Subgroups — C₈₃×#7900 #35235724

Order 21248.

638 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₈₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₈₃	C₂	1	1	yes	maximal, characteristic
3	C₈₃×#2285	C₂	1	1	yes	maximal, characteristic
4	C₈₃×#1279	C₂	1	4	yes	maximal, complemented
5	C₂×#117×C₈₃	—	2	1	no	—
6	C₂²×C₄²×C₈₃	C₂²	1	1	yes	characteristic
7	#432×C₈₃	—	2	2	no	—
8	#432×C₈₃	C₄	1	4	yes	complemented
9	#431×C₈₃	—	2	2	no	—
10	#431×C₈₃	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₈×C₈₃	—	2	1	no	—
12	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₈×C₈₃	—	2	1	no	—
14	C₂×C₂²⋊C₈×C₈₃	—	2	1	no	—
15	C₂×#97×C₈₃	—	2	1	no	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	2	yes	—
17	#384×C₈₃	C₄	1	2	yes	—
18	#382×C₈₃	—	2	1	no	—
19	#380×C₈₃	—	2	1	no	—
20	#380×C₈₃	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₈₃	—	2	1	no	—
22	C₄×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
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	2	no	—
27	C₂²×C₈×C₈₃	—	2	2	no	—
28	#117×C₈₃	—	4	4	no	—
29	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
33	C₂×C₄²×C₈₃	—	2	2	no	—
34	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₃	D₄	1	2	yes	—
40	C₂×C₄²×C₈₃	Q₈	1	2	yes	—
41	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₈₃	—	2	4	no	—
43	C₄⋊C₈×C₈₃	—	2	2	no	—
44	C₄⋊C₈×C₈₃	C₂×C₄	1	4	yes	—
45	C₂²⋊C₈×C₈₃	—	2	4	no	—
46	C₂²⋊C₈×C₈₃	—	2	4	no	—
47	#97×C₈₃	—	2	4	no	—
48	C₄×C₈×C₈₃	—	2	2	no	—
49	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	—
50	#95×C₈₃	—	2	2	no	—
51	#95×C₈₃	—	2	2	no	—
52	C₂⁴×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₃	—	2	4	no	—
54	C₂²×C₄×C₈₃	—	2	2	no	—
55	C₂²×C₄×C₈₃	—	2	2	no	—
56	C₂²×C₄×C₈₃	—	2	4	no	—
57	C₂²×C₄×C₈₃	—	4	2	no	—
58	C₂²×C₄×C₈₃	C₄²	1	2	yes	—
59	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₈₃	—	2	4	no	—
71	C₂×C₈×C₈₃	—	2	4	no	—
72	C₂×C₈×C₈₃	—	4	4	no	—
73	C₂×C₈×C₈₃	—	4	2	no	—
74	C₂×C₈×C₈₃	—	4	2	no	—
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	4	no	—
79	C₄²×C₈₃	—	4	4	no	—
80	C₄²×C₈₃	C₄²	1	2	yes	—
81	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
85	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
86	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
90	C₂³×C₈₃	—	2	2	no	—
91	C₂³×C₈₃	—	2	1	no	—
92	C₂³×C₈₃	—	2	1	no	—
93	C₂³×C₈₃	—	2	2	no	—
94	C₂³×C₈₃	#95	1	2	yes	—
95	C₂³×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₈₃	—	2	2	no	—
97	C₂×C₄×C₈₃	—	2	1	no	—
98	C₂×C₄×C₈₃	—	2	1	no	—
99	C₂×C₄×C₈₃	—	2	2	no	—
100	C₂×C₄×C₈₃	—	2	2	no	—
101	C₂×C₄×C₈₃	—	2	4	no	—
102	C₂×C₄×C₈₃	—	2	4	no	—
103	C₂×C₄×C₈₃	—	2	4	no	—
104	C₂×C₄×C₈₃	—	2	4	no	—
105	C₂×C₄×C₈₃	—	4	2	no	—
106	C₂×C₄×C₈₃	—	4	2	no	—
107	C₂×C₄×C₈₃	—	4	4	no	—
108	C₂×C₄×C₈₃	—	4	4	no	—
109	C₂×C₄×C₈₃	—	4	4	no	—
110	C₂×C₄×C₈₃	#95	1	2	yes	—
111	C₂×C₄×C₈₃	#95	1	2	yes	—
112	C₂×C₄×C₈₃	#95	1	2	yes	—
113	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	#131	1	4	yes	—
120	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×C₄×C₈₃	#137	1	2	yes	—
122	C₈×C₈₃	—	4	4	no	—
123	C₈×C₈₃	—	4	4	no	—
124	C₂²×C₈₃	—	2	2	no	—
125	C₂²×C₈₃	—	2	2	no	—
126	C₂²×C₈₃	—	2	2	no	—
127	C₂²×C₈₃	—	4	2	no	—
128	C₂²×C₈₃	—	4	2	no	—
129	C₂²×C₈₃	C₂×#95	1	1	yes	characteristic
130	C₂²×C₈₃	#408	1	2	yes	—
131	C₂²×C₈₃	#418	1	1	yes	characteristic
132	C₂²×C₈₃	#419	1	1	yes	characteristic
133	C₂²×C₈₃	#421	1	2	yes	—
134	C₄×C₈₃	—	2	2	no	—
135	C₄×C₈₃	—	2	1	no	—
136	C₄×C₈₃	—	2	2	no	—
137	C₄×C₈₃	—	2	1	no	—
138	C₄×C₈₃	—	2	2	no	—
139	C₄×C₈₃	—	8	4	no	—
140	C₄×C₈₃	#405	1	2	yes	—
141	C₄×C₈₃	#417	1	2	yes	—
142	C₄×C₈₃	#424	1	2	yes	—
143	C₄×C₈₃	#428	1	2	yes	—
144	#7900	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₈₃	—	4	2	no	—
146	C₂×C₈₃	#1496	1	2	yes	—
147	C₂×C₈₃	#1497	1	2	yes	—
148	C₂×C₈₃	#1729	1	1	yes	characteristic
149	C₂×C₈₃	#1733	1	1	yes	characteristic
150	C₂×C₈₃	#1736	1	1	yes	characteristic
151	C₂×#432	C₂×C₈₃	1	1	yes	characteristic
152	C₂×#431	C₂×C₈₃	1	1	yes	characteristic
153	#2285	C₂×C₈₃	1	1	yes	characteristic
154	#1279	C₂×C₈₃	1	4	yes	complemented
155	C₈₃	#7900	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×#117	—	2	1	no	—
157	C₂²×C₄²	C₂²×C₈₃	1	1	yes	characteristic
158	#432	—	2	2	no	—
159	#432	C₄×C₈₃	1	4	yes	complemented
160	#431	—	2	2	no	—
161	#431	C₄×C₈₃	1	4	yes	complemented
162	C₂×C₄⋊C₈	—	2	1	no	—
163	C₂×C₄⋊C₈	C₂²×C₈₃	1	2	yes	—
164	C₂×C₂²⋊C₈	—	2	1	no	—
165	C₂×C₂²⋊C₈	—	2	1	no	—
166	C₂×#97	—	2	1	no	—
167	C₂×C₄×C₈	C₂²×C₈₃	1	2	yes	—
168	#384	C₄×C₈₃	1	2	yes	—
169	#382	—	2	1	no	—
170	#380	—	2	1	no	—
171	#380	C₄×C₈₃	1	2	yes	—
172	C₄×C₄⋊C₄	—	2	1	no	—
173	C₄×C₄⋊C₄	C₂²×C₈₃	1	2	yes	—
174	C₂³×C₄	C₂×C₄×C₈₃	1	1	yes	characteristic
175	C₂³×C₄	D₄×C₈₃	1	1	yes	characteristic
176	C₂³×C₄	D₄×C₈₃	1	1	yes	characteristic
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	#117	—	4	4	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	C₂×C₄×C₈₃	1	4	yes	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂×C₄×C₈₃	1	1	yes	characteristic
186	C₂×C₄²	C₂×C₄×C₈₃	1	2	yes	—
187	C₂×C₄²	C₂×C₄×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	D₄×C₈₃	1	2	yes	—
191	C₂×C₄²	Q₈×C₈₃	1	2	yes	—
192	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
193	C₄⋊C₈	—	2	4	no	—
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	C₂×C₄×C₈₃	1	4	yes	—
196	C₂²⋊C₈	—	2	4	no	—
197	C₂²⋊C₈	—	2	4	no	—
198	#97	—	2	4	no	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	C₂×C₄×C₈₃	1	4	yes	—
201	#95	—	2	2	no	—
202	#95	—	2	2	no	—
203	C₂⁴	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₄²×C₈₃	1	2	yes	—
210	C₂²×C₄	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²⋊C₄×C₈₃	1	2	yes	—
214	C₂²×C₄	C₂²⋊C₄×C₈₃	1	2	yes	—
215	C₂²×C₄	C₄⋊C₄×C₈₃	1	2	yes	—
216	C₂²×C₄	C₄⋊C₄×C₈₃	1	2	yes	—
217	C₂²×C₄	C₄⋊C₄×C₈₃	1	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₈₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	4	4	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₄⋊C₄	—	2	8	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₂²⋊C₄	—	4	4	no	—
230	C₄²	—	4	4	no	—
231	C₄²	C₄²×C₈₃	1	2	yes	—
232	C₄²	C₂²⋊C₄×C₈₃	1	2	yes	—
233	C₄²	C₂²⋊C₄×C₈₃	1	2	yes	—
234	C₄²	C₄⋊C₄×C₈₃	1	2	yes	—
235	C₄²	C₄⋊C₄×C₈₃	1	2	yes	—
236	C₄²	C₄⋊C₄×C₈₃	1	2	yes	—
237	C₄²	C₂²×C₄×C₈₃	1	1	yes	characteristic
238	C₄²	C₂²×C₄×C₈₃	1	1	yes	characteristic
239	C₄²	C₂×D₄×C₈₃	1	1	yes	characteristic
240	C₄²	C₂×Q₈×C₈₃	1	1	yes	characteristic
241	C₂³	—	2	2	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	2	no	—
245	C₂³	#95×C₈₃	1	2	yes	—
246	C₂³	C₂×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	#95×C₈₃	1	2	yes	—
262	C₂×C₄	#95×C₈₃	1	2	yes	—
263	C₂×C₄	#95×C₈₃	1	2	yes	—
264	C₂×C₄	C₂×C₄²×C₈₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
266	C₂×C₄	C₂×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊C₄×C₈₃	1	1	yes	characteristic
268	C₂×C₄	C₂×C₄⋊C₄×C₈₃	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄⋊C₄×C₈₃	1	1	yes	characteristic
270	C₂×C₄	#131×C₈₃	1	4	yes	—
271	C₂×C₄	AGL(1,ℤ/8)×C₈₃	1	2	yes	—
272	C₂×C₄	#137×C₈₃	1	2	yes	—
273	C₈	—	4	4	no	—
274	C₈	—	4	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	C₂×#95×C₈₃	1	1	yes	characteristic
281	C₂²	#408×C₈₃	1	2	yes	—
282	C₂²	#418×C₈₃	1	1	yes	characteristic
283	C₂²	#419×C₈₃	1	1	yes	characteristic
284	C₂²	#421×C₈₃	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	8	4	no	—
291	C₄	#405×C₈₃	1	2	yes	—
292	C₄	#417×C₈₃	1	2	yes	—
293	C₄	#424×C₈₃	1	2	yes	—
294	C₄	#428×C₈₃	1	2	yes	—
295	C₂	—	4	2	no	—
296	C₂	C₈₃×#1496	1	2	yes	—
297	C₂	C₈₃×#1497	1	2	yes	—
298	C₂	C₈₃×#1729	1	1	yes	characteristic
299	C₂	C₈₃×#1733	1	1	yes	characteristic
300	C₂	C₈₃×#1736	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	C₂³×C₈₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₈₃	C₂²⋊C₄
LCS	G ▷ C₂×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
8. #432×C₈₃	C₄	semidirect
10. #431×C₈₃	C₄	semidirect
144. #7900	C₈₃	direct (N × H)
154. #1279	C₂×C₈₃	semidirect
155. C₈₃	#7900	direct (N × H)
159. #432	C₄×C₈₃	semidirect
161. #431	C₄×C₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#432×C₈₃	2
C₂×#431×C₈₃	2
C₈₃×#2285	2
C₈₃×#1279	2
#7900	83

Subgroups — C83×#7900 #35235724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#7900 #35235724