Subgroups — #35559150

Order 21248.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24916381	C₂	1	1	yes	maximal, characteristic
2	#24909636	C₂	1	4	yes	maximal, complemented
3	#24909631	C₂	1	4	yes	maximal, complemented
4	#24909235	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19127523	C₂	1	1	yes	maximal, characteristic, complemented
6	#24909229	C₂	1	1	yes	maximal, characteristic, complemented
7	#24909198	C₂	1	1	yes	maximal, characteristic
8	C₂×#19127510	C₂	1	1	yes	maximal, characteristic
9	C₈₃×#3163	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14581814	C₂²	1	1	yes	characteristic
11	#19128467	—	2	1	no	—
12	C₂×#14581732	C₂²	1	1	yes	characteristic
13	C₂×#14581732	C₂²	1	1	yes	characteristic
14	#19127824	—	2	2	no	—
15	#19127819	C₂²	1	4	yes	—
16	C₂×#14581728	C₂²	1	1	yes	characteristic
17	#19127719	—	2	1	no	—
18	#19127523	C₂²	1	2	yes	complemented
19	#19127523	C₂²	1	4	yes	complemented
20	#19127523	C₂²	1	2	yes	complemented
21	#19127522	C₂²	1	4	yes	complemented
22	#19127510	C₂²	1	2	yes	—
23	#19127510	C₂²	1	4	yes	—
24	#19127510	C₂²	1	2	yes	—
25	#544×C₈₃	—	2	1	no	—
26	#501×C₈₃	—	2	2	no	—
27	#499×C₈₃	C₂²	1	2	yes	complemented
28	#498×C₈₃	C₂²	1	2	yes	complemented
29	#428×C₈₃	—	2	1	no	—
30	#427×C₈₃	C₂²	1	1	yes	characteristic
31	C₂×#106×C₈₃	C₂²	1	1	yes	characteristic
32	#405×C₈₃	C₂²	1	1	yes	characteristic
33	C₂×#13050590	—	2	1	no	—
34	#14581814	—	2	1	no	—
35	#14581814	—	2	1	no	—
36	#14581814	Q₈	1	2	yes	complemented
37	#14581814	C₂³	1	2	yes	—
38	C₂×Dic₃₃₂	Q₈	1	1	yes	characteristic, complemented
39	C₂×#13050579	—	2	1	no	—
40	C₂×D₆₆₄	Q₈	1	1	yes	characteristic, complemented
41	#14581800	—	4	2	no	—
42	#14581798	—	2	4	no	—
43	C₂×#13050574	—	2	1	no	—
44	M₁₆×D₈₃	—	4	2	no	—
45	#14581732	—	2	1	no	—
46	#14581732	—	2	1	no	—
47	#14581732	C₂³	1	2	yes	—
48	#14581732	C₂³	1	2	yes	—
49	C₂×C₈₃⋊C₁₆	—	2	1	no	—
50	C₂×C₈₃⋊C₁₆	—	2	1	no	—
51	#14581728	—	2	1	no	—
52	#14581728	C₂³	1	2	yes	—
53	#14581720	—	2	4	no	—
54	C₂×M₁₆×C₈₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₈₃	—	2	2	no	—
57	C₄⋊Q₈×C₈₃	—	4	1	no	—
58	#125×C₈₃	—	2	2	no	—
59	#125×C₈₃	—	4	1	no	—
60	C₂²⋊Q₈×C₈₃	—	4	2	no	—
61	#117×C₈₃	—	2	1	no	—
62	#117×C₈₃	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
64	#107×C₈₃	—	2	2	no	—
65	#107×C₈₃	—	2	1	no	—
66	#106×C₈₃	—	2	2	no	—
67	#106×C₈₃	—	2	1	no	—
68	#106×C₈₃	C₂³	1	1	yes	characteristic
69	#106×C₈₃	C₂³	1	2	yes	—
70	#106×C₈₃	C₂³	1	1	yes	characteristic
71	#97×C₈₃	D₄	1	2	yes	—
72	C₄×C₈×C₈₃	C₂³	1	2	yes	—
73	#13050590	—	2	2	no	—
74	#13050590	—	2	2	no	—
75	#13050590	—	4	1	no	—
76	#13050590	—	4	1	no	—
77	C₂×Dic₁₆₆	—	2	1	no	—
78	C₂×D₃₃₂	—	2	1	no	—
79	C₂×C₂²⋊D₈₃	—	4	1	no	—
80	C₂×C₄×D₈₃	—	4	1	no	—
81	Dic₃₃₂	—	2	1	no	—
82	Dic₃₃₂	C₂×Q₈	1	2	yes	—
83	#13050579	—	2	2	no	—
84	#13050579	—	2	2	no	—
85	D₆₆₄	—	2	1	no	—
86	D₆₆₄	C₂×Q₈	1	2	yes	—
87	#13050574	—	2	2	no	—
88	#13050574	—	4	1	no	—
89	#13050573	—	4	4	no	—
90	C₂×C₈₃⋊C₈	—	4	1	no	—
91	C₈×D₈₃	—	4	4	no	—
92	C₈₃⋊C₁₆	—	2	2	no	—
93	C₈₃⋊C₁₆	—	2	2	no	—
94	C₂×Q₈×C₈₃	—	4	2	no	—
95	C₂²×C₄×C₈₃	—	4	1	no	—
96	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
97	M₁₆×C₈₃	SD₁₆	1	2	yes	—
98	M₁₆×C₈₃	SD₁₆	1	2	yes	—
99	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
102	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
104	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
105	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₈₃	—	2	1	no	—
107	C₄⋊C₄×C₈₃	—	2	2	no	—
108	C₄⋊C₄×C₈₃	—	2	1	no	—
109	C₄⋊C₄×C₈₃	—	2	1	no	—
110	C₄⋊C₄×C₈₃	—	4	2	no	—
111	C₄⋊C₄×C₈₃	—	4	2	no	—
112	C₄⋊C₄×C₈₃	—	4	2	no	—
113	C₄⋊C₄×C₈₃	—	4	1	no	—
114	C₂²⋊C₄×C₈₃	—	2	1	no	—
115	C₂²⋊C₄×C₈₃	—	4	1	no	—
116	C₄²×C₈₃	—	4	1	no	—
117	C₄²×C₈₃	C₂×D₄	1	2	yes	—
118	C₂³×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂²×D₈₃	—	4	1	no	—
120	Q₈×C₈₃	—	16	2	no	—
121	C₂×C₄×C₈₃	—	2	2	no	—
122	C₂×C₄×C₈₃	—	4	1	no	—
123	C₂×C₄×C₈₃	—	4	1	no	—
124	C₂×C₄×C₈₃	—	4	2	no	—
125	C₂×C₄×C₈₃	—	16	1	no	—
126	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
132	C₂²⋊D₈₃	—	4	2	no	—
133	C₂²⋊D₈₃	—	4	2	no	—
134	C₂×Dic₈₃	—	4	1	no	—
135	D₃₃₂	—	2	2	no	—
136	D₃₃₂	—	4	1	no	—
137	C₄×D₈₃	—	4	2	no	—
138	C₄×D₈₃	—	4	2	no	—
139	Dic₁₆₆	—	2	2	no	—
140	Dic₁₆₆	—	4	1	no	—
141	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
142	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
143	C₈×C₈₃	C₂×SD₁₆	1	4	yes	—
144	C₈×C₈₃	C₂²×Q₈	1	1	yes	characteristic
145	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
146	C₈₃⋊C₈	—	4	2	no	—
147	C₂²×C₈₃	—	4	1	no	—
148	C₂²×C₈₃	#476	1	1	yes	characteristic
149	C₂²×C₈₃	#476	1	1	yes	characteristic
150	C₂²×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×D₈₃	—	4	2	no	—
152	C₂×D₈₃	—	16	1	no	—
153	C₄×C₈₃	—	4	2	no	—
154	C₄×C₈₃	—	8	1	no	—
155	C₄×C₈₃	—	8	2	no	—
156	C₄×C₈₃	—	8	1	no	—
157	C₄×C₈₃	#476	1	1	yes	characteristic
158	C₄×C₈₃	#476	1	1	yes	characteristic
159	C₄×C₈₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
160	C₄×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
161	Dic₈₃	—	4	2	no	—
162	#33638	—	83	1	no	maximal, Sylow
163	C₂×C₈₃	—	2	1	no	—
164	C₂×C₈₃	—	2	1	no	—
165	C₂×C₈₃	—	2	1	no	—
166	C₂×C₈₃	C₂×#476	1	1	yes	characteristic
167	D₈₃	—	8	2	no	—
168	#3163	D₈₃	1	1	yes	characteristic, complemented
169	#2957	—	83	1	no	—
170	#2231	—	83	4	no	—
171	#2230	—	83	4	no	—
172	#2155	—	83	1	no	—
173	C₂×#365	—	83	1	no	—
174	#2153	—	83	1	no	—
175	#2145	—	83	1	no	—
176	C₂×#360	—	83	1	no	—
177	C₈₃	#33638	1	1	yes	characteristic, complemented, Sylow
178	C₂×#135	—	83	1	no	—
179	#568	—	166	1	no	—
180	#544	—	2	1	no	—
181	C₂×M₃₂	—	83	1	no	—
182	C₂×M₃₂	—	83	1	no	—
183	#501	—	2	2	no	—
184	#499	C₂×D₈₃	1	2	yes	complemented
185	#498	C₂×D₈₃	1	2	yes	complemented
186	#444	—	166	2	no	—
187	#443	—	83	4	no	—
188	C₂×#108	—	83	1	no	—
189	#428	—	2	1	no	—
190	#427	C₂×D₈₃	1	1	yes	characteristic
191	C₂×#106	C₂×D₈₃	1	1	yes	characteristic
192	#421	—	166	1	no	—
193	#405	C₂×D₈₃	1	1	yes	characteristic
194	#365	—	83	4	no	—
195	#365	—	83	2	no	—
196	#365	—	83	2	no	—
197	#364	—	83	4	no	—
198	#360	—	83	4	no	—
199	#360	—	83	2	no	—
200	#360	—	83	2	no	—
201	C₂×#41	—	166	1	no	—
202	#135	—	83	2	no	—
203	#135	—	83	2	no	—
204	#135	—	166	1	no	—
205	#135	—	166	1	no	—
206	C₂×Q₁₆	—	83	1	no	—
207	C₂×SD₁₆	—	166	1	no	—
208	C₂×D₈	—	83	1	no	—
209	#131	—	332	2	no	—
210	#131	—	166	4	no	—
211	C₂×M₁₆	—	332	2	no	—
212	C₂×M₁₆	—	166	1	no	—
213	C₂×M₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
215	C₄⋊Q₈	—	2	2	no	—
216	C₄⋊Q₈	—	4	1	no	—
217	#125	—	2	2	no	—
218	#125	—	4	1	no	—
219	C₂²⋊Q₈	—	4	2	no	—
220	#117	—	2	1	no	—
221	#117	C₂²⋊D₈₃	1	1	yes	characteristic
222	C₂×C₄⋊C₄	—	2	1	no	—
223	M₃₂	—	83	2	no	—
224	M₃₂	—	83	2	no	—
225	M₃₂	—	166	1	no	—
226	M₃₂	—	166	1	no	—
227	C₂×C₁₆	—	166	1	no	—
228	C₂×C₁₆	—	166	1	no	—
229	#108	—	83	2	no	—
230	#108	—	166	1	no	—
231	#107	—	2	1	no	—
232	#107	—	2	2	no	—
233	#106	—	2	1	no	—
234	#106	—	2	2	no	—
235	#106	C₂²×D₈₃	1	2	yes	—
236	#106	C₂²×D₈₃	1	1	yes	characteristic
237	#106	C₂²×D₈₃	1	1	yes	characteristic
238	#104	—	166	4	no	—
239	#97	C₂²⋊D₈₃	1	2	yes	—
240	C₄×C₈	C₂²×D₈₃	1	2	yes	—
241	#41	—	332	1	no	—
242	#41	—	332	1	no	—
243	#41	—	166	2	no	—
244	#41	—	166	2	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	166	1	no	—
247	C₂×D₄	—	332	1	no	—
248	C₂×D₄	—	166	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	332	1	no	—
251	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
252	Q₁₆	—	83	2	no	—
253	Q₁₆	—	166	1	no	—
254	SD₁₆	—	166	2	no	—
255	SD₁₆	—	166	2	no	—
256	D₈	—	83	2	no	—
257	D₈	—	166	1	no	—
258	M₁₆	—	332	1	no	—
259	M₁₆	—	332	4	no	—
260	M₁₆	—	166	2	no	—
261	M₁₆	#13050577	1	2	yes	—
262	M₁₆	#13050577	1	2	yes	—
263	C₂×C₈	—	332	1	no	—
264	C₂×C₈	—	332	4	no	—
265	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
267	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
268	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
269	C₂×C₈	#13050591	1	1	yes	characteristic
270	C₂×C₈	#13050591	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₄²	—	4	1	no	—
283	C₄²	C₂×C₂²⋊D₈₃	1	2	yes	—
284	C₁₆	—	166	2	no	—
285	C₁₆	—	166	2	no	—
286	C₂³	—	332	1	no	—
287	C₂³	#14581818	1	1	yes	characteristic
288	Q₈	—	16	2	no	—
289	Q₈	—	332	1	no	—
290	Q₈	—	166	2	no	—
291	D₄	—	332	2	no	—
292	D₄	—	332	2	no	—
293	D₄	—	332	1	no	—
294	D₄	—	166	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	16	1	no	—
300	C₂×C₄	—	332	1	no	—
301	C₂×C₄	—	332	2	no	—
302	C₂×C₄	—	332	2	no	—
303	C₂×C₄	#14581768	1	1	yes	characteristic
304	C₂×C₄	#14581768	1	1	yes	characteristic
305	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
306	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
307	C₂×C₄	#14581818	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
309	C₈	—	332	2	no	—
310	C₈	#14581768	1	1	yes	characteristic
311	C₈	#14581768	1	1	yes	characteristic
312	C₈	C₂×#13050577	1	4	yes	—
313	C₈	C₂×Q₈×D₈₃	1	1	yes	characteristic
314	C₈	C₂×#13050591	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	1328	1	no	—
317	C₂²	—	332	2	no	—
318	C₂²	#19127988	1	1	yes	characteristic
319	C₂²	#19127988	1	1	yes	characteristic
320	C₂²	C₂×#14581818	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	1	no	—
325	C₄	—	332	2	no	—
326	C₄	#19127988	1	1	yes	characteristic
327	C₄	#19127988	1	1	yes	characteristic
328	C₄	C₂×#14581768	1	1	yes	characteristic
329	C₄	C₂²×#13050577	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	664	2	no	—
334	C₂	C₂×#19127988	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19127988
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#3163	C₂
socle	C₂×C₈₃	C₂×#476
prime radical	C₂×#14581814	C₂²
hypercenter	#3163	D₈₃
hypocenter	C₈₃	#33638
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #3163

Complements

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

N	H ≅ G/N	type
2. #24909636	C₂	semidirect
3. #24909631	C₂	semidirect
4. #24909235	C₂	semidirect
5. C₂×#19127523	C₂	semidirect
6. #24909229	C₂	semidirect
9. C₈₃×#3163	C₂	semidirect
18. #19127523	C₂²	semidirect
19. #19127523	C₂²	semidirect
20. #19127523	C₂²	semidirect
21. #19127522	C₂²	semidirect
27. #499×C₈₃	C₂²	semidirect
28. #498×C₈₃	C₂²	semidirect
36. #14581814	Q₈	semidirect
38. C₂×Dic₃₃₂	Q₈	semidirect
40. C₂×D₆₆₄	Q₈	semidirect
168. #3163	D₈₃	semidirect
177. C₈₃	#33638	semidirect
184. #499	C₂×D₈₃	semidirect
185. #498	C₂×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24916381	2
#24909636	2
#24909631	2
#24909235	2
C₂×#19127523	2
#24909229	2
#24909198	2
C₂×#19127510	2
C₈₃×#3163	2
#33638	83