Subgroups — #33086354

Order 18688.

608 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	#22618358	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	#22609818	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609817	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	M₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
25	C₄×#11924011	C₄	1	2	yes	complemented
26	C₂×#14556501	C₂²	1	1	yes	characteristic
27	#97×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	2	no	—
41	#14556599	—	2	2	no	—
42	#14556597	—	2	1	no	—
43	#14556597	—	2	1	no	—
44	#14556597	—	2	1	no	—
45	#14556597	—	2	1	no	—
46	C₂×#11924012	C₂³	1	2	yes	—
47	C₂×#11924012	C₂³	1	1	yes	characteristic
48	C₂×#11924011	C₂×C₄	1	2	yes	—
49	C₂×#11924011	C₂×C₄	1	2	yes	—
50	C₂²×#10596831	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
53	#14556556	—	2	1	no	—
54	#14556553	—	2	1	no	—
55	#14556551	—	2	2	no	—
56	#14556550	C₂³	1	2	yes	—
57	#14556548	C₂×C₄	1	2	yes	—
58	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
59	C₄²×D₇₃	C₂×C₄	1	2	yes	—
60	#14556527	—	2	1	no	—
61	#14556527	—	2	1	no	—
62	#14556525	—	2	1	no	—
63	#14556523	—	2	1	no	—
64	#14556509	—	2	2	no	—
65	#14556507	—	2	2	no	—
66	#14556505	—	2	4	no	—
67	#14556502	—	2	2	no	—
68	#14556502	C₂×C₄	1	2	yes	—
69	#14556501	—	2	2	no	—
70	#14556501	C₂×C₄	1	1	yes	characteristic
71	#14556501	C₂×C₄	1	2	yes	—
72	#14556501	C₂×C₄	1	1	yes	characteristic
73	#14556501	C₂×C₄	1	1	yes	characteristic
74	#14556501	C₂×C₄	1	1	yes	characteristic
75	C₄×#10596831	—	2	2	no	—
76	C₄×#10596831	C₂×C₄	1	2	yes	—
77	C₈×Dic₇₃	—	2	2	no	—
78	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
79	C₂×M₁₆×C₇₃	C₂³	1	2	yes	—
80	C₂×M₁₆×C₇₃	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
82	#117×C₇₃	C₂³	1	2	yes	—
83	#117×C₇₃	C₂³	1	1	yes	characteristic
84	#108×C₇₃	—	2	1	no	—
85	#108×C₇₃	—	2	1	no	—
86	#107×C₇₃	—	2	1	no	—
87	#106×C₇₃	—	2	1	no	—
88	#97×C₇₃	C₂×C₄	1	2	yes	—
89	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
90	#11924028	—	2	1	no	—
91	#11924028	—	2	1	no	—
92	#11924028	—	2	1	no	—
93	#11924028	—	2	1	no	—
94	C₂×Dic₁₄₆	#41	1	1	yes	characteristic
95	C₂×D₂₉₂	#41	1	1	yes	characteristic
96	C₂×C₂²⋊D₇₃	#41	1	2	yes	—
97	C₂²×Dic₇₃	—	2	1	no	—
98	C₂×C₄×D₇₃	—	2	2	no	—
99	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
100	#11924012	—	2	1	no	—
101	#11924012	—	2	1	no	—
102	#11924012	—	2	2	no	—
103	#11924012	—	2	2	no	—
104	#11924011	—	2	2	no	—
105	#11924011	—	2	2	no	—
106	#11924011	—	2	4	no	—
107	C₂×#10596831	—	2	2	no	—
108	C₂×#10596831	C₂²×C₄	1	2	yes	—
109	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
110	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
111	C₂×#10596831	#41	1	2	yes	—
112	C₂×#10596831	#41	1	2	yes	—
113	C₈×D₇₃	—	2	2	no	—
114	C₈×D₇₃	—	2	2	no	—
115	C₈×D₇₃	—	2	4	no	—
116	C₄⋊Dic₇₃	—	2	2	no	—
117	#11924007	—	2	2	no	—
118	C₂²⋊Dic₇₃	—	2	2	no	—
119	#11924005	—	2	2	no	—
120	C₄×Dic₇₃	—	2	1	no	—
121	C₄×Dic₇₃	—	2	2	no	—
122	C₄×Dic₇₃	—	2	2	no	—
123	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
126	M₁₆×C₇₃	—	2	1	no	—
127	M₁₆×C₇₃	—	2	1	no	—
128	M₁₆×C₇₃	—	2	2	no	—
129	M₁₆×C₇₃	—	2	2	no	—
130	C₂×C₈×C₇₃	—	2	2	no	—
131	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
132	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
135	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
136	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
137	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
138	C₄⋊C₄×C₇₃	—	2	1	no	—
139	C₄⋊C₄×C₇₃	—	2	2	no	—
140	C₂²⋊C₄×C₇₃	—	2	1	no	—
141	C₂²⋊C₄×C₇₃	—	2	2	no	—
142	C₄²×C₇₃	—	2	2	no	—
143	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
144	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
145	C₂²×D₇₃	#117	1	2	yes	—
146	C₂×C₄×C₇₃	—	2	2	no	—
147	C₂×C₄×C₇₃	—	4	2	no	—
148	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
154	C₂²⋊D₇₃	—	2	2	no	—
155	C₂²⋊D₇₃	—	2	2	no	—
156	C₂×Dic₇₃	—	2	1	no	—
157	C₂×Dic₇₃	—	2	1	no	—
158	C₂×Dic₇₃	—	4	2	no	—
159	C₂×Dic₇₃	—	8	1	no	—
160	C₂×Dic₇₃	#117	1	2	yes	—
161	D₂₉₂	—	2	1	no	—
162	D₂₉₂	—	2	1	no	—
163	C₄×D₇₃	—	2	2	no	—
164	C₄×D₇₃	—	2	2	no	—
165	C₄×D₇₃	—	8	2	no	—
166	Dic₁₄₆	—	2	1	no	—
167	Dic₁₄₆	—	2	1	no	—
168	C₈×C₇₃	—	2	2	no	—
169	C₈×C₇₃	—	4	2	no	—
170	C₈×C₇₃	#117	1	1	yes	characteristic
171	C₈×C₇₃	#117	1	1	yes	characteristic
172	C₈×C₇₃	#117	1	1	yes	characteristic
173	C₈×C₇₃	#117	1	1	yes	characteristic
174	#10596831	—	2	2	no	—
175	#10596831	—	4	2	no	—
176	#10596831	#117	1	2	yes	—
177	#10596831	#117	1	2	yes	—
178	C₂²×C₇₃	—	4	1	no	—
179	C₂²×C₇₃	C₂×#117	1	1	yes	characteristic
180	C₂²×C₇₃	#529	1	1	yes	characteristic
181	C₂²×C₇₃	#529	1	1	yes	characteristic
182	C₂×D₇₃	—	2	2	no	—
183	C₂×D₇₃	—	4	2	no	—
184	C₄×C₇₃	—	4	2	no	—
185	C₄×C₇₃	—	8	2	no	—
186	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
187	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
188	C₄×C₇₃	#529	1	1	yes	characteristic
189	C₄×C₇₃	#529	1	1	yes	characteristic
190	Dic₇₃	—	2	2	no	—
191	Dic₇₃	—	4	1	no	—
192	Dic₇₃	—	4	1	no	—
193	Dic₇₃	—	8	2	no	—
194	#20788	—	73	1	no	maximal, Sylow
195	C₂×C₇₃	—	2	1	no	—
196	C₂×C₇₃	—	2	1	no	—
197	C₂×C₇₃	—	2	1	no	—
198	C₂×C₇₃	#2283	1	1	yes	characteristic
199	D₇₃	—	4	2	no	—
200	#2875	—	73	1	no	—
201	#1841	—	73	2	no	—
202	#1833	—	73	1	no	—
203	#1833	D₇₃	1	1	yes	characteristic, complemented
204	#1827	—	73	2	no	—
205	#1813	—	73	1	no	—
206	#1812	—	73	1	no	—
207	#1773	—	73	1	no	—
208	#1773	—	73	1	no	—
209	#1757	—	73	2	no	—
210	#1747	—	73	2	no	—
211	C₇₃	#20788	1	1	yes	characteristic, complemented, Sylow
212	C₂×#131	—	73	1	no	—
213	C₂×#131	—	73	1	no	—
214	C₄×#41	—	73	1	no	—
215	C₂×#108	—	73	1	no	—
216	C₂×#108	C₂×D₇₃	1	1	yes	characteristic
217	#427	—	73	1	no	—
218	#427	C₂×D₇₃	1	1	yes	characteristic
219	#413	—	73	2	no	—
220	#410	—	73	2	no	—
221	#405	—	73	1	no	—
222	#405	—	73	2	no	—
223	#405	—	73	2	no	—
224	#405	—	73	2	no	—
225	#405	—	73	2	no	—
226	#405	C₂×D₇₃	1	1	yes	characteristic
227	C₄×M₁₆	—	73	1	no	—
228	C₄×M₁₆	—	73	2	no	—
229	C₂×#97	—	73	1	no	—
230	C₂×#97	—	73	2	no	—
231	#344	—	73	2	no	—
232	#344	C₂×D₇₃	1	2	yes	—
233	#339	—	73	2	no	—
234	#339	—	73	2	no	—
235	#338	—	73	2	no	—
236	H(3,ℤ/4)	—	73	2	no	—
237	H(3,ℤ/4)	—	73	2	no	—
238	H(3,ℤ/4)	C₂×D₇₃	1	2	yes	—
239	C₂×#41	—	73	1	no	—
240	#131	—	146	2	no	—
241	#131	—	146	2	no	—
242	#131	—	146	1	no	—
243	#131	—	146	1	no	—
244	#131	—	146	1	no	—
245	#131	—	146	1	no	—
246	C₂×M₁₆	—	73	1	no	—
247	C₂×M₁₆	—	73	2	no	—
248	C₂×M₁₆	—	73	2	no	—
249	C₂×M₁₆	—	73	2	no	—
250	C₂×M₁₆	C₂²×D₇₃	1	2	yes	—
251	C₂×M₁₆	C₂²×D₇₃	1	1	yes	characteristic
252	C₂²×C₈	—	73	1	no	—
253	C₂²×C₈	—	73	2	no	—
254	C₂²×C₈	—	73	2	no	—
255	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
256	C₄×Q₈	—	146	1	no	—
257	C₄×D₄	—	146	2	no	—
258	C₄×D₄	—	146	1	no	—
259	#117	—	73	2	no	—
260	#117	—	73	2	no	—
261	#117	C₂²×D₇₃	1	2	yes	—
262	#117	C₂²×D₇₃	1	1	yes	characteristic
263	C₂×C₄²	—	73	1	no	—
264	C₂×C₄²	—	73	2	no	—
265	#108	—	2	1	no	—
266	#108	—	2	1	no	—
267	#108	—	146	1	no	—
268	#108	—	146	1	no	—
269	#107	—	2	1	no	—
270	#107	—	146	1	no	—
271	#106	—	2	1	no	—
272	#106	—	146	1	no	—
273	#101	—	146	2	no	—
274	#100	—	146	2	no	—
275	#99	—	146	4	no	—
276	#97	—	73	1	no	—
277	#97	—	73	1	no	—
278	#97	—	73	2	no	—
279	#97	—	73	1	no	—
280	#97	—	73	1	no	—
281	#97	—	73	2	no	—
282	#97	—	146	2	no	—
283	#97	—	146	2	no	—
284	#97	C₄×D₇₃	1	2	yes	—
285	C₄×C₈	—	73	2	no	—
286	C₄×C₈	—	73	2	no	—
287	C₄×C₈	—	146	2	no	—
288	C₄×C₈	—	146	2	no	—
289	C₄×C₈	C₄×D₇₃	1	2	yes	—
290	#41	—	146	1	no	—
291	#41	—	146	1	no	—
292	#41	—	146	1	no	—
293	#41	—	146	1	no	—
294	C₂×Q₈	—	73	1	no	—
295	C₂×D₄	—	73	2	no	—
296	C₂×D₄	—	73	1	no	—
297	C₂²×C₄	—	73	2	no	—
298	C₂²×C₄	—	146	2	no	—
299	C₂²×C₄	—	146	1	no	—
300	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
301	M₁₆	—	2	1	no	—
302	M₁₆	—	2	1	no	—
303	M₁₆	—	2	2	no	—
304	M₁₆	—	2	2	no	—
305	M₁₆	—	146	4	no	—
306	M₁₆	—	146	2	no	—
307	M₁₆	—	146	2	no	—
308	M₁₆	—	146	1	no	—
309	M₁₆	—	146	1	no	—
310	M₁₆	—	146	2	no	—
311	M₁₆	—	146	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	73	1	no	—
314	C₂×C₈	—	73	1	no	—
315	C₂×C₈	—	73	2	no	—
316	C₂×C₈	—	73	2	no	—
317	C₂×C₈	—	73	2	no	—
318	C₂×C₈	—	146	4	no	—
319	C₂×C₈	—	146	2	no	—
320	C₂×C₈	—	146	2	no	—
321	C₂×C₈	—	146	2	no	—
322	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
323	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
324	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
325	C₂×C₈	#11924027	1	1	yes	characteristic
326	C₂×C₈	#11924027	1	1	yes	characteristic
327	C₂×C₈	#11924029	1	1	yes	characteristic
328	C₂×C₈	#11924029	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	146	2	no	—
332	C₄⋊C₄	—	146	2	no	—
333	C₂²⋊C₄	—	2	1	no	—
334	C₂²⋊C₄	—	2	2	no	—
335	C₂²⋊C₄	—	146	2	no	—
336	C₂²⋊C₄	—	146	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	73	1	no	—
339	C₄²	—	73	1	no	—
340	C₄²	—	146	2	no	—
341	C₄²	—	146	2	no	—
342	C₄²	—	146	1	no	—
343	C₄²	C₂×C₄×D₇₃	1	2	yes	—
344	C₂³	—	73	2	no	—
345	C₂³	#41×D₇₃	1	1	yes	characteristic
346	Q₈	—	146	1	no	—
347	Q₈	—	146	1	no	—
348	D₄	—	146	2	no	—
349	D₄	—	146	2	no	—
350	D₄	—	146	1	no	—
351	D₄	—	146	1	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	292	2	no	—
355	C₂×C₄	—	584	2	no	—
356	C₂×C₄	—	584	1	no	—
357	C₂×C₄	—	73	2	no	—
358	C₂×C₄	—	146	1	no	—
359	C₂×C₄	—	146	1	no	—
360	C₂×C₄	—	146	2	no	—
361	C₂×C₄	—	146	2	no	—
362	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
363	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
364	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
365	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
366	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
367	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
368	C₈	—	2	2	no	—
369	C₈	—	4	2	no	—
370	C₈	—	292	2	no	—
371	C₈	—	73	2	no	—
372	C₈	—	73	2	no	—
373	C₈	—	146	2	no	—
374	C₈	#14556549	1	1	yes	characteristic
375	C₈	#14556549	1	1	yes	characteristic
376	C₈	#14556549	1	1	yes	characteristic
377	C₈	#14556549	1	1	yes	characteristic
378	C₂²	—	4	1	no	—
379	C₂²	—	292	2	no	—
380	C₂²	—	146	2	no	—
381	C₂²	#117×D₇₃	1	1	yes	characteristic
382	C₂²	#17989116	1	1	yes	characteristic
383	C₂²	#17989122	1	1	yes	characteristic
384	C₄	—	4	2	no	—
385	C₄	—	8	2	no	—
386	C₄	—	292	1	no	—
387	C₄	—	292	1	no	—
388	C₄	—	584	2	no	—
389	C₄	—	146	2	no	—
390	C₄	#117×D₇₃	1	1	yes	characteristic
391	C₄	C₂×#14556549	1	1	yes	characteristic
392	C₄	#17989116	1	1	yes	characteristic
393	C₄	#17989122	1	1	yes	characteristic
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	2	1	no	—
397	C₂	—	292	2	no	—
398	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	#20788
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. #22609818	C₂	semidirect
8. #22609817	C₂	semidirect
10. #22609685	C₂	semidirect
11. C₇₃×#1833	C₂	semidirect
20. #17988544	C₄	semidirect
23. #17988541	C₄	semidirect
25. C₄×#11924011	C₄	semidirect
27. #97×D₇₃	C₄	semidirect
203. #1833	D₇₃	semidirect
211. C₇₃	#20788	semidirect

Maximal subgroups

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

subgroup	index
#22618358	2
#22610079	2
#22610055	2
#22610033	2
#22609978	2
#22609974	2
#22609818	2
#22609817	2
#22609730	2
#22609685	2
C₇₃×#1833	2
#20788	73