Subgroups — #33001902

Order 18688.

566 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×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11924003	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17988366	C₂	1	2	yes	maximal
4	#22608008	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₇₃	C₄	1	1	yes	characteristic
9	C₂²×#11924003	C₂²	1	2	yes	—
10	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×D₇₃	—	2	2	no	—
14	C₄×C₈×D₇₃	—	2	2	no	—
15	C₄⋊#11924003	—	2	2	no	—
16	#17988366	—	2	4	no	—
17	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₁₆×C₇₃	—	2	2	no	—
20	C₂²×C₄×D₇₃	C₈	1	1	yes	characteristic
21	C₂²×C₄×D₇₃	C₈	1	1	yes	characteristic
22	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
24	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₇₃	—	2	2	no	—
26	C₂×C₈×D₇₃	—	2	2	no	—
27	C₂×C₈×D₇₃	—	2	2	no	—
28	C₂×C₈×D₇₃	—	2	1	no	—
29	C₂×C₈×D₇₃	—	2	1	no	—
30	C₂×C₈×D₇₃	—	2	1	no	—
31	C₂×C₈×D₇₃	—	2	1	no	—
32	C₂×C₈×D₇₃	D₄	1	2	yes	—
33	C₂×C₈×D₇₃	Q₈	1	2	yes	—
34	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
35	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
36	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×D₇₃	—	2	2	no	—
38	C₄²×D₇₃	—	2	2	no	—
39	C₂×#11924003	—	2	2	no	—
40	C₂×#11924003	—	2	2	no	—
41	C₂×#11924003	—	2	2	no	—
42	C₄×#10596831	—	2	2	no	—
43	C₈×Dic₇₃	—	2	2	no	—
44	C₈×Dic₇₃	—	2	1	no	—
45	C₈×Dic₇₃	—	2	1	no	—
46	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₇₃	—	2	2	no	—
50	C₂×C₁₆×C₇₃	—	2	2	no	—
51	C₂×C₁₆×C₇₃	—	2	2	no	—
52	C₄×C₈×C₇₃	D₄	1	2	yes	—
53	C₄×C₈×C₇₃	D₄	1	2	yes	—
54	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
58	C₂×C₄×D₇₃	—	2	2	no	—
59	C₂×C₄×D₇₃	—	2	2	no	—
60	C₂×C₄×D₇₃	—	2	2	no	—
61	C₂×C₄×D₇₃	—	2	2	no	—
62	C₂×C₄×D₇₃	—	2	1	no	—
63	C₂×C₄×D₇₃	—	2	1	no	—
64	C₂×C₄×D₇₃	—	2	1	no	—
65	C₂×C₄×D₇₃	—	2	1	no	—
66	C₂×C₄×D₇₃	—	2	2	no	—
67	C₂×C₄×D₇₃	C₁₆	1	4	yes	complemented
68	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
70	C₂×#10596831	—	2	1	no	—
71	C₂×#10596831	—	2	1	no	—
72	C₂×#10596831	—	2	1	no	—
73	C₂×#10596831	—	2	1	no	—
74	C₂×#10596831	C₄⋊C₄	1	2	yes	—
75	C₂×#10596831	M₁₆	1	2	yes	—
76	C₈×D₇₃	—	2	8	no	—
77	C₈×D₇₃	—	2	2	no	—
78	C₈×D₇₃	—	2	2	no	—
79	C₈×D₇₃	—	2	2	no	—
80	C₈×D₇₃	—	2	2	no	—
81	C₄×Dic₇₃	—	2	2	no	—
82	C₄×Dic₇₃	—	2	2	no	—
83	C₄×Dic₇₃	—	2	1	no	—
84	C₄×Dic₇₃	—	2	1	no	—
85	#11924003	—	4	2	no	—
86	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₇₃	—	2	1	no	—
90	C₂×C₈×C₇₃	—	2	1	no	—
91	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
92	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
97	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
98	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
99	C₄²×C₇₃	M₁₆	1	2	yes	—
100	C₁₆×C₇₃	—	4	2	no	—
101	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
102	C₂²×D₇₃	—	2	1	no	—
103	C₂²×D₇₃	—	2	1	no	—
104	C₂²×D₇₃	—	2	1	no	—
105	C₂²×D₇₃	—	2	1	no	—
106	C₂²×D₇₃	C₄⋊C₈	1	2	yes	—
107	C₂²×D₇₃	C₂×C₁₆	1	2	yes	—
108	C₂²×D₇₃	M₃₂	1	2	yes	—
109	C₂×C₄×C₇₃	—	2	1	no	—
110	C₂×C₄×C₇₃	—	2	1	no	—
111	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
112	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
113	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
114	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
115	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂×Dic₇₃	—	2	1	no	—
122	C₂×Dic₇₃	—	2	1	no	—
123	C₂×Dic₇₃	—	2	1	no	—
124	C₂×Dic₇₃	—	2	1	no	—
125	C₂×Dic₇₃	—	2	1	no	—
126	C₂×Dic₇₃	—	2	1	no	—
127	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
128	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
129	C₂×Dic₇₃	M₃₂	1	2	yes	—
130	C₄×D₇₃	—	2	8	no	—
131	C₄×D₇₃	—	2	8	no	—
132	C₄×D₇₃	—	2	2	no	—
133	C₄×D₇₃	—	2	2	no	—
134	C₄×D₇₃	—	2	2	no	—
135	C₄×D₇₃	—	2	2	no	—
136	C₈×C₇₃	—	2	2	no	—
137	C₈×C₇₃	#121	1	1	yes	characteristic
138	C₈×C₇₃	#121	1	1	yes	characteristic
139	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
140	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
141	#10596831	—	2	2	no	—
142	#10596831	—	2	2	no	—
143	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
145	C₂²×C₇₃	#431	1	1	yes	characteristic
146	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
147	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
148	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
149	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
150	C₂×D₇₃	—	2	2	no	—
151	C₂×D₇₃	—	2	2	no	—
152	C₂×D₇₃	—	2	2	no	—
153	C₂×D₇₃	—	2	2	no	—
154	C₂×D₇₃	—	2	2	no	—
155	C₂×D₇₃	—	2	2	no	—
156	C₂×D₇₃	C₄⋊C₁₆	1	4	yes	complemented
157	C₄×C₇₃	—	2	2	no	—
158	C₄×C₇₃	C₂²⋊C₁₆	1	4	yes	complemented
159	C₄×C₇₃	#380	1	1	yes	characteristic
160	C₄×C₇₃	#407	1	1	yes	characteristic
161	C₄×C₇₃	#423	1	1	yes	characteristic
162	C₄×C₇₃	#432	1	1	yes	characteristic
163	Dic₇₃	—	2	2	no	—
164	Dic₇₃	—	2	2	no	—
165	Dic₇₃	—	2	2	no	—
166	#12783	—	73	1	no	maximal, Sylow
167	C₂×C₇₃	#1797	1	1	yes	characteristic
168	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
169	C₂×C₇₃	#2112	1	1	yes	characteristic
170	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
171	C₂×C₇₃	#2150	1	1	yes	characteristic
172	C₂×C₇₃	#2162	1	1	yes	characteristic
173	C₂×C₇₃	#2163	1	1	yes	characteristic
174	D₇₃	—	2	4	no	—
175	C₂²×C₄×C₈	—	73	1	no	—
176	C₂×C₄⋊C₁₆	—	73	1	no	—
177	C₂×C₄⋊C₁₆	D₇₃	1	1	yes	characteristic, complemented
178	C₂×C₂²⋊C₁₆	—	73	2	no	—
179	#1374	—	73	2	no	—
180	C₇₃	#12783	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₈	—	73	1	no	—
182	C₂³×C₈	—	73	1	no	—
183	C₂²×C₄²	—	73	1	no	—
184	C₂²×C₁₆	—	73	2	no	—
185	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
186	C₂×C₄×C₈	—	73	1	no	—
187	C₂×C₄×C₈	—	73	1	no	—
188	C₂×C₄×C₈	—	73	1	no	—
189	C₂×C₄×C₈	—	146	2	no	—
190	C₂×C₄×C₈	—	146	2	no	—
191	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
192	C₄⋊C₁₆	—	2	2	no	—
193	C₄⋊C₁₆	—	146	2	no	—
194	C₂²⋊C₁₆	—	146	4	no	—
195	C₂³×C₄	—	73	1	no	—
196	C₂³×C₄	—	73	1	no	—
197	C₂³×C₄	—	73	1	no	—
198	C₂²×C₈	—	73	1	no	—
199	C₂²×C₈	—	73	1	no	—
200	C₂²×C₈	—	73	2	no	—
201	C₂²×C₈	—	73	2	no	—
202	C₂²×C₈	—	146	2	no	—
203	C₂²×C₈	—	146	2	no	—
204	C₂²×C₈	—	146	2	no	—
205	C₂²×C₈	—	146	1	no	—
206	C₂²×C₈	—	146	1	no	—
207	C₂²×C₈	—	146	1	no	—
208	C₂²×C₈	—	146	1	no	—
209	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
211	C₂×C₄²	—	73	1	no	—
212	C₂×C₄²	—	73	1	no	—
213	C₂×C₄²	—	73	1	no	—
214	C₂×C₄²	—	146	2	no	—
215	C₂×C₄²	—	146	2	no	—
216	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
217	C₂×C₁₆	—	2	2	no	—
218	C₂×C₁₆	—	2	2	no	—
219	C₂×C₁₆	—	2	2	no	—
220	C₂×C₁₆	—	146	2	no	—
221	C₂×C₁₆	—	146	2	no	—
222	C₂×C₁₆	—	146	2	no	—
223	C₄×C₈	—	146	2	no	—
224	C₄×C₈	—	146	1	no	—
225	C₄×C₈	—	146	1	no	—
226	C₄×C₈	—	146	2	no	—
227	C₄×C₈	D₂₉₂	1	2	yes	—
228	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
229	C₂⁴	—	73	1	no	—
230	C₂²×C₄	—	73	1	no	—
231	C₂²×C₄	—	73	1	no	—
232	C₂²×C₄	—	73	1	no	—
233	C₂²×C₄	—	73	4	no	—
234	C₂²×C₄	—	73	2	no	—
235	C₂²×C₄	—	73	2	no	—
236	C₂²×C₄	—	146	2	no	—
237	C₂²×C₄	—	146	2	no	—
238	C₂²×C₄	—	146	2	no	—
239	C₂²×C₄	—	146	2	no	—
240	C₂²×C₄	—	146	2	no	—
241	C₂²×C₄	—	146	1	no	—
242	C₂²×C₄	—	146	1	no	—
243	C₂²×C₄	—	146	1	no	—
244	C₂²×C₄	—	146	1	no	—
245	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
246	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	73	2	no	—
251	C₂×C₈	—	73	2	no	—
252	C₂×C₈	—	146	8	no	—
253	C₂×C₈	—	146	1	no	—
254	C₂×C₈	—	146	1	no	—
255	C₂×C₈	—	146	1	no	—
256	C₂×C₈	—	146	1	no	—
257	C₂×C₈	—	146	2	no	—
258	C₂×C₈	—	146	2	no	—
259	C₂×C₈	—	146	2	no	—
260	C₂×C₈	—	146	2	no	—
261	C₂×C₈	#11924005	1	2	yes	—
262	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
264	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
265	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
266	C₂×C₈	#11924027	1	1	yes	characteristic
267	C₂×C₈	#11924029	1	1	yes	characteristic
268	C₄²	—	146	2	no	—
269	C₄²	—	146	2	no	—
270	C₄²	—	146	1	no	—
271	C₄²	—	146	1	no	—
272	C₄²	#11924005	1	2	yes	—
273	C₄²	#11924011	1	2	yes	—
274	C₁₆	—	4	2	no	—
275	C₁₆	—	292	2	no	—
276	C₂³	—	73	2	no	—
277	C₂³	—	73	2	no	—
278	C₂³	—	73	2	no	—
279	C₂³	—	146	1	no	—
280	C₂³	—	146	1	no	—
281	C₂³	—	146	1	no	—
282	C₂³	—	146	1	no	—
283	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	73	2	no	—
287	C₂×C₄	—	73	2	no	—
288	C₂×C₄	—	73	2	no	—
289	C₂×C₄	—	146	8	no	—
290	C₂×C₄	—	146	1	no	—
291	C₂×C₄	—	146	1	no	—
292	C₂×C₄	—	146	1	no	—
293	C₂×C₄	—	146	1	no	—
294	C₂×C₄	—	146	1	no	—
295	C₂×C₄	—	146	1	no	—
296	C₂×C₄	—	146	2	no	—
297	C₂×C₄	—	146	8	no	—
298	C₂×C₄	—	146	2	no	—
299	C₂×C₄	—	146	2	no	—
300	C₂×C₄	—	146	2	no	—
301	C₂×C₄	#14556503	1	2	yes	—
302	C₂×C₄	#14556503	1	2	yes	—
303	C₂×C₄	C₁₆×D₇₃	1	2	yes	—
304	C₂×C₄	#14556530	1	2	yes	—
305	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
306	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
307	C₂×C₄	#14556549	1	1	yes	characteristic
308	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
309	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
310	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	146	2	no	—
313	C₈	—	146	2	no	—
314	C₈	#14556562	1	1	yes	characteristic
315	C₈	#14556566	1	1	yes	characteristic
316	C₈	#14556567	1	1	yes	characteristic
317	C₈	#14556571	1	1	yes	characteristic
318	C₂²	—	73	4	no	—
319	C₂²	—	146	2	no	—
320	C₂²	—	146	2	no	—
321	C₂²	—	146	2	no	—
322	C₂²	—	146	2	no	—
323	C₂²	—	146	2	no	—
324	C₂²	—	146	2	no	—
325	C₂²	C₂×#14556503	1	1	yes	characteristic
326	C₂²	C₄⋊C₈×D₇₃	1	1	yes	characteristic
327	C₂²	#17988639	1	1	yes	characteristic
328	C₂²	C₂×C₁₆×D₇₃	1	1	yes	characteristic
329	C₂²	M₃₂×D₇₃	1	1	yes	characteristic
330	C₂²	M₃₂×D₇₃	1	1	yes	characteristic
331	C₂²	C₂×#14556530	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	146	2	no	—
334	C₄	—	146	2	no	—
335	C₄	—	146	2	no	—
336	C₄	#17988366	1	4	yes	complemented
337	C₄	#17988444	1	1	yes	characteristic
338	C₄	#17988550	1	1	yes	characteristic
339	C₄	#17988609	1	1	yes	characteristic
340	C₄	#17988643	1	1	yes	characteristic
341	C₂	—	146	4	no	—
342	C₂	#22609908	1	1	yes	characteristic
343	C₂	C₂×#17988366	1	1	yes	characteristic
344	C₂	#22611541	1	1	yes	characteristic
345	C₂	C₄⋊C₁₆×D₇₃	1	1	yes	characteristic
346	C₂	#22611685	1	1	yes	characteristic
347	C₂	#22611729	1	1	yes	characteristic
348	C₂	#22611733	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₂×C₄⋊C₁₆×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₇₃
hypocenter	C₇₃	#12783
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊#11924003	C₂	semidirect
4. #22608008	C₂	semidirect
5. C₂×C₄⋊C₁₆×C₇₃	C₂	semidirect
67. C₂×C₄×D₇₃	C₁₆	semidirect
156. C₂×D₇₃	C₄⋊C₁₆	semidirect
158. C₄×C₇₃	C₂²⋊C₁₆	semidirect
177. C₂×C₄⋊C₁₆	D₇₃	semidirect
180. C₇₃	#12783	semidirect
336. C₄	#17988366	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₇₃	2
C₂×C₄⋊#11924003	2
C₂×#17988366	2
#22608008	2
C₂×C₄⋊C₁₆×C₇₃	2
#12783	73