Subgroups — #33002194

Order 18688.

542 conjugacy classes of subgroups.

The subgroup lattice has 324 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×#17988366	C₂	1	2	yes	maximal
3	#22608009	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608008	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1374	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	4	no	—
14	#17988366	—	2	4	no	—
15	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₇₃	C₈	1	1	yes	characteristic
18	C₂²×C₄×D₇₃	C₈	1	1	yes	characteristic
19	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
21	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₇₃	—	2	2	no	—
23	C₂×C₈×D₇₃	—	2	2	no	—
24	C₂×C₈×D₇₃	—	2	2	no	—
25	C₂×C₈×D₇₃	—	2	1	no	—
26	C₂×C₈×D₇₃	—	2	2	no	—
27	C₂×C₈×D₇₃	—	2	1	no	—
28	C₂×C₈×D₇₃	D₄	1	4	yes	—
29	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
30	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
31	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄²×D₇₃	—	2	2	no	—
33	C₄²×D₇₃	—	2	2	no	—
34	C₂×#11924003	—	2	2	no	—
35	C₂×#11924003	—	2	2	no	—
36	C₂×#11924003	—	2	2	no	—
37	C₄×#10596831	—	2	2	no	—
38	C₈×Dic₇₃	—	2	2	no	—
39	C₈×Dic₇₃	—	2	2	no	—
40	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₁₆×C₇₃	—	2	2	no	—
44	C₂×C₁₆×C₇₃	—	2	2	no	—
45	C₂×C₁₆×C₇₃	—	2	2	no	—
46	C₄×C₈×C₇₃	—	2	2	no	—
47	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
51	C₂×C₄×D₇₃	—	2	2	no	—
52	C₂×C₄×D₇₃	—	2	2	no	—
53	C₂×C₄×D₇₃	—	2	2	no	—
54	C₂×C₄×D₇₃	—	2	2	no	—
55	C₂×C₄×D₇₃	—	2	1	no	—
56	C₂×C₄×D₇₃	—	2	1	no	—
57	C₂×C₄×D₇₃	—	2	1	no	—
58	C₂×C₄×D₇₃	—	2	1	no	—
59	C₂×C₄×D₇₃	—	2	2	no	—
60	C₂×C₄×D₇₃	C₁₆	1	4	yes	complemented
61	C₂×C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
63	C₂×#10596831	—	2	1	no	—
64	C₂×#10596831	—	2	1	no	—
65	C₂×#10596831	—	2	1	no	—
66	C₂×#10596831	—	2	1	no	—
67	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
68	C₂×#10596831	M₁₆	1	2	yes	—
69	C₈×D₇₃	—	2	8	no	—
70	C₈×D₇₃	—	2	4	no	—
71	C₈×D₇₃	—	2	4	no	—
72	C₄×Dic₇₃	—	2	1	no	—
73	C₄×Dic₇₃	—	2	1	no	—
74	C₄×Dic₇₃	—	2	2	no	—
75	C₄×Dic₇₃	—	2	2	no	—
76	#11924003	—	4	2	no	—
77	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₇₃	—	2	1	no	—
81	C₂×C₈×C₇₃	—	2	1	no	—
82	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
83	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₇₃	#41	1	2	yes	—
86	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
87	C₄²×C₇₃	—	2	1	no	—
88	C₄²×C₇₃	—	2	1	no	—
89	C₁₆×C₇₃	—	4	2	no	—
90	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
91	C₂²×D₇₃	—	2	1	no	—
92	C₂²×D₇₃	—	2	1	no	—
93	C₂²×D₇₃	—	2	1	no	—
94	C₂²×D₇₃	—	2	1	no	—
95	C₂²×D₇₃	C₂²⋊C₈	1	2	yes	—
96	C₂²×D₇₃	C₂×C₁₆	1	2	yes	—
97	C₂²×D₇₃	M₃₂	1	2	yes	—
98	C₂×C₄×C₇₃	—	2	1	no	—
99	C₂×C₄×C₇₃	—	2	1	no	—
100	C₂×C₄×C₇₃	—	2	1	no	—
101	C₂×C₄×C₇₃	—	2	1	no	—
102	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
103	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
104	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×Dic₇₃	—	2	1	no	—
111	C₂×Dic₇₃	—	2	1	no	—
112	C₂×Dic₇₃	—	2	1	no	—
113	C₂×Dic₇₃	—	2	1	no	—
114	C₂×Dic₇₃	—	2	1	no	—
115	C₂×Dic₇₃	—	2	1	no	—
116	C₂×Dic₇₃	C₂²⋊C₈	1	2	yes	—
117	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
118	C₂×Dic₇₃	M₃₂	1	2	yes	—
119	C₄×D₇₃	—	2	8	no	—
120	C₄×D₇₃	—	2	8	no	—
121	C₄×D₇₃	—	2	2	no	—
122	C₄×D₇₃	—	2	2	no	—
123	C₄×D₇₃	—	2	2	no	—
124	C₄×D₇₃	—	2	2	no	—
125	C₈×C₇₃	—	2	2	no	—
126	C₈×C₇₃	#123	1	2	yes	—
127	C₈×C₇₃	#123	1	2	yes	—
128	#10596831	—	2	2	no	—
129	#10596831	—	2	2	no	—
130	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₂²×C₇₃	#431	1	1	yes	characteristic
132	C₂²×C₇₃	#431	1	1	yes	characteristic
133	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
134	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
135	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
136	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
137	C₂×D₇₃	—	2	2	no	—
138	C₂×D₇₃	—	2	2	no	—
139	C₂×D₇₃	—	2	2	no	—
140	C₂×D₇₃	—	2	2	no	—
141	C₂×D₇₃	—	2	2	no	—
142	C₂×D₇₃	—	2	2	no	—
143	C₂×D₇₃	C₂²⋊C₁₆	1	4	yes	—
144	C₄×C₇₃	—	2	2	no	—
145	C₄×C₇₃	—	2	2	no	—
146	C₄×C₇₃	#381	1	1	yes	characteristic
147	C₄×C₇₃	#407	1	1	yes	characteristic
148	C₄×C₇₃	#432	1	1	yes	characteristic
149	C₄×C₇₃	#432	1	1	yes	characteristic
150	Dic₇₃	—	2	2	no	—
151	Dic₇₃	—	2	2	no	—
152	Dic₇₃	—	2	2	no	—
153	#12844	—	73	1	no	maximal, Sylow
154	C₂×C₇₃	#1817	1	1	yes	characteristic
155	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
156	C₂×C₇₃	#2112	1	1	yes	characteristic
157	C₂×C₇₃	#2162	1	1	yes	characteristic
158	C₂×C₇₃	#2162	1	1	yes	characteristic
159	C₂×C₇₃	#2163	1	1	yes	characteristic
160	C₂×C₇₃	#2163	1	1	yes	characteristic
161	D₇₃	—	2	4	no	—
162	C₂²×C₄×C₈	—	73	1	no	—
163	C₂×C₂²⋊C₁₆	—	73	2	no	—
164	#1374	—	73	2	no	—
165	#1374	—	73	1	no	—
166	#1374	D₇₃	1	1	yes	characteristic, complemented
167	C₇₃	#12844	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₈	—	73	1	no	—
169	C₂³×C₈	—	73	1	no	—
170	C₂²×C₄²	—	73	1	no	—
171	C₂²×C₁₆	—	73	2	no	—
172	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
173	C₂×C₄×C₈	—	73	1	no	—
174	C₂×C₄×C₈	—	73	1	no	—
175	C₂×C₄×C₈	—	73	1	no	—
176	C₂×C₄×C₈	—	146	4	no	—
177	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
178	C₂²⋊C₁₆	—	146	4	no	—
179	C₂³×C₄	—	73	1	no	—
180	C₂³×C₄	—	73	1	no	—
181	C₂³×C₄	—	73	1	no	—
182	C₂²×C₈	—	73	1	no	—
183	C₂²×C₈	—	73	1	no	—
184	C₂²×C₈	—	73	4	no	—
185	C₂²×C₈	—	146	2	no	—
186	C₂²×C₈	—	146	2	no	—
187	C₂²×C₈	—	146	2	no	—
188	C₂²×C₈	—	146	1	no	—
189	C₂²×C₈	—	146	2	no	—
190	C₂²×C₈	—	146	1	no	—
191	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
193	C₂×C₄²	—	73	1	no	—
194	C₂×C₄²	—	73	1	no	—
195	C₂×C₄²	—	73	1	no	—
196	C₂×C₄²	—	146	2	no	—
197	C₂×C₄²	—	146	2	no	—
198	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
199	C₂×C₁₆	—	2	2	no	—
200	C₂×C₁₆	—	2	2	no	—
201	C₂×C₁₆	—	2	2	no	—
202	C₂×C₁₆	—	146	2	no	—
203	C₂×C₁₆	—	146	2	no	—
204	C₂×C₁₆	—	146	2	no	—
205	C₄×C₈	—	2	2	no	—
206	C₄×C₈	—	146	2	no	—
207	C₄×C₈	—	146	2	no	—
208	C₄×C₈	—	146	2	no	—
209	C₂⁴	—	73	1	no	—
210	C₂²×C₄	—	73	1	no	—
211	C₂²×C₄	—	73	1	no	—
212	C₂²×C₄	—	73	1	no	—
213	C₂²×C₄	—	73	4	no	—
214	C₂²×C₄	—	73	2	no	—
215	C₂²×C₄	—	73	2	no	—
216	C₂²×C₄	—	146	2	no	—
217	C₂²×C₄	—	146	2	no	—
218	C₂²×C₄	—	146	2	no	—
219	C₂²×C₄	—	146	2	no	—
220	C₂²×C₄	—	146	2	no	—
221	C₂²×C₄	—	146	1	no	—
222	C₂²×C₄	—	146	1	no	—
223	C₂²×C₄	—	146	1	no	—
224	C₂²×C₄	—	146	1	no	—
225	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
226	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	73	2	no	—
231	C₂×C₈	—	73	2	no	—
232	C₂×C₈	—	146	8	no	—
233	C₂×C₈	—	146	1	no	—
234	C₂×C₈	—	146	1	no	—
235	C₂×C₈	—	146	1	no	—
236	C₂×C₈	—	146	1	no	—
237	C₂×C₈	—	146	4	no	—
238	C₂×C₈	—	146	4	no	—
239	C₂×C₈	#11924011	1	2	yes	—
240	C₂×C₈	D₄×D₇₃	1	2	yes	—
241	C₂×C₈	#11924027	1	1	yes	characteristic
242	C₂×C₈	#11924028	1	2	yes	—
243	C₂×C₈	#11924029	1	1	yes	characteristic
244	C₄²	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₄²	—	146	2	no	—
247	C₄²	—	146	2	no	—
248	C₄²	—	146	1	no	—
249	C₄²	—	146	1	no	—
250	C₁₆	—	4	2	no	—
251	C₁₆	—	292	2	no	—
252	C₂³	—	73	2	no	—
253	C₂³	—	73	2	no	—
254	C₂³	—	73	2	no	—
255	C₂³	—	146	1	no	—
256	C₂³	—	146	1	no	—
257	C₂³	—	146	1	no	—
258	C₂³	—	146	1	no	—
259	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	73	2	no	—
265	C₂×C₄	—	73	2	no	—
266	C₂×C₄	—	73	2	no	—
267	C₂×C₄	—	146	8	no	—
268	C₂×C₄	—	146	8	no	—
269	C₂×C₄	—	146	1	no	—
270	C₂×C₄	—	146	1	no	—
271	C₂×C₄	—	146	1	no	—
272	C₂×C₄	—	146	1	no	—
273	C₂×C₄	—	146	1	no	—
274	C₂×C₄	—	146	1	no	—
275	C₂×C₄	—	146	2	no	—
276	C₂×C₄	—	146	2	no	—
277	C₂×C₄	—	146	2	no	—
278	C₂×C₄	—	146	2	no	—
279	C₂×C₄	C₁₆×D₇₃	1	2	yes	—
280	C₂×C₄	#14556530	1	2	yes	—
281	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
282	C₂×C₄	#14556548	1	1	yes	characteristic
283	C₂×C₄	#14556549	1	1	yes	characteristic
284	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
285	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
286	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	—	146	2	no	—
289	C₈	—	146	2	no	—
290	C₈	#14556573	1	2	yes	—
291	C₈	#14556576	1	2	yes	—
292	C₂²	—	73	4	no	—
293	C₂²	—	146	2	no	—
294	C₂²	—	146	2	no	—
295	C₂²	—	146	2	no	—
296	C₂²	—	146	2	no	—
297	C₂²	—	146	2	no	—
298	C₂²	—	146	2	no	—
299	C₂²	C₂²⋊C₈×D₇₃	1	1	yes	characteristic
300	C₂²	#17988638	1	1	yes	characteristic
301	C₂²	#17988639	1	1	yes	characteristic
302	C₂²	C₂×C₁₆×D₇₃	1	1	yes	characteristic
303	C₂²	M₃₂×D₇₃	1	1	yes	characteristic
304	C₂²	M₃₂×D₇₃	1	1	yes	characteristic
305	C₂²	C₂×#14556530	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	146	2	no	—
309	C₄	—	146	2	no	—
310	C₄	—	146	2	no	—
311	C₄	#17988449	1	1	yes	characteristic
312	C₄	#17988549	1	1	yes	characteristic
313	C₄	#17988642	1	1	yes	characteristic
314	C₄	#17988643	1	1	yes	characteristic
315	C₂	—	146	4	no	—
316	C₂	#22609998	1	1	yes	characteristic
317	C₂	C₂²⋊C₁₆×D₇₃	1	1	yes	characteristic
318	C₂	#22611540	1	1	yes	characteristic
319	C₂	#22611728	1	1	yes	characteristic
320	C₂	#22611729	1	1	yes	characteristic
321	C₂	#22611732	1	1	yes	characteristic
322	C₂	#22611733	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₇₃×#1374	C₂
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1374	D₇₃
hypocenter	C₇₃	#12844
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
3. #22608009	C₂	semidirect
4. #22608008	C₂	semidirect
5. C₇₃×#1374	C₂	semidirect
60. C₂×C₄×D₇₃	C₁₆	semidirect
166. #1374	D₇₃	semidirect
167. C₇₃	#12844	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×#17988366	2
#22608009	2
#22608008	2
C₇₃×#1374	2
#12844	73