Subgroups — C₈₃×#17548 #35245372

Order 21248.

1102 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₃×#3520	C₂	1	1	yes	maximal, characteristic
2	C₈₃×#2974	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₃×#2275	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₃×#1509	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₃×#1508	C₂	1	2	yes	maximal, complemented
6	C₈₃×#1506	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1504	C₂	1	2	yes	maximal, complemented
8	C₈₃×#1503	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₃×#1501	C₂	1	2	yes	maximal, complemented
10	C₈₃×#1500	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#328×C₈₃	C₂	1	2	yes	maximal, complemented
12	#556×C₈₃	C₄	1	1	yes	characteristic, complemented
13	#554×C₈₃	—	2	1	no	—
14	#552×C₈₃	C₄	1	2	yes	complemented
15	#551×C₈₃	—	2	2	no	—
16	D₄×Q₈×C₈₃	C₄	1	1	yes	characteristic, complemented
17	#548×C₈₃	C₄	1	2	yes	complemented
18	#547×C₈₃	C₄	1	1	yes	characteristic, complemented
19	#547×C₈₃	C₄	1	1	yes	characteristic, complemented
20	#546×C₈₃	—	2	1	no	—
21	#542×C₈₃	—	2	1	no	—
22	#539×C₈₃	—	2	2	no	—
23	#533×C₈₃	C₂²	1	1	yes	characteristic
24	#529×C₈₃	C₂²	1	1	yes	characteristic
25	#525×C₈₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊Q₈×C₈₃	C₂²	1	2	yes	—
27	#520×C₈₃	—	2	1	no	—
28	C₂×C₄×D₄×C₈₃	—	2	1	no	—
29	C₂×C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
30	#436×C₈₃	—	2	1	no	—
31	#435×C₈₃	—	2	2	no	—
32	C₈×D₄×C₈₃	—	2	1	no	—
33	#432×C₈₃	C₂²	1	1	yes	characteristic
34	#431×C₈₃	C₂²	1	1	yes	characteristic
35	#407×C₈₃	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₈×C₈₃	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₈₃	—	2	1	no	—
38	C₄×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₈₃	—	2	2	no	—
40	C₄×C₂²⋊C₄×C₈₃	C₂²	1	2	yes	—
41	#376×C₈₃	C₂²	1	1	yes	characteristic
42	C₂×#95×C₈₃	C₂²	1	2	yes	—
43	#328×C₈₃	C₂²	1	1	yes	characteristic
44	#328×C₈₃	C₂²	1	2	yes	—
45	#328×C₈₃	C₂²	1	2	yes	—
46	#328×C₈₃	C₂²	1	2	yes	—
47	#328×C₈₃	C₂²	1	1	yes	characteristic
48	#328×C₈₃	C₂²	1	2	yes	—
49	#328×C₈₃	C₂²	1	2	yes	—
50	#327×C₈₃	C₂²	1	1	yes	characteristic
51	#327×C₈₃	C₂²	1	1	yes	characteristic
52	#327×C₈₃	C₂²	1	2	yes	—
53	C₂×#41×C₈₃	—	2	1	no	—
54	C₂×#41×C₈₃	C₂×C₄	1	2	yes	complemented
55	C₂²×Q₈×C₈₃	C₂×C₄	1	2	yes	complemented
56	C₂²×D₄×C₈₃	D₄	1	1	yes	characteristic
57	C₂³×C₄×C₈₃	—	2	2	no	—
58	C₂³×C₄×C₈₃	C₂³	1	2	yes	—
59	C₂×M₁₆×C₈₃	—	2	2	no	—
60	C₂²×C₈×C₈₃	—	2	2	no	—
61	C₄⋊Q₈×C₈₃	—	2	1	no	—
62	C₄⋊Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic, complemented
63	C₄⋊Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic, complemented
64	#126×C₈₃	—	2	2	no	—
65	#126×C₈₃	—	2	2	no	—
66	#126×C₈₃	—	2	1	no	—
67	#125×C₈₃	—	2	2	no	—
68	#125×C₈₃	—	2	1	no	—
69	#125×C₈₃	C₂×C₄	1	1	yes	characteristic, complemented
70	#124×C₈₃	—	2	1	no	—
71	#124×C₈₃	—	2	2	no	—
72	#124×C₈₃	C₂×C₄	1	1	yes	characteristic, complemented
73	#123×C₈₃	—	2	2	no	—
74	#123×C₈₃	—	2	2	no	—
75	#123×C₈₃	—	2	1	no	—
76	#123×C₈₃	—	2	2	no	—
77	#123×C₈₃	—	2	2	no	—
78	C₂²⋊Q₈×C₈₃	—	2	2	no	—
79	C₂²⋊Q₈×C₈₃	—	2	2	no	—
80	C₂²⋊Q₈×C₈₃	—	2	1	no	—
81	C₂²⋊Q₈×C₈₃	—	2	2	no	—
82	C₂²⋊Q₈×C₈₃	—	2	1	no	—
83	C₂²⋊Q₈×C₈₃	—	2	2	no	—
84	C₂²⋊Q₈×C₈₃	C₂×C₄	1	2	yes	complemented
85	C₂²⋊Q₈×C₈₃	C₂×C₄	1	2	yes	complemented
86	C₂²⋊Q₈×C₈₃	C₂×C₄	1	2	yes	complemented
87	C₂²⋊Q₈×C₈₃	C₂³	1	1	yes	characteristic
88	C₂²⋊Q₈×C₈₃	C₂³	1	2	yes	—
89	#121×C₈₃	—	2	2	no	—
90	#121×C₈₃	—	2	1	no	—
91	#121×C₈₃	C₂×C₄	1	2	yes	complemented
92	#121×C₈₃	C₂³	1	1	yes	characteristic
93	#120×C₈₃	—	2	2	no	—
94	C₄×Q₈×C₈₃	—	2	1	no	—
95	C₄×Q₈×C₈₃	—	2	1	no	—
96	C₄×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic, complemented
97	C₄×Q₈×C₈₃	C₂×C₄	1	2	yes	complemented
98	C₄×D₄×C₈₃	—	2	8	no	—
99	C₄×D₄×C₈₃	—	2	1	no	—
100	C₄×D₄×C₈₃	—	2	2	no	—
101	C₄×D₄×C₈₃	—	2	2	no	—
102	C₄×D₄×C₈₃	—	2	1	no	—
103	C₄×D₄×C₈₃	—	2	2	no	—
104	C₄×D₄×C₈₃	C₂×C₄	1	1	yes	characteristic
105	C₄×D₄×C₈₃	C₂×C₄	1	1	yes	characteristic
106	C₄×D₄×C₈₃	C₂×C₄	1	2	yes	—
107	C₄×D₄×C₈₃	C₂×C₄	1	1	yes	characteristic, complemented
108	C₄×D₄×C₈₃	D₄	1	1	yes	characteristic
109	C₄×D₄×C₈₃	D₄	1	2	yes	—
110	C₄×D₄×C₈₃	D₄	1	1	yes	characteristic
111	#117×C₈₃	—	2	1	no	—
112	#117×C₈₃	—	2	2	no	—
113	#117×C₈₃	—	2	1	no	—
114	#117×C₈₃	C₂×C₄	1	2	yes	complemented
115	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
116	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
117	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
118	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	complemented
119	C₂×C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
120	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
121	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
122	C₂×C₂²⋊C₄×C₈₃	D₄	1	2	yes	—
123	C₂×C₄²×C₈₃	—	2	1	no	—
124	C₂×C₄²×C₈₃	—	2	1	no	—
125	C₂×C₄²×C₈₃	—	2	2	no	—
126	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
127	C₄⋊C₈×C₈₃	—	2	1	no	—
128	C₄⋊C₈×C₈₃	—	2	1	no	—
129	C₂²⋊C₈×C₈₃	—	2	2	no	—
130	C₂²⋊C₈×C₈₃	—	2	2	no	—
131	C₂²⋊C₈×C₈₃	C₂³	1	1	yes	characteristic
132	C₂²⋊C₈×C₈₃	C₂³	1	2	yes	—
133	C₂²⋊C₈×C₈₃	C₂³	1	1	yes	characteristic
134	#97×C₈₃	—	2	1	no	—
135	C₄×C₈×C₈₃	—	2	1	no	—
136	#95×C₈₃	—	2	2	no	—
137	#95×C₈₃	—	2	1	no	—
138	#95×C₈₃	C₂³	1	1	yes	characteristic
139	#95×C₈₃	C₂³	1	2	yes	—
140	#95×C₈₃	C₂³	1	1	yes	characteristic
141	C₂⁴×C₈₃	C₂×D₄	1	2	yes	—
142	#41×C₈₃	—	4	4	no	—
143	#41×C₈₃	—	8	2	no	—
144	C₂×Q₈×C₈₃	—	2	1	no	—
145	C₂×Q₈×C₈₃	—	2	2	no	—
146	C₂×Q₈×C₈₃	—	8	2	no	—
147	C₂×Q₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
148	C₂×Q₈×C₈₃	C₂²×C₄	1	2	yes	—
149	C₂×D₄×C₈₃	—	2	4	no	—
150	C₂×D₄×C₈₃	—	2	2	no	—
151	C₂×D₄×C₈₃	—	2	1	no	—
152	C₂×D₄×C₈₃	—	2	2	no	—
153	C₂×D₄×C₈₃	C₂²⋊C₄	1	2	yes	—
154	C₂×D₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
155	C₂×D₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
156	C₂×D₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₈₃	—	2	2	no	—
158	C₂²×C₄×C₈₃	—	2	1	no	—
159	C₂²×C₄×C₈₃	—	2	1	no	—
160	C₂²×C₄×C₈₃	—	2	2	no	—
161	C₂²×C₄×C₈₃	—	2	8	no	—
162	C₂²×C₄×C₈₃	—	2	2	no	—
163	C₂²×C₄×C₈₃	—	2	1	no	—
164	C₂²×C₄×C₈₃	—	2	2	no	—
165	C₂²×C₄×C₈₃	—	8	2	no	—
166	C₂²×C₄×C₈₃	—	8	2	no	—
167	C₂²×C₄×C₈₃	—	8	2	no	—
168	C₂²×C₄×C₈₃	C₂²×C₄	1	2	yes	—
169	C₂²×C₄×C₈₃	C₂²×C₄	1	2	yes	—
170	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
176	M₁₆×C₈₃	—	8	2	no	—
177	C₂×C₈×C₈₃	—	2	1	no	—
178	C₂×C₈×C₈₃	—	2	1	no	—
179	C₂×C₈×C₈₃	—	2	2	no	—
180	C₂×C₈×C₈₃	—	8	2	no	—
181	C₄⋊C₄×C₈₃	—	2	4	no	—
182	C₄⋊C₄×C₈₃	—	2	2	no	—
183	C₄⋊C₄×C₈₃	—	2	1	no	—
184	C₄⋊C₄×C₈₃	—	2	2	no	—
185	C₄⋊C₄×C₈₃	—	2	2	no	—
186	C₄⋊C₄×C₈₃	—	2	1	no	—
187	C₄⋊C₄×C₈₃	—	2	1	no	—
188	C₄⋊C₄×C₈₃	—	2	1	no	—
189	C₄⋊C₄×C₈₃	—	2	2	no	—
190	C₄⋊C₄×C₈₃	—	2	1	no	—
191	C₄⋊C₄×C₈₃	—	2	2	no	—
192	C₄⋊C₄×C₈₃	—	2	1	no	—
193	C₄⋊C₄×C₈₃	—	2	1	no	—
194	C₄⋊C₄×C₈₃	—	2	2	no	—
195	C₄⋊C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
196	C₄⋊C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
197	C₄⋊C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
198	C₄⋊C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
199	C₄⋊C₄×C₈₃	C₂²×C₄	1	2	yes	—
200	C₄⋊C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
201	C₂²⋊C₄×C₈₃	—	2	8	no	—
202	C₂²⋊C₄×C₈₃	—	2	2	no	—
203	C₂²⋊C₄×C₈₃	—	2	2	no	—
204	C₂²⋊C₄×C₈₃	—	2	2	no	—
205	C₂²⋊C₄×C₈₃	—	2	2	no	—
206	C₂²⋊C₄×C₈₃	—	2	1	no	—
207	C₂²⋊C₄×C₈₃	—	2	2	no	—
208	C₂²⋊C₄×C₈₃	—	2	1	no	—
209	C₂²⋊C₄×C₈₃	—	2	1	no	—
210	C₂²⋊C₄×C₈₃	—	2	2	no	—
211	C₂²⋊C₄×C₈₃	—	2	1	no	—
212	C₂²⋊C₄×C₈₃	—	2	1	no	—
213	C₂²⋊C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
214	C₂²⋊C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
215	C₂²⋊C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
216	C₂²⋊C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
217	C₄²×C₈₃	—	2	4	no	—
218	C₄²×C₈₃	—	2	2	no	—
219	C₄²×C₈₃	—	2	2	no	—
220	C₄²×C₈₃	—	2	1	no	—
221	C₄²×C₈₃	—	8	1	no	—
222	C₄²×C₈₃	—	8	2	no	—
223	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
224	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
225	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
226	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
227	C₂³×C₈₃	—	2	2	no	—
228	C₂³×C₈₃	—	2	2	no	—
229	C₂³×C₈₃	—	2	4	no	—
230	C₂³×C₈₃	—	2	1	no	—
231	C₂³×C₈₃	—	4	2	no	—
232	C₂³×C₈₃	C₂×C₂²⋊C₄	1	2	yes	—
233	C₂³×C₈₃	C₂×C₂²⋊C₄	1	2	yes	—
234	C₂³×C₈₃	C₂²×D₄	1	1	yes	characteristic
235	Q₈×C₈₃	—	4	1	no	—
236	Q₈×C₈₃	—	4	2	no	—
237	Q₈×C₈₃	—	4	2	no	—
238	Q₈×C₈₃	—	8	2	no	—
239	D₄×C₈₃	—	2	4	no	—
240	D₄×C₈₃	—	2	2	no	—
241	D₄×C₈₃	—	4	1	no	—
242	D₄×C₈₃	—	4	1	no	—
243	D₄×C₈₃	—	4	1	no	—
244	D₄×C₈₃	—	4	4	no	—
245	D₄×C₈₃	—	8	2	no	—
246	D₄×C₈₃	#104	1	4	yes	—
247	C₂×C₄×C₈₃	—	2	1	no	—
248	C₂×C₄×C₈₃	—	2	1	no	—
249	C₂×C₄×C₈₃	—	2	2	no	—
250	C₂×C₄×C₈₃	—	2	2	no	—
251	C₂×C₄×C₈₃	—	2	2	no	—
252	C₂×C₄×C₈₃	—	2	4	no	—
253	C₂×C₄×C₈₃	—	2	1	no	—
254	C₂×C₄×C₈₃	—	2	2	no	—
255	C₂×C₄×C₈₃	—	2	1	no	—
256	C₂×C₄×C₈₃	—	2	2	no	—
257	C₂×C₄×C₈₃	—	2	1	no	—
258	C₂×C₄×C₈₃	—	4	2	no	—
259	C₂×C₄×C₈₃	—	4	2	no	—
260	C₂×C₄×C₈₃	—	4	2	no	—
261	C₂×C₄×C₈₃	—	4	1	no	—
262	C₂×C₄×C₈₃	—	4	2	no	—
263	C₂×C₄×C₈₃	—	4	2	no	—
264	C₂×C₄×C₈₃	—	4	1	no	—
265	C₂×C₄×C₈₃	—	4	2	no	—
266	C₂×C₄×C₈₃	—	4	1	no	—
267	C₂×C₄×C₈₃	—	4	2	no	—
268	C₂×C₄×C₈₃	—	4	1	no	—
269	C₂×C₄×C₈₃	—	4	1	no	—
270	C₂×C₄×C₈₃	—	4	4	no	—
271	C₂×C₄×C₈₃	—	4	2	no	—
272	C₂×C₄×C₈₃	—	8	2	no	—
273	C₂×C₄×C₈₃	—	8	2	no	—
274	C₂×C₄×C₈₃	—	8	8	no	—
275	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	2	yes	—
277	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
278	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	2	yes	—
280	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
281	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
282	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
283	C₈×C₈₃	—	4	1	no	—
284	C₈×C₈₃	—	4	2	no	—
285	C₈×C₈₃	—	4	1	no	—
286	C₂²×C₈₃	—	2	1	no	—
287	C₂²×C₈₃	—	2	2	no	—
288	C₂²×C₈₃	—	2	2	no	—
289	C₂²×C₈₃	—	2	1	no	—
290	C₂²×C₈₃	—	2	2	no	—
291	C₂²×C₈₃	—	2	2	no	—
292	C₂²×C₈₃	—	2	2	no	—
293	C₂²×C₈₃	—	4	4	no	—
294	C₂²×C₈₃	—	4	4	no	—
295	C₂²×C₈₃	—	4	1	no	—
296	C₂²×C₈₃	—	4	1	no	—
297	C₂²×C₈₃	—	4	1	no	—
298	C₂²×C₈₃	C₂×#104	1	4	yes	—
299	C₂²×C₈₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
300	C₂²×C₈₃	#577	1	1	yes	characteristic
301	C₂²×C₈₃	#578	1	1	yes	characteristic
302	C₄×C₈₃	—	2	1	no	—
303	C₄×C₈₃	—	2	1	no	—
304	C₄×C₈₃	—	2	2	no	—
305	C₄×C₈₃	—	2	2	no	—
306	C₄×C₈₃	—	2	1	no	—
307	C₄×C₈₃	—	2	1	no	—
308	C₄×C₈₃	—	2	1	no	—
309	C₄×C₈₃	—	4	2	no	—
310	C₄×C₈₃	—	4	4	no	—
311	C₄×C₈₃	—	4	2	no	—
312	C₄×C₈₃	—	4	2	no	—
313	C₄×C₈₃	—	4	1	no	—
314	C₄×C₈₃	—	4	1	no	—
315	C₄×C₈₃	—	4	1	no	—
316	C₄×C₈₃	—	4	2	no	—
317	C₄×C₈₃	C₂×#104	1	2	yes	—
318	#17548	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
319	C₂×C₈₃	—	2	1	no	—
320	C₂×C₈₃	—	2	1	no	—
321	C₂×C₈₃	—	2	4	no	—
322	C₂×C₈₃	—	4	2	no	—
323	C₂×C₈₃	—	8	1	no	—
324	C₂×C₈₃	#2883	1	1	yes	characteristic
325	C₂×C₈₃	#2898	1	1	yes	characteristic
326	C₂×C₈₃	C₂²×#104	1	1	yes	characteristic
327	#3520	C₂×C₈₃	1	1	yes	characteristic
328	#2974	C₂×C₈₃	1	1	yes	characteristic, complemented
329	#2275	C₂×C₈₃	1	1	yes	characteristic, complemented
330	#1509	C₂×C₈₃	1	1	yes	characteristic, complemented
331	#1508	C₂×C₈₃	1	2	yes	complemented
332	#1506	C₂×C₈₃	1	1	yes	characteristic, complemented
333	#1504	C₂×C₈₃	1	2	yes	complemented
334	#1503	C₂×C₈₃	1	1	yes	characteristic, complemented
335	#1501	C₂×C₈₃	1	2	yes	complemented
336	#1500	C₂×C₈₃	1	1	yes	characteristic, complemented
337	C₂×#328	C₂×C₈₃	1	2	yes	complemented
338	C₈₃	#17548	1	1	yes	characteristic, direct product factor, Sylow
339	#556	C₄×C₈₃	1	1	yes	characteristic, complemented
340	#554	—	2	1	no	—
341	#552	C₄×C₈₃	1	2	yes	complemented
342	#551	—	2	2	no	—
343	D₄×Q₈	C₄×C₈₃	1	1	yes	characteristic, complemented
344	#548	C₄×C₈₃	1	2	yes	complemented
345	#547	C₄×C₈₃	1	1	yes	characteristic, complemented
346	#547	C₄×C₈₃	1	1	yes	characteristic, complemented
347	#546	—	2	1	no	—
348	#542	—	2	1	no	—
349	#539	—	2	2	no	—
350	#533	C₂²×C₈₃	1	1	yes	characteristic
351	#529	C₂²×C₈₃	1	1	yes	characteristic
352	#525	C₂²×C₈₃	1	2	yes	—
353	C₂×C₂²⋊Q₈	C₂²×C₈₃	1	2	yes	—
354	#520	—	2	1	no	—
355	C₂×C₄×D₄	—	2	1	no	—
356	C₂×C₄×D₄	C₂²×C₈₃	1	1	yes	characteristic
357	#436	—	2	1	no	—
358	#435	—	2	2	no	—
359	C₈×D₄	—	2	1	no	—
360	#432	C₂²×C₈₃	1	1	yes	characteristic
361	#431	C₂²×C₈₃	1	1	yes	characteristic
362	#407	C₂²×C₈₃	1	2	yes	—
363	C₂×C₂²⋊C₈	C₂²×C₈₃	1	2	yes	—
364	C₄×C₄⋊C₄	—	2	1	no	—
365	C₄×C₄⋊C₄	C₂²×C₈₃	1	1	yes	characteristic
366	C₄×C₂²⋊C₄	—	2	2	no	—
367	C₄×C₂²⋊C₄	C₂²×C₈₃	1	2	yes	—
368	#376	C₂²×C₈₃	1	1	yes	characteristic
369	C₂×#95	C₂²×C₈₃	1	2	yes	—
370	#328	C₂²×C₈₃	1	1	yes	characteristic
371	#328	C₂²×C₈₃	1	2	yes	—
372	#328	C₂²×C₈₃	1	2	yes	—
373	#328	C₂²×C₈₃	1	2	yes	—
374	#328	C₂²×C₈₃	1	1	yes	characteristic
375	#328	C₂²×C₈₃	1	2	yes	—
376	#328	C₂²×C₈₃	1	2	yes	—
377	#327	C₂²×C₈₃	1	1	yes	characteristic
378	#327	C₂²×C₈₃	1	1	yes	characteristic
379	#327	C₂²×C₈₃	1	2	yes	—
380	C₂×#41	—	2	1	no	—
381	C₂×#41	C₂×C₄×C₈₃	1	2	yes	complemented
382	C₂²×Q₈	C₂×C₄×C₈₃	1	2	yes	complemented
383	C₂²×D₄	D₄×C₈₃	1	1	yes	characteristic
384	C₂³×C₄	—	2	2	no	—
385	C₂³×C₄	C₂³×C₈₃	1	2	yes	—
386	C₂×M₁₆	—	2	2	no	—
387	C₂²×C₈	—	2	2	no	—
388	C₄⋊Q₈	—	2	1	no	—
389	C₄⋊Q₈	C₂×C₄×C₈₃	1	1	yes	characteristic, complemented
390	C₄⋊Q₈	C₂×C₄×C₈₃	1	1	yes	characteristic, complemented
391	#126	—	2	2	no	—
392	#126	—	2	2	no	—
393	#126	—	2	1	no	—
394	#125	—	2	2	no	—
395	#125	—	2	1	no	—
396	#125	C₂×C₄×C₈₃	1	1	yes	characteristic, complemented
397	#124	—	2	1	no	—
398	#124	—	2	2	no	—
399	#124	C₂×C₄×C₈₃	1	1	yes	characteristic, complemented
400	#123	—	2	2	no	—
401	#123	—	2	2	no	—
402	#123	—	2	1	no	—
403	#123	—	2	2	no	—
404	#123	—	2	2	no	—
405	C₂²⋊Q₈	—	2	2	no	—
406	C₂²⋊Q₈	—	2	2	no	—
407	C₂²⋊Q₈	—	2	1	no	—
408	C₂²⋊Q₈	—	2	2	no	—
409	C₂²⋊Q₈	—	2	1	no	—
410	C₂²⋊Q₈	—	2	2	no	—
411	C₂²⋊Q₈	C₂×C₄×C₈₃	1	2	yes	complemented
412	C₂²⋊Q₈	C₂×C₄×C₈₃	1	2	yes	complemented
413	C₂²⋊Q₈	C₂×C₄×C₈₃	1	2	yes	complemented
414	C₂²⋊Q₈	C₂³×C₈₃	1	1	yes	characteristic
415	C₂²⋊Q₈	C₂³×C₈₃	1	2	yes	—
416	#121	—	2	2	no	—
417	#121	—	2	1	no	—
418	#121	C₂×C₄×C₈₃	1	2	yes	complemented
419	#121	C₂³×C₈₃	1	1	yes	characteristic
420	#120	—	2	2	no	—
421	C₄×Q₈	—	2	1	no	—
422	C₄×Q₈	—	2	1	no	—
423	C₄×Q₈	C₂×C₄×C₈₃	1	1	yes	characteristic, complemented
424	C₄×Q₈	C₂×C₄×C₈₃	1	2	yes	complemented
425	C₄×D₄	—	2	8	no	—
426	C₄×D₄	—	2	1	no	—
427	C₄×D₄	—	2	2	no	—
428	C₄×D₄	—	2	2	no	—
429	C₄×D₄	—	2	1	no	—
430	C₄×D₄	—	2	2	no	—
431	C₄×D₄	C₂×C₄×C₈₃	1	1	yes	characteristic
432	C₄×D₄	C₂×C₄×C₈₃	1	1	yes	characteristic
433	C₄×D₄	C₂×C₄×C₈₃	1	2	yes	—
434	C₄×D₄	C₂×C₄×C₈₃	1	1	yes	characteristic, complemented
435	C₄×D₄	D₄×C₈₃	1	1	yes	characteristic
436	C₄×D₄	D₄×C₈₃	1	2	yes	—
437	C₄×D₄	D₄×C₈₃	1	1	yes	characteristic
438	#117	—	2	1	no	—
439	#117	—	2	2	no	—
440	#117	—	2	1	no	—
441	#117	C₂×C₄×C₈₃	1	2	yes	complemented
442	C₂×C₄⋊C₄	—	2	1	no	—
443	C₂×C₄⋊C₄	—	2	2	no	—
444	C₂×C₄⋊C₄	—	2	1	no	—
445	C₂×C₄⋊C₄	C₂×C₄×C₈₃	1	2	yes	complemented
446	C₂×C₄⋊C₄	D₄×C₈₃	1	1	yes	characteristic
447	C₂×C₂²⋊C₄	—	2	2	no	—
448	C₂×C₂²⋊C₄	—	2	2	no	—
449	C₂×C₂²⋊C₄	D₄×C₈₃	1	2	yes	—
450	C₂×C₄²	—	2	1	no	—
451	C₂×C₄²	—	2	1	no	—
452	C₂×C₄²	—	2	2	no	—
453	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
454	C₄⋊C₈	—	2	1	no	—
455	C₄⋊C₈	—	2	1	no	—
456	C₂²⋊C₈	—	2	2	no	—
457	C₂²⋊C₈	—	2	2	no	—
458	C₂²⋊C₈	C₂³×C₈₃	1	1	yes	characteristic
459	C₂²⋊C₈	C₂³×C₈₃	1	2	yes	—
460	C₂²⋊C₈	C₂³×C₈₃	1	1	yes	characteristic
461	#97	—	2	1	no	—
462	C₄×C₈	—	2	1	no	—
463	#95	—	2	2	no	—
464	#95	—	2	1	no	—
465	#95	C₂³×C₈₃	1	1	yes	characteristic
466	#95	C₂³×C₈₃	1	2	yes	—
467	#95	C₂³×C₈₃	1	1	yes	characteristic
468	C₂⁴	C₂×D₄×C₈₃	1	2	yes	—
469	#41	—	4	4	no	—
470	#41	—	8	2	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	2	no	—
473	C₂×Q₈	—	8	2	no	—
474	C₂×Q₈	C₂²×C₄×C₈₃	1	1	yes	characteristic
475	C₂×Q₈	C₂²×C₄×C₈₃	1	2	yes	—
476	C₂×D₄	—	2	4	no	—
477	C₂×D₄	—	2	2	no	—
478	C₂×D₄	—	2	1	no	—
479	C₂×D₄	—	2	2	no	—
480	C₂×D₄	C₂²⋊C₄×C₈₃	1	2	yes	—
481	C₂×D₄	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
482	C₂×D₄	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
483	C₂×D₄	C₂²×C₄×C₈₃	1	1	yes	characteristic
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	2	no	—
488	C₂²×C₄	—	2	8	no	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	2	no	—
492	C₂²×C₄	—	8	2	no	—
493	C₂²×C₄	—	8	2	no	—
494	C₂²×C₄	—	8	2	no	—
495	C₂²×C₄	C₂²×C₄×C₈₃	1	2	yes	—
496	C₂²×C₄	C₂²×C₄×C₈₃	1	2	yes	—
497	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
498	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
499	C₂²×C₄	C₂×D₄×C₈₃	1	2	yes	—
500	C₂²×C₄	C₂×D₄×C₈₃	1	2	yes	—
501	C₂²×C₄	C₂×D₄×C₈₃	1	2	yes	—
502	C₂²×C₄	C₂⁴×C₈₃	1	1	yes	characteristic
503	M₁₆	—	8	2	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	2	no	—
507	C₂×C₈	—	8	2	no	—
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	2	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	2	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
523	C₄⋊C₄	C₂²⋊C₄×C₈₃	1	2	yes	—
524	C₄⋊C₄	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
525	C₄⋊C₄	C₂²×C₄×C₈₃	1	1	yes	characteristic
526	C₄⋊C₄	C₂²×C₄×C₈₃	1	2	yes	—
527	C₄⋊C₄	C₂²×C₄×C₈₃	1	1	yes	characteristic
528	C₂²⋊C₄	—	2	8	no	—
529	C₂²⋊C₄	—	2	2	no	—
530	C₂²⋊C₄	—	2	2	no	—
531	C₂²⋊C₄	—	2	2	no	—
532	C₂²⋊C₄	—	2	2	no	—
533	C₂²⋊C₄	—	2	1	no	—
534	C₂²⋊C₄	—	2	2	no	—
535	C₂²⋊C₄	—	2	1	no	—
536	C₂²⋊C₄	—	2	1	no	—
537	C₂²⋊C₄	—	2	2	no	—
538	C₂²⋊C₄	—	2	1	no	—
539	C₂²⋊C₄	—	2	1	no	—
540	C₂²⋊C₄	C₂²⋊C₄×C₈₃	1	2	yes	—
541	C₂²⋊C₄	C₂²⋊C₄×C₈₃	1	2	yes	—
542	C₂²⋊C₄	C₂²⋊C₄×C₈₃	1	2	yes	—
543	C₂²⋊C₄	C₂²⋊C₄×C₈₃	1	2	yes	—
544	C₄²	—	2	4	no	—
545	C₄²	—	2	2	no	—
546	C₄²	—	2	2	no	—
547	C₄²	—	2	1	no	—
548	C₄²	—	8	1	no	—
549	C₄²	—	8	2	no	—
550	C₄²	C₂²×C₄×C₈₃	1	1	yes	characteristic
551	C₄²	C₂²×C₄×C₈₃	1	1	yes	characteristic
552	C₄²	C₂×D₄×C₈₃	1	1	yes	characteristic
553	C₄²	C₂×D₄×C₈₃	1	1	yes	characteristic
554	C₂³	—	2	2	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	2	4	no	—
557	C₂³	—	2	1	no	—
558	C₂³	—	4	2	no	—
559	C₂³	C₂×C₂²⋊C₄×C₈₃	1	2	yes	—
560	C₂³	C₂×C₂²⋊C₄×C₈₃	1	2	yes	—
561	C₂³	C₂²×D₄×C₈₃	1	1	yes	characteristic
562	Q₈	—	4	1	no	—
563	Q₈	—	4	2	no	—
564	Q₈	—	4	2	no	—
565	Q₈	—	8	2	no	—
566	D₄	—	2	4	no	—
567	D₄	—	2	2	no	—
568	D₄	—	4	1	no	—
569	D₄	—	4	1	no	—
570	D₄	—	4	1	no	—
571	D₄	—	4	4	no	—
572	D₄	—	8	2	no	—
573	D₄	#104×C₈₃	1	4	yes	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	4	2	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	2	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	2	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	2	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	4	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	8	2	no	—
600	C₂×C₄	—	8	2	no	—
601	C₂×C₄	—	8	8	no	—
602	C₂×C₄	C₂×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
603	C₂×C₄	C₂×C₂²⋊C₄×C₈₃	1	2	yes	—
604	C₂×C₄	C₂×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
605	C₂×C₄	C₂×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
606	C₂×C₄	C₂×C₂²⋊C₄×C₈₃	1	2	yes	—
607	C₂×C₄	C₂×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
608	C₂×C₄	C₂³×C₄×C₈₃	1	1	yes	characteristic
609	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
610	C₈	—	4	1	no	—
611	C₈	—	4	2	no	—
612	C₈	—	4	1	no	—
613	C₂²	—	2	1	no	—
614	C₂²	—	2	2	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	2	1	no	—
617	C₂²	—	2	2	no	—
618	C₂²	—	2	2	no	—
619	C₂²	—	2	2	no	—
620	C₂²	—	4	4	no	—
621	C₂²	—	4	4	no	—
622	C₂²	—	4	1	no	—
623	C₂²	—	4	1	no	—
624	C₂²	—	4	1	no	—
625	C₂²	C₂×#104×C₈₃	1	4	yes	—
626	C₂²	C₂²×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
627	C₂²	#577×C₈₃	1	1	yes	characteristic
628	C₂²	#578×C₈₃	1	1	yes	characteristic
629	C₄	—	2	1	no	—
630	C₄	—	2	1	no	—
631	C₄	—	2	2	no	—
632	C₄	—	2	2	no	—
633	C₄	—	2	1	no	—
634	C₄	—	2	1	no	—
635	C₄	—	2	1	no	—
636	C₄	—	4	2	no	—
637	C₄	—	4	4	no	—
638	C₄	—	4	2	no	—
639	C₄	—	4	2	no	—
640	C₄	—	4	1	no	—
641	C₄	—	4	1	no	—
642	C₄	—	4	1	no	—
643	C₄	—	4	2	no	—
644	C₄	C₂×#104×C₈₃	1	2	yes	—
645	C₂	—	2	1	no	—
646	C₂	—	2	1	no	—
647	C₂	—	2	4	no	—
648	C₂	—	4	2	no	—
649	C₂	—	8	1	no	—
650	C₂	C₈₃×#2883	1	1	yes	characteristic
651	C₂	C₈₃×#2898	1	1	yes	characteristic
652	C₂	C₂²×#104×C₈₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₈₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₃
prime radical	C₈₃×#3520	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₃ ◁ C₂×C₄×D₄×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₃×#2974	C₂	semidirect
3. C₈₃×#2275	C₂	semidirect
4. C₈₃×#1509	C₂	semidirect
5. C₈₃×#1508	C₂	semidirect
6. C₈₃×#1506	C₂	semidirect
7. C₈₃×#1504	C₂	semidirect
8. C₈₃×#1503	C₂	semidirect
9. C₈₃×#1501	C₂	semidirect
10. C₈₃×#1500	C₂	semidirect
11. C₂×#328×C₈₃	C₂	semidirect
12. #556×C₈₃	C₄	semidirect
14. #552×C₈₃	C₄	semidirect
16. D₄×Q₈×C₈₃	C₄	semidirect
17. #548×C₈₃	C₄	semidirect
18. #547×C₈₃	C₄	semidirect
19. #547×C₈₃	C₄	semidirect
54. C₂×#41×C₈₃	C₂×C₄	semidirect
55. C₂²×Q₈×C₈₃	C₂×C₄	semidirect
62. C₄⋊Q₈×C₈₃	C₂×C₄	semidirect
63. C₄⋊Q₈×C₈₃	C₂×C₄	semidirect
69. #125×C₈₃	C₂×C₄	semidirect
72. #124×C₈₃	C₂×C₄	semidirect
84. C₂²⋊Q₈×C₈₃	C₂×C₄	semidirect
85. C₂²⋊Q₈×C₈₃	C₂×C₄	semidirect
86. C₂²⋊Q₈×C₈₃	C₂×C₄	semidirect
91. #121×C₈₃	C₂×C₄	semidirect
96. C₄×Q₈×C₈₃	C₂×C₄	semidirect
97. C₄×Q₈×C₈₃	C₂×C₄	semidirect
107. C₄×D₄×C₈₃	C₂×C₄	semidirect
114. #117×C₈₃	C₂×C₄	semidirect
118. C₂×C₄⋊C₄×C₈₃	C₂×C₄	semidirect
318. #17548	C₈₃	direct (N × H)
328. #2974	C₂×C₈₃	semidirect
329. #2275	C₂×C₈₃	semidirect
330. #1509	C₂×C₈₃	semidirect
331. #1508	C₂×C₈₃	semidirect
332. #1506	C₂×C₈₃	semidirect
333. #1504	C₂×C₈₃	semidirect
334. #1503	C₂×C₈₃	semidirect
335. #1501	C₂×C₈₃	semidirect
336. #1500	C₂×C₈₃	semidirect
337. C₂×#328	C₂×C₈₃	semidirect
338. C₈₃	#17548	direct (N × H)
339. #556	C₄×C₈₃	semidirect
341. #552	C₄×C₈₃	semidirect
343. D₄×Q₈	C₄×C₈₃	semidirect
344. #548	C₄×C₈₃	semidirect
345. #547	C₄×C₈₃	semidirect
346. #547	C₄×C₈₃	semidirect
381. C₂×#41	C₂×C₄×C₈₃	semidirect
382. C₂²×Q₈	C₂×C₄×C₈₃	semidirect
389. C₄⋊Q₈	C₂×C₄×C₈₃	semidirect
390. C₄⋊Q₈	C₂×C₄×C₈₃	semidirect
396. #125	C₂×C₄×C₈₃	semidirect
399. #124	C₂×C₄×C₈₃	semidirect
411. C₂²⋊Q₈	C₂×C₄×C₈₃	semidirect
412. C₂²⋊Q₈	C₂×C₄×C₈₃	semidirect
413. C₂²⋊Q₈	C₂×C₄×C₈₃	semidirect
418. #121	C₂×C₄×C₈₃	semidirect
423. C₄×Q₈	C₂×C₄×C₈₃	semidirect
424. C₄×Q₈	C₂×C₄×C₈₃	semidirect
434. C₄×D₄	C₂×C₄×C₈₃	semidirect
441. #117	C₂×C₄×C₈₃	semidirect
445. C₂×C₄⋊C₄	C₂×C₄×C₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₈₃×#3520	2
C₈₃×#2974	2
C₈₃×#2275	2
C₈₃×#1509	2
C₈₃×#1508	2
C₈₃×#1506	2
C₈₃×#1504	2
C₈₃×#1503	2
C₈₃×#1501	2
C₈₃×#1500	2
C₂×#328×C₈₃	2
#17548	83

Subgroups — C83×#17548 #35245372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#17548 #35245372