Subgroups — #33084822

Order 18688.

994 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988611	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556550	C₂	1	1	yes	maximal, characteristic
3	#22609798	C₂	1	2	yes	maximal, complemented
4	C₈×C₄⋊Dic₇₃	C₂	1	2	yes	maximal, complemented
5	#22609786	C₂	1	2	yes	maximal, complemented
6	C₄⋊C₈×Dic₇₃	C₂	1	2	yes	maximal, complemented
7	#22609717	C₂	1	4	yes	maximal
8	C₂×#431×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11924012	C₂²	1	2	yes	—
10	C₂×#14556550	C₄	1	2	yes	—
11	#17988611	C₄	1	8	yes	complemented
12	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
13	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
14	C₂×#14556501	C₂²	1	4	yes	—
15	C₂×C₈×Dic₇₃	C₂²	1	4	yes	—
16	C₄×C₄⋊Dic₇₃	C₄	1	2	yes	—
17	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
19	C₄×C₂²⋊Dic₇₃	C₄	1	2	yes	—
20	C₄×C₂²⋊Dic₇₃	C₂²	1	2	yes	—
21	#17988433	C₄	1	2	yes	—
22	C₄²×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	C₂×#11924012	—	4	2	no	—
32	C₂×#11924012	—	4	2	no	—
33	C₂×#11924012	C₂×C₄	1	8	yes	—
34	C₂²×#10596831	C₂³	1	4	yes	—
35	#14556550	—	2	8	no	—
36	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
38	C₂×C₂²⋊Dic₇₃	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
41	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
42	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
43	C₄⋊#10596831	C₂×C₄	1	8	yes	—
44	C₄⋊#10596831	C₂×C₄	1	8	yes	—
45	#14556501	—	4	4	no	—
46	C₈×Dic₇₃	—	4	4	no	—
47	#14556498	C₂×C₄	1	2	yes	—
48	#14556498	C₂×C₄	1	2	yes	—
49	#14556498	C₂×C₄	1	4	yes	—
50	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
58	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
59	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
62	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
63	C₂²⋊C₈×C₇₃	C₂×C₄	1	8	yes	—
64	C₄×C₈×C₇₃	C₂×C₄	1	8	yes	—
65	C₂²×Dic₇₃	C₂²×C₄	1	4	yes	—
66	C₂²×Dic₇₃	C₂²×C₄	1	4	yes	—
67	#11924012	—	4	8	no	—
68	C₂×#10596831	—	2	4	no	—
69	C₂×#10596831	—	2	4	no	—
70	C₂×#10596831	C₂²×C₄	1	8	yes	—
71	C₄⋊Dic₇₃	—	2	4	no	—
72	C₄⋊Dic₇₃	M₁₆	1	8	yes	complemented
73	C₂²⋊Dic₇₃	—	2	8	no	—
74	C₄×Dic₇₃	—	2	8	no	—
75	C₄×Dic₇₃	—	2	2	no	—
76	C₄×Dic₇₃	—	2	2	no	—
77	C₄×Dic₇₃	—	2	2	no	—
78	C₄×Dic₇₃	—	2	2	no	—
79	C₄×Dic₇₃	—	2	8	no	—
80	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	—	2	4	no	—
82	C₂²×C₄×C₇₃	—	2	4	no	—
83	C₂²×C₄×C₇₃	—	2	1	no	—
84	C₂²×C₄×C₇₃	—	2	1	no	—
85	C₂²×C₄×C₇₃	—	2	1	no	—
86	C₂²×C₄×C₇₃	—	2	1	no	—
87	C₂²×C₄×C₇₃	C₄²	1	4	yes	—
88	C₂²×C₄×C₇₃	C₄²	1	4	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	1	yes	characteristic
92	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
97	C₂×C₈×C₇₃	—	2	2	no	—
98	C₂×C₈×C₇₃	—	2	2	no	—
99	C₂×C₈×C₇₃	—	2	2	no	—
100	C₂×C₈×C₇₃	—	2	2	no	—
101	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
102	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
103	C₄²×C₇₃	C₄²	1	4	yes	—
104	C₄²×C₇₃	C₄²	1	4	yes	—
105	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
107	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
108	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
109	C₂³×C₇₃	—	2	1	no	—
110	C₂³×C₇₃	—	2	1	no	—
111	C₂³×C₇₃	—	2	2	no	—
112	C₂³×C₇₃	—	2	2	no	—
113	C₂³×C₇₃	C₂×C₄²	1	2	yes	—
114	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	—	2	1	no	—
116	C₂×C₄×C₇₃	—	2	1	no	—
117	C₂×C₄×C₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	2	no	—
119	C₂×C₄×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	—	2	2	no	—
121	C₂×C₄×C₇₃	—	2	2	no	—
122	C₂×C₄×C₇₃	—	2	2	no	—
123	C₂×C₄×C₇₃	—	2	8	no	—
124	C₂×C₄×C₇₃	—	2	8	no	—
125	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
130	C₂×C₄×C₇₃	C₂×M₁₆	1	4	yes	—
131	C₂×C₄×C₇₃	#131	1	4	yes	—
132	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
137	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
138	C₂×Dic₇₃	—	2	4	no	—
139	C₂×Dic₇₃	—	2	4	no	—
140	C₂×Dic₇₃	—	2	4	no	—
141	C₂×Dic₇₃	—	2	4	no	—
142	C₂×Dic₇₃	—	2	4	no	—
143	C₂×Dic₇₃	C₂×M₁₆	1	8	yes	—
144	C₈×C₇₃	—	2	4	no	—
145	C₈×C₇₃	—	2	4	no	—
146	#10596831	—	2	8	no	—
147	C₂²×C₇₃	—	2	2	no	—
148	C₂²×C₇₃	—	2	2	no	—
149	C₂²×C₇₃	—	2	2	no	—
150	C₂²×C₇₃	—	4	2	no	—
151	C₂²×C₇₃	—	4	2	no	—
152	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
153	C₂²×C₇₃	#520	1	1	yes	characteristic
154	C₂²×C₇₃	#520	1	1	yes	characteristic
155	C₂²×C₇₃	C₂²×M₁₆	1	2	yes	—
156	C₂²×C₇₃	C₂×#131	1	2	yes	—
157	C₄×C₇₃	—	2	2	no	—
158	C₄×C₇₃	—	2	2	no	—
159	C₄×C₇₃	—	2	2	no	—
160	C₄×C₇₃	C₄×M₁₆	1	4	yes	—
161	C₄×C₇₃	#405	1	4	yes	—
162	C₄×C₇₃	#518	1	1	yes	characteristic
163	C₄×C₇₃	#519	1	1	yes	characteristic
164	C₄×C₇₃	#520	1	1	yes	characteristic
165	C₄×C₇₃	#520	1	1	yes	characteristic
166	Dic₇₃	—	2	8	no	—
167	Dic₇₃	—	4	4	no	—
168	#20550	—	73	1	no	maximal, Sylow
169	C₂×C₇₃	—	4	2	no	—
170	C₂×C₇₃	#2274	1	1	yes	characteristic
171	C₂×C₇₃	C₂×C₄×M₁₆	1	1	yes	characteristic
172	C₂×C₇₃	C₂×#405	1	1	yes	characteristic
173	C₂×C₇₃	#2990	1	2	yes	—
174	C₂×C₇₃	#2991	1	2	yes	—
175	C₂×#431	D₇₃	1	1	yes	characteristic
176	C₂×#423	—	73	1	no	—
177	C₄×#117	—	73	1	no	—
178	#1770	—	73	2	no	—
179	C₈×C₄⋊C₄	—	73	2	no	—
180	#1767	—	73	2	no	—
181	C₄×C₄⋊C₈	—	73	2	no	—
182	#1753	—	73	4	no	—
183	C₇₃	#20550	1	1	yes	characteristic, complemented, Sylow
184	C₂²×M₁₆	—	73	2	no	—
185	C₂×#117	—	73	2	no	—
186	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
187	#431	Dic₇₃	1	8	yes	complemented
188	#423	—	73	8	no	—
189	C₂×C₄⋊C₈	—	73	2	no	—
190	C₂×C₄⋊C₈	—	73	2	no	—
191	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
192	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
193	C₂×C₂²⋊C₈	C₂×D₇₃	1	2	yes	—
194	C₂×#97	—	73	4	no	—
195	C₂×C₄×C₈	—	73	4	no	—
196	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
197	C₄×C₄⋊C₄	—	73	2	no	—
198	C₄×C₄⋊C₄	—	73	1	no	—
199	C₄×C₄⋊C₄	—	73	1	no	—
200	C₄×C₂²⋊C₄	—	73	2	no	—
201	C₄×C₂²⋊C₄	—	73	2	no	—
202	#376	—	73	2	no	—
203	C₄³	—	73	2	no	—
204	#336	—	73	4	no	—
205	#336	—	73	4	no	—
206	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
207	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
208	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
209	C₂×M₁₆	—	292	2	no	—
210	C₂×M₁₆	—	292	2	no	—
211	C₂×M₁₆	—	73	8	no	—
212	C₂²×C₈	—	73	4	no	—
213	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
214	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
215	#117	—	146	8	no	—
216	C₂×C₄⋊C₄	—	73	2	no	—
217	C₂×C₄⋊C₄	—	73	2	no	—
218	C₂×C₂²⋊C₄	—	73	4	no	—
219	C₂×C₄²	—	73	4	no	—
220	C₂×C₄²	—	73	4	no	—
221	C₂×C₄²	—	73	2	no	—
222	C₂×C₄²	—	73	2	no	—
223	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
224	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
225	C₂×C₄²	C₄×D₇₃	1	4	yes	—
226	C₂×C₄²	C₂×Dic₇₃	1	4	yes	—
227	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
229	C₄⋊C₈	—	73	8	no	—
230	C₄⋊C₈	—	73	8	no	—
231	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
232	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
233	C₂²⋊C₈	C₂×Dic₇₃	1	8	yes	—
234	#97	—	292	4	no	—
235	C₄×C₈	—	292	4	no	—
236	C₄×C₈	C₂×Dic₇₃	1	8	yes	—
237	#95	—	73	2	no	—
238	#95	—	73	4	no	—
239	#95	—	73	2	no	—
240	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	73	4	no	—
248	C₂²×C₄	—	73	4	no	—
249	C₂²×C₄	C₄×Dic₇₃	1	4	yes	—
250	C₂²×C₄	C₄×Dic₇₃	1	4	yes	—
251	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
255	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
257	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
258	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
259	M₁₆	—	292	8	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	73	8	no	—
265	C₂×C₈	—	146	4	no	—
266	C₂×C₈	—	146	4	no	—
267	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
268	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
269	C₄⋊C₄	—	73	8	no	—
270	C₄⋊C₄	—	146	4	no	—
271	C₂²⋊C₄	—	146	8	no	—
272	C₄²	—	146	8	no	—
273	C₄²	—	146	8	no	—
274	C₄²	—	146	2	no	—
275	C₄²	—	146	2	no	—
276	C₄²	—	146	2	no	—
277	C₄²	—	146	2	no	—
278	C₄²	C₄×Dic₇₃	1	4	yes	—
279	C₄²	C₄×Dic₇₃	1	4	yes	—
280	C₄²	C₂×C₄×D₇₃	1	2	yes	—
281	C₄²	C₂×C₄×D₇₃	1	2	yes	—
282	C₄²	C₂²×Dic₇₃	1	2	yes	—
283	C₄²	C₂²×Dic₇₃	1	2	yes	—
284	C₂³	—	2	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	2	no	—
287	C₂³	—	2	2	no	—
288	C₂³	C₂×C₄×Dic₇₃	1	2	yes	—
289	C₂³	C₂²×C₄×D₇₃	1	1	yes	characteristic
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	8	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	73	8	no	—
301	C₂×C₄	—	146	4	no	—
302	C₂×C₄	—	146	4	no	—
303	C₂×C₄	—	146	4	no	—
304	C₂×C₄	—	146	4	no	—
305	C₂×C₄	—	146	4	no	—
306	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
307	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
308	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
309	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
310	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
311	C₂×C₄	M₁₆×D₇₃	1	4	yes	—
312	C₂×C₄	#14556597	1	4	yes	—
313	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
314	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
315	C₂×C₄	#14556637	1	1	yes	characteristic
316	C₂×C₄	#14556637	1	1	yes	characteristic
317	C₂×C₄	#14556638	1	1	yes	characteristic
318	C₂×C₄	#14556638	1	1	yes	characteristic
319	C₈	—	2	4	no	—
320	C₈	—	2	4	no	—
321	C₈	—	146	8	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
328	C₂²	#17989067	1	1	yes	characteristic
329	C₂²	#17989067	1	1	yes	characteristic
330	C₂²	C₂×M₁₆×D₇₃	1	2	yes	—
331	C₂²	C₂×#14556597	1	2	yes	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	292	4	no	—
336	C₄	—	146	8	no	—
337	C₄	M₁₆×Dic₇₃	1	4	yes	—
338	C₄	#17988542	1	4	yes	—
339	C₄	#17989061	1	1	yes	characteristic
340	C₄	#17989064	1	1	yes	characteristic
341	C₄	#17989070	1	1	yes	characteristic
342	C₄	#17989070	1	1	yes	characteristic
343	C₂	—	4	2	no	—
344	C₂	#22612287	1	1	yes	characteristic
345	C₂	C₂×M₁₆×Dic₇₃	1	1	yes	characteristic
346	C₂	C₂×#17988542	1	1	yes	characteristic
347	C₂	#22619052	1	2	yes	—
348	C₂	#22619060	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	#20550
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. #22609798	C₂	semidirect
4. C₈×C₄⋊Dic₇₃	C₂	semidirect
5. #22609786	C₂	semidirect
6. C₄⋊C₈×Dic₇₃	C₂	semidirect
11. #17988611	C₄	semidirect
26. #431×C₇₃	C₄	semidirect
72. C₄⋊Dic₇₃	M₁₆	semidirect
183. C₇₃	#20550	semidirect
187. #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×#17988611	2
C₄×#14556550	2
#22609798	2
C₈×C₄⋊Dic₇₃	2
#22609786	2
C₄⋊C₈×Dic₇₃	2
#22609717	2
C₂×#431×C₇₃	2
#20550	73