Subgroups — #35557544

Order 21248.

528 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14581800	C₂	1	1	yes	maximal, characteristic
2	#24909277	C₂	1	2	yes	maximal, complemented
3	#24909276	C₂	1	2	yes	maximal, complemented
4	#24909275	C₂	1	1	yes	maximal, characteristic, complemented
5	#24909274	C₂	1	2	yes	maximal, complemented
6	#364×D₈₃	C₂	1	2	yes	maximal, complemented
7	#24909082	C₂	1	2	yes	maximal
8	#24909066	C₂	1	2	yes	maximal, complemented
9	C₈₃×#2165	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14581800	C₄	1	1	yes	characteristic
11	C₂×#14581800	C₂²	1	1	yes	characteristic
12	C₄×#13050590	C₄	1	1	yes	characteristic
13	M₃₂×D₈₃	C₂²	1	4	yes	—
14	#19127659	C₄	1	1	yes	characteristic
15	#19127659	C₂²	1	1	yes	characteristic
16	#19127658	C₄	1	1	yes	characteristic
17	#19127658	C₄	1	1	yes	characteristic
18	#19127658	C₂²	1	1	yes	characteristic
19	#19127658	C₂²	1	1	yes	characteristic
20	C₄×#13050573	C₄	1	1	yes	characteristic
21	C₄×#13050573	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈₃⋊C₈	C₄	1	1	yes	characteristic
23	C₄×C₈×D₈₃	C₄	1	1	yes	characteristic
24	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
25	#19127522	C₂²	1	2	yes	—
26	#19127521	C₂²	1	4	yes	—
27	#19127483	C₂²	1	2	yes	—
28	#19127482	C₂²	1	4	yes	—
29	#19127479	C₂²	1	2	yes	—
30	#19127478	C₂²	1	4	yes	—
31	C₄×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
32	#364×C₈₃	C₂²	1	2	yes	—
33	#349×C₈₃	C₂²	1	2	yes	—
34	#347×C₈₃	C₂²	1	2	yes	—
35	C₂×#13050590	C₂×C₄	1	1	yes	characteristic
36	#14581800	—	2	4	no	—
37	#14581800	—	2	2	no	—
38	#14581800	—	2	2	no	—
39	C₂×#13050573	C₂×C₄	1	1	yes	characteristic
40	C₂×#13050573	C₂×C₄	1	1	yes	characteristic
41	C₂×#13050573	C₂×C₄	1	1	yes	characteristic
42	C₂×#13050573	C₂³	1	1	yes	characteristic
43	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₈₃	C₂³	1	1	yes	characteristic
49	C₄×Dic₁₆₆	—	2	1	no	—
50	C₄×D₃₃₂	—	2	1	no	—
51	C₄×C₂²⋊D₈₃	—	2	1	no	—
52	C₄×C₂²⋊D₈₃	—	2	1	no	—
53	#14581751	C₂×C₄	1	1	yes	characteristic
54	#14581749	C₂×C₄	1	1	yes	characteristic
55	#14581749	C₂×C₄	1	1	yes	characteristic
56	C₄²×D₈₃	C₂×C₄	1	1	yes	characteristic
57	C₄²×D₈₃	C₂×C₄	1	1	yes	characteristic
58	#14581732	C₂³	1	4	yes	—
59	#14581731	—	2	4	no	—
60	C₁₆×D₈₃	—	2	4	no	—
61	#14581702	C₂×C₄	1	1	yes	characteristic
62	#14581702	C₂×C₄	1	1	yes	characteristic
63	#14581702	C₂×C₄	1	1	yes	characteristic
64	#14581702	C₂³	1	1	yes	characteristic
65	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₈₃	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
74	C₂×M₁₆×C₈₃	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
76	M₃₂×C₈₃	C₂³	1	4	yes	—
77	#97×C₈₃	C₂×C₄	1	1	yes	characteristic
78	#97×C₈₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
80	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
81	#13050590	—	2	2	no	—
82	#13050590	—	2	2	no	—
83	C₂×Dic₁₆₆	M₁₆	1	1	yes	characteristic
84	C₂×D₃₃₂	M₁₆	1	1	yes	characteristic
85	C₂×C₂²⋊D₈₃	M₁₆	1	1	yes	characteristic
86	C₂×C₂²⋊D₈₃	M₁₆	1	1	yes	characteristic
87	C₂×C₄×D₈₃	—	2	1	no	—
88	C₂×C₄×D₈₃	—	2	1	no	—
89	C₂×C₄×D₈₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×D₈₃	C₂²×C₄	1	1	yes	characteristic
91	#13050573	—	2	2	no	—
92	#13050573	—	2	2	no	—
93	#13050573	—	2	2	no	—
94	#13050573	—	2	2	no	—
95	C₂×C₈₃⋊C₈	—	2	1	no	—
96	C₂×C₈₃⋊C₈	—	2	1	no	—
97	C₂×C₈₃⋊C₈	—	2	2	no	—
98	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
102	C₈×D₈₃	—	2	1	no	—
103	C₈×D₈₃	—	2	1	no	—
104	C₈×D₈₃	—	2	2	no	—
105	C₈×D₈₃	M₁₆	1	4	yes	—
106	C₈×D₈₃	#41	1	2	yes	—
107	C₈×D₈₃	#41	1	2	yes	—
108	C₄⋊Dic₈₃	—	2	1	no	—
109	Dic₈₃⋊C₄	—	2	1	no	—
110	Dic₈₃⋊C₄	—	2	1	no	—
111	C₂²⋊Dic₈₃	—	2	1	no	—
112	(C₂×D₈₃)⋊C₄	—	2	1	no	—
113	(C₂×D₈₃)⋊C₄	—	2	1	no	—
114	C₄×Dic₈₃	—	2	1	no	—
115	C₄×Dic₈₃	—	2	1	no	—
116	C₄×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
117	C₄×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
118	C₈₃⋊C₁₆	—	2	4	no	—
119	C₂²×C₄×C₈₃	—	2	1	no	—
120	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
121	M₁₆×C₈₃	—	2	2	no	—
122	M₁₆×C₈₃	—	2	2	no	—
123	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
127	C₄²×C₈₃	—	2	1	no	—
128	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
130	C₁₆×C₈₃	—	2	4	no	—
131	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
132	C₂²×D₈₃	C₂×M₁₆	1	1	yes	characteristic
133	C₂²×D₈₃	C₂×M₁₆	1	1	yes	characteristic
134	C₂×C₄×C₈₃	—	2	1	no	—
135	C₂×C₄×C₈₃	—	2	1	no	—
136	C₂×C₄×C₈₃	—	2	1	no	—
137	C₂×C₄×C₈₃	—	2	1	no	—
138	C₂×C₄×C₈₃	—	4	2	no	—
139	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
140	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
141	C₂²⋊D₈₃	—	2	2	no	—
142	C₂²⋊D₈₃	—	2	2	no	—
143	C₂×Dic₈₃	—	2	1	no	—
144	C₂×Dic₈₃	—	2	1	no	—
145	C₂×Dic₈₃	C₂×M₁₆	1	1	yes	characteristic
146	C₂×Dic₈₃	C₂×M₁₆	1	1	yes	characteristic
147	D₃₃₂	—	2	2	no	—
148	C₄×D₈₃	—	2	1	no	—
149	C₄×D₈₃	—	2	1	no	—
150	C₄×D₈₃	—	4	2	no	—
151	C₄×D₈₃	—	4	2	no	—
152	C₄×D₈₃	#117	1	2	yes	—
153	C₄×D₈₃	C₂×M₁₆	1	2	yes	—
154	Dic₁₆₆	—	2	2	no	—
155	C₈×C₈₃	—	2	1	no	—
156	C₈×C₈₃	—	2	1	no	—
157	C₈×C₈₃	C₂×M₁₆	1	2	yes	—
158	C₈×C₈₃	C₂×#41	1	2	yes	—
159	C₈₃⋊C₈	—	2	1	no	—
160	C₈₃⋊C₈	—	2	1	no	—
161	C₈₃⋊C₈	#117	1	1	yes	characteristic
162	C₈₃⋊C₈	#117	1	1	yes	characteristic
163	C₈₃⋊C₈	C₂×M₁₆	1	2	yes	—
164	C₂²×C₈₃	—	2	1	no	—
165	C₂²×C₈₃	—	4	1	no	—
166	C₂²×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
167	C₂×D₈₃	—	2	1	no	—
168	C₂×D₈₃	—	4	1	no	—
169	C₂×D₈₃	—	4	1	no	—
170	C₂×D₈₃	#432	1	2	yes	—
171	C₄×C₈₃	—	2	2	no	—
172	C₄×C₈₃	—	2	1	no	—
173	C₄×C₈₃	—	4	1	no	—
174	C₄×C₈₃	C₂×#117	1	1	yes	characteristic
175	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
176	Dic₈₃	—	2	1	no	—
177	Dic₈₃	—	4	1	no	—
178	Dic₈₃	—	4	1	no	—
179	Dic₈₃	#432	1	2	yes	—
180	#33428	—	83	1	no	maximal, Sylow
181	C₂×C₈₃	—	2	1	no	—
182	C₂×C₈₃	—	4	1	no	—
183	C₂×C₈₃	C₂×#432	1	1	yes	characteristic
184	D₈₃	—	2	2	no	—
185	D₈₃	—	4	1	no	—
186	C₄×#131	—	83	1	no	—
187	#2166	—	83	2	no	—
188	#2166	—	83	2	no	—
189	#2165	—	83	1	no	—
190	#2165	—	83	2	no	—
191	#2165	D₈₃	1	1	yes	characteristic, complemented
192	C₂×#364	—	83	2	no	—
193	#2115	—	83	2	no	—
194	#2110	—	83	2	no	—
195	C₈₃	#33428	1	1	yes	characteristic, complemented, Sylow
196	C₂×#131	—	83	1	no	—
197	C₂×#131	—	83	1	no	—
198	C₄×#41	—	83	1	no	—
199	C₂×M₃₂	—	83	4	no	—
200	#405	—	83	1	no	—
201	#405	—	83	1	no	—
202	#405	—	83	1	no	—
203	#405	—	83	1	no	—
204	#405	—	83	1	no	—
205	#405	—	83	1	no	—
206	C₄×M₁₆	—	83	1	no	—
207	C₄×M₁₆	—	83	1	no	—
208	C₄×M₁₆	C₂×D₈₃	1	1	yes	characteristic
209	C₂×C₄×C₈	—	83	1	no	—
210	C₂×C₄×C₈	—	83	1	no	—
211	C₂×C₄×C₈	—	83	1	no	—
212	#364	—	83	4	no	—
213	#364	—	83	2	no	—
214	#364	C₂×D₈₃	1	2	yes	—
215	#349	—	83	2	no	—
216	#349	—	83	4	no	—
217	#349	C₂×D₈₃	1	2	yes	—
218	#347	—	83	4	no	—
219	#347	—	83	2	no	—
220	#347	C₂×D₈₃	1	2	yes	—
221	C₂×#41	—	83	1	no	—
222	#131	—	166	4	no	—
223	#131	—	166	2	no	—
224	#131	—	166	2	no	—
225	C₂×M₁₆	—	83	1	no	—
226	C₂×M₁₆	—	83	1	no	—
227	C₂×M₁₆	—	83	1	no	—
228	C₂×M₁₆	—	83	1	no	—
229	C₂×M₁₆	C₄×D₈₃	1	1	yes	characteristic
230	C₂×M₁₆	C₂²×D₈₃	1	1	yes	characteristic
231	C₂²×C₈	—	83	1	no	—
232	C₂²×C₈	—	83	1	no	—
233	C₂²×C₈	—	83	1	no	—
234	C₂²×C₈	—	83	1	no	—
235	C₂²×C₈	—	83	1	no	—
236	C₂²×C₈	—	83	1	no	—
237	C₄×Q₈	—	166	1	no	—
238	C₄×D₄	—	166	1	no	—
239	C₄×D₄	—	166	1	no	—
240	C₄×D₄	—	166	1	no	—
241	#117	—	83	1	no	—
242	#117	—	83	1	no	—
243	#117	—	83	1	no	—
244	C₂×C₄²	—	83	1	no	—
245	C₂×C₄²	—	83	1	no	—
246	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
247	M₃₂	—	83	4	no	—
248	M₃₂	—	166	4	no	—
249	M₃₂	C₂²×D₈₃	1	4	yes	—
250	C₂×C₁₆	—	166	4	no	—
251	#97	—	83	1	no	—
252	#97	—	83	1	no	—
253	#97	—	83	1	no	—
254	#97	—	83	1	no	—
255	#97	C₄×D₈₃	1	1	yes	characteristic
256	#97	C₂²×D₈₃	1	1	yes	characteristic
257	C₄×C₈	—	83	1	no	—
258	C₄×C₈	—	83	1	no	—
259	C₄×C₈	—	83	1	no	—
260	C₄×C₈	—	83	1	no	—
261	C₄×C₈	—	83	1	no	—
262	C₄×C₈	—	83	1	no	—
263	C₄×C₈	—	83	1	no	—
264	C₄×C₈	—	83	1	no	—
265	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
267	#41	—	166	2	no	—
268	#41	—	166	2	no	—
269	C₂×Q₈	—	83	1	no	—
270	C₂×D₄	—	83	1	no	—
271	C₂×D₄	—	83	1	no	—
272	C₂×D₄	—	83	1	no	—
273	C₂²×C₄	—	2	1	no	—
274	C₂²×C₄	—	83	1	no	—
275	C₂²×C₄	—	83	1	no	—
276	C₂²×C₄	—	166	1	no	—
277	C₂²×C₄	—	166	1	no	—
278	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	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	—	83	1	no	—
286	C₂×C₈	—	83	1	no	—
287	C₂×C₈	—	83	4	no	—
288	C₂×C₈	—	83	2	no	—
289	C₂×C₈	—	83	2	no	—
290	C₂×C₈	—	83	1	no	—
291	C₂×C₈	—	83	1	no	—
292	C₂×C₈	—	166	1	no	—
293	C₂×C₈	—	166	1	no	—
294	C₂×C₈	—	166	2	no	—
295	C₂×C₈	—	166	1	no	—
296	C₂×C₈	—	166	1	no	—
297	C₂×C₈	—	166	2	no	—
298	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
299	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
300	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
301	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
302	C₄⋊C₄	—	166	1	no	—
303	C₄⋊C₄	—	166	1	no	—
304	C₄⋊C₄	—	166	1	no	—
305	C₂²⋊C₄	—	166	1	no	—
306	C₂²⋊C₄	—	166	1	no	—
307	C₂²⋊C₄	—	166	1	no	—
308	C₄²	—	2	1	no	—
309	C₄²	—	83	1	no	—
310	C₄²	—	83	1	no	—
311	C₄²	—	166	1	no	—
312	C₄²	—	166	1	no	—
313	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
314	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
315	C₁₆	—	2	4	no	—
316	C₁₆	—	166	4	no	—
317	C₂³	—	83	1	no	—
318	C₂³	—	83	1	no	—
319	C₂³	M₁₆×D₈₃	1	1	yes	characteristic
320	Q₈	—	166	2	no	—
321	D₄	—	166	2	no	—
322	D₄	—	166	2	no	—
323	D₄	—	166	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	332	2	no	—
330	C₂×C₄	—	332	2	no	—
331	C₂×C₄	—	83	1	no	—
332	C₂×C₄	—	83	2	no	—
333	C₂×C₄	—	83	2	no	—
334	C₂×C₄	—	83	1	no	—
335	C₂×C₄	—	166	1	no	—
336	C₂×C₄	—	166	1	no	—
337	C₂×C₄	—	166	1	no	—
338	C₂×C₄	—	166	1	no	—
339	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
340	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	83	1	no	—
344	C₈	—	83	1	no	—
345	C₈	—	83	2	no	—
346	C₈	—	166	1	no	—
347	C₈	—	166	1	no	—
348	C₈	M₁₆×D₈₃	1	2	yes	—
349	C₈	#41×D₈₃	1	2	yes	—
350	C₂²	—	2	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	—	332	1	no	—
353	C₂²	—	332	1	no	—
354	C₂²	—	83	2	no	—
355	C₂²	—	166	1	no	—
356	C₂²	C₂×M₁₆×D₈₃	1	1	yes	characteristic
357	C₄	—	2	2	no	—
358	C₄	—	2	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	332	1	no	—
361	C₄	—	332	1	no	—
362	C₄	—	83	2	no	—
363	C₄	—	166	1	no	—
364	C₄	#117×D₈₃	1	1	yes	characteristic
365	C₄	C₂×M₁₆×D₈₃	1	1	yes	characteristic
366	C₂	—	2	1	no	—
367	C₂	—	4	1	no	—
368	C₂	—	332	1	no	—
369	C₂	—	166	2	no	—
370	C₂	#432×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#2165	C₂
socle	C₂×C₈₃	C₂×#432
prime radical	C₂×#13050590	C₂×C₄
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	#2165	D₈₃
hypocenter	C₈₃	#33428
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ #97 ◁ #2165

Complements

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

N	H ≅ G/N	type
2. #24909277	C₂	semidirect
3. #24909276	C₂	semidirect
4. #24909275	C₂	semidirect
5. #24909274	C₂	semidirect
6. #364×D₈₃	C₂	semidirect
8. #24909066	C₂	semidirect
9. C₈₃×#2165	C₂	semidirect
191. #2165	D₈₃	semidirect
195. C₈₃	#33428	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14581800	2
#24909277	2
#24909276	2
#24909275	2
#24909274	2
#364×D₈₃	2
#24909082	2
#24909066	2
C₈₃×#2165	2
#33428	83