Subgroups — #32972568

Order 18688.

476 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22609999	C₂	1	1	yes	maximal, characteristic
2	#22608158	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1728	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊#10596831	—	2	1	no	—
5	C₂×#14556498	—	2	1	no	—
6	C₂×#14556498	—	2	1	no	—
7	C₂×#14556498	C₄	1	2	yes	—
8	#17988342	—	2	1	no	—
9	#17988342	C₄	1	2	yes	complemented
10	C₂²×M₁₆×C₇₃	—	2	1	no	—
11	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
12	#336×C₇₃	—	2	1	no	—
13	#336×C₇₃	C₄	1	2	yes	complemented
14	C₂³×Dic₇₃	—	2	2	no	—
15	C₂²×#10596831	—	2	2	no	—
16	C₂²⋊#10596831	—	4	2	no	—
17	#14556498	—	2	4	no	—
18	#14556498	—	4	2	no	—
19	#14556498	—	4	1	no	—
20	#14556498	—	4	1	no	—
21	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₇₃	—	2	4	no	—
25	C₂×M₁₆×C₇₃	—	4	2	no	—
26	C₂×M₁₆×C₇₃	—	4	1	no	—
27	C₂×M₁₆×C₇₃	—	4	1	no	—
28	C₂²×C₈×C₇₃	—	2	2	no	—
29	C₂×C₄²×C₇₃	—	4	2	no	—
30	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₇₃	Q₈	1	1	yes	characteristic
34	C₂²×Dic₇₃	—	2	4	no	—
35	C₂²×Dic₇₃	—	4	2	no	—
36	C₂²×Dic₇₃	—	4	2	no	—
37	C₂²×Dic₇₃	—	4	2	no	—
38	C₂²×Dic₇₃	—	4	2	no	—
39	C₂²×Dic₇₃	—	4	2	no	—
40	C₂²×Dic₇₃	—	4	2	no	—
41	C₂×#10596831	—	4	2	no	—
42	C₂×#10596831	—	4	2	no	—
43	C₂×#10596831	—	4	2	no	—
44	C₂⁴×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	—	2	2	no	—
46	C₂²×C₄×C₇₃	—	2	2	no	—
47	C₂²×C₄×C₇₃	—	2	2	no	—
48	C₂²×C₄×C₇₃	—	2	2	no	—
49	C₂²×C₄×C₇₃	—	2	2	no	—
50	C₂²×C₄×C₇₃	—	4	1	no	—
51	C₂²×C₄×C₇₃	—	4	1	no	—
52	C₂²×C₄×C₇₃	—	4	1	no	—
53	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	Q₁₆	1	1	yes	characteristic
63	M₁₆×C₇₃	—	4	4	no	—
64	M₁₆×C₇₃	—	4	4	no	—
65	C₂×C₈×C₇₃	—	2	4	no	—
66	C₂×C₈×C₇₃	—	4	2	no	—
67	C₂×C₈×C₇₃	—	4	2	no	—
68	C₄²×C₇₃	—	4	1	no	—
69	C₄²×C₇₃	—	4	1	no	—
70	C₄²×C₇₃	—	4	1	no	—
71	C₄²×C₇₃	—	4	1	no	—
72	C₂³×C₇₃	—	2	1	no	—
73	C₂³×C₇₃	—	2	1	no	—
74	C₂³×C₇₃	—	2	1	no	—
75	C₂³×C₇₃	—	2	1	no	—
76	C₂³×C₇₃	#95	1	1	yes	characteristic
77	C₂³×C₇₃	#99	1	2	yes	—
78	C₂³×C₇₃	#102	1	1	yes	characteristic
79	C₂³×C₇₃	#103	1	1	yes	characteristic
80	C₂³×C₇₃	#104	1	2	yes	—
81	C₂×C₄×C₇₃	—	2	2	no	—
82	C₂×C₄×C₇₃	—	2	2	no	—
83	C₂×C₄×C₇₃	—	2	2	no	—
84	C₂×C₄×C₇₃	—	2	2	no	—
85	C₂×C₄×C₇₃	—	2	1	no	—
86	C₂×C₄×C₇₃	—	2	1	no	—
87	C₂×C₄×C₇₃	—	2	1	no	—
88	C₂×C₄×C₇₃	—	2	1	no	—
89	C₂×C₄×C₇₃	—	2	1	no	—
90	C₂×C₄×C₇₃	—	2	1	no	—
91	C₂×C₄×C₇₃	—	2	1	no	—
92	C₂×C₄×C₇₃	—	2	1	no	—
93	C₂×C₄×C₇₃	—	2	1	no	—
94	C₂×C₄×C₇₃	—	2	1	no	—
95	C₂×C₄×C₇₃	—	2	1	no	—
96	C₂×C₄×C₇₃	—	2	1	no	—
97	C₂×C₄×C₇₃	—	2	1	no	—
98	C₂×C₄×C₇₃	—	2	1	no	—
99	C₂×C₄×C₇₃	—	2	1	no	—
100	C₂×C₄×C₇₃	—	4	4	no	—
101	C₂×C₄×C₇₃	—	4	4	no	—
102	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
103	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
104	C₂×C₄×C₇₃	#104	1	2	yes	—
105	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
106	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
107	C₂×Dic₇₃	—	4	4	no	—
108	C₂×Dic₇₃	—	4	4	no	—
109	C₂×Dic₇₃	—	4	4	no	—
110	C₂×Dic₇₃	—	8	2	no	—
111	C₂×Dic₇₃	—	8	2	no	—
112	C₂×Dic₇₃	—	8	2	no	—
113	C₂×Dic₇₃	—	8	2	no	—
114	C₈×C₇₃	—	4	4	no	—
115	#10596831	—	8	2	no	—
116	C₂²×C₇₃	—	2	2	no	—
117	C₂²×C₇₃	—	2	1	no	—
118	C₂²×C₇₃	—	2	1	no	—
119	C₂²×C₇₃	—	2	2	no	—
120	C₂²×C₇₃	—	2	2	no	—
121	C₂²×C₇₃	—	2	1	no	—
122	C₂²×C₇₃	—	2	1	no	—
123	C₂²×C₇₃	—	2	2	no	—
124	C₂²×C₇₃	#327	1	2	yes	—
125	C₂²×C₇₃	#328	1	2	yes	—
126	C₂²×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
127	C₂²×C₇₃	#339	1	1	yes	characteristic
128	C₂²×C₇₃	#339	1	1	yes	characteristic
129	C₂²×C₇₃	#340	1	1	yes	characteristic
130	C₂²×C₇₃	#342	1	1	yes	characteristic
131	C₂²×C₇₃	#344	1	1	yes	characteristic
132	C₂²×C₇₃	#344	1	1	yes	characteristic
133	C₄×C₇₃	—	2	2	no	—
134	C₄×C₇₃	—	2	1	no	—
135	C₄×C₇₃	—	2	1	no	—
136	C₄×C₇₃	—	2	1	no	—
137	C₄×C₇₃	—	2	1	no	—
138	C₄×C₇₃	—	4	1	no	—
139	C₄×C₇₃	—	4	1	no	—
140	C₄×C₇₃	—	4	1	no	—
141	Dic₇₃	—	8	4	no	—
142	#7037	—	73	1	no	maximal, Sylow
143	C₂×C₇₃	—	2	2	no	—
144	C₂×C₇₃	—	2	2	no	—
145	C₂×C₇₃	#1284	1	1	yes	characteristic
146	C₂×C₇₃	#1286	1	1	yes	characteristic
147	C₂×C₇₃	#1288	1	1	yes	characteristic
148	C₂×C₇₃	#1294	1	1	yes	characteristic
149	C₂×C₇₃	#1300	1	1	yes	characteristic
150	C₂×C₇₃	#1303	1	1	yes	characteristic
151	C₂×C₇₃	#1306	1	1	yes	characteristic
152	#1817	—	73	1	no	—
153	#1728	D₇₃	1	1	yes	characteristic
154	#1434	—	73	1	no	—
155	C₇₃	#7037	1	1	yes	characteristic, complemented, Sylow
156	C₂²×M₁₆	—	2	1	no	—
157	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
158	C₂×C₂²⋊C₈	—	146	1	no	—
159	C₂×#95	—	73	2	no	—
160	C₂×#95	—	146	1	no	—
161	C₂×#95	—	146	1	no	—
162	#336	—	2	1	no	—
163	#336	—	73	2	no	—
164	#336	—	146	1	no	—
165	#336	Dic₇₃	1	2	yes	complemented
166	C₂³×C₄	—	146	2	no	—
167	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
168	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
169	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	4	2	no	—
172	C₂×M₁₆	—	4	1	no	—
173	C₂×M₁₆	—	4	1	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	146	2	no	—
176	C₂×C₄²	—	4	2	no	—
177	C₂×C₄²	Dic₁₄₆	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
179	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
181	C₂²⋊C₈	—	292	2	no	—
182	#95	—	292	2	no	—
183	#95	—	292	1	no	—
184	#95	—	292	1	no	—
185	#95	—	146	4	no	—
186	C₂⁴	#11924005	1	1	yes	characteristic
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	292	2	no	—
196	C₂²×C₄	—	292	2	no	—
197	C₂²×C₄	—	292	2	no	—
198	C₂²×C₄	—	292	2	no	—
199	C₂²×C₄	—	292	2	no	—
200	C₂²×C₄	—	292	2	no	—
201	C₂²×C₄	—	146	4	no	—
202	C₂²×C₄	C₄×Dic₇₃	1	1	yes	characteristic
203	C₂²×C₄	#11924005	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
205	C₂²×C₄	#11924007	1	1	yes	characteristic
206	C₂²×C₄	#11924007	1	1	yes	characteristic
207	C₂²×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
208	C₂²×C₄	#11924013	1	1	yes	characteristic
209	C₂²×C₄	#11924015	1	1	yes	characteristic
210	C₂²×C₄	#11924016	1	1	yes	characteristic
211	C₂²×C₄	#11924018	1	1	yes	characteristic
212	M₁₆	—	4	4	no	—
213	M₁₆	—	4	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	292	2	no	—
218	C₂×C₈	—	292	2	no	—
219	C₂×C₈	—	292	2	no	—
220	C₄²	—	4	1	no	—
221	C₄²	—	4	1	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	1	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	#14556498	1	1	yes	characteristic
229	C₂³	#14556505	1	2	yes	—
230	C₂³	#14556511	1	1	yes	characteristic
231	C₂³	#14556514	1	1	yes	characteristic
232	C₂³	#14556519	1	2	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	292	4	no	—
255	C₂×C₄	—	292	4	no	—
256	C₂×C₄	—	292	4	no	—
257	C₂×C₄	—	584	2	no	—
258	C₂×C₄	—	584	2	no	—
259	C₂×C₄	—	584	2	no	—
260	C₂×C₄	—	584	2	no	—
261	C₂×C₄	#14556512	1	1	yes	characteristic
262	C₂×C₄	#14556515	1	1	yes	characteristic
263	C₂×C₄	#14556517	1	2	yes	—
264	C₂×C₄	#14556522	1	1	yes	characteristic
265	C₂×C₄	#14556524	1	1	yes	characteristic
266	C₈	—	4	4	no	—
267	C₈	—	584	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	2	no	—
276	C₂²	#17988320	1	2	yes	—
277	C₂²	#17988323	1	2	yes	—
278	C₂²	#17988344	1	1	yes	characteristic
279	C₂²	#17988347	1	1	yes	characteristic
280	C₂²	#17988348	1	1	yes	characteristic
281	C₂²	#17988349	1	1	yes	characteristic
282	C₂²	#17988353	1	1	yes	characteristic
283	C₂²	#17988358	1	1	yes	characteristic
284	C₂²	#17988359	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	584	4	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	2	no	—
296	C₂	#22607785	1	1	yes	characteristic
297	C₂	#22607791	1	1	yes	characteristic
298	C₂	#22607796	1	1	yes	characteristic
299	C₂	#22607810	1	1	yes	characteristic
300	C₂	#22607825	1	1	yes	characteristic
301	C₂	#22607832	1	1	yes	characteristic
302	C₂	#22607840	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556498
commutator	C₂²×C₄×C₇₃	C₄²
Frattini	C₂²×C₄²	C₂×D₇₃
Fitting	C₇₃×#1728	C₂
socle	C₂³×C₇₃	#95
prime radical	C₂⁴×C₇₃	C₂²⋊C₄
cosocle	C₂²×C₄²×C₇₃	C₂²
hypercenter	#1728	D₇₃
hypocenter	C₇₃	#7037
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
9. #17988342	C₄	semidirect
13. #336×C₇₃	C₄	semidirect
155. C₇₃	#7037	semidirect
165. #336	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22609999	2
#22608158	2
C₇₃×#1728	2
#7037	73