Subgroups — #24917309

Order 10624.

376 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128328	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128180	C₂	1	1	yes	maximal, characteristic
3	#19128009	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128007	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128003	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128001	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127995	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127993	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127979	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127742	C₂	1	1	yes	maximal, characteristic, complemented
11	#107×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127726	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127699	C₂	1	1	yes	maximal, characteristic
14	#19127694	C₂	1	1	yes	maximal, characteristic
15	#474×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050589	C₂²	1	1	yes	characteristic
17	C₂×#13050573	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
19	#14581791	C₂²	1	1	yes	characteristic
20	#14581790	—	2	1	no	—
21	#14581784	C₂²	1	1	yes	characteristic
22	#14581783	C₂²	1	1	yes	characteristic
23	#14581782	—	2	1	no	—
24	#14581770	—	2	1	no	—
25	#14581768	—	2	1	no	—
26	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
27	#14581757	C₂²	1	1	yes	characteristic
28	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊D₈₃	—	2	1	no	—
30	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
31	#14581750	C₂²	1	1	yes	characteristic
32	#14581749	C₂²	1	1	yes	characteristic
33	#14581748	—	2	1	no	—
34	C₂×Dic₈₃⋊C₄	—	2	1	no	—
35	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
37	#14581726	C₂²	1	1	yes	characteristic
38	#14581725	C₂²	1	1	yes	characteristic
39	#14581725	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	#14581715	C₂²	1	1	yes	characteristic
49	#14581715	C₂²	1	1	yes	characteristic
50	#14581713	C₂²	1	1	yes	characteristic
51	#14581713	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
54	#107×C₈₃	C₂²	1	1	yes	characteristic
55	#107×C₈₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	#102×C₈₃	C₂²	1	1	yes	characteristic
59	#13050589	—	2	2	no	—
60	#13050589	Q₈	1	4	yes	complemented
61	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₃	—	2	1	no	—
64	C₂²×Dic₈₃	—	2	1	no	—
65	C₂²×Dic₈₃	—	2	1	no	—
66	C₂×C₄×D₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
69	#13050573	—	2	2	no	—
70	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
72	C₈×D₈₃	—	2	2	no	—
73	C₄⋊Dic₈₃	—	2	1	no	—
74	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
86	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₈₃	—	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	—	2	1	no	—
96	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₈₃	—	2	1	no	—
98	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₃	—	2	1	no	—
101	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₃	—	2	1	no	—
105	C₄²×C₈₃	C₂³	1	1	yes	characteristic
106	C₂³×C₈₃	—	2	1	no	—
107	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
108	Q₈×C₈₃	—	8	1	no	—
109	D₄×C₈₃	—	2	1	no	—
110	D₄×C₈₃	C₂×Q₈	1	2	yes	—
111	C₂×C₄×C₈₃	—	2	1	no	—
112	C₂×C₄×C₈₃	—	2	1	no	—
113	C₂×C₄×C₈₃	—	2	1	no	—
114	C₂×C₄×C₈₃	—	8	1	no	—
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	C₂²⋊D₈₃	—	2	4	no	—
119	C₂×Dic₈₃	—	2	1	no	—
120	C₂×Dic₈₃	—	2	4	no	—
121	C₂×Dic₈₃	—	2	1	no	—
122	C₂×Dic₈₃	—	2	1	no	—
123	C₂×Dic₈₃	—	2	1	no	—
124	C₂×Dic₈₃	—	8	1	no	—
125	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
128	C₄×D₈₃	—	8	1	no	—
129	C₄×D₈₃	—	8	1	no	—
130	C₄×D₈₃	C₂×Q₈	1	2	yes	—
131	C₄×D₈₃	#41	1	2	yes	—
132	Dic₁₆₆	—	2	1	no	—
133	Dic₁₆₆	—	8	1	no	—
134	Dic₁₆₆	C₂×Q₈	1	2	yes	—
135	C₈×C₈₃	—	2	1	no	—
136	C₈×C₈₃	—	2	1	no	—
137	C₈₃⋊C₈	—	2	1	no	—
138	C₈₃⋊C₈	—	2	1	no	—
139	C₂²×C₈₃	—	2	2	no	—
140	C₂²×C₈₃	—	4	1	no	—
141	C₂²×C₈₃	—	4	1	no	—
142	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₈₃	—	2	1	no	—
144	C₂×D₈₃	—	2	1	no	—
145	C₂×D₈₃	C₂²⋊Q₈	1	2	yes	—
146	C₄×C₈₃	—	2	1	no	—
147	C₄×C₈₃	—	2	1	no	—
148	C₄×C₈₃	—	4	1	no	—
149	C₄×C₈₃	—	4	1	no	—
150	C₄×C₈₃	—	4	1	no	—
151	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
152	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
153	Dic₈₃	—	2	2	no	—
154	Dic₈₃	—	2	1	no	—
155	Dic₈₃	—	2	1	no	—
156	Dic₈₃	—	4	1	no	—
157	Dic₈₃	—	4	1	no	—
158	Dic₈₃	—	4	1	no	—
159	Dic₈₃	C₂²⋊Q₈	1	2	yes	—
160	C₂×C₈₃	—	4	2	no	—
161	C₂×C₈₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₈₃	#576	1	1	yes	characteristic
163	C₂×C₈₃	#578	1	1	yes	characteristic
164	D₈₃	—	2	2	no	—
165	#3083	—	83	1	no	maximal, Sylow
166	C₈₃	#3083	1	1	yes	characteristic, complemented, Sylow
167	#544	—	83	1	no	—
168	#520	—	83	1	no	—
169	#479	—	83	1	no	—
170	#479	—	83	1	no	—
171	#478	—	83	1	no	—
172	#478	—	83	1	no	—
173	#477	—	83	1	no	—
174	#477	—	83	1	no	—
175	#474	—	83	1	no	—
176	#474	D₈₃	1	1	yes	characteristic, complemented
177	#428	—	83	1	no	—
178	C₂×#107	—	83	1	no	—
179	#424	—	83	1	no	—
180	#417	—	83	1	no	—
181	#416	—	83	1	no	—
182	C₂×#41	—	83	1	no	—
183	C₂×M₁₆	—	83	1	no	—
184	C₂²×C₈	—	83	1	no	—
185	C₄⋊Q₈	—	83	1	no	—
186	C₄⋊Q₈	—	166	1	no	—
187	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
188	#125	—	83	1	no	—
189	#125	—	83	1	no	—
190	#125	—	166	1	no	—
191	C₂²⋊Q₈	—	166	1	no	—
192	C₂²⋊Q₈	—	166	1	no	—
193	C₄×Q₈	—	83	1	no	—
194	C₄×Q₈	—	83	1	no	—
195	C₄×D₄	—	83	1	no	—
196	C₄×D₄	—	166	1	no	—
197	C₄×D₄	—	166	1	no	—
198	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
199	#117	—	83	1	no	—
200	#117	—	83	1	no	—
201	#117	—	166	1	no	—
202	C₂×C₄⋊C₄	—	83	1	no	—
203	C₂×C₄⋊C₄	—	83	1	no	—
204	C₂×C₄⋊C₄	—	166	1	no	—
205	#107	—	83	1	no	—
206	#107	—	83	1	no	—
207	#107	—	83	1	no	—
208	#107	—	83	1	no	—
209	#107	C₂×D₈₃	1	1	yes	characteristic
210	#107	C₂×D₈₃	1	1	yes	characteristic
211	#106	—	83	1	no	—
212	#106	—	83	1	no	—
213	C₄⋊C₈	—	83	1	no	—
214	C₄⋊C₈	—	83	1	no	—
215	C₄⋊C₈	—	83	1	no	—
216	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
217	#103	—	83	1	no	—
218	#103	—	83	1	no	—
219	#103	—	83	1	no	—
220	#103	—	83	1	no	—
221	#102	—	83	1	no	—
222	#102	—	83	1	no	—
223	#102	C₂×D₈₃	1	1	yes	characteristic
224	#102	C₂×D₈₃	1	1	yes	characteristic
225	#41	—	83	4	no	—
226	#41	—	166	2	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	83	1	no	—
229	C₂×Q₈	—	166	1	no	—
230	C₂×D₄	—	166	1	no	—
231	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	83	1	no	—
234	C₂²×C₄	—	166	1	no	—
235	C₂²×C₄	—	166	1	no	—
236	C₂²×C₄	—	166	1	no	—
237	C₂²×C₄	—	166	1	no	—
238	M₁₆	—	166	2	no	—
239	C₂×C₈	—	83	1	no	—
240	C₂×C₈	—	83	1	no	—
241	C₂×C₈	—	166	2	no	—
242	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	83	1	no	—
246	C₄⋊C₄	—	83	1	no	—
247	C₄⋊C₄	—	83	1	no	—
248	C₄⋊C₄	—	83	1	no	—
249	C₄⋊C₄	—	83	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	—	166	1	no	—
257	C₄⋊C₄	—	166	1	no	—
258	C₄⋊C₄	—	166	1	no	—
259	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	83	1	no	—
264	C₂²⋊C₄	—	83	1	no	—
265	C₂²⋊C₄	—	166	1	no	—
266	C₂²⋊C₄	—	166	1	no	—
267	C₄²	—	83	1	no	—
268	C₄²	—	166	1	no	—
269	C₄²	—	166	1	no	—
270	C₄²	—	166	1	no	—
271	C₄²	C₂²×D₈₃	1	1	yes	characteristic
272	C₂³	—	2	1	no	—
273	C₂³	—	83	1	no	—
274	Q₈	—	8	1	no	—
275	Q₈	—	83	2	no	—
276	Q₈	—	664	1	no	—
277	Q₈	—	166	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	166	4	no	—
280	D₄	Q₈×D₈₃	1	2	yes	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	83	1	no	—
286	C₂×C₄	—	83	2	no	—
287	C₂×C₄	—	83	2	no	—
288	C₂×C₄	—	83	1	no	—
289	C₂×C₄	—	83	1	no	—
290	C₂×C₄	—	664	1	no	—
291	C₂×C₄	—	664	1	no	—
292	C₂×C₄	—	664	1	no	—
293	C₂×C₄	—	166	1	no	—
294	C₂×C₄	—	166	1	no	—
295	C₂×C₄	—	166	1	no	—
296	C₂×C₄	—	166	4	no	—
297	C₂×C₄	—	166	1	no	—
298	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
299	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	166	1	no	—
304	C₈	—	166	1	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	83	2	no	—
309	C₂²	—	166	1	no	—
310	C₂²	—	166	1	no	—
311	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	332	1	no	—
318	C₄	—	332	1	no	—
319	C₄	—	332	1	no	—
320	C₄	—	83	2	no	—
321	C₄	—	166	2	no	—
322	C₄	—	166	1	no	—
323	C₄	—	166	1	no	—
324	C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
325	C₄	#41×D₈₃	1	1	yes	characteristic
326	C₂	—	4	2	no	—
327	C₂	—	166	2	no	—
328	C₂	C₂²⋊Q₈×D₈₃	1	1	yes	characteristic
329	C₂	#19128502	1	1	yes	characteristic
330	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	#474×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×#13050589	C₂²
hypercenter	#474	D₈₃
hypocenter	C₈₃	#3083
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #474

Complements

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

N	H ≅ G/N	type
1. #19128328	C₂	semidirect
3. #19128009	C₂	semidirect
4. #19128007	C₂	semidirect
5. #19128003	C₂	semidirect
6. #19128001	C₂	semidirect
7. #19127995	C₂	semidirect
8. #19127993	C₂	semidirect
9. #19127979	C₂	semidirect
10. #19127742	C₂	semidirect
11. #107×D₈₃	C₂	semidirect
12. #19127726	C₂	semidirect
15. #474×C₈₃	C₂	semidirect
60. #13050589	Q₈	semidirect
166. C₈₃	#3083	semidirect
176. #474	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128328	2
#19128180	2
#19128009	2
#19128007	2
#19128003	2
#19128001	2
#19127995	2
#19127993	2
#19127979	2
#19127742	2
#107×D₈₃	2
#19127726	2
#19127699	2
#19127694	2
#474×C₈₃	2
#3083	83