Subgroups — #35562099

Order 21248.

472 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24920452	C₂	1	1	yes	maximal, characteristic
2	#24909697	C₂	1	2	yes	maximal
3	#24909692	C₂	1	1	yes	maximal, characteristic, complemented
4	#24909624	C₂	1	2	yes	maximal, complemented
5	#24909597	C₂	1	1	yes	maximal, characteristic
6	#24909525	C₂	1	2	yes	maximal
7	#24909513	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14581738	C₂	1	2	yes	maximal, complemented
10	C₄×#14581735	C₂	1	1	yes	maximal, characteristic
11	C₈₃×#3383	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128424	—	2	1	no	—
13	C₂×#14581738	C₂²	1	2	yes	—
14	C₂×#14581735	C₂²	1	1	yes	characteristic
15	#19128085	C₂²	1	2	yes	—
16	#19128081	C₂²	1	1	yes	characteristic
17	#19127924	—	2	2	no	—
18	#19127907	—	2	1	no	—
19	C₄×Dic₃₃₂	C₂²	1	2	yes	—
20	C₄×D₆₆₄	C₂²	1	1	yes	characteristic
21	#19127528	C₂²	1	1	yes	characteristic
22	#19127526	C₂²	1	2	yes	—
23	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	1	yes	characteristic
24	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	2	yes	—
25	#19127505	C₂²	1	1	yes	characteristic
26	#19127504	C₂²	1	2	yes	—
27	#19127504	C₂²	1	2	yes	—
28	#19127504	C₂²	1	2	yes	—
29	#19127501	C₂²	1	1	yes	characteristic
30	#19127501	C₂²	1	2	yes	—
31	C₄×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₄×C₈₃⋊C₁₆	C₂²	1	2	yes	—
33	#557×C₈₃	—	2	1	no	—
34	#500×C₈₃	C₂²	1	1	yes	characteristic
35	#499×C₈₃	C₂²	1	2	yes	—
36	#479×C₈₃	—	2	2	no	—
37	#475×C₈₃	—	2	1	no	—
38	C₈×Q₈×C₈₃	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₈₃	C₂²	1	2	yes	—
41	C₂×Dic₃₃₂	—	2	1	no	—
42	C₂×Dic₃₃₂	C₂³	1	2	yes	—
43	C₂×#13050579	—	2	2	no	—
44	C₂×D₆₆₄	C₂³	1	1	yes	characteristic
45	#14581792	—	2	1	no	—
46	#14581787	—	4	2	no	—
47	#14581780	—	2	2	no	—
48	#14581768	—	4	1	no	—
49	C₄×Dic₁₆₆	—	2	2	no	—
50	C₄×D₃₃₂	—	2	1	no	—
51	#14581750	—	4	1	no	—
52	#14581738	—	2	4	no	—
53	#14581735	—	2	2	no	—
54	C₂×C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₈₃⋊C₁₆	C₂³	1	2	yes	—
56	C₂×C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
57	#14581726	C₂³	1	1	yes	characteristic
58	#14581717	—	2	2	no	—
59	#14581714	—	2	1	no	—
60	C₂×Q₁₆×C₈₃	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₈₃	—	2	1	no	—
62	C₄⋊Q₈×C₈₃	—	4	1	no	—
63	#125×C₈₃	—	2	2	no	—
64	#125×C₈₃	—	4	2	no	—
65	C₄×Q₈×C₈₃	—	2	1	no	—
66	C₄×Q₈×C₈₃	—	2	2	no	—
67	C₄×Q₈×C₈₃	—	4	1	no	—
68	C₄×Q₈×C₈₃	D₄	1	1	yes	characteristic
69	#107×C₈₃	—	2	1	no	—
70	#107×C₈₃	C₂³	1	1	yes	characteristic
71	#107×C₈₃	C₂³	1	2	yes	—
72	#106×C₈₃	—	2	2	no	—
73	C₄⋊C₈×C₈₃	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₈₃	D₄	1	2	yes	—
75	#103×C₈₃	—	2	2	no	—
76	#103×C₈₃	—	2	1	no	—
77	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₃	C₂³	1	2	yes	—
79	C₂×Dic₁₆₆	—	2	2	no	—
80	C₂×D₃₃₂	—	2	1	no	—
81	C₂×C₄×D₈₃	—	4	1	no	—
82	Dic₃₃₂	—	2	2	no	—
83	Dic₃₃₂	—	2	2	no	—
84	Dic₃₃₂	—	8	1	no	—
85	#13050579	—	8	2	no	—
86	D₆₆₄	—	2	1	no	—
87	D₆₆₄	—	2	1	no	—
88	C₄⋊Dic₈₃	—	2	1	no	—
89	C₄⋊Dic₈₃	—	4	1	no	—
90	C₄⋊Dic₈₃	—	4	1	no	—
91	Dic₈₃⋊C₄	—	4	2	no	—
92	(C₂×D₈₃)⋊C₄	—	4	2	no	—
93	(C₂×D₈₃)⋊C₄	—	4	2	no	—
94	(C₂×D₈₃)⋊C₄	—	4	1	no	—
95	C₄×Dic₈₃	—	4	2	no	—
96	C₈₃⋊C₁₆	—	2	1	no	—
97	C₈₃⋊C₁₆	—	2	2	no	—
98	C₈₃⋊C₁₆	—	2	1	no	—
99	C₂×Q₈×C₈₃	—	2	1	no	—
100	C₂×Q₈×C₈₃	—	4	1	no	—
101	C₂×Q₈×C₈₃	D₈	1	1	yes	characteristic
102	Q₁₆×C₈₃	—	2	1	no	—
103	Q₁₆×C₈₃	#41	1	2	yes	—
104	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₈₃	—	2	1	no	—
108	C₄⋊C₄×C₈₃	—	2	2	no	—
109	C₄⋊C₄×C₈₃	—	4	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	C₄⋊C₄×C₈₃	D₈	1	2	yes	—
115	C₄⋊C₄×C₈₃	D₈	1	1	yes	characteristic
116	C₄²×C₈₃	—	4	2	no	—
117	C₄²×C₈₃	—	4	1	no	—
118	C₄²×C₈₃	C₂×D₄	1	2	yes	—
119	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×D₈₃	—	4	1	no	—
121	Q₈×C₈₃	—	2	2	no	—
122	Q₈×C₈₃	—	2	2	no	—
123	Q₈×C₈₃	—	4	1	no	—
124	Q₈×C₈₃	—	8	2	no	—
125	C₂×C₄×C₈₃	—	4	1	no	—
126	C₂×C₄×C₈₃	—	4	1	no	—
127	C₂×C₄×C₈₃	—	4	2	no	—
128	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₈₃	—	4	1	no	—
134	C₂×Dic₈₃	—	4	2	no	—
135	D₃₃₂	—	4	1	no	—
136	D₃₃₂	—	4	1	no	—
137	C₄×D₈₃	—	8	2	no	—
138	Dic₁₆₆	—	4	2	no	—
139	Dic₁₆₆	—	4	2	no	—
140	C₈×C₈₃	—	2	2	no	—
141	C₈×C₈₃	—	2	1	no	—
142	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
143	C₈×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
144	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₈₃	—	8	1	no	—
146	C₂×D₈₃	—	8	1	no	—
147	C₂×D₈₃	—	8	1	no	—
148	C₄×C₈₃	—	2	2	no	—
149	C₄×C₈₃	—	2	2	no	—
150	C₄×C₈₃	—	2	1	no	—
151	C₄×C₈₃	—	4	2	no	—
152	C₄×C₈₃	—	8	2	no	—
153	C₄×C₈₃	—	8	1	no	—
154	C₄×C₈₃	#508	1	2	yes	—
155	C₄×C₈₃	#548	1	1	yes	characteristic
156	C₄×C₈₃	#577	1	1	yes	characteristic
157	Dic₈₃	—	8	2	no	—
158	Dic₈₃	—	8	1	no	—
159	#33931	—	83	1	no	maximal, Sylow
160	C₂×C₈₃	#3358	1	1	yes	characteristic
161	C₂×C₈₃	C₂×#508	1	1	yes	characteristic
162	C₂×C₈₃	#3417	1	1	yes	characteristic
163	D₈₃	—	16	1	no	—
164	#3393	—	83	1	no	—
165	#3383	D₈₃	1	1	yes	characteristic, complemented
166	#2242	—	83	2	no	—
167	#2241	—	83	1	no	—
168	#2229	—	83	2	no	—
169	#2224	—	83	1	no	—
170	#2211	—	83	2	no	—
171	#2209	—	83	1	no	—
172	Q₈×C₁₆	—	83	1	no	—
173	C₄×Q₃₂	—	83	2	no	—
174	C₄×SD₃₂	—	83	1	no	—
175	C₈₃	#33931	1	1	yes	characteristic, complemented, Sylow
176	#557	—	2	1	no	—
177	#556	—	166	1	no	—
178	C₂×Q₃₂	—	83	2	no	—
179	C₂×SD₃₂	—	83	1	no	—
180	#500	C₂×D₈₃	1	1	yes	characteristic
181	#499	C₂×D₈₃	1	2	yes	—
182	#495	—	83	2	no	—
183	#494	—	83	1	no	—
184	#479	—	2	2	no	—
185	#475	—	2	1	no	—
186	#464	—	166	2	no	—
187	#461	—	166	1	no	—
188	C₈×Q₈	C₂×D₈₃	1	1	yes	characteristic
189	C₄×Q₁₆	—	83	2	no	—
190	C₄×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
191	C₄×Q₁₆	C₂×D₈₃	1	2	yes	—
192	C₄×D₈	—	83	1	no	—
193	#367	—	83	1	no	—
194	#366	—	83	2	no	—
195	C₄⋊C₁₆	—	83	1	no	—
196	C₄⋊C₁₆	—	83	2	no	—
197	#358	—	83	1	no	—
198	#358	—	83	2	no	—
199	#358	—	83	2	no	—
200	#358	—	83	2	no	—
201	#357	—	83	1	no	—
202	#357	—	83	2	no	—
203	C₄×C₁₆	—	83	1	no	—
204	C₄×C₁₆	—	83	2	no	—
205	C₂×Q₁₆	—	83	2	no	—
206	C₂×Q₁₆	—	166	1	no	—
207	C₂×Q₁₆	C₂²×D₈₃	1	1	yes	characteristic
208	C₂×SD₁₆	—	166	2	no	—
209	C₂×D₈	—	83	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	4	1	no	—
212	C₄⋊Q₈	—	166	1	no	—
213	#126	—	332	2	no	—
214	#125	—	2	2	no	—
215	#125	—	4	2	no	—
216	#124	—	166	2	no	—
217	C₂²⋊Q₈	—	332	1	no	—
218	C₄×Q₈	—	2	1	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	—	4	1	no	—
221	C₄×Q₈	—	166	2	no	—
222	C₄×Q₈	C₂²⋊D₈₃	1	1	yes	characteristic
223	C₄×D₄	—	166	1	no	—
224	#117	—	332	1	no	—
225	Q₃₂	—	166	4	no	—
226	SD₃₂	—	166	2	no	—
227	C₂×C₁₆	—	83	1	no	—
228	C₂×C₁₆	—	83	2	no	—
229	C₂×C₁₆	—	83	1	no	—
230	#107	—	2	1	no	—
231	#107	—	83	1	no	—
232	#107	C₂²×D₈₃	1	2	yes	—
233	#107	C₂²×D₈₃	1	1	yes	characteristic
234	#106	—	2	2	no	—
235	C₄⋊C₈	C₂²⋊D₈₃	1	1	yes	characteristic
236	C₄⋊C₈	C₂²⋊D₈₃	1	2	yes	—
237	#103	—	2	2	no	—
238	#103	—	2	1	no	—
239	#103	—	166	2	no	—
240	#102	—	166	1	no	—
241	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₈₃	1	2	yes	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	166	2	no	—
246	C₂×Q₈	#13050575	1	1	yes	characteristic
247	C₂×D₄	—	166	1	no	—
248	C₂²×C₄	—	332	1	no	—
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	664	1	no	—
251	Q₁₆	—	166	2	no	—
252	Q₁₆	—	166	2	no	—
253	Q₁₆	#13050591	1	2	yes	—
254	SD₁₆	—	664	2	no	—
255	D₈	—	166	1	no	—
256	D₈	—	166	1	no	—
257	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
258	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
259	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	332	1	no	—
268	C₄⋊C₄	—	332	2	no	—
269	C₄⋊C₄	—	332	1	no	—
270	C₄⋊C₄	—	166	1	no	—
271	C₄⋊C₄	#13050575	1	1	yes	characteristic
272	C₄⋊C₄	#13050575	1	2	yes	—
273	C₂²⋊C₄	—	332	2	no	—
274	C₂²⋊C₄	—	332	1	no	—
275	C₂²⋊C₄	—	332	2	no	—
276	C₄²	—	4	2	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	332	2	no	—
279	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
280	C₄²	C₂×C₂²⋊D₈₃	1	2	yes	—
281	C₁₆	—	166	1	no	—
282	C₁₆	—	166	2	no	—
283	C₁₆	—	166	1	no	—
284	C₂³	—	332	1	no	—
285	Q₈	—	2	2	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	4	1	no	—
288	Q₈	—	8	2	no	—
289	Q₈	—	332	2	no	—
290	Q₈	—	332	2	no	—
291	D₄	—	332	1	no	—
292	D₄	—	332	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	332	1	no	—
297	C₂×C₄	—	332	2	no	—
298	C₂×C₄	—	664	2	no	—
299	C₂×C₄	C₂×#13050575	1	2	yes	—
300	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
301	C₂×C₄	C₂×#13050575	1	2	yes	—
302	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	1	no	—
306	C₈	C₂×#13050591	1	1	yes	characteristic
307	C₈	#14581840	1	1	yes	characteristic
308	C₂²	—	664	1	no	—
309	C₂²	—	664	1	no	—
310	C₂²	—	664	1	no	—
311	C₂²	C₂²×#13050575	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	1	no	—
318	C₄	—	664	1	no	—
319	C₄	—	664	2	no	—
320	C₄	#19128131	1	2	yes	—
321	C₄	#19128362	1	1	yes	characteristic
322	C₄	#19128509	1	1	yes	characteristic
323	C₂	—	1328	1	no	—
324	C₂	#24920098	1	1	yes	characteristic
325	C₂	C₂×#19128131	1	1	yes	characteristic
326	C₂	#24920614	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13050575
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#3383	C₂
socle	C₂²×C₈₃	C₂²×D₈
prime radical	C₂×D₆₆₄	C₂³
hypercenter	#3383	D₈₃
hypocenter	C₈₃	#33931
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3383

Complements

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

N	H ≅ G/N	type
3. #24909692	C₂	semidirect
4. #24909624	C₂	semidirect
7. #24909513	C₂	semidirect
8. Q₈×C₈₃⋊C₁₆	C₂	semidirect
9. C₄×#14581738	C₂	semidirect
11. C₈₃×#3383	C₂	semidirect
165. #3383	D₈₃	semidirect
175. C₈₃	#33931	semidirect

Maximal subgroups

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

subgroup	index
#24920452	2
#24909697	2
#24909692	2
#24909624	2
#24909597	2
#24909525	2
#24909513	2
Q₈×C₈₃⋊C₁₆	2
C₄×#14581738	2
C₄×#14581735	2
C₈₃×#3383	2
#33931	83