Subgroups — #35562112

Order 21248.

484 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	#24920476	C₂	1	1	yes	maximal, characteristic
2	#24920355	C₂	1	1	yes	maximal, characteristic, complemented
3	#24909695	C₂	1	2	yes	maximal, complemented
4	#24909687	C₂	1	1	yes	maximal, characteristic
5	#24909621	C₂	1	2	yes	maximal, complemented
6	#24909606	C₂	1	1	yes	maximal, characteristic, complemented
7	#24909526	C₂	1	2	yes	maximal, complemented
8	#24909504	C₂	1	1	yes	maximal, characteristic
9	C₄×Dic₆₆₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#14581737	C₂	1	2	yes	maximal, complemented
11	Q₈×C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128436	—	2	1	no	—
13	#19128429	—	2	1	no	—
14	C₂×Dic₆₆₄	C₂²	1	1	yes	characteristic
15	C₂×#14581737	C₂²	1	2	yes	—
16	#19128106	C₂²	1	1	yes	characteristic
17	#19128103	C₂²	1	2	yes	—
18	#19128085	C₂²	1	2	yes	—
19	#19128078	C₂²	1	1	yes	characteristic
20	#19128009	—	2	2	no	—
21	#19127987	—	2	1	no	—
22	#19127917	—	2	2	no	—
23	#19127902	—	2	1	no	—
24	C₄×Dic₃₃₂	C₂²	1	2	yes	—
25	C₄×Dic₃₃₂	C₂²	1	1	yes	characteristic
26	C₄×Dic₃₃₂	C₂²	1	1	yes	characteristic
27	C₄×D₆₆₄	C₂²	1	2	yes	—
28	#19127529	C₂²	1	2	yes	—
29	#19127527	C₂²	1	1	yes	characteristic
30	#19127506	C₂²	1	1	yes	characteristic
31	#19127506	C₂²	1	1	yes	characteristic
32	#19127506	C₂²	1	2	yes	—
33	#19127503	C₂²	1	2	yes	—
34	#19127503	C₂²	1	2	yes	—
35	#19127503	C₂²	1	2	yes	—
36	C₈×Q₈×C₈₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₈₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₈₃	C₂²	1	2	yes	—
39	C₄×C₁₆×C₈₃	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₃₂	C₂³	1	1	yes	characteristic
42	C₂×Dic₃₃₂	C₂³	1	1	yes	characteristic
43	C₂×#13050579	—	2	2	no	—
44	C₂×D₆₆₄	—	2	1	no	—
45	C₂×D₆₆₄	C₂³	1	2	yes	—
46	#14581792	—	2	1	no	—
47	#14581790	—	4	1	no	—
48	#14581789	—	2	1	no	—
49	#14581784	—	2	2	no	—
50	#14581782	—	4	2	no	—
51	#14581781	—	4	1	no	—
52	#14581780	—	2	2	no	—
53	#14581777	—	4	2	no	—
54	C₄×Dic₁₆₆	—	2	1	no	—
55	C₄×Dic₁₆₆	—	2	1	no	—
56	C₄×Dic₁₆₆	—	2	2	no	—
57	#14581757	—	4	1	no	—
58	C₄×D₃₃₂	—	2	2	no	—
59	#14581754	—	4	1	no	—
60	Dic₆₆₄	—	2	2	no	—
61	#14581737	—	2	4	no	—
62	#14581726	—	2	1	no	—
63	#14581726	C₂³	1	1	yes	characteristic
64	#14581726	C₂³	1	2	yes	—
65	#14581726	C₂³	1	1	yes	characteristic
66	#14581724	—	2	2	no	—
67	#14581717	—	2	1	no	—
68	#14581717	—	2	2	no	—
69	#14581717	—	2	1	no	—
70	#14581714	—	2	2	no	—
71	C₄×Q₈×C₈₃	D₄	1	1	yes	characteristic
72	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
73	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₈₃	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₈₃	D₄	1	2	yes	—
77	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₃	C₂³	1	2	yes	—
79	C₂×Dic₁₆₆	—	2	1	no	—
80	C₂×Dic₁₆₆	—	2	1	no	—
81	C₂×Dic₁₆₆	—	4	1	no	—
82	C₂×D₃₃₂	—	2	2	no	—
83	C₂×D₃₃₂	—	4	1	no	—
84	C₂×D₃₃₂	—	4	1	no	—
85	C₂×C₄×D₈₃	—	4	2	no	—
86	Dic₃₃₂	—	2	1	no	—
87	Dic₃₃₂	—	2	1	no	—
88	Dic₃₃₂	—	2	1	no	—
89	Dic₃₃₂	#41	1	2	yes	—
90	#13050579	—	8	2	no	—
91	D₆₆₄	—	2	2	no	—
92	D₆₆₄	—	2	2	no	—
93	D₆₆₄	—	8	1	no	—
94	C₄⋊Dic₈₃	—	2	1	no	—
95	C₄⋊Dic₈₃	—	2	2	no	—
96	C₄⋊Dic₈₃	—	2	1	no	—
97	C₄⋊Dic₈₃	—	4	1	no	—
98	Dic₈₃⋊C₄	—	4	1	no	—
99	Dic₈₃⋊C₄	—	4	1	no	—
100	Dic₈₃⋊C₄	—	4	2	no	—
101	Dic₈₃⋊C₄	—	4	2	no	—
102	Dic₈₃⋊C₄	—	4	2	no	—
103	(C₂×D₈₃)⋊C₄	—	4	2	no	—
104	(C₂×D₈₃)⋊C₄	—	4	2	no	—
105	(C₂×D₈₃)⋊C₄	—	4	2	no	—
106	C₄×Dic₈₃	—	4	1	no	—
107	C₄×Dic₈₃	—	4	2	no	—
108	C₄×Dic₈₃	—	4	1	no	—
109	C₂×Q₈×C₈₃	D₈	1	1	yes	characteristic
110	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₃	D₈	1	2	yes	—
114	C₄⋊C₄×C₈₃	D₈	1	1	yes	characteristic
115	C₄²×C₈₃	C₂×D₄	1	2	yes	—
116	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₈₃	—	2	2	no	—
118	C₁₆×C₈₃	—	2	1	no	—
119	C₁₆×C₈₃	—	2	1	no	—
120	C₂²×D₈₃	—	4	2	no	—
121	Q₈×C₈₃	—	2	2	no	—
122	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₈₃	—	4	1	no	—
128	C₂×Dic₈₃	—	4	1	no	—
129	C₂×Dic₈₃	—	4	1	no	—
130	C₂×Dic₈₃	—	4	1	no	—
131	C₂×Dic₈₃	—	4	2	no	—
132	D₃₃₂	—	4	2	no	—
133	D₃₃₂	—	4	2	no	—
134	D₃₃₂	—	8	2	no	—
135	D₃₃₂	—	16	1	no	—
136	C₄×D₈₃	—	16	2	no	—
137	Dic₁₆₆	—	2	2	no	—
138	Dic₁₆₆	—	4	1	no	—
139	Dic₁₆₆	—	4	1	no	—
140	Dic₁₆₆	—	4	1	no	—
141	Dic₁₆₆	—	8	2	no	—
142	C₈×C₈₃	—	2	1	no	—
143	C₈×C₈₃	—	2	2	no	—
144	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
145	C₈×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₈₃	—	8	2	no	—
148	C₂×D₈₃	—	8	2	no	—
149	C₂×D₈₃	—	8	2	no	—
150	C₄×C₈₃	—	2	2	no	—
151	C₄×C₈₃	—	2	2	no	—
152	C₄×C₈₃	—	2	1	no	—
153	C₄×C₈₃	#508	1	2	yes	—
154	C₄×C₈₃	#548	1	1	yes	characteristic
155	C₄×C₈₃	#577	1	1	yes	characteristic
156	Dic₈₃	—	4	2	no	—
157	Dic₈₃	—	8	1	no	—
158	Dic₈₃	—	8	2	no	—
159	Dic₈₃	—	8	1	no	—
160	Dic₈₃	—	8	1	no	—
161	#33932	—	83	1	no	maximal, Sylow
162	C₂×C₈₃	#3358	1	1	yes	characteristic
163	C₂×C₈₃	C₂×#508	1	1	yes	characteristic
164	C₂×C₈₃	#3416	1	1	yes	characteristic
165	D₈₃	—	16	2	no	—
166	#3396	—	83	1	no	—
167	#3383	—	83	1	no	—
168	#2242	—	83	2	no	—
169	#2240	—	83	1	no	—
170	#2228	—	83	2	no	—
171	#2225	—	83	1	no	—
172	#2211	—	83	2	no	—
173	#2207	—	83	1	no	—
174	Q₈×C₁₆	D₈₃	1	1	yes	characteristic, complemented
175	C₄×Q₃₂	—	83	1	no	—
176	C₄×SD₃₂	—	83	2	no	—
177	C₈₃	#33932	1	1	yes	characteristic, complemented, Sylow
178	#559	—	166	1	no	—
179	#557	—	166	1	no	—
180	C₂×Q₃₂	—	83	1	no	—
181	C₂×SD₃₂	—	83	2	no	—
182	#500	—	83	1	no	—
183	#499	—	83	2	no	—
184	#495	—	83	2	no	—
185	#493	—	83	1	no	—
186	#479	—	166	2	no	—
187	#475	—	166	1	no	—
188	#463	—	166	2	no	—
189	#460	—	166	1	no	—
190	C₈×Q₈	C₂×D₈₃	1	1	yes	characteristic
191	C₄×Q₁₆	—	83	2	no	—
192	C₄×Q₁₆	—	83	1	no	—
193	C₄×Q₁₆	—	83	1	no	—
194	C₄×D₈	—	83	2	no	—
195	#367	—	83	2	no	—
196	#366	—	83	1	no	—
197	C₄⋊C₁₆	C₂×D₈₃	1	1	yes	characteristic
198	C₄⋊C₁₆	C₂×D₈₃	1	2	yes	—
199	#358	—	83	1	no	—
200	#358	—	83	1	no	—
201	#358	—	83	2	no	—
202	#357	—	83	2	no	—
203	#357	—	83	2	no	—
204	#357	—	83	2	no	—
205	C₄×C₁₆	C₂×D₈₃	1	2	yes	—
206	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
207	C₂×Q₁₆	—	83	1	no	—
208	C₂×Q₁₆	—	83	1	no	—
209	C₂×SD₁₆	—	166	2	no	—
210	C₂×D₈	—	83	2	no	—
211	C₂×D₈	—	166	1	no	—
212	C₄⋊Q₈	—	332	1	no	—
213	C₄⋊Q₈	—	166	1	no	—
214	#127	—	166	1	no	—
215	#125	—	332	2	no	—
216	#125	—	166	2	no	—
217	#124	—	332	1	no	—
218	#124	—	166	2	no	—
219	#123	—	332	2	no	—
220	C₄×Q₈	—	332	1	no	—
221	C₄×Q₈	—	166	1	no	—
222	C₄×Q₈	—	166	2	no	—
223	C₄×Q₈	—	166	1	no	—
224	C₄×Q₈	D₃₃₂	1	1	yes	characteristic
225	C₄×D₄	—	332	1	no	—
226	C₄×D₄	—	166	2	no	—
227	Q₃₂	—	166	2	no	—
228	SD₃₂	—	166	4	no	—
229	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
230	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
231	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
232	#107	—	83	1	no	—
233	#107	—	83	2	no	—
234	#107	—	83	1	no	—
235	#107	—	166	1	no	—
236	#106	—	166	2	no	—
237	C₄⋊C₈	D₃₃₂	1	1	yes	characteristic
238	C₄⋊C₈	D₃₃₂	1	2	yes	—
239	#103	—	166	1	no	—
240	#103	—	166	2	no	—
241	#103	—	166	1	no	—
242	#102	—	166	2	no	—
243	C₄×C₈	C₂²×D₈₃	1	2	yes	—
244	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
245	C₂×Q₈	—	332	1	no	—
246	C₂×Q₈	—	166	1	no	—
247	C₂×Q₈	—	166	1	no	—
248	C₂×Q₈	D₆₆₄	1	1	yes	characteristic
249	C₂×D₄	—	332	1	no	—
250	C₂×D₄	—	332	1	no	—
251	C₂×D₄	—	166	2	no	—
252	C₂²×C₄	—	332	2	no	—
253	Q₁₆	—	83	2	no	—
254	Q₁₆	—	166	1	no	—
255	Q₁₆	—	166	1	no	—
256	Q₁₆	—	166	1	no	—
257	SD₁₆	—	664	2	no	—
258	D₈	—	664	1	no	—
259	D₈	—	166	2	no	—
260	D₈	—	166	2	no	—
261	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
262	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
264	C₄⋊C₄	—	332	2	no	—
265	C₄⋊C₄	—	332	1	no	—
266	C₄⋊C₄	—	332	1	no	—
267	C₄⋊C₄	—	332	2	no	—
268	C₄⋊C₄	—	332	1	no	—
269	C₄⋊C₄	—	332	2	no	—
270	C₄⋊C₄	—	166	1	no	—
271	C₄⋊C₄	—	166	1	no	—
272	C₄⋊C₄	—	166	2	no	—
273	C₄⋊C₄	D₆₆₄	1	1	yes	characteristic
274	C₄⋊C₄	D₆₆₄	1	2	yes	—
275	C₂²⋊C₄	—	332	2	no	—
276	C₂²⋊C₄	—	332	2	no	—
277	C₂²⋊C₄	—	332	2	no	—
278	C₄²	—	332	1	no	—
279	C₄²	—	332	2	no	—
280	C₄²	—	332	1	no	—
281	C₄²	C₂×D₃₃₂	1	2	yes	—
282	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
283	C₁₆	—	2	2	no	—
284	C₁₆	—	2	1	no	—
285	C₁₆	—	2	1	no	—
286	C₂³	—	332	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	332	1	no	—
289	Q₈	—	332	1	no	—
290	Q₈	—	332	1	no	—
291	Q₈	—	664	2	no	—
292	Q₈	—	166	2	no	—
293	D₄	—	1328	1	no	—
294	D₄	—	332	2	no	—
295	D₄	—	332	2	no	—
296	D₄	—	664	2	no	—
297	C₂×C₄	—	1328	2	no	—
298	C₂×C₄	—	332	1	no	—
299	C₂×C₄	—	332	1	no	—
300	C₂×C₄	—	332	1	no	—
301	C₂×C₄	—	332	1	no	—
302	C₂×C₄	—	332	2	no	—
303	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
304	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
305	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
306	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	2	no	—
310	C₈	#41×D₈₃	1	1	yes	characteristic
311	C₈	#14581840	1	1	yes	characteristic
312	C₂²	—	664	2	no	—
313	C₂²	—	664	2	no	—
314	C₂²	—	664	2	no	—
315	C₂²	C₂²×D₆₆₄	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	332	2	no	—
320	C₄	—	664	2	no	—
321	C₄	—	664	1	no	—
322	C₄	—	664	1	no	—
323	C₄	—	664	1	no	—
324	C₄	#19128133	1	2	yes	—
325	C₄	#19128363	1	1	yes	characteristic
326	C₄	#19128513	1	1	yes	characteristic
327	C₂	—	1328	2	no	—
328	C₂	#24920099	1	1	yes	characteristic
329	C₂	C₂×#19128133	1	1	yes	characteristic
330	C₂	#24920611	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₆₄
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	Q₈×C₁₆×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₈
prime radical	#19128078	C₂²
hypercenter	Q₈×C₁₆	D₈₃
hypocenter	C₈₃	#33932
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #24920355	C₂	semidirect
3. #24909695	C₂	semidirect
5. #24909621	C₂	semidirect
6. #24909606	C₂	semidirect
7. #24909526	C₂	semidirect
9. C₄×Dic₆₆₄	C₂	semidirect
10. C₄×#14581737	C₂	semidirect
11. Q₈×C₁₆×C₈₃	C₂	semidirect
174. Q₈×C₁₆	D₈₃	semidirect
177. C₈₃	#33932	semidirect

Maximal subgroups

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

subgroup	index
#24920476	2
#24920355	2
#24909695	2
#24909687	2
#24909621	2
#24909606	2
#24909526	2
#24909504	2
C₄×Dic₆₆₄	2
C₄×#14581737	2
Q₈×C₁₆×C₈₃	2
#33932	83