Subgroups — #33094142

Order 18688.

770 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618866	C₂	1	1	yes	maximal, characteristic, complemented
2	#22618643	C₂	1	1	yes	maximal, characteristic
3	#22610909	C₂	1	1	yes	maximal, characteristic, complemented
4	#22610760	C₂	1	1	yes	maximal, characteristic
5	#22610637	C₂	1	4	yes	maximal, complemented
6	#22610069	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609771	C₂	1	4	yes	maximal, complemented
8	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#3131	C₂	1	1	yes	maximal, characteristic
10	C₂×#14556598	C₂²	1	2	yes	—
11	#17988718	—	2	2	no	—
12	Q₈×#10596831	—	2	2	no	—
13	#17988665	—	2	2	no	—
14	D₄×#10596831	—	2	2	no	—
15	#17988614	C₂²	1	2	yes	—
16	#17988611	C₂²	1	1	yes	characteristic
17	#17988611	C₂²	1	1	yes	characteristic
18	C₂×#14556518	C₂²	1	4	yes	—
19	#17988554	C₂²	1	2	yes	—
20	#17988543	C₂²	1	2	yes	—
21	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
23	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
24	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
25	#17988352	C₂²	1	4	yes	—
26	#17988346	C₂²	1	4	yes	—
27	#533×C₇₃	C₂²	1	1	yes	characteristic
28	#532×C₇₃	C₂²	1	1	yes	characteristic
29	#488×C₇₃	C₄	1	4	yes	complemented
30	#487×C₇₃	C₄	1	2	yes	complemented
31	#486×C₇₃	C₄	1	2	yes	complemented
32	#416×C₇₃	C₂²	1	4	yes	—
33	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
34	#14556598	—	2	8	no	—
35	C₂×#11924012	—	2	2	no	—
36	C₂×#11924012	C₂³	1	2	yes	—
37	C₂×#11924012	C₂³	1	2	yes	—
38	C₂²×#10596831	—	2	2	no	—
39	C₂²×#10596831	C₂³	1	1	yes	characteristic
40	C₂²×#10596831	C₂³	1	1	yes	characteristic
41	C₂×C₄×Dic₇₃	—	2	2	no	—
42	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
43	C₄⋊#10596831	—	2	4	no	—
44	C₄⋊#10596831	—	2	2	no	—
45	C₄⋊#10596831	—	2	2	no	—
46	#14556518	—	2	8	no	—
47	C₂²⋊#10596831	—	2	4	no	—
48	#14556502	—	2	2	no	—
49	C₄×#10596831	—	2	2	no	—
50	C₄×#10596831	D₄	1	2	yes	—
51	C₄×#10596831	D₄	1	2	yes	—
52	C₈×Dic₇₃	—	2	4	no	—
53	C₈×Dic₇₃	—	2	4	no	—
54	C₂×#41×C₇₃	C₂³	1	2	yes	—
55	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
58	#127×C₇₃	C₂×C₄	1	1	yes	characteristic
59	#125×C₇₃	C₂×C₄	1	2	yes	—
60	#124×C₇₃	C₂×C₄	1	2	yes	—
61	C₂²⋊Q₈×C₇₃	—	2	2	no	—
62	#121×C₇₃	—	2	2	no	—
63	C₄×Q₈×C₇₃	—	2	2	no	—
64	C₄×D₄×C₇₃	—	2	2	no	—
65	#117×C₇₃	C₂³	1	2	yes	—
66	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
67	#103×C₇₃	C₂×C₄	1	8	yes	—
68	#102×C₇₃	C₂×C₄	1	8	yes	—
69	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
71	C₂²×Dic₇₃	—	2	2	no	—
72	#11924012	—	2	4	no	—
73	#11924012	—	2	4	no	—
74	#11924012	—	4	4	no	—
75	C₂×#10596831	—	2	2	no	—
76	C₂×#10596831	—	2	2	no	—
77	C₂×#10596831	—	2	2	no	—
78	C₂×#10596831	—	4	4	no	—
79	C₂×#10596831	C₂×D₄	1	2	yes	—
80	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
81	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
82	C₂×#10596831	#41	1	4	yes	—
83	C₄×Dic₇₃	—	2	8	no	—
84	C₄×Dic₇₃	—	2	2	no	—
85	C₄×Dic₇₃	#41	1	4	yes	—
86	#41×C₇₃	—	2	4	no	—
87	#41×C₇₃	—	2	4	no	—
88	C₂×Q₈×C₇₃	—	2	2	no	—
89	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
90	C₂×D₄×C₇₃	—	2	2	no	—
91	C₂×D₄×C₇₃	—	2	2	no	—
92	C₂×D₄×C₇₃	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₇₃	—	2	2	no	—
94	C₂²×C₄×C₇₃	#41	1	2	yes	—
95	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
96	C₂×C₈×C₇₃	—	2	2	no	—
97	C₂×C₈×C₇₃	—	2	2	no	—
98	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
99	C₄⋊C₄×C₇₃	—	2	2	no	—
100	C₄⋊C₄×C₇₃	—	2	4	no	—
101	C₄⋊C₄×C₇₃	—	2	2	no	—
102	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
103	C₂²⋊C₄×C₇₃	—	2	2	no	—
104	C₂²⋊C₄×C₇₃	—	2	4	no	—
105	C₄²×C₇₃	—	2	2	no	—
106	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₇₃	C₂×D₄	1	2	yes	—
109	C₂³×C₇₃	—	2	2	no	—
110	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
111	Q₈×C₇₃	—	2	4	no	—
112	Q₈×C₇₃	—	4	4	no	—
113	D₄×C₇₃	—	2	4	no	—
114	D₄×C₇₃	—	4	4	no	—
115	D₄×C₇₃	—	4	4	no	—
116	C₂×C₄×C₇₃	—	2	4	no	—
117	C₂×C₄×C₇₃	—	2	4	no	—
118	C₂×C₄×C₇₃	—	2	2	no	—
119	C₂×C₄×C₇₃	—	4	2	no	—
120	C₂×C₄×C₇₃	—	4	2	no	—
121	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
126	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
127	C₂×Dic₇₃	—	2	8	no	—
128	C₂×Dic₇₃	—	2	4	no	—
129	C₈×C₇₃	—	2	4	no	—
130	#10596831	—	2	2	no	—
131	#10596831	—	4	2	no	—
132	#10596831	—	4	2	no	—
133	#10596831	#124	1	2	yes	—
134	#10596831	#124	1	2	yes	—
135	C₂²×C₇₃	—	2	2	no	—
136	C₂²×C₇₃	—	4	4	no	—
137	C₂²×C₇₃	—	4	2	no	—
138	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂²×C₇₃	#529	1	2	yes	—
140	C₄×C₇₃	—	2	2	no	—
141	C₄×C₇₃	—	4	4	no	—
142	C₄×C₇₃	—	4	2	no	—
143	C₄×C₇₃	#443	1	4	yes	—
144	C₄×C₇₃	C₄×#41	1	2	yes	—
145	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
146	C₄×C₇₃	#532	1	1	yes	characteristic
147	Dic₇₃	—	2	8	no	—
148	#21658	—	73	1	no	maximal, Sylow
149	C₂×C₇₃	—	2	2	no	—
150	C₂×C₇₃	—	8	2	no	—
151	C₂×C₇₃	C₄×#124	1	1	yes	characteristic
152	C₂×C₇₃	C₂×#443	1	2	yes	—
153	#3131	D₇₃	1	1	yes	characteristic
154	#2960	—	73	1	no	—
155	#2931	—	73	1	no	—
156	#1994	—	73	1	no	—
157	#1962	—	73	1	no	—
158	#1937	—	73	4	no	—
159	#1837	—	73	1	no	—
160	#1764	—	73	4	no	—
161	C₄²×C₈	—	73	1	no	—
162	C₇₃	#21658	1	1	yes	characteristic, complemented, Sylow
163	C₂×#131	—	73	2	no	—
164	#533	C₂×D₇₃	1	1	yes	characteristic
165	#532	C₂×D₇₃	1	1	yes	characteristic
166	#488	Dic₇₃	1	4	yes	complemented
167	#487	Dic₇₃	1	2	yes	complemented
168	#486	Dic₇₃	1	2	yes	complemented
169	#446	—	146	2	no	—
170	C₈×Q₈	—	146	2	no	—
171	#436	—	146	2	no	—
172	C₈×D₄	—	146	2	no	—
173	#424	—	73	2	no	—
174	#423	—	73	1	no	—
175	#423	—	73	1	no	—
176	C₂×#104	—	73	4	no	—
177	#416	C₂×D₇₃	1	4	yes	—
178	#408	—	73	2	no	—
179	#405	—	73	2	no	—
180	C₂×C₄×C₈	—	73	1	no	—
181	C₂×C₄×C₈	—	73	2	no	—
182	C₂×C₄×C₈	—	73	2	no	—
183	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
184	C₄³	—	73	1	no	—
185	#341	—	73	4	no	—
186	#339	—	73	4	no	—
187	C₂×#41	C₂²×D₇₃	1	2	yes	—
188	#131	—	146	8	no	—
189	C₂×M₁₆	—	73	2	no	—
190	C₂×M₁₆	—	73	2	no	—
191	C₂×M₁₆	—	146	2	no	—
192	C₂²×C₈	—	73	1	no	—
193	C₂²×C₈	—	73	1	no	—
194	C₂²×C₈	—	146	2	no	—
195	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
196	C₄⋊Q₈	C₂×Dic₇₃	1	1	yes	characteristic
197	C₄⋊Q₈	C₂×Dic₇₃	1	2	yes	—
198	#127	C₂×Dic₇₃	1	1	yes	characteristic
199	#125	C₂×Dic₇₃	1	2	yes	—
200	#124	C₂×Dic₇₃	1	2	yes	—
201	C₂²⋊Q₈	—	2	2	no	—
202	#121	—	2	2	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×D₄	—	2	2	no	—
205	#117	C₂²×D₇₃	1	2	yes	—
206	C₂×C₄²	—	73	2	no	—
207	C₂×C₄²	—	146	2	no	—
208	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
209	C₄⋊C₈	—	146	4	no	—
210	C₄⋊C₈	—	146	2	no	—
211	C₄⋊C₈	—	146	2	no	—
212	#104	—	146	8	no	—
213	#103	C₂×Dic₇₃	1	8	yes	—
214	#102	C₂×Dic₇₃	1	8	yes	—
215	C₂²⋊C₈	—	146	4	no	—
216	#97	—	146	2	no	—
217	C₄×C₈	—	73	2	no	—
218	C₄×C₈	—	73	2	no	—
219	C₄×C₈	—	146	2	no	—
220	C₄×C₈	—	146	4	no	—
221	C₄×C₈	—	146	4	no	—
222	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
223	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
224	#41	—	2	4	no	—
225	#41	—	2	4	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	C₂²×Dic₇₃	1	2	yes	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	C₂²×Dic₇₃	1	2	yes	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	146	2	no	—
233	C₂²×C₄	#11924027	1	2	yes	—
234	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
235	M₁₆	—	292	4	no	—
236	M₁₆	—	146	4	no	—
237	M₁₆	—	146	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	292	4	no	—
241	C₂×C₈	—	73	1	no	—
242	C₂×C₈	—	73	1	no	—
243	C₂×C₈	—	73	4	no	—
244	C₂×C₈	—	73	2	no	—
245	C₂×C₈	—	146	2	no	—
246	C₂×C₈	—	146	2	no	—
247	C₂×C₈	—	146	2	no	—
248	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	C₂²×Dic₇₃	1	4	yes	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	73	4	no	—
257	C₄²	—	146	8	no	—
258	C₄²	—	146	2	no	—
259	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
260	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
261	C₄²	D₄×D₇₃	1	2	yes	—
262	C₂³	—	2	2	no	—
263	C₂³	C₂×#11924027	1	1	yes	characteristic
264	Q₈	—	2	4	no	—
265	Q₈	—	4	4	no	—
266	D₄	—	2	4	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	146	8	no	—
275	C₂×C₄	—	146	4	no	—
276	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
277	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
278	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
279	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
280	C₂×C₄	#41×D₇₃	1	2	yes	—
281	C₂×C₄	#41×D₇₃	1	2	yes	—
282	C₈	—	2	4	no	—
283	C₈	—	292	2	no	—
284	C₈	—	292	2	no	—
285	C₈	—	73	2	no	—
286	C₈	—	73	2	no	—
287	C₈	—	146	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	2	no	—
291	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
292	C₂²	#17989116	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	4	4	no	—
295	C₄	—	4	2	no	—
296	C₄	—	146	8	no	—
297	C₄	#17988704	1	4	yes	—
298	C₄	#41×Dic₇₃	1	2	yes	—
299	C₄	#124×D₇₃	1	1	yes	characteristic
300	C₄	#17989136	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	8	2	no	—
303	C₂	#124×Dic₇₃	1	1	yes	characteristic
304	C₂	C₂×#17988704	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#3131	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#532×C₇₃	C₂²
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#3131	D₇₃
hypocenter	C₇₃	#21658
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3131

Complements

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

N	H ≅ G/N	type
1. #22618866	C₂	semidirect
3. #22610909	C₂	semidirect
5. #22610637	C₂	semidirect
6. #22610069	C₂	semidirect
7. #22609771	C₂	semidirect
8. C₄×C₈×Dic₇₃	C₂	semidirect
29. #488×C₇₃	C₄	semidirect
30. #487×C₇₃	C₄	semidirect
31. #486×C₇₃	C₄	semidirect
162. C₇₃	#21658	semidirect
166. #488	Dic₇₃	semidirect
167. #487	Dic₇₃	semidirect
168. #486	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22618866	2
#22618643	2
#22610909	2
#22610760	2
#22610637	2
#22610069	2
#22609771	2
C₄×C₈×Dic₇₃	2
C₇₃×#3131	2
#21658	73