Subgroups — #35504366

Order 21248.

608 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24917986	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19127822	C₂	1	1	yes	maximal, characteristic
3	#24908638	C₂	1	2	yes	maximal, complemented
4	#24908634	C₂	1	2	yes	maximal, complemented
5	#24908361	C₂	1	1	yes	maximal, characteristic
6	#24908359	C₂	1	1	yes	maximal, characteristic
7	#24907588	C₂	1	1	yes	maximal, characteristic, complemented
8	#24907516	C₂	1	2	yes	maximal, complemented
9	#24907508	C₂	1	2	yes	maximal, complemented
10	#24907348	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₃×#1833	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14581814	C₂²	1	1	yes	characteristic
13	C₂×#14581800	C₂²	1	2	yes	—
14	#19128254	C₂²	1	2	yes	—
15	#19128109	—	2	2	no	—
16	#19128108	—	2	2	no	—
17	#19127822	—	2	2	no	—
18	#19127822	—	2	2	no	—
19	C₂×#14581727	C₂²	1	1	yes	characteristic
20	#19127739	C₂²	1	1	yes	characteristic
21	#19127737	C₂²	1	1	yes	characteristic
22	#19127737	C₂²	1	2	yes	—
23	#19127719	C₂²	1	2	yes	—
24	#19127717	C₂²	1	2	yes	—
25	C₂×#14581718	C₂²	1	2	yes	—
26	#19127674	C₂²	1	2	yes	—
27	#19127674	C₂²	1	2	yes	—
28	#19127656	C₂²	1	1	yes	characteristic
29	C₂×#14581702	C₂²	1	1	yes	characteristic
30	#19127473	C₂²	1	2	yes	—
31	#19127462	C₂²	1	1	yes	characteristic
32	#19127460	C₂²	1	1	yes	characteristic
33	#19127458	C₂²	1	2	yes	—
34	C₂×#108×C₈₃	C₂²	1	1	yes	characteristic
35	#427×C₈₃	C₂²	1	1	yes	characteristic
36	#405×C₈₃	C₂²	1	1	yes	characteristic
37	#344×C₈₃	C₂²	1	2	yes	—
38	H(3,ℤ/4)×C₈₃	C₂²	1	2	yes	—
39	C₂×#13050590	C₂³	1	2	yes	—
40	#14581814	—	2	1	no	—
41	#14581814	—	2	1	no	—
42	#14581814	—	2	1	no	—
43	#14581814	—	2	1	no	—
44	C₂×Dic₃₃₂	Q₈	1	1	yes	characteristic, complemented
45	C₂×#13050579	Q₈	1	2	yes	complemented
46	C₂×D₆₆₄	Q₈	1	1	yes	characteristic, complemented
47	#14581800	—	2	2	no	—
48	#14581800	—	2	2	no	—
49	#14581800	—	2	2	no	—
50	#14581800	—	2	2	no	—
51	C₂×#13050574	C₂³	1	1	yes	characteristic
52	C₂×#13050573	—	2	2	no	—
53	C₂²×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
54	C₂×C₈×D₈₃	—	2	2	no	—
55	#14581791	—	2	2	no	—
56	#14581783	—	2	2	no	—
57	#14581769	—	4	2	no	—
58	C₄×Dic₁₆₆	—	4	2	no	—
59	#14581751	C₂³	1	1	yes	characteristic
60	#14581751	C₂³	1	2	yes	—
61	C₂×C₄×Dic₈₃	C₂³	1	1	yes	characteristic
62	#14581727	—	2	2	no	—
63	#14581726	—	2	1	no	—
64	#14581725	—	2	2	no	—
65	#14581725	—	2	1	no	—
66	#14581724	—	2	1	no	—
67	#14581723	—	2	2	no	—
68	#14581723	—	2	1	no	—
69	#14581720	—	4	2	no	—
70	#14581718	—	2	4	no	—
71	#14581718	—	4	2	no	—
72	#14581706	—	4	2	no	—
73	#14581706	—	4	2	no	—
74	#14581702	—	2	1	no	—
75	#14581702	D₄	1	2	yes	—
76	#14581702	D₄	1	2	yes	—
77	C₈×Dic₈₃	—	2	1	no	—
78	C₂×M₁₆×C₈₃	C₂³	1	2	yes	—
79	C₂×M₁₆×C₈₃	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
81	#117×C₈₃	C₂³	1	1	yes	characteristic
82	#117×C₈₃	C₂³	1	2	yes	—
83	#108×C₈₃	—	2	1	no	—
84	#108×C₈₃	—	2	1	no	—
85	#107×C₈₃	—	2	1	no	—
86	#106×C₈₃	—	2	1	no	—
87	#97×C₈₃	—	2	1	no	—
88	C₄×C₈×C₈₃	—	2	1	no	—
89	#13050590	—	2	2	no	—
90	#13050590	—	2	2	no	—
91	#13050590	—	2	2	no	—
92	#13050590	—	2	2	no	—
93	C₂×Dic₁₆₆	—	4	2	no	—
94	C₂×Dic₁₆₆	C₂×Q₈	1	2	yes	—
95	C₂×D₃₃₂	C₂×Q₈	1	2	yes	—
96	C₂×C₂²⋊D₈₃	—	2	2	no	—
97	C₂²×Dic₈₃	—	2	1	no	—
98	C₂×C₄×D₈₃	—	2	2	no	—
99	Dic₃₃₂	—	2	1	no	—
100	Dic₃₃₂	—	2	1	no	—
101	#13050579	—	2	2	no	—
102	#13050579	—	2	2	no	—
103	D₆₆₄	—	2	1	no	—
104	D₆₆₄	—	2	1	no	—
105	#13050574	—	2	1	no	—
106	#13050574	—	2	1	no	—
107	#13050573	—	4	2	no	—
108	#13050573	—	4	2	no	—
109	C₂×C₈₃⋊C₈	—	2	1	no	—
110	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
111	C₂×C₈₃⋊C₈	#41	1	2	yes	—
112	C₂×C₈₃⋊C₈	#41	1	1	yes	characteristic
113	C₂×C₈₃⋊C₈	#41	1	1	yes	characteristic
114	C₈×D₈₃	—	4	2	no	—
115	C₈×D₈₃	—	4	2	no	—
116	C₄⋊Dic₈₃	—	2	2	no	—
117	C₄⋊Dic₈₃	—	2	1	no	—
118	C₄⋊Dic₈₃	—	4	2	no	—
119	Dic₈₃⋊C₄	—	4	2	no	—
120	Dic₈₃⋊C₄	—	4	2	no	—
121	C₂²⋊Dic₈₃	—	2	2	no	—
122	C₂²⋊Dic₈₃	—	2	1	no	—
123	C₄×Dic₈₃	—	2	1	no	—
124	C₄×Dic₈₃	—	2	1	no	—
125	C₄×Dic₈₃	—	2	2	no	—
126	C₄×Dic₈₃	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
128	M₁₆×C₈₃	—	2	2	no	—
129	M₁₆×C₈₃	—	2	2	no	—
130	M₁₆×C₈₃	—	2	1	no	—
131	M₁₆×C₈₃	—	2	1	no	—
132	C₂×C₈×C₈₃	—	2	2	no	—
133	C₂×C₈×C₈₃	—	2	1	no	—
134	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
137	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
138	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
139	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
140	C₄⋊C₄×C₈₃	—	2	2	no	—
141	C₄⋊C₄×C₈₃	—	2	1	no	—
142	C₂²⋊C₄×C₈₃	—	2	2	no	—
143	C₂²⋊C₄×C₈₃	—	2	1	no	—
144	C₄²×C₈₃	—	2	1	no	—
145	C₄²×C₈₃	—	2	2	no	—
146	C₂³×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂²×D₈₃	—	2	2	no	—
148	C₂×C₄×C₈₃	—	4	2	no	—
149	C₂×C₄×C₈₃	—	4	1	no	—
150	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
152	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²⋊D₈₃	—	4	2	no	—
157	C₂²⋊D₈₃	—	4	2	no	—
158	C₂×Dic₈₃	—	2	2	no	—
159	C₂×Dic₈₃	—	2	2	no	—
160	C₂×Dic₈₃	—	4	2	no	—
161	C₂×Dic₈₃	—	4	1	no	—
162	C₂×Dic₈₃	—	8	1	no	—
163	D₃₃₂	—	2	2	no	—
164	D₃₃₂	—	2	2	no	—
165	C₄×D₈₃	—	4	2	no	—
166	C₄×D₈₃	—	4	2	no	—
167	Dic₁₆₆	—	2	2	no	—
168	Dic₁₆₆	—	2	2	no	—
169	Dic₁₆₆	—	8	4	no	—
170	C₈×C₈₃	—	4	2	no	—
171	C₈×C₈₃	—	4	1	no	—
172	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
173	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
174	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
175	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
176	C₈₃⋊C₈	—	4	1	no	—
177	C₈₃⋊C₈	#124	1	2	yes	—
178	C₈₃⋊C₈	#124	1	2	yes	—
179	C₂²×C₈₃	—	4	1	no	—
180	C₂²×C₈₃	#535	1	1	yes	characteristic
181	C₂²×C₈₃	#554	1	1	yes	characteristic
182	C₂²×C₈₃	#555	1	1	yes	characteristic
183	C₂×D₈₃	—	4	2	no	—
184	C₂×D₈₃	—	8	2	no	—
185	C₄×C₈₃	—	8	2	no	—
186	C₄×C₈₃	—	8	1	no	—
187	C₄×C₈₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
188	C₄×C₈₃	C₂×#124	1	1	yes	characteristic
189	C₄×C₈₃	#554	1	1	yes	characteristic
190	C₄×C₈₃	#555	1	1	yes	characteristic
191	Dic₈₃	—	4	2	no	—
192	Dic₈₃	—	4	2	no	—
193	Dic₈₃	—	8	2	no	—
194	Dic₈₃	—	8	1	no	—
195	#29155	—	83	1	no	maximal, Sylow
196	C₂×C₈₃	—	2	1	no	—
197	C₂×C₈₃	—	2	1	no	—
198	C₂×C₈₃	—	2	1	no	—
199	C₂×C₈₃	#2563	1	1	yes	characteristic
200	D₈₃	—	8	2	no	—
201	#3162	—	83	1	no	—
202	C₂×#444	—	83	1	no	—
203	#2031	—	83	2	no	—
204	#2031	—	83	2	no	—
205	#1976	—	83	1	no	—
206	#1976	—	83	1	no	—
207	#1833	—	83	1	no	—
208	#1833	D₈₃	1	1	yes	characteristic, complemented
209	#1814	—	83	2	no	—
210	#1812	—	83	2	no	—
211	#1773	—	83	1	no	—
212	C₈₃	#29155	1	1	yes	characteristic, complemented, Sylow
213	C₂×#135	—	83	1	no	—
214	C₂×#131	—	83	2	no	—
215	#533	—	83	2	no	—
216	#501	—	166	2	no	—
217	#501	—	166	2	no	—
218	#444	—	166	2	no	—
219	#444	—	166	2	no	—
220	C₂×#108	—	83	1	no	—
221	C₂×#108	C₂×D₈₃	1	1	yes	characteristic
222	#427	—	83	1	no	—
223	#427	—	83	1	no	—
224	#427	—	83	2	no	—
225	#427	C₂×D₈₃	1	1	yes	characteristic
226	#421	—	83	2	no	—
227	#421	—	83	2	no	—
228	C₂×#104	—	83	2	no	—
229	#410	—	83	2	no	—
230	#410	—	83	2	no	—
231	#405	—	83	1	no	—
232	#405	C₂×D₈₃	1	1	yes	characteristic
233	C₂×#97	—	83	1	no	—
234	#344	—	83	2	no	—
235	#344	C₂×D₈₃	1	2	yes	—
236	#339	—	83	1	no	—
237	#339	—	83	1	no	—
238	H(3,ℤ/4)	—	83	2	no	—
239	H(3,ℤ/4)	C₂×D₈₃	1	2	yes	—
240	C₂×#41	—	83	2	no	—
241	#135	—	166	1	no	—
242	#135	—	166	1	no	—
243	#135	—	166	1	no	—
244	#135	—	166	1	no	—
245	C₂×Q₁₆	—	83	1	no	—
246	C₂×SD₁₆	—	83	2	no	—
247	C₂×D₈	—	83	1	no	—
248	#131	—	166	2	no	—
249	#131	—	166	2	no	—
250	#131	—	166	2	no	—
251	#131	—	166	2	no	—
252	C₂×M₁₆	—	83	1	no	—
253	C₂×M₁₆	—	166	2	no	—
254	C₂×M₁₆	C₂²×D₈₃	1	2	yes	—
255	C₂×M₁₆	C₂²×D₈₃	1	1	yes	characteristic
256	C₂²×C₈	—	83	1	no	—
257	C₂²×C₈	—	166	2	no	—
258	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
259	C₄⋊Q₈	—	166	2	no	—
260	#125	—	166	2	no	—
261	C₂²⋊Q₈	—	332	2	no	—
262	C₄×Q₈	—	332	2	no	—
263	#117	—	83	2	no	—
264	#117	—	83	1	no	—
265	#117	C₂²×D₈₃	1	2	yes	—
266	#117	C₂²×D₈₃	1	1	yes	characteristic
267	C₂×C₄²	—	83	1	no	—
268	#108	—	2	1	no	—
269	#108	—	2	1	no	—
270	#108	—	166	2	no	—
271	#107	—	2	1	no	—
272	#107	—	166	1	no	—
273	#107	—	166	2	no	—
274	#107	—	166	1	no	—
275	#106	—	2	1	no	—
276	#106	—	166	1	no	—
277	#106	—	166	2	no	—
278	#106	—	166	1	no	—
279	#104	—	332	2	no	—
280	#104	—	332	2	no	—
281	#104	—	166	4	no	—
282	#99	—	332	2	no	—
283	#99	—	332	2	no	—
284	#97	—	2	1	no	—
285	#97	—	83	2	no	—
286	#97	—	83	2	no	—
287	#97	—	166	1	no	—
288	C₄×C₈	—	2	1	no	—
289	C₄×C₈	—	166	1	no	—
290	#41	—	166	2	no	—
291	#41	—	166	2	no	—
292	#41	—	166	2	no	—
293	#41	—	166	2	no	—
294	C₂×Q₈	—	332	2	no	—
295	C₂×Q₈	—	83	2	no	—
296	C₂×D₄	—	83	2	no	—
297	C₂×D₄	—	166	2	no	—
298	C₂²×C₄	—	166	2	no	—
299	C₂²×C₄	—	166	1	no	—
300	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
301	Q₁₆	—	166	1	no	—
302	Q₁₆	—	166	1	no	—
303	SD₁₆	—	166	2	no	—
304	SD₁₆	—	166	2	no	—
305	D₈	—	166	1	no	—
306	D₈	—	166	1	no	—
307	M₁₆	—	2	2	no	—
308	M₁₆	—	2	1	no	—
309	M₁₆	—	2	2	no	—
310	M₁₆	—	2	1	no	—
311	M₁₆	—	332	2	no	—
312	M₁₆	—	332	2	no	—
313	M₁₆	—	166	1	no	—
314	M₁₆	—	166	1	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	332	2	no	—
318	C₂×C₈	—	332	2	no	—
319	C₂×C₈	—	83	2	no	—
320	C₂×C₈	—	83	1	no	—
321	C₂×C₈	—	83	1	no	—
322	C₂×C₈	—	83	2	no	—
323	C₂×C₈	—	166	1	no	—
324	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
325	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
326	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
327	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
328	C₂×C₈	#13050590	1	1	yes	characteristic
329	C₂×C₈	#13050590	1	1	yes	characteristic
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	1	no	—
332	C₄⋊C₄	—	332	2	no	—
333	C₄⋊C₄	—	332	2	no	—
334	C₄⋊C₄	—	332	2	no	—
335	C₄⋊C₄	—	166	2	no	—
336	C₄⋊C₄	—	166	1	no	—
337	C₂²⋊C₄	—	2	1	no	—
338	C₂²⋊C₄	—	2	2	no	—
339	C₂²⋊C₄	—	166	1	no	—
340	C₂²⋊C₄	—	166	2	no	—
341	C₄²	—	2	1	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	83	2	no	—
344	C₄²	—	166	1	no	—
345	C₄²	—	166	2	no	—
346	C₄²	—	166	1	no	—
347	C₂³	—	166	2	no	—
348	C₂³	#14581838	1	1	yes	characteristic
349	Q₈	—	664	4	no	—
350	Q₈	—	166	2	no	—
351	Q₈	—	166	2	no	—
352	D₄	—	332	2	no	—
353	D₄	—	332	2	no	—
354	D₄	—	166	2	no	—
355	D₄	—	166	2	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	332	2	no	—
359	C₂×C₄	—	332	2	no	—
360	C₂×C₄	—	332	1	no	—
361	C₂×C₄	—	332	2	no	—
362	C₂×C₄	—	664	1	no	—
363	C₂×C₄	—	166	2	no	—
364	C₂×C₄	—	166	2	no	—
365	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
366	C₂×C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
367	C₂×C₄	#41×D₈₃	1	1	yes	characteristic
368	C₂×C₄	#41×D₈₃	1	1	yes	characteristic
369	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
370	C₂×C₄	#14581838	1	1	yes	characteristic
371	C₈	—	4	1	no	—
372	C₈	—	4	2	no	—
373	C₈	—	332	1	no	—
374	C₈	—	83	2	no	—
375	C₈	—	83	2	no	—
376	C₈	#14581770	1	1	yes	characteristic
377	C₈	#14581770	1	1	yes	characteristic
378	C₈	#14581770	1	1	yes	characteristic
379	C₈	#14581770	1	1	yes	characteristic
380	C₂²	—	4	1	no	—
381	C₂²	—	332	2	no	—
382	C₂²	—	664	2	no	—
383	C₂²	#19128269	1	1	yes	characteristic
384	C₂²	#19128407	1	1	yes	characteristic
385	C₂²	#19128414	1	1	yes	characteristic
386	C₄	—	8	2	no	—
387	C₄	—	8	1	no	—
388	C₄	—	332	2	no	—
389	C₄	—	332	2	no	—
390	C₄	—	664	2	no	—
391	C₄	—	664	1	no	—
392	C₄	C₂×#14581770	1	1	yes	characteristic
393	C₄	#124×D₈₃	1	1	yes	characteristic
394	C₄	#19128407	1	1	yes	characteristic
395	C₄	#19128414	1	1	yes	characteristic
396	C₂	—	2	1	no	—
397	C₂	—	2	1	no	—
398	C₂	—	2	1	no	—
399	C₂	—	664	2	no	—
400	C₂	#24912635	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19128414
commutator, cosocle	C₂²×C₄×C₈₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	C₈₃×#1833	C₂
socle	C₂×C₈₃	#2563
prime radical	C₂×#14581814	C₂²
hypercenter	#1833	D₈₃
hypocenter	C₈₃	#29155
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1833

Complements

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

N	H ≅ G/N	type
1. #24917986	C₂	semidirect
3. #24908638	C₂	semidirect
4. #24908634	C₂	semidirect
7. #24907588	C₂	semidirect
8. #24907516	C₂	semidirect
9. #24907508	C₂	semidirect
10. #24907348	C₂	semidirect
11. C₈₃×#1833	C₂	semidirect
44. C₂×Dic₃₃₂	Q₈	semidirect
45. C₂×#13050579	Q₈	semidirect
46. C₂×D₆₆₄	Q₈	semidirect
208. #1833	D₈₃	semidirect
212. C₈₃	#29155	semidirect

Maximal subgroups

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

subgroup	index
#24917986	2
C₂×#19127822	2
#24908638	2
#24908634	2
#24908361	2
#24908359	2
#24907588	2
#24907516	2
#24907508	2
#24907348	2
C₈₃×#1833	2
#29155	83