Subgroups — C₈₃×#61837 #35289661

Order 21248.

1928 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₃×#3523	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₃×#3386	C₂	1	6	yes	maximal, complemented
4	Q₈×D₈×C₈₃	C₂	1	2	yes	maximal, complemented
5	C₈₃×#3367	C₂	1	6	yes	maximal, complemented
6	C₈₃×#3359	C₂	1	2	yes	maximal, complemented
7	C₈₃×#3239	C₂	1	3	yes	maximal, complemented
8	C₈₃×#3230	C₂	1	3	yes	maximal, complemented
9	C₈₃×#3110	C₂	1	3	yes	maximal, complemented
10	C₈₃×#3105	C₂	1	3	yes	maximal
11	C₈₃×#2984	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₈₃	—	2	1	no	—
13	#557×C₈₃	C₂²	1	1	yes	characteristic, complemented
14	#557×C₈₃	C₂²	1	3	yes	complemented
15	#556×C₈₃	C₂²	1	3	yes	complemented
16	#552×C₈₃	—	2	3	no	—
17	D₄×Q₈×C₈₃	—	2	1	no	—
18	D₄×Q₈×C₈₃	—	2	1	no	—
19	D₄×Q₈×C₈₃	—	2	3	no	—
20	D₄×Q₈×C₈₃	C₂²	1	2	yes	—
21	D₄×Q₈×C₈₃	C₂²	1	6	yes	—
22	D₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic, complemented
23	#541×C₈₃	—	2	3	no	—
24	#533×C₈₃	C₂²	1	3	yes	—
25	C₂×C₄⋊Q₈×C₈₃	C₂²	1	3	yes	—
26	#529×C₈₃	C₂²	1	3	yes	—
27	C₂×C₂²⋊Q₈×C₈₃	—	2	3	no	—
28	#519×C₈₃	—	2	1	no	—
29	C₂×C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₈₃	C₂²	1	3	yes	—
31	#501×C₈₃	C₂²	1	6	yes	complemented
32	#500×C₈₃	C₂²	1	6	yes	complemented
33	#489×C₈₃	C₂²	1	6	yes	complemented
34	#487×C₈₃	C₂²	1	6	yes	complemented
35	#484×C₈₃	C₂²	1	3	yes	—
36	#484×C₈₃	C₂²	1	6	yes	—
37	#481×C₈₃	C₂²	1	3	yes	—
38	#478×C₈₃	C₂²	1	6	yes	—
39	#477×C₈₃	C₂²	1	12	yes	complemented
40	#475×C₈₃	C₂²	1	6	yes	complemented
41	#474×C₈₃	C₂²	1	6	yes	—
42	#474×C₈₃	C₂²	1	6	yes	complemented
43	#463×C₈₃	C₂²	1	6	yes	—
44	#461×C₈₃	C₂²	1	6	yes	—
45	#450×C₈₃	C₂²	1	3	yes	—
46	#447×C₈₃	C₂²	1	1	yes	characteristic
47	#446×C₈₃	C₂²	1	6	yes	complemented
48	C₈×Q₈×C₈₃	C₂²	1	2	yes	complemented
49	#440×C₈₃	C₂²	1	12	yes	—
50	C₄×SD₁₆×C₈₃	C₂²	1	6	yes	—
51	C₄×D₈×C₈₃	C₂²	1	6	yes	—
52	#432×C₈₃	C₂²	1	3	yes	—
53	#431×C₈₃	C₂²	1	3	yes	—
54	C₂²×Q₈×C₈₃	—	2	1	no	—
55	C₂²×Q₈×C₈₃	—	2	6	no	—
56	C₂²×Q₈×C₈₃	—	2	2	no	—
57	C₂²×Q₈×C₈₃	—	2	3	no	—
58	C₂²×Q₈×C₈₃	—	4	8	no	—
59	C₂²×Q₈×C₈₃	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₈₃	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₈₃	—	2	3	no	—
62	C₂×SD₁₆×C₈₃	C₂³	1	6	yes	—
63	C₂×D₈×C₈₃	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₈₃	—	2	3	no	—
65	C₄⋊Q₈×C₈₃	—	2	3	no	—
66	C₄⋊Q₈×C₈₃	—	2	3	no	—
67	C₄⋊Q₈×C₈₃	C₂³	1	3	yes	—
68	C₄⋊Q₈×C₈₃	C₂³	1	3	yes	—
69	C₄⋊Q₈×C₈₃	C₂³	1	6	yes	—
70	C₄⋊Q₈×C₈₃	C₂³	1	6	yes	—
71	#126×C₈₃	—	2	6	no	—
72	#126×C₈₃	—	2	3	no	—
73	#125×C₈₃	—	2	6	no	—
74	#125×C₈₃	—	2	3	no	—
75	#125×C₈₃	—	2	3	no	—
76	#125×C₈₃	C₂³	1	3	yes	—
77	#124×C₈₃	C₂³	1	3	yes	—
78	#123×C₈₃	—	2	3	no	—
79	C₂²⋊Q₈×C₈₃	—	2	3	no	—
80	C₂²⋊Q₈×C₈₃	—	2	3	no	—
81	C₂²⋊Q₈×C₈₃	—	2	6	no	—
82	C₂²⋊Q₈×C₈₃	—	2	6	no	—
83	C₂²⋊Q₈×C₈₃	—	2	12	no	—
84	C₂²⋊Q₈×C₈₃	—	2	3	no	—
85	C₂²⋊Q₈×C₈₃	C₂³	1	3	yes	—
86	#121×C₈₃	C₂³	1	1	yes	characteristic
87	C₄×Q₈×C₈₃	—	2	1	no	—
88	C₄×Q₈×C₈₃	—	2	3	no	—
89	C₄×Q₈×C₈₃	—	2	3	no	—
90	C₄×Q₈×C₈₃	D₄	1	1	yes	characteristic
91	C₄×Q₈×C₈₃	D₄	1	3	yes	—
92	C₄×Q₈×C₈₃	C₂³	1	3	yes	—
93	C₄×Q₈×C₈₃	C₂³	1	6	yes	—
94	C₄×Q₈×C₈₃	C₂³	1	1	yes	characteristic
95	C₄×Q₈×C₈₃	C₂³	1	3	yes	—
96	C₄×D₄×C₈₃	—	2	3	no	—
97	C₄×D₄×C₈₃	—	2	3	no	—
98	C₄×D₄×C₈₃	—	2	3	no	—
99	C₄×D₄×C₈₃	C₂³	1	6	yes	—
100	C₄×D₄×C₈₃	C₂³	1	6	yes	—
101	C₄×D₄×C₈₃	C₂³	1	3	yes	—
102	#117×C₈₃	—	2	3	no	—
103	#117×C₈₃	—	2	3	no	—
104	#117×C₈₃	—	2	3	no	—
105	C₂×C₄⋊C₄×C₈₃	—	2	3	no	—
106	C₂×C₄⋊C₄×C₈₃	D₄	1	3	yes	—
107	C₂×C₄⋊C₄×C₈₃	C₂³	1	3	yes	—
108	C₂×C₂²⋊C₄×C₈₃	—	2	3	no	—
109	C₂×C₄²×C₈₃	C₂³	1	3	yes	—
110	#107×C₈₃	C₂³	1	12	yes	—
111	#107×C₈₃	C₂³	1	6	yes	—
112	#106×C₈₃	C₂³	1	6	yes	—
113	C₄⋊C₈×C₈₃	C₂³	1	6	yes	—
114	C₄⋊C₈×C₈₃	C₂³	1	6	yes	—
115	#103×C₈₃	C₂³	1	6	yes	—
116	#103×C₈₃	C₂³	1	12	yes	—
117	#102×C₈₃	C₂³	1	6	yes	—
118	#102×C₈₃	C₂³	1	6	yes	—
119	#102×C₈₃	C₂³	1	2	yes	—
120	C₂²⋊C₈×C₈₃	C₂³	1	3	yes	—
121	C₂²⋊C₈×C₈₃	C₂³	1	1	yes	characteristic
122	#97×C₈₃	C₂³	1	6	yes	—
123	C₄×C₈×C₈₃	C₂³	1	6	yes	—
124	C₂⁴×C₈₃	—	2	1	no	—
125	C₂×Q₈×C₈₃	—	2	4	no	—
126	C₂×Q₈×C₈₃	—	2	3	no	—
127	C₂×Q₈×C₈₃	—	2	6	no	—
128	C₂×Q₈×C₈₃	—	2	6	no	—
129	C₂×Q₈×C₈₃	—	4	12	no	—
130	C₂×Q₈×C₈₃	—	4	24	no	—
131	C₂×Q₈×C₈₃	—	4	4	no	—
132	C₂×Q₈×C₈₃	—	4	8	no	—
133	C₂×Q₈×C₈₃	C₂×D₄	1	3	yes	—
134	C₂×Q₈×C₈₃	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₈₃	C₂⁴	1	3	yes	—
136	C₂×D₄×C₈₃	—	2	4	no	—
137	C₂×D₄×C₈₃	—	2	1	no	—
138	C₂×D₄×C₈₃	—	2	1	no	—
139	C₂×D₄×C₈₃	C₂⁴	1	1	yes	characteristic
140	C₂×D₄×C₈₃	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₈₃	—	2	3	no	—
142	C₂²×C₄×C₈₃	—	2	3	no	—
143	C₂²×C₄×C₈₃	—	2	6	no	—
144	C₂²×C₄×C₈₃	—	2	6	no	—
145	C₂²×C₄×C₈₃	—	4	12	no	—
146	C₂²×C₄×C₈₃	C₂×D₄	1	3	yes	—
147	C₂²×C₄×C₈₃	C₂×D₄	1	3	yes	—
148	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
149	SD₁₆×C₈₃	—	2	12	no	—
150	D₈×C₈₃	—	2	4	no	—
151	C₂×C₈×C₈₃	C₂⁴	1	2	yes	—
152	C₂×C₈×C₈₃	C₂⁴	1	6	yes	—
153	C₄⋊C₄×C₈₃	—	2	6	no	—
154	C₄⋊C₄×C₈₃	—	2	6	no	—
155	C₄⋊C₄×C₈₃	—	2	3	no	—
156	C₄⋊C₄×C₈₃	—	2	3	no	—
157	C₄⋊C₄×C₈₃	—	2	3	no	—
158	C₄⋊C₄×C₈₃	—	2	3	no	—
159	C₄⋊C₄×C₈₃	—	2	3	no	—
160	C₄⋊C₄×C₈₃	—	2	3	no	—
161	C₄⋊C₄×C₈₃	—	2	3	no	—
162	C₄⋊C₄×C₈₃	—	2	6	no	—
163	C₄⋊C₄×C₈₃	—	2	3	no	—
164	C₄⋊C₄×C₈₃	C₂×D₄	1	3	yes	—
165	C₄⋊C₄×C₈₃	C₂×D₄	1	3	yes	—
166	C₄⋊C₄×C₈₃	C₂×D₄	1	6	yes	—
167	C₄⋊C₄×C₈₃	C₂⁴	1	1	yes	characteristic
168	C₄⋊C₄×C₈₃	C₂⁴	1	3	yes	—
169	C₄⋊C₄×C₈₃	C₂⁴	1	6	yes	—
170	C₂²⋊C₄×C₈₃	—	2	3	no	—
171	C₂²⋊C₄×C₈₃	—	2	6	no	—
172	C₂²⋊C₄×C₈₃	—	2	6	no	—
173	C₂²⋊C₄×C₈₃	—	2	1	no	—
174	C₂²⋊C₄×C₈₃	—	2	3	no	—
175	C₂²⋊C₄×C₈₃	—	2	3	no	—
176	C₂²⋊C₄×C₈₃	—	2	3	no	—
177	C₄²×C₈₃	—	2	6	no	—
178	C₄²×C₈₃	—	2	3	no	—
179	C₄²×C₈₃	C₂×D₄	1	3	yes	—
180	C₄²×C₈₃	C₂×D₄	1	3	yes	—
181	C₄²×C₈₃	C₂⁴	1	3	yes	—
182	C₄²×C₈₃	C₂⁴	1	3	yes	—
183	C₂³×C₈₃	—	2	4	no	—
184	C₂³×C₈₃	—	2	2	no	—
185	C₂³×C₈₃	—	2	1	no	—
186	C₂³×C₈₃	—	4	2	no	—
187	C₂³×C₈₃	—	4	2	no	—
188	C₂³×C₈₃	C₂²×D₄	1	1	yes	characteristic
189	Q₈×C₈₃	—	2	6	no	—
190	Q₈×C₈₃	—	2	6	no	—
191	Q₈×C₈₃	—	4	12	no	—
192	Q₈×C₈₃	—	4	6	no	—
193	Q₈×C₈₃	—	4	12	no	—
194	Q₈×C₈₃	AGL(1,ℤ/8)	1	4	yes	—
195	D₄×C₈₃	—	2	4	no	—
196	D₄×C₈₃	—	2	2	no	—
197	D₄×C₈₃	—	2	2	no	—
198	D₄×C₈₃	—	4	2	no	—
199	D₄×C₈₃	2₋¹⁺⁴	1	4	yes	—
200	C₂×C₄×C₈₃	—	2	6	no	—
201	C₂×C₄×C₈₃	—	2	6	no	—
202	C₂×C₄×C₈₃	—	2	2	no	—
203	C₂×C₄×C₈₃	—	2	3	no	—
204	C₂×C₄×C₈₃	—	2	1	no	—
205	C₂×C₄×C₈₃	—	2	3	no	—
206	C₂×C₄×C₈₃	—	2	6	no	—
207	C₂×C₄×C₈₃	—	4	12	no	—
208	C₂×C₄×C₈₃	—	4	6	no	—
209	C₂×C₄×C₈₃	—	4	12	no	—
210	C₂×C₄×C₈₃	—	4	6	no	—
211	C₂×C₄×C₈₃	C₂²×D₄	1	3	yes	—
212	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₈₃	C₂²×D₄	1	3	yes	—
214	C₂×C₄×C₈₃	C₂²×D₄	1	3	yes	—
215	C₂×C₄×C₈₃	C₂²×D₄	1	3	yes	—
216	C₂×C₄×C₈₃	C₂⁵	1	1	yes	characteristic
217	C₈×C₈₃	—	2	2	no	—
218	C₈×C₈₃	—	2	6	no	—
219	C₂²×C₈₃	—	2	1	no	—
220	C₂²×C₈₃	—	2	1	no	—
221	C₂²×C₈₃	—	2	4	no	—
222	C₂²×C₈₃	—	4	8	no	—
223	C₂²×C₈₃	—	4	2	no	—
224	C₂²×C₈₃	—	4	2	no	—
225	C₂²×C₈₃	—	4	1	no	—
226	C₂²×C₈₃	—	4	1	no	—
227	C₂²×C₈₃	—	4	1	no	—
228	C₂²×C₈₃	#578	1	2	yes	—
229	C₂²×C₈₃	C₂³×D₄	1	1	yes	characteristic
230	C₄×C₈₃	—	2	1	no	—
231	C₄×C₈₃	—	2	3	no	—
232	C₄×C₈₃	—	2	3	no	—
233	C₄×C₈₃	—	2	3	no	—
234	C₄×C₈₃	—	4	3	no	—
235	C₄×C₈₃	—	4	6	no	—
236	C₄×C₈₃	—	4	3	no	—
237	C₄×C₈₃	—	4	1	no	—
238	C₄×C₈₃	C₂×AGL(1,ℤ/8)	1	6	yes	—
239	C₄×C₈₃	C₂×2₋¹⁺⁴	1	2	yes	—
240	#61837	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
241	C₂×C₈₃	—	2	2	no	—
242	C₂×C₈₃	—	4	4	no	—
243	C₂×C₈₃	—	8	1	no	—
244	C₂×C₈₃	#3486	1	1	yes	characteristic
245	C₂×C₈₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
246	C₂×C₈₃	C₂×#578	1	1	yes	characteristic
247	#3523	C₂×C₈₃	1	1	yes	characteristic, complemented
248	C₂×D₄×Q₈	C₂×C₈₃	1	1	yes	characteristic, complemented
249	#3386	C₂×C₈₃	1	6	yes	complemented
250	Q₈×D₈	C₂×C₈₃	1	2	yes	complemented
251	#3367	C₂×C₈₃	1	6	yes	complemented
252	#3359	C₂×C₈₃	1	2	yes	complemented
253	#3239	C₂×C₈₃	1	3	yes	complemented
254	#3230	C₂×C₈₃	1	3	yes	complemented
255	#3110	C₂×C₈₃	1	3	yes	complemented
256	#3105	C₂×C₈₃	1	3	yes	—
257	#2984	C₂×C₈₃	1	1	yes	characteristic, complemented
258	C₈₃	#61837	1	1	yes	characteristic, direct product factor, Sylow
259	C₂³×Q₈	—	2	1	no	—
260	#557	C₂²×C₈₃	1	1	yes	characteristic, complemented
261	#557	C₂²×C₈₃	1	3	yes	complemented
262	#556	C₂²×C₈₃	1	3	yes	complemented
263	#552	—	2	3	no	—
264	D₄×Q₈	—	2	1	no	—
265	D₄×Q₈	—	2	1	no	—
266	D₄×Q₈	—	2	3	no	—
267	D₄×Q₈	C₂²×C₈₃	1	2	yes	—
268	D₄×Q₈	C₂²×C₈₃	1	6	yes	—
269	D₄×Q₈	C₂²×C₈₃	1	1	yes	characteristic, complemented
270	#541	—	2	3	no	—
271	#533	C₂²×C₈₃	1	3	yes	—
272	C₂×C₄⋊Q₈	C₂²×C₈₃	1	3	yes	—
273	#529	C₂²×C₈₃	1	3	yes	—
274	C₂×C₂²⋊Q₈	—	2	3	no	—
275	#519	—	2	1	no	—
276	C₂×C₄×Q₈	C₂²×C₈₃	1	1	yes	characteristic
277	C₂×C₄×D₄	C₂²×C₈₃	1	3	yes	—
278	#501	C₂²×C₈₃	1	6	yes	complemented
279	#500	C₂²×C₈₃	1	6	yes	complemented
280	#489	C₂²×C₈₃	1	6	yes	complemented
281	#487	C₂²×C₈₃	1	6	yes	complemented
282	#484	C₂²×C₈₃	1	3	yes	—
283	#484	C₂²×C₈₃	1	6	yes	—
284	#481	C₂²×C₈₃	1	3	yes	—
285	#478	C₂²×C₈₃	1	6	yes	—
286	#477	C₂²×C₈₃	1	12	yes	complemented
287	#475	C₂²×C₈₃	1	6	yes	complemented
288	#474	C₂²×C₈₃	1	6	yes	—
289	#474	C₂²×C₈₃	1	6	yes	complemented
290	#463	C₂²×C₈₃	1	6	yes	—
291	#461	C₂²×C₈₃	1	6	yes	—
292	#450	C₂²×C₈₃	1	3	yes	—
293	#447	C₂²×C₈₃	1	1	yes	characteristic
294	#446	C₂²×C₈₃	1	6	yes	complemented
295	C₈×Q₈	C₂²×C₈₃	1	2	yes	complemented
296	#440	C₂²×C₈₃	1	12	yes	—
297	C₄×SD₁₆	C₂²×C₈₃	1	6	yes	—
298	C₄×D₈	C₂²×C₈₃	1	6	yes	—
299	#432	C₂²×C₈₃	1	3	yes	—
300	#431	C₂²×C₈₃	1	3	yes	—
301	C₂²×Q₈	—	2	1	no	—
302	C₂²×Q₈	—	2	6	no	—
303	C₂²×Q₈	—	2	2	no	—
304	C₂²×Q₈	—	2	3	no	—
305	C₂²×Q₈	—	4	8	no	—
306	C₂²×Q₈	D₄×C₈₃	1	1	yes	characteristic
307	C₂²×D₄	C₂³×C₈₃	1	1	yes	characteristic
308	C₂³×C₄	—	2	3	no	—
309	C₂×SD₁₆	C₂³×C₈₃	1	6	yes	—
310	C₂×D₈	C₂³×C₈₃	1	2	yes	—
311	C₄⋊Q₈	—	2	3	no	—
312	C₄⋊Q₈	—	2	3	no	—
313	C₄⋊Q₈	—	2	3	no	—
314	C₄⋊Q₈	C₂³×C₈₃	1	3	yes	—
315	C₄⋊Q₈	C₂³×C₈₃	1	6	yes	—
316	C₄⋊Q₈	C₂³×C₈₃	1	6	yes	—
317	C₄⋊Q₈	C₂³×C₈₃	1	3	yes	—
318	#126	—	2	6	no	—
319	#126	—	2	3	no	—
320	#125	—	2	6	no	—
321	#125	—	2	3	no	—
322	#125	—	2	3	no	—
323	#125	C₂³×C₈₃	1	3	yes	—
324	#124	C₂³×C₈₃	1	3	yes	—
325	#123	—	2	3	no	—
326	C₂²⋊Q₈	—	2	3	no	—
327	C₂²⋊Q₈	—	2	3	no	—
328	C₂²⋊Q₈	—	2	6	no	—
329	C₂²⋊Q₈	—	2	6	no	—
330	C₂²⋊Q₈	—	2	12	no	—
331	C₂²⋊Q₈	—	2	3	no	—
332	C₂²⋊Q₈	C₂³×C₈₃	1	3	yes	—
333	#121	C₂³×C₈₃	1	1	yes	characteristic
334	C₄×Q₈	—	2	1	no	—
335	C₄×Q₈	—	2	3	no	—
336	C₄×Q₈	—	2	3	no	—
337	C₄×Q₈	D₄×C₈₃	1	1	yes	characteristic
338	C₄×Q₈	D₄×C₈₃	1	3	yes	—
339	C₄×Q₈	C₂³×C₈₃	1	3	yes	—
340	C₄×Q₈	C₂³×C₈₃	1	6	yes	—
341	C₄×Q₈	C₂³×C₈₃	1	1	yes	characteristic
342	C₄×Q₈	C₂³×C₈₃	1	3	yes	—
343	C₄×D₄	—	2	3	no	—
344	C₄×D₄	—	2	3	no	—
345	C₄×D₄	—	2	3	no	—
346	C₄×D₄	C₂³×C₈₃	1	6	yes	—
347	C₄×D₄	C₂³×C₈₃	1	6	yes	—
348	C₄×D₄	C₂³×C₈₃	1	3	yes	—
349	#117	—	2	3	no	—
350	#117	—	2	3	no	—
351	#117	—	2	3	no	—
352	C₂×C₄⋊C₄	—	2	3	no	—
353	C₂×C₄⋊C₄	D₄×C₈₃	1	3	yes	—
354	C₂×C₄⋊C₄	C₂³×C₈₃	1	3	yes	—
355	C₂×C₂²⋊C₄	—	2	3	no	—
356	C₂×C₄²	C₂³×C₈₃	1	3	yes	—
357	#107	C₂³×C₈₃	1	6	yes	—
358	#107	C₂³×C₈₃	1	12	yes	—
359	#106	C₂³×C₈₃	1	6	yes	—
360	C₄⋊C₈	C₂³×C₈₃	1	6	yes	—
361	C₄⋊C₈	C₂³×C₈₃	1	6	yes	—
362	#103	C₂³×C₈₃	1	6	yes	—
363	#103	C₂³×C₈₃	1	12	yes	—
364	#102	C₂³×C₈₃	1	6	yes	—
365	#102	C₂³×C₈₃	1	6	yes	—
366	#102	C₂³×C₈₃	1	2	yes	—
367	C₂²⋊C₈	C₂³×C₈₃	1	1	yes	characteristic
368	C₂²⋊C₈	C₂³×C₈₃	1	3	yes	—
369	#97	C₂³×C₈₃	1	6	yes	—
370	C₄×C₈	C₂³×C₈₃	1	6	yes	—
371	C₂⁴	—	2	1	no	—
372	C₂×Q₈	—	2	4	no	—
373	C₂×Q₈	—	2	3	no	—
374	C₂×Q₈	—	2	6	no	—
375	C₂×Q₈	—	2	6	no	—
376	C₂×Q₈	—	4	12	no	—
377	C₂×Q₈	—	4	24	no	—
378	C₂×Q₈	—	4	8	no	—
379	C₂×Q₈	—	4	4	no	—
380	C₂×Q₈	C₂×D₄×C₈₃	1	3	yes	—
381	C₂×Q₈	C₂×D₄×C₈₃	1	1	yes	characteristic
382	C₂×Q₈	C₂⁴×C₈₃	1	3	yes	—
383	C₂×D₄	—	2	4	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	C₂⁴×C₈₃	1	1	yes	characteristic
387	C₂×D₄	C₂⁴×C₈₃	1	2	yes	—
388	C₂²×C₄	—	2	3	no	—
389	C₂²×C₄	—	2	3	no	—
390	C₂²×C₄	—	2	6	no	—
391	C₂²×C₄	—	2	6	no	—
392	C₂²×C₄	—	4	12	no	—
393	C₂²×C₄	C₂×D₄×C₈₃	1	3	yes	—
394	C₂²×C₄	C₂×D₄×C₈₃	1	3	yes	—
395	C₂²×C₄	C₂⁴×C₈₃	1	1	yes	characteristic
396	SD₁₆	—	2	12	no	—
397	D₈	—	2	4	no	—
398	C₂×C₈	C₂⁴×C₈₃	1	2	yes	—
399	C₂×C₈	C₂⁴×C₈₃	1	6	yes	—
400	C₄⋊C₄	—	2	6	no	—
401	C₄⋊C₄	—	2	6	no	—
402	C₄⋊C₄	—	2	3	no	—
403	C₄⋊C₄	—	2	3	no	—
404	C₄⋊C₄	—	2	3	no	—
405	C₄⋊C₄	—	2	3	no	—
406	C₄⋊C₄	—	2	3	no	—
407	C₄⋊C₄	—	2	3	no	—
408	C₄⋊C₄	—	2	3	no	—
409	C₄⋊C₄	—	2	6	no	—
410	C₄⋊C₄	—	2	3	no	—
411	C₄⋊C₄	C₂×D₄×C₈₃	1	3	yes	—
412	C₄⋊C₄	C₂×D₄×C₈₃	1	3	yes	—
413	C₄⋊C₄	C₂×D₄×C₈₃	1	6	yes	—
414	C₄⋊C₄	C₂⁴×C₈₃	1	1	yes	characteristic
415	C₄⋊C₄	C₂⁴×C₈₃	1	3	yes	—
416	C₄⋊C₄	C₂⁴×C₈₃	1	6	yes	—
417	C₂²⋊C₄	—	2	3	no	—
418	C₂²⋊C₄	—	2	6	no	—
419	C₂²⋊C₄	—	2	6	no	—
420	C₂²⋊C₄	—	2	1	no	—
421	C₂²⋊C₄	—	2	3	no	—
422	C₂²⋊C₄	—	2	3	no	—
423	C₂²⋊C₄	—	2	3	no	—
424	C₄²	—	2	6	no	—
425	C₄²	—	2	3	no	—
426	C₄²	C₂×D₄×C₈₃	1	3	yes	—
427	C₄²	C₂×D₄×C₈₃	1	3	yes	—
428	C₄²	C₂⁴×C₈₃	1	3	yes	—
429	C₄²	C₂⁴×C₈₃	1	3	yes	—
430	C₂³	—	2	4	no	—
431	C₂³	—	2	2	no	—
432	C₂³	—	2	1	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	2	no	—
435	C₂³	C₂²×D₄×C₈₃	1	1	yes	characteristic
436	Q₈	—	2	6	no	—
437	Q₈	—	2	6	no	—
438	Q₈	—	4	12	no	—
439	Q₈	—	4	12	no	—
440	Q₈	—	4	6	no	—
441	Q₈	AGL(1,ℤ/8)×C₈₃	1	4	yes	—
442	D₄	—	2	4	no	—
443	D₄	—	2	2	no	—
444	D₄	—	2	2	no	—
445	D₄	—	4	2	no	—
446	D₄	2₋¹⁺⁴×C₈₃	1	4	yes	—
447	C₂×C₄	—	2	6	no	—
448	C₂×C₄	—	2	6	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	3	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	3	no	—
453	C₂×C₄	—	2	6	no	—
454	C₂×C₄	—	4	12	no	—
455	C₂×C₄	—	4	6	no	—
456	C₂×C₄	—	4	12	no	—
457	C₂×C₄	—	4	6	no	—
458	C₂×C₄	C₂²×D₄×C₈₃	1	3	yes	—
459	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₈₃	1	3	yes	—
461	C₂×C₄	C₂²×D₄×C₈₃	1	3	yes	—
462	C₂×C₄	C₂²×D₄×C₈₃	1	3	yes	—
463	C₂×C₄	C₂⁵×C₈₃	1	1	yes	characteristic
464	C₈	—	2	2	no	—
465	C₈	—	2	6	no	—
466	C₂²	—	2	1	no	—
467	C₂²	—	2	1	no	—
468	C₂²	—	2	4	no	—
469	C₂²	—	4	8	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	1	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	4	1	no	—
475	C₂²	#578×C₈₃	1	2	yes	—
476	C₂²	C₂³×D₄×C₈₃	1	1	yes	characteristic
477	C₄	—	2	3	no	—
478	C₄	—	2	3	no	—
479	C₄	—	2	3	no	—
480	C₄	—	2	1	no	—
481	C₄	—	4	6	no	—
482	C₄	—	4	3	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	3	no	—
485	C₄	C₂×AGL(1,ℤ/8)×C₈₃	1	6	yes	—
486	C₄	C₂×2₋¹⁺⁴×C₈₃	1	2	yes	—
487	C₂	—	2	2	no	—
488	C₂	—	4	4	no	—
489	C₂	—	8	1	no	—
490	C₂	C₈₃×#3486	1	1	yes	characteristic
491	C₂	C₂²×AGL(1,ℤ/8)×C₈₃	1	1	yes	characteristic
492	C₂	C₂×#578×C₈₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₃
prime radical	#447×C₈₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₃ ◁ C₂×C₄×Q₈×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₃×#3523	C₂	semidirect
2. C₂×D₄×Q₈×C₈₃	C₂	semidirect
3. C₈₃×#3386	C₂	semidirect
4. Q₈×D₈×C₈₃	C₂	semidirect
5. C₈₃×#3367	C₂	semidirect
6. C₈₃×#3359	C₂	semidirect
7. C₈₃×#3239	C₂	semidirect
8. C₈₃×#3230	C₂	semidirect
9. C₈₃×#3110	C₂	semidirect
11. C₈₃×#2984	C₂	semidirect
13. #557×C₈₃	C₂²	semidirect
14. #557×C₈₃	C₂²	semidirect
15. #556×C₈₃	C₂²	semidirect
22. D₄×Q₈×C₈₃	C₂²	semidirect
31. #501×C₈₃	C₂²	semidirect
32. #500×C₈₃	C₂²	semidirect
33. #489×C₈₃	C₂²	semidirect
34. #487×C₈₃	C₂²	semidirect
39. #477×C₈₃	C₂²	semidirect
40. #475×C₈₃	C₂²	semidirect
42. #474×C₈₃	C₂²	semidirect
47. #446×C₈₃	C₂²	semidirect
48. C₈×Q₈×C₈₃	C₂²	semidirect
240. #61837	C₈₃	direct (N × H)
247. #3523	C₂×C₈₃	semidirect
248. C₂×D₄×Q₈	C₂×C₈₃	semidirect
249. #3386	C₂×C₈₃	semidirect
250. Q₈×D₈	C₂×C₈₃	semidirect
251. #3367	C₂×C₈₃	semidirect
252. #3359	C₂×C₈₃	semidirect
253. #3239	C₂×C₈₃	semidirect
254. #3230	C₂×C₈₃	semidirect
255. #3110	C₂×C₈₃	semidirect
257. #2984	C₂×C₈₃	semidirect
258. C₈₃	#61837	direct (N × H)
260. #557	C₂²×C₈₃	semidirect
261. #557	C₂²×C₈₃	semidirect
262. #556	C₂²×C₈₃	semidirect
269. D₄×Q₈	C₂²×C₈₃	semidirect
278. #501	C₂²×C₈₃	semidirect
279. #500	C₂²×C₈₃	semidirect
280. #489	C₂²×C₈₃	semidirect
281. #487	C₂²×C₈₃	semidirect
286. #477	C₂²×C₈₃	semidirect
287. #475	C₂²×C₈₃	semidirect
289. #474	C₂²×C₈₃	semidirect
294. #446	C₂²×C₈₃	semidirect
295. C₈×Q₈	C₂²×C₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₈₃×#3523	2
C₂×D₄×Q₈×C₈₃	2
C₈₃×#3386	2
Q₈×D₈×C₈₃	2
C₈₃×#3367	2
C₈₃×#3359	2
C₈₃×#3239	2
C₈₃×#3230	2
C₈₃×#3110	2
C₈₃×#3105	2
C₈₃×#2984	2
#61837	83

Subgroups — C83×#61837 #35289661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#61837 #35289661