Subgroups — #24917293

Order 10624.

398 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128331	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128173	C₂	1	1	yes	maximal, characteristic
3	#19128003	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128000	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127998	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127992	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127989	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127981	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127980	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127742	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127736	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127726	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127704	C₂	1	1	yes	maximal, characteristic
14	#102×D₈₃	C₂	1	1	yes	maximal, characteristic
15	#478×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₈₃	C₂²	1	1	yes	characteristic
17	C₂×#13050573	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
19	#14581792	C₂²	1	1	yes	characteristic, complemented
20	#14581791	—	2	1	no	—
21	#14581791	C₂²	1	1	yes	characteristic, complemented
22	#14581783	C₂²	1	1	yes	characteristic, complemented
23	#14581782	—	2	1	no	—
24	#14581772	—	2	1	no	—
25	#14581770	—	2	1	no	—
26	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
27	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
28	#14581754	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	#14581750	C₂²	1	1	yes	characteristic
33	#14581749	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
35	C₂×(C₂×D₈₃)⋊C₄	—	2	1	no	—
36	C₂²⋊C₄×D₈₃	—	2	1	no	—
37	#14581725	C₂²	1	1	yes	characteristic, complemented
38	#14581725	C₂²	1	1	yes	characteristic, complemented
39	#14581725	C₂²	1	1	yes	characteristic, complemented
40	#14581724	C₂²	1	1	yes	characteristic, complemented
41	#14581724	C₂²	1	1	yes	characteristic, complemented
42	#14581723	C₂²	1	1	yes	characteristic, complemented
43	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
44	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
45	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
46	#14581714	C₂²	1	1	yes	characteristic
47	#14581714	C₂²	1	1	yes	characteristic
48	#14581713	C₂²	1	1	yes	characteristic
49	#14581713	C₂²	1	1	yes	characteristic
50	#14581712	C₂²	1	1	yes	characteristic
51	#14581712	C₂²	1	1	yes	characteristic
52	#125×C₈₃	C₂²	1	1	yes	characteristic, complemented
53	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
54	#107×C₈₃	C₂²	1	1	yes	characteristic, complemented
55	#106×C₈₃	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic, complemented
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	#102×C₈₃	C₂²	1	1	yes	characteristic
59	C₂³×D₈₃	—	2	1	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₃	—	2	1	no	—
64	D₄×D₈₃	—	2	2	no	—
65	D₄×D₈₃	Q₈	1	4	yes	complemented
66	C₂²×Dic₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
71	#13050573	—	2	2	no	—
72	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₈×D₈₃	—	2	2	no	—
75	C₄⋊Dic₈₃	—	2	1	no	—
76	C₄⋊Dic₈₃	—	2	1	no	—
77	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	C₂²⋊Dic₈₃	—	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	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
91	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₈₃	—	2	1	no	—
97	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₃	—	2	1	no	—
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	—	2	1	no	—
108	C₂²×D₈₃	—	2	4	no	—
109	C₂²×D₈₃	—	8	1	no	—
110	C₂²×D₈₃	—	8	1	no	—
111	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
112	D₄×C₈₃	—	2	1	no	—
113	D₄×C₈₃	C₂×Q₈	1	2	yes	—
114	C₂×C₄×C₈₃	—	2	1	no	—
115	C₂×C₄×C₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	—	8	1	no	—
118	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₈₃	—	2	4	no	—
122	C₂×Dic₈₃	—	2	1	no	—
123	C₂×Dic₈₃	—	2	1	no	—
124	C₂×Dic₈₃	—	2	1	no	—
125	C₂×Dic₈₃	—	8	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	D₃₃₂	—	2	1	no	—
130	D₃₃₂	C₂×Q₈	1	2	yes	—
131	C₄×D₈₃	—	8	1	no	—
132	C₄×D₈₃	—	8	1	no	—
133	C₄×D₈₃	—	8	1	no	—
134	C₄×D₈₃	C₂×Q₈	1	2	yes	—
135	C₄×D₈₃	#41	1	2	yes	—
136	Dic₁₆₆	—	8	1	no	—
137	Dic₁₆₆	—	8	1	no	—
138	C₈×C₈₃	—	2	1	no	—
139	C₈×C₈₃	—	2	1	no	—
140	C₈₃⋊C₈	—	2	1	no	—
141	C₈₃⋊C₈	—	2	1	no	—
142	C₂²×C₈₃	—	2	2	no	—
143	C₂²×C₈₃	—	4	1	no	—
144	C₂²×C₈₃	—	4	1	no	—
145	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₈₃	—	2	1	no	—
147	C₂×D₈₃	—	2	1	no	—
148	C₂×D₈₃	—	2	2	no	—
149	C₂×D₈₃	—	4	1	no	—
150	C₂×D₈₃	—	4	1	no	—
151	C₂×D₈₃	—	8	4	no	—
152	C₂×D₈₃	C₂²⋊Q₈	1	2	yes	—
153	C₄×C₈₃	—	2	1	no	—
154	C₄×C₈₃	—	2	1	no	—
155	C₄×C₈₃	—	4	1	no	—
156	C₄×C₈₃	—	4	1	no	—
157	C₄×C₈₃	—	4	1	no	—
158	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
159	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
160	Dic₈₃	—	2	1	no	—
161	Dic₈₃	—	2	1	no	—
162	Dic₈₃	—	4	1	no	—
163	Dic₈₃	—	4	1	no	—
164	Dic₈₃	—	4	1	no	—
165	Dic₈₃	C₂²⋊Q₈	1	2	yes	—
166	C₂×C₈₃	—	4	2	no	—
167	C₂×C₈₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
168	C₂×C₈₃	#576	1	1	yes	characteristic
169	C₂×C₈₃	#577	1	1	yes	characteristic
170	D₈₃	—	2	2	no	—
171	D₈₃	—	4	2	no	—
172	#3081	—	83	1	no	maximal, Sylow
173	C₈₃	#3081	1	1	yes	characteristic, complemented, Sylow
174	#544	—	83	1	no	—
175	#518	—	83	1	no	—
176	#478	—	83	1	no	—
177	#478	—	83	1	no	—
178	#478	—	83	1	no	—
179	#478	D₈₃	1	1	yes	characteristic, complemented
180	#476	—	83	1	no	—
181	#476	—	83	1	no	—
182	#474	—	83	1	no	—
183	#474	—	83	1	no	—
184	#428	—	83	1	no	—
185	#427	—	83	1	no	—
186	#424	—	83	1	no	—
187	#418	—	83	1	no	—
188	C₂×#102	—	83	1	no	—
189	C₂²×D₄	—	83	1	no	—
190	C₂×M₁₆	—	83	1	no	—
191	C₂²×C₈	—	83	1	no	—
192	C₄⋊Q₈	—	83	1	no	—
193	C₄⋊Q₈	—	83	1	no	—
194	C₄⋊Q₈	—	166	1	no	—
195	#125	—	83	1	no	—
196	#125	—	166	1	no	—
197	#125	C₂×D₈₃	1	1	yes	characteristic, complemented
198	C₂²⋊Q₈	—	166	1	no	—
199	C₂²⋊Q₈	—	166	1	no	—
200	C₄×D₄	—	83	1	no	—
201	C₄×D₄	—	83	1	no	—
202	C₄×D₄	—	83	1	no	—
203	C₄×D₄	—	166	1	no	—
204	C₄×D₄	—	166	1	no	—
205	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
206	#117	—	83	1	no	—
207	#117	—	83	1	no	—
208	#117	—	83	1	no	—
209	C₂×C₄⋊C₄	—	83	1	no	—
210	C₂×C₂²⋊C₄	—	166	1	no	—
211	C₂×C₂²⋊C₄	—	166	1	no	—
212	#107	—	83	1	no	—
213	#107	—	83	1	no	—
214	#107	—	83	1	no	—
215	#107	C₂×D₈₃	1	1	yes	characteristic, complemented
216	#106	—	83	1	no	—
217	#106	—	83	1	no	—
218	#106	—	83	1	no	—
219	#106	C₂×D₈₃	1	1	yes	characteristic, complemented
220	C₄⋊C₈	—	83	1	no	—
221	C₄⋊C₈	—	83	1	no	—
222	C₄⋊C₈	—	83	1	no	—
223	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic, complemented
224	#102	—	83	1	no	—
225	#102	—	83	1	no	—
226	#102	—	83	1	no	—
227	#102	—	83	1	no	—
228	#102	—	83	1	no	—
229	#102	—	83	1	no	—
230	#102	C₂×D₈₃	1	1	yes	characteristic
231	#102	C₂×D₈₃	1	1	yes	characteristic
232	C₂⁴	—	166	1	no	—
233	C₂×Q₈	—	166	1	no	—
234	C₂×Q₈	—	166	1	no	—
235	C₂×D₄	—	83	4	no	—
236	C₂×D₄	—	83	1	no	—
237	C₂×D₄	—	166	2	no	—
238	C₂×D₄	—	166	1	no	—
239	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	83	1	no	—
242	C₂²×C₄	—	166	1	no	—
243	C₂²×C₄	—	166	1	no	—
244	C₂²×C₄	—	166	1	no	—
245	C₂²×C₄	—	166	1	no	—
246	M₁₆	—	166	2	no	—
247	C₂×C₈	—	83	1	no	—
248	C₂×C₈	—	83	1	no	—
249	C₂×C₈	—	166	2	no	—
250	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	83	1	no	—
255	C₄⋊C₄	—	83	1	no	—
256	C₄⋊C₄	—	83	1	no	—
257	C₄⋊C₄	—	83	1	no	—
258	C₄⋊C₄	—	83	1	no	—
259	C₄⋊C₄	—	166	1	no	—
260	C₄⋊C₄	—	166	1	no	—
261	C₄⋊C₄	—	166	1	no	—
262	C₄⋊C₄	—	166	1	no	—
263	C₄⋊C₄	—	166	1	no	—
264	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
266	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	83	1	no	—
269	C₂²⋊C₄	—	83	1	no	—
270	C₂²⋊C₄	—	166	1	no	—
271	C₂²⋊C₄	—	166	1	no	—
272	C₂²⋊C₄	—	166	1	no	—
273	C₂²⋊C₄	—	166	1	no	—
274	C₂²⋊C₄	—	166	1	no	—
275	C₄²	—	83	1	no	—
276	C₄²	—	166	1	no	—
277	C₄²	—	166	1	no	—
278	C₄²	C₂²×D₈₃	1	1	yes	characteristic
279	C₂³	—	2	1	no	—
280	C₂³	—	83	1	no	—
281	C₂³	—	664	1	no	—
282	C₂³	—	664	1	no	—
283	C₂³	—	166	4	no	—
284	C₂³	—	166	1	no	—
285	Q₈	—	664	1	no	—
286	Q₈	—	664	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	83	2	no	—
289	D₄	—	166	4	no	—
290	D₄	—	166	1	no	—
291	D₄	Q₈×D₈₃	1	2	yes	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	—	83	1	no	—
297	C₂×C₄	—	83	2	no	—
298	C₂×C₄	—	83	2	no	—
299	C₂×C₄	—	83	1	no	—
300	C₂×C₄	—	83	1	no	—
301	C₂×C₄	—	664	1	no	—
302	C₂×C₄	—	664	1	no	—
303	C₂×C₄	—	664	1	no	—
304	C₂×C₄	—	664	1	no	—
305	C₂×C₄	—	166	1	no	—
306	C₂×C₄	—	166	1	no	—
307	C₂×C₄	—	166	1	no	—
308	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
309	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	166	1	no	—
314	C₈	—	166	1	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	332	1	no	—
319	C₂²	—	332	1	no	—
320	C₂²	—	83	2	no	—
321	C₂²	—	664	4	no	—
322	C₂²	—	166	2	no	—
323	C₂²	—	166	1	no	—
324	C₂²	—	166	1	no	—
325	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	332	1	no	—
332	C₄	—	332	1	no	—
333	C₄	—	332	1	no	—
334	C₄	—	83	2	no	—
335	C₄	—	166	1	no	—
336	C₄	—	166	1	no	—
337	C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
338	C₄	#41×D₈₃	1	1	yes	characteristic
339	C₂	—	4	2	no	—
340	C₂	—	332	2	no	—
341	C₂	—	166	2	no	—
342	C₂	C₂²⋊Q₈×D₈₃	1	1	yes	characteristic
343	C₂	#19128505	1	1	yes	characteristic
344	C₂	#19128508	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#478×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×D₄×D₈₃	C₂²
hypercenter	#478	D₈₃
hypocenter	C₈₃	#3081
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #19128331	C₂	semidirect
3. #19128003	C₂	semidirect
4. #19128000	C₂	semidirect
5. #19127998	C₂	semidirect
6. #19127992	C₂	semidirect
7. #19127989	C₂	semidirect
8. #19127981	C₂	semidirect
9. #19127980	C₂	semidirect
10. #19127742	C₂	semidirect
11. #19127736	C₂	semidirect
12. #19127726	C₂	semidirect
15. #478×C₈₃	C₂	semidirect
19. #14581792	C₂²	semidirect
21. #14581791	C₂²	semidirect
22. #14581783	C₂²	semidirect
37. #14581725	C₂²	semidirect
38. #14581725	C₂²	semidirect
39. #14581725	C₂²	semidirect
40. #14581724	C₂²	semidirect
41. #14581724	C₂²	semidirect
42. #14581723	C₂²	semidirect
43. C₄⋊(C₈₃⋊C₈)	C₂²	semidirect
44. Dic₈₃⋊C₈	C₂²	semidirect
45. Dic₈₃⋊C₈	C₂²	semidirect
52. #125×C₈₃	C₂²	semidirect
54. #107×C₈₃	C₂²	semidirect
55. #106×C₈₃	C₂²	semidirect
56. C₄⋊C₈×C₈₃	C₂²	semidirect
65. D₄×D₈₃	Q₈	semidirect
173. C₈₃	#3081	semidirect
179. #478	D₈₃	semidirect
197. #125	C₂×D₈₃	semidirect
215. #107	C₂×D₈₃	semidirect
219. #106	C₂×D₈₃	semidirect
223. C₄⋊C₈	C₂×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128331	2
#19128173	2
#19128003	2
#19128000	2
#19127998	2
#19127992	2
#19127989	2
#19127981	2
#19127980	2
#19127742	2
#19127736	2
#19127726	2
#19127704	2
#102×D₈₃	2
#478×C₈₃	2
#3081	83