Subgroups — #24920231

Order 10624.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128436	C₂	1	1	yes	maximal, characteristic
2	#19128067	C₂	1	1	yes	maximal, characteristic
3	#19128063	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128058	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128005	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127998	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127988	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127923	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127923	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127901	C₂	1	1	yes	maximal, characteristic
11	#19127832	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127806	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127799	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13050577	C₂	1	1	yes	maximal, characteristic, complemented
15	#557×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050580	C₂²	1	1	yes	characteristic
17	C₂×#13050577	C₂²	1	1	yes	characteristic
18	C₂×#13050577	C₂²	1	1	yes	characteristic
19	#14581789	C₂²	1	1	yes	characteristic
20	#14581781	—	2	1	no	—
21	#14581780	C₂²	1	1	yes	characteristic
22	#14581780	C₂²	1	1	yes	characteristic
23	#14581777	—	2	1	no	—
24	#14581777	—	2	1	no	—
25	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
26	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
27	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
28	#14581754	—	2	1	no	—
29	#14581725	C₂²	1	1	yes	characteristic
30	#14581723	C₂²	1	1	yes	characteristic
31	#14581723	C₂²	1	1	yes	characteristic
32	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
33	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
34	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
35	#14581716	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	#14581712	C₂²	1	1	yes	characteristic
40	#14581712	C₂²	1	1	yes	characteristic
41	#14581712	C₂²	1	1	yes	characteristic
42	#14581712	C₂²	1	1	yes	characteristic
43	#14581712	C₂²	1	1	yes	characteristic
44	#14581712	C₂²	1	1	yes	characteristic
45	#14581703	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	#125×C₈₃	—	2	1	no	—
51	#125×C₈₃	—	2	1	no	—
52	#125×C₈₃	C₂²	1	1	yes	characteristic
53	#125×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₈₃	—	2	1	no	—
55	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
57	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
58	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
60	C₂×D₃₃₂	—	2	1	no	—
61	C₂×D₃₃₂	—	2	1	no	—
62	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
63	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₈₃	—	2	1	no	—
65	C₂×C₄×D₈₃	—	2	1	no	—
66	#13050580	—	2	2	no	—
67	#13050577	—	2	2	no	—
68	#13050577	—	2	2	no	—
69	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
74	Dic₈₃⋊C₄	—	2	1	no	—
75	(C₂×D₈₃)⋊C₄	—	2	1	no	—
76	(C₂×D₈₃)⋊C₄	—	2	1	no	—
77	(C₂×D₈₃)⋊C₄	—	2	1	no	—
78	(C₂×D₈₃)⋊C₄	—	2	1	no	—
79	(C₂×D₈₃)⋊C₄	—	2	1	no	—
80	(C₂×D₈₃)⋊C₄	—	2	1	no	—
81	C₄×Dic₈₃	—	2	1	no	—
82	C₂×Q₈×C₈₃	—	2	1	no	—
83	C₂×Q₈×C₈₃	D₄	1	1	yes	characteristic
84	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
91	C₄⋊C₄×C₈₃	—	2	1	no	—
92	C₄⋊C₄×C₈₃	—	2	1	no	—
93	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
94	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
95	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
96	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₄²×C₈₃	—	2	1	no	—
100	C₄²×C₈₃	—	2	1	no	—
101	C₄²×C₈₃	—	2	1	no	—
102	C₄²×C₈₃	C₂³	1	1	yes	characteristic
103	C₄²×C₈₃	C₂³	1	1	yes	characteristic
104	C₄²×C₈₃	C₂³	1	1	yes	characteristic
105	C₂²×D₈₃	—	2	1	no	—
106	C₂²×D₈₃	—	2	1	no	—
107	Q₈×C₈₃	—	2	1	no	—
108	Q₈×C₈₃	—	4	1	no	—
109	Q₈×C₈₃	—	4	2	no	—
110	Q₈×C₈₃	#41	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	—	2	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	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
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	D₃₃₂	—	2	1	no	—
125	D₃₃₂	—	2	1	no	—
126	D₃₃₂	—	2	1	no	—
127	D₃₃₂	—	2	1	no	—
128	D₃₃₂	—	4	2	no	—
129	D₃₃₂	—	8	1	no	—
130	C₄×D₈₃	—	8	1	no	—
131	C₄×D₈₃	—	8	1	no	—
132	Dic₁₆₆	—	2	1	no	—
133	Dic₁₆₆	—	2	1	no	—
134	C₈₃⋊C₈	—	2	1	no	—
135	C₈₃⋊C₈	—	2	1	no	—
136	C₈₃⋊C₈	—	2	1	no	—
137	C₈₃⋊C₈	—	2	1	no	—
138	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₈₃	—	4	1	no	—
140	C₂×D₈₃	—	4	1	no	—
141	C₂×D₈₃	—	4	1	no	—
142	C₂×D₈₃	—	4	1	no	—
143	C₂×D₈₃	—	4	1	no	—
144	C₂×D₈₃	—	4	1	no	—
145	C₄×C₈₃	—	2	2	no	—
146	C₄×C₈₃	—	2	1	no	—
147	C₄×C₈₃	—	2	1	no	—
148	C₄×C₈₃	—	2	1	no	—
149	C₄×C₈₃	—	2	1	no	—
150	C₄×C₈₃	—	2	1	no	—
151	C₄×C₈₃	—	2	1	no	—
152	C₄×C₈₃	—	4	1	no	—
153	C₄×C₈₃	—	4	1	no	—
154	C₄×C₈₃	—	4	1	no	—
155	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
156	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
157	Dic₈₃	—	4	1	no	—
158	Dic₈₃	—	4	1	no	—
159	C₂×C₈₃	#548	1	1	yes	characteristic
160	C₂×C₈₃	#575	1	1	yes	characteristic
161	C₂×C₈₃	#577	1	1	yes	characteristic
162	D₈₃	—	8	1	no	—
163	D₈₃	—	8	1	no	—
164	#3371	—	83	1	no	maximal, Sylow
165	C₈₃	#3371	1	1	yes	characteristic, complemented, Sylow
166	#559	—	83	1	no	—
167	#557	D₈₃	1	1	yes	characteristic, complemented
168	#490	—	83	1	no	—
169	#489	—	83	1	no	—
170	#488	—	83	1	no	—
171	#479	—	83	1	no	—
172	#478	—	83	1	no	—
173	#476	—	83	1	no	—
174	#464	—	83	1	no	—
175	#464	—	83	1	no	—
176	#460	—	83	1	no	—
177	#446	—	83	1	no	—
178	#441	—	83	1	no	—
179	#440	—	83	1	no	—
180	C₄×SD₁₆	—	83	1	no	—
181	C₂×Q₁₆	—	83	1	no	—
182	C₂×SD₁₆	—	83	1	no	—
183	C₂×SD₁₆	—	83	1	no	—
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
186	#127	—	83	1	no	—
187	#125	—	2	1	no	—
188	#125	—	2	1	no	—
189	#125	C₂×D₈₃	1	1	yes	characteristic
190	#125	C₂×D₈₃	1	1	yes	characteristic
191	#124	—	83	1	no	—
192	#124	—	83	1	no	—
193	#124	—	166	1	no	—
194	#123	—	166	1	no	—
195	#123	—	166	1	no	—
196	C₄×Q₈	—	2	1	no	—
197	C₄×Q₈	—	83	1	no	—
198	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
199	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
200	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
201	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
202	C₄×D₄	—	83	1	no	—
203	C₄×D₄	—	83	1	no	—
204	C₄×D₄	—	166	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	#103	—	83	1	no	—
212	#103	—	83	1	no	—
213	#103	—	83	1	no	—
214	#103	—	83	1	no	—
215	#102	—	83	1	no	—
216	#102	—	83	1	no	—
217	#102	—	83	1	no	—
218	#102	—	83	1	no	—
219	#102	—	83	1	no	—
220	#102	—	83	1	no	—
221	#97	—	83	1	no	—
222	#97	—	83	1	no	—
223	C₄×C₈	—	83	1	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	83	1	no	—
226	C₂×Q₈	C₂²⋊D₈₃	1	1	yes	characteristic
227	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
228	C₂×D₄	—	83	1	no	—
229	C₂×D₄	—	83	1	no	—
230	C₂×D₄	—	166	1	no	—
231	C₂×D₄	—	166	1	no	—
232	C₂²×C₄	—	166	1	no	—
233	C₂²×C₄	—	166	1	no	—
234	Q₁₆	—	166	2	no	—
235	SD₁₆	—	166	2	no	—
236	SD₁₆	—	166	2	no	—
237	C₂×C₈	—	83	1	no	—
238	C₂×C₈	—	83	1	no	—
239	C₂×C₈	—	83	1	no	—
240	C₂×C₈	—	83	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	83	1	no	—
250	C₄⋊C₄	—	166	1	no	—
251	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
252	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
253	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
257	C₂²⋊C₄	—	166	1	no	—
258	C₂²⋊C₄	—	166	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	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	166	1	no	—
267	C₄²	C₂²×D₈₃	1	1	yes	characteristic
268	C₄²	C₂²×D₈₃	1	1	yes	characteristic
269	C₄²	C₂²×D₈₃	1	1	yes	characteristic
270	C₂³	—	166	1	no	—
271	C₂³	—	166	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	4	1	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	166	1	no	—
276	Q₈	—	166	1	no	—
277	Q₈	#13050591	1	2	yes	—
278	D₄	—	332	2	no	—
279	D₄	—	664	1	no	—
280	D₄	—	166	1	no	—
281	D₄	—	166	1	no	—
282	D₄	—	166	1	no	—
283	D₄	—	166	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	664	1	no	—
289	C₂×C₄	—	664	1	no	—
290	C₂×C₄	—	166	1	no	—
291	C₂×C₄	—	166	1	no	—
292	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
293	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
294	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
296	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
299	C₈	—	166	1	no	—
300	C₈	—	166	1	no	—
301	C₈	—	166	1	no	—
302	C₈	—	166	1	no	—
303	C₂²	—	332	1	no	—
304	C₂²	—	332	1	no	—
305	C₂²	—	332	1	no	—
306	C₂²	—	332	1	no	—
307	C₂²	—	332	1	no	—
308	C₂²	—	332	1	no	—
309	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	332	1	no	—
321	C₄	—	332	1	no	—
322	C₄	C₂×#13050591	1	1	yes	characteristic
323	C₄	#14581840	1	1	yes	characteristic
324	C₂	—	664	1	no	—
325	C₂	—	664	1	no	—
326	C₂	#19128362	1	1	yes	characteristic
327	C₂	#19128499	1	1	yes	characteristic
328	C₂	#19128509	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#557×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#14581789	C₂²
hypercenter	#557	D₈₃
hypocenter	C₈₃	#3371
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

Complements

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

N	H ≅ G/N	type
3. #19128063	C₂	semidirect
4. #19128058	C₂	semidirect
5. #19128005	C₂	semidirect
6. #19127998	C₂	semidirect
7. #19127988	C₂	semidirect
8. #19127923	C₂	semidirect
9. #19127923	C₂	semidirect
11. #19127832	C₂	semidirect
12. #19127806	C₂	semidirect
13. #19127799	C₂	semidirect
14. C₄×#13050577	C₂	semidirect
15. #557×C₈₃	C₂	semidirect
165. C₈₃	#3371	semidirect
167. #557	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128436	2
#19128067	2
#19128063	2
#19128058	2
#19128005	2
#19127998	2
#19127988	2
#19127923	2
#19127901	2
#19127832	2
#19127806	2
#19127799	2
C₄×#13050577	2
#557×C₈₃	2
#3371	83