Subgroups — #35435283

Order 21248.

616 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24916367	C₂	1	1	yes	maximal, characteristic
2	#24909043	C₂	1	1	yes	maximal, characteristic, complemented
3	#24909038	C₂	1	1	yes	maximal, characteristic, complemented
4	#24908914	C₂	1	2	yes	maximal, complemented
5	#24908769	C₂	1	1	yes	maximal, characteristic
6	#24908750	C₂	1	1	yes	maximal, characteristic
7	#24908632	C₂	1	2	yes	maximal, complemented
8	#24907300	C₂	1	2	yes	maximal, complemented
9	#24907264	C₂	1	2	yes	maximal, complemented
10	#24907218	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128501	C₂²	1	1	yes	characteristic
13	C₂×#14581798	C₂²	1	2	yes	—
14	#19127824	—	2	2	no	—
15	#19127819	—	2	2	no	—
16	#19127750	C₂²	1	1	yes	characteristic
17	#19127719	C₂²	1	2	yes	—
18	C₂×#14581720	C₂²	1	2	yes	—
19	#19127698	C₂²	1	2	yes	—
20	#19127683	C₂²	1	2	yes	—
21	#19127674	C₂²	1	2	yes	—
22	#19127656	C₂²	1	2	yes	—
23	#19127473	C₂²	1	2	yes	—
24	#19127471	C₂²	1	2	yes	—
25	#19127466	C₂²	1	2	yes	—
26	#19127461	C₂²	1	2	yes	—
27	C₈₃⋊#338	C₂²	1	1	yes	characteristic
28	#19127458	C₂²	1	1	yes	characteristic
29	#19127457	C₂²	1	2	yes	—
30	#533×C₈₃	C₂²	1	1	yes	characteristic
31	#446×C₈₃	—	2	2	no	—
32	C₈×Q₈×C₈₃	—	2	2	no	—
33	#424×C₈₃	C₂²	1	1	yes	characteristic
34	#424×C₈₃	C₂²	1	1	yes	characteristic
35	#423×C₈₃	C₂²	1	1	yes	characteristic
36	#405×C₈₃	C₂²	1	1	yes	characteristic
37	#405×C₈₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
39	C₂×#13050590	C₂³	1	2	yes	—
40	#14581825	—	2	2	no	—
41	#14581820	—	2	2	no	—
42	#14581814	—	2	2	no	—
43	#14581798	—	2	2	no	—
44	#14581798	—	2	2	no	—
45	#14581798	—	2	2	no	—
46	#14581798	—	2	2	no	—
47	C₂×#13050574	C₂³	1	2	yes	—
48	C₂×#13050574	C₂³	1	2	yes	—
49	C₂×#13050573	—	2	2	no	—
50	C₂×C₈×D₈₃	—	2	2	no	—
51	#14581751	C₂³	1	2	yes	—
52	#14581728	—	2	2	no	—
53	#14581720	—	2	4	no	—
54	#14581720	—	2	2	no	—
55	#14581720	—	2	2	no	—
56	#14581717	—	2	2	no	—
57	#14581714	—	2	2	no	—
58	#14581710	—	2	2	no	—
59	#14581708	—	2	2	no	—
60	#14581706	—	2	4	no	—
61	#14581702	—	2	2	no	—
62	C₈×Dic₈₃	—	2	2	no	—
63	C₂×M₁₆×C₈₃	C₂³	1	1	yes	characteristic
64	C₂×M₁₆×C₈₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₈₃	—	2	1	no	—
68	#125×C₈₃	—	2	1	no	—
69	C₂²⋊Q₈×C₈₃	—	2	2	no	—
70	C₄×Q₈×C₈₃	—	2	2	no	—
71	#117×C₈₃	C₂³	1	1	yes	characteristic
72	#117×C₈₃	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₈₃	—	2	2	no	—
75	C₄⋊C₈×C₈₃	—	2	2	no	—
76	C₄⋊C₈×C₈₃	—	2	2	no	—
77	#97×C₈₃	—	2	1	no	—
78	#97×C₈₃	D₄	1	2	yes	—
79	C₄×C₈×C₈₃	—	2	1	no	—
80	C₄×C₈×C₈₃	—	2	1	no	—
81	C₄×C₈×C₈₃	—	2	1	no	—
82	C₄×C₈×C₈₃	D₄	1	2	yes	—
83	#13050590	—	2	2	no	—
84	#13050590	—	2	2	no	—
85	#13050590	—	2	2	no	—
86	#13050590	—	2	2	no	—
87	C₂×Dic₁₆₆	#41	1	2	yes	—
88	C₂×D₃₃₂	#41	1	2	yes	—
89	C₂×C₂²⋊D₈₃	—	2	2	no	—
90	C₂×C₄×D₈₃	—	2	2	no	—
91	Dic₃₃₂	—	2	2	no	—
92	#13050579	—	2	4	no	—
93	D₆₆₄	—	2	2	no	—
94	#13050574	—	2	2	no	—
95	#13050574	—	2	2	no	—
96	#13050574	—	2	2	no	—
97	#13050574	—	2	2	no	—
98	#13050573	—	4	2	no	—
99	#13050573	—	4	2	no	—
100	C₂×C₈₃⋊C₈	—	2	2	no	—
101	C₂×C₈₃⋊C₈	—	2	2	no	—
102	C₈×D₈₃	—	4	2	no	—
103	C₈×D₈₃	—	4	2	no	—
104	C₄⋊Dic₈₃	—	2	2	no	—
105	C₂²⋊Dic₈₃	—	2	2	no	—
106	C₄×Dic₈₃	—	2	2	no	—
107	C₂×Q₈×C₈₃	—	2	2	no	—
108	C₂²×C₄×C₈₃	—	2	1	no	—
109	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
110	M₁₆×C₈₃	—	2	2	no	—
111	M₁₆×C₈₃	—	2	2	no	—
112	C₂×C₈×C₈₃	—	2	1	no	—
113	C₂×C₈×C₈₃	—	2	1	no	—
114	C₂×C₈×C₈₃	—	2	2	no	—
115	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₈₃	#41	1	2	yes	—
119	C₂×C₈×C₈₃	#41	1	2	yes	—
120	C₄⋊C₄×C₈₃	—	2	4	no	—
121	C₄⋊C₄×C₈₃	—	2	2	no	—
122	C₄⋊C₄×C₈₃	—	2	1	no	—
123	C₄⋊C₄×C₈₃	—	2	1	no	—
124	C₂²⋊C₄×C₈₃	—	2	1	no	—
125	C₂²⋊C₄×C₈₃	#41	1	2	yes	—
126	C₄²×C₈₃	—	2	1	no	—
127	C₄²×C₈₃	—	2	1	no	—
128	C₄²×C₈₃	C₂×D₄	1	2	yes	—
129	C₄²×C₈₃	#41	1	1	yes	characteristic
130	C₄²×C₈₃	#41	1	1	yes	characteristic
131	C₂³×C₈₃	C₂×#41	1	1	yes	characteristic
132	C₂²×D₈₃	—	2	2	no	—
133	Q₈×C₈₃	—	4	4	no	—
134	C₂×C₄×C₈₃	—	2	1	no	—
135	C₂×C₄×C₈₃	—	2	1	no	—
136	C₂×C₄×C₈₃	—	2	2	no	—
137	C₂×C₄×C₈₃	—	2	2	no	—
138	C₂×C₄×C₈₃	—	4	2	no	—
139	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₈₃	C₂×#41	1	2	yes	—
141	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₈₃	C₂×#41	1	2	yes	—
143	C₂²⋊D₈₃	—	4	2	no	—
144	C₂²⋊D₈₃	—	4	2	no	—
145	C₂×Dic₈₃	—	2	2	no	—
146	C₂×Dic₈₃	—	4	2	no	—
147	D₃₃₂	—	2	2	no	—
148	D₃₃₂	—	2	2	no	—
149	C₄×D₈₃	—	4	2	no	—
150	C₄×D₈₃	—	4	2	no	—
151	Dic₁₆₆	—	2	2	no	—
152	Dic₁₆₆	—	2	2	no	—
153	C₈×C₈₃	—	2	2	no	—
154	C₈×C₈₃	—	4	1	no	—
155	C₈×C₈₃	—	4	1	no	—
156	C₈×C₈₃	#124	1	2	yes	—
157	C₈×C₈₃	#124	1	1	yes	characteristic
158	C₈×C₈₃	#124	1	1	yes	characteristic
159	C₈₃⋊C₈	—	4	2	no	—
160	C₈₃⋊C₈	—	4	2	no	—
161	C₂²×C₈₃	—	4	1	no	—
162	C₂²×C₈₃	#525	1	1	yes	characteristic
163	C₂²×C₈₃	#529	1	2	yes	—
164	C₂×D₈₃	—	4	2	no	—
165	C₂×D₈₃	—	8	2	no	—
166	C₄×C₈₃	—	2	2	no	—
167	C₄×C₈₃	—	4	1	no	—
168	C₄×C₈₃	—	4	2	no	—
169	C₄×C₈₃	—	4	2	no	—
170	C₄×C₈₃	#525	1	1	yes	characteristic
171	C₄×C₈₃	C₂×#124	1	1	yes	characteristic
172	C₄×C₈₃	#529	1	2	yes	—
173	Dic₈₃	—	4	2	no	—
174	Dic₈₃	—	8	2	no	—
175	#23833	—	83	1	no	maximal, Sylow
176	C₂×C₈₃	—	2	2	no	—
177	C₂×C₈₃	—	2	1	no	—
178	C₂×C₈₃	#2398	1	1	yes	characteristic
179	D₈₃	—	8	2	no	—
180	#2960	D₈₃	1	1	yes	characteristic, complemented
181	#2955	—	83	1	no	—
182	#2103	—	83	1	no	—
183	#2102	—	83	1	no	—
184	#2080	—	83	2	no	—
185	#2053	—	83	1	no	—
186	#2050	—	83	1	no	—
187	#2031	—	83	2	no	—
188	#1764	—	83	2	no	—
189	#1757	—	83	2	no	—
190	#1747	—	83	1	no	—
191	C₈₃	#23833	1	1	yes	characteristic, complemented, Sylow
192	#575	—	83	1	no	—
193	C₂×#131	—	83	2	no	—
194	#533	C₂×D₈₃	1	1	yes	characteristic
195	#446	—	2	2	no	—
196	C₈×Q₈	—	2	2	no	—
197	#444	—	166	2	no	—
198	#443	—	166	2	no	—
199	#430	—	83	1	no	—
200	#424	C₂×D₈₃	1	1	yes	characteristic
201	#424	C₂×D₈₃	1	1	yes	characteristic
202	#423	C₂×D₈₃	1	1	yes	characteristic
203	#421	—	83	2	no	—
204	C₂×#104	—	83	2	no	—
205	#416	—	83	2	no	—
206	#413	—	83	2	no	—
207	#410	—	83	2	no	—
208	#405	—	83	2	no	—
209	#405	C₂×D₈₃	1	1	yes	characteristic
210	#405	C₂×D₈₃	1	1	yes	characteristic
211	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
212	#344	—	83	2	no	—
213	#344	—	83	2	no	—
214	#341	—	83	2	no	—
215	#339	—	83	2	no	—
216	#338	—	83	1	no	—
217	H(3,ℤ/4)	—	83	1	no	—
218	H(3,ℤ/4)	—	83	2	no	—
219	C₂×#41	—	83	2	no	—
220	#137	—	166	2	no	—
221	AGL(1,ℤ/8)	—	166	2	no	—
222	#135	—	166	2	no	—
223	#131	—	166	2	no	—
224	#131	—	166	2	no	—
225	#131	—	166	2	no	—
226	#131	—	166	2	no	—
227	C₂×M₁₆	—	83	2	no	—
228	C₂×M₁₆	—	83	2	no	—
229	C₂×M₁₆	—	166	2	no	—
230	C₂×M₁₆	C₂²×D₈₃	1	1	yes	characteristic
231	C₂×M₁₆	C₂²×D₈₃	1	1	yes	characteristic
232	C₂²×C₈	—	166	2	no	—
233	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
235	C₄⋊Q₈	—	2	1	no	—
236	#125	—	2	1	no	—
237	C₂²⋊Q₈	—	2	2	no	—
238	C₄×Q₈	—	2	2	no	—
239	#117	—	83	2	no	—
240	#117	C₂²×D₈₃	1	1	yes	characteristic
241	#117	C₂²×D₈₃	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
243	#108	—	166	2	no	—
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	2	2	no	—
247	#104	—	166	4	no	—
248	#104	—	166	2	no	—
249	#104	—	166	2	no	—
250	#103	—	166	2	no	—
251	#102	—	166	2	no	—
252	#101	—	166	2	no	—
253	#100	—	166	2	no	—
254	#99	—	166	4	no	—
255	#97	—	2	1	no	—
256	#97	—	166	2	no	—
257	#97	C₂²⋊D₈₃	1	2	yes	—
258	C₄×C₈	—	2	1	no	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	—	166	2	no	—
262	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
263	#41	—	166	2	no	—
264	#41	—	166	2	no	—
265	#41	—	166	2	no	—
266	#41	—	166	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	83	2	no	—
269	C₂×D₄	—	83	2	no	—
270	C₂×D₄	—	166	2	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	166	2	no	—
273	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
274	Q₁₆	—	166	2	no	—
275	SD₁₆	—	166	4	no	—
276	D₈	—	166	2	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	332	2	no	—
280	M₁₆	—	332	2	no	—
281	M₁₆	—	166	2	no	—
282	M₁₆	—	166	2	no	—
283	M₁₆	—	166	2	no	—
284	M₁₆	—	166	2	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	332	2	no	—
289	C₂×C₈	—	332	2	no	—
290	C₂×C₈	—	166	2	no	—
291	C₂×C₈	—	166	2	no	—
292	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
293	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
294	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
295	C₂×C₈	#13050591	1	2	yes	—
296	C₂×C₈	#13050591	1	2	yes	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	166	2	no	—
302	C₂²⋊C₄	—	2	1	no	—
303	C₂²⋊C₄	—	166	2	no	—
304	C₂²⋊C₄	#13050590	1	2	yes	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	1	no	—
307	C₄²	—	166	2	no	—
308	C₄²	C₂×C₂²⋊D₈₃	1	2	yes	—
309	C₄²	#13050590	1	1	yes	characteristic
310	C₄²	#13050590	1	1	yes	characteristic
311	C₂³	—	166	2	no	—
312	C₂³	C₂×#13050590	1	1	yes	characteristic
313	Q₈	—	4	4	no	—
314	Q₈	—	166	2	no	—
315	Q₈	—	166	2	no	—
316	D₄	—	332	2	no	—
317	D₄	—	332	2	no	—
318	D₄	—	166	2	no	—
319	D₄	—	166	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	332	2	no	—
326	C₂×C₄	—	332	2	no	—
327	C₂×C₄	—	332	2	no	—
328	C₂×C₄	—	166	2	no	—
329	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
330	C₂×C₄	#41×D₈₃	1	2	yes	—
331	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
332	C₂×C₄	C₂×#13050591	1	2	yes	—
333	C₈	—	2	2	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	332	2	no	—
337	C₈	—	332	2	no	—
338	C₈	#14581781	1	2	yes	—
339	C₈	#14581781	1	1	yes	characteristic
340	C₈	#14581781	1	1	yes	characteristic
341	C₂²	—	4	1	no	—
342	C₂²	—	332	2	no	—
343	C₂²	—	664	2	no	—
344	C₂²	#19128212	1	1	yes	characteristic
345	C₂²	#19128238	1	2	yes	—
346	C₄	—	2	2	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	4	2	no	—
350	C₄	—	332	2	no	—
351	C₄	—	664	2	no	—
352	C₄	#19128208	1	1	yes	characteristic
353	C₄	C₂×#14581781	1	1	yes	characteristic
354	C₄	#19128238	1	2	yes	—
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	664	2	no	—
358	C₂	#24910746	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14581781
commutator, cosocle	C₂²×C₄×C₈₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	C₈₃×#2960	C₂
socle	C₂×C₈₃	#2398
prime radical	#19128501	C₂²
hypercenter	#2960	D₈₃
hypocenter	C₈₃	#23833
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
2. #24909043	C₂	semidirect
3. #24909038	C₂	semidirect
4. #24908914	C₂	semidirect
7. #24908632	C₂	semidirect
8. #24907300	C₂	semidirect
9. #24907264	C₂	semidirect
10. #24907218	C₂	semidirect
11. C₈₃×#2960	C₂	semidirect
180. #2960	D₈₃	semidirect
191. C₈₃	#23833	semidirect

Maximal subgroups

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

subgroup	index
#24916367	2
#24909043	2
#24909038	2
#24908914	2
#24908769	2
#24908750	2
#24908632	2
#24907300	2
#24907264	2
#24907218	2
C₈₃×#2960	2
#23833	83