Subgroups — C₇₃×#10713 #32920117

Order 18688.

1470 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#117×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊C₈×C₇₃	C₂	1	2	yes	maximal
3	C₂×#336×C₇₃	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
5	C₂³×C₈×C₇₃	C₂²	1	4	yes	—
6	C₂×#117×C₇₃	—	2	4	no	—
7	C₂×#117×C₇₃	—	2	8	no	—
8	C₂²×C₄⋊C₄×C₇₃	C₄	1	2	yes	—
9	C₂²×C₂²⋊C₄×C₇₃	C₄	1	2	yes	—
10	C₂²×C₄²×C₇₃	C₂²	1	2	yes	—
11	C₂×C₂²⋊C₈×C₇₃	—	2	4	no	—
12	C₂×C₂²⋊C₈×C₇₃	—	2	4	no	—
13	C₂×C₂²⋊C₈×C₇₃	C₄	1	8	yes	complemented
14	#336×C₇₃	—	2	8	no	—
15	C₂⁵×C₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₇₃	—	2	4	no	—
17	C₂³×C₄×C₇₃	—	2	4	no	—
18	C₂³×C₄×C₇₃	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₇₃	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₇₃	D₄	1	2	yes	—
22	C₂³×C₄×C₇₃	D₄	1	4	yes	—
23	C₂³×C₄×C₇₃	Q₈	1	2	yes	—
24	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₇₃	—	2	4	no	—
26	C₂²×C₈×C₇₃	—	2	4	no	—
27	C₂²×C₈×C₇₃	—	2	8	no	—
28	C₂²×C₈×C₇₃	—	2	8	no	—
29	C₂²×C₈×C₇₃	C₂×C₄	1	8	yes	—
30	#117×C₇₃	—	2	32	no	—
31	C₂×C₄⋊C₄×C₇₃	—	2	8	no	—
32	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
33	C₂×C₂²⋊C₄×C₇₃	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₇₃	—	2	8	no	—
35	C₂×C₄²×C₇₃	—	2	4	no	—
36	C₂×C₄²×C₇₃	—	2	4	no	—
37	C₂×C₄²×C₇₃	D₄	1	8	yes	—
38	C₂²⋊C₈×C₇₃	—	2	16	no	—
39	C₂⁴×C₇₃	—	2	2	no	—
40	C₂⁴×C₇₃	—	2	2	no	—
41	C₂⁴×C₇₃	—	2	8	no	—
42	C₂⁴×C₇₃	C₄²	1	2	yes	—
43	C₂⁴×C₇₃	C₂²⋊C₄	1	2	yes	—
44	C₂⁴×C₇₃	C₄⋊C₄	1	2	yes	—
45	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	—	2	2	no	—
47	C₂²×C₄×C₇₃	—	2	2	no	—
48	C₂²×C₄×C₇₃	—	2	16	no	—
49	C₂²×C₄×C₇₃	—	2	8	no	—
50	C₂²×C₄×C₇₃	—	2	8	no	—
51	C₂²×C₄×C₇₃	—	2	8	no	—
52	C₂²×C₄×C₇₃	—	2	8	no	—
53	C₂²×C₄×C₇₃	—	2	4	no	—
54	C₂²×C₄×C₇₃	—	2	4	no	—
55	C₂²×C₄×C₇₃	—	2	16	no	—
56	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
57	C₂²×C₄×C₇₃	C₂²⋊C₄	1	8	yes	—
58	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₇₃	C₂²⋊C₄	1	8	yes	—
60	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₇₃	C₄⋊C₄	1	8	yes	—
62	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	#41	1	4	yes	—
68	C₂×C₈×C₇₃	—	2	8	no	—
69	C₂×C₈×C₇₃	—	2	16	no	—
70	C₂×C₈×C₇₃	—	4	4	no	—
71	C₂×C₈×C₇₃	—	4	8	no	—
72	C₂×C₈×C₇₃	—	4	4	no	—
73	C₄⋊C₄×C₇₃	—	2	16	no	—
74	C₂²⋊C₄×C₇₃	—	2	16	no	—
75	C₄²×C₇₃	—	2	16	no	—
76	C₂³×C₇₃	—	2	4	no	—
77	C₂³×C₇₃	—	2	2	no	—
78	C₂³×C₇₃	—	2	4	no	—
79	C₂³×C₇₃	—	2	2	no	—
80	C₂³×C₇₃	—	2	8	no	—
81	C₂³×C₇₃	—	2	8	no	—
82	C₂³×C₇₃	—	2	8	no	—
83	C₂³×C₇₃	—	4	8	no	—
84	C₂³×C₇₃	—	4	8	no	—
85	C₂³×C₇₃	#95	1	4	yes	—
86	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
87	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂³×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂³×C₇₃	#117	1	2	yes	—
91	C₂³×C₇₃	#131	1	8	yes	—
92	C₂×C₄×C₇₃	—	2	4	no	—
93	C₂×C₄×C₇₃	—	2	2	no	—
94	C₂×C₄×C₇₃	—	2	4	no	—
95	C₂×C₄×C₇₃	—	2	2	no	—
96	C₂×C₄×C₇₃	—	2	16	no	—
97	C₂×C₄×C₇₃	—	2	16	no	—
98	C₂×C₄×C₇₃	—	2	8	no	—
99	C₂×C₄×C₇₃	—	2	16	no	—
100	C₂×C₄×C₇₃	—	4	8	no	—
101	C₂×C₄×C₇₃	—	4	8	no	—
102	C₂×C₄×C₇₃	#95	1	4	yes	—
103	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
104	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₇₃	C₄×D₄	1	8	yes	—
106	C₂×C₄×C₇₃	#120	1	4	yes	—
107	C₂×C₄×C₇₃	#121	1	4	yes	—
108	C₂×C₄×C₇₃	C₂²⋊Q₈	1	4	yes	—
109	C₂×C₄×C₇₃	#123	1	4	yes	—
110	C₈×C₇₃	—	4	8	no	—
111	C₂²×C₇₃	—	2	4	no	—
112	C₂²×C₇₃	—	2	8	no	—
113	C₂²×C₇₃	—	4	8	no	—
114	C₂²×C₇₃	—	4	4	no	—
115	C₂²×C₇₃	—	4	8	no	—
116	C₂²×C₇₃	—	4	4	no	—
117	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
118	C₂²×C₇₃	C₄×C₂²⋊C₄	1	2	yes	—
119	C₂²×C₇₃	#379	1	1	yes	characteristic
120	C₂²×C₇₃	#380	1	2	yes	—
121	C₂²×C₇₃	#381	1	1	yes	characteristic
122	C₂²×C₇₃	#405	1	4	yes	—
123	C₂²×C₇₃	#408	1	8	yes	—
124	C₂²×C₇₃	#408	1	4	yes	—
125	C₂²×C₇₃	#424	1	4	yes	—
126	C₂²×C₇₃	#433	1	8	yes	—
127	C₄×C₇₃	—	2	4	no	—
128	C₄×C₇₃	—	4	8	no	—
129	C₄×C₇₃	#385	1	4	yes	—
130	C₄×C₇₃	#386	1	4	yes	—
131	#10713	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₇₃	—	4	4	no	—
133	C₂×C₇₃	C₂²⋊#95	1	1	yes	characteristic
134	C₂×C₇₃	#1729	1	2	yes	—
135	C₂×C₇₃	#1750	1	4	yes	—
136	C₂×C₇₃	#1780	1	2	yes	—
137	C₂×C₇₃	#1798	1	4	yes	—
138	C₂×C₇₃	#1818	1	2	yes	—
139	C₂²×#117	C₂×C₇₃	1	1	yes	characteristic
140	C₂²×C₂²⋊C₈	C₂×C₇₃	1	2	yes	—
141	C₂×#336	C₂×C₇₃	1	4	yes	complemented
142	C₇₃	#10713	1	1	yes	characteristic, direct product factor, Sylow
143	C₂⁴×C₄	C₂²×C₇₃	1	1	yes	characteristic
144	C₂³×C₈	C₂²×C₇₃	1	4	yes	—
145	C₂×#117	—	2	4	no	—
146	C₂×#117	—	2	8	no	—
147	C₂²×C₄⋊C₄	C₄×C₇₃	1	2	yes	—
148	C₂²×C₂²⋊C₄	C₄×C₇₃	1	2	yes	—
149	C₂²×C₄²	C₂²×C₇₃	1	2	yes	—
150	C₂×C₂²⋊C₈	—	2	4	no	—
151	C₂×C₂²⋊C₈	—	2	4	no	—
152	C₂×C₂²⋊C₈	C₄×C₇₃	1	8	yes	complemented
153	#336	—	2	8	no	—
154	C₂⁵	C₂×C₄×C₇₃	1	1	yes	characteristic
155	C₂³×C₄	—	2	4	no	—
156	C₂³×C₄	—	2	4	no	—
157	C₂³×C₄	C₂×C₄×C₇₃	1	4	yes	—
158	C₂³×C₄	C₂×C₄×C₇₃	1	1	yes	characteristic
159	C₂³×C₄	C₂×C₄×C₇₃	1	4	yes	—
160	C₂³×C₄	D₄×C₇₃	1	2	yes	—
161	C₂³×C₄	D₄×C₇₃	1	4	yes	—
162	C₂³×C₄	Q₈×C₇₃	1	2	yes	—
163	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
164	C₂²×C₈	—	2	8	no	—
165	C₂²×C₈	—	2	4	no	—
166	C₂²×C₈	—	2	4	no	—
167	C₂²×C₈	—	2	8	no	—
168	C₂²×C₈	C₂×C₄×C₇₃	1	8	yes	—
169	#117	—	2	32	no	—
170	C₂×C₄⋊C₄	—	2	8	no	—
171	C₂×C₄⋊C₄	—	2	4	no	—
172	C₂×C₂²⋊C₄	—	2	4	no	—
173	C₂×C₂²⋊C₄	—	2	8	no	—
174	C₂×C₄²	—	2	4	no	—
175	C₂×C₄²	—	2	4	no	—
176	C₂×C₄²	D₄×C₇₃	1	8	yes	—
177	C₂²⋊C₈	—	2	16	no	—
178	C₂⁴	—	2	2	no	—
179	C₂⁴	—	2	2	no	—
180	C₂⁴	—	2	8	no	—
181	C₂⁴	C₄²×C₇₃	1	2	yes	—
182	C₂⁴	C₂²⋊C₄×C₇₃	1	2	yes	—
183	C₂⁴	C₄⋊C₄×C₇₃	1	2	yes	—
184	C₂⁴	C₂²×C₄×C₇₃	1	1	yes	characteristic
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	16	no	—
188	C₂²×C₄	—	2	8	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	16	no	—
195	C₂²×C₄	C₄²×C₇₃	1	2	yes	—
196	C₂²×C₄	C₂²⋊C₄×C₇₃	1	8	yes	—
197	C₂²×C₄	C₂²⋊C₄×C₇₃	1	2	yes	—
198	C₂²×C₄	C₂²⋊C₄×C₇₃	1	8	yes	—
199	C₂²×C₄	C₄⋊C₄×C₇₃	1	2	yes	—
200	C₂²×C₄	C₄⋊C₄×C₇₃	1	8	yes	—
201	C₂²×C₄	C₂²×C₄×C₇₃	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
205	C₂²×C₄	C₂×Q₈×C₇₃	1	1	yes	characteristic
206	C₂²×C₄	#41×C₇₃	1	4	yes	—
207	C₂×C₈	—	2	8	no	—
208	C₂×C₈	—	2	16	no	—
209	C₂×C₈	—	4	4	no	—
210	C₂×C₈	—	4	8	no	—
211	C₂×C₈	—	4	4	no	—
212	C₄⋊C₄	—	2	16	no	—
213	C₂²⋊C₄	—	2	16	no	—
214	C₄²	—	2	16	no	—
215	C₂³	—	2	4	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	8	no	—
220	C₂³	—	2	8	no	—
221	C₂³	—	2	8	no	—
222	C₂³	—	4	8	no	—
223	C₂³	—	4	8	no	—
224	C₂³	#95×C₇₃	1	4	yes	—
225	C₂³	C₂×C₄²×C₇₃	1	1	yes	characteristic
226	C₂³	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
227	C₂³	C₂×C₂²⋊C₄×C₇₃	1	1	yes	characteristic
228	C₂³	C₂×C₄⋊C₄×C₇₃	1	1	yes	characteristic
229	C₂³	#117×C₇₃	1	2	yes	—
230	C₂³	#131×C₇₃	1	8	yes	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	16	no	—
236	C₂×C₄	—	2	16	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	16	no	—
239	C₂×C₄	—	4	8	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	#95×C₇₃	1	4	yes	—
242	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	2	yes	—
244	C₂×C₄	C₄×D₄×C₇₃	1	8	yes	—
245	C₂×C₄	#120×C₇₃	1	4	yes	—
246	C₂×C₄	#121×C₇₃	1	4	yes	—
247	C₂×C₄	C₂²⋊Q₈×C₇₃	1	4	yes	—
248	C₂×C₄	#123×C₇₃	1	4	yes	—
249	C₈	—	4	8	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	8	no	—
252	C₂²	—	4	8	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	8	no	—
255	C₂²	—	4	4	no	—
256	C₂²	C₂×#95×C₇₃	1	1	yes	characteristic
257	C₂²	C₄×C₂²⋊C₄×C₇₃	1	2	yes	—
258	C₂²	#379×C₇₃	1	1	yes	characteristic
259	C₂²	#380×C₇₃	1	2	yes	—
260	C₂²	#381×C₇₃	1	1	yes	characteristic
261	C₂²	#405×C₇₃	1	4	yes	—
262	C₂²	#408×C₇₃	1	4	yes	—
263	C₂²	#408×C₇₃	1	8	yes	—
264	C₂²	#424×C₇₃	1	4	yes	—
265	C₂²	#433×C₇₃	1	8	yes	—
266	C₄	—	2	4	no	—
267	C₄	—	4	8	no	—
268	C₄	#385×C₇₃	1	4	yes	—
269	C₄	#386×C₇₃	1	4	yes	—
270	C₂	—	4	4	no	—
271	C₂	C₇₃×C₂²⋊#95	1	1	yes	characteristic
272	C₂	C₇₃×#1729	1	2	yes	—
273	C₂	C₇₃×#1750	1	4	yes	—
274	C₂	C₇₃×#1780	1	2	yes	—
275	C₂	C₇₃×#1798	1	4	yes	—
276	C₂	C₇₃×#1818	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₇₃	C₂³
commutator	C₂³	C₂×C₄²×C₇₃
Frattini, cosocle	C₂³×C₄	C₂³×C₇₃
socle	C₂⁴×C₇₃	C₂²×C₄
prime radical	C₂⁵×C₇₃	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#336×C₇₃	C₂	semidirect
13. C₂×C₂²⋊C₈×C₇₃	C₄	semidirect
131. #10713	C₇₃	direct (N × H)
141. C₂×#336	C₂×C₇₃	semidirect
142. C₇₃	#10713	direct (N × H)
152. C₂×C₂²⋊C₈	C₄×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#117×C₇₃	2
C₂²×C₂²⋊C₈×C₇₃	2
C₂×#336×C₇₃	2
#10713	73

Subgroups — C73×#10713 #32920117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#10713 #32920117