
#335×D₇₃	2
#22608317	2
#22608213	2
C₇₃×#1591	2
#17898	73