Subgroups — D₇₃⋊#1591 #34018482

Order 18688.

670 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₇₃⋊#446	C₂	1	2	yes	maximal
2	#22625571	C₂	1	2	yes	maximal, complemented
3	#22625570	C₂	1	2	yes	maximal, complemented
4	#22625569	C₂	1	4	yes	maximal, complemented
5	#22625476	C₂	1	1	yes	maximal, characteristic
6	#22625220	C₂	1	2	yes	maximal
7	C₈×#11924036	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
9	#17989639	—	2	4	no	—
10	#17989638	C₂²	1	4	yes	—
11	#17989637	C₂²	1	4	yes	—
12	#17989588	—	2	2	no	—
13	#17989588	C₂²	1	2	yes	—
14	#17989546	—	2	2	no	—
15	C₄×#11924036	C₂²	1	1	yes	characteristic
16	C₄×#11924036	C₂²	1	1	yes	characteristic
17	#17989451	C₂²	1	2	yes	—
18	#17989450	C₂²	1	2	yes	—
19	#17989437	C₂²	1	4	yes	—
20	Q₈⋊#10596833	C₂²	1	4	yes	—
21	C₈×#10596833	C₂²	1	2	yes	—
22	C₈×#10596833	C₂²	1	2	yes	—
23	C₂×Q₁₆×D₇₃	C₄	1	1	yes	characteristic
24	C₄×Q₈×D₇₃	C₂²	1	2	yes	—
25	C₄×Dic₂₉₂	C₄	1	1	yes	characteristic
26	Q₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
27	#17988681	C₂²	1	1	yes	characteristic
28	C₄×#11924018	C₄	1	2	yes	—
29	#17988679	C₂²	1	2	yes	—
30	#107×D₇₃	C₄	1	1	yes	characteristic
31	#103×D₇₃	C₄	1	2	yes	—
32	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
33	C₄×Q₁₆×C₇₃	C₄	1	1	yes	characteristic
34	C₂×#11924036	—	2	2	no	—
35	C₂×#11924036	—	2	1	no	—
36	C₂×#11924036	D₄	1	1	yes	characteristic
37	C₂×#11924036	D₄	1	1	yes	characteristic
38	#14556658	—	2	4	no	—
39	#14556658	—	2	2	no	—
40	C₄⋊#10596833	—	2	4	no	—
41	C₄⋊#10596833	C₂³	1	4	yes	—
42	#14556646	—	2	4	no	—
43	#14556646	—	2	4	no	—
44	C₄×#10596833	—	2	1	no	—
45	C₄×#10596833	D₄	1	2	yes	—
46	C₄×#10596833	C₂³	1	2	yes	—
47	C₄×#10596833	C₂³	1	2	yes	—
48	C₂×Q₈×D₇₃	C₂×C₄	1	2	yes	—
49	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
50	C₂×#11924018	C₂×C₄	1	2	yes	—
51	Q₁₆×D₇₃	—	2	4	no	—
52	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
54	Q₈×Dic₇₃	C₂³	1	2	yes	—
55	C₄×Dic₁₄₆	—	2	2	no	—
56	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
57	#14556556	—	2	2	no	—
58	#14556556	C₂³	1	2	yes	—
59	C₄⋊C₄×D₇₃	—	2	2	no	—
60	C₄⋊C₄×D₇₃	C₂×C₄	1	2	yes	—
61	C₄²×D₇₃	—	2	2	no	—
62	C₄²×D₇₃	C₂³	1	1	yes	characteristic
63	#14556525	C₂×C₄	1	1	yes	characteristic
64	#14556524	C₂×C₄	1	2	yes	—
65	#14556516	C₂×C₄	1	2	yes	—
66	#14556515	C₂×C₄	1	2	yes	—
67	#14556514	C₂×C₄	1	2	yes	—
68	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
70	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
71	C₂×Q₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
72	C₄×Q₈×C₇₃	C₂×C₄	1	2	yes	—
73	#107×C₇₃	C₂×C₄	1	1	yes	characteristic
74	#103×C₇₃	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
76	#11924036	—	2	4	no	—
77	#11924036	—	2	2	no	—
78	#11924036	—	4	4	no	—
79	#11924036	Q₁₆	1	4	yes	complemented
80	C₂×#10596833	—	2	2	no	—
81	C₂×#10596833	—	2	4	no	—
82	C₂×#10596833	C₂×D₄	1	2	yes	—
83	C₂×#10596833	C₂×D₄	1	2	yes	—
84	C₂×Dic₁₄₆	—	2	2	no	—
85	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
86	Q₈×D₇₃	—	2	4	no	—
87	Q₈×D₇₃	—	2	4	no	—
88	C₂×C₄×D₇₃	—	2	2	no	—
89	C₂×C₄×D₇₃	—	2	2	no	—
90	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
93	Dic₂₉₂	—	2	2	no	—
94	#11924018	—	2	4	no	—
95	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
96	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
97	C₈×D₇₃	—	2	2	no	—
98	C₈×D₇₃	M₁₆	1	4	yes	—
99	C₄⋊Dic₇₃	—	2	2	no	—
100	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
101	#11924007	—	2	2	no	—
102	#11924007	—	2	2	no	—
103	#11924007	—	2	2	no	—
104	C₄×Dic₇₃	—	2	2	no	—
105	C₄×Dic₇₃	—	2	2	no	—
106	C₄×Dic₇₃	—	2	2	no	—
107	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
108	C₄×Dic₇₃	#41	1	1	yes	characteristic
109	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
110	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
111	Q₁₆×C₇₃	—	2	2	no	—
112	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₄⋊C₄×C₇₃	—	2	2	no	—
115	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
116	C₄²×C₇₃	—	2	2	no	—
117	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
119	Q₈×C₇₃	—	2	2	no	—
120	Q₈×C₇₃	—	2	2	no	—
121	C₂×C₄×C₇₃	—	2	2	no	—
122	C₂×C₄×C₇₃	—	2	2	no	—
123	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×Dic₇₃	—	2	2	no	—
127	C₂×Dic₇₃	—	2	2	no	—
128	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
129	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₇₃	C₂×#41	1	1	yes	characteristic
131	C₄×D₇₃	—	2	2	no	—
132	C₄×D₇₃	—	4	4	no	—
133	C₄×D₇₃	—	4	4	no	—
134	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
135	C₄×D₇₃	#131	1	2	yes	—
136	C₄×D₇₃	C₂×Q₁₆	1	2	yes	—
137	C₄×D₇₃	#135	1	2	yes	—
138	Dic₁₄₆	—	2	2	no	—
139	Dic₁₄₆	—	2	2	no	—
140	Dic₁₄₆	—	4	4	no	—
141	#10596833	—	2	4	no	—
142	#10596833	—	2	2	no	—
143	#10596833	—	4	4	no	—
144	#10596833	C₂×Q₁₆	1	4	yes	complemented
145	C₈×C₇₃	—	2	1	no	—
146	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
147	#10596831	—	2	1	no	—
148	#10596831	C₂×M₁₆	1	2	yes	—
149	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
150	C₂×D₇₃	#436	1	2	yes	—
151	C₂×D₇₃	C₄×Q₁₆	1	2	yes	—
152	C₂×D₇₃	#443	1	2	yes	—
153	C₄×C₇₃	—	2	1	no	—
154	C₄×C₇₃	—	4	2	no	—
155	C₄×C₇₃	—	4	2	no	—
156	C₄×C₇₃	C₄×Q₁₆	1	2	yes	—
157	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
158	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
159	Dic₇₃	—	2	1	no	—
160	Dic₇₃	—	4	2	no	—
161	Dic₇₃	—	4	2	no	—
162	Dic₇₃	#436	1	2	yes	—
163	Dic₇₃	C₂²×Q₁₆	1	1	yes	characteristic
164	Dic₇₃	C₂×#135	1	1	yes	characteristic
165	C₂×#1591	—	73	1	no	maximal, Sylow
166	C₂×C₇₃	C₂×#436	1	1	yes	characteristic
167	C₂×C₇₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
168	C₂×C₇₃	C₂×#443	1	1	yes	characteristic
169	D₇₃	#1591	1	4	yes	complemented
170	C₂×#446	—	73	2	no	—
171	C₂×C₄×Q₁₆	—	73	1	no	—
172	#1591	—	73	4	no	—
173	#1591	—	73	2	no	—
174	#1591	—	73	2	no	—
175	C₂×#335	—	73	1	no	—
176	C₂×Q₈⋊C₈	—	73	2	no	—
177	C₂×C₈²	—	73	1	no	—
178	C₇₃	C₂×#1591	1	1	yes	characteristic, complemented, Sylow
179	C₂²×Q₁₆	—	73	1	no	—
180	C₂×C₄×Q₈	—	73	2	no	—
181	#446	—	73	4	no	—
182	#446	—	73	4	no	—
183	#446	—	146	4	no	—
184	C₄×Q₁₆	—	73	1	no	—
185	C₄×Q₁₆	—	73	2	no	—
186	C₄×Q₁₆	—	73	1	no	—
187	C₄×Q₁₆	—	73	2	no	—
188	C₄×Q₁₆	—	73	1	no	—
189	C₄×Q₁₆	#64408	1	1	yes	characteristic
190	C₂×#107	—	73	1	no	—
191	C₂×C₄⋊C₈	—	73	2	no	—
192	C₂×C₄⋊C₈	—	146	2	no	—
193	C₂×#103	—	73	2	no	—
194	C₂×#97	—	146	2	no	—
195	C₂×C₄×C₈	—	73	1	no	—
196	C₂×C₄×C₈	—	73	1	no	—
197	C₂×C₄×C₈	—	73	1	no	—
198	#335	—	73	2	no	—
199	#335	—	73	2	no	—
200	Q₈⋊C₈	—	73	4	no	—
201	Q₈⋊C₈	—	73	4	no	—
202	C₈²	—	73	2	no	—
203	C₈²	—	73	2	no	—
204	C₂²×Q₈	—	73	2	no	—
205	C₂×Q₁₆	—	73	2	no	—
206	C₂×Q₁₆	—	73	1	no	—
207	C₂×Q₁₆	—	146	4	no	—
208	C₂×Q₁₆	C₂×#64408	1	1	yes	characteristic
209	C₂²×C₈	—	73	1	no	—
210	C₂²×C₈	—	73	1	no	—
211	C₂²×C₈	—	73	1	no	—
212	C₂²×C₈	—	73	1	no	—
213	C₂²×C₈	—	146	2	no	—
214	C₂²×C₈	—	146	1	no	—
215	C₄×Q₈	—	73	2	no	—
216	C₄×Q₈	—	73	2	no	—
217	C₄×Q₈	—	73	2	no	—
218	C₄×Q₈	—	146	2	no	—
219	C₄×Q₈	—	146	2	no	—
220	C₄×Q₈	C₂×#64408	1	2	yes	—
221	C₂×C₄⋊C₄	—	73	2	no	—
222	C₂×C₄⋊C₄	—	146	2	no	—
223	C₂×C₄²	—	73	1	no	—
224	C₂×C₄²	—	146	2	no	—
225	#107	—	73	1	no	—
226	#107	—	73	2	no	—
227	#107	C₂×#64408	1	1	yes	characteristic
228	C₄⋊C₈	—	73	4	no	—
229	C₄⋊C₈	—	146	2	no	—
230	C₄⋊C₈	—	146	4	no	—
231	C₄⋊C₈	—	146	4	no	—
232	#103	—	73	2	no	—
233	#103	—	73	2	no	—
234	#103	—	73	2	no	—
235	#103	C₂×#64408	1	2	yes	—
236	#97	—	146	4	no	—
237	#97	—	146	4	no	—
238	C₄×C₈	—	73	2	no	—
239	C₄×C₈	—	73	2	no	—
240	C₄×C₈	—	73	1	no	—
241	C₄×C₈	—	73	1	no	—
242	C₄×C₈	—	73	2	no	—
243	C₄×C₈	—	73	1	no	—
244	C₄×C₈	—	146	1	no	—
245	C₄×C₈	C₂×#64408	1	1	yes	characteristic
246	C₂×Q₈	—	73	2	no	—
247	C₂×Q₈	—	146	2	no	—
248	C₂×Q₈	—	146	4	no	—
249	C₂×Q₈	—	146	4	no	—
250	C₂×Q₈	C₂²×#64408	1	2	yes	—
251	C₂²×C₄	—	73	1	no	—
252	C₂²×C₄	—	73	1	no	—
253	C₂²×C₄	—	73	1	no	—
254	C₂²×C₄	—	146	2	no	—
255	C₂²×C₄	—	146	2	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	146	4	no	—
258	Q₁₆	—	146	2	no	—
259	C₂×C₈	—	292	4	no	—
260	C₂×C₈	—	73	2	no	—
261	C₂×C₈	—	73	4	no	—
262	C₂×C₈	—	73	1	no	—
263	C₂×C₈	—	73	4	no	—
264	C₂×C₈	—	73	2	no	—
265	C₂×C₈	—	73	1	no	—
266	C₂×C₈	—	146	2	no	—
267	C₂×C₈	—	146	4	no	—
268	C₂×C₈	—	146	2	no	—
269	C₂×C₈	—	146	2	no	—
270	C₂×C₈	—	146	4	no	—
271	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
272	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	73	2	no	—
275	C₄⋊C₄	—	146	2	no	—
276	C₄⋊C₄	—	146	2	no	—
277	C₄⋊C₄	—	146	2	no	—
278	C₄⋊C₄	—	146	2	no	—
279	C₄⋊C₄	C₂²×#64408	1	2	yes	—
280	C₄²	—	2	2	no	—
281	C₄²	—	73	1	no	—
282	C₄²	—	73	1	no	—
283	C₄²	—	73	1	no	—
284	C₄²	—	146	2	no	—
285	C₄²	—	146	2	no	—
286	C₄²	—	146	2	no	—
287	C₄²	C₂²×#64408	1	1	yes	characteristic
288	C₂³	—	73	1	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	2	2	no	—
291	Q₈	—	292	4	no	—
292	Q₈	—	146	2	no	—
293	Q₈	—	146	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	292	4	no	—
297	C₂×C₄	—	292	4	no	—
298	C₂×C₄	—	73	1	no	—
299	C₂×C₄	—	73	1	no	—
300	C₂×C₄	—	73	2	no	—
301	C₂×C₄	—	73	2	no	—
302	C₂×C₄	—	73	2	no	—
303	C₂×C₄	—	73	2	no	—
304	C₂×C₄	—	73	1	no	—
305	C₂×C₄	—	146	2	no	—
306	C₂×C₄	—	146	2	no	—
307	C₂×C₄	—	146	2	no	—
308	C₂×C₄	D₄×#64408	1	1	yes	characteristic
309	C₂×C₄	D₄×#64408	1	1	yes	characteristic
310	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	292	4	no	—
313	C₈	—	73	4	no	—
314	C₈	—	73	2	no	—
315	C₈	—	146	2	no	—
316	C₈	—	146	4	no	—
317	C₈	—	146	1	no	—
318	C₈	D₇₃⋊M₁₆	1	2	yes	—
319	C₂²	—	73	2	no	—
320	C₂²	—	73	2	no	—
321	C₂²	—	73	2	no	—
322	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	292	2	no	—
327	C₄	—	292	2	no	—
328	C₄	—	73	2	no	—
329	C₄	—	73	1	no	—
330	C₄	—	73	1	no	—
331	C₄	—	146	1	no	—
332	C₄	Q₁₆×#64408	1	2	yes	—
333	C₄	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
334	C₄	D₇₃⋊#131	1	1	yes	characteristic
335	C₂	—	73	4	no	—
336	C₂	D₇₃⋊#436	1	1	yes	characteristic
337	C₂	C₂×Q₁₆×#64408	1	1	yes	characteristic
338	C₂	D₇₃⋊#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₄×Q₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂²×D₇₃	C₄×D₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	C₄×Q₁₆	#64408
hypocenter	C₇₃	C₂×#1591
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #22625571	C₂	semidirect
3. #22625570	C₂	semidirect
4. #22625569	C₂	semidirect
79. #11924036	Q₁₆	semidirect
144. #10596833	C₂×Q₁₆	semidirect
169. D₇₃	#1591	semidirect
178. C₇₃	C₂×#1591	semidirect

Maximal subgroups

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

subgroup	index
D₇₃⋊#446	2
#22625571	2
#22625570	2
#22625569	2
#22625476	2
#22625220	2
C₈×#11924036	2
C₄×Q₁₆×D₇₃	2
C₂×#1591	73

Subgroups — D73⋊#1591 #34018482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇₃⋊#1591 #34018482