Subgroups — C₇₃×#9645 #32919049

Order 18688.

558 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	C₄×#124×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Q₈⋊C₈×C₇₃	C₂	1	2	yes	maximal, complemented
5	C₂×D₄⋊C₈×C₇₃	C₂	1	2	yes	maximal
6	C₂×#124×C₇₃	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
11	#391×C₇₃	C₄	1	1	yes	characteristic
12	#390×C₇₃	C₄	1	1	yes	characteristic
13	#388×C₇₃	—	2	2	no	—
14	#382×C₇₃	C₄	1	1	yes	characteristic
15	C₄×C₂²⋊C₄×C₇₃	—	2	2	no	—
16	C₄³×C₇₃	C₂²	1	1	yes	characteristic
17	#336×C₇₃	—	2	2	no	—
18	#336×C₇₃	—	2	2	no	—
19	Q₈⋊C₈×C₇₃	—	2	4	no	—
20	D₄⋊C₈×C₇₃	—	2	4	no	—
21	C₂²×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₇₃	—	2	1	no	—
24	C₂²×C₈×C₇₃	—	2	2	no	—
25	C₂²×C₈×C₇₃	—	2	2	no	—
26	#124×C₇₃	—	2	2	no	—
27	#124×C₇₃	—	2	2	no	—
28	C₄×Q₈×C₇₃	—	2	2	no	—
29	C₄×Q₈×C₇₃	—	2	2	no	—
30	C₄×D₄×C₇₃	—	2	2	no	—
31	C₄×D₄×C₇₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
38	C₂×C₄²×C₇₃	—	2	1	no	—
39	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₃	D₄	1	2	yes	—
42	C₂×C₄²×C₇₃	D₄	1	2	yes	—
43	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₇₃	—	2	2	no	—
45	C₄⋊C₈×C₇₃	—	2	2	no	—
46	C₄×C₈×C₇₃	—	2	2	no	—
47	C₄×C₈×C₇₃	D₄	1	4	yes	—
48	#95×C₇₃	—	2	2	no	—
49	#95×C₇₃	—	2	2	no	—
50	C₂⁴×C₇₃	—	2	1	no	—
51	C₂×Q₈×C₇₃	—	2	1	no	—
52	C₂×Q₈×C₇₃	—	2	2	no	—
53	C₂×Q₈×C₇₃	—	2	2	no	—
54	C₂×Q₈×C₇₃	M₁₆	1	2	yes	—
55	C₂×D₄×C₇₃	—	2	2	no	—
56	C₂×D₄×C₇₃	—	2	1	no	—
57	C₂×D₄×C₇₃	—	2	2	no	—
58	C₂×D₄×C₇₃	M₁₆	1	2	yes	—
59	C₂²×C₄×C₇₃	—	2	1	no	—
60	C₂²×C₄×C₇₃	—	2	1	no	—
61	C₂²×C₄×C₇₃	—	2	1	no	—
62	C₂²×C₄×C₇₃	—	4	1	no	—
63	C₂²×C₄×C₇₃	—	4	1	no	—
64	C₂²×C₄×C₇₃	—	4	4	no	—
65	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₇₃	—	2	4	no	—
71	C₂×C₈×C₇₃	—	4	4	no	—
72	C₂×C₈×C₇₃	—	4	2	no	—
73	C₂×C₈×C₇₃	—	4	4	no	—
74	C₄⋊C₄×C₇₃	—	2	2	no	—
75	C₄⋊C₄×C₇₃	—	2	2	no	—
76	C₄⋊C₄×C₇₃	—	4	2	no	—
77	C₂²⋊C₄×C₇₃	—	2	8	no	—
78	C₂²⋊C₄×C₇₃	—	4	4	no	—
79	C₂²⋊C₄×C₇₃	—	4	2	no	—
80	C₄²×C₇₃	—	2	1	no	—
81	C₄²×C₇₃	—	2	1	no	—
82	C₄²×C₇₃	—	2	2	no	—
83	C₄²×C₇₃	—	2	4	no	—
84	C₄²×C₇₃	—	4	2	no	—
85	C₄²×C₇₃	—	4	1	no	—
86	C₄²×C₇₃	—	4	1	no	—
87	C₄²×C₇₃	C₂×D₄	1	2	yes	—
88	C₄²×C₇₃	#41	1	2	yes	—
89	C₂³×C₇₃	—	2	2	no	—
90	C₂³×C₇₃	—	4	1	no	—
91	C₂³×C₇₃	—	4	1	no	—
92	C₂³×C₇₃	—	4	2	no	—
93	C₂³×C₇₃	—	4	2	no	—
94	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	Q₈×C₇₃	—	2	4	no	—
96	Q₈×C₇₃	—	4	2	no	—
97	D₄×C₇₃	—	2	4	no	—
98	D₄×C₇₃	—	4	2	no	—
99	C₂×C₄×C₇₃	—	2	2	no	—
100	C₂×C₄×C₇₃	—	2	2	no	—
101	C₂×C₄×C₇₃	—	2	2	no	—
102	C₂×C₄×C₇₃	—	2	2	no	—
103	C₂×C₄×C₇₃	—	2	2	no	—
104	C₂×C₄×C₇₃	—	2	2	no	—
105	C₂×C₄×C₇₃	—	2	2	no	—
106	C₂×C₄×C₇₃	—	2	2	no	—
107	C₂×C₄×C₇₃	—	4	1	no	—
108	C₂×C₄×C₇₃	—	4	1	no	—
109	C₂×C₄×C₇₃	—	4	1	no	—
110	C₂×C₄×C₇₃	—	4	1	no	—
111	C₂×C₄×C₇₃	—	4	2	no	—
112	C₂×C₄×C₇₃	—	4	2	no	—
113	C₂×C₄×C₇₃	—	8	2	no	—
114	C₂×C₄×C₇₃	—	8	2	no	—
115	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₇₃	#121	1	2	yes	—
117	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
118	C₂×C₄×C₇₃	#127	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×C₄×C₇₃	#131	1	2	yes	—
122	C₂×C₄×C₇₃	#135	1	4	yes	—
123	C₈×C₇₃	—	4	4	no	—
124	C₈×C₇₃	—	8	2	no	—
125	C₂²×C₇₃	—	4	2	no	—
126	C₂²×C₇₃	—	4	4	no	—
127	C₂²×C₇₃	—	8	1	no	—
128	C₂²×C₇₃	—	8	1	no	—
129	C₂²×C₇₃	—	8	2	no	—
130	C₂²×C₇₃	#390	1	1	yes	characteristic
131	C₂²×C₇₃	#407	1	1	yes	characteristic
132	C₂²×C₇₃	#408	1	1	yes	characteristic
133	C₂²×C₇₃	#416	1	2	yes	—
134	C₂²×C₇₃	#421	1	2	yes	—
135	C₄×C₇₃	—	2	2	no	—
136	C₄×C₇₃	—	4	2	no	—
137	C₄×C₇₃	—	4	2	no	—
138	C₄×C₇₃	—	4	2	no	—
139	C₄×C₇₃	—	8	1	no	—
140	C₄×C₇₃	—	8	1	no	—
141	C₄×C₇₃	#435	1	2	yes	—
142	C₄×C₇₃	#435	1	2	yes	—
143	C₄×C₇₃	#488	1	2	yes	—
144	C₄×C₇₃	#495	1	2	yes	—
145	#9645	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂×C₇₃	—	8	2	no	—
147	C₂×C₇₃	#1482	1	2	yes	—
148	C₂×C₇₃	#1488	1	2	yes	—
149	C₂×C₇₃	#1955	1	1	yes	characteristic
150	C₂×C₇₃	#1958	1	1	yes	characteristic
151	C₂×C₇₃	#1963	1	1	yes	characteristic
152	C₄×#124	C₂×C₇₃	1	1	yes	characteristic
153	#1831	C₂×C₇₃	1	1	yes	characteristic, complemented
154	#1744	C₂×C₇₃	1	1	yes	characteristic, complemented
155	C₂×Q₈⋊C₈	C₂×C₇₃	1	2	yes	complemented
156	C₂×D₄⋊C₈	C₂×C₇₃	1	2	yes	—
157	C₇₃	#9645	1	1	yes	characteristic, direct product factor, Sylow
158	C₂×#124	C₄×C₇₃	1	1	yes	characteristic
159	C₂×C₄×Q₈	C₂²×C₇₃	1	1	yes	characteristic
160	C₂×C₄×D₄	C₂²×C₇₃	1	1	yes	characteristic
161	C₂×C₄⋊C₈	C₂²×C₇₃	1	2	yes	—
162	C₂×C₄×C₈	C₂²×C₇₃	1	2	yes	—
163	#391	C₄×C₇₃	1	1	yes	characteristic
164	#390	C₄×C₇₃	1	1	yes	characteristic
165	#388	—	2	2	no	—
166	#382	C₄×C₇₃	1	1	yes	characteristic
167	C₄×C₂²⋊C₄	—	2	2	no	—
168	C₄³	C₂²×C₇₃	1	1	yes	characteristic
169	#336	—	2	2	no	—
170	#336	—	2	2	no	—
171	Q₈⋊C₈	—	2	4	no	—
172	D₄⋊C₈	—	2	4	no	—
173	C₂²×Q₈	C₂×C₄×C₇₃	1	1	yes	characteristic
174	C₂²×D₄	C₂×C₄×C₇₃	1	1	yes	characteristic
175	C₂³×C₄	—	2	1	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	#124	—	2	2	no	—
179	#124	—	2	2	no	—
180	C₄×Q₈	—	2	2	no	—
181	C₄×Q₈	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₄×D₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₇₃	1	1	yes	characteristic
186	C₂×C₄⋊C₄	C₂×C₄×C₇₃	1	1	yes	characteristic
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	—	2	1	no	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂×C₄×C₇₃	1	1	yes	characteristic
192	C₂×C₄²	C₂×C₄×C₇₃	1	1	yes	characteristic
193	C₂×C₄²	D₄×C₇₃	1	2	yes	—
194	C₂×C₄²	D₄×C₇₃	1	2	yes	—
195	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	D₄×C₇₃	1	4	yes	—
200	#95	—	2	2	no	—
201	#95	—	2	2	no	—
202	C₂⁴	—	2	1	no	—
203	C₂×Q₈	—	2	1	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	M₁₆×C₇₃	1	2	yes	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	M₁₆×C₇₃	1	2	yes	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	C₂²⋊C₄×C₇₃	1	2	yes	—
218	C₂²×C₄	C₂²⋊C₄×C₇₃	1	2	yes	—
219	C₂²×C₄	C₂²×C₄×C₇₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	4	4	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	4	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	2	8	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	C₂×D₄×C₇₃	1	2	yes	—
240	C₄²	#41×C₇₃	1	2	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	4	1	no	—
243	C₂³	—	4	1	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	C₂×C₂²⋊C₄×C₇₃	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	Q₈	—	4	2	no	—
249	D₄	—	2	4	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	C₄×D₄×C₇₃	1	2	yes	—
268	C₂×C₄	#121×C₇₃	1	2	yes	—
269	C₂×C₄	#124×C₇₃	1	1	yes	characteristic
270	C₂×C₄	#127×C₇₃	1	1	yes	characteristic
271	C₂×C₄	C₂×M₁₆×C₇₃	1	1	yes	characteristic
272	C₂×C₄	C₂×M₁₆×C₇₃	1	1	yes	characteristic
273	C₂×C₄	#131×C₇₃	1	2	yes	—
274	C₂×C₄	#135×C₇₃	1	4	yes	—
275	C₈	—	4	4	no	—
276	C₈	—	8	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	8	1	no	—
280	C₂²	—	8	1	no	—
281	C₂²	—	8	2	no	—
282	C₂²	#390×C₇₃	1	1	yes	characteristic
283	C₂²	#407×C₇₃	1	1	yes	characteristic
284	C₂²	#408×C₇₃	1	1	yes	characteristic
285	C₂²	#416×C₇₃	1	2	yes	—
286	C₂²	#421×C₇₃	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	2	no	—
290	C₄	—	4	2	no	—
291	C₄	—	8	1	no	—
292	C₄	—	8	1	no	—
293	C₄	#435×C₇₃	1	2	yes	—
294	C₄	#435×C₇₃	1	2	yes	—
295	C₄	#488×C₇₃	1	2	yes	—
296	C₄	#495×C₇₃	1	2	yes	—
297	C₂	—	8	2	no	—
298	C₂	C₇₃×#1482	1	2	yes	—
299	C₂	C₇₃×#1488	1	2	yes	—
300	C₂	C₇₃×#1955	1	1	yes	characteristic
301	C₂	C₇₃×#1958	1	1	yes	characteristic
302	C₂	C₇₃×#1963	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
socle	C₂³×C₇₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₇₃	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇₃ ◁ C₄³×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₃×#1831	C₂	semidirect
3. C₇₃×#1744	C₂	semidirect
4. C₂×Q₈⋊C₈×C₇₃	C₂	semidirect
145. #9645	C₇₃	direct (N × H)
153. #1831	C₂×C₇₃	semidirect
154. #1744	C₂×C₇₃	semidirect
155. C₂×Q₈⋊C₈	C₂×C₇₃	semidirect
157. C₇₃	#9645	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#124×C₇₃	2
C₇₃×#1831	2
C₇₃×#1744	2
C₂×Q₈⋊C₈×C₇₃	2
C₂×D₄⋊C₈×C₇₃	2
#9645	73

Subgroups — C73×#9645 #32919049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#9645 #32919049