Subgroups — #33094191

Order 18688.

730 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	#22618866	C₂	1	1	yes	maximal, characteristic, complemented
2	#22618643	C₂	1	1	yes	maximal, characteristic
3	#22610913	C₂	1	1	yes	maximal, characteristic, complemented
4	#22610764	C₂	1	1	yes	maximal, characteristic
5	#22610637	C₂	1	4	yes	maximal, complemented
6	#22610072	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609778	C₂	1	4	yes	maximal, complemented
8	#22609619	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#3135	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×#14556501	C₂²	1	2	yes	—
22	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
24	#17988433	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	#491×C₇₃	C₄	1	2	yes	complemented
30	#490×C₇₃	C₄	1	2	yes	complemented
31	#489×C₇₃	C₄	1	4	yes	complemented
32	#416×C₇₃	C₂²	1	4	yes	—
33	C₂×#97×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	C₂³	1	2	yes	—
42	C₄⋊#10596831	—	2	4	no	—
43	C₄⋊#10596831	—	2	2	no	—
44	C₄⋊#10596831	—	2	2	no	—
45	#14556518	—	2	8	no	—
46	C₂²⋊#10596831	—	2	4	no	—
47	#14556502	—	2	2	no	—
48	#14556501	—	2	4	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	#14556498	—	2	2	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	#97×C₇₃	C₂×C₄	1	2	yes	—
70	#97×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	2	no	—
84	C₄×Dic₇₃	#41	1	4	yes	—
85	#41×C₇₃	—	2	4	no	—
86	#41×C₇₃	—	2	4	no	—
87	C₂×Q₈×C₇₃	—	2	2	no	—
88	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
89	C₂×D₄×C₇₃	—	2	2	no	—
90	C₂×D₄×C₇₃	—	2	2	no	—
91	C₂×D₄×C₇₃	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₇₃	—	2	2	no	—
93	C₂²×C₄×C₇₃	#41	1	2	yes	—
94	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
95	C₂×C₈×C₇₃	—	2	2	no	—
96	C₂×C₈×C₇₃	—	2	2	no	—
97	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
98	C₄⋊C₄×C₇₃	—	2	2	no	—
99	C₄⋊C₄×C₇₃	—	2	4	no	—
100	C₄⋊C₄×C₇₃	—	2	2	no	—
101	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
102	C₂²⋊C₄×C₇₃	—	2	2	no	—
103	C₂²⋊C₄×C₇₃	—	2	4	no	—
104	C₄²×C₇₃	—	2	2	no	—
105	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₇₃	C₂×D₄	1	2	yes	—
108	C₂³×C₇₃	—	2	2	no	—
109	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
110	Q₈×C₇₃	—	2	4	no	—
111	Q₈×C₇₃	—	4	4	no	—
112	D₄×C₇₃	—	2	4	no	—
113	D₄×C₇₃	—	4	4	no	—
114	D₄×C₇₃	—	4	4	no	—
115	C₂×C₄×C₇₃	—	2	4	no	—
116	C₂×C₄×C₇₃	—	2	2	no	—
117	C₂×C₄×C₇₃	—	4	2	no	—
118	C₂×C₄×C₇₃	—	4	2	no	—
119	C₂×C₄×C₇₃	—	4	2	no	—
120	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
125	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
126	C₂×Dic₇₃	—	2	4	no	—
127	C₂×Dic₇₃	—	4	4	no	—
128	C₈×C₇₃	—	2	4	no	—
129	#10596831	—	2	2	no	—
130	#10596831	—	4	2	no	—
131	#10596831	—	4	2	no	—
132	#10596831	#124	1	2	yes	—
133	#10596831	#124	1	2	yes	—
134	C₂²×C₇₃	—	2	2	no	—
135	C₂²×C₇₃	—	4	4	no	—
136	C₂²×C₇₃	—	4	2	no	—
137	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₇₃	#529	1	2	yes	—
139	C₄×C₇₃	—	2	2	no	—
140	C₄×C₇₃	—	2	2	no	—
141	C₄×C₇₃	—	4	4	no	—
142	C₄×C₇₃	—	4	2	no	—
143	C₄×C₇₃	C₄×#41	1	2	yes	—
144	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
145	C₄×C₇₃	#532	1	1	yes	characteristic
146	Dic₇₃	—	4	4	no	—
147	#21663	—	73	1	no	maximal, Sylow
148	C₂×C₇₃	—	2	2	no	—
149	C₂×C₇₃	—	8	2	no	—
150	C₂×C₇₃	C₄×#124	1	1	yes	characteristic
151	C₂×C₇₃	#2955	1	2	yes	—
152	#3135	D₇₃	1	1	yes	characteristic
153	#2960	—	73	1	no	—
154	#2931	—	73	1	no	—
155	#1995	—	73	1	no	—
156	#1963	—	73	1	no	—
157	#1937	—	73	4	no	—
158	#1838	—	73	1	no	—
159	#1765	—	73	4	no	—
160	#1726	—	73	1	no	—
161	C₇₃	#21663	1	1	yes	characteristic, complemented, Sylow
162	C₂×#131	—	73	2	no	—
163	#533	C₂×D₇₃	1	1	yes	characteristic
164	#532	C₂×D₇₃	1	1	yes	characteristic
165	#491	Dic₇₃	1	2	yes	complemented
166	#490	Dic₇₃	1	2	yes	complemented
167	#489	Dic₇₃	1	4	yes	complemented
168	#446	—	146	2	no	—
169	C₈×Q₈	—	146	2	no	—
170	#436	—	146	2	no	—
171	C₈×D₄	—	146	2	no	—
172	#424	—	73	2	no	—
173	#423	—	73	1	no	—
174	#423	—	73	1	no	—
175	C₂×#104	—	73	4	no	—
176	#416	C₂×D₇₃	1	4	yes	—
177	#408	—	73	2	no	—
178	#405	—	73	2	no	—
179	C₂×#97	—	73	2	no	—
180	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
181	C₂×C₄×C₈	—	73	1	no	—
182	C₂×C₄×C₈	—	73	2	no	—
183	#376	—	73	1	no	—
184	#341	—	73	4	no	—
185	#339	—	73	4	no	—
186	C₂×#41	C₂²×D₇₃	1	2	yes	—
187	#131	—	146	8	no	—
188	C₂×M₁₆	—	73	2	no	—
189	C₂×M₁₆	—	73	2	no	—
190	C₂×M₁₆	—	146	2	no	—
191	C₂²×C₈	—	73	1	no	—
192	C₂²×C₈	—	73	1	no	—
193	C₂²×C₈	—	146	2	no	—
194	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
195	C₄⋊Q₈	C₂×Dic₇₃	1	1	yes	characteristic
196	C₄⋊Q₈	C₂×Dic₇₃	1	2	yes	—
197	#127	C₂×Dic₇₃	1	1	yes	characteristic
198	#125	C₂×Dic₇₃	1	2	yes	—
199	#124	C₂×Dic₇₃	1	2	yes	—
200	C₂²⋊Q₈	—	2	2	no	—
201	#121	—	2	2	no	—
202	C₄×Q₈	—	2	2	no	—
203	C₄×D₄	—	2	2	no	—
204	#117	C₂²×D₇₃	1	2	yes	—
205	C₂×C₄²	—	73	2	no	—
206	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
207	C₄⋊C₈	—	146	4	no	—
208	C₄⋊C₈	—	146	2	no	—
209	C₄⋊C₈	—	146	2	no	—
210	#104	—	146	8	no	—
211	#103	C₂×Dic₇₃	1	8	yes	—
212	#102	C₂×Dic₇₃	1	8	yes	—
213	C₂²⋊C₈	—	146	4	no	—
214	#97	—	146	2	no	—
215	#97	—	146	4	no	—
216	#97	C₂×Dic₇₃	1	2	yes	—
217	#97	C₂×Dic₇₃	1	2	yes	—
218	C₄×C₈	—	73	2	no	—
219	C₄×C₈	—	73	2	no	—
220	C₄×C₈	—	146	2	no	—
221	C₄×C₈	—	146	4	no	—
222	#95	—	146	2	no	—
223	#41	—	2	4	no	—
224	#41	—	2	4	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	C₂²×Dic₇₃	1	2	yes	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×Dic₇₃	1	2	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	146	2	no	—
232	C₂²×C₄	#11924027	1	2	yes	—
233	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
234	M₁₆	—	292	4	no	—
235	M₁₆	—	146	4	no	—
236	M₁₆	—	146	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	292	4	no	—
240	C₂×C₈	—	73	1	no	—
241	C₂×C₈	—	73	1	no	—
242	C₂×C₈	—	73	4	no	—
243	C₂×C₈	—	73	2	no	—
244	C₂×C₈	—	146	2	no	—
245	C₂×C₈	—	146	2	no	—
246	C₂×C₈	—	146	2	no	—
247	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	C₂²×Dic₇₃	1	4	yes	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	73	4	no	—
256	C₄²	—	146	2	no	—
257	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
258	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
259	C₄²	D₄×D₇₃	1	2	yes	—
260	C₂³	—	2	2	no	—
261	C₂³	C₂×#11924027	1	1	yes	characteristic
262	Q₈	—	2	4	no	—
263	Q₈	—	4	4	no	—
264	D₄	—	2	4	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	292	4	no	—
273	C₂×C₄	—	146	4	no	—
274	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
275	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
276	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
278	C₂×C₄	#41×D₇₃	1	2	yes	—
279	C₂×C₄	#41×D₇₃	1	2	yes	—
280	C₈	—	2	4	no	—
281	C₈	—	292	2	no	—
282	C₈	—	292	2	no	—
283	C₈	—	73	2	no	—
284	C₈	—	73	2	no	—
285	C₈	—	146	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	2	no	—
289	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
290	C₂²	#17989116	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	4	no	—
294	C₄	—	4	2	no	—
295	C₄	—	292	4	no	—
296	C₄	#41×Dic₇₃	1	2	yes	—
297	C₄	#124×D₇₃	1	1	yes	characteristic
298	C₄	#17989136	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	8	2	no	—
301	C₂	#124×Dic₇₃	1	1	yes	characteristic
302	C₂	#22618834	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×#3135	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#532×C₇₃	C₂²
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#3135	D₇₃
hypocenter	C₇₃	#21663
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3135

Complements

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

N	H ≅ G/N	type
1. #22618866	C₂	semidirect
3. #22610913	C₂	semidirect
5. #22610637	C₂	semidirect
6. #22610072	C₂	semidirect
7. #22609778	C₂	semidirect
8. #22609619	C₂	semidirect
29. #491×C₇₃	C₄	semidirect
30. #490×C₇₃	C₄	semidirect
31. #489×C₇₃	C₄	semidirect
161. C₇₃	#21663	semidirect
165. #491	Dic₇₃	semidirect
166. #490	Dic₇₃	semidirect
167. #489	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
#22610913	2
#22610764	2
#22610637	2
#22610072	2
#22609778	2
#22609619	2
C₇₃×#3135	2
#21663	73