Subgroups — C₇₃×#22612 #32932016

Order 18688.

822 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	C₂²×#107×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#2345	C₂	1	2	yes	maximal
3	C₇₃×#1942	C₂	1	2	yes	maximal, complemented
4	C₇₃×#1921	C₂	1	4	yes	maximal, complemented
5	C₇₃×#1824	C₂	1	2	yes	maximal
6	C₇₃×#1808	C₂	1	1	yes	maximal, characteristic
7	C₄×#107×C₇₃	C₂	1	2	yes	maximal, complemented
8	C₈×C₂²⋊C₄×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
11	C₂×#107×C₇₃	C₄	1	8	yes	complemented
12	C₂×#107×C₇₃	C₂²	1	2	yes	—
13	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×#107×C₇₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
18	#389×C₇₃	C₂²	1	4	yes	—
19	#387×C₇₃	—	2	4	no	—
20	#380×C₇₃	—	2	2	no	—
21	#380×C₇₃	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₇₃	C₂²	1	4	yes	—
23	C₄×C₂²⋊C₄×C₇₃	—	2	2	no	—
24	C₄×C₂²⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
25	#340×C₇₃	C₂²	1	4	yes	—
26	#340×C₇₃	C₂²	1	4	yes	—
27	#340×C₇₃	C₂²	1	2	yes	—
28	#336×C₇₃	C₂²	1	2	yes	—
29	C₂³×C₄×C₇₃	—	2	2	no	—
30	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	—	2	4	no	—
32	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	8	yes	—
40	C₂×C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
43	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₇₃	D₄	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₇₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₃	—	2	4	no	—
47	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
48	#107×C₇₃	—	2	4	no	—
49	#107×C₇₃	C₂×C₄	1	8	yes	—
50	#107×C₇₃	C₂×C₄	1	4	yes	—
51	#107×C₇₃	C₂×C₄	1	4	yes	—
52	C₂²⋊C₈×C₇₃	—	2	2	no	—
53	C₄×C₈×C₇₃	—	2	4	no	—
54	#95×C₇₃	—	2	4	no	—
55	#95×C₇₃	—	2	4	no	—
56	#95×C₇₃	—	2	2	no	—
57	#95×C₇₃	D₄	1	2	yes	—
58	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	—	2	2	no	—
60	C₂²×C₄×C₇₃	—	2	2	no	—
61	C₂²×C₄×C₇₃	—	2	2	no	—
62	C₂²×C₄×C₇₃	—	2	2	no	—
63	C₂²×C₄×C₇₃	—	2	8	no	—
64	C₂²×C₄×C₇₃	—	2	4	no	—
65	C₂²×C₄×C₇₃	—	8	4	no	—
66	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₇₃	#41	1	2	yes	—
71	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₇₃	—	2	2	no	—
73	C₂×C₈×C₇₃	—	2	8	no	—
74	C₂×C₈×C₇₃	—	2	2	no	—
75	C₂×C₈×C₇₃	—	2	2	no	—
76	C₂×C₈×C₇₃	—	2	2	no	—
77	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₇₃	#41	1	4	yes	—
80	C₄⋊C₄×C₇₃	—	2	8	no	—
81	C₄⋊C₄×C₇₃	—	2	2	no	—
82	C₄⋊C₄×C₇₃	—	2	2	no	—
83	C₄⋊C₄×C₇₃	—	2	4	no	—
84	C₄⋊C₄×C₇₃	—	2	4	no	—
85	C₄⋊C₄×C₇₃	—	8	4	no	—
86	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
87	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
88	C₂²⋊C₄×C₇₃	—	2	2	no	—
89	C₂²⋊C₄×C₇₃	—	4	4	no	—
90	C₂²⋊C₄×C₇₃	Q₁₆	1	4	yes	—
91	C₄²×C₇₃	—	2	2	no	—
92	C₄²×C₇₃	—	2	2	no	—
93	C₄²×C₇₃	—	8	4	no	—
94	C₂³×C₇₃	—	2	2	no	—
95	C₂³×C₇₃	—	2	2	no	—
96	C₂³×C₇₃	C₄×D₄	1	2	yes	—
97	C₂³×C₇₃	C₂×Q₁₆	1	2	yes	—
98	C₂³×C₇₃	#135	1	2	yes	—
99	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
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	4	no	—
106	C₂×C₄×C₇₃	—	2	4	no	—
107	C₂×C₄×C₇₃	—	2	4	no	—
108	C₂×C₄×C₇₃	—	2	2	no	—
109	C₂×C₄×C₇₃	—	2	2	no	—
110	C₂×C₄×C₇₃	—	2	2	no	—
111	C₂×C₄×C₇₃	—	4	4	no	—
112	C₂×C₄×C₇₃	—	4	4	no	—
113	C₂×C₄×C₇₃	—	4	4	no	—
114	C₂×C₄×C₇₃	—	4	4	no	—
115	C₂×C₄×C₇₃	—	4	4	no	—
116	C₂×C₄×C₇₃	—	8	8	no	—
117	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₇₃	C₂×Q₁₆	1	4	yes	—
119	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
121	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₇₃	2₋¹⁺⁴	1	2	yes	—
123	C₈×C₇₃	—	2	2	no	—
124	C₈×C₇₃	—	4	2	no	—
125	C₈×C₇₃	#117	1	4	yes	—
126	C₂²×C₇₃	—	2	4	no	—
127	C₂²×C₇₃	—	2	4	no	—
128	C₂²×C₇₃	—	2	2	no	—
129	C₂²×C₇₃	—	2	2	no	—
130	C₂²×C₇₃	C₄×Q₁₆	1	4	yes	—
131	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂²×C₇₃	#547	1	2	yes	—
133	C₂²×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
134	C₂²×C₇₃	C₂×#135	1	1	yes	characteristic
135	C₂²×C₇₃	#576	1	2	yes	—
136	C₄×C₇₃	—	2	4	no	—
137	C₄×C₇₃	—	2	2	no	—
138	C₄×C₇₃	—	4	2	no	—
139	C₄×C₇₃	—	4	4	no	—
140	C₄×C₇₃	—	4	4	no	—
141	C₄×C₇₃	—	8	4	no	—
142	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
143	C₄×C₇₃	#519	1	1	yes	characteristic
144	C₄×C₇₃	#552	1	2	yes	—
145	#22612	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂×C₇₃	—	2	4	no	—
147	C₂×C₇₃	#2344	1	1	yes	characteristic
148	C₂×C₇₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
149	C₂×C₇₃	#2948	1	1	yes	characteristic
150	C₂×C₇₃	#3338	1	2	yes	—
151	C₂×C₇₃	#3339	1	2	yes	—
152	C₂²×#107	C₂×C₇₃	1	1	yes	characteristic
153	#2345	C₂×C₇₃	1	2	yes	—
154	#1942	C₂×C₇₃	1	2	yes	complemented
155	#1921	C₂×C₇₃	1	4	yes	complemented
156	#1824	C₂×C₇₃	1	2	yes	—
157	#1808	C₂×C₇₃	1	1	yes	characteristic
158	C₄×#107	C₂×C₇₃	1	2	yes	complemented
159	C₈×C₂²⋊C₄	C₂×C₇₃	1	1	yes	characteristic
160	C₇₃	#22612	1	1	yes	characteristic, direct product factor, Sylow
161	C₂³×C₈	C₂²×C₇₃	1	1	yes	characteristic
162	C₂²×C₄⋊C₄	C₂²×C₇₃	1	2	yes	—
163	C₂×#107	C₄×C₇₃	1	8	yes	complemented
164	C₂×#107	C₂²×C₇₃	1	2	yes	—
165	C₂×#107	C₂²×C₇₃	1	2	yes	—
166	C₂×#107	C₂²×C₇₃	1	1	yes	characteristic
167	C₂×#107	C₂²×C₇₃	1	1	yes	characteristic
168	C₂×C₂²⋊C₈	C₂²×C₇₃	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂²×C₇₃	1	2	yes	—
170	#389	C₂²×C₇₃	1	4	yes	—
171	#387	—	2	4	no	—
172	#380	—	2	2	no	—
173	#380	C₂²×C₇₃	1	2	yes	—
174	C₄×C₄⋊C₄	C₂²×C₇₃	1	4	yes	—
175	C₄×C₂²⋊C₄	—	2	2	no	—
176	C₄×C₂²⋊C₄	C₂²×C₇₃	1	1	yes	characteristic
177	#340	C₂²×C₇₃	1	2	yes	—
178	#340	C₂²×C₇₃	1	4	yes	—
179	#340	C₂²×C₇₃	1	4	yes	—
180	#336	C₂²×C₇₃	1	2	yes	—
181	C₂³×C₄	—	2	2	no	—
182	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
183	C₂²×C₈	—	2	4	no	—
184	C₂²×C₈	C₂×C₄×C₇₃	1	4	yes	—
185	C₂²×C₈	C₂³×C₇₃	1	2	yes	—
186	C₂²×C₈	C₂³×C₇₃	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₇₃	1	1	yes	characteristic
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	—	2	4	no	—
191	C₂×C₄⋊C₄	C₂×C₄×C₇₃	1	8	yes	—
192	C₂×C₄⋊C₄	C₂³×C₇₃	1	4	yes	—
193	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
194	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	D₄×C₇₃	1	1	yes	characteristic
197	C₂×C₂²⋊C₄	D₄×C₇₃	1	1	yes	characteristic
198	C₂×C₄²	—	2	4	no	—
199	C₂×C₄²	C₂³×C₇₃	1	2	yes	—
200	#107	—	2	4	no	—
201	#107	C₂×C₄×C₇₃	1	8	yes	—
202	#107	C₂×C₄×C₇₃	1	4	yes	—
203	#107	C₂×C₄×C₇₃	1	4	yes	—
204	C₂²⋊C₈	—	2	2	no	—
205	C₄×C₈	—	2	4	no	—
206	#95	—	2	4	no	—
207	#95	—	2	4	no	—
208	#95	—	2	2	no	—
209	#95	D₄×C₇₃	1	2	yes	—
210	C₂⁴	C₂×D₄×C₇₃	1	1	yes	characteristic
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	8	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	8	4	no	—
218	C₂²×C₄	C₂²×C₄×C₇₃	1	2	yes	—
219	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
220	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
222	C₂²×C₄	#41×C₇₃	1	2	yes	—
223	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	C₂²×C₄×C₇₃	1	2	yes	—
230	C₂×C₈	C₂²×C₄×C₇₃	1	2	yes	—
231	C₂×C₈	#41×C₇₃	1	4	yes	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	8	4	no	—
238	C₄⋊C₄	C₂²×C₄×C₇₃	1	4	yes	—
239	C₄⋊C₄	C₂²×C₄×C₇₃	1	4	yes	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	4	4	no	—
242	C₂²⋊C₄	Q₁₆×C₇₃	1	4	yes	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	8	4	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	C₄×D₄×C₇₃	1	2	yes	—
249	C₂³	C₂×Q₁₆×C₇₃	1	2	yes	—
250	C₂³	#135×C₇₃	1	2	yes	—
251	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
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	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	8	8	no	—
269	C₂×C₄	C₄×D₄×C₇₃	1	2	yes	—
270	C₂×C₄	C₂×Q₁₆×C₇₃	1	4	yes	—
271	C₂×C₄	C₂³×C₄×C₇₃	1	1	yes	characteristic
272	C₂×C₄	C₂×#41×C₇₃	1	2	yes	—
273	C₂×C₄	C₂×#41×C₇₃	1	1	yes	characteristic
274	C₂×C₄	2₋¹⁺⁴×C₇₃	1	2	yes	—
275	C₈	—	2	2	no	—
276	C₈	—	4	2	no	—
277	C₈	#117×C₇₃	1	4	yes	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	C₄×Q₁₆×C₇₃	1	4	yes	—
283	C₂²	C₂×C₄×D₄×C₇₃	1	1	yes	characteristic
284	C₂²	#547×C₇₃	1	2	yes	—
285	C₂²	C₂²×Q₁₆×C₇₃	1	1	yes	characteristic
286	C₂²	C₂×#135×C₇₃	1	1	yes	characteristic
287	C₂²	#576×C₇₃	1	2	yes	—
288	C₄	—	2	4	no	—
289	C₄	—	2	2	no	—
290	C₄	—	4	2	no	—
291	C₄	—	4	4	no	—
292	C₄	—	4	4	no	—
293	C₄	—	8	4	no	—
294	C₄	C₂×#117×C₇₃	1	1	yes	characteristic
295	C₄	#519×C₇₃	1	1	yes	characteristic
296	C₄	#552×C₇₃	1	2	yes	—
297	C₂	—	2	4	no	—
298	C₂	C₇₃×#2344	1	1	yes	characteristic
299	C₂	C₂×C₄×Q₁₆×C₇₃	1	1	yes	characteristic
300	C₂	C₇₃×#2948	1	1	yes	characteristic
301	C₂	C₇₃×#3338	1	2	yes	—
302	C₂	C₇₃×#3339	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₃
prime radical	C₂⁴×C₇₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₄×C₂²⋊C₄×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇₃×#1942	C₂	semidirect
4. C₇₃×#1921	C₂	semidirect
7. C₄×#107×C₇₃	C₂	semidirect
11. C₂×#107×C₇₃	C₄	semidirect
145. #22612	C₇₃	direct (N × H)
154. #1942	C₂×C₇₃	semidirect
155. #1921	C₂×C₇₃	semidirect
158. C₄×#107	C₂×C₇₃	semidirect
160. C₇₃	#22612	direct (N × H)
163. C₂×#107	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×#107×C₇₃	2
C₇₃×#2345	2
C₇₃×#1942	2
C₇₃×#1921	2
C₇₃×#1824	2
C₇₃×#1808	2
C₄×#107×C₇₃	2
C₈×C₂²⋊C₄×C₇₃	2
#22612	73

Subgroups — C73×#22612 #32932016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#22612 #32932016