Subgroups — #24918915

Order 10624.

322 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128312	C₂	1	1	yes	maximal, characteristic
2	#19128309	C₂	1	1	yes	maximal, characteristic
3	#19128010	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128009	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128008	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128006	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127958	C₂	1	1	yes	maximal, characteristic
8	#19127956	C₂	1	1	yes	maximal, characteristic
9	#19127941	C₂	1	1	yes	maximal, characteristic
10	#19127939	C₂	1	1	yes	maximal, characteristic
11	#19127909	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127909	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127908	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127726	C₂	1	1	yes	maximal, characteristic
15	#462×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050580	C₂²	1	1	yes	characteristic
17	C₂×#13050580	C₂²	1	1	yes	characteristic
18	C₂×#13050573	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
20	#14581790	C₂²	1	1	yes	characteristic
21	#14581787	—	2	1	no	—
22	#14581787	—	2	1	no	—
23	#14581787	—	2	1	no	—
24	#14581786	—	2	1	no	—
25	#14581784	—	2	1	no	—
26	#14581784	C₂²	1	1	yes	characteristic
27	#14581783	C₂²	1	1	yes	characteristic
28	#14581782	—	2	1	no	—
29	#14581782	—	2	1	no	—
30	#14581782	—	2	1	no	—
31	#14581772	C₂²	1	1	yes	characteristic
32	#14581768	C₂²	1	1	yes	characteristic
33	#14581768	C₂²	1	1	yes	characteristic
34	#14581768	C₂²	1	1	yes	characteristic
35	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
36	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
37	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
38	#14581749	C₂²	1	1	yes	characteristic
39	#14581726	C₂²	1	1	yes	characteristic
40	#14581725	C₂²	1	1	yes	characteristic
41	#14581724	C₂²	1	1	yes	characteristic
42	#14581723	C₂²	1	1	yes	characteristic
43	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
46	#14581717	C₂²	1	1	yes	characteristic
47	#14581717	C₂²	1	1	yes	characteristic
48	#14581716	C₂²	1	1	yes	characteristic
49	#14581716	C₂²	1	1	yes	characteristic
50	#14581716	C₂²	1	1	yes	characteristic
51	#14581716	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	#103×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
60	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
61	#13050580	—	4	1	no	—
62	#13050580	—	4	1	no	—
63	#13050573	—	4	1	no	—
64	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
65	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
66	C₈×D₈₃	—	4	1	no	—
67	C₄⋊Dic₈₃	—	2	1	no	—
68	C₄⋊Dic₈₃	—	2	1	no	—
69	C₄⋊Dic₈₃	—	2	1	no	—
70	C₄⋊Dic₈₃	—	2	1	no	—
71	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
74	Dic₈₃⋊C₄	—	2	1	no	—
75	Dic₈₃⋊C₄	—	2	1	no	—
76	Dic₈₃⋊C₄	—	2	1	no	—
77	Dic₈₃⋊C₄	—	2	1	no	—
78	Dic₈₃⋊C₄	—	2	1	no	—
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
84	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
85	(C₂×D₈₃)⋊C₄	—	2	1	no	—
86	(C₂×D₈₃)⋊C₄	—	2	1	no	—
87	(C₂×D₈₃)⋊C₄	—	2	1	no	—
88	(C₂×D₈₃)⋊C₄	—	2	1	no	—
89	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
90	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
91	C₄×Dic₈₃	—	2	1	no	—
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
98	Q₁₆×C₈₃	—	4	1	no	—
99	Q₁₆×C₈₃	—	4	1	no	—
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	—	2	1	no	—
103	C₄⋊C₄×C₈₃	—	2	1	no	—
104	C₄⋊C₄×C₈₃	—	2	1	no	—
105	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
106	C₄²×C₈₃	—	2	1	no	—
107	C₄²×C₈₃	C₂³	1	1	yes	characteristic
108	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₈₃	—	2	1	no	—
110	Q₈×C₈₃	—	2	1	no	—
111	Q₈×C₈₃	—	2	1	no	—
112	Q₈×C₈₃	—	2	1	no	—
113	Q₈×C₈₃	—	2	1	no	—
114	Q₈×C₈₃	—	2	1	no	—
115	C₂×C₄×C₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	—	2	1	no	—
118	C₂×C₄×C₈₃	—	2	1	no	—
119	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₈₃	—	2	1	no	—
123	C₂×Dic₈₃	—	2	1	no	—
124	C₂×Dic₈₃	—	2	1	no	—
125	C₂×Dic₈₃	—	2	1	no	—
126	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
129	C₄×D₈₃	—	2	1	no	—
130	C₄×D₈₃	—	2	1	no	—
131	Dic₁₆₆	—	2	1	no	—
132	Dic₁₆₆	—	2	1	no	—
133	C₈×C₈₃	—	2	1	no	—
134	C₈×C₈₃	—	2	1	no	—
135	C₈₃⋊C₈	—	2	1	no	—
136	C₈₃⋊C₈	—	2	1	no	—
137	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×D₈₃	—	2	1	no	—
139	C₂×D₈₃	—	2	1	no	—
140	C₂×D₈₃	—	2	1	no	—
141	C₄×C₈₃	—	2	1	no	—
142	C₄×C₈₃	—	2	1	no	—
143	C₄×C₈₃	—	4	1	no	—
144	C₄×C₈₃	—	4	1	no	—
145	C₄×C₈₃	—	4	1	no	—
146	C₄×C₈₃	—	4	1	no	—
147	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
148	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₈₃	—	2	1	no	—
150	Dic₈₃	—	2	1	no	—
151	Dic₈₃	—	2	1	no	—
152	Dic₈₃	—	4	1	no	—
153	Dic₈₃	—	4	1	no	—
154	Dic₈₃	—	4	1	no	—
155	Dic₈₃	—	4	1	no	—
156	C₂×C₈₃	#537	1	1	yes	characteristic
157	C₂×C₈₃	#578	1	1	yes	characteristic
158	C₂×C₈₃	#578	1	1	yes	characteristic
159	D₈₃	—	4	1	no	—
160	#3261	—	83	1	no	maximal, Sylow
161	C₈₃	#3261	1	1	yes	characteristic, complemented, Sylow
162	#541	—	83	1	no	—
163	#541	—	83	1	no	—
164	#479	—	83	1	no	—
165	#479	—	83	1	no	—
166	#479	—	83	1	no	—
167	#479	—	83	1	no	—
168	#470	—	83	1	no	—
169	#470	—	83	1	no	—
170	#467	—	83	1	no	—
171	#467	—	83	1	no	—
172	#462	—	83	1	no	—
173	#462	—	83	1	no	—
174	#462	—	83	1	no	—
175	#462	D₈₃	1	1	yes	characteristic, complemented
176	#424	—	83	1	no	—
177	C₂×Q₁₆	—	83	1	no	—
178	C₂×Q₁₆	—	83	1	no	—
179	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
180	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
181	C₂×M₁₆	—	83	1	no	—
182	C₂²×C₈	—	83	1	no	—
183	C₄⋊Q₈	—	83	1	no	—
184	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
185	#126	—	166	1	no	—
186	#126	—	166	1	no	—
187	#126	—	166	1	no	—
188	#126	—	166	1	no	—
189	#125	—	83	1	no	—
190	#125	—	83	1	no	—
191	#125	—	166	1	no	—
192	#125	—	166	1	no	—
193	#125	—	166	1	no	—
194	#125	—	166	1	no	—
195	C₂²⋊Q₈	—	83	1	no	—
196	C₂²⋊Q₈	—	83	1	no	—
197	C₂²⋊Q₈	—	83	1	no	—
198	C₂²⋊Q₈	—	83	1	no	—
199	C₄×Q₈	—	83	1	no	—
200	C₄×Q₈	—	83	1	no	—
201	C₄×Q₈	—	83	1	no	—
202	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
203	#117	—	83	1	no	—
204	#107	—	83	1	no	—
205	#107	—	83	1	no	—
206	#106	—	83	1	no	—
207	#106	—	83	1	no	—
208	C₄⋊C₈	—	83	1	no	—
209	C₄⋊C₈	—	83	1	no	—
210	C₄⋊C₈	—	83	1	no	—
211	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
212	#103	—	83	1	no	—
213	#103	—	83	1	no	—
214	#103	—	83	1	no	—
215	#103	—	83	1	no	—
216	#103	—	83	1	no	—
217	#103	—	83	1	no	—
218	#103	C₂×D₈₃	1	1	yes	characteristic
219	#103	C₂×D₈₃	1	1	yes	characteristic
220	C₂×Q₈	—	83	1	no	—
221	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
222	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
223	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
224	C₂²×C₄	—	83	1	no	—
225	Q₁₆	—	4	1	no	—
226	Q₁₆	—	4	1	no	—
227	Q₁₆	—	332	1	no	—
228	Q₁₆	—	332	1	no	—
229	M₁₆	—	332	1	no	—
230	C₂×C₈	—	332	1	no	—
231	C₂×C₈	—	83	1	no	—
232	C₂×C₈	—	83	1	no	—
233	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	83	1	no	—
239	C₄⋊C₄	—	83	1	no	—
240	C₄⋊C₄	—	83	1	no	—
241	C₄⋊C₄	—	83	1	no	—
242	C₄⋊C₄	—	83	1	no	—
243	C₄⋊C₄	—	166	1	no	—
244	C₄⋊C₄	—	166	1	no	—
245	C₄⋊C₄	—	166	1	no	—
246	C₄⋊C₄	—	166	1	no	—
247	C₄⋊C₄	—	166	1	no	—
248	C₄⋊C₄	—	166	1	no	—
249	C₄⋊C₄	—	166	1	no	—
250	C₄⋊C₄	—	166	1	no	—
251	C₄⋊C₄	—	166	1	no	—
252	C₄⋊C₄	—	166	1	no	—
253	C₄⋊C₄	—	166	1	no	—
254	C₄⋊C₄	—	166	1	no	—
255	C₄⋊C₄	—	166	1	no	—
256	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
257	C₂²⋊C₄	—	83	1	no	—
258	C₂²⋊C₄	—	83	1	no	—
259	C₂²⋊C₄	—	166	1	no	—
260	C₂²⋊C₄	—	166	1	no	—
261	C₂²⋊C₄	—	166	1	no	—
262	C₂²⋊C₄	—	166	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	83	1	no	—
265	C₄²	—	166	1	no	—
266	C₄²	—	166	1	no	—
267	C₄²	—	166	1	no	—
268	C₄²	C₂²×D₈₃	1	1	yes	characteristic
269	C₂³	—	83	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	166	1	no	—
277	Q₈	—	166	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	83	1	no	—
283	C₂×C₄	—	83	1	no	—
284	C₂×C₄	—	83	1	no	—
285	C₂×C₄	—	166	1	no	—
286	C₂×C₄	—	166	1	no	—
287	C₂×C₄	—	166	1	no	—
288	C₂×C₄	—	166	1	no	—
289	C₂×C₄	—	166	1	no	—
290	C₂×C₄	—	166	1	no	—
291	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
292	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	166	1	no	—
297	C₈	—	166	1	no	—
298	C₂²	—	166	1	no	—
299	C₂²	—	166	1	no	—
300	C₂²	—	166	1	no	—
301	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	332	1	no	—
309	C₄	—	332	1	no	—
310	C₄	—	332	1	no	—
311	C₄	—	332	1	no	—
312	C₄	—	166	1	no	—
313	C₄	—	166	1	no	—
314	C₄	—	166	1	no	—
315	C₄	#14581837	1	1	yes	characteristic
316	C₄	#14581839	1	1	yes	characteristic
317	C₂	—	332	1	no	—
318	C₂	#19128280	1	1	yes	characteristic
319	C₂	#19128514	1	1	yes	characteristic
320	C₂	#19128517	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#462×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂²×D₈₃	C₂×D₄
hypercenter	#462	D₈₃
hypocenter	C₈₃	#3261
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

Complements

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

N	H ≅ G/N	type
3. #19128010	C₂	semidirect
4. #19128009	C₂	semidirect
5. #19128008	C₂	semidirect
6. #19128006	C₂	semidirect
11. #19127909	C₂	semidirect
12. #19127909	C₂	semidirect
13. #19127908	C₂	semidirect
15. #462×C₈₃	C₂	semidirect
161. C₈₃	#3261	semidirect
175. #462	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128312	2
#19128309	2
#19128010	2
#19128009	2
#19128008	2
#19128006	2
#19127958	2
#19127956	2
#19127941	2
#19127939	2
#19127909	2
#19127908	2
#19127726	2
#462×C₈₃	2
#3261	83