Subgroups — #35559332

Order 21248.

610 conjugacy classes of subgroups.

The subgroup lattice has 336 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	#24909658	C₂	1	2	yes	maximal, complemented
3	#24909607	C₂	1	2	yes	maximal, complemented
4	#24909505	C₂	1	2	yes	maximal, complemented
5	#24909391	C₂	1	2	yes	maximal
6	C₄×#14581735	C₂	1	4	yes	maximal, complemented
7	#24909262	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₆₆₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₃₃₂	—	2	1	no	—
11	C₂×#14581735	C₂²	1	4	yes	—
12	C₄×D₆₆₄	—	2	1	no	—
13	C₄×D₆₆₄	—	2	1	no	—
14	C₄×D₆₆₄	C₂²	1	2	yes	—
15	C₄×D₆₆₄	C₂²	1	2	yes	—
16	C₂×#14581726	C₂²	1	1	yes	characteristic
17	C₂×#14581714	—	2	1	no	—
18	#19127528	C₂²	1	4	yes	complemented
19	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	2	yes	complemented
20	#19127505	C₂²	1	4	yes	complemented
21	#19127501	C₂²	1	4	yes	—
22	C₂²⋊(C₈₃⋊C₁₆)	C₂²	1	2	yes	—
23	C₄×C₈₃⋊C₁₆	C₂²	1	2	yes	complemented
24	#533×C₈₃	—	2	1	no	—
25	#500×C₈₃	C₂²	1	1	yes	characteristic, complemented
26	#494×C₈₃	C₂²	1	1	yes	characteristic, complemented
27	#487×C₈₃	—	2	1	no	—
28	#467×C₈₃	—	2	1	no	—
29	#467×C₈₃	C₂²	1	2	yes	—
30	C₄×Q₁₆×C₈₃	—	2	1	no	—
31	C₄×Q₁₆×C₈₃	C₂²	1	2	yes	complemented
32	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
33	C₂²×D₃₃₂	—	2	1	no	—
34	C₂²×C₄×D₈₃	—	4	1	no	—
35	C₂×D₆₆₄	—	2	4	no	—
36	C₂×D₆₆₄	—	2	1	no	—
37	C₂×D₆₆₄	C₂³	1	2	yes	—
38	C₄×D₃₃₂	—	2	2	no	—
39	C₄×D₃₃₂	—	2	2	no	—
40	C₄×D₃₃₂	—	4	1	no	—
41	C₄×D₃₃₂	—	4	1	no	—
42	C₂×C₄⋊Dic₈₃	—	2	1	no	—
43	C₂×(C₂×D₈₃)⋊C₄	—	4	1	no	—
44	#14581735	—	2	8	no	—
45	C₂×C₈₃⋊C₁₆	C₂³	1	4	yes	—
46	#14581726	—	2	1	no	—
47	#14581726	C₂³	1	2	yes	—
48	#14581714	—	2	2	no	—
49	#14581714	—	2	2	no	—
50	C₂×Q₁₆×C₈₃	—	2	1	no	—
51	C₂×Q₁₆×C₈₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
54	C₄⋊Q₈×C₈₃	—	2	1	no	—
55	C₄⋊Q₈×C₈₃	—	2	1	no	—
56	#125×C₈₃	—	4	1	no	—
57	C₂²⋊Q₈×C₈₃	—	2	2	no	—
58	C₂²⋊Q₈×C₈₃	—	4	1	no	—
59	C₄×Q₈×C₈₃	—	2	2	no	—
60	C₄×Q₈×C₈₃	—	4	1	no	—
61	#117×C₈₃	—	4	1	no	—
62	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
63	#107×C₈₃	—	2	1	no	—
64	#107×C₈₃	C₂³	1	2	yes	—
65	#103×C₈₃	—	2	2	no	—
66	#103×C₈₃	—	2	2	no	—
67	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
68	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
71	C₂³×D₈₃	—	4	1	no	—
72	C₂×D₃₃₂	—	2	4	no	—
73	C₂×D₃₃₂	—	2	2	no	—
74	C₂×D₃₃₂	—	4	2	no	—
75	C₂×D₃₃₂	—	4	1	no	—
76	C₂²×Dic₈₃	—	4	1	no	—
77	C₂×C₄×D₈₃	—	4	2	no	—
78	C₂×C₄×D₈₃	—	4	2	no	—
79	C₂×C₄×D₈₃	—	8	4	no	—
80	D₆₆₄	—	2	4	no	—
81	D₆₆₄	—	2	2	no	—
82	D₆₆₄	#41	1	4	yes	—
83	C₄⋊Dic₈₃	—	2	2	no	—
84	C₄⋊Dic₈₃	—	4	1	no	—
85	(C₂×D₈₃)⋊C₄	—	4	2	no	—
86	(C₂×D₈₃)⋊C₄	—	4	2	no	—
87	C₈₃⋊C₁₆	—	2	4	no	—
88	C₂×Q₈×C₈₃	—	2	2	no	—
89	C₂×Q₈×C₈₃	—	4	1	no	—
90	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
91	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
93	Q₁₆×C₈₃	—	2	4	no	—
94	Q₁₆×C₈₃	—	4	2	no	—
95	C₂×C₈×C₈₃	—	2	2	no	—
96	C₂×C₈×C₈₃	—	4	1	no	—
97	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₈₃	—	2	2	no	—
102	C₄⋊C₄×C₈₃	—	4	1	no	—
103	C₄⋊C₄×C₈₃	—	4	1	no	—
104	C₄⋊C₄×C₈₃	—	4	2	no	—
105	C₄⋊C₄×C₈₃	—	4	2	no	—
106	C₄⋊C₄×C₈₃	—	4	1	no	—
107	C₂²⋊C₄×C₈₃	—	4	2	no	—
108	C₄²×C₈₃	—	4	2	no	—
109	C₄²×C₈₃	D₈	1	1	yes	characteristic
110	C₄²×C₈₃	D₈	1	1	yes	characteristic
111	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
113	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
114	C₂²×D₈₃	—	4	4	no	—
115	C₂²×D₈₃	—	4	2	no	—
116	C₂²×D₈₃	—	8	2	no	—
117	C₂²×D₈₃	—	8	2	no	—
118	Q₈×C₈₃	—	4	2	no	—
119	Q₈×C₈₃	—	4	2	no	—
120	Q₈×C₈₃	—	8	2	no	—
121	C₂×C₄×C₈₃	—	2	1	no	—
122	C₂×C₄×C₈₃	—	2	1	no	—
123	C₂×C₄×C₈₃	—	4	1	no	—
124	C₂×C₄×C₈₃	—	4	2	no	—
125	C₂×C₄×C₈₃	—	4	2	no	—
126	C₂×C₄×C₈₃	SD₃₂	1	4	yes	complemented
127	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₈₃	—	4	2	no	—
134	C₂×Dic₈₃	—	8	2	no	—
135	D₃₃₂	—	2	4	no	—
136	D₃₃₂	—	4	4	no	—
137	D₃₃₂	—	4	2	no	—
138	C₄×D₈₃	—	8	4	no	—
139	C₄×D₈₃	—	8	4	no	—
140	C₈×C₈₃	—	2	1	no	—
141	C₈×C₈₃	—	2	1	no	—
142	C₈×C₈₃	—	2	1	no	—
143	C₈×C₈₃	C₂×#41	1	2	yes	—
144	C₂²×C₈₃	—	2	1	no	—
145	C₂²×C₈₃	—	2	1	no	—
146	C₂²×C₈₃	C₂×SD₃₂	1	2	yes	—
147	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₈₃	—	4	4	no	—
149	C₂×D₈₃	—	8	8	no	—
150	C₂×D₈₃	—	8	2	no	—
151	C₂×D₈₃	—	8	2	no	—
152	C₄×C₈₃	—	2	1	no	—
153	C₄×C₈₃	—	2	1	no	—
154	C₄×C₈₃	—	2	1	no	—
155	C₄×C₈₃	—	8	2	no	—
156	C₄×C₈₃	—	8	2	no	—
157	C₄×C₈₃	C₂×SD₃₂	1	2	yes	—
158	C₄×C₈₃	C₂×SD₃₂	1	2	yes	—
159	C₄×C₈₃	#525	1	1	yes	characteristic
160	C₄×C₈₃	#575	1	1	yes	characteristic
161	Dic₈₃	—	8	2	no	—
162	#33661	—	83	1	no	maximal, Sylow
163	C₂×C₈₃	—	2	2	no	—
164	C₂×C₈₃	#3100	1	1	yes	characteristic
165	C₂×C₈₃	C₂²×SD₃₂	1	1	yes	characteristic
166	C₂×C₈₃	#3414	1	1	yes	characteristic
167	D₈₃	—	8	4	no	—
168	#3170	D₈₃	1	1	yes	characteristic, complemented
169	C₂×C₄×D₈	—	83	1	no	—
170	#2235	—	83	2	no	—
171	#2226	—	83	2	no	—
172	#2208	—	83	2	no	—
173	#2187	—	83	2	no	—
174	C₄×SD₃₂	—	83	4	no	—
175	#2162	—	83	1	no	—
176	C₈₃	#33661	1	1	yes	characteristic, complemented, Sylow
177	C₂²×D₈	—	83	1	no	—
178	#533	—	2	1	no	—
179	C₂×C₄×D₄	—	166	1	no	—
180	C₂×SD₃₂	—	83	4	no	—
181	#500	C₂×D₈₃	1	1	yes	characteristic, complemented
182	#494	C₂×D₈₃	1	1	yes	characteristic, complemented
183	#487	—	2	1	no	—
184	#467	—	2	1	no	—
185	#467	C₂×D₈₃	1	2	yes	—
186	C₄×Q₁₆	—	2	1	no	—
187	C₄×Q₁₆	C₂×D₈₃	1	2	yes	complemented
188	C₄×D₈	—	83	2	no	—
189	C₄×D₈	—	83	2	no	—
190	C₄×D₈	—	166	1	no	—
191	C₄×D₈	—	166	1	no	—
192	C₂×#107	—	83	1	no	—
193	C₂×#102	—	166	1	no	—
194	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
195	#367	—	83	4	no	—
196	C₄⋊C₁₆	—	83	2	no	—
197	#358	—	83	4	no	—
198	#357	—	83	4	no	—
199	C₂²⋊C₁₆	—	83	2	no	—
200	C₄×C₁₆	—	83	2	no	—
201	C₂²×D₄	—	166	1	no	—
202	C₂³×C₄	—	332	1	no	—
203	C₂×Q₁₆	—	2	1	no	—
204	C₂×Q₁₆	C₂²×D₈₃	1	2	yes	—
205	C₂×D₈	—	83	2	no	—
206	C₂×D₈	—	166	4	no	—
207	C₂×D₈	—	166	1	no	—
208	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	#125	—	4	1	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	C₂²⋊Q₈	—	4	1	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×Q₈	—	4	1	no	—
217	C₄×D₄	—	332	1	no	—
218	C₄×D₄	—	332	1	no	—
219	C₄×D₄	—	166	2	no	—
220	C₄×D₄	—	166	2	no	—
221	#117	—	4	1	no	—
222	C₂×C₄⋊C₄	—	166	1	no	—
223	C₂×C₂²⋊C₄	—	332	1	no	—
224	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
225	SD₃₂	—	166	8	no	—
226	C₂×C₁₆	—	83	4	no	—
227	#107	—	2	1	no	—
228	#107	—	83	2	no	—
229	#107	—	166	1	no	—
230	#107	C₂²×D₈₃	1	2	yes	—
231	#103	—	2	2	no	—
232	#103	—	2	2	no	—
233	#102	—	166	2	no	—
234	#102	—	166	2	no	—
235	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
236	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
239	C₂⁴	—	332	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×D₄	—	332	2	no	—
243	C₂×D₄	—	332	1	no	—
244	C₂×D₄	—	166	4	no	—
245	C₂×D₄	—	166	2	no	—
246	C₂²×C₄	—	332	1	no	—
247	C₂²×C₄	—	332	2	no	—
248	C₂²×C₄	—	332	2	no	—
249	C₂²×C₄	—	664	4	no	—
250	C₂²×C₄	#13050575	1	1	yes	characteristic
251	C₂²×C₄	#13050575	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
253	Q₁₆	—	2	4	no	—
254	Q₁₆	—	4	2	no	—
255	D₈	—	83	4	no	—
256	D₈	—	166	4	no	—
257	D₈	—	166	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	332	1	no	—
271	C₄⋊C₄	—	166	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	332	2	no	—
274	C₂²⋊C₄	—	332	2	no	—
275	C₄²	—	4	2	no	—
276	C₄²	#13050575	1	1	yes	characteristic
277	C₄²	#13050575	1	1	yes	characteristic
278	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
279	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
280	C₁₆	—	166	4	no	—
281	C₂³	—	332	4	no	—
282	C₂³	—	332	2	no	—
283	C₂³	—	664	2	no	—
284	C₂³	—	664	2	no	—
285	C₂³	C₂×#13050575	1	1	yes	characteristic
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	8	2	no	—
289	D₄	—	332	4	no	—
290	D₄	—	332	2	no	—
291	D₄	—	166	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	332	2	no	—
298	C₂×C₄	—	664	2	no	—
299	C₂×C₄	—	664	4	no	—
300	C₂×C₄	—	664	4	no	—
301	C₂×C₄	#14581735	1	4	yes	complemented
302	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
303	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
304	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
305	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
306	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	#41×D₈₃	1	2	yes	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	332	4	no	—
315	C₂²	—	664	8	no	—
316	C₂²	—	664	2	no	—
317	C₂²	—	664	2	no	—
318	C₂²	C₂×#14581735	1	2	yes	—
319	C₂²	C₂²×#13050575	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	664	2	no	—
326	C₄	C₂×#14581735	1	2	yes	—
327	C₄	C₂×#14581735	1	2	yes	—
328	C₄	#19128208	1	1	yes	characteristic
329	C₄	#19128499	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	664	4	no	—
332	C₂	#24917459	1	1	yes	characteristic
333	C₂	C₂²×#14581735	1	1	yes	characteristic
334	C₂	#24920597	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#13050575
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#3170	C₂
socle	C₂²×C₈₃	C₂²×D₈
prime radical	C₂²×D₆₆₄	C₂²
hypercenter	#3170	D₈₃
hypocenter	C₈₃	#33661
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #24909658	C₂	semidirect
3. #24909607	C₂	semidirect
4. #24909505	C₂	semidirect
6. C₄×#14581735	C₂	semidirect
7. #24909262	C₂	semidirect
8. C₈₃×#3170	C₂	semidirect
18. #19127528	C₂²	semidirect
19. C₄⋊(C₈₃⋊C₁₆)	C₂²	semidirect
20. #19127505	C₂²	semidirect
23. C₄×C₈₃⋊C₁₆	C₂²	semidirect
25. #500×C₈₃	C₂²	semidirect
26. #494×C₈₃	C₂²	semidirect
31. C₄×Q₁₆×C₈₃	C₂²	semidirect
126. C₂×C₄×C₈₃	SD₃₂	semidirect
168. #3170	D₈₃	semidirect
176. C₈₃	#33661	semidirect
181. #500	C₂×D₈₃	semidirect
182. #494	C₂×D₈₃	semidirect
187. C₄×Q₁₆	C₂×D₈₃	semidirect
301. C₂×C₄	#14581735	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
#24909658	2
#24909607	2
#24909505	2
#24909391	2
C₄×#14581735	2
#24909262	2
C₈₃×#3170	2
#33661	83