Subgroups — #35320995

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×#19127526	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19127501	C₂	1	1	yes	maximal, characteristic
4	C₂×#19127501	C₂	1	1	yes	maximal, characteristic
5	#24905554	C₂	1	1	yes	maximal, characteristic, complemented
6	#24905541	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1942	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₆₆₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₃₃₂	—	2	1	no	—
10	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₄×D₆₆₄	—	2	2	no	—
13	C₄×D₆₆₄	—	2	2	no	—
14	C₂×#14581726	C₂²	1	1	yes	characteristic
15	C₂×#14581714	—	2	1	no	—
16	#19127526	—	2	2	no	—
17	#19127501	—	2	2	no	—
18	#19127501	C₄	1	4	yes	complemented
19	C₂×#107×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×#107×C₈₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
22	#389×C₈₃	—	2	1	no	—
23	#340×C₈₃	—	2	1	no	—
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×(C₂×D₈₃)⋊C₄	—	4	1	no	—
36	C₂×C₈₃⋊C₁₆	—	2	1	no	—
37	C₂×C₈₃⋊C₁₆	—	2	1	no	—
38	C₂×C₈₃⋊C₁₆	—	2	1	no	—
39	C₂×C₈₃⋊C₁₆	—	2	1	no	—
40	C₂×C₈₃⋊C₁₆	—	2	1	no	—
41	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
42	#14581726	—	2	1	no	—
43	#14581726	D₄	1	2	yes	—
44	#14581714	—	2	4	no	—
45	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
48	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
49	C₂×C₄⋊C₄×C₈₃	—	4	1	no	—
50	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
51	#107×C₈₃	—	2	1	no	—
52	#107×C₈₃	—	2	1	no	—
53	#107×C₈₃	—	2	1	no	—
54	#107×C₈₃	C₂×C₄	1	2	yes	complemented
55	C₄×C₈×C₈₃	—	2	1	no	—
56	C₄×C₈×C₈₃	—	2	1	no	—
57	#95×C₈₃	—	4	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	C₂×C₄×D₈₃	—	4	4	no	—
67	C₂×C₄×D₈₃	—	8	2	no	—
68	C₂×C₄×D₈₃	—	8	2	no	—
69	D₆₆₄	—	2	2	no	—
70	D₆₆₄	—	2	2	no	—
71	D₆₆₄	—	2	2	no	—
72	D₆₆₄	C₄⋊C₄	1	4	yes	complemented
73	C₄⋊Dic₈₃	—	2	2	no	—
74	C₄⋊Dic₈₃	—	4	1	no	—
75	(C₂×D₈₃)⋊C₄	—	4	4	no	—
76	C₈₃⋊C₁₆	—	2	2	no	—
77	C₈₃⋊C₁₆	—	4	1	no	—
78	C₂²×C₄×C₈₃	—	4	1	no	—
79	C₂²×C₄×C₈₃	—	4	1	no	—
80	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₈₃	—	2	1	no	—
84	C₂×C₈×C₈₃	—	2	1	no	—
85	C₂×C₈×C₈₃	—	2	1	no	—
86	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
91	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
92	C₄⋊C₄×C₈₃	—	2	2	no	—
93	C₄⋊C₄×C₈₃	—	4	1	no	—
94	C₄⋊C₄×C₈₃	—	4	1	no	—
95	C₄⋊C₄×C₈₃	—	4	1	no	—
96	C₄⋊C₄×C₈₃	—	8	2	no	—
97	C₄²×C₈₃	—	2	1	no	—
98	C₄²×C₈₃	—	2	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	—	4	2	no	—
113	C₂×C₄×C₈₃	—	8	1	no	—
114	C₂×C₄×C₈₃	—	8	1	no	—
115	C₂×C₄×C₈₃	—	8	1	no	—
116	C₂×C₄×C₈₃	—	8	1	no	—
117	C₂×C₄×C₈₃	—	8	1	no	—
118	C₂×C₄×C₈₃	D₁₆	1	2	yes	—
119	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	#120	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
123	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
125	C₂×Dic₈₃	—	4	2	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×D₁₆	1	1	yes	characteristic
142	C₂²×C₈₃	#508	1	1	yes	characteristic
143	C₂²×C₈₃	#509	1	1	yes	characteristic
144	C₂²×C₈₃	#510	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	—	8	2	no	—
155	C₄×C₈₃	—	16	1	no	—
156	C₄×C₈₃	#385	1	1	yes	characteristic
157	C₄×C₈₃	#440	1	1	yes	characteristic
158	C₄×C₈₃	#452	1	1	yes	characteristic
159	C₄×C₈₃	#478	1	1	yes	characteristic
160	Dic₈₃	—	8	2	no	—
161	#12935	—	83	1	no	maximal, Sylow
162	C₂×C₈₃	#1862	1	1	yes	characteristic
163	C₂×C₈₃	C₄×D₁₆	1	1	yes	characteristic
164	C₂×C₈₃	#2175	1	1	yes	characteristic
165	C₂×C₈₃	#2184	1	1	yes	characteristic
166	C₂×C₈₃	#2189	1	1	yes	characteristic
167	C₂×C₈₃	#2224	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×#366	—	83	1	no	—
172	C₂×#357	—	83	1	no	—
173	C₂×#357	—	83	1	no	—
174	#1942	D₈₃	1	1	yes	characteristic, complemented
175	#1380	—	83	1	no	—
176	#1374	—	83	1	no	—
177	C₈₃	#12935	1	1	yes	characteristic, complemented, Sylow
178	C₂²×D₈	—	83	1	no	—
179	C₂×C₄×D₄	—	166	1	no	—
180	C₂²×C₁₆	—	83	1	no	—
181	C₂²×C₁₆	—	83	1	no	—
182	C₄×D₈	—	166	2	no	—
183	C₄×D₈	—	166	2	no	—
184	C₂×#107	—	83	1	no	—
185	C₂×#107	C₂×D₈₃	1	1	yes	characteristic
186	C₂×#107	C₂×D₈₃	1	1	yes	characteristic
187	C₂×#102	—	166	1	no	—
188	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
189	#389	—	2	1	no	—
190	#366	—	166	2	no	—
191	#357	—	83	4	no	—
192	#357	—	166	2	no	—
193	#340	—	2	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	C₂²⋊D₈₃	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	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	4	1	no	—
209	C₂×C₄⋊C₄	—	166	1	no	—
210	C₂×C₂²⋊C₄	—	332	1	no	—
211	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
212	C₂×C₁₆	—	83	2	no	—
213	C₂×C₁₆	—	166	1	no	—
214	C₂×C₁₆	—	166	1	no	—
215	C₂×C₁₆	—	166	1	no	—
216	C₂×C₁₆	—	166	1	no	—
217	C₂×C₁₆	—	166	1	no	—
218	#107	—	2	1	no	—
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	—	83	2	no	—
222	#107	—	166	1	no	—
223	#107	C₄×D₈₃	1	2	yes	complemented
224	#102	—	166	4	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	#95	—	4	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	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	332	4	no	—
238	C₂²×C₄	—	332	1	no	—
239	C₂²×C₄	—	664	2	no	—
240	C₂²×C₄	—	664	2	no	—
241	C₂²×C₄	#13050575	1	1	yes	characteristic
242	C₂²×C₄	#13050575	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₈₃	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	#13050590	1	1	yes	characteristic
256	C₂×C₈	#13050591	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	8	2	no	—
262	C₄⋊C₄	—	332	1	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	—	332	1	no	—
268	C₁₆	—	166	2	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×#13050575	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	—	4	2	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	8	1	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	332	2	no	—
293	C₂×C₄	—	664	8	no	—
294	C₂×C₄	—	664	1	no	—
295	C₂×C₄	—	664	1	no	—
296	C₂×C₄	#14581733	1	2	yes	—
297	C₂×C₄	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
298	C₂×C₄	#14581760	1	1	yes	characteristic
299	C₂×C₄	#14581768	1	1	yes	characteristic
300	C₂×C₄	#14581812	1	1	yes	characteristic
301	C₂×C₄	#14581817	1	1	yes	characteristic
302	C₂×C₄	#14581823	1	1	yes	characteristic
303	C₈	—	4	1	no	—
304	C₈	C₄⋊C₄×D₈₃	1	1	yes	characteristic
305	C₈	#14581754	1	1	yes	characteristic
306	C₈	#14581770	1	1	yes	characteristic
307	C₈	#14581777	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×#13050575	1	1	yes	characteristic
316	C₂²	#19127834	1	1	yes	characteristic
317	C₂²	#19127978	1	1	yes	characteristic
318	C₂²	C₂×#14581733	1	1	yes	characteristic
319	C₂²	#19128131	1	1	yes	characteristic
320	C₂²	#19128136	1	1	yes	characteristic
321	C₂²	#19128142	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	4	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	16	1	no	—
326	C₄	—	664	2	no	—
327	C₄	#19127572	1	1	yes	characteristic
328	C₄	#19127799	1	1	yes	characteristic
329	C₄	#19127861	1	1	yes	characteristic
330	C₄	#19127998	1	1	yes	characteristic
331	C₂	—	664	4	no	—
332	C₂	#24907709	1	1	yes	characteristic
333	C₂	C₄×#14581733	1	1	yes	characteristic
334	C₂	#24909325	1	1	yes	characteristic
335	C₂	#24909374	1	1	yes	characteristic
336	C₂	#24909401	1	1	yes	characteristic
337	C₂	#24909597	1	1	yes	characteristic
338	C₂	#24909617	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13050575
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1942	C₂
socle	C₂³×C₈₃	C₂×D₈
prime radical	C₂²×D₆₆₄	C₂²
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1942	D₈₃
hypocenter	C₈₃	#12935
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1942

Complements

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

N	H ≅ G/N	type
2. C₂×#19127526	C₂	semidirect
5. #24905554	C₂	semidirect
6. #24905541	C₂	semidirect
7. C₈₃×#1942	C₂	semidirect
18. #19127501	C₄	semidirect
41. C₂×C₈₃⋊C₁₆	C₂×C₄	semidirect
54. #107×C₈₃	C₂×C₄	semidirect
72. D₆₆₄	C₄⋊C₄	semidirect
174. #1942	D₈₃	semidirect
177. C₈₃	#12935	semidirect
223. #107	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×#19127526	2
C₂×#19127501	2
#24905554	2
#24905541	2
C₈₃×#1942	2
#12935	83