Subgroups — #35304275

Order 21248.

516 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	#24918022	C₂	1	1	yes	maximal, characteristic
2	#24908440	C₂	1	1	yes	maximal, characteristic, complemented
3	#24908291	C₂	1	1	yes	maximal, characteristic
4	#24906219	C₂	1	1	yes	maximal, characteristic, complemented
5	#24905919	C₂	1	1	yes	maximal, characteristic, complemented
6	#24905915	C₂	1	1	yes	maximal, characteristic
7	C₄²×C₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128258	C₂²	1	1	yes	characteristic
9	#19128251	C₂²	1	1	yes	characteristic
10	#19128099	C₄	1	1	yes	characteristic, complemented
11	#19128074	C₄	1	1	yes	characteristic, complemented
12	#19128052	C₄	1	1	yes	characteristic, complemented
13	#19128048	C₄	1	1	yes	characteristic, complemented
14	#19127930	—	2	2	no	—
15	C₄×#13050579	—	2	2	no	—
16	#19127725	C₂²	1	1	yes	characteristic
17	#19127725	C₂²	1	1	yes	characteristic
18	C₂×#14581720	—	2	2	no	—
19	#19127461	—	2	2	no	—
20	#19127452	—	2	2	no	—
21	#19127433	—	2	2	no	—
22	#19127430	—	2	2	no	—
23	C₂×C₄×C₈×C₈₃	—	2	2	no	—
24	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
26	C₄³×C₈₃	C₂²	1	1	yes	characteristic
27	C₂×#13050590	—	2	1	no	—
28	C₂×#13050579	—	2	2	no	—
29	C₂×#13050574	—	2	1	no	—
30	C₂×#13050574	—	2	1	no	—
31	#14581792	C₂×C₄	1	1	yes	characteristic
32	#14581792	C₂×C₄	1	1	yes	characteristic
33	#14581792	C₂×C₄	1	1	yes	characteristic
34	#14581789	C₂×C₄	1	1	yes	characteristic
35	#14581784	—	2	1	no	—
36	#14581780	—	2	1	no	—
37	#14581771	—	2	1	no	—
38	#14581771	—	2	1	no	—
39	#14581766	—	2	1	no	—
40	#14581766	—	2	1	no	—
41	C₄×Dic₁₆₆	—	2	1	no	—
42	C₄×Dic₁₆₆	—	2	1	no	—
43	C₄×D₃₃₂	—	2	1	no	—
44	C₄×D₃₃₂	—	2	1	no	—
45	#14581751	—	2	1	no	—
46	#14581724	—	2	2	no	—
47	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
48	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
49	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
50	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
51	#14581720	—	4	4	no	—
52	#14581717	—	2	2	no	—
53	#14581714	—	2	2	no	—
54	C₂²×C₈×C₈₃	—	2	1	no	—
55	C₂²×C₈×C₈₃	D₄	1	2	yes	—
56	C₂×C₄²×C₈₃	—	2	1	no	—
57	C₂×C₄²×C₈₃	—	2	1	no	—
58	C₂×C₄²×C₈₃	D₄	1	2	yes	—
59	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₈₃	—	2	8	no	—
61	C₄×C₈×C₈₃	—	2	1	no	—
62	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₈₃	D₄	1	2	yes	—
65	C₄×C₈×C₈₃	D₄	1	2	yes	—
66	#13050590	—	4	2	no	—
67	#13050590	—	4	2	no	—
68	C₂×Dic₁₆₆	—	2	1	no	—
69	C₂×Dic₁₆₆	—	2	1	no	—
70	C₂×Dic₁₆₆	—	2	1	no	—
71	C₂×D₃₃₂	—	2	1	no	—
72	C₂×D₃₃₂	—	2	1	no	—
73	C₂×D₃₃₂	—	2	1	no	—
74	C₂×C₂²⋊D₈₃	—	4	1	no	—
75	C₂×C₄×D₈₃	—	4	1	no	—
76	#13050579	—	4	4	no	—
77	#13050574	—	4	2	no	—
78	#13050574	—	4	2	no	—
79	C₂×C₈₃⋊C₈	—	4	1	no	—
80	C₂×C₈₃⋊C₈	—	4	1	no	—
81	C₄⋊Dic₈₃	—	2	1	no	—
82	C₄⋊Dic₈₃	—	2	1	no	—
83	C₄⋊Dic₈₃	—	2	1	no	—
84	C₄⋊Dic₈₃	—	2	1	no	—
85	C₄⋊Dic₈₃	—	2	1	no	—
86	C₄⋊Dic₈₃	—	2	1	no	—
87	Dic₈₃⋊C₄	—	4	1	no	—
88	Dic₈₃⋊C₄	—	4	1	no	—
89	C₂²⋊Dic₈₃	—	4	1	no	—
90	(C₂×D₈₃)⋊C₄	—	4	1	no	—
91	(C₂×D₈₃)⋊C₄	—	4	1	no	—
92	C₄×Dic₈₃	—	4	1	no	—
93	C₂²×C₄×C₈₃	—	2	2	no	—
94	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
97	C₂×C₈×C₈₃	—	2	4	no	—
98	C₂×C₈×C₈₃	—	2	2	no	—
99	C₂×C₈×C₈₃	—	2	4	no	—
100	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
101	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
102	C₄²×C₈₃	—	2	4	no	—
103	C₄²×C₈₃	—	2	4	no	—
104	C₄²×C₈₃	—	2	2	no	—
105	C₄²×C₈₃	SD₁₆	1	4	yes	complemented
106	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₈₃	#41	1	1	yes	characteristic
110	C₂³×C₈₃	#121	1	1	yes	characteristic
111	C₂²×D₈₃	—	4	1	no	—
112	C₂×C₄×C₈₃	—	2	4	no	—
113	C₂×C₄×C₈₃	—	2	4	no	—
114	C₂×C₄×C₈₃	—	2	4	no	—
115	C₂×C₄×C₈₃	—	2	2	no	—
116	C₂×C₄×C₈₃	—	2	2	no	—
117	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
122	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
124	C₂×C₄×C₈₃	#135	1	2	yes	—
125	C₂²⋊D₈₃	—	8	2	no	—
126	C₂×Dic₈₃	—	4	1	no	—
127	C₂×Dic₈₃	—	4	1	no	—
128	C₂×Dic₈₃	—	4	1	no	—
129	D₃₃₂	—	4	2	no	—
130	D₃₃₂	—	4	2	no	—
131	D₃₃₂	—	4	2	no	—
132	C₄×D₈₃	—	8	1	no	—
133	C₄×D₈₃	—	8	1	no	—
134	Dic₁₆₆	—	4	2	no	—
135	Dic₁₆₆	—	4	2	no	—
136	Dic₁₆₆	—	4	2	no	—
137	C₈×C₈₃	—	2	4	no	—
138	C₈×C₈₃	—	2	2	no	—
139	C₈×C₈₃	#104	1	4	yes	complemented
140	C₈₃⋊C₈	—	8	1	no	—
141	C₈₃⋊C₈	—	8	1	no	—
142	C₂²×C₈₃	—	2	1	no	—
143	C₂²×C₈₃	—	2	1	no	—
144	C₂²×C₈₃	#390	1	1	yes	characteristic
145	C₂²×C₈₃	#465	1	2	yes	—
146	C₂×D₈₃	—	8	1	no	—
147	C₂×D₈₃	—	8	1	no	—
148	C₂×D₈₃	—	8	1	no	—
149	C₄×C₈₃	—	2	8	no	—
150	C₄×C₈₃	—	2	1	no	—
151	C₄×C₈₃	—	2	1	no	—
152	C₄×C₈₃	C₂×#104	1	2	yes	—
153	C₄×C₈₃	C₄×SD₁₆	1	2	yes	—
154	C₄×C₈₃	#443	1	2	yes	—
155	C₄×C₈₃	#492	1	1	yes	characteristic
156	C₄×C₈₃	#495	1	1	yes	characteristic
157	Dic₈₃	—	8	1	no	—
158	Dic₈₃	—	8	1	no	—
159	Dic₈₃	—	8	1	no	—
160	#9843	—	83	1	no	maximal, Sylow
161	C₂×C₈₃	—	2	2	no	—
162	C₂×C₈₃	#1962	1	1	yes	characteristic
163	C₂×C₈₃	#1968	1	1	yes	characteristic
164	C₂×C₈₃	#1974	1	1	yes	characteristic
165	D₈₃	—	16	1	no	—
166	#3167	—	83	1	no	—
167	#1994	—	83	1	no	—
168	#1962	—	83	1	no	—
169	C₄²×C₈	D₈₃	1	1	yes	characteristic, complemented
170	#1566	—	83	1	no	—
171	#1493	—	83	1	no	—
172	#1492	—	83	1	no	—
173	C₈₃	#9843	1	1	yes	characteristic, complemented, Sylow
174	#533	—	83	1	no	—
175	#532	—	83	1	no	—
176	#498	—	83	1	no	—
177	#492	—	83	1	no	—
178	#487	—	83	1	no	—
179	#486	—	83	1	no	—
180	#465	—	166	2	no	—
181	C₄×SD₁₆	—	166	2	no	—
182	#423	—	83	1	no	—
183	#423	—	83	1	no	—
184	C₂×#104	—	166	2	no	—
185	C₂×C₄×C₈	—	2	2	no	—
186	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
187	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
188	C₄³	C₂×D₈₃	1	1	yes	characteristic
189	#339	—	166	2	no	—
190	#334	—	166	2	no	—
191	Q₈⋊C₈	—	166	2	no	—
192	D₄⋊C₈	—	166	2	no	—
193	C₂×#41	—	166	1	no	—
194	C₂×SD₁₆	—	166	2	no	—
195	C₂×M₁₆	—	166	1	no	—
196	C₂×M₁₆	—	166	1	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
199	C₄⋊Q₈	—	83	1	no	—
200	C₄⋊Q₈	—	83	1	no	—
201	C₄⋊Q₈	—	83	1	no	—
202	#127	—	83	1	no	—
203	#125	—	166	1	no	—
204	#124	—	166	1	no	—
205	C₂²⋊Q₈	—	166	1	no	—
206	C₂²⋊Q₈	—	166	1	no	—
207	#121	—	166	1	no	—
208	#121	—	166	1	no	—
209	C₄×Q₈	—	166	1	no	—
210	C₄×Q₈	—	166	1	no	—
211	C₄×D₄	—	166	1	no	—
212	C₄×D₄	—	166	1	no	—
213	#117	—	166	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	D₃₃₂	1	2	yes	—
217	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
218	#106	—	166	2	no	—
219	C₄⋊C₈	—	166	1	no	—
220	C₄⋊C₈	—	166	1	no	—
221	C₄⋊C₈	—	166	1	no	—
222	C₄⋊C₈	—	166	1	no	—
223	#104	—	332	4	no	—
224	#103	—	166	2	no	—
225	#102	—	166	2	no	—
226	C₄×C₈	—	2	8	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
229	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
230	C₄×C₈	D₃₃₂	1	2	yes	—
231	C₄×C₈	D₃₃₂	1	2	yes	—
232	#41	—	332	2	no	—
233	#41	—	332	2	no	—
234	C₂×Q₈	—	166	1	no	—
235	C₂×Q₈	—	166	1	no	—
236	C₂×Q₈	—	166	1	no	—
237	C₂×D₄	—	332	1	no	—
238	C₂×D₄	—	166	1	no	—
239	C₂×D₄	—	166	1	no	—
240	C₂×D₄	—	166	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	332	1	no	—
243	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
245	C₂²×C₄	#13050590	1	1	yes	characteristic
246	SD₁₆	—	332	4	no	—
247	M₁₆	—	332	2	no	—
248	M₁₆	—	332	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	332	1	no	—
253	C₂×C₈	—	332	1	no	—
254	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
255	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
256	C₄⋊C₄	—	332	1	no	—
257	C₄⋊C₄	—	332	1	no	—
258	C₄⋊C₄	—	166	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	C₂²⋊C₄	—	332	1	no	—
265	C₂²⋊C₄	—	332	1	no	—
266	C₂²⋊C₄	—	332	1	no	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	4	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	332	1	no	—
271	C₄²	#13050579	1	4	yes	complemented
272	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
273	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
274	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
275	C₄²	#13050590	1	1	yes	characteristic
276	C₂³	—	332	1	no	—
277	C₂³	#14581766	1	1	yes	characteristic
278	Q₈	—	332	2	no	—
279	Q₈	—	332	2	no	—
280	Q₈	—	332	2	no	—
281	D₄	—	332	2	no	—
282	D₄	—	332	2	no	—
283	D₄	—	332	2	no	—
284	D₄	—	664	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	332	1	no	—
291	C₂×C₄	—	332	1	no	—
292	C₂×C₄	—	332	1	no	—
293	C₂×C₄	—	664	1	no	—
294	C₂×C₄	—	664	1	no	—
295	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
296	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
297	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
298	C₂×C₄	#14581766	1	1	yes	characteristic
299	C₂×C₄	#14581780	1	1	yes	characteristic
300	C₂×C₄	#14581789	1	1	yes	characteristic
301	C₂×C₄	C₂×#13050579	1	2	yes	—
302	C₂×C₄	#14581814	1	2	yes	—
303	C₈	—	2	4	no	—
304	C₈	—	2	2	no	—
305	C₈	—	664	1	no	—
306	C₈	—	664	1	no	—
307	C₈	#14581720	1	4	yes	complemented
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	664	1	no	—
311	C₂²	—	664	1	no	—
312	C₂²	—	664	1	no	—
313	C₂²	#19127603	1	1	yes	characteristic
314	C₂²	#19127930	1	2	yes	—
315	C₄	—	2	8	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	664	1	no	—
319	C₄	—	664	1	no	—
320	C₄	—	664	1	no	—
321	C₄	C₂×#14581720	1	2	yes	—
322	C₄	C₄×#13050579	1	2	yes	—
323	C₄	#19127819	1	2	yes	—
324	C₄	#19128074	1	1	yes	characteristic
325	C₄	#19128085	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	1328	1	no	—
328	C₂	#24908291	1	1	yes	characteristic
329	C₂	#24908317	1	1	yes	characteristic
330	C₂	#24908351	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581789
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₄²×C₈×C₈₃	C₂
socle	C₂²×C₈₃	#390
prime radical	#19128251	C₂²
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₄²×C₈	D₈₃
hypocenter	C₈₃	#9843
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
2. #24908440	C₂	semidirect
4. #24906219	C₂	semidirect
5. #24905919	C₂	semidirect
7. C₄²×C₈×C₈₃	C₂	semidirect
10. #19128099	C₄	semidirect
11. #19128074	C₄	semidirect
12. #19128052	C₄	semidirect
13. #19128048	C₄	semidirect
105. C₄²×C₈₃	SD₁₆	semidirect
139. C₈×C₈₃	#104	semidirect
169. C₄²×C₈	D₈₃	semidirect
173. C₈₃	#9843	semidirect
271. C₄²	#13050579	semidirect
307. C₈	#14581720	semidirect

Maximal subgroups

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

subgroup	index
#24918022	2
#24908440	2
#24908291	2
#24906219	2
#24905919	2
#24905915	2
C₄²×C₈×C₈₃	2
#9843	83