Subgroups — C₈₃×#61882 #35289706

Order 21248.

1750 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₃×#3507	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#556×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₃×#3394	C₂	1	4	yes	maximal, complemented
4	C₈₃×#3393	C₂	1	4	yes	maximal, complemented
5	C₈₃×#3364	C₂	1	4	yes	maximal, complemented
6	C₈₃×#3360	C₂	1	4	yes	maximal, complemented
7	C₈₃×#3211	C₂	1	4	yes	maximal, complemented
8	C₈₃×#3209	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₃×#3206	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₃×#3108	C₂	1	4	yes	maximal, complemented
11	C₈₃×#3101	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈₃×#3100	C₂	1	1	yes	maximal, characteristic
13	C₈₃×#2982	C₂	1	1	yes	maximal, characteristic, complemented
14	#559×C₈₃	C₂²	1	2	yes	complemented
15	#557×C₈₃	C₂²	1	2	yes	complemented
16	#556×C₈₃	—	2	2	no	—
17	#556×C₈₃	—	2	2	no	—
18	#556×C₈₃	—	2	2	no	—
19	#556×C₈₃	C₂²	1	4	yes	—
20	#556×C₈₃	C₂²	1	4	yes	—
21	#556×C₈₃	C₂²	1	2	yes	complemented
22	#554×C₈₃	C₂²	1	2	yes	complemented
23	#552×C₈₃	—	2	2	no	—
24	#539×C₈₃	—	2	2	no	—
25	#536×C₈₃	—	2	1	no	—
26	#533×C₈₃	C₂²	1	1	yes	characteristic
27	#532×C₈₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
29	#529×C₈₃	C₂²	1	4	yes	—
30	C₂×#126×C₈₃	—	2	2	no	—
31	C₂×#124×C₈₃	C₂²	1	2	yes	—
32	C₂×C₂²⋊Q₈×C₈₃	—	2	1	no	—
33	#520×C₈₃	—	2	1	no	—
34	C₂×C₄×Q₈×C₈₃	C₂²	1	2	yes	—
35	C₂×C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
37	C₂×#117×C₈₃	—	2	1	no	—
38	#495×C₈₃	C₂²	1	8	yes	—
39	#494×C₈₃	C₂²	1	2	yes	complemented
40	#492×C₈₃	C₂²	1	2	yes	complemented
41	#488×C₈₃	C₂²	1	8	yes	—
42	#487×C₈₃	C₂²	1	2	yes	complemented
43	#487×C₈₃	C₂²	1	2	yes	complemented
44	#483×C₈₃	C₂²	1	2	yes	—
45	#480×C₈₃	C₂²	1	2	yes	—
46	#479×C₈₃	C₂²	1	8	yes	complemented
47	#476×C₈₃	C₂²	1	2	yes	—
48	#474×C₈₃	C₂²	1	2	yes	—
49	#464×C₈₃	C₂²	1	8	yes	—
50	#464×C₈₃	C₂²	1	8	yes	complemented
51	#463×C₈₃	C₂²	1	8	yes	—
52	#462×C₈₃	C₂²	1	4	yes	complemented
53	#461×C₈₃	C₂²	1	2	yes	—
54	#461×C₈₃	C₂²	1	4	yes	complemented
55	#459×C₈₃	C₂²	1	2	yes	—
56	#451×C₈₃	C₂²	1	4	yes	—
57	#450×C₈₃	C₂²	1	2	yes	—
58	#448×C₈₃	C₂²	1	4	yes	—
59	#447×C₈₃	C₂²	1	2	yes	—
60	C₈×Q₈×C₈₃	C₂²	1	4	yes	complemented
61	C₈×Q₈×C₈₃	C₂²	1	4	yes	complemented
62	C₄×Q₁₆×C₈₃	C₂²	1	8	yes	complemented
63	C₄×SD₁₆×C₈₃	C₂²	1	4	yes	—
64	C₄×SD₁₆×C₈₃	C₂²	1	8	yes	complemented
65	C₄×D₈×C₈₃	C₂²	1	4	yes	—
66	#431×C₈₃	C₂²	1	1	yes	characteristic
67	#431×C₈₃	C₂²	1	1	yes	characteristic
68	#431×C₈₃	C₂²	1	4	yes	—
69	C₂×#41×C₈₃	—	2	1	no	—
70	C₂²×Q₈×C₈₃	D₄	1	1	yes	characteristic
71	C₂²×Q₈×C₈₃	C₂³	1	2	yes	—
72	C₂²×D₄×C₈₃	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₈₃	—	2	1	no	—
74	C₂×Q₁₆×C₈₃	C₂³	1	8	yes	—
75	C₂×SD₁₆×C₈₃	C₂³	1	8	yes	—
76	C₂×SD₁₆×C₈₃	C₂³	1	4	yes	—
77	C₂×D₈×C₈₃	C₂³	1	4	yes	—
78	C₄⋊Q₈×C₈₃	—	2	2	no	—
79	C₄⋊Q₈×C₈₃	—	2	1	no	—
80	C₄⋊Q₈×C₈₃	—	2	1	no	—
81	C₄⋊Q₈×C₈₃	C₂³	1	2	yes	—
82	C₄⋊Q₈×C₈₃	C₂³	1	2	yes	—
83	C₄⋊Q₈×C₈₃	C₂³	1	1	yes	characteristic
84	C₄⋊Q₈×C₈₃	C₂³	1	1	yes	characteristic
85	C₄⋊Q₈×C₈₃	C₂³	1	1	yes	characteristic
86	#127×C₈₃	C₂³	1	1	yes	characteristic
87	#126×C₈₃	—	2	8	no	—
88	#126×C₈₃	—	2	8	no	—
89	#126×C₈₃	—	2	4	no	—
90	#126×C₈₃	—	2	4	no	—
91	#125×C₈₃	—	2	4	no	—
92	#125×C₈₃	—	2	1	no	—
93	#125×C₈₃	C₂³	1	4	yes	—
94	#124×C₈₃	—	2	2	no	—
95	#124×C₈₃	—	2	4	no	—
96	#124×C₈₃	—	2	1	no	—
97	#124×C₈₃	C₂³	1	8	yes	—
98	#124×C₈₃	C₂³	1	4	yes	—
99	#123×C₈₃	—	2	4	no	—
100	#123×C₈₃	—	2	4	no	—
101	C₂²⋊Q₈×C₈₃	—	2	4	no	—
102	C₂²⋊Q₈×C₈₃	—	2	4	no	—
103	C₂²⋊Q₈×C₈₃	—	2	2	no	—
104	C₂²⋊Q₈×C₈₃	—	2	2	no	—
105	C₂²⋊Q₈×C₈₃	—	2	1	no	—
106	C₂²⋊Q₈×C₈₃	—	2	4	no	—
107	C₂²⋊Q₈×C₈₃	C₂³	1	2	yes	—
108	#121×C₈₃	—	2	1	no	—
109	#121×C₈₃	C₂³	1	2	yes	—
110	C₄×Q₈×C₈₃	—	2	2	no	—
111	C₄×Q₈×C₈₃	—	2	4	no	—
112	C₄×Q₈×C₈₃	—	2	1	no	—
113	C₄×Q₈×C₈₃	D₄	1	2	yes	—
114	C₄×Q₈×C₈₃	D₄	1	2	yes	—
115	C₄×Q₈×C₈₃	C₂³	1	4	yes	—
116	C₄×Q₈×C₈₃	C₂³	1	8	yes	—
117	C₄×Q₈×C₈₃	C₂³	1	2	yes	—
118	C₄×Q₈×C₈₃	C₂³	1	2	yes	—
119	C₄×Q₈×C₈₃	C₂³	1	2	yes	—
120	C₄×D₄×C₈₃	—	2	4	no	—
121	C₄×D₄×C₈₃	—	2	2	no	—
122	C₄×D₄×C₈₃	—	2	1	no	—
123	C₄×D₄×C₈₃	—	2	1	no	—
124	C₄×D₄×C₈₃	—	2	1	no	—
125	C₄×D₄×C₈₃	C₂³	1	4	yes	—
126	C₄×D₄×C₈₃	C₂³	1	2	yes	—
127	C₄×D₄×C₈₃	C₂³	1	2	yes	—
128	C₄×D₄×C₈₃	C₂³	1	2	yes	—
129	#117×C₈₃	—	2	1	no	—
130	#117×C₈₃	—	2	4	no	—
131	#117×C₈₃	—	2	4	no	—
132	#117×C₈₃	—	2	2	no	—
133	#117×C₈₃	—	2	4	no	—
134	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
135	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
136	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
137	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
138	C₂×C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
139	C₂×C₄⋊C₄×C₈₃	D₄	1	2	yes	—
140	C₂×C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
141	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
142	C₂×C₂²⋊C₄×C₈₃	—	2	1	no	—
143	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
144	C₂×C₄²×C₈₃	—	2	2	no	—
145	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
146	C₂×C₄²×C₈₃	C₂³	1	2	yes	—
147	#107×C₈₃	C₂³	1	4	yes	—
148	#106×C₈₃	C₂³	1	4	yes	—
149	C₄⋊C₈×C₈₃	C₂³	1	8	yes	—
150	C₄⋊C₈×C₈₃	C₂³	1	2	yes	—
151	C₄⋊C₈×C₈₃	C₂³	1	2	yes	—
152	#103×C₈₃	C₂³	1	8	yes	—
153	#103×C₈₃	C₂³	1	8	yes	—
154	#103×C₈₃	C₂³	1	4	yes	—
155	#102×C₈₃	C₂³	1	4	yes	—
156	#102×C₈₃	C₂³	1	4	yes	—
157	#102×C₈₃	C₂³	1	4	yes	—
158	C₂²⋊C₈×C₈₃	C₂³	1	2	yes	—
159	C₂²⋊C₈×C₈₃	C₂³	1	2	yes	—
160	C₄×C₈×C₈₃	C₂³	1	2	yes	—
161	C₄×C₈×C₈₃	C₂³	1	2	yes	—
162	C₄×C₈×C₈₃	C₂³	1	8	yes	—
163	C₂⁴×C₈₃	—	2	1	no	—
164	#41×C₈₃	—	4	4	no	—
165	C₂×Q₈×C₈₃	—	2	4	no	—
166	C₂×Q₈×C₈₃	—	2	2	no	—
167	C₂×Q₈×C₈₃	—	2	1	no	—
168	C₂×Q₈×C₈₃	—	2	1	no	—
169	C₂×Q₈×C₈₃	—	4	4	no	—
170	C₂×Q₈×C₈₃	C₂×D₄	1	2	yes	—
171	C₂×Q₈×C₈₃	C₂×D₄	1	2	yes	—
172	C₂×Q₈×C₈₃	C₂⁴	1	4	yes	—
173	C₂×Q₈×C₈₃	C₂⁴	1	2	yes	—
174	C₂×D₄×C₈₃	—	2	2	no	—
175	C₂×D₄×C₈₃	—	2	1	no	—
176	C₂×D₄×C₈₃	—	2	1	no	—
177	C₂×D₄×C₈₃	—	2	1	no	—
178	C₂×D₄×C₈₃	—	4	2	no	—
179	C₂×D₄×C₈₃	C₂⁴	1	2	yes	—
180	C₂×D₄×C₈₃	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₈₃	—	2	1	no	—
182	C₂²×C₄×C₈₃	—	2	2	no	—
183	C₂²×C₄×C₈₃	—	2	2	no	—
184	C₂²×C₄×C₈₃	—	2	2	no	—
185	C₂²×C₄×C₈₃	—	2	2	no	—
186	C₂²×C₄×C₈₃	—	2	4	no	—
187	C₂²×C₄×C₈₃	—	4	4	no	—
188	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
189	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
190	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
193	Q₁₆×C₈₃	—	4	8	no	—
194	SD₁₆×C₈₃	—	4	8	no	—
195	SD₁₆×C₈₃	—	4	4	no	—
196	D₈×C₈₃	—	4	4	no	—
197	C₂×C₈×C₈₃	C₂⁴	1	4	yes	—
198	C₂×C₈×C₈₃	C₂⁴	1	4	yes	—
199	C₄⋊C₄×C₈₃	—	2	1	no	—
200	C₄⋊C₄×C₈₃	—	2	2	no	—
201	C₄⋊C₄×C₈₃	—	2	2	no	—
202	C₄⋊C₄×C₈₃	—	2	4	no	—
203	C₄⋊C₄×C₈₃	—	2	8	no	—
204	C₄⋊C₄×C₈₃	—	2	2	no	—
205	C₄⋊C₄×C₈₃	—	2	2	no	—
206	C₄⋊C₄×C₈₃	—	2	1	no	—
207	C₄⋊C₄×C₈₃	—	2	1	no	—
208	C₄⋊C₄×C₈₃	—	2	2	no	—
209	C₄⋊C₄×C₈₃	—	2	4	no	—
210	C₄⋊C₄×C₈₃	—	2	1	no	—
211	C₄⋊C₄×C₈₃	—	2	1	no	—
212	C₄⋊C₄×C₈₃	—	2	2	no	—
213	C₄⋊C₄×C₈₃	—	2	4	no	—
214	C₄⋊C₄×C₈₃	C₂×D₄	1	2	yes	—
215	C₄⋊C₄×C₈₃	C₂×D₄	1	2	yes	—
216	C₄⋊C₄×C₈₃	C₂×D₄	1	4	yes	—
217	C₄⋊C₄×C₈₃	C₂×D₄	1	4	yes	—
218	C₄⋊C₄×C₈₃	C₂⁴	1	2	yes	—
219	C₄⋊C₄×C₈₃	C₂⁴	1	2	yes	—
220	C₄⋊C₄×C₈₃	C₂⁴	1	2	yes	—
221	C₂²⋊C₄×C₈₃	—	2	2	no	—
222	C₂²⋊C₄×C₈₃	—	2	8	no	—
223	C₂²⋊C₄×C₈₃	—	2	2	no	—
224	C₂²⋊C₄×C₈₃	—	2	2	no	—
225	C₂²⋊C₄×C₈₃	—	2	8	no	—
226	C₂²⋊C₄×C₈₃	—	2	2	no	—
227	C₂²⋊C₄×C₈₃	—	2	4	no	—
228	C₂²⋊C₄×C₈₃	—	2	4	no	—
229	C₂²⋊C₄×C₈₃	—	2	2	no	—
230	C₂²⋊C₄×C₈₃	—	2	4	no	—
231	C₄²×C₈₃	—	2	8	no	—
232	C₄²×C₈₃	—	2	4	no	—
233	C₄²×C₈₃	—	2	2	no	—
234	C₄²×C₈₃	C₂×D₄	1	4	yes	—
235	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
236	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
237	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
238	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
239	C₄²×C₈₃	C₂⁴	1	4	yes	—
240	C₂³×C₈₃	—	2	2	no	—
241	C₂³×C₈₃	—	2	2	no	—
242	C₂³×C₈₃	—	4	2	no	—
243	C₂³×C₈₃	—	4	2	no	—
244	C₂³×C₈₃	—	4	2	no	—
245	C₂³×C₈₃	C₂²×D₄	1	1	yes	characteristic
246	Q₈×C₈₃	—	2	4	no	—
247	Q₈×C₈₃	—	2	4	no	—
248	Q₈×C₈₃	—	2	4	no	—
249	Q₈×C₈₃	—	2	8	no	—
250	Q₈×C₈₃	—	4	4	no	—
251	Q₈×C₈₃	—	4	2	no	—
252	Q₈×C₈₃	—	8	1	no	—
253	D₄×C₈₃	—	2	2	no	—
254	D₄×C₈₃	—	2	4	no	—
255	D₄×C₈₃	—	2	2	no	—
256	D₄×C₈₃	—	4	2	no	—
257	D₄×C₈₃	—	4	4	no	—
258	D₄×C₈₃	—	4	2	no	—
259	D₄×C₈₃	—	8	1	no	—
260	C₂×C₄×C₈₃	—	2	4	no	—
261	C₂×C₄×C₈₃	—	2	2	no	—
262	C₂×C₄×C₈₃	—	2	2	no	—
263	C₂×C₄×C₈₃	—	2	2	no	—
264	C₂×C₄×C₈₃	—	2	4	no	—
265	C₂×C₄×C₈₃	—	2	2	no	—
266	C₂×C₄×C₈₃	—	2	2	no	—
267	C₂×C₄×C₈₃	—	4	2	no	—
268	C₂×C₄×C₈₃	—	4	1	no	—
269	C₂×C₄×C₈₃	—	4	4	no	—
270	C₂×C₄×C₈₃	—	4	2	no	—
271	C₂×C₄×C₈₃	—	4	4	no	—
272	C₂×C₄×C₈₃	—	4	4	no	—
273	C₂×C₄×C₈₃	—	4	4	no	—
274	C₂×C₄×C₈₃	—	8	4	no	—
275	C₂×C₄×C₈₃	#135	1	4	yes	—
276	C₂×C₄×C₈₃	C₂²×D₄	1	4	yes	—
277	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
278	C₂×C₄×C₈₃	C₂²×D₄	1	4	yes	—
279	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
280	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
281	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₈₃	C₂⁵	1	1	yes	characteristic
284	C₈×C₈₃	—	2	4	no	—
285	C₈×C₈₃	—	2	4	no	—
286	C₂²×C₈₃	—	2	1	no	—
287	C₂²×C₈₃	—	2	1	no	—
288	C₂²×C₈₃	—	4	2	no	—
289	C₂²×C₈₃	—	4	2	no	—
290	C₂²×C₈₃	—	4	2	no	—
291	C₂²×C₈₃	—	4	2	no	—
292	C₂²×C₈₃	—	4	2	no	—
293	C₂²×C₈₃	—	4	2	no	—
294	C₂²×C₈₃	—	8	4	no	—
295	C₂²×C₈₃	C₂×#135	1	2	yes	—
296	C₂²×C₈₃	C₂³×D₄	1	1	yes	characteristic
297	C₄×C₈₃	—	2	4	no	—
298	C₄×C₈₃	—	2	1	no	—
299	C₄×C₈₃	—	2	1	no	—
300	C₄×C₈₃	—	2	1	no	—
301	C₄×C₈₃	—	2	4	no	—
302	C₄×C₈₃	—	4	2	no	—
303	C₄×C₈₃	—	4	4	no	—
304	C₄×C₈₃	—	4	2	no	—
305	C₄×C₈₃	—	4	2	no	—
306	C₄×C₈₃	—	4	2	no	—
307	C₄×C₈₃	C₂×#135	1	2	yes	—
308	C₄×C₈₃	C₂×#135	1	2	yes	—
309	C₄×C₈₃	#585	1	2	yes	—
310	#61882	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
311	C₂×C₈₃	—	2	2	no	—
312	C₂×C₈₃	—	8	2	no	—
313	C₂×C₈₃	—	8	2	no	—
314	C₂×C₈₃	#3487	1	1	yes	characteristic
315	C₂×C₈₃	C₂²×#135	1	1	yes	characteristic
316	C₂×C₈₃	#3586	1	1	yes	characteristic
317	#3507	C₂×C₈₃	1	1	yes	characteristic, complemented
318	C₂×#556	C₂×C₈₃	1	1	yes	characteristic, complemented
319	#3394	C₂×C₈₃	1	4	yes	complemented
320	#3393	C₂×C₈₃	1	4	yes	complemented
321	#3364	C₂×C₈₃	1	4	yes	complemented
322	#3360	C₂×C₈₃	1	4	yes	complemented
323	#3211	C₂×C₈₃	1	4	yes	complemented
324	#3209	C₂×C₈₃	1	1	yes	characteristic, complemented
325	#3206	C₂×C₈₃	1	1	yes	characteristic, complemented
326	#3108	C₂×C₈₃	1	4	yes	complemented
327	#3101	C₂×C₈₃	1	1	yes	characteristic, complemented
328	#3100	C₂×C₈₃	1	1	yes	characteristic
329	#2982	C₂×C₈₃	1	1	yes	characteristic, complemented
330	C₈₃	#61882	1	1	yes	characteristic, direct product factor, Sylow
331	#559	C₂²×C₈₃	1	2	yes	complemented
332	#557	C₂²×C₈₃	1	2	yes	complemented
333	#556	—	2	2	no	—
334	#556	—	2	2	no	—
335	#556	—	2	2	no	—
336	#556	C₂²×C₈₃	1	4	yes	—
337	#556	C₂²×C₈₃	1	4	yes	—
338	#556	C₂²×C₈₃	1	2	yes	complemented
339	#554	C₂²×C₈₃	1	2	yes	complemented
340	#552	—	2	2	no	—
341	#539	—	2	2	no	—
342	#536	—	2	1	no	—
343	#533	C₂²×C₈₃	1	1	yes	characteristic
344	#532	C₂²×C₈₃	1	1	yes	characteristic
345	C₂×C₄⋊Q₈	C₂²×C₈₃	1	1	yes	characteristic
346	#529	C₂²×C₈₃	1	4	yes	—
347	C₂×#126	—	2	2	no	—
348	C₂×#124	C₂²×C₈₃	1	2	yes	—
349	C₂×C₂²⋊Q₈	—	2	1	no	—
350	#520	—	2	1	no	—
351	C₂×C₄×Q₈	C₂²×C₈₃	1	2	yes	—
352	C₂×C₄×Q₈	C₂²×C₈₃	1	1	yes	characteristic
353	C₂×C₄×D₄	C₂²×C₈₃	1	1	yes	characteristic
354	C₂×#117	—	2	1	no	—
355	#495	C₂²×C₈₃	1	8	yes	—
356	#494	C₂²×C₈₃	1	2	yes	complemented
357	#492	C₂²×C₈₃	1	2	yes	complemented
358	#488	C₂²×C₈₃	1	8	yes	—
359	#487	C₂²×C₈₃	1	2	yes	complemented
360	#487	C₂²×C₈₃	1	2	yes	complemented
361	#483	C₂²×C₈₃	1	2	yes	—
362	#480	C₂²×C₈₃	1	2	yes	—
363	#479	C₂²×C₈₃	1	8	yes	complemented
364	#476	C₂²×C₈₃	1	2	yes	—
365	#474	C₂²×C₈₃	1	2	yes	—
366	#464	C₂²×C₈₃	1	8	yes	—
367	#464	C₂²×C₈₃	1	8	yes	complemented
368	#463	C₂²×C₈₃	1	8	yes	—
369	#462	C₂²×C₈₃	1	4	yes	complemented
370	#461	C₂²×C₈₃	1	2	yes	—
371	#461	C₂²×C₈₃	1	4	yes	complemented
372	#459	C₂²×C₈₃	1	2	yes	—
373	#451	C₂²×C₈₃	1	4	yes	—
374	#450	C₂²×C₈₃	1	2	yes	—
375	#448	C₂²×C₈₃	1	4	yes	—
376	#447	C₂²×C₈₃	1	2	yes	—
377	C₈×Q₈	C₂²×C₈₃	1	4	yes	complemented
378	C₈×Q₈	C₂²×C₈₃	1	4	yes	complemented
379	C₄×Q₁₆	C₂²×C₈₃	1	8	yes	complemented
380	C₄×SD₁₆	C₂²×C₈₃	1	4	yes	—
381	C₄×SD₁₆	C₂²×C₈₃	1	8	yes	complemented
382	C₄×D₈	C₂²×C₈₃	1	4	yes	—
383	#431	C₂²×C₈₃	1	1	yes	characteristic
384	#431	C₂²×C₈₃	1	1	yes	characteristic
385	#431	C₂²×C₈₃	1	4	yes	—
386	C₂×#41	—	2	1	no	—
387	C₂²×Q₈	D₄×C₈₃	1	1	yes	characteristic
388	C₂²×Q₈	C₂³×C₈₃	1	2	yes	—
389	C₂²×D₄	C₂³×C₈₃	1	1	yes	characteristic
390	C₂³×C₄	—	2	1	no	—
391	C₂×Q₁₆	C₂³×C₈₃	1	8	yes	—
392	C₂×SD₁₆	C₂³×C₈₃	1	4	yes	—
393	C₂×SD₁₆	C₂³×C₈₃	1	8	yes	—
394	C₂×D₈	C₂³×C₈₃	1	4	yes	—
395	C₄⋊Q₈	—	2	2	no	—
396	C₄⋊Q₈	—	2	1	no	—
397	C₄⋊Q₈	—	2	1	no	—
398	C₄⋊Q₈	C₂³×C₈₃	1	2	yes	—
399	C₄⋊Q₈	C₂³×C₈₃	1	2	yes	—
400	C₄⋊Q₈	C₂³×C₈₃	1	1	yes	characteristic
401	C₄⋊Q₈	C₂³×C₈₃	1	1	yes	characteristic
402	C₄⋊Q₈	C₂³×C₈₃	1	1	yes	characteristic
403	#127	C₂³×C₈₃	1	1	yes	characteristic
404	#126	—	2	8	no	—
405	#126	—	2	8	no	—
406	#126	—	2	4	no	—
407	#126	—	2	4	no	—
408	#125	—	2	4	no	—
409	#125	—	2	1	no	—
410	#125	C₂³×C₈₃	1	4	yes	—
411	#124	—	2	2	no	—
412	#124	—	2	4	no	—
413	#124	—	2	1	no	—
414	#124	C₂³×C₈₃	1	8	yes	—
415	#124	C₂³×C₈₃	1	4	yes	—
416	#123	—	2	4	no	—
417	#123	—	2	4	no	—
418	C₂²⋊Q₈	—	2	4	no	—
419	C₂²⋊Q₈	—	2	4	no	—
420	C₂²⋊Q₈	—	2	2	no	—
421	C₂²⋊Q₈	—	2	2	no	—
422	C₂²⋊Q₈	—	2	1	no	—
423	C₂²⋊Q₈	—	2	4	no	—
424	C₂²⋊Q₈	C₂³×C₈₃	1	2	yes	—
425	#121	—	2	1	no	—
426	#121	C₂³×C₈₃	1	2	yes	—
427	C₄×Q₈	—	2	2	no	—
428	C₄×Q₈	—	2	4	no	—
429	C₄×Q₈	—	2	1	no	—
430	C₄×Q₈	D₄×C₈₃	1	2	yes	—
431	C₄×Q₈	D₄×C₈₃	1	2	yes	—
432	C₄×Q₈	C₂³×C₈₃	1	8	yes	—
433	C₄×Q₈	C₂³×C₈₃	1	4	yes	—
434	C₄×Q₈	C₂³×C₈₃	1	2	yes	—
435	C₄×Q₈	C₂³×C₈₃	1	2	yes	—
436	C₄×Q₈	C₂³×C₈₃	1	2	yes	—
437	C₄×D₄	—	2	4	no	—
438	C₄×D₄	—	2	2	no	—
439	C₄×D₄	—	2	1	no	—
440	C₄×D₄	—	2	1	no	—
441	C₄×D₄	—	2	1	no	—
442	C₄×D₄	C₂³×C₈₃	1	2	yes	—
443	C₄×D₄	C₂³×C₈₃	1	2	yes	—
444	C₄×D₄	C₂³×C₈₃	1	4	yes	—
445	C₄×D₄	C₂³×C₈₃	1	2	yes	—
446	#117	—	2	1	no	—
447	#117	—	2	4	no	—
448	#117	—	2	4	no	—
449	#117	—	2	2	no	—
450	#117	—	2	4	no	—
451	C₂×C₄⋊C₄	—	2	1	no	—
452	C₂×C₄⋊C₄	—	2	2	no	—
453	C₂×C₄⋊C₄	—	2	1	no	—
454	C₂×C₄⋊C₄	—	2	2	no	—
455	C₂×C₄⋊C₄	D₄×C₈₃	1	1	yes	characteristic
456	C₂×C₄⋊C₄	D₄×C₈₃	1	2	yes	—
457	C₂×C₄⋊C₄	C₂³×C₈₃	1	1	yes	characteristic
458	C₂×C₂²⋊C₄	—	2	2	no	—
459	C₂×C₂²⋊C₄	—	2	1	no	—
460	C₂×C₂²⋊C₄	—	2	2	no	—
461	C₂×C₄²	—	2	2	no	—
462	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
463	C₂×C₄²	C₂³×C₈₃	1	2	yes	—
464	#107	C₂³×C₈₃	1	4	yes	—
465	#106	C₂³×C₈₃	1	4	yes	—
466	C₄⋊C₈	C₂³×C₈₃	1	2	yes	—
467	C₄⋊C₈	C₂³×C₈₃	1	2	yes	—
468	C₄⋊C₈	C₂³×C₈₃	1	8	yes	—
469	#103	C₂³×C₈₃	1	4	yes	—
470	#103	C₂³×C₈₃	1	8	yes	—
471	#103	C₂³×C₈₃	1	8	yes	—
472	#102	C₂³×C₈₃	1	4	yes	—
473	#102	C₂³×C₈₃	1	4	yes	—
474	#102	C₂³×C₈₃	1	4	yes	—
475	C₂²⋊C₈	C₂³×C₈₃	1	2	yes	—
476	C₂²⋊C₈	C₂³×C₈₃	1	2	yes	—
477	C₄×C₈	C₂³×C₈₃	1	2	yes	—
478	C₄×C₈	C₂³×C₈₃	1	2	yes	—
479	C₄×C₈	C₂³×C₈₃	1	8	yes	—
480	C₂⁴	—	2	1	no	—
481	#41	—	4	4	no	—
482	C₂×Q₈	—	2	4	no	—
483	C₂×Q₈	—	2	2	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	—	4	4	no	—
487	C₂×Q₈	C₂×D₄×C₈₃	1	2	yes	—
488	C₂×Q₈	C₂×D₄×C₈₃	1	2	yes	—
489	C₂×Q₈	C₂⁴×C₈₃	1	4	yes	—
490	C₂×Q₈	C₂⁴×C₈₃	1	2	yes	—
491	C₂×D₄	—	2	2	no	—
492	C₂×D₄	—	2	1	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	—	2	1	no	—
495	C₂×D₄	—	4	2	no	—
496	C₂×D₄	C₂⁴×C₈₃	1	2	yes	—
497	C₂×D₄	C₂⁴×C₈₃	1	2	yes	—
498	C₂²×C₄	—	2	1	no	—
499	C₂²×C₄	—	2	2	no	—
500	C₂²×C₄	—	2	2	no	—
501	C₂²×C₄	—	2	2	no	—
502	C₂²×C₄	—	2	2	no	—
503	C₂²×C₄	—	2	4	no	—
504	C₂²×C₄	—	4	4	no	—
505	C₂²×C₄	C₂×D₄×C₈₃	1	2	yes	—
506	C₂²×C₄	C₂×D₄×C₈₃	1	2	yes	—
507	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
508	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
509	C₂²×C₄	C₂⁴×C₈₃	1	1	yes	characteristic
510	Q₁₆	—	4	8	no	—
511	SD₁₆	—	4	8	no	—
512	SD₁₆	—	4	4	no	—
513	D₈	—	4	4	no	—
514	C₂×C₈	C₂⁴×C₈₃	1	4	yes	—
515	C₂×C₈	C₂⁴×C₈₃	1	4	yes	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	2	no	—
518	C₄⋊C₄	—	2	2	no	—
519	C₄⋊C₄	—	2	4	no	—
520	C₄⋊C₄	—	2	8	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	—	2	2	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	2	no	—
526	C₄⋊C₄	—	2	4	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	1	no	—
529	C₄⋊C₄	—	2	2	no	—
530	C₄⋊C₄	—	2	4	no	—
531	C₄⋊C₄	C₂×D₄×C₈₃	1	2	yes	—
532	C₄⋊C₄	C₂×D₄×C₈₃	1	2	yes	—
533	C₄⋊C₄	C₂×D₄×C₈₃	1	4	yes	—
534	C₄⋊C₄	C₂×D₄×C₈₃	1	4	yes	—
535	C₄⋊C₄	C₂⁴×C₈₃	1	2	yes	—
536	C₄⋊C₄	C₂⁴×C₈₃	1	2	yes	—
537	C₄⋊C₄	C₂⁴×C₈₃	1	2	yes	—
538	C₂²⋊C₄	—	2	2	no	—
539	C₂²⋊C₄	—	2	8	no	—
540	C₂²⋊C₄	—	2	2	no	—
541	C₂²⋊C₄	—	2	2	no	—
542	C₂²⋊C₄	—	2	8	no	—
543	C₂²⋊C₄	—	2	2	no	—
544	C₂²⋊C₄	—	2	4	no	—
545	C₂²⋊C₄	—	2	4	no	—
546	C₂²⋊C₄	—	2	2	no	—
547	C₂²⋊C₄	—	2	4	no	—
548	C₄²	—	2	8	no	—
549	C₄²	—	2	4	no	—
550	C₄²	—	2	2	no	—
551	C₄²	C₂×D₄×C₈₃	1	4	yes	—
552	C₄²	C₂×D₄×C₈₃	1	1	yes	characteristic
553	C₄²	C₂×D₄×C₈₃	1	1	yes	characteristic
554	C₄²	C₂⁴×C₈₃	1	1	yes	characteristic
555	C₄²	C₂⁴×C₈₃	1	1	yes	characteristic
556	C₄²	C₂⁴×C₈₃	1	4	yes	—
557	C₂³	—	2	2	no	—
558	C₂³	—	2	2	no	—
559	C₂³	—	4	2	no	—
560	C₂³	—	4	2	no	—
561	C₂³	—	4	2	no	—
562	C₂³	C₂²×D₄×C₈₃	1	1	yes	characteristic
563	Q₈	—	2	4	no	—
564	Q₈	—	2	4	no	—
565	Q₈	—	2	4	no	—
566	Q₈	—	2	8	no	—
567	Q₈	—	4	4	no	—
568	Q₈	—	4	2	no	—
569	Q₈	—	8	1	no	—
570	D₄	—	2	2	no	—
571	D₄	—	2	4	no	—
572	D₄	—	2	2	no	—
573	D₄	—	4	2	no	—
574	D₄	—	4	4	no	—
575	D₄	—	4	2	no	—
576	D₄	—	8	1	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	4	2	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	4	no	—
587	C₂×C₄	—	4	2	no	—
588	C₂×C₄	—	4	4	no	—
589	C₂×C₄	—	4	4	no	—
590	C₂×C₄	—	4	4	no	—
591	C₂×C₄	—	8	4	no	—
592	C₂×C₄	#135×C₈₃	1	4	yes	—
593	C₂×C₄	C₂²×D₄×C₈₃	1	4	yes	—
594	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
595	C₂×C₄	C₂²×D₄×C₈₃	1	4	yes	—
596	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
597	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
598	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
599	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
600	C₂×C₄	C₂⁵×C₈₃	1	1	yes	characteristic
601	C₈	—	2	4	no	—
602	C₈	—	2	4	no	—
603	C₂²	—	2	1	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	8	4	no	—
612	C₂²	C₂×#135×C₈₃	1	2	yes	—
613	C₂²	C₂³×D₄×C₈₃	1	1	yes	characteristic
614	C₄	—	2	4	no	—
615	C₄	—	2	4	no	—
616	C₄	—	2	1	no	—
617	C₄	—	2	1	no	—
618	C₄	—	2	1	no	—
619	C₄	—	4	2	no	—
620	C₄	—	4	2	no	—
621	C₄	—	4	2	no	—
622	C₄	—	4	4	no	—
623	C₄	—	4	2	no	—
624	C₄	C₂×#135×C₈₃	1	2	yes	—
625	C₄	C₂×#135×C₈₃	1	2	yes	—
626	C₄	#585×C₈₃	1	2	yes	—
627	C₂	—	2	2	no	—
628	C₂	—	8	2	no	—
629	C₂	—	8	2	no	—
630	C₂	C₈₃×#3487	1	1	yes	characteristic
631	C₂	C₂²×#135×C₈₃	1	1	yes	characteristic
632	C₂	C₈₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₃
prime radical	C₈₃×#3100	C₂
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×#3507	C₂	semidirect
2. C₂×#556×C₈₃	C₂	semidirect
3. C₈₃×#3394	C₂	semidirect
4. C₈₃×#3393	C₂	semidirect
5. C₈₃×#3364	C₂	semidirect
6. C₈₃×#3360	C₂	semidirect
7. C₈₃×#3211	C₂	semidirect
8. C₈₃×#3209	C₂	semidirect
9. C₈₃×#3206	C₂	semidirect
10. C₈₃×#3108	C₂	semidirect
11. C₈₃×#3101	C₂	semidirect
13. C₈₃×#2982	C₂	semidirect
14. #559×C₈₃	C₂²	semidirect
15. #557×C₈₃	C₂²	semidirect
21. #556×C₈₃	C₂²	semidirect
22. #554×C₈₃	C₂²	semidirect
39. #494×C₈₃	C₂²	semidirect
40. #492×C₈₃	C₂²	semidirect
42. #487×C₈₃	C₂²	semidirect
43. #487×C₈₃	C₂²	semidirect
46. #479×C₈₃	C₂²	semidirect
50. #464×C₈₃	C₂²	semidirect
52. #462×C₈₃	C₂²	semidirect
54. #461×C₈₃	C₂²	semidirect
60. C₈×Q₈×C₈₃	C₂²	semidirect
61. C₈×Q₈×C₈₃	C₂²	semidirect
62. C₄×Q₁₆×C₈₃	C₂²	semidirect
64. C₄×SD₁₆×C₈₃	C₂²	semidirect
310. #61882	C₈₃	direct (N × H)
317. #3507	C₂×C₈₃	semidirect
318. C₂×#556	C₂×C₈₃	semidirect
319. #3394	C₂×C₈₃	semidirect
320. #3393	C₂×C₈₃	semidirect
321. #3364	C₂×C₈₃	semidirect
322. #3360	C₂×C₈₃	semidirect
323. #3211	C₂×C₈₃	semidirect
324. #3209	C₂×C₈₃	semidirect
325. #3206	C₂×C₈₃	semidirect
326. #3108	C₂×C₈₃	semidirect
327. #3101	C₂×C₈₃	semidirect
329. #2982	C₂×C₈₃	semidirect
330. C₈₃	#61882	direct (N × H)
331. #559	C₂²×C₈₃	semidirect
332. #557	C₂²×C₈₃	semidirect
338. #556	C₂²×C₈₃	semidirect
339. #554	C₂²×C₈₃	semidirect
356. #494	C₂²×C₈₃	semidirect
357. #492	C₂²×C₈₃	semidirect
359. #487	C₂²×C₈₃	semidirect
360. #487	C₂²×C₈₃	semidirect
363. #479	C₂²×C₈₃	semidirect
367. #464	C₂²×C₈₃	semidirect
369. #462	C₂²×C₈₃	semidirect
371. #461	C₂²×C₈₃	semidirect
377. C₈×Q₈	C₂²×C₈₃	semidirect
378. C₈×Q₈	C₂²×C₈₃	semidirect
379. C₄×Q₁₆	C₂²×C₈₃	semidirect
381. C₄×SD₁₆	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×#3507	2
C₂×#556×C₈₃	2
C₈₃×#3394	2
C₈₃×#3393	2
C₈₃×#3364	2
C₈₃×#3360	2
C₈₃×#3211	2
C₈₃×#3209	2
C₈₃×#3206	2
C₈₃×#3108	2
C₈₃×#3101	2
C₈₃×#3100	2
C₈₃×#2982	2
#61882	83

Subgroups — C83×#61882 #35289706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#61882 #35289706