Subgroups — #35304143

Order 21248.

642 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128075	C₂	1	1	yes	maximal, characteristic
2	#24908425	C₂	1	1	yes	maximal, characteristic, complemented
3	#24908269	C₂	1	1	yes	maximal, characteristic
4	#24908268	C₂	1	1	yes	maximal, characteristic
5	#24907619	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#13050577	—	2	2	no	—
9	C₂×#14581789	C₂²	1	1	yes	characteristic
10	#19128075	—	2	2	no	—
11	#19128075	C₄	1	4	yes	complemented
12	C₂×#14581723	—	2	2	no	—
13	C₂×#14581714	—	2	2	no	—
14	C₂×#14581712	—	2	2	no	—
15	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
17	#19127571	C₂²	1	1	yes	characteristic
18	#19127463	—	2	2	no	—
19	C₈×C₈₃⋊C₈	—	2	2	no	—
20	C₂×C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
22	#384×C₈₃	C₂²	1	1	yes	characteristic
23	#340×C₈₃	—	2	2	no	—
24	C₂²×D₃₃₂	—	2	1	no	—
25	C₂²×D₃₃₂	—	2	1	no	—
26	C₂²×D₃₃₂	—	2	1	no	—
27	C₂×#13050577	—	2	8	no	—
28	C₂×#13050577	—	4	4	no	—
29	C₂×#13050577	—	4	4	no	—
30	C₂²×C₈₃⋊C₈	D₄	1	2	yes	—
31	C₂²×C₈₃⋊C₈	D₄	1	2	yes	—
32	#14581789	—	2	1	no	—
33	#14581789	—	2	1	no	—
34	#14581789	—	2	1	no	—
35	#14581789	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₈₃	—	2	1	no	—
37	C₂×C₄⋊Dic₈₃	—	2	1	no	—
38	C₂×C₄⋊Dic₈₃	—	2	1	no	—
39	#14581723	—	4	4	no	—
40	#14581714	—	4	4	no	—
41	#14581712	—	4	4	no	—
42	C₄×C₈₃⋊C₈	—	2	1	no	—
43	C₄×C₈₃⋊C₈	—	2	1	no	—
44	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
45	C₄×C₈₃⋊C₈	D₄	1	2	yes	—
46	C₂²×Q₈×C₈₃	—	2	1	no	—
47	C₂²×C₈×C₈₃	D₄	1	2	yes	—
48	C₄⋊Q₈×C₈₃	—	2	1	no	—
49	C₄⋊Q₈×C₈₃	—	2	1	no	—
50	C₄⋊Q₈×C₈₃	—	2	1	no	—
51	C₄⋊Q₈×C₈₃	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
57	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₈₃	—	2	1	no	—
59	C₄×C₈×C₈₃	—	2	1	no	—
60	C₂³×D₈₃	—	4	1	no	—
61	C₂×D₃₃₂	—	2	2	no	—
62	C₂×D₃₃₂	—	2	2	no	—
63	C₂×D₃₃₂	—	2	2	no	—
64	C₂×D₃₃₂	—	4	2	no	—
65	C₂×D₃₃₂	—	4	2	no	—
66	C₂×D₃₃₂	—	4	2	no	—
67	C₂×D₃₃₂	—	4	2	no	—
68	C₂×D₃₃₂	—	4	2	no	—
69	C₂×D₃₃₂	—	4	2	no	—
70	C₂×D₃₃₂	—	4	1	no	—
71	C₂×D₃₃₂	—	4	1	no	—
72	C₂×D₃₃₂	—	4	1	no	—
73	C₂²×Dic₈₃	—	4	1	no	—
74	#13050577	—	4	16	no	—
75	C₂×C₈₃⋊C₈	—	2	2	no	—
76	C₂×C₈₃⋊C₈	—	2	2	no	—
77	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	4	yes	—
78	C₂×C₈₃⋊C₈	D₈	1	4	yes	—
79	C₂×C₈₃⋊C₈	SD₁₆	1	4	yes	—
80	C₂×C₈₃⋊C₈	SD₁₆	1	4	yes	complemented
81	C₄⋊Dic₈₃	—	4	2	no	—
82	C₄⋊Dic₈₃	—	4	2	no	—
83	C₄⋊Dic₈₃	—	4	2	no	—
84	C₂×Q₈×C₈₃	—	2	2	no	—
85	C₂×Q₈×C₈₃	—	4	1	no	—
86	C₂×Q₈×C₈₃	—	4	2	no	—
87	C₂×Q₈×C₈₃	—	4	2	no	—
88	C₂²×C₄×C₈₃	—	4	1	no	—
89	C₂²×C₄×C₈₃	—	4	1	no	—
90	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₈₃	—	2	2	no	—
94	C₂×C₈×C₈₃	—	2	2	no	—
95	C₂×C₈×C₈₃	SD₁₆	1	4	yes	complemented
96	C₄⋊C₄×C₈₃	—	2	2	no	—
97	C₄⋊C₄×C₈₃	—	2	2	no	—
98	C₄⋊C₄×C₈₃	—	4	1	no	—
99	C₄⋊C₄×C₈₃	—	4	1	no	—
100	C₄⋊C₄×C₈₃	—	4	2	no	—
101	C₄⋊C₄×C₈₃	—	4	2	no	—
102	C₄⋊C₄×C₈₃	—	4	2	no	—
103	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₈₃	#41	1	1	yes	characteristic
106	C₄²×C₈₃	#41	1	1	yes	characteristic
107	C₂³×C₈₃	#127	1	1	yes	characteristic
108	C₂²×D₈₃	—	4	2	no	—
109	C₂²×D₈₃	—	8	1	no	—
110	C₂²×D₈₃	—	8	1	no	—
111	C₂²×D₈₃	—	8	1	no	—
112	C₂²×D₈₃	—	8	1	no	—
113	C₂²×D₈₃	—	8	1	no	—
114	C₂²×D₈₃	—	8	1	no	—
115	Q₈×C₈₃	—	4	4	no	—
116	Q₈×C₈₃	—	8	2	no	—
117	C₂×C₄×C₈₃	—	4	2	no	—
118	C₂×C₄×C₈₃	—	8	1	no	—
119	C₂×C₄×C₈₃	—	8	1	no	—
120	C₂×C₄×C₈₃	—	8	1	no	—
121	C₂×C₄×C₈₃	—	8	1	no	—
122	C₂×C₄×C₈₃	—	8	1	no	—
123	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
127	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
129	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
131	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
132	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
133	C₂×C₄×C₈₃	#135	1	2	yes	—
134	C₂×C₄×C₈₃	#135	1	2	yes	—
135	C₂×Dic₈₃	—	8	1	no	—
136	C₂×Dic₈₃	—	8	1	no	—
137	C₂×Dic₈₃	—	8	1	no	—
138	D₃₃₂	—	4	4	no	—
139	D₃₃₂	—	4	4	no	—
140	D₃₃₂	—	4	4	no	—
141	D₃₃₂	—	8	2	no	—
142	D₃₃₂	—	8	2	no	—
143	D₃₃₂	—	8	2	no	—
144	C₈×C₈₃	—	2	4	no	—
145	C₈₃⋊C₈	—	2	4	no	—
146	C₈₃⋊C₈	#102	1	8	yes	complemented
147	C₂²×C₈₃	#390	1	1	yes	characteristic
148	C₂²×C₈₃	#492	1	1	yes	characteristic
149	C₂²×C₈₃	#492	1	1	yes	characteristic
150	C₂²×C₈₃	#492	1	1	yes	characteristic
151	C₂²×C₈₃	#493	1	1	yes	characteristic
152	C₂²×C₈₃	#495	1	1	yes	characteristic
153	C₂²×C₈₃	#495	1	1	yes	characteristic
154	C₂×D₈₃	—	8	2	no	—
155	C₂×D₈₃	—	8	2	no	—
156	C₂×D₈₃	—	8	2	no	—
157	C₂×D₈₃	—	16	1	no	—
158	C₂×D₈₃	—	16	1	no	—
159	C₂×D₈₃	—	16	1	no	—
160	C₂×D₈₃	—	16	1	no	—
161	C₄×C₈₃	—	8	2	no	—
162	C₄×C₈₃	—	16	1	no	—
163	C₄×C₈₃	C₂×#102	1	2	yes	—
164	C₄×C₈₃	C₄×SD₁₆	1	2	yes	—
165	C₄×C₈₃	C₄×SD₁₆	1	2	yes	—
166	C₄×C₈₃	#465	1	2	yes	—
167	C₄×C₈₃	#465	1	2	yes	—
168	C₄×C₈₃	#486	1	2	yes	—
169	Dic₈₃	—	16	1	no	—
170	#9820	—	83	1	no	maximal, Sylow
171	C₂×C₈₃	#1705	1	1	yes	characteristic
172	C₂×C₈₃	#1706	1	1	yes	characteristic
173	C₂×C₈₃	#1706	1	1	yes	characteristic
174	C₂×C₈₃	#1708	1	1	yes	characteristic
175	C₂×C₈₃	#1957	1	1	yes	characteristic
176	C₂×C₈₃	#1968	1	1	yes	characteristic
177	C₂×C₈₃	#1968	1	1	yes	characteristic
178	D₈₃	—	16	2	no	—
179	C₂×#492	—	83	1	no	—
180	#1990	—	83	1	no	—
181	#1990	D₈₃	1	1	yes	characteristic, complemented
182	#1957	—	83	1	no	—
183	#1957	—	83	1	no	—
184	#1844	—	83	1	no	—
185	C₂×C₈²	—	83	1	no	—
186	C₈₃	#9820	1	1	yes	characteristic, complemented, Sylow
187	C₂²×SD₁₆	—	166	2	no	—
188	C₂×C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
189	C₂×#127	—	83	1	no	—
190	#492	—	83	4	no	—
191	#492	—	166	2	no	—
192	C₂×#106	—	166	2	no	—
193	C₂×#102	—	166	2	no	—
194	C₂×#102	—	166	2	no	—
195	C₂×C₄×C₈	—	83	1	no	—
196	C₂×C₄×C₈	—	83	1	no	—
197	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
198	#384	—	83	1	no	—
199	#384	C₂×D₈₃	1	1	yes	characteristic
200	#340	—	2	2	no	—
201	#340	—	166	2	no	—
202	C₈²	—	166	2	no	—
203	C₂²×Q₈	—	2	1	no	—
204	C₂²×D₄	—	166	1	no	—
205	C₂²×D₄	—	166	1	no	—
206	C₂²×D₄	—	166	1	no	—
207	C₂×SD₁₆	—	332	4	no	—
208	C₂×SD₁₆	—	332	4	no	—
209	C₂×SD₁₆	—	166	8	no	—
210	C₂²×C₈	—	83	2	no	—
211	C₂²×C₈	—	83	2	no	—
212	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	C₄×D₈₃	1	2	yes	—
217	#127	—	83	2	no	—
218	#127	—	166	1	no	—
219	#127	—	166	1	no	—
220	#127	—	166	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	166	1	no	—
227	C₂×C₄⋊C₄	—	166	1	no	—
228	C₂×C₄⋊C₄	—	166	1	no	—
229	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
230	#106	—	332	4	no	—
231	#102	—	332	4	no	—
232	#102	—	332	4	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	—	83	2	no	—
236	C₄×C₈	—	83	2	no	—
237	C₄×C₈	—	166	1	no	—
238	C₄×C₈	—	166	1	no	—
239	C₂⁴	—	332	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×D₄	—	332	1	no	—
245	C₂×D₄	—	332	1	no	—
246	C₂×D₄	—	332	1	no	—
247	C₂×D₄	—	332	2	no	—
248	C₂×D₄	—	332	2	no	—
249	C₂×D₄	—	332	2	no	—
250	C₂×D₄	—	332	2	no	—
251	C₂×D₄	—	332	2	no	—
252	C₂×D₄	—	332	2	no	—
253	C₂×D₄	—	166	2	no	—
254	C₂×D₄	—	166	2	no	—
255	C₂×D₄	—	166	2	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	332	1	no	—
259	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
260	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
262	SD₁₆	—	332	16	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	83	4	no	—
266	C₂×C₈	—	83	4	no	—
267	C₂×C₈	—	83	4	no	—
268	C₂×C₈	—	83	4	no	—
269	C₂×C₈	—	166	2	no	—
270	C₂×C₈	—	166	2	no	—
271	C₂×C₈	#13050577	1	4	yes	complemented
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	332	2	no	—
280	C₄⋊C₄	—	332	2	no	—
281	C₄⋊C₄	—	332	2	no	—
282	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
283	C₄²	D₄×D₈₃	1	1	yes	characteristic
284	C₄²	#13050589	1	1	yes	characteristic
285	C₄²	#13050590	1	1	yes	characteristic
286	C₂³	—	332	2	no	—
287	C₂³	—	664	1	no	—
288	C₂³	—	664	1	no	—
289	C₂³	—	664	1	no	—
290	C₂³	—	664	1	no	—
291	C₂³	—	664	1	no	—
292	C₂³	—	664	1	no	—
293	C₂³	#14581788	1	1	yes	characteristic
294	Q₈	—	4	4	no	—
295	Q₈	—	8	2	no	—
296	D₄	—	332	4	no	—
297	D₄	—	332	4	no	—
298	D₄	—	332	4	no	—
299	D₄	—	664	2	no	—
300	D₄	—	664	2	no	—
301	D₄	—	664	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	8	1	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	—	664	1	no	—
309	C₂×C₄	—	664	1	no	—
310	C₂×C₄	—	664	1	no	—
311	C₂×C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
312	C₂×C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
313	C₂×C₄	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
314	C₂×C₄	#14581764	1	1	yes	characteristic
315	C₂×C₄	#14581765	1	1	yes	characteristic
316	C₂×C₄	#14581778	1	1	yes	characteristic
317	C₂×C₄	D₈×D₈₃	1	2	yes	—
318	C₂×C₄	SD₁₆×D₈₃	1	2	yes	—
319	C₂×C₄	SD₁₆×D₈₃	1	2	yes	—
320	C₂×C₄	C₂×#13050577	1	2	yes	—
321	C₂×C₄	#14581812	1	2	yes	—
322	C₂×C₄	#14581815	1	2	yes	—
323	C₈	—	2	4	no	—
324	C₈	—	83	8	no	—
325	C₈	—	166	4	no	—
326	C₂²	—	1328	1	no	—
327	C₂²	—	1328	1	no	—
328	C₂²	—	1328	1	no	—
329	C₂²	—	1328	1	no	—
330	C₂²	—	664	2	no	—
331	C₂²	—	664	2	no	—
332	C₂²	—	664	2	no	—
333	C₂²	#19127601	1	1	yes	characteristic
334	C₂²	#19128072	1	1	yes	characteristic
335	C₂²	#19128072	1	1	yes	characteristic
336	C₂²	#19128075	1	1	yes	characteristic
337	C₂²	#19128076	1	1	yes	characteristic
338	C₂²	#19128083	1	1	yes	characteristic
339	C₂²	#19128086	1	1	yes	characteristic
340	C₄	—	8	2	no	—
341	C₄	—	16	1	no	—
342	C₄	—	1328	1	no	—
343	C₄	#102×D₈₃	1	2	yes	—
344	C₄	#19127786	1	2	yes	—
345	C₄	C₄×#13050577	1	2	yes	—
346	C₄	#19127932	1	2	yes	—
347	C₄	#19127933	1	2	yes	—
348	C₄	#19128046	1	2	yes	—
349	C₂	—	1328	2	no	—
350	C₂	#24907051	1	1	yes	characteristic
351	C₂	#24907054	1	1	yes	characteristic
352	C₂	#24907056	1	1	yes	characteristic
353	C₂	#24907066	1	1	yes	characteristic
354	C₂	#24908267	1	1	yes	characteristic
355	C₂	#24908314	1	1	yes	characteristic
356	C₂	#24908320	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581788
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1990	C₂
socle	C₂³×C₈₃	#127
prime radical	C₂×#14581789	C₂²
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1990	D₈₃
hypocenter	C₈₃	#9820
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
2. #24908425	C₂	semidirect
5. #24907619	C₂	semidirect
6. C₂×C₈×C₈₃⋊C₈	C₂	semidirect
7. C₈₃×#1990	C₂	semidirect
11. #19128075	C₄	semidirect
80. C₂×C₈₃⋊C₈	SD₁₆	semidirect
95. C₂×C₈×C₈₃	SD₁₆	semidirect
146. C₈₃⋊C₈	#102	semidirect
181. #1990	D₈₃	semidirect
186. C₈₃	#9820	semidirect
271. C₂×C₈	#13050577	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128075	2
#24908425	2
#24908269	2
#24908268	2
#24907619	2
C₂×C₈×C₈₃⋊C₈	2
C₈₃×#1990	2
#9820	83