Subgroups — #35557878

Order 21248.

668 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128139	C₂	1	1	yes	maximal, characteristic
2	#24916379	C₂	1	1	yes	maximal, characteristic, complemented
3	#24916314	C₂	1	1	yes	maximal, characteristic, complemented
4	#24909360	C₂	1	8	yes	maximal, complemented
5	C₂×#19127513	C₂	1	1	yes	maximal, characteristic, complemented
6	#24909198	C₂	1	1	yes	maximal, characteristic, complemented
7	#24909067	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₃×#2155	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14581814	C₂²	1	1	yes	characteristic
10	C₂²×D₆₆₄	C₂²	1	1	yes	characteristic
11	#19128467	—	2	1	no	—
12	#19128175	—	2	1	no	—
13	#19128139	C₄	1	4	yes	complemented
14	#19128139	C₂²	1	4	yes	—
15	C₂×#14581737	C₄	1	2	yes	complemented
16	C₂×D₁₃₂₈	C₄	1	2	yes	complemented
17	C₂×#14581732	C₂²	1	1	yes	characteristic
18	#19127822	C₂²	1	4	yes	complemented
19	#19127817	—	2	2	no	—
20	#19127812	C₂²	1	4	yes	complemented
21	#19127783	—	2	2	no	—
22	#19127717	—	2	1	no	—
23	#19127705	—	2	1	no	—
24	#19127656	C₂²	1	1	yes	characteristic
25	#19127513	C₂²	1	4	yes	complemented
26	#19127510	C₂²	1	4	yes	complemented
27	#19127478	C₂²	1	4	yes	complemented
28	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
29	C₂×#108×C₈₃	C₂²	1	1	yes	characteristic
30	#427×C₈₃	C₂²	1	1	yes	characteristic
31	#365×C₈₃	C₂²	1	4	yes	complemented
32	C₂×#13050590	—	2	1	no	—
33	C₂²×D₃₃₂	—	2	1	no	—
34	#14581814	—	2	1	no	—
35	#14581814	—	2	1	no	—
36	#14581814	C₂×C₄	1	2	yes	complemented
37	#14581814	C₂³	1	2	yes	—
38	C₂×Dic₃₃₂	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂×#13050579	—	2	1	no	—
40	C₂×D₆₆₄	—	2	1	no	—
41	C₂×D₆₆₄	—	2	1	no	—
42	C₂×D₆₆₄	—	2	1	no	—
43	C₂×D₆₆₄	C₂×C₄	1	2	yes	—
44	C₂×D₆₆₄	C₂×C₄	1	2	yes	—
45	C₂×D₆₆₄	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×D₆₆₄	C₂³	1	2	yes	—
47	#14581800	—	2	4	no	—
48	#14581798	—	4	2	no	—
49	C₂×#13050574	D₄	1	1	yes	characteristic
50	M₁₆×D₈₃	—	4	2	no	—
51	C₄×D₃₃₂	—	4	2	no	—
52	#14581754	—	2	4	no	—
53	#14581751	D₄	1	1	yes	characteristic
54	C₂²⋊C₄×D₈₃	—	4	2	no	—
55	#14581737	C₂×C₄	1	8	yes	complemented
56	D₁₃₂₈	C₂×C₄	1	8	yes	complemented
57	#14581732	—	2	1	no	—
58	#14581732	C₂³	1	2	yes	—
59	C₂×C₈₃⋊C₁₆	—	2	1	no	—
60	#14581718	—	2	4	no	—
61	#14581712	—	2	4	no	—
62	#14581702	C₂³	1	2	yes	—
63	C₈×Dic₈₃	D₄	1	2	yes	—
64	C₂×M₁₆×C₈₃	—	2	1	no	—
65	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
66	#117×C₈₃	—	2	1	no	—
67	M₃₂×C₈₃	C₂×C₄	1	2	yes	complemented
68	M₃₂×C₈₃	C₂³	1	2	yes	—
69	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	complemented
70	#108×C₈₃	—	2	1	no	—
71	#108×C₈₃	C₂³	1	2	yes	—
72	#107×C₈₃	—	2	1	no	—
73	#106×C₈₃	C₂³	1	2	yes	—
74	C₂³×D₈₃	—	4	1	no	—
75	#13050590	—	2	2	no	—
76	#13050590	—	2	2	no	—
77	#13050590	—	4	1	no	—
78	#13050590	—	4	1	no	—
79	C₂×Dic₁₆₆	—	2	1	no	—
80	C₂×D₃₃₂	—	2	2	no	—
81	C₂×D₃₃₂	—	2	2	no	—
82	C₂×D₃₃₂	—	2	1	no	—
83	C₂×D₃₃₂	—	2	2	no	—
84	C₂×D₃₃₂	—	4	1	no	—
85	C₂×D₃₃₂	—	4	1	no	—
86	C₂×D₃₃₂	—	4	1	no	—
87	C₂×C₂²⋊D₈₃	—	4	1	no	—
88	C₂×C₄×D₈₃	—	4	1	no	—
89	C₂×C₄×D₈₃	—	4	4	no	—
90	Dic₃₃₂	—	2	1	no	—
91	Dic₃₃₂	C₂²×C₄	1	2	yes	—
92	#13050579	—	2	2	no	—
93	#13050579	—	2	2	no	—
94	D₆₆₄	—	2	2	no	—
95	D₆₆₄	—	2	2	no	—
96	D₆₆₄	—	2	1	no	—
97	D₆₆₄	—	2	2	no	—
98	D₆₆₄	C₂²×C₄	1	4	yes	—
99	D₆₆₄	C₂²×C₄	1	2	yes	—
100	#13050574	—	2	1	no	—
101	#13050574	—	2	1	no	—
102	#13050573	—	4	4	no	—
103	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
104	C₈×D₈₃	—	4	4	no	—
105	C₄⋊Dic₈₃	—	2	1	no	—
106	C₂²⋊Dic₈₃	—	2	1	no	—
107	(C₂×D₈₃)⋊C₄	—	4	4	no	—
108	C₄×Dic₈₃	C₂×D₄	1	2	yes	—
109	C₈₃⋊C₁₆	—	2	2	no	—
110	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
111	M₁₆×C₈₃	—	2	2	no	—
112	M₁₆×C₈₃	—	4	1	no	—
113	C₂×C₈×C₈₃	—	4	1	no	—
114	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
118	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
119	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₈₃	—	2	2	no	—
121	C₂²⋊C₄×C₈₃	—	4	1	no	—
122	C₄²×C₈₃	—	4	1	no	—
123	C₁₆×C₈₃	C₂²×C₄	1	4	yes	—
124	C₂³×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂²×D₈₃	—	4	2	no	—
126	C₂²×D₈₃	—	4	1	no	—
127	C₂²×D₈₃	—	4	2	no	—
128	C₂²×D₈₃	—	4	2	no	—
129	C₂²×D₈₃	—	16	2	no	—
130	C₂×C₄×C₈₃	—	4	2	no	—
131	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₈₃	—	4	2	no	—
138	C₂²⋊D₈₃	—	4	2	no	—
139	C₂×Dic₈₃	—	2	2	no	—
140	C₂×Dic₈₃	—	4	1	no	—
141	D₃₃₂	—	2	4	no	—
142	D₃₃₂	—	2	2	no	—
143	D₃₃₂	—	4	2	no	—
144	D₃₃₂	—	4	2	no	—
145	D₃₃₂	—	4	2	no	—
146	D₃₃₂	—	4	1	no	—
147	C₄×D₈₃	—	4	2	no	—
148	C₄×D₈₃	—	4	2	no	—
149	C₄×D₈₃	—	16	4	no	—
150	Dic₁₆₆	—	2	2	no	—
151	Dic₁₆₆	—	4	1	no	—
152	C₈×C₈₃	—	4	2	no	—
153	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
154	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
155	C₈×C₈₃	C₂³×C₄	1	1	yes	characteristic
156	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
157	C₈₃⋊C₈	—	2	2	no	—
158	C₂²×C₈₃	—	4	1	no	—
159	C₂²×C₈₃	#442	1	1	yes	characteristic
160	C₂²×C₈₃	#442	1	1	yes	characteristic
161	C₂²×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×D₈₃	—	4	2	no	—
163	C₂×D₈₃	—	4	4	no	—
164	C₂×D₈₃	—	8	4	no	—
165	C₂×D₈₃	—	16	1	no	—
166	C₂×D₈₃	—	16	2	no	—
167	C₂×D₈₃	—	16	2	no	—
168	C₄×C₈₃	—	8	2	no	—
169	C₄×C₈₃	#442	1	1	yes	characteristic
170	C₄×C₈₃	#442	1	1	yes	characteristic
171	C₄×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
172	C₄×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	Dic₈₃	—	4	2	no	—
174	Dic₈₃	—	4	2	no	—
175	#33471	—	83	1	no	maximal, Sylow
176	C₂×C₈₃	—	2	1	no	—
177	C₂×C₈₃	—	2	1	no	—
178	C₂×C₈₃	—	2	1	no	—
179	C₂×C₈₃	C₂×#442	1	1	yes	characteristic
180	D₈₃	—	8	4	no	—
181	D₈₃	—	8	2	no	—
182	C₂×#509	—	83	1	no	—
183	#2957	—	83	1	no	—
184	#2948	—	83	1	no	—
185	AGL(1,ℤ/16)	—	83	8	no	—
186	#2155	D₈₃	1	1	yes	characteristic, complemented
187	C₂×#361	—	83	1	no	—
188	#2145	—	83	1	no	—
189	#2110	—	83	1	no	—
190	C₈₃	#33471	1	1	yes	characteristic, complemented, Sylow
191	C₂×#135	—	83	1	no	—
192	C₂²×D₈	—	83	1	no	—
193	#568	—	166	1	no	—
194	#518	—	166	1	no	—
195	#509	—	83	4	no	—
196	#509	—	83	4	no	—
197	C₂×SD₃₂	—	83	2	no	—
198	C₂×D₁₆	—	83	2	no	—
199	C₂×M₃₂	—	83	1	no	—
200	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
201	#444	—	83	4	no	—
202	#443	—	166	2	no	—
203	#442	—	83	4	no	—
204	C₄×D₈	—	166	2	no	—
205	C₂×#108	C₂×D₈₃	1	1	yes	characteristic
206	#427	C₂×D₈₃	1	1	yes	characteristic
207	#421	—	166	1	no	—
208	#418	—	166	1	no	—
209	#405	—	83	1	no	—
210	#365	C₂×D₈₃	1	4	yes	complemented
211	#361	—	83	4	no	—
212	#360	—	83	4	no	—
213	#347	—	83	4	no	—
214	C₂×#41	—	166	1	no	—
215	C₂²×D₄	—	166	1	no	—
216	#135	—	83	2	no	—
217	#135	—	83	2	no	—
218	#135	—	166	1	no	—
219	#135	—	166	1	no	—
220	C₂×Q₁₆	—	83	1	no	—
221	C₂×SD₁₆	—	166	1	no	—
222	C₂×D₈	—	83	2	no	—
223	C₂×D₈	—	83	2	no	—
224	C₂×D₈	—	83	1	no	—
225	C₂×D₈	—	83	2	no	—
226	C₂×D₈	—	166	1	no	—
227	C₂×D₈	—	166	1	no	—
228	C₂×D₈	—	166	1	no	—
229	#131	—	332	2	no	—
230	#131	—	166	4	no	—
231	C₂×M₁₆	—	2	1	no	—
232	C₂×M₁₆	—	332	2	no	—
233	C₂×M₁₆	—	83	1	no	—
234	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
235	C₄×D₄	—	332	2	no	—
236	C₄×D₄	—	166	4	no	—
237	#117	—	2	1	no	—
238	#117	—	83	1	no	—
239	C₂×C₂²⋊C₄	—	332	2	no	—
240	SD₃₂	—	83	8	no	—
241	D₁₆	—	83	8	no	—
242	M₃₂	—	83	2	no	—
243	M₃₂	—	166	1	no	—
244	M₃₂	C₄×D₈₃	1	2	yes	complemented
245	M₃₂	C₂²×D₈₃	1	2	yes	—
246	C₂×C₁₆	—	166	1	no	—
247	C₂×C₁₆	C₄×D₈₃	1	2	yes	complemented
248	#108	—	2	1	no	—
249	#108	C₂²×D₈₃	1	2	yes	—
250	#107	—	2	1	no	—
251	#106	C₂²×D₈₃	1	2	yes	—
252	#104	—	166	4	no	—
253	#102	—	166	4	no	—
254	#97	—	83	2	no	—
255	C₄×C₈	—	83	2	no	—
256	C₂⁴	—	332	1	no	—
257	#41	—	332	1	no	—
258	#41	—	332	1	no	—
259	#41	—	166	2	no	—
260	#41	—	166	2	no	—
261	C₂×Q₈	—	166	1	no	—
262	C₂×D₄	—	332	1	no	—
263	C₂×D₄	—	332	1	no	—
264	C₂×D₄	—	332	1	no	—
265	C₂×D₄	—	332	1	no	—
266	C₂×D₄	—	166	2	no	—
267	C₂×D₄	—	166	2	no	—
268	C₂×D₄	—	166	2	no	—
269	C₂×D₄	—	166	1	no	—
270	C₂²×C₄	—	332	4	no	—
271	C₂²×C₄	—	332	1	no	—
272	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
273	Q₁₆	—	83	2	no	—
274	Q₁₆	—	166	1	no	—
275	SD₁₆	—	166	2	no	—
276	SD₁₆	—	166	2	no	—
277	D₈	—	83	2	no	—
278	D₈	—	83	4	no	—
279	D₈	—	166	2	no	—
280	D₈	—	166	2	no	—
281	D₈	—	166	2	no	—
282	D₈	—	166	1	no	—
283	M₁₆	—	2	2	no	—
284	M₁₆	—	4	1	no	—
285	M₁₆	—	332	4	no	—
286	M₁₆	—	166	1	no	—
287	M₁₆	—	166	1	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	332	4	no	—
290	C₂×C₈	—	83	2	no	—
291	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
293	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
294	C₂×C₈	#13050591	1	1	yes	characteristic
295	C₂×C₈	#13050591	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	166	1	no	—
299	C₂²⋊C₄	—	4	1	no	—
300	C₂²⋊C₄	—	332	4	no	—
301	C₂²⋊C₄	—	166	1	no	—
302	C₄²	—	4	1	no	—
303	C₄²	—	83	2	no	—
304	C₁₆	—	166	2	no	—
305	C₁₆	C₂×C₄×D₈₃	1	4	yes	—
306	C₂³	—	1328	2	no	—
307	C₂³	—	332	1	no	—
308	C₂³	—	332	2	no	—
309	C₂³	—	332	2	no	—
310	C₂³	—	332	2	no	—
311	C₂³	#14581819	1	1	yes	characteristic
312	Q₈	—	332	1	no	—
313	Q₈	—	166	2	no	—
314	D₄	—	332	2	no	—
315	D₄	—	332	2	no	—
316	D₄	—	332	2	no	—
317	D₄	—	332	2	no	—
318	D₄	—	332	1	no	—
319	D₄	—	332	2	no	—
320	D₄	—	166	4	no	—
321	D₄	—	166	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	1328	4	no	—
324	C₂×C₄	—	332	1	no	—
325	C₂×C₄	—	332	2	no	—
326	C₂×C₄	—	332	2	no	—
327	C₂×C₄	—	166	2	no	—
328	C₂×C₄	#14581754	1	1	yes	characteristic
329	C₂×C₄	#14581754	1	1	yes	characteristic
330	C₂×C₄	#14581819	1	1	yes	characteristic
331	C₂×C₄	#14581819	1	1	yes	characteristic
332	C₂×C₄	#14581819	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
334	C₈	—	4	2	no	—
335	C₈	—	166	2	no	—
336	C₈	#14581754	1	1	yes	characteristic
337	C₈	#14581754	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₈₃	1	1	yes	characteristic
339	C₈	C₂×#13050591	1	1	yes	characteristic
340	C₂²	—	4	1	no	—
341	C₂²	—	1328	2	no	—
342	C₂²	—	1328	2	no	—
343	C₂²	—	1328	1	no	—
344	C₂²	—	332	2	no	—
345	C₂²	—	332	4	no	—
346	C₂²	—	664	4	no	—
347	C₂²	#19127812	1	1	yes	characteristic
348	C₂²	#19127812	1	1	yes	characteristic
349	C₂²	C₂×#14581819	1	1	yes	characteristic
350	C₄	—	8	2	no	—
351	C₄	—	332	2	no	—
352	C₄	—	332	2	no	—
353	C₄	#19127812	1	1	yes	characteristic
354	C₄	#19127812	1	1	yes	characteristic
355	C₄	C₂×#14581754	1	1	yes	characteristic
356	C₄	C₂×#14581819	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	664	4	no	—
361	C₂	—	664	2	no	—
362	C₂	C₂×#19127812	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19127812
commutator	C₈×C₈₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#2155	C₂
socle	C₂×C₈₃	C₂×#442
prime radical	C₂×#19128139	C₂
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	#2155	D₈₃
hypocenter	C₈₃	#33471
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2155

Complements

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

N	H ≅ G/N	type
2. #24916379	C₂	semidirect
3. #24916314	C₂	semidirect
4. #24909360	C₂	semidirect
5. C₂×#19127513	C₂	semidirect
6. #24909198	C₂	semidirect
7. #24909067	C₂	semidirect
8. C₈₃×#2155	C₂	semidirect
13. #19128139	C₄	semidirect
15. C₂×#14581737	C₄	semidirect
16. C₂×D₁₃₂₈	C₄	semidirect
18. #19127822	C₂²	semidirect
20. #19127812	C₂²	semidirect
25. #19127513	C₂²	semidirect
26. #19127510	C₂²	semidirect
27. #19127478	C₂²	semidirect
31. #365×C₈₃	C₂²	semidirect
36. #14581814	C₂×C₄	semidirect
38. C₂×Dic₃₃₂	C₂×C₄	semidirect
45. C₂×D₆₆₄	C₂×C₄	semidirect
55. #14581737	C₂×C₄	semidirect
56. D₁₃₂₈	C₂×C₄	semidirect
67. M₃₂×C₈₃	C₂×C₄	semidirect
69. C₂×C₁₆×C₈₃	C₂×C₄	semidirect
186. #2155	D₈₃	semidirect
190. C₈₃	#33471	semidirect
210. #365	C₂×D₈₃	semidirect
244. M₃₂	C₄×D₈₃	semidirect
247. C₂×C₁₆	C₄×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128139	2
#24916379	2
#24916314	2
#24909360	2
C₂×#19127513	2
#24909198	2
#24909067	2
C₈₃×#2155	2
#33471	83