Subgroups — C₇₃×#10722 #32920126

Order 18688.

1518 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#380×C₇₃	C₂	1	2	yes	maximal
3	C₂×#340×C₇₃	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₇₃	—	2	4	no	—
6	C₂²×M₁₆×C₇₃	—	2	8	no	—
7	C₂²×M₁₆×C₇₃	C₄	1	4	yes	complemented
8	C₂³×C₈×C₇₃	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₇₃	C₂²	1	4	yes	—
10	C₂²×C₂²⋊C₄×C₇₃	—	2	2	no	—
11	#380×C₇₃	—	2	4	no	—
12	#380×C₇₃	C₄	1	8	yes	complemented
13	C₂×#95×C₇₃	—	2	2	no	—
14	#340×C₇₃	—	2	8	no	—
15	C₂⁵×C₇₃	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₇₃	—	2	4	no	—
17	C₂³×C₄×C₇₃	—	2	4	no	—
18	C₂³×C₄×C₇₃	—	2	4	no	—
19	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₇₃	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
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	C₂×M₁₆×C₇₃	—	2	32	no	—
26	C₂×M₁₆×C₇₃	—	2	8	no	—
27	C₂×M₁₆×C₇₃	—	2	16	no	—
28	C₂²×C₈×C₇₃	—	2	4	no	—
29	C₂²×C₈×C₇₃	—	2	4	no	—
30	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₇₃	D₄	1	8	yes	—
32	C₂×C₄⋊C₄×C₇₃	—	2	8	no	—
33	C₂×C₄⋊C₄×C₇₃	—	2	8	no	—
34	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
35	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	8	yes	—
36	C₂×C₂²⋊C₄×C₇₃	—	2	8	no	—
37	C₂×C₂²⋊C₄×C₇₃	—	4	8	no	—
38	#95×C₇₃	—	2	8	no	—
39	C₂⁴×C₇₃	—	2	4	no	—
40	C₂⁴×C₇₃	—	2	4	no	—
41	C₂⁴×C₇₃	—	2	4	no	—
42	C₂⁴×C₇₃	C₂²⋊C₄	1	4	yes	—
43	C₂⁴×C₇₃	C₄⋊C₄	1	2	yes	—
44	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	—	2	4	no	—
46	C₂²×C₄×C₇₃	—	2	4	no	—
47	C₂²×C₄×C₇₃	—	2	4	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	—	4	8	no	—
54	C₂²×C₄×C₇₃	—	4	8	no	—
55	C₂²×C₄×C₇₃	—	4	4	no	—
56	C₂²×C₄×C₇₃	—	4	4	no	—
57	C₂²×C₄×C₇₃	C₄²	1	4	yes	—
58	C₂²×C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₇₃	C₂²⋊C₄	1	4	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	1	yes	characteristic
63	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
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	M₁₆×C₇₃	—	2	32	no	—
69	C₂×C₈×C₇₃	—	2	4	no	—
70	C₂×C₈×C₇₃	—	2	4	no	—
71	C₂×C₈×C₇₃	—	2	16	no	—
72	C₂×C₈×C₇₃	C₂²⋊C₄	1	8	yes	complemented
73	C₄⋊C₄×C₇₃	—	2	16	no	—
74	C₄⋊C₄×C₇₃	—	4	4	no	—
75	C₄⋊C₄×C₇₃	—	4	4	no	—
76	C₂²⋊C₄×C₇₃	—	4	16	no	—
77	C₂³×C₇₃	—	2	4	no	—
78	C₂³×C₇₃	—	2	8	no	—
79	C₂³×C₇₃	—	2	8	no	—
80	C₂³×C₇₃	—	2	4	no	—
81	C₂³×C₇₃	—	2	4	no	—
82	C₂³×C₇₃	—	2	8	no	—
83	C₂³×C₇₃	—	4	8	no	—
84	C₂³×C₇₃	—	4	4	no	—
85	C₂³×C₇₃	—	4	4	no	—
86	C₂³×C₇₃	#95	1	4	yes	—
87	C₂³×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
88	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂³×C₇₃	#121	1	4	yes	—
90	C₂³×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
91	C₂³×C₇₃	#137	1	4	yes	—
92	C₂×C₄×C₇₃	—	2	4	no	—
93	C₂×C₄×C₇₃	—	2	8	no	—
94	C₂×C₄×C₇₃	—	2	8	no	—
95	C₂×C₄×C₇₃	—	2	4	no	—
96	C₂×C₄×C₇₃	—	2	4	no	—
97	C₂×C₄×C₇₃	—	2	16	no	—
98	C₂×C₄×C₇₃	—	4	8	no	—
99	C₂×C₄×C₇₃	—	4	8	no	—
100	C₂×C₄×C₇₃	—	4	8	no	—
101	C₂×C₄×C₇₃	—	8	8	no	—
102	C₂×C₄×C₇₃	—	8	8	no	—
103	C₂×C₄×C₇₃	#95	1	4	yes	—
104	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₇₃	#117	1	2	yes	—
109	C₂×C₄×C₇₃	C₄×D₄	1	8	yes	—
110	C₂×C₄×C₇₃	#120	1	4	yes	—
111	C₂×C₄×C₇₃	C₂²⋊Q₈	1	4	yes	—
112	C₂×C₄×C₇₃	#123	1	4	yes	—
113	C₈×C₇₃	—	2	8	no	—
114	C₂²×C₇₃	—	2	4	no	—
115	C₂²×C₇₃	—	2	8	no	—
116	C₂²×C₇₃	—	4	8	no	—
117	C₂²×C₇₃	—	4	8	no	—
118	C₂²×C₇₃	—	4	8	no	—
119	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
120	C₂²×C₇₃	#380	1	2	yes	—
121	C₂²×C₇₃	#386	1	4	yes	—
122	C₂²×C₇₃	#417	1	4	yes	—
123	C₂²×C₇₃	#418	1	2	yes	—
124	C₂²×C₇₃	#419	1	2	yes	—
125	C₂²×C₇₃	#428	1	4	yes	—
126	C₂²×C₇₃	#468	1	8	yes	—
127	C₂²×C₇₃	#469	1	4	yes	—
128	C₂²×C₇₃	#470	1	4	yes	—
129	C₄×C₇₃	—	2	4	no	—
130	C₄×C₇₃	—	8	8	no	—
131	C₄×C₇₃	C₄×C₂²⋊C₄	1	2	yes	—
132	C₄×C₇₃	#379	1	1	yes	characteristic
133	C₄×C₇₃	#381	1	1	yes	characteristic
134	C₄×C₇₃	#385	1	4	yes	—
135	#10722	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₇₃	—	4	4	no	—
137	C₂×C₇₃	C₂²⋊#95	1	1	yes	characteristic
138	C₂×C₇₃	#1736	1	2	yes	—
139	C₂×C₇₃	#1809	1	4	yes	—
140	C₂×C₇₃	#1894	1	4	yes	—
141	C₂×C₇₃	#1895	1	4	yes	—
142	C₂³×M₁₆	C₂×C₇₃	1	1	yes	characteristic
143	C₂×#380	C₂×C₇₃	1	2	yes	—
144	C₂×#340	C₂×C₇₃	1	4	yes	complemented
145	C₇₃	#10722	1	1	yes	characteristic, direct product factor, Sylow
146	C₂⁴×C₄	C₂²×C₇₃	1	1	yes	characteristic
147	C₂²×M₁₆	—	2	4	no	—
148	C₂²×M₁₆	—	2	8	no	—
149	C₂²×M₁₆	C₄×C₇₃	1	4	yes	complemented
150	C₂³×C₈	C₂²×C₇₃	1	2	yes	—
151	C₂²×C₄⋊C₄	C₂²×C₇₃	1	4	yes	—
152	C₂²×C₂²⋊C₄	—	2	2	no	—
153	#380	—	2	4	no	—
154	#380	C₄×C₇₃	1	8	yes	complemented
155	C₂×#95	—	2	2	no	—
156	#340	—	2	8	no	—
157	C₂⁵	D₄×C₇₃	1	1	yes	characteristic
158	C₂³×C₄	—	2	4	no	—
159	C₂³×C₄	—	2	4	no	—
160	C₂³×C₄	—	2	4	no	—
161	C₂³×C₄	C₂×C₄×C₇₃	1	2	yes	—
162	C₂³×C₄	C₂×C₄×C₇₃	1	4	yes	—
163	C₂³×C₄	D₄×C₇₃	1	1	yes	characteristic
164	C₂³×C₄	D₄×C₇₃	1	4	yes	—
165	C₂³×C₄	Q₈×C₇₃	1	2	yes	—
166	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
167	C₂×M₁₆	—	2	32	no	—
168	C₂×M₁₆	—	2	8	no	—
169	C₂×M₁₆	—	2	16	no	—
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	—	2	4	no	—
172	C₂²×C₈	C₂×C₄×C₇₃	1	4	yes	—
173	C₂²×C₈	D₄×C₇₃	1	8	yes	—
174	C₂×C₄⋊C₄	—	2	8	no	—
175	C₂×C₄⋊C₄	—	2	8	no	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	C₂×C₄×C₇₃	1	8	yes	—
178	C₂×C₂²⋊C₄	—	2	8	no	—
179	C₂×C₂²⋊C₄	—	4	8	no	—
180	#95	—	2	8	no	—
181	C₂⁴	—	2	4	no	—
182	C₂⁴	—	2	4	no	—
183	C₂⁴	—	2	4	no	—
184	C₂⁴	C₂²⋊C₄×C₇₃	1	4	yes	—
185	C₂⁴	C₄⋊C₄×C₇₃	1	2	yes	—
186	C₂⁴	C₂×D₄×C₇₃	1	1	yes	characteristic
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	16	no	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	4	8	no	—
196	C₂²×C₄	—	4	8	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	C₄²×C₇₃	1	4	yes	—
200	C₂²×C₄	C₂²⋊C₄×C₇₃	1	4	yes	—
201	C₂²×C₄	C₂²⋊C₄×C₇₃	1	4	yes	—
202	C₂²×C₄	C₄⋊C₄×C₇₃	1	2	yes	—
203	C₂²×C₄	C₄⋊C₄×C₇₃	1	8	yes	—
204	C₂²×C₄	C₂²×C₄×C₇₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₇₃	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
208	C₂²×C₄	C₂×Q₈×C₇₃	1	1	yes	characteristic
209	C₂²×C₄	#41×C₇₃	1	4	yes	—
210	M₁₆	—	2	32	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	16	no	—
214	C₂×C₈	C₂²⋊C₄×C₇₃	1	8	yes	complemented
215	C₄⋊C₄	—	2	16	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₂²⋊C₄	—	4	16	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	2	8	no	—
221	C₂³	—	2	8	no	—
222	C₂³	—	2	4	no	—
223	C₂³	—	2	4	no	—
224	C₂³	—	2	8	no	—
225	C₂³	—	4	8	no	—
226	C₂³	—	4	4	no	—
227	C₂³	—	4	4	no	—
228	C₂³	#95×C₇₃	1	4	yes	—
229	C₂³	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
230	C₂³	C₂×C₄⋊C₄×C₇₃	1	1	yes	characteristic
231	C₂³	#121×C₇₃	1	4	yes	—
232	C₂³	AGL(1,ℤ/8)×C₇₃	1	4	yes	—
233	C₂³	#137×C₇₃	1	4	yes	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	16	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	8	8	no	—
244	C₂×C₄	—	8	8	no	—
245	C₂×C₄	#95×C₇₃	1	4	yes	—
246	C₂×C₄	C₂×C₄²×C₇₃	1	1	yes	characteristic
247	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
248	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	1	yes	characteristic
249	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	2	yes	—
250	C₂×C₄	#117×C₇₃	1	2	yes	—
251	C₂×C₄	C₄×D₄×C₇₃	1	8	yes	—
252	C₂×C₄	#120×C₇₃	1	4	yes	—
253	C₂×C₄	C₂²⋊Q₈×C₇₃	1	4	yes	—
254	C₂×C₄	#123×C₇₃	1	4	yes	—
255	C₈	—	2	8	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	8	no	—
258	C₂²	—	4	8	no	—
259	C₂²	—	4	8	no	—
260	C₂²	—	4	8	no	—
261	C₂²	C₂×#95×C₇₃	1	1	yes	characteristic
262	C₂²	#380×C₇₃	1	2	yes	—
263	C₂²	#386×C₇₃	1	4	yes	—
264	C₂²	#417×C₇₃	1	4	yes	—
265	C₂²	#418×C₇₃	1	2	yes	—
266	C₂²	#419×C₇₃	1	2	yes	—
267	C₂²	#428×C₇₃	1	4	yes	—
268	C₂²	#468×C₇₃	1	8	yes	—
269	C₂²	#469×C₇₃	1	4	yes	—
270	C₂²	#470×C₇₃	1	4	yes	—
271	C₄	—	2	4	no	—
272	C₄	—	8	8	no	—
273	C₄	C₄×C₂²⋊C₄×C₇₃	1	2	yes	—
274	C₄	#379×C₇₃	1	1	yes	characteristic
275	C₄	#381×C₇₃	1	1	yes	characteristic
276	C₄	#385×C₇₃	1	4	yes	—
277	C₂	—	4	4	no	—
278	C₂	C₇₃×C₂²⋊#95	1	1	yes	characteristic
279	C₂	C₇₃×#1736	1	2	yes	—
280	C₂	C₇₃×#1809	1	4	yes	—
281	C₂	C₇₃×#1894	1	4	yes	—
282	C₂	C₇₃×#1895	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₇₃
Frattini, cosocle	C₂³×C₄	C₂³×C₇₃
prime radical	C₂⁵×C₇₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇₃ ◁ C₂⁴×C₄×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#340×C₇₃	C₂	semidirect
7. C₂²×M₁₆×C₇₃	C₄	semidirect
12. #380×C₇₃	C₄	semidirect
72. C₂×C₈×C₇₃	C₂²⋊C₄	semidirect
135. #10722	C₇₃	direct (N × H)
144. C₂×#340	C₂×C₇₃	semidirect
145. C₇₃	#10722	direct (N × H)
149. C₂²×M₁₆	C₄×C₇₃	semidirect
154. #380	C₄×C₇₃	semidirect
214. C₂×C₈	C₂²⋊C₄×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₇₃	2
C₂×#380×C₇₃	2
C₂×#340×C₇₃	2
#10722	73

Subgroups — C73×#10722 #32920126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#10722 #32920126