Subgroups — #35375836

Order 21248.

656 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24916478	C₂	1	2	yes	maximal, complemented
2	C₄×#14581814	C₂	1	1	yes	maximal, characteristic
3	#24906361	C₂	1	2	yes	maximal, complemented
4	#24906301	C₂	1	4	yes	maximal, complemented
5	C₈×D₆₆₄	C₂	1	2	yes	maximal, complemented
6	#24906226	C₂	1	1	yes	maximal, characteristic, complemented
7	#24905898	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14581814	C₄	1	1	yes	characteristic
10	C₄×#13050590	C₂²	1	2	yes	—
11	#19127831	C₂²	1	4	yes	—
12	C₄×Dic₃₃₂	C₄	1	1	yes	characteristic
13	C₄×Dic₃₃₂	C₂²	1	1	yes	characteristic
14	C₄×#13050579	C₄	1	2	yes	—
15	C₄×#13050579	C₂²	1	2	yes	—
16	C₄×D₆₆₄	C₄	1	1	yes	characteristic
17	C₄×D₆₆₄	C₂²	1	1	yes	characteristic
18	#19127770	—	2	4	no	—
19	C₈×D₃₃₂	C₂²	1	4	yes	—
20	#19127758	C₂²	1	2	yes	—
21	#19127756	—	2	2	no	—
22	#19127739	C₄	1	1	yes	characteristic
23	#19127698	C₄	1	2	yes	—
24	#97×D₈₃	—	2	2	no	—
25	#19127454	C₂²	1	2	yes	—
26	#19127452	C₂²	1	2	yes	—
27	#19127433	C₂²	1	4	yes	—
28	#19127430	C₂²	1	4	yes	—
29	#431×C₈₃	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
32	#322×C₈₃	C₂²	1	2	yes	—
33	C₈²×C₈₃	C₂²	1	2	yes	—
34	C₂×#13050590	C₂×C₄	1	2	yes	—
35	#14581814	—	2	4	no	—
36	C₂×Dic₃₃₂	C₂×C₄	1	1	yes	characteristic
37	C₂×#13050579	C₂×C₄	1	2	yes	—
38	C₂×D₆₆₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₈₃	—	2	4	no	—
40	C₄×Dic₁₆₆	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₆₆	C₂³	1	2	yes	—
42	C₄×D₃₃₂	C₂×C₄	1	2	yes	—
43	C₄×D₃₃₂	C₂³	1	2	yes	—
44	C₄×C₂²⋊D₈₃	—	2	2	no	—
45	C₄×C₂²⋊D₈₃	—	2	2	no	—
46	#14581751	C₂×C₄	1	2	yes	—
47	#14581749	—	2	2	no	—
48	C₄²×D₈₃	—	2	2	no	—
49	#14581726	C₂×C₄	1	1	yes	characteristic
50	#14581726	C₂×C₄	1	1	yes	characteristic
51	#14581724	C₂×C₄	1	2	yes	—
52	C₄⋊(C₈₃⋊C₈)	C₂³	1	4	yes	—
53	Dic₈₃⋊C₈	—	2	4	no	—
54	#14581717	C₂×C₄	1	2	yes	—
55	#14581717	C₂×C₄	1	2	yes	—
56	#14581714	C₂×C₄	1	2	yes	—
57	#14581714	C₂×C₄	1	2	yes	—
58	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
59	(C₂×D₈₃)⋊C₈	—	2	4	no	—
60	#14581703	—	2	2	no	—
61	#14581702	—	2	4	no	—
62	C₂×M₁₆×C₈₃	D₄	1	2	yes	—
63	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₈₃	—	2	1	no	—
67	C₂²⋊C₈×C₈₃	—	2	1	no	—
68	#97×C₈₃	D₄	1	1	yes	characteristic
69	#97×C₈₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₈₃	C₂³	1	2	yes	—
73	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₈₃	C₂³	1	2	yes	—
76	#13050590	—	2	4	no	—
77	#13050590	—	4	2	no	—
78	C₂×Dic₁₆₆	C₂²×C₄	1	2	yes	—
79	C₂×D₃₃₂	C₂²×C₄	1	2	yes	—
80	C₂×C₂²⋊D₈₃	—	2	2	no	—
81	C₂×C₄×D₈₃	—	2	2	no	—
82	C₂×C₄×D₈₃	—	2	2	no	—
83	C₂×C₄×D₈₃	—	2	2	no	—
84	Dic₃₃₂	—	2	2	no	—
85	#13050579	—	2	4	no	—
86	D₆₆₄	M₁₆	1	4	yes	complemented
87	C₂×C₈₃⋊C₈	—	2	4	no	—
88	C₈×D₈₃	—	4	8	no	—
89	C₄⋊Dic₈₃	C₂²×C₄	1	2	yes	—
90	C₄⋊Dic₈₃	C₂²×C₄	1	2	yes	—
91	Dic₈₃⋊C₄	—	2	2	no	—
92	Dic₈₃⋊C₄	—	2	2	no	—
93	C₂²⋊Dic₈₃	—	2	2	no	—
94	(C₂×D₈₃)⋊C₄	—	2	2	no	—
95	(C₂×D₈₃)⋊C₄	—	2	2	no	—
96	C₄×Dic₈₃	—	2	2	no	—
97	C₄×Dic₈₃	—	2	2	no	—
98	C₄×Dic₈₃	—	2	2	no	—
99	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
102	M₁₆×C₈₃	—	2	4	no	—
103	C₂×C₈×C₈₃	—	2	2	no	—
104	C₂×C₈×C₈₃	—	2	2	no	—
105	C₂×C₈×C₈₃	—	4	1	no	—
106	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
112	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₈₃	#41	1	1	yes	characteristic
115	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
116	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₈₃	—	2	2	no	—
118	C₂×C₄×C₈₃	—	2	1	no	—
119	C₂×C₄×C₈₃	—	2	1	no	—
120	C₂×C₄×C₈₃	—	2	2	no	—
121	C₂×C₄×C₈₃	—	4	1	no	—
122	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
128	C₂²⋊D₈₃	—	4	4	no	—
129	C₂×Dic₈₃	—	2	2	no	—
130	C₂×Dic₈₃	—	2	2	no	—
131	C₂×Dic₈₃	—	2	2	no	—
132	D₃₃₂	—	2	2	no	—
133	D₃₃₂	C₂×M₁₆	1	4	yes	—
134	C₄×D₈₃	—	4	4	no	—
135	C₄×D₈₃	—	4	4	no	—
136	C₄×D₈₃	—	8	2	no	—
137	Dic₁₆₆	—	2	2	no	—
138	Dic₁₆₆	—	2	2	no	—
139	C₈×C₈₃	—	2	1	no	—
140	C₈×C₈₃	—	2	1	no	—
141	C₈×C₈₃	—	2	1	no	—
142	C₈×C₈₃	—	2	2	no	—
143	C₈×C₈₃	—	4	2	no	—
144	C₈×C₈₃	C₂×M₁₆	1	2	yes	—
145	C₈×C₈₃	#135	1	4	yes	complemented
146	C₈₃⋊C₈	—	4	4	no	—
147	C₂²×C₈₃	—	2	1	no	—
148	C₂²×C₈₃	—	2	1	no	—
149	C₂²×C₈₃	—	2	1	no	—
150	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂×D₈₃	—	2	4	no	—
152	C₂×D₈₃	—	4	2	no	—
153	C₂×D₈₃	—	4	2	no	—
154	C₄×C₈₃	—	2	1	no	—
155	C₄×C₈₃	—	2	1	no	—
156	C₄×C₈₃	—	2	1	no	—
157	C₄×C₈₃	—	2	1	no	—
158	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
159	C₄×C₈₃	#568	1	1	yes	characteristic
160	C₄×C₈₃	C₂×#135	1	2	yes	—
161	Dic₈₃	—	4	2	no	—
162	Dic₈₃	—	4	2	no	—
163	Dic₈₃	—	4	2	no	—
164	#18035	—	83	1	no	maximal, Sylow
165	C₂×C₈₃	—	4	1	no	—
166	C₂×C₈₃	D₄×M₁₆	1	1	yes	characteristic
167	C₂×C₈₃	C₄×#135	1	1	yes	characteristic
168	C₂×C₈₃	#2957	1	1	yes	characteristic
169	D₈₃	—	4	4	no	—
170	#2970	—	83	2	no	—
171	C₄×#135	—	83	1	no	—
172	#1591	—	83	2	no	—
173	#1582	—	83	4	no	—
174	C₈×D₈	—	83	2	no	—
175	#1568	—	83	1	no	—
176	#1488	—	83	2	no	—
177	C₈×M₁₆	D₈₃	1	1	yes	characteristic, complemented
178	C₈₃	#18035	1	1	yes	characteristic, complemented, Sylow
179	C₂×#135	—	83	1	no	—
180	C₄×#41	—	83	2	no	—
181	#446	—	83	4	no	—
182	C₄×Q₁₆	—	83	1	no	—
183	C₄×Q₁₆	—	83	1	no	—
184	C₄×SD₁₆	—	83	2	no	—
185	C₄×SD₁₆	—	83	2	no	—
186	C₄×D₈	—	83	1	no	—
187	C₄×D₈	—	83	1	no	—
188	#435	—	166	4	no	—
189	C₈×D₄	—	83	4	no	—
190	#432	—	83	2	no	—
191	#432	—	166	2	no	—
192	#431	C₂×D₈₃	1	1	yes	characteristic
193	#427	—	83	1	no	—
194	#416	—	83	2	no	—
195	C₄×M₁₆	C₂×D₈₃	1	1	yes	characteristic
196	C₂×#97	—	166	2	no	—
197	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
198	#335	—	83	2	no	—
199	#334	—	83	2	no	—
200	Q₈⋊C₈	—	83	4	no	—
201	D₄⋊C₈	—	83	4	no	—
202	#322	C₂×D₈₃	1	2	yes	—
203	C₈²	C₂×D₈₃	1	2	yes	—
204	C₂×#41	—	83	2	no	—
205	#135	—	166	4	no	—
206	C₂×Q₁₆	—	83	1	no	—
207	C₂×SD₁₆	—	83	2	no	—
208	C₂×D₈	—	83	1	no	—
209	C₂×M₁₆	D₃₃₂	1	2	yes	—
210	C₂²×C₈	—	166	4	no	—
211	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
212	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
213	C₄×Q₈	—	83	2	no	—
214	C₄×Q₈	—	83	2	no	—
215	C₄×D₄	—	83	2	no	—
216	C₄×D₄	—	83	2	no	—
217	C₄×D₄	—	166	2	no	—
218	C₄×D₄	—	166	2	no	—
219	#117	—	83	2	no	—
220	#117	—	166	2	no	—
221	C₂×C₄²	—	166	2	no	—
222	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
223	#107	—	83	1	no	—
224	#107	—	83	1	no	—
225	#106	—	83	2	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	83	4	no	—
228	C₄⋊C₈	—	166	4	no	—
229	#103	—	83	2	no	—
230	#103	—	83	2	no	—
231	#102	—	83	2	no	—
232	#102	—	83	2	no	—
233	C₂²⋊C₈	—	2	1	no	—
234	C₂²⋊C₈	—	166	2	no	—
235	C₂²⋊C₈	—	166	4	no	—
236	#97	—	166	4	no	—
237	#97	—	166	2	no	—
238	#97	D₃₃₂	1	1	yes	characteristic
239	#97	D₃₃₂	1	1	yes	characteristic
240	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
241	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₈₃	1	2	yes	—
243	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₈₃	1	2	yes	—
246	#41	—	332	2	no	—
247	#41	—	166	4	no	—
248	C₂×Q₈	—	83	2	no	—
249	C₂×D₄	—	83	2	no	—
250	C₂×D₄	—	166	2	no	—
251	C₂²×C₄	—	166	2	no	—
252	C₂²×C₄	—	166	2	no	—
253	C₂²×C₄	—	166	2	no	—
254	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
256	C₂²×C₄	#13050590	1	1	yes	characteristic
257	Q₁₆	—	166	2	no	—
258	SD₁₆	—	166	4	no	—
259	D₈	—	83	4	no	—
260	M₁₆	—	2	4	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	332	8	no	—
265	C₂×C₈	—	166	4	no	—
266	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
270	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
271	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
272	C₄⋊C₄	—	83	2	no	—
273	C₄⋊C₄	—	83	2	no	—
274	C₄⋊C₄	—	166	2	no	—
275	C₄⋊C₄	—	166	2	no	—
276	C₂²⋊C₄	—	166	2	no	—
277	C₂²⋊C₄	—	166	2	no	—
278	C₂²⋊C₄	—	166	2	no	—
279	C₄²	—	166	2	no	—
280	C₄²	—	166	2	no	—
281	C₄²	—	166	2	no	—
282	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
283	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
284	C₄²	#13050590	1	1	yes	characteristic
285	C₄²	C₂³×D₈₃	1	1	yes	characteristic
286	C₂³	—	166	2	no	—
287	C₂³	C₄×D₃₃₂	1	1	yes	characteristic
288	Q₈	—	166	2	no	—
289	Q₈	—	166	2	no	—
290	D₄	—	332	4	no	—
291	D₄	—	83	4	no	—
292	D₄	—	166	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	332	4	no	—
298	C₂×C₄	—	332	4	no	—
299	C₂×C₄	—	664	2	no	—
300	C₂×C₄	—	166	2	no	—
301	C₂×C₄	—	166	2	no	—
302	C₂×C₄	—	166	2	no	—
303	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
304	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
305	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
308	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	2	no	—
313	C₈	—	4	2	no	—
314	C₈	—	332	4	no	—
315	C₈	M₁₆×D₈₃	1	2	yes	—
316	C₈	#14581814	1	4	yes	complemented
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	332	2	no	—
321	C₂²	—	332	2	no	—
322	C₂²	—	166	4	no	—
323	C₂²	C₂×C₄×D₃₃₂	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	332	2	no	—
329	C₄	—	332	2	no	—
330	C₄	—	332	2	no	—
331	C₄	C₂×M₁₆×D₈₃	1	1	yes	characteristic
332	C₄	#19128467	1	1	yes	characteristic
333	C₄	C₂×#14581814	1	2	yes	—
334	C₂	—	4	1	no	—
335	C₂	—	332	4	no	—
336	C₂	M₁₆×D₃₃₂	1	1	yes	characteristic
337	C₂	C₄×#14581814	1	1	yes	characteristic
338	C₂	#24916381	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₄²	C₂³×D₈₃
Fitting	C₈×M₁₆×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄×D₄
prime radical	C₂×#14581814	C₄
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	C₈×M₁₆	D₈₃
hypocenter	C₈₃	#18035
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #24916478	C₂	semidirect
3. #24906361	C₂	semidirect
4. #24906301	C₂	semidirect
5. C₈×D₆₆₄	C₂	semidirect
6. #24906226	C₂	semidirect
7. #24905898	C₂	semidirect
8. C₈×M₁₆×C₈₃	C₂	semidirect
86. D₆₆₄	M₁₆	semidirect
145. C₈×C₈₃	#135	semidirect
177. C₈×M₁₆	D₈₃	semidirect
178. C₈₃	#18035	semidirect
316. C₈	#14581814	semidirect

Maximal subgroups

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

subgroup	index
#24916478	2
C₄×#14581814	2
#24906361	2
#24906301	2
C₈×D₆₆₄	2
#24906226	2
#24905898	2
C₈×M₁₆×C₈₃	2
#18035	83