Subgroups — #33094188

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	#22620276	C₂	1	1	yes	maximal, characteristic
2	#22618865	C₂	1	1	yes	maximal, characteristic, complemented
3	#22610911	C₂	1	1	yes	maximal, characteristic, complemented
4	#22610764	C₂	1	1	yes	maximal, characteristic
5	#22610638	C₂	1	4	yes	maximal, complemented
6	#22610071	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609773	C₂	1	4	yes	maximal, complemented
8	#22609617	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#2931	C₂	1	1	yes	maximal, characteristic
10	#17989144	C₂²	1	1	yes	characteristic
11	#17988955	C₄	1	2	yes	complemented
12	#17988951	C₄	1	2	yes	complemented
13	#17988946	C₄	1	4	yes	complemented
14	#17988717	—	2	2	no	—
15	C₈×Dic₁₄₆	—	2	2	no	—
16	#17988613	C₂²	1	2	yes	—
17	#17988611	C₂²	1	1	yes	characteristic
18	C₂×#14556518	C₂²	1	4	yes	—
19	#17988583	C₂²	1	4	yes	—
20	#17988542	C₂²	1	2	yes	—
21	C₂×#14556502	C₂²	1	1	yes	characteristic
22	C₂×#14556501	C₂²	1	2	yes	—
23	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
24	#17988433	C₂²	1	1	yes	characteristic
25	#17988351	C₂²	1	4	yes	—
26	#17988348	C₂²	1	4	yes	—
27	C₂×#131×C₇₃	C₂²	1	2	yes	—
28	#532×C₇₃	C₂²	1	1	yes	characteristic
29	#436×C₇₃	—	2	2	no	—
30	C₈×D₄×C₇₃	—	2	2	no	—
31	#423×C₇₃	C₂²	1	1	yes	characteristic
32	#408×C₇₃	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
34	C₂×#11924012	C₂³	1	2	yes	—
35	C₂²×#10596831	C₂³	1	2	yes	—
36	#14556591	C₂×C₄	1	2	yes	—
37	#14556583	C₂×C₄	1	2	yes	—
38	#14556570	—	2	2	no	—
39	C₄×Dic₁₄₆	—	2	2	no	—
40	#14556550	C₂³	1	2	yes	—
41	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
42	C₄⋊#10596831	—	2	2	no	—
43	#14556520	—	2	4	no	—
44	#14556518	—	2	8	no	—
45	#14556515	C₂×C₄	1	8	yes	—
46	#14556512	C₂×C₄	1	8	yes	—
47	#14556502	C₂×C₄	1	2	yes	—
48	#14556502	C₂×C₄	1	2	yes	—
49	#14556501	—	2	2	no	—
50	#14556501	—	2	4	no	—
51	C₈×Dic₇₃	—	2	4	no	—
52	C₈×Dic₇₃	—	2	2	no	—
53	#14556498	—	2	2	no	—
54	C₂×#41×C₇₃	C₂³	1	2	yes	—
55	#131×C₇₃	—	2	8	no	—
56	C₂×M₁₆×C₇₃	—	2	2	no	—
57	C₂×M₁₆×C₇₃	C₂³	1	2	yes	—
58	C₂²×C₈×C₇₃	—	2	2	no	—
59	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
62	#127×C₇₃	C₂×C₄	1	1	yes	characteristic
63	#124×C₇₃	C₂×C₄	1	2	yes	—
64	#121×C₇₃	—	2	2	no	—
65	C₄×D₄×C₇₃	—	2	2	no	—
66	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₇₃	—	2	2	no	—
68	C₂²⋊C₈×C₇₃	—	2	4	no	—
69	C₄×C₈×C₇₃	D₄	1	2	yes	—
70	C₄×C₈×C₇₃	D₄	1	2	yes	—
71	C₂×Dic₁₄₆	—	2	2	no	—
72	C₂²×Dic₇₃	—	2	2	no	—
73	#11924012	—	2	4	no	—
74	C₂×#10596831	—	2	2	no	—
75	C₂×#10596831	—	2	2	no	—
76	C₂×#10596831	—	2	2	no	—
77	C₂×#10596831	C₂²×C₄	1	4	yes	—
78	C₄⋊Dic₇₃	—	2	2	no	—
79	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
80	#11924007	—	2	4	no	—
81	C₂²⋊Dic₇₃	—	2	2	no	—
82	C₄×Dic₇₃	—	2	2	no	—
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	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₇₃	—	2	2	no	—
89	C₂×D₄×C₇₃	—	2	2	no	—
90	C₂×D₄×C₇₃	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₇₃	—	2	2	no	—
92	C₂²×C₄×C₇₃	#41	1	2	yes	—
93	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
94	M₁₆×C₇₃	—	2	4	no	—
95	M₁₆×C₇₃	—	4	4	no	—
96	C₂×C₈×C₇₃	—	2	2	no	—
97	C₂×C₈×C₇₃	—	2	2	no	—
98	C₂×C₈×C₇₃	—	4	4	no	—
99	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₇₃	#41	1	4	yes	—
103	C₄⋊C₄×C₇₃	—	2	2	no	—
104	C₂²⋊C₄×C₇₃	—	2	4	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	D₄×C₇₃	—	2	4	no	—
112	D₄×C₇₃	—	4	4	no	—
113	D₄×C₇₃	—	4	4	no	—
114	C₂×C₄×C₇₃	—	2	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	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
123	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
124	C₂×Dic₇₃	—	2	4	no	—
125	C₂×Dic₇₃	—	2	4	no	—
126	C₂×Dic₇₃	—	4	4	no	—
127	Dic₁₄₆	—	4	4	no	—
128	C₈×C₇₃	—	2	2	no	—
129	C₈×C₇₃	—	4	2	no	—
130	C₈×C₇₃	#124	1	2	yes	—
131	C₈×C₇₃	#124	1	2	yes	—
132	#10596831	—	2	4	no	—
133	#10596831	—	4	2	no	—
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	2	no	—
142	C₄×C₇₃	C₄×#41	1	2	yes	—
143	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
144	C₄×C₇₃	#532	1	1	yes	characteristic
145	Dic₇₃	—	4	4	no	—
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	—	73	1	no	—
153	#2960	—	73	1	no	—
154	#2931	D₇₃	1	1	yes	characteristic
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	C₂×D₇₃	1	2	yes	—
163	#533	—	73	1	no	—
164	#532	C₂×D₇₃	1	1	yes	characteristic
165	#491	—	73	2	no	—
166	#490	—	73	2	no	—
167	#489	—	73	4	no	—
168	#446	—	146	2	no	—
169	C₈×Q₈	—	146	2	no	—
170	#436	—	2	2	no	—
171	C₈×D₄	—	2	2	no	—
172	#424	—	73	2	no	—
173	#423	—	73	1	no	—
174	#423	C₂×D₇₃	1	1	yes	characteristic
175	C₂×#104	—	73	4	no	—
176	#416	—	73	4	no	—
177	#408	C₂×D₇₃	1	2	yes	—
178	#405	—	73	2	no	—
179	C₂×#97	—	73	1	no	—
180	C₂×#97	—	73	2	no	—
181	C₂×C₄×C₈	—	73	2	no	—
182	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
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	—	2	8	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	73	2	no	—
190	C₂×M₁₆	C₂²×D₇₃	1	2	yes	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	73	2	no	—
193	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
195	C₄⋊Q₈	—	73	2	no	—
196	C₄⋊Q₈	C₄×D₇₃	1	1	yes	characteristic
197	#127	C₄×D₇₃	1	1	yes	characteristic
198	#125	—	73	2	no	—
199	#124	C₄×D₇₃	1	2	yes	—
200	C₂²⋊Q₈	—	146	2	no	—
201	#121	—	2	2	no	—
202	C₄×Q₈	—	146	2	no	—
203	C₄×D₄	—	2	2	no	—
204	#117	—	73	2	no	—
205	C₂×C₄²	—	73	2	no	—
206	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	—	146	4	no	—
209	C₄⋊C₈	—	146	2	no	—
210	#104	—	146	8	no	—
211	#103	—	73	8	no	—
212	#102	—	73	8	no	—
213	C₂²⋊C₈	—	2	4	no	—
214	#97	—	73	2	no	—
215	#97	—	73	2	no	—
216	#97	—	146	2	no	—
217	#97	—	146	4	no	—
218	C₄×C₈	—	146	4	no	—
219	C₄×C₈	—	146	2	no	—
220	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
221	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
222	#95	—	146	2	no	—
223	#41	—	2	4	no	—
224	#41	—	2	4	no	—
225	C₂×Q₈	—	146	2	no	—
226	C₂×Q₈	C₂×C₄×D₇₃	1	2	yes	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂×C₄×D₇₃	1	2	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	146	2	no	—
232	C₂²×C₄	#11924028	1	2	yes	—
233	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
234	M₁₆	—	2	4	no	—
235	M₁₆	—	4	4	no	—
236	M₁₆	—	146	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	4	no	—
240	C₂×C₈	—	73	4	no	—
241	C₂×C₈	—	146	2	no	—
242	C₂×C₈	—	146	2	no	—
243	C₂×C₈	—	146	2	no	—
244	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
245	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
246	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
247	C₂×C₈	#11924027	1	4	yes	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	73	4	no	—
250	C₄⋊C₄	—	146	4	no	—
251	C₄⋊C₄	—	146	2	no	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	—	146	2	no	—
254	C₄²	—	73	4	no	—
255	C₄²	—	146	2	no	—
256	C₄²	—	146	2	no	—
257	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
258	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
259	C₄²	C₂×C₂²⋊D₇₃	1	2	yes	—
260	C₂³	—	2	2	no	—
261	C₂³	C₂×#11924028	1	1	yes	characteristic
262	Q₈	—	2	4	no	—
263	Q₈	—	292	4	no	—
264	D₄	—	2	4	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	292	4	no	—
272	C₂×C₄	—	146	4	no	—
273	C₂×C₄	—	146	4	no	—
274	C₂×C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
275	C₂×C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
276	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
278	C₂×C₄	#41×D₇₃	1	2	yes	—
279	C₂×C₄	C₂×#11924027	1	2	yes	—
280	C₈	—	2	2	no	—
281	C₈	—	4	2	no	—
282	C₈	—	292	2	no	—
283	C₈	—	146	4	no	—
284	C₈	#14556578	1	2	yes	—
285	C₈	#14556578	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	2	no	—
289	C₂²	C₂×C₄×C₂²⋊D₇₃	1	1	yes	characteristic
290	C₂²	#17989119	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	2	no	—
294	C₄	—	292	4	no	—
295	C₄	—	292	4	no	—
296	C₄	C₄×#11924027	1	2	yes	—
297	C₄	C₂×#14556578	1	1	yes	characteristic
298	C₄	#17989133	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	8	2	no	—
301	C₂	C₄×#14556578	1	1	yes	characteristic
302	C₂	#22618832	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#2931	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#532×C₇₃	C₂²
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#2931	D₇₃
hypocenter	C₇₃	#21663
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
2. #22618865	C₂	semidirect
3. #22610911	C₂	semidirect
5. #22610638	C₂	semidirect
6. #22610071	C₂	semidirect
7. #22609773	C₂	semidirect
8. #22609617	C₂	semidirect
11. #17988955	C₄	semidirect
12. #17988951	C₄	semidirect
13. #17988946	C₄	semidirect
161. C₇₃	#21663	semidirect

Maximal subgroups

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

subgroup	index
#22620276	2
#22618865	2
#22610911	2
#22610764	2
#22610638	2
#22610071	2
#22609773	2
#22609617	2
C₇₃×#2931	2
#21663	73