Subgroups — C₈₃×#7891 #35235715

Order 21248.

770 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₈₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₈₃×#1279	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₈₃	—	2	1	no	—
6	C₂×#117×C₈₃	C₄	1	2	yes	—
7	C₂²×C₄²×C₈₃	C₂²	1	1	yes	characteristic
8	#431×C₈₃	—	2	2	no	—
9	#431×C₈₃	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₈×C₈₃	—	2	1	no	—
11	C₂×C₄⋊C₈×C₈₃	—	2	2	no	—
12	C₂×C₄⋊C₈×C₈₃	—	2	1	no	—
13	C₂×C₄⋊C₈×C₈₃	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₈₃	—	2	1	no	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₈₃	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
19	C₄×C₂²⋊C₄×C₈₃	—	2	1	no	—
20	C₄×C₂²⋊C₄×C₈₃	—	2	1	no	—
21	#376×C₈₃	—	2	1	no	—
22	C₄³×C₈₃	—	2	1	no	—
23	C₂³×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₈₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₈₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₈₃	—	2	4	no	—
27	C₂²×C₈×C₈₃	—	2	2	no	—
28	C₂²×C₈×C₈₃	—	2	2	no	—
29	C₂²×C₈×C₈₃	—	4	4	no	—
30	#117×C₈₃	—	2	4	no	—
31	#117×C₈₃	C₈	1	8	yes	complemented
32	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	4	yes	—
33	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
34	C₂×C₄²×C₈₃	—	2	2	no	—
35	C₂×C₄²×C₈₃	—	2	2	no	—
36	C₂×C₄²×C₈₃	—	2	2	no	—
37	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₃	D₄	1	2	yes	—
43	C₂×C₄²×C₈₃	Q₈	1	2	yes	—
44	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₈₃	—	2	4	no	—
46	C₄⋊C₈×C₈₃	—	2	4	no	—
47	C₄⋊C₈×C₈₃	—	2	2	no	—
48	C₄⋊C₈×C₈₃	C₂×C₄	1	4	yes	—
49	C₂²⋊C₈×C₈₃	—	2	4	no	—
50	C₄×C₈×C₈₃	—	2	2	no	—
51	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	—
52	#95×C₈₃	—	2	2	no	—
53	#95×C₈₃	—	2	2	no	—
54	C₂⁴×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₃	—	2	2	no	—
56	C₂²×C₄×C₈₃	—	2	2	no	—
57	C₂²×C₄×C₈₃	—	2	4	no	—
58	C₂²×C₄×C₈₃	—	4	2	no	—
59	C₂²×C₄×C₈₃	C₄²	1	2	yes	—
60	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₈₃	C₂×C₈	1	4	yes	—
69	C₂²×C₄×C₈₃	M₁₆	1	4	yes	—
70	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₈₃	—	2	4	no	—
74	C₂×C₈×C₈₃	—	2	4	no	—
75	C₂×C₈×C₈₃	—	4	4	no	—
76	C₂×C₈×C₈₃	—	4	8	no	—
77	C₂×C₈×C₈₃	—	4	4	no	—
78	C₄⋊C₄×C₈₃	—	2	4	no	—
79	C₄⋊C₄×C₈₃	C₂×C₈	1	8	yes	complemented
80	C₂²⋊C₄×C₈₃	—	2	8	no	—
81	C₄²×C₈₃	—	2	8	no	—
82	C₄²×C₈₃	—	4	4	no	—
83	C₄²×C₈₃	—	4	2	no	—
84	C₄²×C₈₃	—	4	2	no	—
85	C₄²×C₈₃	C₄²	1	2	yes	—
86	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
89	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
90	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
91	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
95	C₂³×C₈₃	—	2	2	no	—
96	C₂³×C₈₃	—	2	2	no	—
97	C₂³×C₈₃	#95	1	2	yes	—
98	C₂³×C₈₃	C₂²⋊C₈	1	4	yes	—
99	C₂³×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₃	—	2	2	no	—
101	C₂×C₄×C₈₃	—	2	2	no	—
102	C₂×C₄×C₈₃	—	2	2	no	—
103	C₂×C₄×C₈₃	—	2	2	no	—
104	C₂×C₄×C₈₃	—	2	2	no	—
105	C₂×C₄×C₈₃	—	2	4	no	—
106	C₂×C₄×C₈₃	—	2	8	no	—
107	C₂×C₄×C₈₃	—	4	2	no	—
108	C₂×C₄×C₈₃	—	4	2	no	—
109	C₂×C₄×C₈₃	—	4	4	no	—
110	C₂×C₄×C₈₃	—	4	4	no	—
111	C₂×C₄×C₈₃	#95	1	2	yes	—
112	C₂×C₄×C₈₃	#95	1	2	yes	—
113	C₂×C₄×C₈₃	#95	1	2	yes	—
114	C₂×C₄×C₈₃	C₄×C₈	1	4	yes	—
115	C₂×C₄×C₈₃	#97	1	4	yes	—
116	C₂×C₄×C₈₃	C₂²⋊C₈	1	4	yes	—
117	C₂×C₄×C₈₃	C₄⋊C₈	1	8	yes	—
118	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₈₃	#137	1	2	yes	—
128	C₈×C₈₃	—	4	4	no	—
129	C₈×C₈₃	—	4	4	no	—
130	C₂²×C₈₃	—	2	8	no	—
131	C₂²×C₈₃	—	2	2	no	—
132	C₂²×C₈₃	—	2	2	no	—
133	C₂²×C₈₃	#336	1	4	yes	—
134	C₂²×C₈₃	C₂×#95	1	1	yes	characteristic
135	C₂²×C₈₃	C₂×C₂²⋊C₈	1	2	yes	—
136	C₂²×C₈₃	#417	1	2	yes	—
137	C₂²×C₈₃	#421	1	2	yes	—
138	C₄×C₈₃	—	2	1	no	—
139	C₄×C₈₃	—	2	2	no	—
140	C₄×C₈₃	—	2	1	no	—
141	C₄×C₈₃	—	2	2	no	—
142	C₄×C₈₃	—	4	8	no	—
143	C₄×C₈₃	#336	1	4	yes	—
144	C₄×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
145	C₄×C₈₃	C₂×#97	1	1	yes	characteristic
146	C₄×C₈₃	#418	1	1	yes	characteristic
147	C₄×C₈₃	#419	1	1	yes	characteristic
148	C₄×C₈₃	C₂×C₄⋊C₈	1	2	yes	—
149	C₄×C₈₃	#428	1	2	yes	—
150	#7891	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₈₃	—	2	4	no	—
152	C₂×C₈₃	#1477	1	4	yes	—
153	C₂×C₈₃	C₂×#336	1	1	yes	characteristic
154	C₂×C₈₃	#1733	1	1	yes	characteristic
155	C₂×C₈₃	#1736	1	1	yes	characteristic
156	C₂×#431	C₂×C₈₃	1	1	yes	characteristic
157	C₂²×C₄⋊C₈	C₂×C₈₃	1	1	yes	characteristic
158	C₄×#117	C₂×C₈₃	1	1	yes	characteristic
159	#1279	C₂×C₈₃	1	4	yes	complemented
160	C₈₃	#7891	1	1	yes	characteristic, direct product factor, Sylow
161	C₂³×C₈	—	2	1	no	—
162	C₂×#117	C₄×C₈₃	1	2	yes	—
163	C₂²×C₄²	C₂²×C₈₃	1	1	yes	characteristic
164	#431	—	2	2	no	—
165	#431	C₄×C₈₃	1	4	yes	complemented
166	C₂×C₄⋊C₈	—	2	1	no	—
167	C₂×C₄⋊C₈	—	2	2	no	—
168	C₂×C₄⋊C₈	—	2	1	no	—
169	C₂×C₄⋊C₈	C₄×C₈₃	1	4	yes	complemented
170	C₂×C₄⋊C₈	C₂²×C₈₃	1	2	yes	—
171	C₂×C₂²⋊C₈	—	2	1	no	—
172	C₂×C₄×C₈	C₂²×C₈₃	1	2	yes	—
173	C₄×C₄⋊C₄	C₄×C₈₃	1	2	yes	—
174	C₄×C₄⋊C₄	C₂²×C₈₃	1	2	yes	—
175	C₄×C₂²⋊C₄	—	2	1	no	—
176	C₄×C₂²⋊C₄	—	2	1	no	—
177	#376	—	2	1	no	—
178	C₄³	—	2	1	no	—
179	C₂³×C₄	C₂×C₄×C₈₃	1	1	yes	characteristic
180	C₂³×C₄	D₄×C₈₃	1	1	yes	characteristic
181	C₂³×C₄	D₄×C₈₃	1	1	yes	characteristic
182	C₂²×C₈	—	2	4	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	4	4	no	—
186	#117	—	2	4	no	—
187	#117	C₈×C₈₃	1	8	yes	complemented
188	C₂×C₄⋊C₄	C₂×C₄×C₈₃	1	4	yes	—
189	C₂×C₂²⋊C₄	—	2	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	C₂×C₄×C₈₃	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₈₃	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₈₃	1	2	yes	—
196	C₂×C₄²	D₄×C₈₃	1	1	yes	characteristic
197	C₂×C₄²	D₄×C₈₃	1	1	yes	characteristic
198	C₂×C₄²	D₄×C₈₃	1	2	yes	—
199	C₂×C₄²	Q₈×C₈₃	1	2	yes	—
200	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
201	C₄⋊C₈	—	2	4	no	—
202	C₄⋊C₈	—	2	4	no	—
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	C₂×C₄×C₈₃	1	4	yes	—
205	C₂²⋊C₈	—	2	4	no	—
206	C₄×C₈	—	2	2	no	—
207	C₄×C₈	C₂×C₄×C₈₃	1	4	yes	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	C₂⁴	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₄²×C₈₃	1	2	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²⋊C₄×C₈₃	1	2	yes	—
219	C₂²×C₄	C₂²⋊C₄×C₈₃	1	2	yes	—
220	C₂²×C₄	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
221	C₂²×C₄	C₄⋊C₄×C₈₃	1	2	yes	—
222	C₂²×C₄	C₄⋊C₄×C₈₃	1	2	yes	—
223	C₂²×C₄	C₄⋊C₄×C₈₃	1	2	yes	—
224	C₂²×C₄	C₂×C₈×C₈₃	1	4	yes	—
225	C₂²×C₄	M₁₆×C₈₃	1	4	yes	—
226	C₂²×C₄	C₂²×C₄×C₈₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	4	8	no	—
232	C₂×C₈	—	4	4	no	—
233	C₂×C₈	—	4	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	C₂×C₈×C₈₃	1	8	yes	complemented
236	C₂²⋊C₄	—	2	8	no	—
237	C₄²	—	2	8	no	—
238	C₄²	—	4	4	no	—
239	C₄²	—	4	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	C₄²×C₈₃	1	2	yes	—
242	C₄²	C₂²⋊C₄×C₈₃	1	2	yes	—
243	C₄²	C₂²⋊C₄×C₈₃	1	2	yes	—
244	C₄²	C₄⋊C₄×C₈₃	1	2	yes	—
245	C₄²	C₄⋊C₄×C₈₃	1	2	yes	—
246	C₄²	C₄⋊C₄×C₈₃	1	2	yes	—
247	C₄²	C₂²×C₄×C₈₃	1	1	yes	characteristic
248	C₄²	C₂²×C₄×C₈₃	1	1	yes	characteristic
249	C₄²	C₂×D₄×C₈₃	1	1	yes	characteristic
250	C₄²	C₂×Q₈×C₈₃	1	1	yes	characteristic
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	#95×C₈₃	1	2	yes	—
254	C₂³	C₂²⋊C₈×C₈₃	1	4	yes	—
255	C₂³	C₂×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	#95×C₈₃	1	2	yes	—
268	C₂×C₄	#95×C₈₃	1	2	yes	—
269	C₂×C₄	#95×C₈₃	1	2	yes	—
270	C₂×C₄	C₄×C₈×C₈₃	1	4	yes	—
271	C₂×C₄	#97×C₈₃	1	4	yes	—
272	C₂×C₄	C₂²⋊C₈×C₈₃	1	4	yes	—
273	C₂×C₄	C₄⋊C₈×C₈₃	1	8	yes	—
274	C₂×C₄	C₂×C₄²×C₈₃	1	1	yes	characteristic
275	C₂×C₄	C₂×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
276	C₂×C₄	C₂×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄⋊C₄×C₈₃	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄⋊C₄×C₈₃	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊C₄×C₈₃	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₈×C₈₃	1	2	yes	—
281	C₂×C₄	C₂×M₁₆×C₈₃	1	2	yes	—
282	C₂×C₄	AGL(1,ℤ/8)×C₈₃	1	2	yes	—
283	C₂×C₄	#137×C₈₃	1	2	yes	—
284	C₈	—	4	4	no	—
285	C₈	—	4	4	no	—
286	C₂²	—	2	8	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	#336×C₈₃	1	4	yes	—
290	C₂²	C₂×#95×C₈₃	1	1	yes	characteristic
291	C₂²	C₂×C₂²⋊C₈×C₈₃	1	2	yes	—
292	C₂²	#417×C₈₃	1	2	yes	—
293	C₂²	#421×C₈₃	1	2	yes	—
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	8	no	—
299	C₄	#336×C₈₃	1	4	yes	—
300	C₄	C₂×C₄×C₈×C₈₃	1	1	yes	characteristic
301	C₄	C₂×#97×C₈₃	1	1	yes	characteristic
302	C₄	#418×C₈₃	1	1	yes	characteristic
303	C₄	#419×C₈₃	1	1	yes	characteristic
304	C₄	C₂×C₄⋊C₈×C₈₃	1	2	yes	—
305	C₄	#428×C₈₃	1	2	yes	—
306	C₂	—	2	4	no	—
307	C₂	C₈₃×#1477	1	4	yes	—
308	C₂	C₂×#336×C₈₃	1	1	yes	characteristic
309	C₂	C₈₃×#1733	1	1	yes	characteristic
310	C₂	C₈₃×#1736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₈₃	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₈₃
Frattini, cosocle	C₂×C₄²	C₂³×C₈₃
socle	C₂³×C₈₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₈₃	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₈₃ ◁ C₂²×C₄²×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₈₃×#1279	C₂	semidirect
9. #431×C₈₃	C₄	semidirect
13. C₂×C₄⋊C₈×C₈₃	C₄	semidirect
31. #117×C₈₃	C₈	semidirect
79. C₄⋊C₄×C₈₃	C₂×C₈	semidirect
150. #7891	C₈₃	direct (N × H)
159. #1279	C₂×C₈₃	semidirect
160. C₈₃	#7891	direct (N × H)
165. #431	C₄×C₈₃	semidirect
169. C₂×C₄⋊C₈	C₄×C₈₃	semidirect
187. #117	C₈×C₈₃	semidirect
235. C₄⋊C₄	C₂×C₈×C₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431×C₈₃	2
C₂²×C₄⋊C₈×C₈₃	2
C₄×#117×C₈₃	2
C₈₃×#1279	2
#7891	83

Subgroups — C83×#7891 #35235715

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#7891 #35235715