Subgroups — C₈₃×#22269 #35250093

Order 21248.

1222 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₈₃	C₂	1	1	yes	maximal, characteristic
2	C₄×#127×C₈₃	C₂	1	2	yes	maximal, complemented
3	C₈₃×#1957	C₂	1	2	yes	maximal, complemented
4	C₈₃×#1879	C₂	1	8	yes	maximal, complemented
5	C₈₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₈₃	C₂²	1	4	yes	—
8	C₂×#127×C₈₃	C₂²	1	2	yes	—
9	C₂×C₄×D₄×C₈₃	—	2	2	no	—
10	C₂×C₄×D₄×C₈₃	—	2	2	no	—
11	C₂×C₄×D₄×C₈₃	C₂²	1	4	yes	—
12	#493×C₈₃	C₄	1	8	yes	complemented
13	C₂×#102×C₈₃	C₂²	1	8	yes	—
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	#390×C₈₃	—	2	2	no	—
17	#390×C₈₃	—	2	2	no	—
18	#390×C₈₃	C₂²	1	4	yes	—
19	#384×C₈₃	C₂²	1	2	yes	—
20	C₄³×C₈₃	C₂²	1	1	yes	characteristic
21	#340×C₈₃	C₂²	1	4	yes	—
22	#336×C₈₃	C₂²	1	4	yes	—
23	C₂²×D₄×C₈₃	—	2	2	no	—
24	C₂²×D₄×C₈₃	—	2	2	no	—
25	C₂²×D₄×C₈₃	C₂³	1	4	yes	—
26	C₂³×C₄×C₈₃	—	2	4	no	—
27	C₂×D₈×C₈₃	—	4	8	no	—
28	C₂×D₈×C₈₃	C₂×C₄	1	16	yes	complemented
29	C₂²×C₈×C₈₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₈₃	C₂³	1	2	yes	—
31	#127×C₈₃	—	2	4	no	—
32	#127×C₈₃	C₂×C₄	1	8	yes	complemented
33	C₄×D₄×C₈₃	—	4	8	no	—
34	C₄×D₄×C₈₃	—	4	8	no	—
35	C₄×D₄×C₈₃	—	4	8	no	—
36	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₈₃	C₂³	1	4	yes	—
39	C₂×C₂²⋊C₄×C₈₃	—	2	4	no	—
40	C₂×C₂²⋊C₄×C₈₃	—	2	4	no	—
41	C₂×C₂²⋊C₄×C₈₃	—	2	4	no	—
42	C₂×C₄²×C₈₃	D₄	1	2	yes	—
43	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
47	#102×C₈₃	—	4	8	no	—
48	C₄×C₈×C₈₃	—	2	2	no	—
49	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	complemented
50	C₂⁴×C₈₃	—	2	4	no	—
51	C₂×D₄×C₈₃	—	2	8	no	—
52	C₂×D₄×C₈₃	—	2	8	no	—
53	C₂×D₄×C₈₃	—	2	4	no	—
54	C₂×D₄×C₈₃	—	2	8	no	—
55	C₂×D₄×C₈₃	—	2	8	no	—
56	C₂×D₄×C₈₃	—	4	4	no	—
57	C₂×D₄×C₈₃	—	4	4	no	—
58	C₂×D₄×C₈₃	—	4	4	no	—
59	C₂×D₄×C₈₃	—	4	4	no	—
60	C₂×D₄×C₈₃	C₂²×C₄	1	8	yes	—
61	C₂²×C₄×C₈₃	—	2	4	no	—
62	C₂²×C₄×C₈₃	—	4	8	no	—
63	C₂²×C₄×C₈₃	—	4	8	no	—
64	C₂²×C₄×C₈₃	—	4	8	no	—
65	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
70	D₈×C₈₃	—	4	16	no	—
71	C₂×C₈×C₈₃	—	2	4	no	—
72	C₂×C₈×C₈₃	—	2	2	no	—
73	C₂×C₈×C₈₃	—	2	4	no	—
74	C₂×C₈×C₈₃	C₂²×C₄	1	4	yes	—
75	C₄⋊C₄×C₈₃	—	2	8	no	—
76	C₄⋊C₄×C₈₃	—	4	4	no	—
77	C₄⋊C₄×C₈₃	—	4	4	no	—
78	C₂²⋊C₄×C₈₃	—	4	8	no	—
79	C₂²⋊C₄×C₈₃	—	4	8	no	—
80	C₂²⋊C₄×C₈₃	—	4	8	no	—
81	C₄²×C₈₃	—	2	2	no	—
82	C₄²×C₈₃	—	2	4	no	—
83	C₄²×C₈₃	—	2	4	no	—
84	C₄²×C₈₃	D₈	1	4	yes	complemented
85	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
86	C₄²×C₈₃	#41	1	2	yes	—
87	C₂³×C₈₃	—	2	8	no	—
88	C₂³×C₈₃	—	4	4	no	—
89	C₂³×C₈₃	—	4	4	no	—
90	C₂³×C₈₃	—	4	4	no	—
91	C₂³×C₈₃	—	4	4	no	—
92	C₂³×C₈₃	—	4	4	no	—
93	C₂³×C₈₃	—	4	4	no	—
94	C₂³×C₈₃	C₂²×D₄	1	1	yes	characteristic
95	D₄×C₈₃	—	2	16	no	—
96	D₄×C₈₃	—	4	8	no	—
97	D₄×C₈₃	—	4	16	no	—
98	D₄×C₈₃	—	4	16	no	—
99	C₂×C₄×C₈₃	—	2	2	no	—
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	—	4	4	no	—
105	C₂×C₄×C₈₃	—	4	4	no	—
106	C₂×C₄×C₈₃	—	4	4	no	—
107	C₂×C₄×C₈₃	—	8	16	no	—
108	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₈₃	C₂×D₈	1	4	yes	—
111	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₈₃	#135	1	2	yes	—
113	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	4	yes	—
114	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₈₃	H(5,𝔽₂)	1	4	yes	—
117	C₈×C₈₃	—	2	4	no	—
118	C₈×C₈₃	—	4	2	no	—
119	C₂²×C₈₃	—	4	8	no	—
120	C₂²×C₈₃	—	4	8	no	—
121	C₂²×C₈₃	—	4	8	no	—
122	C₂²×C₈₃	—	8	4	no	—
123	C₂²×C₈₃	—	8	4	no	—
124	C₂²×C₈₃	—	8	4	no	—
125	C₂²×C₈₃	—	8	4	no	—
126	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂²×C₈₃	#535	1	2	yes	—
128	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
129	C₂²×C₈₃	C₂×#135	1	1	yes	characteristic
130	C₂²×C₈₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
131	C₄×C₈₃	—	2	4	no	—
132	C₄×C₈₃	—	4	2	no	—
133	C₄×C₈₃	—	8	4	no	—
134	C₄×C₈₃	C₄×D₈	1	4	yes	—
135	C₄×C₈₃	#442	1	4	yes	—
136	C₄×C₈₃	#518	1	2	yes	—
137	C₄×C₈₃	#540	1	2	yes	—
138	#22269	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂×C₈₃	—	8	8	no	—
140	C₂×C₈₃	#2328	1	1	yes	characteristic
141	C₂×C₈₃	C₂×C₄×D₈	1	1	yes	characteristic
142	C₂×C₈₃	C₂×#442	1	1	yes	characteristic
143	C₂×C₈₃	#3205	1	2	yes	—
144	C₂×C₈₃	#3207	1	2	yes	—
145	C₂×#493	C₂×C₈₃	1	1	yes	characteristic
146	C₄×#127	C₂×C₈₃	1	2	yes	complemented
147	#1957	C₂×C₈₃	1	2	yes	complemented
148	#1879	C₂×C₈₃	1	8	yes	complemented
149	#1834	C₂×C₈₃	1	1	yes	characteristic, complemented
150	#1744	C₂×C₈₃	1	1	yes	characteristic, complemented
151	C₈₃	#22269	1	1	yes	characteristic, direct product factor, Sylow
152	C₂²×D₈	C₂²×C₈₃	1	4	yes	—
153	C₂×#127	C₂²×C₈₃	1	2	yes	—
154	C₂×C₄×D₄	—	2	2	no	—
155	C₂×C₄×D₄	—	2	2	no	—
156	C₂×C₄×D₄	C₂²×C₈₃	1	4	yes	—
157	#493	C₄×C₈₃	1	8	yes	complemented
158	C₂×#102	C₂²×C₈₃	1	8	yes	—
159	C₂×C₄×C₈	C₂²×C₈₃	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂²×C₈₃	1	1	yes	characteristic
161	#390	—	2	2	no	—
162	#390	—	2	2	no	—
163	#390	C₂²×C₈₃	1	4	yes	—
164	#384	C₂²×C₈₃	1	2	yes	—
165	C₄³	C₂²×C₈₃	1	1	yes	characteristic
166	#340	C₂²×C₈₃	1	4	yes	—
167	#336	C₂²×C₈₃	1	4	yes	—
168	C₂²×D₄	—	2	2	no	—
169	C₂²×D₄	—	2	2	no	—
170	C₂²×D₄	C₂³×C₈₃	1	4	yes	—
171	C₂³×C₄	—	2	4	no	—
172	C₂×D₈	—	4	8	no	—
173	C₂×D₈	C₂×C₄×C₈₃	1	16	yes	complemented
174	C₂²×C₈	C₂³×C₈₃	1	2	yes	—
175	C₂²×C₈	C₂³×C₈₃	1	2	yes	—
176	#127	—	2	4	no	—
177	#127	C₂×C₄×C₈₃	1	8	yes	complemented
178	C₄×D₄	—	4	8	no	—
179	C₄×D₄	—	4	8	no	—
180	C₄×D₄	—	4	8	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	C₂³×C₈₃	1	4	yes	—
184	C₂×C₂²⋊C₄	—	2	4	no	—
185	C₂×C₂²⋊C₄	—	2	4	no	—
186	C₂×C₂²⋊C₄	—	2	4	no	—
187	C₂×C₄²	D₄×C₈₃	1	2	yes	—
188	C₂×C₄²	D₄×C₈₃	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₈₃	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₈₃	1	2	yes	—
191	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
192	#102	—	4	8	no	—
193	C₄×C₈	—	2	2	no	—
194	C₄×C₈	C₂×C₄×C₈₃	1	4	yes	complemented
195	C₂⁴	—	2	4	no	—
196	C₂×D₄	—	2	8	no	—
197	C₂×D₄	—	2	8	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	8	no	—
200	C₂×D₄	—	2	8	no	—
201	C₂×D₄	—	4	4	no	—
202	C₂×D₄	—	4	4	no	—
203	C₂×D₄	—	4	4	no	—
204	C₂×D₄	—	4	4	no	—
205	C₂×D₄	C₂²×C₄×C₈₃	1	8	yes	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	8	no	—
208	C₂²×C₄	—	4	8	no	—
209	C₂²×C₄	—	4	8	no	—
210	C₂²×C₄	C₂×D₄×C₈₃	1	2	yes	—
211	C₂²×C₄	C₂×D₄×C₈₃	1	2	yes	—
212	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
214	C₂²×C₄	C₂⁴×C₈₃	1	1	yes	characteristic
215	D₈	—	4	16	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	C₂²×C₄×C₈₃	1	4	yes	—
220	C₄⋊C₄	—	2	8	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₂²⋊C₄	—	4	8	no	—
224	C₂²⋊C₄	—	4	8	no	—
225	C₂²⋊C₄	—	4	8	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	4	no	—
228	C₄²	—	2	4	no	—
229	C₄²	D₈×C₈₃	1	4	yes	complemented
230	C₄²	C₂²×C₄×C₈₃	1	2	yes	—
231	C₄²	#41×C₈₃	1	2	yes	—
232	C₂³	—	2	8	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	4	no	—
235	C₂³	—	4	4	no	—
236	C₂³	—	4	4	no	—
237	C₂³	—	4	4	no	—
238	C₂³	—	4	4	no	—
239	C₂³	C₂²×D₄×C₈₃	1	1	yes	characteristic
240	D₄	—	2	16	no	—
241	D₄	—	4	8	no	—
242	D₄	—	4	16	no	—
243	D₄	—	4	16	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	8	16	no	—
253	C₂×C₄	C₄×D₄×C₈₃	1	2	yes	—
254	C₂×C₄	C₄×D₄×C₈₃	1	2	yes	—
255	C₂×C₄	C₂×D₈×C₈₃	1	4	yes	—
256	C₂×C₄	C₂×D₈×C₈₃	1	2	yes	—
257	C₂×C₄	#135×C₈₃	1	2	yes	—
258	C₂×C₄	AGL(1,ℤ/8)×C₈₃	1	4	yes	—
259	C₂×C₄	C₂³×C₄×C₈₃	1	1	yes	characteristic
260	C₂×C₄	C₂×#41×C₈₃	1	1	yes	characteristic
261	C₂×C₄	H(5,𝔽₂)×C₈₃	1	4	yes	—
262	C₈	—	2	4	no	—
263	C₈	—	4	2	no	—
264	C₂²	—	4	8	no	—
265	C₂²	—	4	8	no	—
266	C₂²	—	4	8	no	—
267	C₂²	—	8	4	no	—
268	C₂²	—	8	4	no	—
269	C₂²	—	8	4	no	—
270	C₂²	—	8	4	no	—
271	C₂²	C₂×C₄×D₄×C₈₃	1	1	yes	characteristic
272	C₂²	#535×C₈₃	1	2	yes	—
273	C₂²	C₂²×D₈×C₈₃	1	1	yes	characteristic
274	C₂²	C₂×#135×C₈₃	1	1	yes	characteristic
275	C₂²	C₂×AGL(1,ℤ/8)×C₈₃	1	2	yes	—
276	C₄	—	2	4	no	—
277	C₄	—	4	2	no	—
278	C₄	—	8	4	no	—
279	C₄	C₄×D₈×C₈₃	1	4	yes	—
280	C₄	#442×C₈₃	1	4	yes	—
281	C₄	#518×C₈₃	1	2	yes	—
282	C₄	#540×C₈₃	1	2	yes	—
283	C₂	—	8	8	no	—
284	C₂	C₈₃×#2328	1	1	yes	characteristic
285	C₂	C₂×C₄×D₈×C₈₃	1	1	yes	characteristic
286	C₂	C₂×#442×C₈₃	1	1	yes	characteristic
287	C₂	C₈₃×#3205	1	2	yes	—
288	C₂	C₈₃×#3207	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₈₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₃
prime radical	C₂×#493×C₈₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₃ ◁ C₄³×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#127×C₈₃	C₂	semidirect
3. C₈₃×#1957	C₂	semidirect
4. C₈₃×#1879	C₂	semidirect
5. C₈₃×#1834	C₂	semidirect
6. C₈₃×#1744	C₂	semidirect
12. #493×C₈₃	C₄	semidirect
28. C₂×D₈×C₈₃	C₂×C₄	semidirect
32. #127×C₈₃	C₂×C₄	semidirect
49. C₄×C₈×C₈₃	C₂×C₄	semidirect
84. C₄²×C₈₃	D₈	semidirect
138. #22269	C₈₃	direct (N × H)
146. C₄×#127	C₂×C₈₃	semidirect
147. #1957	C₂×C₈₃	semidirect
148. #1879	C₂×C₈₃	semidirect
149. #1834	C₂×C₈₃	semidirect
150. #1744	C₂×C₈₃	semidirect
151. C₈₃	#22269	direct (N × H)
157. #493	C₄×C₈₃	semidirect
173. C₂×D₈	C₂×C₄×C₈₃	semidirect
177. #127	C₂×C₄×C₈₃	semidirect
194. C₄×C₈	C₂×C₄×C₈₃	semidirect
229. C₄²	D₈×C₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493×C₈₃	2
C₄×#127×C₈₃	2
C₈₃×#1957	2
C₈₃×#1879	2
C₈₃×#1834	2
C₈₃×#1744	2
#22269	83

Subgroups — C83×#22269 #35250093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#22269 #35250093