Subgroups — #35321013

Order 21248.

490 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	C₂×C₄×D₆₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19127503	C₂	1	1	yes	maximal, characteristic
4	C₂×#19127503	C₂	1	1	yes	maximal, characteristic
5	#24908196	C₂	1	1	yes	maximal, characteristic, complemented
6	#24905555	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₆₆₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₃₃₂	—	2	1	no	—
10	C₄×D₆₆₄	—	2	2	no	—
11	C₄×D₆₆₄	—	2	2	no	—
12	C₂×#14581726	C₂²	1	1	yes	characteristic
13	C₂×#14581726	C₂²	1	1	yes	characteristic
14	C₂×#14581726	C₂²	1	1	yes	characteristic
15	C₂×#14581714	—	2	1	no	—
16	#19127599	—	2	1	no	—
17	#19127529	—	2	2	no	—
18	#19127503	—	2	2	no	—
19	#19127503	C₄	1	4	yes	complemented
20	#19127464	—	2	1	no	—
21	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
24	C₂²×D₃₃₂	—	2	1	no	—
25	C₂²×C₄×D₈₃	—	4	1	no	—
26	C₂×D₆₆₄	—	2	1	no	—
27	C₂×D₆₆₄	—	2	1	no	—
28	C₂×D₆₆₄	—	2	1	no	—
29	C₂×D₆₆₄	C₂×C₄	1	2	yes	—
30	C₂×D₆₆₄	D₄	1	2	yes	—
31	C₂×D₆₆₄	Q₈	1	2	yes	—
32	C₄×D₃₃₂	—	2	4	no	—
33	C₄×D₃₃₂	—	4	2	no	—
34	C₂×C₄⋊Dic₈₃	—	2	1	no	—
35	C₂×C₄⋊Dic₈₃	—	2	1	no	—
36	C₂×C₄⋊Dic₈₃	—	2	1	no	—
37	C₂×Dic₈₃⋊C₄	—	4	1	no	—
38	C₂×(C₂×D₈₃)⋊C₄	—	4	1	no	—
39	#14581726	—	2	1	no	—
40	#14581726	—	2	1	no	—
41	#14581726	—	2	1	no	—
42	#14581726	—	2	1	no	—
43	#14581726	C₂×C₄	1	2	yes	complemented
44	#14581726	D₄	1	2	yes	—
45	#14581714	—	2	4	no	—
46	C₈₃⋊#95	—	4	1	no	—
47	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
50	C₂×C₁₆×C₈₃	—	2	1	no	—
51	C₂×C₁₆×C₈₃	—	2	1	no	—
52	C₂×C₁₆×C₈₃	—	2	1	no	—
53	C₂×C₁₆×C₈₃	—	2	1	no	—
54	C₂×C₁₆×C₈₃	—	2	1	no	—
55	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	complemented
56	C₄×C₈×C₈₃	—	2	1	no	—
57	C₄×C₈×C₈₃	—	2	1	no	—
58	C₂³×D₈₃	—	4	1	no	—
59	C₂×D₃₃₂	—	2	2	no	—
60	C₂×D₃₃₂	—	2	2	no	—
61	C₂×D₃₃₂	—	2	2	no	—
62	C₂×D₃₃₂	—	4	1	no	—
63	C₂×D₃₃₂	—	4	1	no	—
64	C₂×D₃₃₂	—	4	1	no	—
65	C₂²×Dic₈₃	—	4	1	no	—
66	C₂²×Dic₈₃	—	4	1	no	—
67	C₂²×Dic₈₃	—	4	1	no	—
68	C₂×C₄×D₈₃	—	4	4	no	—
69	C₂×C₄×D₈₃	—	8	2	no	—
70	C₂×C₄×D₈₃	—	8	2	no	—
71	D₆₆₄	—	2	2	no	—
72	D₆₆₄	—	2	2	no	—
73	D₆₆₄	—	2	2	no	—
74	D₆₆₄	C₄⋊C₄	1	4	yes	complemented
75	C₄⋊Dic₈₃	—	2	2	no	—
76	C₄⋊Dic₈₃	—	2	2	no	—
77	C₄⋊Dic₈₃	—	4	1	no	—
78	C₄⋊Dic₈₃	—	4	1	no	—
79	C₄⋊Dic₈₃	—	4	1	no	—
80	C₄⋊Dic₈₃	—	4	1	no	—
81	Dic₈₃⋊C₄	—	8	2	no	—
82	(C₂×D₈₃)⋊C₄	—	4	4	no	—
83	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₈₃	—	2	1	no	—
87	C₂×C₈×C₈₃	—	2	1	no	—
88	C₂×C₈×C₈₃	—	2	1	no	—
89	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
94	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
95	C₄²×C₈₃	—	2	1	no	—
96	C₄²×C₈₃	—	2	1	no	—
97	C₁₆×C₈₃	—	2	2	no	—
98	C₁₆×C₈₃	—	4	1	no	—
99	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
100	C₂²×D₈₃	—	4	2	no	—
101	C₂²×D₈₃	—	4	2	no	—
102	C₂²×D₈₃	—	4	2	no	—
103	C₂²×D₈₃	—	8	1	no	—
104	C₂²×D₈₃	—	8	1	no	—
105	C₂²×D₈₃	—	8	1	no	—
106	C₂²×D₈₃	—	8	1	no	—
107	C₂×C₄×C₈₃	—	2	1	no	—
108	C₂×C₄×C₈₃	—	2	1	no	—
109	C₂×C₄×C₈₃	—	2	1	no	—
110	C₂×C₄×C₈₃	—	2	1	no	—
111	C₂×C₄×C₈₃	—	2	1	no	—
112	C₂×C₄×C₈₃	SD₃₂	1	2	yes	—
113	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₃	#120	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
117	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
119	C₂×Dic₈₃	—	4	2	no	—
120	C₂×Dic₈₃	—	4	2	no	—
121	C₂×Dic₈₃	—	8	1	no	—
122	C₂×Dic₈₃	—	8	1	no	—
123	C₂×Dic₈₃	—	8	1	no	—
124	C₂×Dic₈₃	—	8	1	no	—
125	C₂×Dic₈₃	—	8	1	no	—
126	C₂×Dic₈₃	—	8	1	no	—
127	C₂×Dic₈₃	—	8	1	no	—
128	D₃₃₂	—	2	4	no	—
129	D₃₃₂	—	4	2	no	—
130	D₃₃₂	—	4	2	no	—
131	D₃₃₂	—	4	2	no	—
132	C₄×D₈₃	—	8	8	no	—
133	C₈×C₈₃	—	4	1	no	—
134	C₈×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
136	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
137	C₈×C₈₃	#123	1	1	yes	characteristic
138	C₂²×C₈₃	C₄×D₈	1	1	yes	characteristic
139	C₂²×C₈₃	#447	1	1	yes	characteristic
140	C₂²×C₈₃	#474	1	1	yes	characteristic
141	C₂²×C₈₃	C₂×SD₃₂	1	1	yes	characteristic
142	C₂²×C₈₃	#508	1	1	yes	characteristic
143	C₂²×C₈₃	#509	1	1	yes	characteristic
144	C₂²×C₈₃	#509	1	1	yes	characteristic
145	C₂×D₈₃	—	4	4	no	—
146	C₂×D₈₃	—	8	2	no	—
147	C₂×D₈₃	—	8	2	no	—
148	C₂×D₈₃	—	8	2	no	—
149	C₂×D₈₃	—	8	2	no	—
150	C₂×D₈₃	—	8	2	no	—
151	C₂×D₈₃	—	8	2	no	—
152	C₄×C₈₃	—	2	2	no	—
153	C₄×C₈₃	—	4	1	no	—
154	C₄×C₈₃	#385	1	1	yes	characteristic
155	C₄×C₈₃	#440	1	1	yes	characteristic
156	C₄×C₈₃	#452	1	1	yes	characteristic
157	C₄×C₈₃	#478	1	1	yes	characteristic
158	Dic₈₃	—	8	2	no	—
159	Dic₈₃	—	8	2	no	—
160	Dic₈₃	—	16	1	no	—
161	#12937	—	83	1	no	maximal, Sylow
162	C₂×C₈₃	#1862	1	1	yes	characteristic
163	C₂×C₈₃	C₄×SD₃₂	1	1	yes	characteristic
164	C₂×C₈₃	#2177	1	1	yes	characteristic
165	C₂×C₈₃	#2186	1	1	yes	characteristic
166	C₂×C₈₃	#2187	1	1	yes	characteristic
167	C₂×C₈₃	#2226	1	1	yes	characteristic
168	C₂×C₈₃	#2228	1	1	yes	characteristic
169	D₈₃	—	8	4	no	—
170	C₂×C₄×D₈	—	83	1	no	—
171	C₂×#367	—	83	1	no	—
172	C₂×#357	—	83	1	no	—
173	C₂×#357	—	83	1	no	—
174	#1942	—	83	1	no	—
175	#1380	—	83	1	no	—
176	#1374	D₈₃	1	1	yes	characteristic, complemented
177	C₈₃	#12937	1	1	yes	characteristic, complemented, Sylow
178	C₂²×D₈	—	83	1	no	—
179	C₂×C₄×D₄	—	166	1	no	—
180	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
181	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
182	C₄×D₈	—	166	2	no	—
183	C₄×D₈	—	166	2	no	—
184	C₂×#107	—	83	1	no	—
185	C₂×#107	—	83	1	no	—
186	C₂×#107	—	83	1	no	—
187	C₂×#102	—	166	1	no	—
188	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
189	#389	—	166	1	no	—
190	#367	—	166	2	no	—
191	#357	—	83	4	no	—
192	#357	—	166	2	no	—
193	#340	—	166	1	no	—
194	C₂²×D₄	—	166	1	no	—
195	C₂³×C₄	—	332	1	no	—
196	C₂×D₈	—	83	2	no	—
197	C₂×D₈	—	83	2	no	—
198	C₂×D₈	—	83	2	no	—
199	C₂×D₈	—	166	1	no	—
200	C₂×D₈	—	166	1	no	—
201	C₂×D₈	—	166	1	no	—
202	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
204	C₄×D₄	—	332	2	no	—
205	C₄×D₄	—	166	4	no	—
206	C₂×C₄⋊C₄	—	332	1	no	—
207	C₂×C₄⋊C₄	—	166	1	no	—
208	C₂×C₄⋊C₄	—	166	1	no	—
209	C₂×C₄⋊C₄	—	166	1	no	—
210	C₂×C₂²⋊C₄	—	332	1	no	—
211	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	C₄×D₈₃	1	2	yes	complemented
218	#107	—	83	2	no	—
219	#107	—	83	2	no	—
220	#107	—	166	1	no	—
221	#107	—	166	1	no	—
222	#107	—	166	1	no	—
223	#107	—	166	1	no	—
224	#102	—	166	4	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	#95	—	332	1	no	—
228	C₂⁴	—	332	1	no	—
229	C₂×D₄	—	332	1	no	—
230	C₂×D₄	—	332	1	no	—
231	C₂×D₄	—	332	1	no	—
232	C₂×D₄	—	166	2	no	—
233	C₂×D₄	—	166	2	no	—
234	C₂×D₄	—	166	2	no	—
235	C₂²×C₄	—	332	4	no	—
236	C₂²×C₄	—	332	1	no	—
237	C₂²×C₄	—	332	1	no	—
238	C₂²×C₄	—	332	1	no	—
239	C₂²×C₄	—	664	2	no	—
240	C₂²×C₄	—	664	2	no	—
241	C₂²×C₄	D₆₆₄	1	1	yes	characteristic
242	C₂²×C₄	D₆₆₄	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
244	D₈	—	83	4	no	—
245	D₈	—	166	2	no	—
246	D₈	—	166	2	no	—
247	D₈	—	166	2	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
252	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
253	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
254	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
255	C₂×C₈	#13050589	1	1	yes	characteristic
256	C₂×C₈	#13050590	1	1	yes	characteristic
257	C₄⋊C₄	—	332	1	no	—
258	C₄⋊C₄	—	332	1	no	—
259	C₄⋊C₄	—	332	1	no	—
260	C₄⋊C₄	—	332	1	no	—
261	C₄⋊C₄	—	664	2	no	—
262	C₄⋊C₄	—	166	2	no	—
263	C₄⋊C₄	—	166	2	no	—
264	C₂²⋊C₄	—	332	4	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₁₆	—	2	2	no	—
268	C₁₆	—	4	1	no	—
269	C₂³	—	332	2	no	—
270	C₂³	—	332	2	no	—
271	C₂³	—	332	2	no	—
272	C₂³	—	664	1	no	—
273	C₂³	—	664	1	no	—
274	C₂³	—	664	1	no	—
275	C₂³	—	664	1	no	—
276	C₂³	C₂×D₆₆₄	1	1	yes	characteristic
277	D₄	—	332	2	no	—
278	D₄	—	332	2	no	—
279	D₄	—	332	2	no	—
280	D₄	—	166	4	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	332	2	no	—
287	C₂×C₄	—	332	2	no	—
288	C₂×C₄	—	664	8	no	—
289	C₂×C₄	—	664	1	no	—
290	C₂×C₄	—	664	1	no	—
291	C₂×C₄	—	664	1	no	—
292	C₂×C₄	—	664	1	no	—
293	C₂×C₄	—	664	1	no	—
294	C₂×C₄	—	664	1	no	—
295	C₂×C₄	—	664	1	no	—
296	C₂×C₄	#14581737	1	2	yes	—
297	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
298	C₂×C₄	#14581762	1	1	yes	characteristic
299	C₂×C₄	#14581772	1	1	yes	characteristic
300	C₂×C₄	#14581814	1	1	yes	characteristic
301	C₂×C₄	#14581820	1	1	yes	characteristic
302	C₂×C₄	#14581825	1	1	yes	characteristic
303	C₈	—	4	1	no	—
304	C₈	C₄⋊C₄×D₈₃	1	1	yes	characteristic
305	C₈	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
306	C₈	#14581770	1	1	yes	characteristic
307	C₈	#14581774	1	1	yes	characteristic
308	C₂²	—	332	4	no	—
309	C₂²	—	664	2	no	—
310	C₂²	—	664	2	no	—
311	C₂²	—	664	2	no	—
312	C₂²	—	664	2	no	—
313	C₂²	—	664	2	no	—
314	C₂²	—	664	2	no	—
315	C₂²	C₄×D₆₆₄	1	1	yes	characteristic
316	C₂²	#19127837	1	1	yes	characteristic
317	C₂²	#19127982	1	1	yes	characteristic
318	C₂²	C₂×#14581737	1	1	yes	characteristic
319	C₂²	#19128133	1	1	yes	characteristic
320	C₂²	#19128139	1	1	yes	characteristic
321	C₂²	#19128139	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	4	1	no	—
324	C₄	—	1328	1	no	—
325	C₄	—	664	2	no	—
326	C₄	—	664	2	no	—
327	C₄	#19127575	1	1	yes	characteristic
328	C₄	#19127803	1	1	yes	characteristic
329	C₄	#19127865	1	1	yes	characteristic
330	C₄	#19128002	1	1	yes	characteristic
331	C₂	—	664	4	no	—
332	C₂	#24907713	1	1	yes	characteristic
333	C₂	C₄×#14581737	1	1	yes	characteristic
334	C₂	#24909340	1	1	yes	characteristic
335	C₂	#24909388	1	1	yes	characteristic
336	C₂	#24909394	1	1	yes	characteristic
337	C₂	#24909611	1	1	yes	characteristic
338	C₂	#24909621	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₆₄
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1374	C₂
socle	C₂³×C₈₃	C₂×D₈
prime radical	C₂²×D₆₆₄	C₂²
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1374	D₈₃
hypocenter	C₈₃	#12937
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
2. C₂×#19127529	C₂	semidirect
5. #24908196	C₂	semidirect
6. #24905555	C₂	semidirect
7. C₈₃×#1374	C₂	semidirect
19. #19127503	C₄	semidirect
43. #14581726	C₂×C₄	semidirect
55. C₂×C₁₆×C₈₃	C₂×C₄	semidirect
74. D₆₆₄	C₄⋊C₄	semidirect
176. #1374	D₈₃	semidirect
177. C₈₃	#12937	semidirect
217. C₂×C₁₆	C₄×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₆₆₄	2
C₂×#19127529	2
C₂×#19127503	2
#24908196	2
#24905555	2
C₈₃×#1374	2
#12937	83