Subgroups — #35562128

Order 21248.

508 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₆₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#24909699	C₂	1	2	yes	maximal, complemented
3	#24909690	C₂	1	1	yes	maximal, characteristic, complemented
4	#24909615	C₂	1	2	yes	maximal, complemented
5	#24909600	C₂	1	1	yes	maximal, characteristic, complemented
6	#24909515	C₂	1	2	yes	maximal
7	#24909505	C₂	1	1	yes	maximal, characteristic, complemented
8	#24909372	C₂	1	1	yes	maximal, characteristic, complemented
9	#24909325	C₂	1	2	yes	maximal, complemented
10	C₄×#14581733	C₂	1	1	yes	maximal, characteristic
11	C₈₃×#3393	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×D₃₃₂	—	2	1	no	—
13	C₂×#14581735	C₂²	1	2	yes	—
14	C₂×#14581733	C₂²	1	1	yes	characteristic
15	#19128106	C₂²	1	1	yes	characteristic
16	#19128106	C₂²	1	2	yes	—
17	#19127982	—	2	2	no	—
18	#19127982	—	2	1	no	—
19	C₄×D₆₆₄	C₂²	1	2	yes	—
20	C₄×D₆₆₄	C₂²	1	1	yes	characteristic
21	#19127526	C₂²	1	2	yes	—
22	#19127526	C₂²	1	1	yes	characteristic
23	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	1	yes	characteristic
24	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	2	yes	—
25	#19127505	C₂²	1	2	yes	—
26	#19127505	C₂²	1	2	yes	—
27	#19127505	C₂²	1	2	yes	—
28	#19127502	C₂²	1	2	yes	—
29	#19127502	C₂²	1	1	yes	characteristic
30	#19127501	C₂²	1	1	yes	characteristic
31	#19127477	C₂²	1	2	yes	—
32	C₄×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
33	#556×C₈₃	—	2	1	no	—
34	#495×C₈₃	C₂²	1	2	yes	—
35	#494×C₈₃	C₂²	1	1	yes	characteristic
36	#464×C₈₃	—	2	2	no	—
37	#461×C₈₃	—	2	1	no	—
38	C₈×Q₈×C₈₃	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₈₃	C₂²	1	2	yes	—
40	C₄×D₈×C₈₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₈₃	—	4	1	no	—
42	C₂×D₆₆₄	C₂³	1	1	yes	characteristic
43	#14581792	—	2	1	no	—
44	#14581792	—	2	2	no	—
45	#14581772	—	4	1	no	—
46	#14581772	—	4	2	no	—
47	C₄×D₃₃₂	—	2	1	no	—
48	C₄×D₃₃₂	—	2	2	no	—
49	#14581735	—	2	4	no	—
50	#14581733	—	2	2	no	—
51	C₂×C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₈₃⋊C₁₆	C₂³	1	2	yes	—
53	C₂×C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
54	#14581726	—	2	1	no	—
55	#14581726	—	2	2	no	—
56	#14581726	C₂³	1	2	yes	—
57	#14581726	C₂³	1	1	yes	characteristic
58	#14581714	—	2	1	no	—
59	#14581714	—	2	2	no	—
60	C₂×Q₁₆×C₈₃	—	2	1	no	—
61	C₂×Q₁₆×C₈₃	C₂³	1	2	yes	—
62	C₂×SD₁₆×C₈₃	—	2	2	no	—
63	C₂×D₈×C₈₃	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₈₃	—	2	1	no	—
65	#126×C₈₃	—	4	2	no	—
66	#124×C₈₃	—	2	2	no	—
67	C₂²⋊Q₈×C₈₃	—	4	1	no	—
68	C₄×Q₈×C₈₃	—	2	2	no	—
69	C₄×Q₈×C₈₃	D₄	1	1	yes	characteristic
70	C₄×D₄×C₈₃	—	2	1	no	—
71	#117×C₈₃	—	4	1	no	—
72	#107×C₈₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₈₃	D₄	1	2	yes	—
74	C₄⋊C₈×C₈₃	D₄	1	1	yes	characteristic
75	#103×C₈₃	—	2	2	no	—
76	#102×C₈₃	—	2	1	no	—
77	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₃	C₂³	1	2	yes	—
79	C₂×Dic₁₆₆	—	4	2	no	—
80	C₂×Dic₁₆₆	—	4	1	no	—
81	Q₈×D₈₃	—	8	4	no	—
82	C₂×D₃₃₂	—	2	1	no	—
83	C₂×C₄×D₈₃	—	4	2	no	—
84	C₂×C₄×D₈₃	—	4	1	no	—
85	D₆₆₄	—	2	1	no	—
86	D₆₆₄	#41	1	2	yes	—
87	C₄⋊Dic₈₃	—	2	2	no	—
88	C₄⋊Dic₈₃	—	2	1	no	—
89	C₄⋊Dic₈₃	—	4	2	no	—
90	C₄⋊Dic₈₃	—	4	1	no	—
91	(C₂×D₈₃)⋊C₄	—	4	2	no	—
92	(C₂×D₈₃)⋊C₄	—	4	1	no	—
93	C₈₃⋊C₁₆	—	2	1	no	—
94	C₈₃⋊C₁₆	—	2	2	no	—
95	C₈₃⋊C₁₆	—	2	1	no	—
96	C₂×Q₈×C₈₃	—	2	2	no	—
97	C₂×Q₈×C₈₃	D₈	1	1	yes	characteristic
98	C₂×D₄×C₈₃	—	2	1	no	—
99	C₂²×C₄×C₈₃	—	4	1	no	—
100	Q₁₆×C₈₃	—	2	2	no	—
101	Q₁₆×C₈₃	—	2	2	no	—
102	Q₁₆×C₈₃	—	8	1	no	—
103	SD₁₆×C₈₃	—	8	2	no	—
104	D₈×C₈₃	—	2	1	no	—
105	D₈×C₈₃	—	2	1	no	—
106	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₈₃	—	2	1	no	—
110	C₄⋊C₄×C₈₃	—	4	1	no	—
111	C₄⋊C₄×C₈₃	—	4	2	no	—
112	C₄⋊C₄×C₈₃	—	4	1	no	—
113	C₄⋊C₄×C₈₃	D₈	1	1	yes	characteristic
114	C₄⋊C₄×C₈₃	D₈	1	2	yes	—
115	C₂²⋊C₄×C₈₃	—	4	2	no	—
116	C₂²⋊C₄×C₈₃	—	4	1	no	—
117	C₂²⋊C₄×C₈₃	—	4	2	no	—
118	C₄²×C₈₃	—	4	2	no	—
119	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₈₃	C₂×D₄	1	2	yes	—
121	C₂³×C₈₃	—	4	1	no	—
122	C₂²×D₈₃	—	4	1	no	—
123	Q₈×C₈₃	—	2	2	no	—
124	Q₈×C₈₃	—	4	2	no	—
125	Q₈×C₈₃	—	4	2	no	—
126	D₄×C₈₃	—	4	1	no	—
127	D₄×C₈₃	—	4	1	no	—
128	C₂×C₄×C₈₃	—	4	1	no	—
129	C₂×C₄×C₈₃	—	4	2	no	—
130	C₂×C₄×C₈₃	—	8	2	no	—
131	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₈₃	—	4	1	no	—
137	C₂×Dic₈₃	—	4	2	no	—
138	D₃₃₂	—	2	2	no	—
139	D₃₃₂	—	4	1	no	—
140	C₄×D₈₃	—	8	4	no	—
141	C₄×D₈₃	—	8	2	no	—
142	Dic₁₆₆	—	8	4	no	—
143	Dic₁₆₆	—	8	2	no	—
144	C₈×C₈₃	—	2	2	no	—
145	C₈×C₈₃	—	2	1	no	—
146	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
147	C₈×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂²×C₈₃	—	8	1	no	—
149	C₂²×C₈₃	—	8	1	no	—
150	C₂²×C₈₃	—	8	1	no	—
151	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
152	C₂×D₈₃	—	4	2	no	—
153	C₂×D₈₃	—	8	1	no	—
154	C₂×D₈₃	—	8	1	no	—
155	C₄×C₈₃	—	2	2	no	—
156	C₄×C₈₃	—	2	2	no	—
157	C₄×C₈₃	—	2	1	no	—
158	C₄×C₈₃	—	8	2	no	—
159	C₄×C₈₃	—	8	1	no	—
160	C₄×C₈₃	#509	1	2	yes	—
161	C₄×C₈₃	#548	1	1	yes	characteristic
162	C₄×C₈₃	#577	1	1	yes	characteristic
163	Dic₈₃	—	8	1	no	—
164	Dic₈₃	—	8	2	no	—
165	#33934	—	83	1	no	maximal, Sylow
166	C₂×C₈₃	—	16	1	no	—
167	C₂×C₈₃	#3358	1	1	yes	characteristic
168	C₂×C₈₃	C₂×#509	1	1	yes	characteristic
169	C₂×C₈₃	#3417	1	1	yes	characteristic
170	D₈₃	—	8	2	no	—
171	#3393	D₈₃	1	1	yes	characteristic, complemented
172	Q₈×D₈	—	83	1	no	—
173	#2243	—	83	2	no	—
174	#2241	—	83	1	no	—
175	#2227	—	83	2	no	—
176	#2224	—	83	1	no	—
177	#2210	—	83	2	no	—
178	#2208	—	83	1	no	—
179	#2183	—	83	1	no	—
180	#2175	—	83	2	no	—
181	C₄×D₁₆	—	83	1	no	—
182	C₈₃	#33934	1	1	yes	characteristic, complemented, Sylow
183	#556	—	2	1	no	—
184	D₄×Q₈	—	166	1	no	—
185	C₂×SD₃₂	—	83	2	no	—
186	C₂×D₁₆	—	83	1	no	—
187	#500	—	83	1	no	—
188	#500	—	83	2	no	—
189	#495	C₂×D₈₃	1	2	yes	—
190	#494	C₂×D₈₃	1	1	yes	characteristic
191	#474	—	166	1	no	—
192	#474	—	166	2	no	—
193	#464	—	2	2	no	—
194	#461	—	2	1	no	—
195	C₈×Q₈	C₂×D₈₃	1	1	yes	characteristic
196	C₄×Q₁₆	C₂×D₈₃	1	2	yes	—
197	C₄×D₈	—	83	1	no	—
198	C₄×D₈	—	83	2	no	—
199	C₄×D₈	C₂×D₈₃	1	1	yes	characteristic
200	#366	—	83	1	no	—
201	#366	—	83	2	no	—
202	C₄⋊C₁₆	—	83	1	no	—
203	C₄⋊C₁₆	—	83	2	no	—
204	#358	—	83	2	no	—
205	#358	—	83	2	no	—
206	#358	—	83	2	no	—
207	#357	—	83	1	no	—
208	#357	—	83	2	no	—
209	#357	—	83	1	no	—
210	#346	—	83	2	no	—
211	C₄×C₁₆	—	83	1	no	—
212	C₂²×Q₈	—	332	1	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	C₂²×D₈₃	1	2	yes	—
215	C₂×SD₁₆	—	2	2	no	—
216	C₂×D₈	—	83	1	no	—
217	C₂×D₈	C₂²×D₈₃	1	1	yes	characteristic
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	166	1	no	—
220	C₄⋊Q₈	—	166	2	no	—
221	#126	—	4	2	no	—
222	#124	—	2	2	no	—
223	C₂²⋊Q₈	—	4	1	no	—
224	C₂²⋊Q₈	—	332	1	no	—
225	C₂²⋊Q₈	—	332	2	no	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×Q₈	C₂²⋊D₈₃	1	1	yes	characteristic
228	C₄×D₄	—	2	1	no	—
229	C₄×D₄	—	166	1	no	—
230	C₄×D₄	—	166	2	no	—
231	#117	—	4	1	no	—
232	SD₃₂	—	166	4	no	—
233	D₁₆	—	166	2	no	—
234	C₂×C₁₆	—	83	1	no	—
235	C₂×C₁₆	—	83	1	no	—
236	C₂×C₁₆	—	83	2	no	—
237	#107	—	83	1	no	—
238	#107	—	83	2	no	—
239	#107	—	166	1	no	—
240	#107	—	166	2	no	—
241	#107	C₂²×D₈₃	1	1	yes	characteristic
242	C₄⋊C₈	C₂²⋊D₈₃	1	1	yes	characteristic
243	C₄⋊C₈	C₂²⋊D₈₃	1	2	yes	—
244	#103	—	2	2	no	—
245	#102	—	2	1	no	—
246	#102	—	166	1	no	—
247	#102	—	166	2	no	—
248	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₈₃	1	2	yes	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	332	2	no	—
252	C₂×Q₈	—	332	1	no	—
253	C₂×Q₈	—	664	4	no	—
254	C₂×Q₈	#13050575	1	1	yes	characteristic
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	166	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	332	2	no	—
259	C₂²×C₄	—	332	1	no	—
260	Q₁₆	—	2	2	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	8	1	no	—
263	SD₁₆	—	8	2	no	—
264	D₈	—	2	1	no	—
265	D₈	—	2	1	no	—
266	D₈	—	83	2	no	—
267	D₈	—	166	1	no	—
268	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
269	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
270	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	332	2	no	—
276	C₄⋊C₄	—	332	1	no	—
277	C₄⋊C₄	—	166	1	no	—
278	C₄⋊C₄	—	166	2	no	—
279	C₄⋊C₄	#13050575	1	1	yes	characteristic
280	C₄⋊C₄	#13050575	1	2	yes	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	332	2	no	—
285	C₂²⋊C₄	—	332	1	no	—
286	C₄²	—	4	2	no	—
287	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₈₃	1	2	yes	—
289	C₁₆	—	166	1	no	—
290	C₁₆	—	166	2	no	—
291	C₁₆	—	166	1	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	332	1	no	—
294	Q₈	—	2	2	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	4	2	no	—
297	Q₈	—	664	4	no	—
298	Q₈	—	664	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	332	1	no	—
302	D₄	—	166	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	332	2	no	—
307	C₂×C₄	—	332	1	no	—
308	C₂×C₄	—	664	4	no	—
309	C₂×C₄	—	664	2	no	—
310	C₂×C₄	C₂×#13050575	1	2	yes	—
311	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
312	C₂×C₄	C₂×#13050575	1	2	yes	—
313	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	2	1	no	—
317	C₈	#41×D₈₃	1	1	yes	characteristic
318	C₈	#14581839	1	1	yes	characteristic
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	332	2	no	—
323	C₂²	—	664	1	no	—
324	C₂²	—	664	1	no	—
325	C₂²	C₂²×#13050575	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	2	no	—
331	C₄	—	664	2	no	—
332	C₄	—	664	1	no	—
333	C₄	#19128136	1	2	yes	—
334	C₄	#19128358	1	1	yes	characteristic
335	C₄	#19128512	1	1	yes	characteristic
336	C₂	—	16	1	no	—
337	C₂	—	664	2	no	—
338	C₂	#24920094	1	1	yes	characteristic
339	C₂	C₂×#19128136	1	1	yes	characteristic
340	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×#3393	C₂
socle	C₂²×C₈₃	C₂²×D₈
prime radical	C₂×#14581733	C₂²
hypercenter	#3393	D₈₃
hypocenter	C₈₃	#33934
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3393

Complements

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

N	H ≅ G/N	type
1. Q₈×D₆₆₄	C₂	semidirect
2. #24909699	C₂	semidirect
3. #24909690	C₂	semidirect
4. #24909615	C₂	semidirect
5. #24909600	C₂	semidirect
7. #24909505	C₂	semidirect
8. #24909372	C₂	semidirect
9. #24909325	C₂	semidirect
11. C₈₃×#3393	C₂	semidirect
171. #3393	D₈₃	semidirect
182. C₈₃	#33934	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₆₆₄	2
#24909699	2
#24909690	2
#24909615	2
#24909600	2
#24909515	2
#24909505	2
#24909372	2
#24909325	2
C₄×#14581733	2
C₈₃×#3393	2
#33934	83