Subgroups — #33073946

Order 18688.

710 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620879	C₂	1	2	yes	maximal
2	#22620517	C₂	1	1	yes	maximal, characteristic
3	#22620473	C₂	1	2	yes	maximal
4	#22609607	C₂	1	4	yes	maximal, complemented
5	#22609548	C₂	1	2	yes	maximal, complemented
6	#22609529	C₂	1	2	yes	maximal, complemented
7	C₈×#11924012	C₂	1	1	yes	maximal, characteristic
8	C₂×#500×C₇₃	C₂	1	1	yes	maximal, characteristic
9	#17989144	C₂²	1	2	yes	—
10	#17988993	—	2	4	no	—
11	#17988992	C₂²	1	1	yes	characteristic
12	#17988990	C₂²	1	4	yes	—
13	#17988983	C₂²	1	4	yes	—
14	#17988967	C₂²	1	1	yes	characteristic
15	#17988966	C₂²	1	4	yes	—
16	#17988938	C₂²	1	2	yes	—
17	#17988937	C₂²	1	4	yes	—
18	#17988828	—	2	2	no	—
19	#17988794	—	2	4	no	—
20	#17988743	—	2	4	no	—
21	C₄×Dic₂₉₂	—	2	2	no	—
22	#17988640	C₂²	1	1	yes	characteristic
23	#17988627	—	2	4	no	—
24	C₄×#11924012	C₂²	1	1	yes	characteristic
25	#17988309	C₂²	1	2	yes	—
26	C₈×#10596831	C₂²	1	2	yes	—
27	C₂×C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
28	#500×C₇₃	—	2	2	no	—
29	#500×C₇₃	C₂²	1	2	yes	—
30	#500×C₇₃	C₂²	1	2	yes	—
31	C₂×#107×C₇₃	—	2	2	no	—
32	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
33	C₂×Dic₂₉₂	—	2	2	no	—
34	C₂×#11924018	—	2	8	no	—
35	C₂×#11924012	D₄	1	2	yes	—
36	#14556591	C₂³	1	2	yes	—
37	#14556591	C₂³	1	2	yes	—
38	#14556583	—	2	2	no	—
39	#14556570	—	2	2	no	—
40	#14556570	—	2	2	no	—
41	C₄×Dic₁₄₆	—	2	2	no	—
42	C₄×Dic₁₄₆	—	2	2	no	—
43	#14556550	—	2	2	no	—
44	#14556525	—	2	2	no	—
45	#14556524	—	2	8	no	—
46	#14556522	—	2	8	no	—
47	C₄⋊#10596831	D₄	1	2	yes	—
48	#14556516	—	2	4	no	—
49	#14556515	—	2	8	no	—
50	C₂²⋊#10596831	D₄	1	2	yes	—
51	#14556502	D₄	1	2	yes	—
52	C₄×#10596831	C₂³	1	2	yes	—
53	C₄×#10596831	C₂³	1	2	yes	—
54	C₂²×Q₈×C₇₃	—	2	2	no	—
55	C₂²×C₈×C₇₃	D₄	1	2	yes	—
56	C₄⋊Q₈×C₇₃	—	2	2	no	—
57	C₄⋊Q₈×C₇₃	—	2	2	no	—
58	C₄⋊Q₈×C₇₃	—	2	2	no	—
59	C₄⋊Q₈×C₇₃	C₂³	1	4	yes	—
60	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
61	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
62	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
63	#107×C₇₃	—	2	4	no	—
64	#107×C₇₃	—	2	4	no	—
65	C₄×C₈×C₇₃	D₄	1	2	yes	—
66	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
68	C₂×Dic₁₄₆	—	2	2	no	—
69	C₂×Dic₁₄₆	—	2	2	no	—
70	Dic₂₉₂	—	4	4	no	—
71	#11924018	—	8	8	no	—
72	#11924012	Q₁₆	1	8	yes	complemented
73	C₂×#10596831	C₂×D₄	1	4	yes	—
74	C₂×#10596831	C₂×D₄	1	4	yes	—
75	C₄⋊Dic₇₃	—	2	2	no	—
76	C₄⋊Dic₇₃	—	2	2	no	—
77	C₄⋊Dic₇₃	—	2	2	no	—
78	C₄⋊Dic₇₃	—	2	2	no	—
79	#11924007	—	4	2	no	—
80	#11924007	—	4	2	no	—
81	C₂²⋊Dic₇₃	—	4	2	no	—
82	C₄×Dic₇₃	—	4	2	no	—
83	C₂×Q₈×C₇₃	—	2	4	no	—
84	C₂×Q₈×C₇₃	—	4	2	no	—
85	C₂×Q₈×C₇₃	—	8	4	no	—
86	C₂²×C₄×C₇₃	—	4	2	no	—
87	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₇₃	—	2	4	no	—
91	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
93	C₄⋊C₄×C₇₃	—	2	4	no	—
94	C₄⋊C₄×C₇₃	—	2	4	no	—
95	C₄⋊C₄×C₇₃	—	4	2	no	—
96	C₄⋊C₄×C₇₃	—	4	2	no	—
97	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
101	C₂³×C₇₃	#127	1	1	yes	characteristic
102	Q₈×C₇₃	—	4	8	no	—
103	Q₈×C₇₃	—	8	4	no	—
104	C₂×C₄×C₇₃	—	2	2	no	—
105	C₂×C₄×C₇₃	—	2	2	no	—
106	C₂×C₄×C₇₃	—	4	4	no	—
107	C₂×C₄×C₇₃	—	8	4	no	—
108	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
109	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
110	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂×Q₁₆	1	4	yes	—
112	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×Dic₇₃	—	4	2	no	—
116	C₂×Dic₇₃	—	4	2	no	—
117	Dic₁₄₆	—	4	4	no	—
118	Dic₁₄₆	—	4	4	no	—
119	C₈×C₇₃	—	2	2	no	—
120	C₈×C₇₃	#137	1	4	yes	—
121	#10596831	—	2	4	no	—
122	#10596831	C₂×Q₁₆	1	8	yes	—
123	C₂²×C₇₃	—	2	1	no	—
124	C₂²×C₇₃	—	2	1	no	—
125	C₂²×C₇₃	#494	1	2	yes	—
126	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
127	C₄×C₇₃	—	2	1	no	—
128	C₄×C₇₃	—	2	1	no	—
129	C₄×C₇₃	—	8	4	no	—
130	C₄×C₇₃	#494	1	2	yes	—
131	C₄×C₇₃	C₂²×Q₁₆	1	2	yes	—
132	C₄×C₇₃	C₂×#137	1	2	yes	—
133	C₄×C₇₃	#576	1	2	yes	—
134	Dic₇₃	—	8	2	no	—
135	Dic₇₃	—	8	2	no	—
136	#19633	—	73	1	no	maximal, Sylow
137	C₂×C₇₃	—	2	2	no	—
138	C₂×C₇₃	C₂×#494	1	1	yes	characteristic
139	C₂×C₇₃	#3152	1	1	yes	characteristic
140	C₂×C₇₃	#3155	1	1	yes	characteristic
141	#3209	—	73	2	no	—
142	#3170	—	73	1	no	—
143	#3165	—	73	2	no	—
144	C₂×#500	D₇₃	1	1	yes	characteristic
145	#1723	—	73	4	no	—
146	#1713	—	73	2	no	—
147	#1707	—	73	2	no	—
148	C₈×M₁₆	—	73	1	no	—
149	C₇₃	#19633	1	1	yes	characteristic, complemented, Sylow
150	#533	—	73	2	no	—
151	C₂×C₄⋊Q₈	C₂×D₇₃	1	2	yes	—
152	#501	—	146	4	no	—
153	#500	—	2	2	no	—
154	#500	—	73	4	no	—
155	#500	—	73	1	no	—
156	#500	C₂×D₇₃	1	2	yes	—
157	#500	C₂×D₇₃	1	2	yes	—
158	#498	—	73	4	no	—
159	#494	—	73	4	no	—
160	#494	—	73	1	no	—
161	#487	—	73	4	no	—
162	#487	—	73	2	no	—
163	#467	—	146	2	no	—
164	#462	—	146	4	no	—
165	#451	—	146	4	no	—
166	C₄×Q₁₆	—	146	2	no	—
167	#431	—	73	1	no	—
168	#428	—	146	4	no	—
169	C₂×#107	—	2	2	no	—
170	C₄×M₁₆	—	73	1	no	—
171	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
172	#322	—	73	2	no	—
173	C₈²	—	73	2	no	—
174	C₂²×Q₈	—	2	2	no	—
175	C₂×Q₁₆	—	146	8	no	—
176	C₂×Q₁₆	—	146	2	no	—
177	C₂×M₁₆	—	73	2	no	—
178	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	—	2	2	no	—
182	C₄⋊Q₈	—	73	2	no	—
183	C₄⋊Q₈	—	73	2	no	—
184	C₄⋊Q₈	C₂²×D₇₃	1	4	yes	—
185	#125	—	146	2	no	—
186	C₂²⋊Q₈	—	146	2	no	—
187	C₂²⋊Q₈	—	146	2	no	—
188	C₄×Q₈	—	146	2	no	—
189	C₄×Q₈	—	146	2	no	—
190	#117	—	146	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
194	#107	—	2	4	no	—
195	#107	—	2	4	no	—
196	#107	—	146	8	no	—
197	#107	—	146	2	no	—
198	#106	—	146	8	no	—
199	C₄⋊C₈	—	73	2	no	—
200	#103	—	146	8	no	—
201	#103	—	146	4	no	—
202	C₂²⋊C₈	—	73	2	no	—
203	#97	—	73	2	no	—
204	C₄×C₈	—	73	2	no	—
205	C₄×C₈	—	73	2	no	—
206	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
207	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	8	4	no	—
212	C₂×Q₈	—	146	2	no	—
213	C₂×Q₈	—	146	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
218	Q₁₆	—	292	4	no	—
219	Q₁₆	—	584	8	no	—
220	M₁₆	—	73	8	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	73	4	no	—
223	C₂×C₈	—	73	4	no	—
224	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
225	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	292	2	no	—
231	C₄⋊C₄	—	292	2	no	—
232	C₄⋊C₄	—	146	2	no	—
233	C₄⋊C₄	—	146	2	no	—
234	C₄⋊C₄	—	146	2	no	—
235	C₄⋊C₄	—	146	2	no	—
236	C₂²⋊C₄	—	292	2	no	—
237	C₄²	—	292	2	no	—
238	C₄²	D₄×D₇₃	1	1	yes	characteristic
239	C₄²	D₄×D₇₃	1	1	yes	characteristic
240	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
241	C₄²	C₂³×D₇₃	1	1	yes	characteristic
242	C₂³	#14556587	1	1	yes	characteristic
243	Q₈	—	4	8	no	—
244	Q₈	—	8	4	no	—
245	Q₈	—	292	4	no	—
246	Q₈	—	292	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	—	292	2	no	—
252	C₂×C₄	—	292	2	no	—
253	C₂×C₄	#14556587	1	1	yes	characteristic
254	C₂×C₄	#14556587	1	1	yes	characteristic
255	C₂×C₄	#14556587	1	1	yes	characteristic
256	C₂×C₄	Q₁₆×D₇₃	1	4	yes	—
257	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	73	8	no	—
262	C₈	—	146	4	no	—
263	C₈	#14556622	1	4	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#17988965	1	2	yes	—
267	C₂²	C₂×#14556587	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	8	4	no	—
271	C₄	—	584	2	no	—
272	C₄	—	584	2	no	—
273	C₄	#17988965	1	2	yes	—
274	C₄	C₂×Q₁₆×D₇₃	1	2	yes	—
275	C₄	C₂×#14556622	1	2	yes	—
276	C₄	#17989388	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	C₂×#17988965	1	1	yes	characteristic
279	C₂	#22620388	1	1	yes	characteristic
280	C₂	#22620412	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556587
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×#500×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#127
prime radical	C₂³×C₇₃	#127
hypercenter	C₂×#500	D₇₃
hypocenter	C₇₃	#19633
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
4. #22609607	C₂	semidirect
5. #22609548	C₂	semidirect
6. #22609529	C₂	semidirect
72. #11924012	Q₁₆	semidirect
149. C₇₃	#19633	semidirect

Maximal subgroups

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

subgroup	index
#22620879	2
#22620517	2
#22620473	2
#22609607	2
#22609548	2
#22609529	2
C₈×#11924012	2
C₂×#500×C₇₃	2
#19633	73