Subgroups — #33086345

Order 18688.

616 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618353	C₂	1	1	yes	maximal, characteristic
2	#22610079	C₂	1	2	yes	maximal, complemented
3	#22610055	C₂	1	1	yes	maximal, characteristic, complemented
4	#22610033	C₂	1	2	yes	maximal
5	#22609978	C₂	1	1	yes	maximal, characteristic
6	#22609974	C₂	1	1	yes	maximal, characteristic
7	#22609813	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609812	C₂	1	1	yes	maximal, characteristic, complemented
9	#22609730	C₂	1	2	yes	maximal
10	#22609685	C₂	1	2	yes	maximal, complemented
11	C₇₃×#1833	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556599	C₂²	1	1	yes	characteristic
13	C₂×#14556597	C₂²	1	1	yes	characteristic
14	#17989055	C₂²	1	1	yes	characteristic
15	C₂×#14556527	C₂²	1	1	yes	characteristic
16	#17988625	C₂²	1	1	yes	characteristic
17	#17988569	C₂²	1	2	yes	—
18	#17988560	C₂²	1	2	yes	—
19	#17988545	C₂²	1	2	yes	—
20	#17988544	C₄	1	2	yes	complemented
21	#17988543	C₂²	1	2	yes	—
22	#17988543	C₂²	1	1	yes	characteristic
23	#17988541	C₄	1	2	yes	complemented
24	#17988539	C₄	1	2	yes	complemented
25	M₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
27	C₄×C₈×D₇₃	C₄	1	2	yes	complemented
28	#17988358	C₂²	1	2	yes	—
29	#17988348	C₂²	1	2	yes	—
30	#17988346	C₂²	1	2	yes	—
31	#17988345	C₂²	1	2	yes	—
32	#17988343	C₂²	1	2	yes	—
33	#17988343	C₂²	1	2	yes	—
34	C₂×#108×C₇₃	C₂²	1	1	yes	characteristic
35	#427×C₇₃	C₂²	1	1	yes	characteristic
36	#405×C₇₃	C₂²	1	1	yes	characteristic
37	#344×C₇₃	C₂²	1	2	yes	—
38	H(3,ℤ/4)×C₇₃	C₂²	1	2	yes	—
39	C₂×#11924028	C₂³	1	1	yes	characteristic
40	#14556599	—	2	1	no	—
41	#14556599	—	2	1	no	—
42	#14556599	—	2	1	no	—
43	#14556599	—	2	1	no	—
44	#14556597	—	2	1	no	—
45	#14556597	—	2	1	no	—
46	#14556597	—	2	1	no	—
47	#14556597	—	2	1	no	—
48	C₂×#11924012	C₂³	1	2	yes	—
49	C₂×#11924012	C₂³	1	1	yes	characteristic
50	C₂×#11924011	C₂×C₄	1	2	yes	—
51	C₂×#11924011	C₂×C₄	1	2	yes	—
52	C₂²×#10596831	C₂³	1	1	yes	characteristic
53	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
54	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
55	#14556556	—	2	1	no	—
56	#14556553	—	2	1	no	—
57	#14556551	—	2	2	no	—
58	#14556550	C₂³	1	2	yes	—
59	#14556548	C₂×C₄	1	2	yes	—
60	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
61	C₄²×D₇₃	C₂×C₄	1	2	yes	—
62	#14556527	—	2	1	no	—
63	#14556527	—	2	1	no	—
64	#14556525	—	2	1	no	—
65	#14556523	—	2	1	no	—
66	#14556509	—	2	2	no	—
67	#14556507	—	2	2	no	—
68	#14556505	—	2	4	no	—
69	#14556502	—	2	2	no	—
70	#14556502	C₂×C₄	1	2	yes	—
71	#14556501	—	2	2	no	—
72	#14556501	C₂×C₄	1	2	yes	—
73	C₄×#10596831	—	2	2	no	—
74	C₄×#10596831	C₂×C₄	1	2	yes	—
75	C₈×Dic₇₃	—	2	2	no	—
76	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
77	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
78	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
79	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
81	C₂×M₁₆×C₇₃	C₂³	1	2	yes	—
82	C₂×M₁₆×C₇₃	C₂³	1	1	yes	characteristic
83	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
84	#117×C₇₃	C₂³	1	2	yes	—
85	#117×C₇₃	C₂³	1	1	yes	characteristic
86	#108×C₇₃	—	2	1	no	—
87	#108×C₇₃	—	2	1	no	—
88	#107×C₇₃	—	2	1	no	—
89	#106×C₇₃	—	2	1	no	—
90	#97×C₇₃	C₂×C₄	1	2	yes	—
91	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
92	#11924028	—	2	1	no	—
93	#11924028	—	2	1	no	—
94	#11924028	—	2	1	no	—
95	#11924028	—	2	1	no	—
96	C₂×Dic₁₄₆	#41	1	1	yes	characteristic
97	C₂×D₂₉₂	#41	1	1	yes	characteristic
98	C₂×C₂²⋊D₇₃	#41	1	2	yes	—
99	C₂²×Dic₇₃	—	2	1	no	—
100	C₂×C₄×D₇₃	—	2	2	no	—
101	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
102	#11924012	—	2	1	no	—
103	#11924012	—	2	1	no	—
104	#11924012	—	2	2	no	—
105	#11924012	—	2	2	no	—
106	#11924011	—	2	2	no	—
107	#11924011	—	2	2	no	—
108	#11924011	—	2	2	no	—
109	#11924011	—	2	2	no	—
110	C₂×#10596831	—	2	2	no	—
111	C₂×#10596831	C₂²×C₄	1	2	yes	—
112	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
113	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
114	C₂×#10596831	#41	1	1	yes	characteristic
115	C₂×#10596831	#41	1	1	yes	characteristic
116	C₂×#10596831	#41	1	1	yes	characteristic
117	C₂×#10596831	#41	1	1	yes	characteristic
118	C₈×D₇₃	—	2	2	no	—
119	C₈×D₇₃	—	2	2	no	—
120	C₈×D₇₃	—	2	2	no	—
121	C₈×D₇₃	—	2	2	no	—
122	C₄⋊Dic₇₃	—	2	2	no	—
123	#11924007	—	2	2	no	—
124	C₂²⋊Dic₇₃	—	2	2	no	—
125	#11924005	—	2	2	no	—
126	C₄×Dic₇₃	—	2	1	no	—
127	C₄×Dic₇₃	—	2	2	no	—
128	C₄×Dic₇₃	—	2	2	no	—
129	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
130	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
132	M₁₆×C₇₃	—	2	1	no	—
133	M₁₆×C₇₃	—	2	1	no	—
134	M₁₆×C₇₃	—	2	2	no	—
135	M₁₆×C₇₃	—	2	2	no	—
136	C₂×C₈×C₇₃	—	2	2	no	—
137	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
138	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
141	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
142	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
143	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
144	C₄⋊C₄×C₇₃	—	2	1	no	—
145	C₄⋊C₄×C₇₃	—	2	2	no	—
146	C₂²⋊C₄×C₇₃	—	2	1	no	—
147	C₂²⋊C₄×C₇₃	—	2	2	no	—
148	C₄²×C₇₃	—	2	2	no	—
149	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
150	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
151	C₂²×D₇₃	#117	1	2	yes	—
152	C₂×C₄×C₇₃	—	2	2	no	—
153	C₂×C₄×C₇₃	—	4	2	no	—
154	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
160	C₂²⋊D₇₃	—	2	2	no	—
161	C₂²⋊D₇₃	—	2	2	no	—
162	C₂×Dic₇₃	—	2	1	no	—
163	C₂×Dic₇₃	—	2	1	no	—
164	C₂×Dic₇₃	—	4	2	no	—
165	C₂×Dic₇₃	—	4	2	no	—
166	C₂×Dic₇₃	#117	1	2	yes	—
167	D₂₉₂	—	2	1	no	—
168	D₂₉₂	—	2	1	no	—
169	C₄×D₇₃	—	2	2	no	—
170	C₄×D₇₃	—	2	2	no	—
171	C₄×D₇₃	—	4	4	no	—
172	Dic₁₄₆	—	2	1	no	—
173	Dic₁₄₆	—	2	1	no	—
174	C₈×C₇₃	—	2	2	no	—
175	C₈×C₇₃	—	4	2	no	—
176	C₈×C₇₃	#117	1	1	yes	characteristic
177	C₈×C₇₃	#117	1	1	yes	characteristic
178	C₈×C₇₃	#117	1	1	yes	characteristic
179	C₈×C₇₃	#117	1	1	yes	characteristic
180	#10596831	—	2	2	no	—
181	#10596831	—	4	2	no	—
182	#10596831	#117	1	1	yes	characteristic
183	#10596831	#117	1	1	yes	characteristic
184	#10596831	#117	1	1	yes	characteristic
185	#10596831	#117	1	1	yes	characteristic
186	C₂²×C₇₃	—	4	1	no	—
187	C₂²×C₇₃	C₂×#117	1	1	yes	characteristic
188	C₂²×C₇₃	#529	1	1	yes	characteristic
189	C₂²×C₇₃	#529	1	1	yes	characteristic
190	C₂×D₇₃	—	2	2	no	—
191	C₂×D₇₃	—	4	2	no	—
192	C₄×C₇₃	—	4	2	no	—
193	C₄×C₇₃	—	8	2	no	—
194	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
195	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
196	C₄×C₇₃	#529	1	1	yes	characteristic
197	C₄×C₇₃	#529	1	1	yes	characteristic
198	Dic₇₃	—	2	2	no	—
199	Dic₇₃	—	2	2	no	—
200	Dic₇₃	—	4	1	no	—
201	Dic₇₃	—	8	2	no	—
202	#20787	—	73	1	no	maximal, Sylow
203	C₂×C₇₃	—	2	1	no	—
204	C₂×C₇₃	—	2	1	no	—
205	C₂×C₇₃	—	2	1	no	—
206	C₂×C₇₃	#2283	1	1	yes	characteristic
207	D₇₃	—	4	2	no	—
208	C₄×#131	—	73	1	no	—
209	#1841	—	73	2	no	—
210	#1833	—	73	1	no	—
211	#1833	D₇₃	1	1	yes	characteristic, complemented
212	#1827	—	73	2	no	—
213	#1813	—	73	1	no	—
214	#1812	—	73	1	no	—
215	#1772	—	73	1	no	—
216	#1772	—	73	1	no	—
217	#1757	—	73	2	no	—
218	#1747	—	73	2	no	—
219	C₇₃	#20787	1	1	yes	characteristic, complemented, Sylow
220	C₂×#131	—	73	1	no	—
221	C₂×#131	—	73	1	no	—
222	C₄×#41	—	73	1	no	—
223	C₂×#108	—	73	1	no	—
224	C₂×#108	C₂×D₇₃	1	1	yes	characteristic
225	#427	—	73	1	no	—
226	#427	C₂×D₇₃	1	1	yes	characteristic
227	#413	—	73	2	no	—
228	#410	—	73	2	no	—
229	#405	—	73	1	no	—
230	#405	—	73	2	no	—
231	#405	—	73	2	no	—
232	#405	—	73	2	no	—
233	#405	—	73	2	no	—
234	#405	C₂×D₇₃	1	1	yes	characteristic
235	C₄×M₁₆	—	73	1	no	—
236	C₄×M₁₆	—	73	2	no	—
237	C₂×C₄×C₈	—	73	1	no	—
238	C₂×C₄×C₈	—	73	2	no	—
239	#344	—	73	2	no	—
240	#344	C₂×D₇₃	1	2	yes	—
241	#339	—	73	2	no	—
242	#339	—	73	2	no	—
243	#338	—	73	2	no	—
244	H(3,ℤ/4)	—	73	2	no	—
245	H(3,ℤ/4)	—	73	2	no	—
246	H(3,ℤ/4)	C₂×D₇₃	1	2	yes	—
247	C₂×#41	—	73	1	no	—
248	#131	—	146	1	no	—
249	#131	—	146	1	no	—
250	#131	—	146	1	no	—
251	#131	—	146	1	no	—
252	#131	—	146	1	no	—
253	#131	—	146	1	no	—
254	#131	—	146	1	no	—
255	#131	—	146	1	no	—
256	C₂×M₁₆	—	73	1	no	—
257	C₂×M₁₆	—	73	2	no	—
258	C₂×M₁₆	—	73	2	no	—
259	C₂×M₁₆	—	73	2	no	—
260	C₂×M₁₆	C₂²×D₇₃	1	2	yes	—
261	C₂×M₁₆	C₂²×D₇₃	1	1	yes	characteristic
262	C₂²×C₈	—	73	1	no	—
263	C₂²×C₈	—	73	2	no	—
264	C₂²×C₈	—	73	2	no	—
265	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
266	C₄×Q₈	—	146	1	no	—
267	C₄×D₄	—	146	2	no	—
268	C₄×D₄	—	146	1	no	—
269	#117	—	73	2	no	—
270	#117	—	73	2	no	—
271	#117	C₂²×D₇₃	1	2	yes	—
272	#117	C₂²×D₇₃	1	1	yes	characteristic
273	C₂×C₄²	—	73	1	no	—
274	C₂×C₄²	—	73	2	no	—
275	#108	—	2	1	no	—
276	#108	—	2	1	no	—
277	#108	—	146	1	no	—
278	#108	—	146	1	no	—
279	#107	—	2	1	no	—
280	#107	—	146	1	no	—
281	#106	—	2	1	no	—
282	#106	—	146	1	no	—
283	#101	—	146	2	no	—
284	#100	—	146	2	no	—
285	#99	—	146	4	no	—
286	#97	—	73	2	no	—
287	#97	—	73	2	no	—
288	#97	—	146	2	no	—
289	#97	—	146	2	no	—
290	#97	C₄×D₇₃	1	2	yes	—
291	C₄×C₈	—	73	1	no	—
292	C₄×C₈	—	73	1	no	—
293	C₄×C₈	—	73	2	no	—
294	C₄×C₈	—	73	1	no	—
295	C₄×C₈	—	73	1	no	—
296	C₄×C₈	—	73	2	no	—
297	C₄×C₈	—	146	2	no	—
298	C₄×C₈	—	146	2	no	—
299	C₄×C₈	C₄×D₇₃	1	2	yes	—
300	#41	—	146	1	no	—
301	#41	—	146	1	no	—
302	#41	—	146	1	no	—
303	#41	—	146	1	no	—
304	C₂×Q₈	—	73	1	no	—
305	C₂×D₄	—	73	2	no	—
306	C₂×D₄	—	73	1	no	—
307	C₂²×C₄	—	73	2	no	—
308	C₂²×C₄	—	146	2	no	—
309	C₂²×C₄	—	146	1	no	—
310	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
311	M₁₆	—	2	1	no	—
312	M₁₆	—	2	1	no	—
313	M₁₆	—	2	2	no	—
314	M₁₆	—	2	2	no	—
315	M₁₆	—	146	2	no	—
316	M₁₆	—	146	2	no	—
317	M₁₆	—	146	2	no	—
318	M₁₆	—	146	2	no	—
319	M₁₆	—	146	1	no	—
320	M₁₆	—	146	1	no	—
321	M₁₆	—	146	2	no	—
322	M₁₆	—	146	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	73	1	no	—
325	C₂×C₈	—	73	1	no	—
326	C₂×C₈	—	73	1	no	—
327	C₂×C₈	—	73	1	no	—
328	C₂×C₈	—	73	1	no	—
329	C₂×C₈	—	73	1	no	—
330	C₂×C₈	—	73	2	no	—
331	C₂×C₈	—	146	2	no	—
332	C₂×C₈	—	146	2	no	—
333	C₂×C₈	—	146	2	no	—
334	C₂×C₈	—	146	2	no	—
335	C₂×C₈	—	146	2	no	—
336	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
337	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
338	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
339	C₂×C₈	#11924027	1	1	yes	characteristic
340	C₂×C₈	#11924027	1	1	yes	characteristic
341	C₂×C₈	#11924029	1	1	yes	characteristic
342	C₂×C₈	#11924029	1	1	yes	characteristic
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	146	2	no	—
346	C₄⋊C₄	—	146	2	no	—
347	C₂²⋊C₄	—	2	1	no	—
348	C₂²⋊C₄	—	2	2	no	—
349	C₂²⋊C₄	—	146	2	no	—
350	C₂²⋊C₄	—	146	2	no	—
351	C₄²	—	2	2	no	—
352	C₄²	—	73	1	no	—
353	C₄²	—	73	1	no	—
354	C₄²	—	146	2	no	—
355	C₄²	—	146	2	no	—
356	C₄²	—	146	1	no	—
357	C₄²	C₂×C₄×D₇₃	1	2	yes	—
358	C₂³	—	73	2	no	—
359	C₂³	#41×D₇₃	1	1	yes	characteristic
360	Q₈	—	146	1	no	—
361	Q₈	—	146	1	no	—
362	D₄	—	146	2	no	—
363	D₄	—	146	2	no	—
364	D₄	—	146	1	no	—
365	D₄	—	146	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	292	4	no	—
369	C₂×C₄	—	292	2	no	—
370	C₂×C₄	—	292	2	no	—
371	C₂×C₄	—	73	2	no	—
372	C₂×C₄	—	146	1	no	—
373	C₂×C₄	—	146	1	no	—
374	C₂×C₄	—	146	2	no	—
375	C₂×C₄	—	146	2	no	—
376	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
377	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
378	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
379	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
380	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
381	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
382	C₈	—	2	2	no	—
383	C₈	—	4	2	no	—
384	C₈	—	292	2	no	—
385	C₈	—	73	1	no	—
386	C₈	—	73	1	no	—
387	C₈	—	73	1	no	—
388	C₈	—	73	1	no	—
389	C₈	—	146	2	no	—
390	C₈	#14556549	1	1	yes	characteristic
391	C₈	#14556549	1	1	yes	characteristic
392	C₈	#14556549	1	1	yes	characteristic
393	C₈	#14556549	1	1	yes	characteristic
394	C₂²	—	4	1	no	—
395	C₂²	—	292	2	no	—
396	C₂²	—	146	2	no	—
397	C₂²	#117×D₇₃	1	1	yes	characteristic
398	C₂²	#17989116	1	1	yes	characteristic
399	C₂²	#17989122	1	1	yes	characteristic
400	C₄	—	4	2	no	—
401	C₄	—	8	2	no	—
402	C₄	—	292	1	no	—
403	C₄	—	584	2	no	—
404	C₄	—	146	2	no	—
405	C₄	—	146	2	no	—
406	C₄	#117×D₇₃	1	1	yes	characteristic
407	C₄	C₂×#14556549	1	1	yes	characteristic
408	C₄	#17989116	1	1	yes	characteristic
409	C₄	#17989122	1	1	yes	characteristic
410	C₂	—	2	1	no	—
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	292	2	no	—
414	C₂	#22612324	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1833	C₂
socle	C₂×C₇₃	#2283
prime radical	C₂×#11924028	C₂³
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1833	D₇₃
hypocenter	C₇₃	#20787
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ #117 ◁ #1833

Complements

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

N	H ≅ G/N	type
2. #22610079	C₂	semidirect
3. #22610055	C₂	semidirect
7. #22609813	C₂	semidirect
8. #22609812	C₂	semidirect
10. #22609685	C₂	semidirect
11. C₇₃×#1833	C₂	semidirect
20. #17988544	C₄	semidirect
23. #17988541	C₄	semidirect
24. #17988539	C₄	semidirect
27. C₄×C₈×D₇₃	C₄	semidirect
211. #1833	D₇₃	semidirect
219. C₇₃	#20787	semidirect

Maximal subgroups

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

subgroup	index
#22618353	2
#22610079	2
#22610055	2
#22610033	2
#22609978	2
#22609974	2
#22609813	2
#22609812	2
#22609730	2
#22609685	2
C₇₃×#1833	2
#20787	73