Subgroups — #32981998

Order 18688.

504 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618495	C₂	1	1	yes	maximal, characteristic
2	#22608616	C₂	1	2	yes	maximal, complemented
3	#22608245	C₂	1	1	yes	maximal, characteristic
4	#22607789	C₂	1	2	yes	maximal, complemented
5	C₇₃×#2628	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924012	—	2	1	no	—
7	#17988640	C₂²	1	2	yes	—
8	#17988614	—	2	4	no	—
9	#17988611	—	2	1	no	—
10	#17988611	C₂²	1	2	yes	—
11	#17988333	—	2	4	no	—
12	#17988313	—	2	1	no	—
13	#17988311	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
16	#400×C₇₃	C₄	1	2	yes	—
17	#399×C₇₃	C₄	1	2	yes	—
18	#395×C₇₃	—	2	1	no	—
19	#393×C₇₃	—	2	1	no	—
20	#389×C₇₃	—	2	1	no	—
21	#382×C₇₃	C₂²	1	1	yes	characteristic
22	#380×C₇₃	—	2	1	no	—
23	C₂×#11924012	—	2	2	no	—
24	C₂×#11924012	—	2	2	no	—
25	C₂×#11924012	—	4	2	no	—
26	C₂×#11924012	—	4	2	no	—
27	C₂²×#10596831	—	2	2	no	—
28	C₄⋊#10596831	—	2	4	no	—
29	C₄⋊#10596831	—	2	2	no	—
30	C₄⋊#10596831	—	2	2	no	—
31	C₄⋊#10596831	D₄	1	4	yes	—
32	C₂²⋊#10596831	—	2	2	no	—
33	C₄×#10596831	—	2	2	no	—
34	C₂²×Q₈×C₇₃	—	2	1	no	—
35	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
36	C₄⋊Q₈×C₇₃	—	2	2	no	—
37	C₄⋊Q₈×C₇₃	—	2	2	no	—
38	#117×C₇₃	—	2	2	no	—
39	#117×C₇₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₇₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
51	#95×C₇₃	—	2	2	no	—
52	#95×C₇₃	—	2	2	no	—
53	#95×C₇₃	—	2	1	no	—
54	#95×C₇₃	C₂×C₄	1	2	yes	—
55	#11924012	—	4	4	no	—
56	#11924012	—	4	2	no	—
57	#11924012	—	4	2	no	—
58	C₂×#10596831	—	2	4	no	—
59	C₂×#10596831	—	4	4	no	—
60	C₂×#10596831	—	4	2	no	—
61	C₂⁴×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×Q₈×C₇₃	—	4	2	no	—
63	C₂×Q₈×C₇₃	—	8	2	no	—
64	C₂²×C₄×C₇₃	—	2	1	no	—
65	C₂²×C₄×C₇₃	—	2	1	no	—
66	C₂²×C₄×C₇₃	—	2	1	no	—
67	C₂²×C₄×C₇₃	—	2	1	no	—
68	C₂²×C₄×C₇₃	—	2	2	no	—
69	C₂²×C₄×C₇₃	—	2	2	no	—
70	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	SD₁₆	1	2	yes	—
74	C₂²×C₄×C₇₃	Q₁₆	1	2	yes	—
75	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₄⋊C₄×C₇₃	—	2	2	no	—
79	C₄⋊C₄×C₇₃	—	2	2	no	—
80	C₄⋊C₄×C₇₃	—	2	4	no	—
81	C₄⋊C₄×C₇₃	—	4	1	no	—
82	C₄⋊C₄×C₇₃	—	4	1	no	—
83	C₄⋊C₄×C₇₃	—	4	1	no	—
84	C₄⋊C₄×C₇₃	—	4	1	no	—
85	C₄⋊C₄×C₇₃	—	8	2	no	—
86	C₂²⋊C₄×C₇₃	—	2	2	no	—
87	C₂²⋊C₄×C₇₃	—	2	2	no	—
88	C₂²⋊C₄×C₇₃	—	8	2	no	—
89	C₄²×C₇₃	—	2	1	no	—
90	C₄²×C₇₃	—	2	1	no	—
91	C₄²×C₇₃	C₂×D₄	1	2	yes	—
92	C₄²×C₇₃	#41	1	2	yes	—
93	C₂³×C₇₃	—	2	1	no	—
94	C₂³×C₇₃	—	2	1	no	—
95	C₂³×C₇₃	—	4	1	no	—
96	C₂³×C₇₃	—	4	1	no	—
97	C₂³×C₇₃	#103	1	2	yes	—
98	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	Q₈×C₇₃	—	8	4	no	—
100	C₂×C₄×C₇₃	—	2	1	no	—
101	C₂×C₄×C₇₃	—	2	1	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	2	no	—
106	C₂×C₄×C₇₃	—	2	1	no	—
107	C₂×C₄×C₇₃	—	2	1	no	—
108	C₂×C₄×C₇₃	—	4	1	no	—
109	C₂×C₄×C₇₃	—	4	1	no	—
110	C₂×C₄×C₇₃	—	4	1	no	—
111	C₂×C₄×C₇₃	—	4	1	no	—
112	C₂×C₄×C₇₃	—	4	2	no	—
113	C₂×C₄×C₇₃	—	4	2	no	—
114	C₂×C₄×C₇₃	—	4	2	no	—
115	C₂×C₄×C₇₃	—	4	2	no	—
116	C₂×C₄×C₇₃	—	4	2	no	—
117	C₂×C₄×C₇₃	—	4	2	no	—
118	C₂×C₄×C₇₃	#103	1	2	yes	—
119	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
121	C₂×C₄×C₇₃	#121	1	2	yes	—
122	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
125	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
127	#10596831	—	4	4	no	—
128	#10596831	—	8	2	no	—
129	C₂²×C₇₃	—	2	1	no	—
130	C₂²×C₇₃	—	2	1	no	—
131	C₂²×C₇₃	—	2	2	no	—
132	C₂²×C₇₃	—	4	2	no	—
133	C₂²×C₇₃	—	4	1	no	—
134	C₂²×C₇₃	—	4	1	no	—
135	C₂²×C₇₃	—	8	1	no	—
136	C₂²×C₇₃	#386	1	1	yes	characteristic
137	C₂²×C₇₃	C₂×#103	1	1	yes	characteristic
138	C₂²×C₇₃	#417	1	1	yes	characteristic
139	C₄×C₇₃	—	2	2	no	—
140	C₄×C₇₃	—	2	2	no	—
141	C₄×C₇₃	—	2	2	no	—
142	C₄×C₇₃	—	4	1	no	—
143	C₄×C₇₃	—	4	1	no	—
144	C₄×C₇₃	—	8	2	no	—
145	C₄×C₇₃	—	8	2	no	—
146	C₄×C₇₃	#450	1	1	yes	characteristic
147	C₄×C₇₃	#451	1	1	yes	characteristic
148	C₄×C₇₃	#483	1	1	yes	characteristic
149	C₄×C₇₃	#484	1	1	yes	characteristic
150	#9193	—	73	1	no	maximal, Sylow
151	C₂×C₇₃	—	2	1	no	—
152	C₂×C₇₃	—	2	1	no	—
153	C₂×C₇₃	—	4	2	no	—
154	C₂×C₇₃	#1872	1	1	yes	characteristic
155	C₂×C₇₃	#1910	1	1	yes	characteristic
156	C₂×C₇₃	#1915	1	1	yes	characteristic
157	#2905	—	73	1	no	—
158	#2628	D₇₃	1	1	yes	characteristic
159	#1549	—	73	2	no	—
160	#1458	—	73	1	no	—
161	#1286	—	73	2	no	—
162	C₇₃	#9193	1	1	yes	characteristic, complemented, Sylow
163	C₂²×M₁₆	—	146	1	no	—
164	C₂×C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
165	C₂×#117	C₂×D₇₃	1	1	yes	characteristic
166	#431	—	73	2	no	—
167	#424	—	146	4	no	—
168	#423	—	73	2	no	—
169	#423	—	146	1	no	—
170	#400	Dic₇₃	1	2	yes	—
171	#399	Dic₇₃	1	2	yes	—
172	#395	—	2	1	no	—
173	#393	—	2	1	no	—
174	#389	—	2	1	no	—
175	#382	C₂×D₇₃	1	1	yes	characteristic
176	#380	—	2	1	no	—
177	#332	—	146	4	no	—
178	#324	—	146	1	no	—
179	#323	—	146	1	no	—
180	C₂²×Q₈	—	2	1	no	—
181	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
182	C₂×M₁₆	—	292	2	no	—
183	C₂×M₁₆	—	292	2	no	—
184	C₂×M₁₆	—	146	2	no	—
185	C₂×M₁₆	—	146	2	no	—
186	C₂²×C₈	—	146	2	no	—
187	C₄⋊Q₈	—	2	2	no	—
188	C₄⋊Q₈	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
195	C₂×C₄⋊C₄	C₂×Dic₇₃	1	1	yes	characteristic
196	C₂×C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	C₂×Dic₇₃	1	1	yes	characteristic
199	C₂×C₂²⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
202	C₄⋊C₈	—	73	4	no	—
203	C₄⋊C₈	—	146	4	no	—
204	C₄⋊C₈	—	146	2	no	—
205	C₄⋊C₈	—	146	2	no	—
206	C₂²⋊C₈	—	146	2	no	—
207	C₄×C₈	—	146	2	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	#95	—	2	1	no	—
211	#95	C₂×Dic₇₃	1	2	yes	—
212	C₂⁴	C₂²⋊Dic₇₃	1	1	yes	characteristic
213	C₂×Q₈	—	4	2	no	—
214	C₂×Q₈	—	8	2	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
224	C₂²×C₄	#11924015	1	2	yes	—
225	C₂²×C₄	#11924018	1	2	yes	—
226	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
229	M₁₆	—	292	4	no	—
230	M₁₆	—	292	2	no	—
231	M₁₆	—	292	2	no	—
232	C₂×C₈	—	292	4	no	—
233	C₂×C₈	—	292	2	no	—
234	C₂×C₈	—	146	4	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	4	1	no	—
239	C₄⋊C₄	—	4	1	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	8	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	—	8	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	D₄×D₇₃	1	2	yes	—
249	C₄²	#11924027	1	2	yes	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	4	1	no	—
253	C₂³	—	4	1	no	—
254	C₂³	#14556515	1	2	yes	—
255	C₂³	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
256	Q₈	—	8	4	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	#14556515	1	2	yes	—
276	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
277	C₂×C₄	#14556559	1	1	yes	characteristic
278	C₂×C₄	#14556563	1	2	yes	—
279	C₂×C₄	#14556572	1	1	yes	characteristic
280	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
281	C₂×C₄	C₂×#11924018	1	1	yes	characteristic
282	C₂×C₄	#14556616	1	1	yes	characteristic
283	C₂×C₄	#14556621	1	1	yes	characteristic
284	C₈	—	292	4	no	—
285	C₈	—	584	2	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	8	1	no	—
293	C₂²	#17988467	1	1	yes	characteristic
294	C₂²	C₂×#14556515	1	1	yes	characteristic
295	C₂²	#17988588	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	8	2	no	—
302	C₄	—	8	2	no	—
303	C₄	#17988737	1	1	yes	characteristic
304	C₄	#17988742	1	1	yes	characteristic
305	C₄	#17988915	1	1	yes	characteristic
306	C₄	#17988920	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	4	2	no	—
310	C₂	#22610239	1	1	yes	characteristic
311	C₂	#22610459	1	1	yes	characteristic
312	C₂	#22610484	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17988467
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#2628	C₂
socle	C₂²×C₇₃	#386
prime radical	C₂⁴×C₇₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#2628	D₇₃
hypocenter	C₇₃	#9193
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2628

Complements

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

N	H ≅ G/N	type
2. #22608616	C₂	semidirect
4. #22607789	C₂	semidirect
162. C₇₃	#9193	semidirect

Maximal subgroups

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

subgroup	index
#22618495	2
#22608616	2
#22608245	2
#22607789	2
C₇₃×#2628	2
#9193	73