Subgroups — #33239064

Order 18688.

682 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
2	#22611732	C₂	1	4	yes	maximal, complemented
3	#22611728	C₂	1	4	yes	maximal, complemented
4	#22611686	C₂	1	2	yes	maximal, complemented
5	#22611541	C₂	1	2	yes	maximal
6	C₄×#14556531	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×D₇₃	C₄	1	1	yes	characteristic
11	C₂×#14556531	C₂²	1	2	yes	—
12	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×D₇₃	C₄	1	2	yes	—
16	C₄×C₈×D₇₃	C₄	1	2	yes	—
17	C₄×C₈×D₇₃	C₂²	1	2	yes	—
18	C₄×C₈×D₇₃	C₂²	1	2	yes	—
19	#17988405	C₂²	1	8	yes	complemented
20	#17988366	C₂²	1	8	yes	—
21	#17988363	C₂²	1	2	yes	complemented
22	C₄×#11924003	C₂²	1	2	yes	complemented
23	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
25	#346×C₇₃	C₂²	1	2	yes	complemented
26	C₄×C₁₆×C₇₃	C₂²	1	2	yes	complemented
27	C₂²×C₄×D₇₃	C₈	1	2	yes	—
28	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
30	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₇₃	—	2	4	no	—
32	C₂×C₈×D₇₃	—	2	4	no	—
33	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₇₃	C₂³	1	2	yes	—
37	C₂×C₄×Dic₇₃	C₈	1	2	yes	—
38	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×D₇₃	C₈	1	2	yes	—
40	C₄²×D₇₃	C₈	1	2	yes	—
41	C₄²×D₇₃	C₂×C₄	1	2	yes	—
42	C₄²×D₇₃	C₂×C₄	1	2	yes	—
43	#14556531	—	2	4	no	—
44	C₂×#11924003	C₂³	1	4	yes	—
45	C₄×#10596831	C₂×C₄	1	2	yes	—
46	C₄×#10596831	C₂×C₄	1	2	yes	—
47	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
48	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
49	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
50	C₈×Dic₇₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₇₃	—	2	4	no	—
55	C₂×C₁₆×C₇₃	C₂³	1	4	yes	—
56	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
60	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
63	C₂×C₄×D₇₃	—	2	2	no	—
64	C₂×C₄×D₇₃	—	2	2	no	—
65	C₂×C₄×D₇₃	—	2	8	no	—
66	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
67	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
68	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
69	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
70	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
71	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
72	C₂×#10596831	—	2	2	no	—
73	C₂×#10596831	—	2	2	no	—
74	C₂×#10596831	C₂²×C₄	1	2	yes	—
75	C₂×#10596831	C₂²×C₄	1	2	yes	—
76	C₈×D₇₃	—	2	4	no	—
77	C₈×D₇₃	—	2	4	no	—
78	C₈×D₇₃	—	2	4	no	—
79	C₈×D₇₃	—	2	4	no	—
80	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
81	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
82	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
83	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
84	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
85	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
86	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
87	#11924003	—	2	4	no	—
88	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
89	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₇₃	M₁₆	1	4	yes	—
91	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₇₃	#41	1	4	yes	—
95	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
96	C₄²×C₇₃	C₂×C₈	1	2	yes	—
97	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₇₃	—	2	4	no	—
100	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
101	C₂²×D₇₃	—	2	2	no	—
102	C₂²×D₇₃	—	2	4	no	—
103	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₇₃	—	2	4	no	—
105	C₂×C₄×C₇₃	#117	1	2	yes	—
106	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×Dic₇₃	—	2	2	no	—
112	C₂×Dic₇₃	—	2	4	no	—
113	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
114	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
115	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
116	C₄×D₇₃	—	2	4	no	—
117	C₄×D₇₃	—	2	2	no	—
118	C₄×D₇₃	—	2	2	no	—
119	C₄×D₇₃	—	2	4	no	—
120	C₄×D₇₃	—	2	4	no	—
121	C₄×D₇₃	—	2	4	no	—
122	C₄×D₇₃	M₃₂	1	8	yes	complemented
123	C₈×C₇₃	#117	1	2	yes	—
124	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
125	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
126	C₈×C₇₃	C₂×#41	1	2	yes	—
127	#10596831	—	2	2	no	—
128	#10596831	—	2	2	no	—
129	C₂²×C₇₃	—	2	2	no	—
130	C₂²×C₇₃	#431	1	2	yes	—
131	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
132	C₂×D₇₃	—	2	8	no	—
133	C₂×D₇₃	—	2	2	no	—
134	C₂×D₇₃	—	2	2	no	—
135	C₂×D₇₃	C₂×M₃₂	1	4	yes	—
136	C₄×C₇₃	—	2	2	no	—
137	C₄×C₇₃	#431	1	2	yes	—
138	C₄×C₇₃	C₂×M₃₂	1	4	yes	—
139	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
140	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
141	Dic₇₃	—	2	2	no	—
142	Dic₇₃	—	2	2	no	—
143	Dic₇₃	C₂×M₃₂	1	4	yes	—
144	#33418	—	73	1	no	maximal, Sylow
145	C₂×C₇₃	—	2	2	no	—
146	C₂×C₇₃	C₂×#431	1	1	yes	characteristic
147	C₂×C₇₃	C₂²×M₃₂	1	2	yes	—
148	D₇₃	—	2	4	no	—
149	C₂²×C₄×C₈	—	73	1	no	—
150	#2163	—	73	4	no	—
151	#2162	—	73	4	no	—
152	#2150	—	73	2	no	—
153	#2112	—	73	2	no	—
154	C₄×M₃₂	—	73	1	no	—
155	C₄×M₃₂	D₇₃	1	1	yes	characteristic, complemented
156	C₇₃	#33418	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₈	—	73	1	no	—
158	C₂³×C₈	—	73	1	no	—
159	C₂²×C₄²	—	73	1	no	—
160	C₂×M₃₂	—	73	2	no	—
161	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
162	C₂×C₄×C₈	—	73	1	no	—
163	C₂×C₄×C₈	—	73	1	no	—
164	C₂×C₄×C₈	—	73	1	no	—
165	C₂×C₄×C₈	—	73	2	no	—
166	C₂×C₄×C₈	—	73	2	no	—
167	C₂×C₄×C₈	—	73	2	no	—
168	C₂×C₄×C₈	—	73	2	no	—
169	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
170	C₄⋊C₁₆	—	73	8	no	—
171	C₂²⋊C₁₆	—	73	8	no	—
172	#346	—	73	2	no	—
173	#346	C₂×D₇₃	1	2	yes	complemented
174	C₄×C₁₆	—	73	2	no	—
175	C₄×C₁₆	C₂×D₇₃	1	2	yes	complemented
176	C₂³×C₄	—	73	2	no	—
177	C₂³×C₄	—	73	1	no	—
178	C₂²×C₈	—	73	1	no	—
179	C₂²×C₈	—	73	1	no	—
180	C₂²×C₈	—	73	2	no	—
181	C₂²×C₈	—	73	2	no	—
182	C₂²×C₈	—	73	2	no	—
183	C₂²×C₈	—	73	2	no	—
184	C₂²×C₈	—	146	4	no	—
185	C₂²×C₈	—	146	4	no	—
186	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
188	C₂×C₄²	—	73	2	no	—
189	C₂×C₄²	—	73	1	no	—
190	C₂×C₄²	—	73	2	no	—
191	C₂×C₄²	—	73	2	no	—
192	C₂×C₄²	—	73	2	no	—
193	C₂×C₄²	—	73	2	no	—
194	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
195	M₃₂	—	2	4	no	—
196	M₃₂	—	146	4	no	—
197	C₂×C₁₆	—	73	4	no	—
198	C₂×C₁₆	C₂²×D₇₃	1	4	yes	—
199	C₄×C₈	—	73	2	no	—
200	C₄×C₈	—	73	2	no	—
201	C₄×C₈	—	73	2	no	—
202	C₄×C₈	—	73	2	no	—
203	C₄×C₈	—	73	2	no	—
204	C₄×C₈	—	73	2	no	—
205	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
206	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
209	C₂⁴	—	73	1	no	—
210	C₂²×C₄	—	73	2	no	—
211	C₂²×C₄	—	73	2	no	—
212	C₂²×C₄	—	73	1	no	—
213	C₂²×C₄	—	73	2	no	—
214	C₂²×C₄	—	73	2	no	—
215	C₂²×C₄	—	73	2	no	—
216	C₂²×C₄	—	73	2	no	—
217	C₂²×C₄	—	73	2	no	—
218	C₂²×C₄	—	146	8	no	—
219	C₂²×C₄	—	146	2	no	—
220	C₂²×C₄	—	146	2	no	—
221	C₂²×C₄	C₈×D₇₃	1	2	yes	—
222	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
223	C₂×C₈	—	73	2	no	—
224	C₂×C₈	—	73	2	no	—
225	C₂×C₈	—	146	2	no	—
226	C₂×C₈	—	146	2	no	—
227	C₂×C₈	—	146	4	no	—
228	C₂×C₈	—	146	4	no	—
229	C₂×C₈	—	146	4	no	—
230	C₂×C₈	—	146	4	no	—
231	C₂×C₈	#11924011	1	4	yes	—
232	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
235	C₂×C₈	#11924028	1	4	yes	—
236	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
237	C₄²	—	73	1	no	—
238	C₄²	—	73	2	no	—
239	C₄²	—	73	2	no	—
240	C₄²	—	73	2	no	—
241	C₄²	—	73	1	no	—
242	C₄²	—	73	2	no	—
243	C₄²	—	73	2	no	—
244	C₄²	C₈×D₇₃	1	2	yes	—
245	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
246	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
247	C₁₆	—	2	4	no	—
248	C₁₆	—	146	4	no	—
249	C₂³	—	73	2	no	—
250	C₂³	—	146	4	no	—
251	C₂³	—	146	2	no	—
252	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	73	2	no	—
255	C₂×C₄	—	73	8	no	—
256	C₂×C₄	—	73	2	no	—
257	C₂×C₄	—	73	2	no	—
258	C₂×C₄	—	146	4	no	—
259	C₂×C₄	—	146	4	no	—
260	C₂×C₄	—	146	4	no	—
261	C₂×C₄	—	146	2	no	—
262	C₂×C₄	—	146	4	no	—
263	C₂×C₄	—	146	4	no	—
264	C₂×C₄	—	146	2	no	—
265	C₂×C₄	—	146	2	no	—
266	C₂×C₄	#14556548	1	2	yes	—
267	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
268	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
269	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
270	C₂×C₄	C₂×#11924011	1	2	yes	—
271	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
272	C₈	—	146	2	no	—
273	C₈	—	146	2	no	—
274	C₈	#14556548	1	2	yes	—
275	C₈	C₂×#11924011	1	2	yes	—
276	C₈	C₂×#11924011	1	2	yes	—
277	C₈	C₂×#11924028	1	2	yes	—
278	C₂²	—	2	2	no	—
279	C₂²	—	73	4	no	—
280	C₂²	—	146	8	no	—
281	C₂²	—	146	2	no	—
282	C₂²	—	146	2	no	—
283	C₂²	#17988638	1	2	yes	—
284	C₂²	C₂²×C₈×D₇₃	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	73	4	no	—
287	C₄	—	146	2	no	—
288	C₄	—	146	2	no	—
289	C₄	#17988638	1	2	yes	—
290	C₄	M₃₂×D₇₃	1	4	yes	—
291	C₄	C₂×#14556548	1	1	yes	characteristic
292	C₄	C₂²×#11924011	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	146	4	no	—
295	C₂	C₂×#17988638	1	1	yes	characteristic
296	C₂	C₂×M₃₂×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₇₃
commutator, socle	C₂²×C₇₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₄×M₃₂×C₇₃	C₂
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₄×M₃₂	D₇₃
hypocenter	C₇₃	#33418
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #22611732	C₂	semidirect
3. #22611728	C₂	semidirect
4. #22611686	C₂	semidirect
6. C₄×#14556531	C₂	semidirect
7. C₄×M₃₂×C₇₃	C₂	semidirect
19. #17988405	C₂²	semidirect
21. #17988363	C₂²	semidirect
22. C₄×#11924003	C₂²	semidirect
25. #346×C₇₃	C₂²	semidirect
26. C₄×C₁₆×C₇₃	C₂²	semidirect
122. C₄×D₇₃	M₃₂	semidirect
155. C₄×M₃₂	D₇₃	semidirect
156. C₇₃	#33418	semidirect
173. #346	C₂×D₇₃	semidirect
175. C₄×C₁₆	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₇₃	2
#22611732	2
#22611728	2
#22611686	2
#22611541	2
C₄×#14556531	2
C₄×M₃₂×C₇₃	2
#33418	73