Subgroups — #35392339

Order 21248.

858 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24918584	C₂	1	2	yes	maximal, complemented
2	#24918042	C₂	1	1	yes	maximal, characteristic
3	#24917920	C₂	1	2	yes	maximal, complemented
4	#24907133	C₂	1	4	yes	maximal, complemented
5	#24907080	C₂	1	2	yes	maximal, complemented
6	#24907056	C₂	1	2	yes	maximal, complemented
7	C₈×#13050574	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#500×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14581823	—	2	4	no	—
10	#19128251	C₂²	1	2	yes	—
11	#19128106	C₂²	1	1	yes	characteristic
12	#19128104	C₂²	1	4	yes	—
13	#19128093	—	2	4	no	—
14	#19128080	C₂²	1	4	yes	—
15	#19128078	C₂²	1	1	yes	characteristic
16	#19128075	C₂²	1	4	yes	—
17	#19128049	C₂²	1	4	yes	—
18	#19128048	C₂²	1	2	yes	—
19	#19128014	—	2	4	no	—
20	#19127978	—	2	4	no	—
21	#19127937	—	2	2	no	—
22	C₄×D₆₆₄	—	2	2	no	—
23	#19127754	C₂²	1	1	yes	characteristic
24	C₄×#13050574	C₂²	1	1	yes	characteristic
25	#19127423	C₂²	1	2	yes	—
26	C₈×C₈₃⋊C₈	C₂²	1	2	yes	—
27	C₂×C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
28	#500×C₈₃	—	2	2	no	—
29	#500×C₈₃	C₂²	1	2	yes	—
30	#500×C₈₃	C₂²	1	2	yes	—
31	C₂×#107×C₈₃	—	2	2	no	—
32	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
33	C₂×#13050590	—	2	2	no	—
34	#14581823	—	4	16	no	—
35	C₂×#13050580	—	2	8	no	—
36	C₂×#13050577	—	2	8	no	—
37	C₂×D₆₆₄	—	2	2	no	—
38	C₂×#13050574	D₄	1	2	yes	—
39	#14581792	C₂³	1	2	yes	—
40	#14581789	C₂³	1	2	yes	—
41	#14581780	—	2	2	no	—
42	#14581766	—	2	2	no	—
43	#14581766	—	2	2	no	—
44	C₄×D₃₃₂	—	2	2	no	—
45	C₄×D₃₃₂	—	2	2	no	—
46	#14581726	—	2	2	no	—
47	#14581725	—	2	8	no	—
48	C₄⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
49	#14581714	—	2	4	no	—
50	#14581712	—	2	8	no	—
51	C₂²⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
52	#14581703	D₄	1	2	yes	—
53	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
54	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
55	C₂²×Q₈×C₈₃	—	2	2	no	—
56	C₂²×C₈×C₈₃	D₄	1	2	yes	—
57	C₄⋊Q₈×C₈₃	—	2	2	no	—
58	C₄⋊Q₈×C₈₃	—	2	2	no	—
59	C₄⋊Q₈×C₈₃	—	2	2	no	—
60	C₄⋊Q₈×C₈₃	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
63	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
64	#107×C₈₃	—	2	4	no	—
65	#107×C₈₃	—	2	4	no	—
66	C₄×C₈×C₈₃	D₄	1	2	yes	—
67	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
69	#13050590	—	4	8	no	—
70	C₂×Dic₁₆₆	—	2	2	no	—
71	C₂×D₃₃₂	—	2	2	no	—
72	C₂×D₃₃₂	—	2	2	no	—
73	C₂×D₃₃₂	—	2	2	no	—
74	C₂×C₂²⋊D₈₃	—	4	2	no	—
75	C₂×C₄×D₈₃	—	4	2	no	—
76	#13050580	—	4	16	no	—
77	#13050577	—	4	16	no	—
78	D₆₆₄	—	4	4	no	—
79	#13050574	D₈	1	8	yes	complemented
80	C₂×C₈₃⋊C₈	C₂×D₄	1	4	yes	—
81	C₂×C₈₃⋊C₈	C₂×D₄	1	4	yes	—
82	C₄⋊Dic₈₃	—	2	2	no	—
83	C₄⋊Dic₈₃	—	2	2	no	—
84	(C₂×D₈₃)⋊C₄	—	4	2	no	—
85	(C₂×D₈₃)⋊C₄	—	4	2	no	—
86	C₂×Q₈×C₈₃	—	2	4	no	—
87	C₂×Q₈×C₈₃	—	4	2	no	—
88	C₂×Q₈×C₈₃	—	4	8	no	—
89	C₂²×C₄×C₈₃	—	4	2	no	—
90	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₈₃	—	2	4	no	—
94	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
96	C₄⋊C₄×C₈₃	—	2	4	no	—
97	C₄⋊C₄×C₈₃	—	2	4	no	—
98	C₄⋊C₄×C₈₃	—	4	2	no	—
99	C₄⋊C₄×C₈₃	—	4	2	no	—
100	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
104	C₂³×C₈₃	#127	1	1	yes	characteristic
105	C₂²×D₈₃	—	4	2	no	—
106	Q₈×C₈₃	—	4	8	no	—
107	Q₈×C₈₃	—	8	4	no	—
108	C₂×C₄×C₈₃	—	2	2	no	—
109	C₂×C₄×C₈₃	—	2	2	no	—
110	C₂×C₄×C₈₃	—	4	4	no	—
111	C₂×C₄×C₈₃	—	8	4	no	—
112	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
113	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
114	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₂×D₈	1	4	yes	—
116	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
119	C₂²⋊D₈₃	—	8	4	no	—
120	C₂×Dic₈₃	—	4	2	no	—
121	D₃₃₂	—	4	4	no	—
122	D₃₃₂	—	4	4	no	—
123	D₃₃₂	—	4	4	no	—
124	C₄×D₈₃	—	8	4	no	—
125	Dic₁₆₆	—	4	4	no	—
126	C₈×C₈₃	—	2	2	no	—
127	C₈×C₈₃	#137	1	4	yes	complemented
128	C₈₃⋊C₈	—	2	4	no	—
129	C₈₃⋊C₈	C₂×D₈	1	8	yes	—
130	C₂²×C₈₃	—	2	1	no	—
131	C₂²×C₈₃	—	2	1	no	—
132	C₂²×C₈₃	#493	1	2	yes	—
133	C₂²×C₈₃	C₂×#127	1	1	yes	characteristic
134	C₂×D₈₃	—	8	2	no	—
135	C₂×D₈₃	—	8	2	no	—
136	C₂×D₈₃	—	8	2	no	—
137	C₄×C₈₃	—	2	1	no	—
138	C₄×C₈₃	—	2	1	no	—
139	C₄×C₈₃	—	8	4	no	—
140	C₄×C₈₃	#493	1	2	yes	—
141	C₄×C₈₃	C₂²×D₈	1	2	yes	—
142	C₄×C₈₃	C₂×#137	1	2	yes	—
143	C₄×C₈₃	#578	1	2	yes	—
144	Dic₈₃	—	8	2	no	—
145	#19630	—	83	1	no	maximal, Sylow
146	C₂×C₈₃	—	2	2	no	—
147	C₂×C₈₃	C₂×#493	1	1	yes	characteristic
148	C₂×C₈₃	#3152	1	1	yes	characteristic
149	C₂×C₈₃	#3156	1	1	yes	characteristic
150	D₈₃	—	16	2	no	—
151	#3226	—	83	2	no	—
152	#3169	—	83	1	no	—
153	C₂×#500	D₈₃	1	1	yes	characteristic, complemented
154	#3152	—	83	2	no	—
155	#1722	—	83	4	no	—
156	#1713	—	83	2	no	—
157	#1706	—	83	2	no	—
158	C₈×M₁₆	—	83	1	no	—
159	C₈₃	#19630	1	1	yes	characteristic, complemented, Sylow
160	C₂×#137	—	166	4	no	—
161	#532	—	83	2	no	—
162	C₂×C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
163	#500	—	2	2	no	—
164	#500	—	83	4	no	—
165	#500	—	83	1	no	—
166	#500	C₂×D₈₃	1	2	yes	—
167	#500	C₂×D₈₃	1	2	yes	—
168	#497	—	166	4	no	—
169	#494	—	83	4	no	—
170	#493	—	83	1	no	—
171	#492	—	83	4	no	—
172	#486	—	83	4	no	—
173	#486	—	83	2	no	—
174	#480	—	166	4	no	—
175	#474	—	166	4	no	—
176	#466	—	166	2	no	—
177	C₄×D₈	—	166	2	no	—
178	#431	—	83	1	no	—
179	C₂×#107	—	2	2	no	—
180	C₄×M₁₆	—	83	1	no	—
181	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
182	#322	—	83	2	no	—
183	C₈²	—	83	2	no	—
184	C₂×#41	—	166	2	no	—
185	C₂²×Q₈	—	2	2	no	—
186	#137	—	332	16	no	—
187	C₂×Q₁₆	—	166	8	no	—
188	C₂×SD₁₆	—	166	8	no	—
189	C₂×D₈	—	166	2	no	—
190	C₂×M₁₆	—	83	2	no	—
191	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	83	2	no	—
196	C₄⋊Q₈	C₂²×D₈₃	1	4	yes	—
197	#127	—	83	2	no	—
198	#124	—	166	2	no	—
199	#121	—	166	2	no	—
200	#121	—	166	2	no	—
201	C₄×D₄	—	166	2	no	—
202	C₄×D₄	—	166	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
206	#107	—	2	4	no	—
207	#107	—	2	4	no	—
208	#107	—	166	8	no	—
209	#107	—	166	2	no	—
210	C₄⋊C₈	—	83	2	no	—
211	#102	—	166	8	no	—
212	#102	—	166	4	no	—
213	C₂²⋊C₈	—	83	2	no	—
214	#97	—	83	2	no	—
215	C₄×C₈	—	83	2	no	—
216	C₄×C₈	—	83	2	no	—
217	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
218	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
220	#41	—	332	8	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	4	8	no	—
224	C₂×Q₈	—	166	2	no	—
225	C₂×D₄	—	332	2	no	—
226	C₂×D₄	—	166	2	no	—
227	C₂×D₄	—	166	2	no	—
228	C₂×D₄	—	166	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	332	2	no	—
231	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
234	Q₁₆	—	332	16	no	—
235	SD₁₆	—	332	16	no	—
236	D₈	—	332	4	no	—
237	M₁₆	—	83	8	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	83	4	no	—
240	C₂×C₈	—	83	4	no	—
241	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
242	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	166	2	no	—
248	C₄⋊C₄	—	166	2	no	—
249	C₂²⋊C₄	—	332	2	no	—
250	C₂²⋊C₄	—	332	2	no	—
251	C₄²	D₄×D₈₃	1	1	yes	characteristic
252	C₄²	D₄×D₈₃	1	1	yes	characteristic
253	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
254	C₄²	C₂³×D₈₃	1	1	yes	characteristic
255	C₂³	—	332	2	no	—
256	C₂³	#14581788	1	1	yes	characteristic
257	Q₈	—	4	8	no	—
258	Q₈	—	8	4	no	—
259	Q₈	—	332	4	no	—
260	D₄	—	332	4	no	—
261	D₄	—	332	4	no	—
262	D₄	—	332	4	no	—
263	D₄	—	664	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	—	332	2	no	—
269	C₂×C₄	—	664	4	no	—
270	C₂×C₄	#14581788	1	1	yes	characteristic
271	C₂×C₄	#14581788	1	1	yes	characteristic
272	C₂×C₄	#14581788	1	1	yes	characteristic
273	C₂×C₄	D₈×D₈₃	1	4	yes	—
274	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	83	8	no	—
279	C₈	—	166	4	no	—
280	C₈	#14581823	1	4	yes	complemented
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	664	2	no	—
284	C₂²	—	664	2	no	—
285	C₂²	—	664	2	no	—
286	C₂²	#19128076	1	2	yes	—
287	C₂²	C₂×#14581788	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	8	4	no	—
291	C₄	—	664	2	no	—
292	C₄	#19128076	1	2	yes	—
293	C₄	C₂×D₈×D₈₃	1	2	yes	—
294	C₄	C₂×#14581823	1	2	yes	—
295	C₄	#19128514	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	1328	2	no	—
298	C₂	C₂×#19128076	1	1	yes	characteristic
299	C₂	#24917920	1	1	yes	characteristic
300	C₂	#24917952	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14581788
commutator, cosocle	C₄²×C₈₃	C₂⁴
Frattini	C₄²	C₂³×D₈₃
Fitting	C₂×#500×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#127
prime radical	#24918042	C₂
hypercenter	C₂×#500	D₈₃
hypocenter	C₈₃	#19630
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
1. #24918584	C₂	semidirect
3. #24917920	C₂	semidirect
4. #24907133	C₂	semidirect
5. #24907080	C₂	semidirect
6. #24907056	C₂	semidirect
7. C₈×#13050574	C₂	semidirect
8. C₂×#500×C₈₃	C₂	semidirect
79. #13050574	D₈	semidirect
127. C₈×C₈₃	#137	semidirect
153. C₂×#500	D₈₃	semidirect
159. C₈₃	#19630	semidirect
280. C₈	#14581823	semidirect

Maximal subgroups

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

subgroup	index
#24918584	2
#24918042	2
#24917920	2
#24907133	2
#24907080	2
#24907056	2
C₈×#13050574	2
C₂×#500×C₈₃	2
#19630	83