Subgroups — C₈₃×#8289 #35236113

Order 21248.

410 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#106×C₈₃	C₂	1	1	yes	maximal, characteristic
2	C₈₃×#1770	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₄×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#335×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#322×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₈₃×#1279	C₂	1	1	yes	maximal, characteristic
7	C₈₃×#1279	C₂	1	1	yes	maximal, characteristic
8	C₂×#106×C₈₃	C₄	1	2	yes	—
9	C₂×C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
11	C₂×#97×C₈₃	—	2	1	no	—
12	C₂×C₄×C₈×C₈₃	—	2	1	no	—
13	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
18	#340×C₈₃	C₄	1	2	yes	—
19	#336×C₈₃	—	2	1	no	—
20	#336×C₈₃	—	2	1	no	—
21	#335×C₈₃	C₄	1	4	yes	complemented
22	#322×C₈₃	C₄	1	4	yes	complemented
23	C₂²×C₈×C₈₃	—	2	1	no	—
24	C₂²×C₈×C₈₃	—	2	1	no	—
25	C₂²×C₈×C₈₃	—	2	1	no	—
26	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₈₃	—	2	1	no	—
33	C₂×C₄²×C₈₃	—	2	1	no	—
34	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
35	#106×C₈₃	—	2	4	no	—
36	C₄⋊C₈×C₈₃	—	2	1	no	—
37	C₄⋊C₈×C₈₃	—	2	1	no	—
38	C₄⋊C₈×C₈₃	C₂×C₄	1	2	yes	—
39	C₄⋊C₈×C₈₃	C₂×C₄	1	2	yes	—
40	#97×C₈₃	—	4	2	no	—
41	C₄×C₈×C₈₃	—	2	1	no	—
42	C₄×C₈×C₈₃	—	4	2	no	—
43	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₈₃	Q₈	1	2	yes	—
48	#95×C₈₃	—	2	1	no	—
49	#95×C₈₃	—	2	1	no	—
50	C₂²×C₄×C₈₃	—	2	2	no	—
51	C₂²×C₄×C₈₃	—	2	2	no	—
52	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
55	C₂×C₈×C₈₃	—	2	4	no	—
56	C₂×C₈×C₈₃	—	2	2	no	—
57	C₂×C₈×C₈₃	—	2	2	no	—
58	C₂×C₈×C₈₃	—	2	2	no	—
59	C₂×C₈×C₈₃	—	2	1	no	—
60	C₂×C₈×C₈₃	—	2	1	no	—
61	C₂×C₈×C₈₃	—	2	1	no	—
62	C₂×C₈×C₈₃	—	2	1	no	—
63	C₂×C₈×C₈₃	—	2	2	no	—
64	C₂×C₈×C₈₃	—	2	2	no	—
65	C₂×C₈×C₈₃	—	4	1	no	—
66	C₂×C₈×C₈₃	—	4	1	no	—
67	C₂×C₈×C₈₃	—	4	1	no	—
68	C₂×C₈×C₈₃	—	4	1	no	—
69	C₂×C₈×C₈₃	C₄²	1	2	yes	—
70	C₂×C₈×C₈₃	C₄²	1	2	yes	—
71	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
72	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
73	C₄⋊C₄×C₈₃	—	2	2	no	—
74	C₄⋊C₄×C₈₃	—	2	2	no	—
75	C₄⋊C₄×C₈₃	—	2	2	no	—
76	C₄⋊C₄×C₈₃	M₁₆	1	4	yes	—
77	C₄²×C₈₃	—	4	1	no	—
78	C₄²×C₈₃	—	4	1	no	—
79	C₄²×C₈₃	—	4	1	no	—
80	C₄²×C₈₃	—	4	1	no	—
81	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
84	C₄²×C₈₃	#41	1	1	yes	characteristic
85	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₈₃	—	2	2	no	—
87	C₂×C₄×C₈₃	—	2	4	no	—
88	C₂×C₄×C₈₃	—	4	2	no	—
89	C₂×C₄×C₈₃	—	4	2	no	—
90	C₂×C₄×C₈₃	—	4	2	no	—
91	C₂×C₄×C₈₃	—	4	2	no	—
92	C₂×C₄×C₈₃	—	4	2	no	—
93	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
96	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₈₃	#131	1	2	yes	—
101	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
104	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
105	C₈×C₈₃	—	2	4	no	—
106	C₈×C₈₃	—	2	2	no	—
107	C₈×C₈₃	—	2	2	no	—
108	C₈×C₈₃	—	2	2	no	—
109	C₈×C₈₃	—	2	2	no	—
110	C₈×C₈₃	—	4	2	no	—
111	C₈×C₈₃	—	4	2	no	—
112	C₂²×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₈₃	#435	1	2	yes	—
114	C₂²×C₈₃	#440	1	1	yes	characteristic
115	C₂²×C₈₃	#440	1	1	yes	characteristic
116	C₂²×C₈₃	#443	1	2	yes	—
117	C₄×C₈₃	—	2	1	no	—
118	C₄×C₈₃	—	2	1	no	—
119	C₄×C₈₃	—	4	4	no	—
120	C₄×C₈₃	—	8	2	no	—
121	C₄×C₈₃	C₄×M₁₆	1	1	yes	characteristic
122	C₄×C₈₃	#405	1	1	yes	characteristic
123	C₄×C₈₃	#428	1	1	yes	characteristic
124	C₄×C₈₃	#428	1	1	yes	characteristic
125	C₄×C₈₃	#441	1	1	yes	characteristic
126	C₄×C₈₃	#442	1	1	yes	characteristic
127	C₄×C₈₃	#446	1	2	yes	—
128	#8289	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₈₃	#1585	1	2	yes	—
130	C₂×C₈₃	#1588	1	2	yes	—
131	C₂×C₈₃	#1771	1	1	yes	characteristic
132	C₂×C₈₃	#1772	1	1	yes	characteristic
133	C₂×C₈₃	#1776	1	1	yes	characteristic
134	C₄×#106	C₂×C₈₃	1	1	yes	characteristic
135	#1770	C₂×C₈₃	1	1	yes	characteristic
136	C₈×C₄⋊C₄	C₂×C₈₃	1	1	yes	characteristic
137	C₂×#335	C₂×C₈₃	1	1	yes	characteristic
138	C₂×#322	C₂×C₈₃	1	1	yes	characteristic
139	#1279	C₂×C₈₃	1	1	yes	characteristic
140	#1279	C₂×C₈₃	1	1	yes	characteristic
141	C₈₃	#8289	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×#106	C₄×C₈₃	1	2	yes	—
143	C₂×C₄⋊C₈	C₂²×C₈₃	1	1	yes	characteristic
144	C₂×C₄⋊C₈	C₂²×C₈₃	1	1	yes	characteristic
145	C₂×#97	—	2	1	no	—
146	C₂×C₄×C₈	—	2	1	no	—
147	C₂×C₄×C₈	C₂²×C₈₃	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂²×C₈₃	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂²×C₈₃	1	1	yes	characteristic
150	C₄×C₄⋊C₄	C₂²×C₈₃	1	1	yes	characteristic
151	C₄×C₄⋊C₄	C₂²×C₈₃	1	1	yes	characteristic
152	#340	C₄×C₈₃	1	2	yes	—
153	#336	—	2	1	no	—
154	#336	—	2	1	no	—
155	#335	C₄×C₈₃	1	4	yes	complemented
156	#322	C₄×C₈₃	1	4	yes	complemented
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	2	1	no	—
160	C₂²×C₈	C₂×C₄×C₈₃	1	1	yes	characteristic
161	C₂²×C₈	C₂×C₄×C₈₃	1	1	yes	characteristic
162	C₂²×C₈	D₄×C₈₃	1	1	yes	characteristic
163	C₂²×C₈	D₄×C₈₃	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂×C₄×C₈₃	1	2	yes	—
165	C₂×C₄⋊C₄	C₂×C₄×C₈₃	1	2	yes	—
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
169	#106	—	2	4	no	—
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	C₂×C₄×C₈₃	1	2	yes	—
173	C₄⋊C₈	C₂×C₄×C₈₃	1	2	yes	—
174	#97	—	4	2	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	—	4	2	no	—
177	C₄×C₈	C₂×C₄×C₈₃	1	2	yes	—
178	C₄×C₈	C₂×C₄×C₈₃	1	2	yes	—
179	C₄×C₈	C₂×C₄×C₈₃	1	2	yes	—
180	C₄×C₈	C₂×C₄×C₈₃	1	2	yes	—
181	C₄×C₈	Q₈×C₈₃	1	2	yes	—
182	#95	—	2	1	no	—
183	#95	—	2	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	C₂²×C₄×C₈₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
188	C₂²×C₄	#41×C₈₃	1	1	yes	characteristic
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	C₄²×C₈₃	1	2	yes	—
204	C₂×C₈	C₄²×C₈₃	1	2	yes	—
205	C₂×C₈	C₄⋊C₄×C₈₃	1	2	yes	—
206	C₂×C₈	C₄⋊C₄×C₈₃	1	2	yes	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	M₁₆×C₈₃	1	4	yes	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	4	1	no	—
215	C₄²	C₂²×C₄×C₈₃	1	1	yes	characteristic
216	C₄²	C₂²×C₄×C₈₃	1	1	yes	characteristic
217	C₄²	C₂×Q₈×C₈₃	1	1	yes	characteristic
218	C₄²	#41×C₈₃	1	1	yes	characteristic
219	C₂³	C₄×D₄×C₈₃	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	C₂×C₄²×C₈₃	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊C₄×C₈₃	1	1	yes	characteristic
229	C₂×C₄	#117×C₈₃	1	1	yes	characteristic
230	C₂×C₄	C₄×D₄×C₈₃	1	1	yes	characteristic
231	C₂×C₄	C₄×Q₈×C₈₃	1	1	yes	characteristic
232	C₂×C₄	C₄×Q₈×C₈₃	1	1	yes	characteristic
233	C₂×C₄	C₂×M₁₆×C₈₃	1	2	yes	—
234	C₂×C₄	#131×C₈₃	1	2	yes	—
235	C₂×C₄	AGL(1,ℤ/8)×C₈₃	1	1	yes	characteristic
236	C₂×C₄	AGL(1,ℤ/8)×C₈₃	1	1	yes	characteristic
237	C₂×C₄	#137×C₈₃	1	1	yes	characteristic
238	C₂×C₄	#137×C₈₃	1	1	yes	characteristic
239	C₈	—	2	4	no	—
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	—	4	2	no	—
245	C₈	—	4	2	no	—
246	C₂²	C₄×C₄⋊C₄×C₈₃	1	1	yes	characteristic
247	C₂²	#435×C₈₃	1	2	yes	—
248	C₂²	#440×C₈₃	1	1	yes	characteristic
249	C₂²	#440×C₈₃	1	1	yes	characteristic
250	C₂²	#443×C₈₃	1	2	yes	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	2	no	—
255	C₄	C₄×M₁₆×C₈₃	1	1	yes	characteristic
256	C₄	#405×C₈₃	1	1	yes	characteristic
257	C₄	#428×C₈₃	1	1	yes	characteristic
258	C₄	#428×C₈₃	1	1	yes	characteristic
259	C₄	#441×C₈₃	1	1	yes	characteristic
260	C₄	#442×C₈₃	1	1	yes	characteristic
261	C₄	#446×C₈₃	1	2	yes	—
262	C₂	C₈₃×#1585	1	2	yes	—
263	C₂	C₈₃×#1588	1	2	yes	—
264	C₂	C₈₃×#1771	1	1	yes	characteristic
265	C₂	C₈₃×#1772	1	1	yes	characteristic
266	C₂	C₈₃×#1776	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
21. #335×C₈₃	C₄	semidirect
22. #322×C₈₃	C₄	semidirect
128. #8289	C₈₃	direct (N × H)
141. C₈₃	#8289	direct (N × H)
155. #335	C₄×C₈₃	semidirect
156. #322	C₄×C₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#106×C₈₃	2
C₈₃×#1770	2
C₈×C₄⋊C₄×C₈₃	2
C₂×#335×C₈₃	2
C₂×#322×C₈₃	2
C₈₃×#1279	2
#8289	83

Subgroups — C83×#8289 #35236113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#8289 #35236113