Subgroups — C₇₃×#23505 #32932909

Order 18688.

2134 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#418×C₇₃	C₂	1	2	yes	maximal, complemented
3	C₂×#418×C₇₃	C₂	1	2	yes	maximal, complemented
4	C₇₃×#2296	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#2041	C₂	1	8	yes	maximal, complemented
6	C₇₃×#1728	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₇₃	—	2	2	no	—
8	C₂³×D₄×C₇₃	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
10	C₂×#127×C₇₃	—	2	8	no	—
11	C₂×#127×C₇₃	—	2	2	no	—
12	C₂×#127×C₇₃	C₂²	1	4	yes	—
13	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
16	#418×C₇₃	—	2	8	no	—
17	#418×C₇₃	—	2	8	no	—
18	C₂×#102×C₇₃	C₂²	1	8	yes	complemented
19	C₂×#102×C₇₃	C₂²	1	8	yes	complemented
20	#389×C₇₃	C₂²	1	4	yes	complemented
21	#385×C₇₃	—	2	2	no	—
22	C₄×C₄⋊C₄×C₇₃	—	2	2	no	—
23	#336×C₇₃	C₂²	1	4	yes	complemented
24	C₂⁵×C₇₃	—	2	2	no	—
25	C₂²×D₄×C₇₃	—	2	16	no	—
26	C₂²×D₄×C₇₃	—	2	4	no	—
27	C₂²×D₄×C₇₃	—	2	2	no	—
28	C₂²×D₄×C₇₃	—	2	4	no	—
29	C₂²×D₄×C₇₃	—	2	8	no	—
30	C₂²×D₄×C₇₃	—	2	4	no	—
31	C₂²×D₄×C₇₃	—	2	4	no	—
32	C₂²×D₄×C₇₃	—	2	4	no	—
33	C₂²×D₄×C₇₃	—	4	16	no	—
34	C₂²×D₄×C₇₃	D₄	1	8	yes	—
35	C₂²×D₄×C₇₃	C₂³	1	4	yes	—
36	C₂³×C₄×C₇₃	D₄	1	2	yes	—
37	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₇₃	—	2	4	no	—
39	C₂×M₁₆×C₇₃	—	2	4	no	—
40	C₂×M₁₆×C₇₃	—	2	4	no	—
41	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
42	#127×C₇₃	—	2	16	no	—
43	#127×C₇₃	—	2	16	no	—
44	#117×C₇₃	—	2	2	no	—
45	#117×C₇₃	—	2	4	no	—
46	#117×C₇₃	D₄	1	4	yes	complemented
47	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
48	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
50	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
51	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
52	C₂×C₄²×C₇₃	—	2	1	no	—
53	C₂×C₄²×C₇₃	—	2	4	no	—
54	C₂×C₄²×C₇₃	—	2	1	no	—
55	C₂×C₄²×C₇₃	D₄	1	2	yes	—
56	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
57	#102×C₇₃	—	2	16	no	—
58	#102×C₇₃	—	2	16	no	—
59	#95×C₇₃	—	2	4	no	—
60	C₂⁴×C₇₃	—	2	8	no	—
61	C₂⁴×C₇₃	—	2	4	no	—
62	C₂⁴×C₇₃	—	4	16	no	—
63	C₂⁴×C₇₃	—	4	4	no	—
64	C₂⁴×C₇₃	—	4	4	no	—
65	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₂×D₄×C₇₃	—	2	32	no	—
67	C₂×D₄×C₇₃	—	2	8	no	—
68	C₂×D₄×C₇₃	—	2	16	no	—
69	C₂×D₄×C₇₃	—	2	16	no	—
70	C₂×D₄×C₇₃	—	2	4	no	—
71	C₂×D₄×C₇₃	—	2	4	no	—
72	C₂×D₄×C₇₃	—	2	4	no	—
73	C₂×D₄×C₇₃	—	2	4	no	—
74	C₂×D₄×C₇₃	—	2	4	no	—
75	C₂×D₄×C₇₃	—	4	64	no	—
76	C₂×D₄×C₇₃	—	4	16	no	—
77	C₂×D₄×C₇₃	—	4	8	no	—
78	C₂×D₄×C₇₃	—	4	4	no	—
79	C₂×D₄×C₇₃	—	4	8	no	—
80	C₂×D₄×C₇₃	—	4	4	no	—
81	C₂×D₄×C₇₃	—	4	16	no	—
82	C₂×D₄×C₇₃	C₂×D₄	1	8	yes	—
83	C₂²×C₄×C₇₃	—	2	4	no	—
84	C₂²×C₄×C₇₃	—	2	8	no	—
85	C₂²×C₄×C₇₃	—	2	2	no	—
86	C₂²×C₄×C₇₃	—	2	2	no	—
87	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
88	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
90	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₇₃	#41	1	4	yes	—
92	C₂²×C₄×C₇₃	#41	1	2	yes	—
93	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
94	M₁₆×C₇₃	—	4	8	no	—
95	C₂×C₈×C₇₃	—	2	4	no	—
96	C₂×C₈×C₇₃	—	2	4	no	—
97	C₂×C₈×C₇₃	—	2	4	no	—
98	C₄⋊C₄×C₇₃	—	2	2	no	—
99	C₄⋊C₄×C₇₃	—	2	4	no	—
100	C₄⋊C₄×C₇₃	—	4	8	no	—
101	C₄⋊C₄×C₇₃	C₂×D₄	1	4	yes	complemented
102	C₂²⋊C₄×C₇₃	—	2	4	no	—
103	C₂²⋊C₄×C₇₃	—	4	2	no	—
104	C₄²×C₇₃	—	2	4	no	—
105	C₄²×C₇₃	—	2	4	no	—
106	C₄²×C₇₃	—	2	4	no	—
107	C₄²×C₇₃	—	4	2	no	—
108	C₂³×C₇₃	—	2	4	no	—
109	C₂³×C₇₃	—	2	8	no	—
110	C₂³×C₇₃	—	4	32	no	—
111	C₂³×C₇₃	—	4	4	no	—
112	C₂³×C₇₃	—	4	4	no	—
113	C₂³×C₇₃	—	4	16	no	—
114	C₂³×C₇₃	—	4	16	no	—
115	C₂³×C₇₃	—	8	16	no	—
116	C₂³×C₇₃	—	8	16	no	—
117	C₂³×C₇₃	#120	1	2	yes	—
118	C₂³×C₇₃	#121	1	2	yes	—
119	C₂³×C₇₃	#124	1	2	yes	—
120	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
121	D₄×C₇₃	—	2	8	no	—
122	D₄×C₇₃	—	2	8	no	—
123	D₄×C₇₃	—	4	64	no	—
124	D₄×C₇₃	—	4	4	no	—
125	D₄×C₇₃	—	4	4	no	—
126	D₄×C₇₃	—	4	16	no	—
127	C₂×C₄×C₇₃	—	2	16	no	—
128	C₂×C₄×C₇₃	—	2	4	no	—
129	C₂×C₄×C₇₃	—	2	4	no	—
130	C₂×C₄×C₇₃	—	2	4	no	—
131	C₂×C₄×C₇₃	—	4	4	no	—
132	C₂×C₄×C₇₃	—	4	2	no	—
133	C₂×C₄×C₇₃	—	4	4	no	—
134	C₂×C₄×C₇₃	#120	1	2	yes	—
135	C₂×C₄×C₇₃	#121	1	8	yes	—
136	C₂×C₄×C₇₃	#121	1	2	yes	—
137	C₂×C₄×C₇₃	#123	1	8	yes	—
138	C₂×C₄×C₇₃	#124	1	2	yes	—
139	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	8	yes	—
140	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
142	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
143	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
144	C₈×C₇₃	—	4	4	no	—
145	C₂²×C₇₃	—	2	4	no	—
146	C₂²×C₇₃	—	2	4	no	—
147	C₂²×C₇₃	—	2	4	no	—
148	C₂²×C₇₃	—	4	8	no	—
149	C₂²×C₇₃	—	4	16	no	—
150	C₂²×C₇₃	—	8	32	no	—
151	C₂²×C₇₃	—	8	8	no	—
152	C₂²×C₇₃	—	8	8	no	—
153	C₂²×C₇₃	#394	1	4	yes	—
154	C₂²×C₇₃	C₂×#120	1	1	yes	characteristic
155	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
156	C₂²×C₇₃	C₂×#124	1	1	yes	characteristic
157	C₂²×C₇₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
158	C₄×C₇₃	—	2	8	no	—
159	C₄×C₇₃	—	4	4	no	—
160	C₄×C₇₃	—	8	2	no	—
161	C₄×C₇₃	#394	1	4	yes	—
162	C₄×C₇₃	C₂×#121	1	2	yes	—
163	C₄×C₇₃	C₂×#123	1	2	yes	—
164	#23505	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂×C₇₃	—	2	4	no	—
166	C₂×C₇₃	—	8	8	no	—
167	C₂×C₇₃	C₂×#394	1	1	yes	characteristic
168	C₂×C₇₃	#3003	1	2	yes	—
169	C₂×C₇₃	#3038	1	2	yes	—
170	C₂×C₇₃	#3138	1	2	yes	—
171	C₂²×#127	C₂×C₇₃	1	1	yes	characteristic
172	C₂×#418	C₂×C₇₃	1	2	yes	complemented
173	C₂×#418	C₂×C₇₃	1	2	yes	complemented
174	#2296	C₂×C₇₃	1	1	yes	characteristic, complemented
175	#2041	C₂×C₇₃	1	8	yes	complemented
176	#1728	C₂×C₇₃	1	1	yes	characteristic, complemented
177	C₇₃	#23505	1	1	yes	characteristic, direct product factor, Sylow
178	C₂³×D₄	—	2	2	no	—
179	C₂³×D₄	C₂²×C₇₃	1	2	yes	—
180	C₂²×M₁₆	C₂²×C₇₃	1	2	yes	—
181	C₂×#127	—	2	8	no	—
182	C₂×#127	—	2	2	no	—
183	C₂×#127	C₂²×C₇₃	1	4	yes	—
184	C₂×#117	C₂²×C₇₃	1	1	yes	characteristic
185	C₂×#117	C₂²×C₇₃	1	1	yes	characteristic
186	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
187	#418	—	2	8	no	—
188	#418	—	2	8	no	—
189	C₂×#102	C₂²×C₇₃	1	8	yes	complemented
190	C₂×#102	C₂²×C₇₃	1	8	yes	complemented
191	#389	C₂²×C₇₃	1	4	yes	complemented
192	#385	—	2	2	no	—
193	C₄×C₄⋊C₄	—	2	2	no	—
194	#336	C₂²×C₇₃	1	4	yes	complemented
195	C₂⁵	—	2	2	no	—
196	C₂²×D₄	—	2	16	no	—
197	C₂²×D₄	—	2	4	no	—
198	C₂²×D₄	—	2	2	no	—
199	C₂²×D₄	—	2	4	no	—
200	C₂²×D₄	—	2	8	no	—
201	C₂²×D₄	—	2	4	no	—
202	C₂²×D₄	—	2	4	no	—
203	C₂²×D₄	—	2	4	no	—
204	C₂²×D₄	—	4	16	no	—
205	C₂²×D₄	D₄×C₇₃	1	8	yes	—
206	C₂²×D₄	C₂³×C₇₃	1	4	yes	—
207	C₂³×C₄	D₄×C₇₃	1	2	yes	—
208	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
209	C₂×M₁₆	—	2	4	no	—
210	C₂×M₁₆	—	2	4	no	—
211	C₂×M₁₆	—	2	4	no	—
212	C₂²×C₈	C₂³×C₇₃	1	4	yes	—
213	#127	—	2	16	no	—
214	#127	—	2	16	no	—
215	#117	—	2	2	no	—
216	#117	—	2	4	no	—
217	#117	D₄×C₇₃	1	4	yes	complemented
218	C₂×C₄⋊C₄	—	2	4	no	—
219	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
220	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
221	C₂×C₂²⋊C₄	—	2	1	no	—
222	C₂×C₂²⋊C₄	—	2	1	no	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	—	2	4	no	—
225	C₂×C₄²	—	2	1	no	—
226	C₂×C₄²	D₄×C₇₃	1	2	yes	—
227	C₂×C₄²	C₂³×C₇₃	1	2	yes	—
228	#102	—	2	16	no	—
229	#102	—	2	16	no	—
230	#95	—	2	4	no	—
231	C₂⁴	—	2	8	no	—
232	C₂⁴	—	2	4	no	—
233	C₂⁴	—	4	16	no	—
234	C₂⁴	—	4	4	no	—
235	C₂⁴	—	4	4	no	—
236	C₂⁴	C₂×D₄×C₇₃	1	1	yes	characteristic
237	C₂×D₄	—	2	32	no	—
238	C₂×D₄	—	2	8	no	—
239	C₂×D₄	—	2	16	no	—
240	C₂×D₄	—	2	16	no	—
241	C₂×D₄	—	2	4	no	—
242	C₂×D₄	—	2	4	no	—
243	C₂×D₄	—	2	4	no	—
244	C₂×D₄	—	2	4	no	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	—	4	64	no	—
247	C₂×D₄	—	4	16	no	—
248	C₂×D₄	—	4	4	no	—
249	C₂×D₄	—	4	8	no	—
250	C₂×D₄	—	4	8	no	—
251	C₂×D₄	—	4	4	no	—
252	C₂×D₄	—	4	16	no	—
253	C₂×D₄	C₂×D₄×C₇₃	1	8	yes	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	2	8	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
259	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
261	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
262	C₂²×C₄	#41×C₇₃	1	4	yes	—
263	C₂²×C₄	#41×C₇₃	1	2	yes	—
264	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
265	M₁₆	—	4	8	no	—
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	2	4	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	4	8	no	—
272	C₄⋊C₄	C₂×D₄×C₇₃	1	4	yes	complemented
273	C₂²⋊C₄	—	2	4	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₄²	—	2	4	no	—
276	C₄²	—	2	4	no	—
277	C₄²	—	2	4	no	—
278	C₄²	—	4	2	no	—
279	C₂³	—	2	4	no	—
280	C₂³	—	2	8	no	—
281	C₂³	—	4	32	no	—
282	C₂³	—	4	16	no	—
283	C₂³	—	4	4	no	—
284	C₂³	—	4	4	no	—
285	C₂³	—	4	16	no	—
286	C₂³	—	8	16	no	—
287	C₂³	—	8	16	no	—
288	C₂³	#120×C₇₃	1	2	yes	—
289	C₂³	#121×C₇₃	1	2	yes	—
290	C₂³	#124×C₇₃	1	2	yes	—
291	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
292	D₄	—	2	8	no	—
293	D₄	—	2	8	no	—
294	D₄	—	4	64	no	—
295	D₄	—	4	16	no	—
296	D₄	—	4	4	no	—
297	D₄	—	4	4	no	—
298	C₂×C₄	—	2	16	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	#120×C₇₃	1	2	yes	—
306	C₂×C₄	#121×C₇₃	1	2	yes	—
307	C₂×C₄	#121×C₇₃	1	8	yes	—
308	C₂×C₄	#123×C₇₃	1	8	yes	—
309	C₂×C₄	#124×C₇₃	1	2	yes	—
310	C₂×C₄	AGL(1,ℤ/8)×C₇₃	1	8	yes	—
311	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₄×C₇₃	1	2	yes	—
313	C₂×C₄	C₂×#41×C₇₃	1	2	yes	—
314	C₂×C₄	C₂×#41×C₇₃	1	1	yes	characteristic
315	C₈	—	4	4	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	4	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	4	16	no	—
320	C₂²	—	4	8	no	—
321	C₂²	—	8	32	no	—
322	C₂²	—	8	8	no	—
323	C₂²	—	8	8	no	—
324	C₂²	#394×C₇₃	1	4	yes	—
325	C₂²	C₂×#120×C₇₃	1	1	yes	characteristic
326	C₂²	C₂×#121×C₇₃	1	1	yes	characteristic
327	C₂²	C₂×#124×C₇₃	1	1	yes	characteristic
328	C₂²	C₂×AGL(1,ℤ/8)×C₇₃	1	4	yes	—
329	C₄	—	2	8	no	—
330	C₄	—	4	4	no	—
331	C₄	—	8	2	no	—
332	C₄	#394×C₇₃	1	4	yes	—
333	C₄	C₂×#121×C₇₃	1	2	yes	—
334	C₄	C₂×#123×C₇₃	1	2	yes	—
335	C₂	—	2	4	no	—
336	C₂	—	8	8	no	—
337	C₂	C₂×#394×C₇₃	1	1	yes	characteristic
338	C₂	C₇₃×#3003	1	2	yes	—
339	C₂	C₇₃×#3038	1	2	yes	—
340	C₂	C₇₃×#3138	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₃
prime radical	C₂²×#127×C₇₃	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₂²×C₄²×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#418×C₇₃	C₂	semidirect
3. C₂×#418×C₇₃	C₂	semidirect
4. C₇₃×#2296	C₂	semidirect
5. C₇₃×#2041	C₂	semidirect
6. C₇₃×#1728	C₂	semidirect
18. C₂×#102×C₇₃	C₂²	semidirect
19. C₂×#102×C₇₃	C₂²	semidirect
20. #389×C₇₃	C₂²	semidirect
23. #336×C₇₃	C₂²	semidirect
46. #117×C₇₃	D₄	semidirect
101. C₄⋊C₄×C₇₃	C₂×D₄	semidirect
164. #23505	C₇₃	direct (N × H)
172. C₂×#418	C₂×C₇₃	semidirect
173. C₂×#418	C₂×C₇₃	semidirect
174. #2296	C₂×C₇₃	semidirect
175. #2041	C₂×C₇₃	semidirect
176. #1728	C₂×C₇₃	semidirect
177. C₇₃	#23505	direct (N × H)
189. C₂×#102	C₂²×C₇₃	semidirect
190. C₂×#102	C₂²×C₇₃	semidirect
191. #389	C₂²×C₇₃	semidirect
194. #336	C₂²×C₇₃	semidirect
217. #117	D₄×C₇₃	semidirect
272. C₄⋊C₄	C₂×D₄×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#127×C₇₃	2
C₂×#418×C₇₃	2
C₇₃×#2296	2
C₇₃×#2041	2
C₇₃×#1728	2
#23505	73

Subgroups — C73×#23505 #32932909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#23505 #32932909