Subgroups — #35320479

Order 21248.

474 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127783	C₂	1	1	yes	maximal, characteristic
2	#24909218	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19127515	C₂	1	1	yes	maximal, characteristic
4	C₂×#19127513	C₂	1	1	yes	maximal, characteristic
5	#24907633	C₂	1	1	yes	maximal, characteristic, complemented
6	#24905571	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1387	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₆₆₄	C₂²	1	1	yes	characteristic
9	C₂×#14581754	—	2	1	no	—
10	C₂×#14581732	C₂²	1	1	yes	characteristic
11	#19127783	—	2	1	no	—
12	#19127783	—	2	1	no	—
13	#19127783	C₄	1	2	yes	—
14	#19127783	C₄	1	2	yes	—
15	C₂×#14581728	C₂²	1	1	yes	characteristic
16	C₂×#14581727	—	2	1	no	—
17	#19127724	—	2	1	no	—
18	C₂×#14581712	—	2	1	no	—
19	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
20	Dic₈₃⋊C₁₆	—	2	1	no	—
21	Dic₈₃⋊C₁₆	—	2	1	no	—
22	#19127515	—	2	2	no	—
23	#19127513	—	2	2	no	—
24	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
25	C₂×#108×C₈₃	C₂²	1	1	yes	characteristic
26	C₂×#107×C₈₃	C₂²	1	1	yes	characteristic
27	C₂²×D₃₃₂	—	2	1	no	—
28	C₂²×C₄×D₈₃	—	4	1	no	—
29	C₂×D₆₆₄	—	2	1	no	—
30	C₂×D₆₆₄	—	2	1	no	—
31	C₂×D₆₆₄	—	2	1	no	—
32	C₂×D₆₆₄	C₂×C₄	1	2	yes	—
33	C₂×D₆₆₄	D₄	1	2	yes	—
34	C₂×D₆₆₄	Q₈	1	2	yes	—
35	C₂×#13050574	—	2	1	no	—
36	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
37	#14581754	—	2	2	no	—
38	#14581754	—	2	2	no	—
39	#14581754	—	4	1	no	—
40	#14581754	—	4	1	no	—
41	C₂×(C₂×D₈₃)⋊C₄	—	4	1	no	—
42	C₂×C₄×Dic₈₃	D₄	1	1	yes	characteristic
43	#14581732	—	2	1	no	—
44	#14581732	—	2	1	no	—
45	C₂×C₈₃⋊C₁₆	—	2	1	no	—
46	#14581728	—	2	1	no	—
47	#14581728	—	2	1	no	—
48	#14581727	—	4	2	no	—
49	Dic₈₃⋊C₈	—	4	1	no	—
50	Dic₈₃⋊C₈	—	4	1	no	—
51	#14581712	—	2	2	no	—
52	#14581712	—	2	2	no	—
53	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
54	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₈₃	D₄	1	1	yes	characteristic
56	C₈×Dic₈₃	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₈₃	—	2	1	no	—
58	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
60	M₃₂×C₈₃	—	2	1	no	—
61	M₃₂×C₈₃	—	2	1	no	—
62	C₂×C₁₆×C₈₃	—	2	1	no	—
63	#108×C₈₃	—	2	1	no	—
64	#108×C₈₃	—	2	1	no	—
65	#107×C₈₃	—	2	1	no	—
66	#107×C₈₃	C₂×C₄	1	2	yes	—
67	C₂³×D₈₃	—	4	1	no	—
68	C₂×D₃₃₂	—	2	2	no	—
69	C₂×D₃₃₂	—	2	2	no	—
70	C₂×D₃₃₂	—	2	2	no	—
71	C₂×D₃₃₂	—	4	1	no	—
72	C₂×D₃₃₂	—	4	1	no	—
73	C₂×D₃₃₂	—	4	1	no	—
74	C₂²×Dic₈₃	SD₁₆	1	1	yes	characteristic
75	C₂²×Dic₈₃	SD₁₆	1	1	yes	characteristic
76	C₂×C₄×D₈₃	—	4	2	no	—
77	C₂×C₄×D₈₃	—	4	2	no	—
78	C₂×C₄×D₈₃	—	8	4	no	—
79	D₆₆₄	—	2	2	no	—
80	D₆₆₄	—	2	2	no	—
81	D₆₆₄	—	2	2	no	—
82	D₆₆₄	C₄⋊C₄	1	4	yes	—
83	#13050574	—	4	1	no	—
84	#13050574	—	4	1	no	—
85	C₂×C₈₃⋊C₈	—	2	1	no	—
86	C₂×C₈₃⋊C₈	—	2	1	no	—
87	C₂×C₈₃⋊C₈	—	4	1	no	—
88	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
90	(C₂×D₈₃)⋊C₄	—	4	2	no	—
91	(C₂×D₈₃)⋊C₄	—	4	2	no	—
92	C₄×Dic₈₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×Dic₈₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×Dic₈₃	D₈	1	1	yes	characteristic
95	C₄×Dic₈₃	D₈	1	1	yes	characteristic
96	C₈₃⋊C₁₆	—	4	1	no	—
97	C₂²×C₄×C₈₃	—	4	1	no	—
98	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
99	M₁₆×C₈₃	—	4	1	no	—
100	M₁₆×C₈₃	—	4	1	no	—
101	C₂×C₈×C₈₃	—	4	1	no	—
102	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
106	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
107	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
108	C₄⋊C₄×C₈₃	—	2	2	no	—
109	C₄⋊C₄×C₈₃	—	4	1	no	—
110	C₁₆×C₈₃	—	4	1	no	—
111	C₂³×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
112	C₂²×D₈₃	—	4	2	no	—
113	C₂²×D₈₃	—	4	2	no	—
114	C₂²×D₈₃	—	4	2	no	—
115	C₂²×D₈₃	—	8	2	no	—
116	C₂²×D₈₃	—	8	2	no	—
117	C₂×C₄×C₈₃	—	4	2	no	—
118	C₂×C₄×C₈₃	—	8	2	no	—
119	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
125	C₂×Dic₈₃	—	2	2	no	—
126	C₂×Dic₈₃	—	4	1	no	—
127	C₂×Dic₈₃	#102	1	1	yes	characteristic
128	C₂×Dic₈₃	#102	1	1	yes	characteristic
129	C₂×Dic₈₃	#102	1	1	yes	characteristic
130	C₂×Dic₈₃	#102	1	1	yes	characteristic
131	D₃₃₂	—	2	4	no	—
132	D₃₃₂	—	4	2	no	—
133	D₃₃₂	—	4	2	no	—
134	D₃₃₂	—	4	2	no	—
135	C₄×D₈₃	—	8	4	no	—
136	C₄×D₈₃	—	8	4	no	—
137	C₈×C₈₃	—	8	1	no	—
138	C₈×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₈×C₈₃	#117	1	1	yes	characteristic
140	C₈×C₈₃	#121	1	1	yes	characteristic
141	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
142	C₈₃⋊C₈	—	2	1	no	—
143	C₈₃⋊C₈	—	2	1	no	—
144	C₈₃⋊C₈	—	8	1	no	—
145	C₂²×C₈₃	—	2	1	no	—
146	C₂²×C₈₃	—	2	1	no	—
147	C₂²×C₈₃	C₂×#102	1	1	yes	characteristic
148	C₂²×C₈₃	#461	1	1	yes	characteristic
149	C₂²×C₈₃	#476	1	1	yes	characteristic
150	C₂×D₈₃	—	4	4	no	—
151	C₂×D₈₃	—	8	4	no	—
152	C₂×D₈₃	—	8	4	no	—
153	C₂×D₈₃	—	8	2	no	—
154	C₂×D₈₃	—	8	2	no	—
155	C₄×C₈₃	—	8	2	no	—
156	C₄×C₈₃	#380	1	1	yes	characteristic
157	C₄×C₈₃	#417	1	1	yes	characteristic
158	C₄×C₈₃	#459	1	1	yes	characteristic
159	C₄×C₈₃	#474	1	1	yes	characteristic
160	Dic₈₃	—	2	1	no	—
161	Dic₈₃	—	2	1	no	—
162	Dic₈₃	—	2	1	no	—
163	Dic₈₃	#361	1	2	yes	complemented
164	#12818	—	83	1	no	maximal, Sylow
165	C₂×C₈₃	—	2	1	no	—
166	C₂×C₈₃	—	2	1	no	—
167	C₂×C₈₃	#1802	1	1	yes	characteristic
168	C₂×C₈₃	C₂×#361	1	1	yes	characteristic
169	C₂×C₈₃	#2148	1	1	yes	characteristic
170	D₈₃	—	8	4	no	—
171	C₂×C₄×D₈	—	83	1	no	—
172	#2150	—	83	1	no	—
173	C₂×#361	—	83	1	no	—
174	C₂×#361	—	83	1	no	—
175	#1848	—	83	1	no	—
176	#1387	—	83	1	no	—
177	#1387	D₈₃	1	1	yes	characteristic, complemented
178	C₈₃	#12818	1	1	yes	characteristic, complemented, Sylow
179	C₂²×D₈	—	83	1	no	—
180	C₂×C₄×D₄	—	166	1	no	—
181	C₂×M₃₂	—	83	1	no	—
182	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
183	C₄×D₈	—	83	2	no	—
184	C₄×D₈	—	83	2	no	—
185	C₄×D₈	—	166	1	no	—
186	C₄×D₈	—	166	1	no	—
187	C₂×#108	—	83	1	no	—
188	C₂×#108	—	166	1	no	—
189	C₂×#108	C₂×D₈₃	1	1	yes	characteristic
190	C₂×#107	C₂×D₈₃	1	1	yes	characteristic
191	#423	—	166	1	no	—
192	C₂×#102	—	166	1	no	—
193	C₂×C₄×C₈	—	83	1	no	—
194	C₄⋊C₁₆	—	166	1	no	—
195	C₄⋊C₁₆	—	166	1	no	—
196	#361	—	166	2	no	—
197	#361	—	166	2	no	—
198	C₂²×D₄	—	166	1	no	—
199	C₂³×C₄	—	332	1	no	—
200	C₂×D₈	—	83	2	no	—
201	C₂×D₈	—	83	2	no	—
202	C₂×D₈	—	83	2	no	—
203	C₂×D₈	—	166	1	no	—
204	C₂×D₈	—	166	1	no	—
205	C₂×D₈	—	166	1	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	—	166	1	no	—
208	C₂²×C₈	—	83	1	no	—
209	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
210	C₄×D₄	—	332	1	no	—
211	C₄×D₄	—	332	1	no	—
212	C₄×D₄	—	166	2	no	—
213	C₄×D₄	—	166	2	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₂²⋊C₄	—	332	1	no	—
216	C₂×C₄²	—	83	1	no	—
217	M₃₂	—	2	1	no	—
218	M₃₂	—	2	1	no	—
219	M₃₂	—	166	1	no	—
220	M₃₂	—	166	1	no	—
221	C₂×C₁₆	—	2	1	no	—
222	C₂×C₁₆	—	166	1	no	—
223	#108	—	2	1	no	—
224	#108	—	2	1	no	—
225	#108	—	332	2	no	—
226	#108	—	166	1	no	—
227	#108	—	166	1	no	—
228	#107	—	2	1	no	—
229	#107	C₄×D₈₃	1	2	yes	—
230	C₄⋊C₈	—	332	1	no	—
231	C₄⋊C₈	—	332	1	no	—
232	#102	—	166	2	no	—
233	#102	—	166	2	no	—
234	C₄×C₈	—	83	1	no	—
235	C₄×C₈	—	83	1	no	—
236	C₄×C₈	—	83	1	no	—
237	C₄×C₈	—	83	1	no	—
238	C₂⁴	—	332	1	no	—
239	C₂×D₄	—	332	1	no	—
240	C₂×D₄	—	332	1	no	—
241	C₂×D₄	—	332	1	no	—
242	C₂×D₄	—	166	2	no	—
243	C₂×D₄	—	166	2	no	—
244	C₂×D₄	—	166	2	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	332	2	no	—
247	C₂²×C₄	—	332	2	no	—
248	C₂²×C₄	—	83	1	no	—
249	C₂²×C₄	—	83	1	no	—
250	C₂²×C₄	—	664	4	no	—
251	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
252	D₈	—	83	4	no	—
253	D₈	—	166	2	no	—
254	D₈	—	166	2	no	—
255	D₈	—	166	2	no	—
256	M₁₆	—	4	1	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	332	1	no	—
259	M₁₆	—	332	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	332	1	no	—
262	C₂×C₈	—	83	1	no	—
263	C₂×C₈	—	83	1	no	—
264	C₂×C₈	—	166	1	no	—
265	C₂×C₈	—	166	1	no	—
266	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
267	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
268	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
269	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
270	C₂×C₈	#13050590	1	1	yes	characteristic
271	C₂×C₈	#13050590	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	332	2	no	—
275	C₂²⋊C₄	—	332	2	no	—
276	C₄²	—	83	1	no	—
277	C₄²	—	83	1	no	—
278	C₄²	—	83	1	no	—
279	C₄²	—	83	1	no	—
280	C₁₆	—	4	1	no	—
281	C₁₆	—	332	1	no	—
282	C₂³	—	332	2	no	—
283	C₂³	—	332	2	no	—
284	C₂³	—	332	2	no	—
285	C₂³	—	664	2	no	—
286	C₂³	—	664	2	no	—
287	C₂³	SD₁₆×D₈₃	1	1	yes	characteristic
288	D₄	—	332	2	no	—
289	D₄	—	332	2	no	—
290	D₄	—	332	2	no	—
291	D₄	—	166	4	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	—	332	1	no	—
295	C₂×C₄	—	83	1	no	—
296	C₂×C₄	—	83	1	no	—
297	C₂×C₄	—	83	1	no	—
298	C₂×C₄	—	83	1	no	—
299	C₂×C₄	—	664	4	no	—
300	C₂×C₄	—	664	4	no	—
301	C₂×C₄	—	166	2	no	—
302	C₂×C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
303	C₂×C₄	#14581765	1	1	yes	characteristic
304	C₂×C₄	#14581770	1	1	yes	characteristic
305	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
306	C₂×C₄	#14581819	1	1	yes	characteristic
307	C₂×C₄	#14581824	1	1	yes	characteristic
308	C₈	—	8	1	no	—
309	C₈	—	664	1	no	—
310	C₈	—	166	1	no	—
311	C₈	—	166	1	no	—
312	C₈	C₄⋊C₄×D₈₃	1	1	yes	characteristic
313	C₈	#14581749	1	1	yes	characteristic
314	C₈	#14581765	1	1	yes	characteristic
315	C₈	#14581770	1	1	yes	characteristic
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	332	4	no	—
319	C₂²	—	664	4	no	—
320	C₂²	—	664	4	no	—
321	C₂²	—	664	2	no	—
322	C₂²	—	664	2	no	—
323	C₂²	#102×D₈₃	1	1	yes	characteristic
324	C₂²	#19127905	1	1	yes	characteristic
325	C₂²	#19127990	1	1	yes	characteristic
326	C₄	—	8	2	no	—
327	C₄	—	83	2	no	—
328	C₄	—	166	1	no	—
329	C₄	—	166	1	no	—
330	C₄	—	166	1	no	—
331	C₄	#19127560	1	1	yes	characteristic
332	C₄	#19127701	1	1	yes	characteristic
333	C₄	#19127895	1	1	yes	characteristic
334	C₄	#19127980	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	664	4	no	—
338	C₂	#24907462	1	1	yes	characteristic
339	C₂	#361×D₈₃	1	1	yes	characteristic
340	C₂	#24909211	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#102×D₈₃
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1387	C₂
socle	C₂²×C₈₃	C₂×#102
prime radical	C₂²×D₆₆₄	C₂²
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1387	D₈₃
hypocenter	C₈₃	#12818
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1387

Complements

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

N	H ≅ G/N	type
2. #24909218	C₂	semidirect
5. #24907633	C₂	semidirect
6. #24905571	C₂	semidirect
7. C₈₃×#1387	C₂	semidirect
163. Dic₈₃	#361	semidirect
177. #1387	D₈₃	semidirect
178. C₈₃	#12818	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127783	2
#24909218	2
C₂×#19127515	2
C₂×#19127513	2
#24907633	2
#24905571	2
C₈₃×#1387	2
#12818	83