Subgroups — #33084743

Order 18688.

922 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988640	C₂	1	1	yes	maximal, characteristic
2	#22612338	C₂	1	1	yes	maximal, characteristic
3	#22609801	C₂	1	2	yes	maximal, complemented
4	#22609798	C₂	1	2	yes	maximal, complemented
5	#22609796	C₂	1	2	yes	maximal, complemented
6	C₈×C₄⋊Dic₇₃	C₂	1	2	yes	maximal, complemented
7	#22609699	C₂	1	4	yes	maximal
8	C₂×#431×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556550	C₂²	1	2	yes	—
10	#17988640	C₄	1	8	yes	complemented
11	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊#10596831	C₂²	1	2	yes	—
14	C₂×#14556501	C₂²	1	4	yes	—
15	C₂×C₄×#10596831	C₂²	1	2	yes	—
16	C₂×C₈×Dic₇₃	C₂²	1	4	yes	—
17	#17988457	C₄	1	2	yes	—
18	#17988453	C₄	1	2	yes	—
19	#17988447	C₄	1	2	yes	—
20	#17988447	C₄	1	2	yes	—
21	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
22	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
23	#17988341	C₂²	1	4	yes	—
24	#17988341	C₂²	1	4	yes	—
25	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
26	#431×C₇₃	C₄	1	8	yes	complemented
27	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₈×C₇₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
31	C₂²×#10596831	C₂³	1	2	yes	—
32	C₂²×#10596831	C₂³	1	2	yes	—
33	#14556550	—	4	2	no	—
34	#14556550	—	4	2	no	—
35	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊Dic₇₃	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₇₃	C₂³	1	4	yes	—
40	C₄⋊#10596831	C₂×C₄	1	4	yes	—
41	C₄⋊#10596831	C₂×C₄	1	4	yes	—
42	C₂²⋊#10596831	C₂×C₄	1	8	yes	—
43	#14556501	—	4	4	no	—
44	C₄×#10596831	C₂×C₄	1	8	yes	—
45	C₈×Dic₇₃	—	4	4	no	—
46	#14556498	C₂×C₄	1	4	yes	—
47	#14556498	C₂×C₄	1	4	yes	—
48	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
56	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
57	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
60	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
61	C₂²⋊C₈×C₇₃	C₂×C₄	1	8	yes	—
62	C₄×C₈×C₇₃	C₂×C₄	1	8	yes	—
63	C₂²×Dic₇₃	C₂²×C₄	1	8	yes	—
64	C₂×#10596831	—	2	2	no	—
65	C₂×#10596831	—	2	2	no	—
66	C₂×#10596831	—	2	2	no	—
67	C₂×#10596831	—	2	2	no	—
68	C₂×#10596831	C₂²×C₄	1	4	yes	—
69	C₂×#10596831	C₂²×C₄	1	4	yes	—
70	C₄⋊Dic₇₃	—	2	8	no	—
71	C₄⋊Dic₇₃	—	2	8	no	—
72	C₄⋊Dic₇₃	—	4	4	no	—
73	C₂²⋊Dic₇₃	—	4	4	no	—
74	C₄×Dic₇₃	—	2	4	no	—
75	C₄×Dic₇₃	—	2	4	no	—
76	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	—	2	4	no	—
78	C₂²×C₄×C₇₃	—	2	4	no	—
79	C₂²×C₄×C₇₃	—	2	1	no	—
80	C₂²×C₄×C₇₃	—	2	1	no	—
81	C₂²×C₄×C₇₃	—	2	1	no	—
82	C₂²×C₄×C₇₃	—	2	1	no	—
83	C₂²×C₄×C₇₃	C₄²	1	4	yes	—
84	C₂²×C₄×C₇₃	C₄²	1	4	yes	—
85	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
93	C₂×C₈×C₇₃	—	2	2	no	—
94	C₂×C₈×C₇₃	—	2	2	no	—
95	C₂×C₈×C₇₃	—	2	2	no	—
96	C₂×C₈×C₇₃	—	2	2	no	—
97	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
98	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
99	C₄²×C₇₃	C₄²	1	4	yes	—
100	C₄²×C₇₃	C₄²	1	4	yes	—
101	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
103	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
105	C₂³×C₇₃	—	2	1	no	—
106	C₂³×C₇₃	—	2	1	no	—
107	C₂³×C₇₃	—	2	2	no	—
108	C₂³×C₇₃	—	2	2	no	—
109	C₂³×C₇₃	C₂×C₄²	1	2	yes	—
110	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₃	—	2	1	no	—
112	C₂×C₄×C₇₃	—	2	1	no	—
113	C₂×C₄×C₇₃	—	2	2	no	—
114	C₂×C₄×C₇₃	—	2	2	no	—
115	C₂×C₄×C₇₃	—	2	2	no	—
116	C₂×C₄×C₇₃	—	2	2	no	—
117	C₂×C₄×C₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	2	no	—
119	C₂×C₄×C₇₃	—	2	8	no	—
120	C₂×C₄×C₇₃	—	2	8	no	—
121	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₇₃	#131	1	4	yes	—
127	C₂×C₄×C₇₃	#131	1	4	yes	—
128	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
134	C₂×Dic₇₃	—	2	8	no	—
135	C₂×Dic₇₃	—	2	8	no	—
136	C₂×Dic₇₃	—	2	8	no	—
137	C₈×C₇₃	—	2	4	no	—
138	C₈×C₇₃	—	2	4	no	—
139	#10596831	—	2	4	no	—
140	#10596831	—	2	4	no	—
141	C₂²×C₇₃	—	2	2	no	—
142	C₂²×C₇₃	—	2	2	no	—
143	C₂²×C₇₃	—	2	2	no	—
144	C₂²×C₇₃	—	4	2	no	—
145	C₂²×C₇₃	—	4	2	no	—
146	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
147	C₂²×C₇₃	#518	1	1	yes	characteristic
148	C₂²×C₇₃	#519	1	1	yes	characteristic
149	C₂²×C₇₃	C₂×#131	1	2	yes	—
150	C₂²×C₇₃	C₂×#131	1	2	yes	—
151	C₄×C₇₃	—	2	2	no	—
152	C₄×C₇₃	—	2	2	no	—
153	C₄×C₇₃	—	2	2	no	—
154	C₄×C₇₃	#405	1	4	yes	—
155	C₄×C₇₃	#405	1	4	yes	—
156	C₄×C₇₃	#520	1	1	yes	characteristic
157	C₄×C₇₃	#520	1	1	yes	characteristic
158	C₄×C₇₃	#520	1	1	yes	characteristic
159	C₄×C₇₃	#520	1	1	yes	characteristic
160	Dic₇₃	—	4	8	no	—
161	#20538	—	73	1	no	maximal, Sylow
162	C₂×C₇₃	—	4	2	no	—
163	C₂×C₇₃	#2274	1	1	yes	characteristic
164	C₂×C₇₃	C₂×#405	1	1	yes	characteristic
165	C₂×C₇₃	C₂×#405	1	1	yes	characteristic
166	C₂×C₇₃	#2979	1	2	yes	—
167	C₂×C₇₃	#2986	1	2	yes	—
168	C₂×#431	—	73	1	no	—
169	C₂×#431	D₇₃	1	1	yes	characteristic
170	#2285	—	73	1	no	—
171	#1770	—	73	2	no	—
172	#1770	—	73	2	no	—
173	C₈×C₄⋊C₄	—	73	2	no	—
174	C₈×C₄⋊C₄	—	73	2	no	—
175	#1750	—	73	4	no	—
176	C₇₃	#20538	1	1	yes	characteristic, complemented, Sylow
177	C₂×#117	—	73	2	no	—
178	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
179	#431	—	73	8	no	—
180	#431	Dic₇₃	1	8	yes	complemented
181	C₂×C₄⋊C₈	—	73	1	no	—
182	C₂×C₄⋊C₈	—	73	1	no	—
183	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
184	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
185	C₂×C₂²⋊C₈	—	73	2	no	—
186	C₂×C₂²⋊C₈	C₂×D₇₃	1	2	yes	—
187	C₂×#97	—	73	4	no	—
188	C₂×C₄×C₈	—	73	4	no	—
189	C₂×C₄×C₈	—	73	2	no	—
190	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
191	#384	—	73	2	no	—
192	#382	—	73	2	no	—
193	#380	—	73	2	no	—
194	#380	—	73	2	no	—
195	C₄×C₄⋊C₄	—	73	2	no	—
196	C₄×C₄⋊C₄	—	73	2	no	—
197	#336	—	73	4	no	—
198	#336	—	73	4	no	—
199	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
200	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
201	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
202	C₂²×C₈	—	73	2	no	—
203	C₂²×C₈	—	73	2	no	—
204	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
205	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
206	#117	—	292	2	no	—
207	#117	—	292	2	no	—
208	C₂×C₄⋊C₄	—	73	4	no	—
209	C₂×C₄⋊C₄	—	73	4	no	—
210	C₂×C₄⋊C₄	—	73	4	no	—
211	C₂×C₂²⋊C₄	—	73	4	no	—
212	C₂×C₄²	—	73	4	no	—
213	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
214	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
215	C₂×C₄²	C₄×D₇₃	1	4	yes	—
216	C₂×C₄²	C₂×Dic₇₃	1	4	yes	—
217	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
218	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
219	C₄⋊C₈	—	73	4	no	—
220	C₄⋊C₈	—	73	4	no	—
221	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
222	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
223	C₂²⋊C₈	—	73	8	no	—
224	C₂²⋊C₈	C₂×Dic₇₃	1	8	yes	—
225	#97	—	292	4	no	—
226	C₄×C₈	—	292	4	no	—
227	C₄×C₈	—	73	8	no	—
228	C₄×C₈	C₂×Dic₇₃	1	8	yes	—
229	#95	—	73	4	no	—
230	#95	—	73	4	no	—
231	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	73	8	no	—
239	C₂²×C₄	C₄×Dic₇₃	1	4	yes	—
240	C₂²×C₄	C₄×Dic₇₃	1	4	yes	—
241	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
245	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
247	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
248	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	73	4	no	—
254	C₂×C₈	—	73	4	no	—
255	C₂×C₈	—	146	2	no	—
256	C₂×C₈	—	146	2	no	—
257	C₂×C₈	—	146	2	no	—
258	C₂×C₈	—	146	2	no	—
259	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
260	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
261	C₄⋊C₄	—	292	4	no	—
262	C₄⋊C₄	—	146	8	no	—
263	C₄⋊C₄	—	146	8	no	—
264	C₂²⋊C₄	—	292	4	no	—
265	C₄²	—	146	4	no	—
266	C₄²	—	146	4	no	—
267	C₄²	C₄×Dic₇₃	1	4	yes	—
268	C₄²	C₄×Dic₇₃	1	4	yes	—
269	C₄²	C₂×C₄×D₇₃	1	2	yes	—
270	C₄²	C₂×C₄×D₇₃	1	2	yes	—
271	C₄²	C₂²×Dic₇₃	1	2	yes	—
272	C₄²	C₂²×Dic₇₃	1	2	yes	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	C₂×C₄×Dic₇₃	1	2	yes	—
278	C₂³	C₂²×C₄×D₇₃	1	1	yes	characteristic
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	146	8	no	—
290	C₂×C₄	—	146	8	no	—
291	C₂×C₄	—	146	8	no	—
292	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
293	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
294	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
295	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
296	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
297	C₂×C₄	#14556597	1	4	yes	—
298	C₂×C₄	#14556599	1	4	yes	—
299	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
300	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
301	C₂×C₄	#14556637	1	1	yes	characteristic
302	C₂×C₄	#14556637	1	1	yes	characteristic
303	C₂×C₄	#14556638	1	1	yes	characteristic
304	C₂×C₄	#14556638	1	1	yes	characteristic
305	C₈	—	2	4	no	—
306	C₈	—	2	4	no	—
307	C₈	—	146	4	no	—
308	C₈	—	146	4	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
315	C₂²	#17989061	1	1	yes	characteristic
316	C₂²	#17989064	1	1	yes	characteristic
317	C₂²	C₂×#14556597	1	2	yes	—
318	C₂²	C₂×#14556599	1	2	yes	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	292	8	no	—
323	C₄	#17988542	1	4	yes	—
324	C₄	#17988545	1	4	yes	—
325	C₄	#17989067	1	1	yes	characteristic
326	C₄	#17989067	1	1	yes	characteristic
327	C₄	#17989070	1	1	yes	characteristic
328	C₄	#17989070	1	1	yes	characteristic
329	C₂	—	4	2	no	—
330	C₂	#22612287	1	1	yes	characteristic
331	C₂	C₂×#17988542	1	1	yes	characteristic
332	C₂	C₂×#17988545	1	1	yes	characteristic
333	C₂	#22618997	1	2	yes	—
334	C₂	#22619032	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₃
commutator	C₂²×C₇₃	C₂²×C₄²
Fitting	C₂×#431×C₇₃	C₂
socle	C₂³×C₇₃	C₂³×C₄
prime radical	C₂⁴×C₇₃	C₂²×C₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#431	D₇₃
hypocenter	C₇₃	#20538
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #22609801	C₂	semidirect
4. #22609798	C₂	semidirect
5. #22609796	C₂	semidirect
6. C₈×C₄⋊Dic₇₃	C₂	semidirect
10. #17988640	C₄	semidirect
26. #431×C₇₃	C₄	semidirect
176. C₇₃	#20538	semidirect
180. #431	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988640	2
#22612338	2
#22609801	2
#22609798	2
#22609796	2
C₈×C₄⋊Dic₇₃	2
#22609699	2
C₂×#431×C₇₃	2
#20538	73