Subgroups — #24918731

Order 10624.

308 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128313	C₂	1	1	yes	maximal, characteristic
2	#19128107	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128100	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128071	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128053	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128042	C₂	1	1	yes	maximal, characteristic
7	#19128042	C₂	1	1	yes	maximal, characteristic
8	#19128036	C₂	1	1	yes	maximal, characteristic
9	#19128031	C₂	1	1	yes	maximal, characteristic
10	#19128005	C₂	1	1	yes	maximal, characteristic, complemented
11	#19128005	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127993	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127983	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127762	C₂	1	1	yes	maximal, characteristic
15	#544×C₈₃	C₂	1	1	yes	maximal, characteristic
16	#14581792	C₂²	1	1	yes	characteristic
17	#14581785	—	2	1	no	—
18	#14581785	—	2	1	no	—
19	#14581784	C₂²	1	1	yes	characteristic
20	#14581782	—	2	1	no	—
21	#14581782	—	2	1	no	—
22	#14581771	C₂²	1	1	yes	characteristic
23	#14581771	C₂²	1	1	yes	characteristic
24	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
26	#14581725	C₂²	1	1	yes	characteristic
27	#14581725	C₂²	1	1	yes	characteristic
28	#14581725	C₂²	1	1	yes	characteristic
29	#14581725	C₂²	1	1	yes	characteristic
30	#14581723	C₂²	1	1	yes	characteristic
31	#14581723	C₂²	1	1	yes	characteristic
32	#14581723	C₂²	1	1	yes	characteristic
33	#14581723	C₂²	1	1	yes	characteristic
34	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
35	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
36	#14581715	C₂²	1	1	yes	characteristic
37	#14581715	C₂²	1	1	yes	characteristic
38	#14581715	C₂²	1	1	yes	characteristic
39	#14581715	C₂²	1	1	yes	characteristic
40	#14581715	C₂²	1	1	yes	characteristic
41	#14581715	C₂²	1	1	yes	characteristic
42	#14581715	C₂²	1	1	yes	characteristic
43	#14581715	C₂²	1	1	yes	characteristic
44	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
45	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
46	#14581703	C₂²	1	1	yes	characteristic
47	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
48	C₄⋊Q₈×C₈₃	—	2	1	no	—
49	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
50	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
51	#125×C₈₃	—	2	1	no	—
52	#125×C₈₃	C₂²	1	1	yes	characteristic
53	#125×C₈₃	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₈₃	—	2	1	no	—
55	C₂²⋊Q₈×C₈₃	—	2	1	no	—
56	#117×C₈₃	C₂²	1	1	yes	characteristic
57	#117×C₈₃	C₂²	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
60	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
62	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
63	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
64	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
65	C₄⋊Dic₈₃	—	2	1	no	—
66	C₄⋊Dic₈₃	—	2	1	no	—
67	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
69	Dic₈₃⋊C₄	—	2	1	no	—
70	Dic₈₃⋊C₄	—	2	1	no	—
71	Dic₈₃⋊C₄	—	2	1	no	—
72	Dic₈₃⋊C₄	—	2	1	no	—
73	Dic₈₃⋊C₄	—	2	1	no	—
74	Dic₈₃⋊C₄	—	2	1	no	—
75	C₂²⋊Dic₈₃	—	2	1	no	—
76	C₂²⋊Dic₈₃	—	2	1	no	—
77	C₄×Dic₈₃	—	2	1	no	—
78	C₄×Dic₈₃	—	2	1	no	—
79	C₂×Q₈×C₈₃	—	2	1	no	—
80	C₂×Q₈×C₈₃	—	2	1	no	—
81	C₂²×C₄×C₈₃	—	2	1	no	—
82	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₈₃	—	2	1	no	—
84	C₄⋊C₄×C₈₃	—	2	1	no	—
85	C₄⋊C₄×C₈₃	—	2	1	no	—
86	C₄⋊C₄×C₈₃	—	2	1	no	—
87	C₄⋊C₄×C₈₃	—	2	1	no	—
88	C₄⋊C₄×C₈₃	—	2	1	no	—
89	C₄⋊C₄×C₈₃	—	2	1	no	—
90	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
91	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
92	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
94	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
96	C₂²⋊C₄×C₈₃	—	2	1	no	—
97	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
98	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
99	C₄²×C₈₃	—	2	1	no	—
100	C₄²×C₈₃	C₂³	1	1	yes	characteristic
101	C₄²×C₈₃	C₂³	1	1	yes	characteristic
102	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
103	Q₈×C₈₃	—	8	1	no	—
104	Q₈×C₈₃	—	8	1	no	—
105	C₂×C₄×C₈₃	—	2	1	no	—
106	C₂×C₄×C₈₃	—	2	1	no	—
107	C₂×C₄×C₈₃	—	2	1	no	—
108	C₂×C₄×C₈₃	—	2	1	no	—
109	C₂×C₄×C₈₃	—	2	1	no	—
110	C₂×C₄×C₈₃	—	2	1	no	—
111	C₂×C₄×C₈₃	—	8	1	no	—
112	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₈₃	—	2	1	no	—
119	C₂×Dic₈₃	—	2	1	no	—
120	C₂×Dic₈₃	—	2	1	no	—
121	C₂×Dic₈₃	—	2	1	no	—
122	Dic₁₆₆	—	2	1	no	—
123	Dic₁₆₆	—	2	1	no	—
124	Dic₁₆₆	—	2	1	no	—
125	Dic₁₆₆	—	2	1	no	—
126	C₈₃⋊C₈	—	2	1	no	—
127	C₈₃⋊C₈	—	2	1	no	—
128	C₈₃⋊C₈	—	2	1	no	—
129	C₈₃⋊C₈	—	2	1	no	—
130	C₂²×C₈₃	—	2	1	no	—
131	C₂²×C₈₃	—	2	1	no	—
132	C₂²×C₈₃	—	2	1	no	—
133	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
134	C₄×C₈₃	—	2	1	no	—
135	C₄×C₈₃	—	2	1	no	—
136	C₄×C₈₃	—	2	1	no	—
137	C₄×C₈₃	—	2	1	no	—
138	C₄×C₈₃	—	2	1	no	—
139	C₄×C₈₃	—	4	1	no	—
140	C₄×C₈₃	—	4	1	no	—
141	C₄×C₈₃	—	4	1	no	—
142	C₄×C₈₃	—	4	1	no	—
143	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
144	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
145	Dic₈₃	—	4	1	no	—
146	Dic₈₃	—	4	1	no	—
147	Dic₈₃	—	4	1	no	—
148	Dic₈₃	—	4	1	no	—
149	C₂×C₈₃	—	4	1	no	—
150	C₂×C₈₃	#536	1	1	yes	characteristic
151	C₂×C₈₃	#577	1	1	yes	characteristic
152	C₂×C₈₃	#578	1	1	yes	characteristic
153	#3242	—	83	1	no	maximal, Sylow
154	C₈₃	#3242	1	1	yes	characteristic, complemented, Sylow
155	#544	D₈₃	1	1	yes	characteristic
156	#541	—	83	1	no	—
157	#501	—	83	1	no	—
158	#499	—	83	1	no	—
159	#491	—	83	1	no	—
160	#487	—	83	1	no	—
161	#485	—	83	1	no	—
162	#485	—	83	1	no	—
163	#484	—	83	1	no	—
164	#483	—	83	1	no	—
165	#479	—	83	1	no	—
166	#479	—	83	1	no	—
167	#477	—	83	1	no	—
168	#475	—	83	1	no	—
169	#433	—	83	1	no	—
170	C₄⋊Q₈	—	2	1	no	—
171	C₄⋊Q₈	—	83	1	no	—
172	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
173	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
174	#126	—	166	1	no	—
175	#126	—	166	1	no	—
176	#125	—	2	1	no	—
177	#125	—	83	1	no	—
178	#125	—	166	1	no	—
179	#125	—	166	1	no	—
180	#125	C₂×D₈₃	1	1	yes	characteristic
181	#125	C₂×D₈₃	1	1	yes	characteristic
182	C₂²⋊Q₈	—	2	1	no	—
183	C₂²⋊Q₈	—	2	1	no	—
184	C₂²⋊Q₈	—	83	1	no	—
185	C₂²⋊Q₈	—	83	1	no	—
186	C₄×Q₈	—	83	1	no	—
187	C₄×Q₈	—	83	1	no	—
188	#117	C₂×D₈₃	1	1	yes	characteristic
189	#117	C₂×D₈₃	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
191	#107	—	83	1	no	—
192	#107	—	83	1	no	—
193	#107	—	83	1	no	—
194	#107	—	83	1	no	—
195	#106	—	83	1	no	—
196	#106	—	83	1	no	—
197	#106	—	83	1	no	—
198	#106	—	83	1	no	—
199	C₄⋊C₈	—	83	1	no	—
200	C₄⋊C₈	—	83	1	no	—
201	#103	—	83	1	no	—
202	#103	—	83	1	no	—
203	#103	—	83	1	no	—
204	#103	—	83	1	no	—
205	#103	—	83	1	no	—
206	#103	—	83	1	no	—
207	#103	—	83	1	no	—
208	#103	—	83	1	no	—
209	C₂²⋊C₈	—	83	1	no	—
210	C₂²⋊C₈	—	83	1	no	—
211	#97	—	83	1	no	—
212	C₄×C₈	—	83	1	no	—
213	C₂×Q₈	—	2	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	83	1	no	—
216	C₂×Q₈	—	83	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
219	C₂×C₈	—	83	1	no	—
220	C₂×C₈	—	83	1	no	—
221	C₂×C₈	—	83	1	no	—
222	C₂×C₈	—	83	1	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	83	1	no	—
231	C₄⋊C₄	—	83	1	no	—
232	C₄⋊C₄	—	166	1	no	—
233	C₄⋊C₄	—	166	1	no	—
234	C₄⋊C₄	—	166	1	no	—
235	C₄⋊C₄	—	166	1	no	—
236	C₄⋊C₄	—	166	1	no	—
237	C₄⋊C₄	—	166	1	no	—
238	C₄⋊C₄	—	166	1	no	—
239	C₄⋊C₄	—	166	1	no	—
240	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
241	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
242	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
243	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
244	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
245	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	166	1	no	—
248	C₂²⋊C₄	—	166	1	no	—
249	C₂²⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
250	C₂²⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
251	C₄²	—	2	1	no	—
252	C₄²	—	166	1	no	—
253	C₄²	—	166	1	no	—
254	C₄²	C₂²×D₈₃	1	1	yes	characteristic
255	C₄²	C₂²×D₈₃	1	1	yes	characteristic
256	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
257	Q₈	—	8	1	no	—
258	Q₈	—	8	1	no	—
259	Q₈	—	166	1	no	—
260	Q₈	—	166	1	no	—
261	Q₈	—	166	1	no	—
262	Q₈	—	166	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	166	1	no	—
271	C₂×C₄	—	166	1	no	—
272	C₂×C₄	—	166	1	no	—
273	C₂×C₄	—	166	1	no	—
274	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
275	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
276	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
277	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
279	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
280	C₈	—	166	1	no	—
281	C₈	—	166	1	no	—
282	C₈	—	166	1	no	—
283	C₈	—	166	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	332	1	no	—
298	C₄	—	332	1	no	—
299	C₄	—	332	1	no	—
300	C₄	—	332	1	no	—
301	C₄	#14581840	1	1	yes	characteristic
302	C₄	#14581840	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	#19128275	1	1	yes	characteristic
305	C₂	#19128512	1	1	yes	characteristic
306	C₂	#19128516	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#544×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂³×C₈₃	C₂×D₄
hypercenter	#544	D₈₃
hypocenter	C₈₃	#3242
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #544

Complements

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

N	H ≅ G/N	type
2. #19128107	C₂	semidirect
3. #19128100	C₂	semidirect
4. #19128071	C₂	semidirect
5. #19128053	C₂	semidirect
10. #19128005	C₂	semidirect
11. #19128005	C₂	semidirect
12. #19127993	C₂	semidirect
13. #19127983	C₂	semidirect
154. C₈₃	#3242	semidirect

Maximal subgroups

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

subgroup	index
#19128313	2
#19128107	2
#19128100	2
#19128071	2
#19128053	2
#19128042	2
#19128036	2
#19128031	2
#19128005	2
#19127993	2
#19127983	2
#19127762	2
#544×C₈₃	2
#3242	83