Subgroups — #35559093

Order 21248.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24918002	C₂	1	1	yes	maximal, characteristic, complemented
2	#24909620	C₂	1	2	yes	maximal, complemented
3	#24909620	C₂	1	2	yes	maximal, complemented
4	#24909610	C₂	1	2	yes	maximal, complemented
5	#24909600	C₂	1	2	yes	maximal, complemented
6	#24909252	C₂	1	1	yes	maximal, characteristic, complemented
7	#24909245	C₂	1	1	yes	maximal, characteristic, complemented
8	#24909222	C₂	1	1	yes	maximal, characteristic, complemented
9	#24909179	C₂	1	1	yes	maximal, characteristic
10	C₂×#19127502	C₂	1	1	yes	maximal, characteristic
11	C₈₃×#2948	C₂	1	1	yes	maximal, characteristic
12	#19128329	—	2	1	no	—
13	C₂×#14581732	C₂²	1	1	yes	characteristic
14	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	#19128110	—	2	2	no	—
16	#19128106	C₂²	1	2	yes	complemented
17	#19128103	C₂²	1	2	yes	complemented
18	#19127743	—	2	1	no	—
19	#19127739	C₂²	1	1	yes	characteristic
20	C₂×#14581726	C₂²	1	1	yes	characteristic
21	#19127528	C₂²	1	1	yes	characteristic, complemented
22	#19127528	C₂²	1	2	yes	complemented
23	#19127528	C₂²	1	1	yes	characteristic, complemented
24	#19127526	C₂²	1	1	yes	characteristic, complemented
25	#19127526	C₂²	1	2	yes	complemented
26	#19127526	C₂²	1	1	yes	characteristic, complemented
27	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	2	yes	complemented
28	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	2	yes	complemented
29	#19127502	C₂²	1	2	yes	—
30	#19127502	C₂²	1	2	yes	—
31	#19127502	C₂²	1	2	yes	—
32	#19127502	C₂²	1	2	yes	—
33	C₂²×D₈×C₈₃	C₂²	1	1	yes	characteristic
34	#518×C₈₃	—	2	1	no	—
35	#442×C₈₃	—	2	2	no	—
36	C₄×D₈×C₈₃	C₂²	1	2	yes	—
37	C₄×D₈×C₈₃	C₂²	1	2	yes	—
38	#427×C₈₃	C₂²	1	1	yes	characteristic
39	#418×C₈₃	—	2	1	no	—
40	#405×C₈₃	C₂²	1	1	yes	characteristic
41	#14581792	—	2	2	no	—
42	#14581791	—	4	1	no	—
43	#14581784	—	2	2	no	—
44	#14581783	—	4	1	no	—
45	#14581769	—	4	2	no	—
46	#14581751	—	2	1	no	—
47	C₂×C₄⋊Dic₈₃	—	2	1	no	—
48	#14581732	—	2	2	no	—
49	C₂×C₈₃⋊C₁₆	—	2	2	no	—
50	C₂×C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₈₃⋊C₁₆	C₂³	1	2	yes	—
53	#14581726	—	2	2	no	—
54	#14581726	—	2	1	no	—
55	#14581726	C₂³	1	1	yes	characteristic
56	#14581726	C₂³	1	2	yes	—
57	#14581726	C₂³	1	1	yes	characteristic
58	#14581724	—	2	2	no	—
59	#14581724	—	2	1	no	—
60	C₂²×D₄×C₈₃	—	2	1	no	—
61	C₂×D₈×C₈₃	—	2	1	no	—
62	C₂×D₈×C₈₃	—	2	2	no	—
63	C₂×D₈×C₈₃	Q₈	1	4	yes	complemented
64	C₂×D₈×C₈₃	C₂³	1	2	yes	—
65	C₂×M₁₆×C₈₃	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₈₃	—	2	2	no	—
68	C₄×D₄×C₈₃	—	2	2	no	—
69	C₄×D₄×C₈₃	—	4	1	no	—
70	C₄×D₄×C₈₃	—	4	1	no	—
71	#117×C₈₃	—	2	1	no	—
72	#117×C₈₃	D₄	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₈₃	—	4	2	no	—
74	#107×C₈₃	C₂³	1	1	yes	characteristic
75	#107×C₈₃	C₂³	1	1	yes	characteristic
76	#106×C₈₃	—	2	1	no	—
77	#102×C₈₃	—	2	2	no	—
78	#102×C₈₃	—	2	2	no	—
79	#97×C₈₃	D₄	1	2	yes	—
80	C₄×C₈×C₈₃	C₂³	1	2	yes	—
81	C₂×Dic₁₆₆	—	4	2	no	—
82	C₂²×Dic₈₃	—	4	1	no	—
83	C₄⋊Dic₈₃	—	2	1	no	—
84	C₄⋊Dic₈₃	—	2	2	no	—
85	C₄⋊Dic₈₃	—	2	1	no	—
86	C₄⋊Dic₈₃	—	4	2	no	—
87	C₄⋊Dic₈₃	—	4	1	no	—
88	Dic₈₃⋊C₄	—	4	2	no	—
89	Dic₈₃⋊C₄	—	4	2	no	—
90	C₂²⋊Dic₈₃	—	4	1	no	—
91	C₄×Dic₈₃	—	4	1	no	—
92	C₈₃⋊C₁₆	—	2	1	no	—
93	C₈₃⋊C₁₆	—	2	1	no	—
94	C₈₃⋊C₁₆	—	2	2	no	—
95	C₂⁴×C₈₃	—	4	1	no	—
96	C₂×D₄×C₈₃	—	2	4	no	—
97	C₂×D₄×C₈₃	—	2	2	no	—
98	C₂×D₄×C₈₃	—	4	1	no	—
99	C₂×D₄×C₈₃	—	4	2	no	—
100	C₂²×C₄×C₈₃	—	4	2	no	—
101	C₂²×C₄×C₈₃	—	4	2	no	—
102	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
103	D₈×C₈₃	—	2	4	no	—
104	D₈×C₈₃	—	2	2	no	—
105	D₈×C₈₃	C₂×Q₈	1	4	yes	—
106	M₁₆×C₈₃	—	2	2	no	—
107	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
110	C₂×C₈×C₈₃	#41	1	2	yes	—
111	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₈₃	—	2	1	no	—
113	C₄⋊C₄×C₈₃	—	2	1	no	—
114	C₄⋊C₄×C₈₃	D₈	1	1	yes	characteristic
115	C₄⋊C₄×C₈₃	D₈	1	1	yes	characteristic
116	C₂²⋊C₄×C₈₃	—	4	1	no	—
117	C₂²⋊C₄×C₈₃	—	4	2	no	—
118	C₂²⋊C₄×C₈₃	—	4	2	no	—
119	C₂²⋊C₄×C₈₃	D₈	1	2	yes	—
120	C₄²×C₈₃	—	4	1	no	—
121	C₄²×C₈₃	C₂×D₄	1	2	yes	—
122	C₂³×C₈₃	—	4	2	no	—
123	C₂³×C₈₃	—	4	4	no	—
124	C₂³×C₈₃	—	16	1	no	—
125	C₂³×C₈₃	—	16	1	no	—
126	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
127	D₄×C₈₃	—	2	4	no	—
128	D₄×C₈₃	—	4	2	no	—
129	D₄×C₈₃	—	4	4	no	—
130	C₂×C₄×C₈₃	—	4	1	no	—
131	C₂×C₄×C₈₃	—	4	1	no	—
132	C₂×C₄×C₈₃	—	16	2	no	—
133	C₂×C₄×C₈₃	—	16	2	no	—
134	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₈₃	#137	1	2	yes	—
139	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×Dic₈₃	—	4	1	no	—
141	C₂×Dic₈₃	—	4	1	no	—
142	C₂×Dic₈₃	—	4	2	no	—
143	C₂×Dic₈₃	—	16	1	no	—
144	Dic₁₆₆	—	16	2	no	—
145	C₈×C₈₃	—	2	2	no	—
146	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
147	C₈×C₈₃	C₂²×Q₈	1	1	yes	characteristic
148	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
149	C₂²×C₈₃	—	2	1	no	—
150	C₂²×C₈₃	—	2	1	no	—
151	C₂²×C₈₃	—	4	4	no	—
152	C₂²×C₈₃	—	8	2	no	—
153	C₂²×C₈₃	—	8	2	no	—
154	C₂²×C₈₃	—	16	4	no	—
155	C₂²×C₈₃	#474	1	2	yes	—
156	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
157	C₄×C₈₃	—	2	2	no	—
158	C₄×C₈₃	—	2	2	no	—
159	C₄×C₈₃	—	8	1	no	—
160	C₄×C₈₃	—	8	1	no	—
161	C₄×C₈₃	#474	1	2	yes	—
162	C₄×C₈₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
163	C₄×C₈₃	C₂×#137	1	1	yes	characteristic
164	Dic₈₃	—	8	1	no	—
165	Dic₈₃	—	8	2	no	—
166	Dic₈₃	—	8	1	no	—
167	#33631	—	83	1	no	maximal, Sylow
168	C₂×C₈₃	—	2	2	no	—
169	C₂×C₈₃	—	8	4	no	—
170	C₂×C₈₃	C₂×#474	1	1	yes	characteristic
171	C₂×C₈₃	#3415	1	1	yes	characteristic
172	C₂×C₈₃	#3416	1	1	yes	characteristic
173	#3164	—	83	1	no	—
174	#2948	D₈₃	1	1	yes	characteristic
175	#2228	—	83	2	no	—
176	#2228	—	83	2	no	—
177	#2226	—	83	2	no	—
178	#2224	—	83	2	no	—
179	#2159	—	83	1	no	—
180	#2158	—	83	1	no	—
181	#2151	—	83	1	no	—
182	#2140	—	83	1	no	—
183	C₂×#357	—	83	1	no	—
184	C₈₃	#33631	1	1	yes	characteristic, complemented, Sylow
185	C₂²×D₈	C₂×D₈₃	1	1	yes	characteristic
186	#544	—	166	1	no	—
187	#518	—	2	1	no	—
188	C₂×M₃₂	—	83	1	no	—
189	C₂²×C₁₆	—	83	1	no	—
190	#501	—	166	2	no	—
191	#500	—	83	2	no	—
192	#499	—	83	2	no	—
193	#442	—	2	2	no	—
194	C₄×D₈	C₂×D₈₃	1	2	yes	—
195	C₄×D₈	C₂×D₈₃	1	2	yes	—
196	#428	—	166	1	no	—
197	#427	—	83	1	no	—
198	#427	C₂×D₈₃	1	1	yes	characteristic
199	C₂×#107	—	83	1	no	—
200	#418	—	2	1	no	—
201	#405	C₂×D₈₃	1	1	yes	characteristic
202	#367	—	83	1	no	—
203	#367	—	83	1	no	—
204	#367	—	83	2	no	—
205	#366	—	83	1	no	—
206	#366	—	83	1	no	—
207	#366	—	83	2	no	—
208	C₄⋊C₁₆	—	83	2	no	—
209	C₄⋊C₁₆	—	83	2	no	—
210	#357	—	83	2	no	—
211	#357	—	83	2	no	—
212	#357	—	83	2	no	—
213	#357	—	83	2	no	—
214	C₂²×D₄	—	2	1	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	2	2	no	—
217	C₂×D₈	Dic₁₆₆	1	4	yes	complemented
218	C₂×D₈	C₂²×D₈₃	1	2	yes	—
219	C₂×M₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
221	C₄⋊Q₈	—	332	1	no	—
222	C₄⋊Q₈	—	166	2	no	—
223	#125	—	332	1	no	—
224	#125	—	166	2	no	—
225	C₂²⋊Q₈	—	332	2	no	—
226	C₄×D₄	—	2	2	no	—
227	C₄×D₄	—	2	2	no	—
228	C₄×D₄	—	4	1	no	—
229	C₄×D₄	—	4	1	no	—
230	#117	—	2	1	no	—
231	#117	—	166	1	no	—
232	#117	C₂²⋊D₈₃	1	1	yes	characteristic
233	C₂×C₄⋊C₄	—	166	1	no	—
234	C₂×C₂²⋊C₄	—	4	2	no	—
235	M₃₂	—	166	2	no	—
236	C₂×C₁₆	—	83	1	no	—
237	C₂×C₁₆	—	83	2	no	—
238	C₂×C₁₆	—	83	1	no	—
239	C₂×C₁₆	—	166	2	no	—
240	#107	—	83	2	no	—
241	#107	—	83	1	no	—
242	#107	—	83	1	no	—
243	#107	—	166	1	no	—
244	#107	—	166	2	no	—
245	#107	C₂²×D₈₃	1	1	yes	characteristic
246	#107	C₂²×D₈₃	1	1	yes	characteristic
247	#106	—	2	1	no	—
248	#106	—	166	1	no	—
249	#106	—	166	2	no	—
250	#102	—	2	2	no	—
251	#102	—	2	2	no	—
252	#97	C₂²⋊D₈₃	1	2	yes	—
253	C₄×C₈	C₂²×D₈₃	1	2	yes	—
254	C₂⁴	—	4	1	no	—
255	C₂×Q₈	—	332	2	no	—
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	332	1	no	—
263	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
264	D₈	—	2	4	no	—
265	D₈	—	2	2	no	—
266	D₈	C₂×Dic₁₆₆	1	4	yes	—
267	M₁₆	—	2	2	no	—
268	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
269	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
270	C₂×C₈	C₂×Dic₁₆₆	1	2	yes	—
271	C₂×C₈	#13050590	1	2	yes	—
272	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	332	1	no	—
276	C₄⋊C₄	—	332	2	no	—
277	C₄⋊C₄	—	332	2	no	—
278	C₄⋊C₄	—	332	2	no	—
279	C₄⋊C₄	—	166	1	no	—
280	C₄⋊C₄	—	166	2	no	—
281	C₄⋊C₄	—	166	1	no	—
282	C₄⋊C₄	#13050575	1	1	yes	characteristic
283	C₄⋊C₄	#13050575	1	1	yes	characteristic
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	—	332	1	no	—
288	C₂²⋊C₄	#13050575	1	2	yes	—
289	C₄²	—	4	1	no	—
290	C₄²	—	332	1	no	—
291	C₄²	C₂×C₂²⋊D₈₃	1	2	yes	—
292	C₁₆	—	166	1	no	—
293	C₁₆	—	166	1	no	—
294	C₁₆	—	166	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	4	no	—
297	C₂³	—	16	1	no	—
298	C₂³	—	16	1	no	—
299	C₂³	C₂×#13050575	1	1	yes	characteristic
300	Q₈	—	1328	2	no	—
301	D₄	—	2	4	no	—
302	D₄	—	4	4	no	—
303	D₄	—	4	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	16	2	no	—
307	C₂×C₄	—	16	2	no	—
308	C₂×C₄	—	1328	1	no	—
309	C₂×C₄	—	332	1	no	—
310	C₂×C₄	—	332	1	no	—
311	C₂×C₄	—	332	2	no	—
312	C₂×C₄	#14581771	1	2	yes	—
313	C₂×C₄	C₂×#13050575	1	2	yes	—
314	C₂×C₄	C₂×#13050575	1	2	yes	—
315	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
316	C₂×C₄	#14581822	1	2	yes	—
317	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	#14581771	1	2	yes	—
320	C₈	C₂²×Dic₁₆₆	1	1	yes	characteristic
321	C₈	C₂×#13050590	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	16	4	no	—
328	C₂²	#19127981	1	2	yes	—
329	C₂²	C₂²×#13050575	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	664	2	no	—
335	C₄	—	664	1	no	—
336	C₄	—	664	1	no	—
337	C₄	#19127981	1	2	yes	—
338	C₄	C₂×#14581771	1	1	yes	characteristic
339	C₄	C₂×#14581822	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	8	4	no	—
342	C₂	C₂×#19127981	1	1	yes	characteristic
343	C₂	#24920602	1	1	yes	characteristic
344	C₂	#24920610	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13050575
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#2948	C₂
socle	C₂²×C₈₃	C₂²×D₈
prime radical	C₂²×D₈×C₈₃	C₂²
hypercenter	#2948	D₈₃
hypocenter	C₈₃	#33631
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2948

Complements

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

N	H ≅ G/N	type
1. #24918002	C₂	semidirect
2. #24909620	C₂	semidirect
3. #24909620	C₂	semidirect
4. #24909610	C₂	semidirect
5. #24909600	C₂	semidirect
6. #24909252	C₂	semidirect
7. #24909245	C₂	semidirect
8. #24909222	C₂	semidirect
16. #19128106	C₂²	semidirect
17. #19128103	C₂²	semidirect
21. #19127528	C₂²	semidirect
22. #19127528	C₂²	semidirect
23. #19127528	C₂²	semidirect
24. #19127526	C₂²	semidirect
25. #19127526	C₂²	semidirect
26. #19127526	C₂²	semidirect
27. C₄⋊(C₈₃⋊C₁₆)	C₂²	semidirect
28. C₄⋊(C₈₃⋊C₁₆)	C₂²	semidirect
63. C₂×D₈×C₈₃	Q₈	semidirect
184. C₈₃	#33631	semidirect
217. C₂×D₈	Dic₁₆₆	semidirect

Maximal subgroups

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

subgroup	index
#24918002	2
#24909620	2
#24909610	2
#24909600	2
#24909252	2
#24909245	2
#24909222	2
#24909179	2
C₂×#19127502	2
C₈₃×#2948	2
#33631	83