Subgroups — C₂×C₈₃×#3211 #35288600

Order 21248.

2294 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#124×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₃×#3211	C₂	1	16	yes	maximal, direct product factor
4	C₂×#495×C₈₃	C₂	1	2	yes	maximal, complemented
5	C₂×#488×C₈₃	C₂	1	2	yes	maximal, complemented
6	C₂×#464×C₈₃	C₂	1	2	yes	maximal, complemented
7	C₂×#463×C₈₃	C₂	1	2	yes	maximal, complemented
8	C₂×#451×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#448×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#450×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#447×C₈₃	C₂	1	1	yes	maximal, characteristic
12	C₂×#431×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₈₃	C₂²	1	1	yes	characteristic
15	C₂²×Q₁₆×C₈₃	C₂²	1	2	yes	complemented
16	C₂²×SD₁₆×C₈₃	C₂²	1	2	yes	—
17	C₂²×SD₁₆×C₈₃	C₂²	1	2	yes	complemented
18	C₂²×D₈×C₈₃	C₂²	1	2	yes	—
19	#529×C₈₃	—	2	4	no	—
20	#529×C₈₃	C₂²	1	8	yes	complemented
21	C₂×#126×C₈₃	—	2	1	no	—
22	C₂×#125×C₈₃	C₂²	1	1	yes	characteristic, complemented
23	C₂×#124×C₈₃	—	2	1	no	—
24	C₂×#124×C₈₃	—	2	1	no	—
25	C₂×#124×C₈₃	—	2	4	no	—
26	C₂×#124×C₈₃	C₂²	1	2	yes	—
27	C₂×#124×C₈₃	C₂²	1	2	yes	—
28	C₂×#124×C₈₃	C₂²	1	8	yes	complemented
29	C₂×#124×C₈₃	C₂²	1	1	yes	characteristic, complemented
30	C₂×#123×C₈₃	—	2	1	no	—
31	C₂×C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
32	C₂×#121×C₈₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₄×D₄×C₈₃	—	2	1	no	—
35	C₂×C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic, complemented
36	C₂×#117×C₈₃	—	2	1	no	—
37	C₂²×C₂²⋊C₄×C₈₃	—	2	1	no	—
38	C₂²×C₂²⋊C₄×C₈₃	—	2	1	no	—
39	C₂²×C₄²×C₈₃	C₂²	1	1	yes	characteristic
40	#495×C₈₃	C₂²	1	16	yes	complemented
41	#488×C₈₃	C₂²	1	16	yes	complemented
42	#464×C₈₃	C₂²	1	16	yes	complemented
43	#463×C₈₃	C₂²	1	16	yes	complemented
44	#451×C₈₃	C₂²	1	8	yes	complemented
45	#450×C₈₃	C₂²	1	8	yes	complemented
46	#448×C₈₃	C₂²	1	8	yes	complemented
47	#447×C₈₃	C₂²	1	8	yes	—
48	#431×C₈₃	C₂²	1	8	yes	complemented
49	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	complemented
50	C₂×#103×C₈₃	C₂²	1	2	yes	complemented
51	C₂×#103×C₈₃	C₂²	1	2	yes	complemented
52	C₂×#102×C₈₃	C₂²	1	2	yes	—
53	C₂×#102×C₈₃	C₂²	1	2	yes	—
54	C₂×C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
55	C₂×C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₂×C₄×C₈×C₈₃	C₂²	1	2	yes	complemented
57	C₂⁵×C₈₃	—	2	1	no	—
58	C₂²×Q₈×C₈₃	—	2	2	no	—
59	C₂²×Q₈×C₈₃	—	2	1	no	—
60	C₂²×Q₈×C₈₃	—	4	4	no	—
61	C₂²×Q₈×C₈₃	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₈₃	C₂³	1	2	yes	—
63	C₂²×Q₈×C₈₃	C₂³	1	4	yes	—
64	C₂²×D₄×C₈₃	—	2	1	no	—
65	C₂²×D₄×C₈₃	—	2	2	no	—
66	C₂²×D₄×C₈₃	—	2	1	no	—
67	C₂²×D₄×C₈₃	—	4	4	no	—
68	C₂²×D₄×C₈₃	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₈₃	C₂³	1	2	yes	—
70	C₂²×D₄×C₈₃	C₂³	1	4	yes	—
71	C₂³×C₄×C₈₃	—	2	1	no	—
72	C₂³×C₄×C₈₃	—	2	1	no	—
73	C₂³×C₄×C₈₃	D₄	1	1	yes	characteristic
74	C₂³×C₄×C₈₃	D₄	1	1	yes	characteristic
75	C₂³×C₄×C₈₃	C₂³	1	1	yes	characteristic
76	C₂×Q₁₆×C₈₃	—	4	4	no	—
77	C₂×Q₁₆×C₈₃	C₂³	1	8	yes	complemented
78	C₂×SD₁₆×C₈₃	—	4	4	no	—
79	C₂×SD₁₆×C₈₃	—	4	4	no	—
80	C₂×SD₁₆×C₈₃	C₂³	1	8	yes	—
81	C₂×SD₁₆×C₈₃	C₂³	1	8	yes	complemented
82	C₂×D₈×C₈₃	—	4	4	no	—
83	C₂×D₈×C₈₃	C₂³	1	8	yes	—
84	C₂²×C₈×C₈₃	C₂³	1	2	yes	—
85	C₂²×C₈×C₈₃	C₂³	1	2	yes	—
86	#126×C₈₃	—	2	8	no	—
87	#125×C₈₃	—	2	2	no	—
88	#125×C₈₃	C₂³	1	4	yes	complemented
89	#124×C₈₃	—	2	4	no	—
90	#124×C₈₃	—	2	4	no	—
91	#124×C₈₃	—	2	8	no	—
92	#124×C₈₃	—	2	8	no	—
93	#124×C₈₃	—	2	2	no	—
94	#124×C₈₃	C₂³	1	8	yes	—
95	#124×C₈₃	C₂³	1	8	yes	—
96	#124×C₈₃	C₂³	1	4	yes	complemented
97	#123×C₈₃	—	2	8	no	—
98	C₂²⋊Q₈×C₈₃	—	2	2	no	—
99	C₂²⋊Q₈×C₈₃	C₂³	1	4	yes	—
100	#121×C₈₃	—	2	2	no	—
101	#121×C₈₃	C₂³	1	4	yes	—
102	C₄×Q₈×C₈₃	—	2	2	no	—
103	C₄×Q₈×C₈₃	C₂³	1	4	yes	complemented
104	C₄×D₄×C₈₃	—	2	8	no	—
105	C₄×D₄×C₈₃	—	2	2	no	—
106	C₄×D₄×C₈₃	C₂³	1	4	yes	complemented
107	#117×C₈₃	—	2	8	no	—
108	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
109	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
110	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
111	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
112	C₂×C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
114	C₂×C₂²⋊C₄×C₈₃	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
119	C₂×C₂²⋊C₄×C₈₃	—	2	8	no	—
120	C₂×C₂²⋊C₄×C₈₃	—	2	8	no	—
121	C₂×C₂²⋊C₄×C₈₃	—	2	1	no	—
122	C₂×C₂²⋊C₄×C₈₃	—	2	1	no	—
123	C₂×C₂²⋊C₄×C₈₃	—	2	1	no	—
124	C₂×C₂²⋊C₄×C₈₃	—	2	1	no	—
125	C₂×C₄²×C₈₃	—	2	1	no	—
126	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
127	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
128	C₂×C₄²×C₈₃	D₄	1	4	yes	—
129	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₈₃	C₂³	1	4	yes	—
132	C₄⋊C₈×C₈₃	C₂³	1	8	yes	complemented
133	#103×C₈₃	C₂³	1	8	yes	complemented
134	#103×C₈₃	C₂³	1	8	yes	complemented
135	#102×C₈₃	C₂³	1	8	yes	—
136	#102×C₈₃	C₂³	1	8	yes	—
137	C₂²⋊C₈×C₈₃	C₂³	1	4	yes	—
138	C₂²⋊C₈×C₈₃	C₂³	1	4	yes	—
139	C₄×C₈×C₈₃	C₂³	1	8	yes	complemented
140	C₂⁴×C₈₃	—	2	4	no	—
141	C₂⁴×C₈₃	—	2	1	no	—
142	C₂⁴×C₈₃	—	2	2	no	—
143	C₂⁴×C₈₃	—	4	4	no	—
144	C₂⁴×C₈₃	—	4	4	no	—
145	C₂⁴×C₈₃	—	4	4	no	—
146	C₂⁴×C₈₃	C₂×D₄	1	1	yes	characteristic
147	C₂×Q₈×C₈₃	—	2	1	no	—
148	C₂×Q₈×C₈₃	—	2	4	no	—
149	C₂×Q₈×C₈₃	—	2	2	no	—
150	C₂×Q₈×C₈₃	—	2	4	no	—
151	C₂×Q₈×C₈₃	—	2	4	no	—
152	C₂×Q₈×C₈₃	—	2	4	no	—
153	C₂×Q₈×C₈₃	—	4	8	no	—
154	C₂×Q₈×C₈₃	—	4	4	no	—
155	C₂×Q₈×C₈₃	—	4	8	no	—
156	C₂×Q₈×C₈₃	C₂⁴	1	4	yes	—
157	C₂×Q₈×C₈₃	C₂⁴	1	2	yes	—
158	C₂×D₄×C₈₃	—	2	4	no	—
159	C₂×D₄×C₈₃	—	2	1	no	—
160	C₂×D₄×C₈₃	—	2	4	no	—
161	C₂×D₄×C₈₃	—	2	2	no	—
162	C₂×D₄×C₈₃	—	2	4	no	—
163	C₂×D₄×C₈₃	—	2	4	no	—
164	C₂×D₄×C₈₃	—	2	4	no	—
165	C₂×D₄×C₈₃	—	4	8	no	—
166	C₂×D₄×C₈₃	—	4	4	no	—
167	C₂×D₄×C₈₃	—	4	8	no	—
168	C₂×D₄×C₈₃	—	4	4	no	—
169	C₂×D₄×C₈₃	C₂⁴	1	4	yes	—
170	C₂×D₄×C₈₃	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₈₃	—	2	4	no	—
172	C₂²×C₄×C₈₃	—	2	4	no	—
173	C₂²×C₄×C₈₃	—	2	4	no	—
174	C₂²×C₄×C₈₃	—	2	1	no	—
175	C₂²×C₄×C₈₃	—	2	1	no	—
176	C₂²×C₄×C₈₃	—	2	2	no	—
177	C₂²×C₄×C₈₃	—	2	1	no	—
178	C₂²×C₄×C₈₃	—	2	4	no	—
179	C₂²×C₄×C₈₃	—	4	2	no	—
180	C₂²×C₄×C₈₃	—	4	4	no	—
181	C₂²×C₄×C₈₃	—	4	4	no	—
182	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₈₃	C₂×D₄	1	4	yes	—
184	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₈₃	C₂×D₄	1	4	yes	—
190	C₂²×C₄×C₈₃	C₂⁴	1	2	yes	—
191	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
192	Q₁₆×C₈₃	—	4	8	no	—
193	SD₁₆×C₈₃	—	4	8	no	—
194	SD₁₆×C₈₃	—	4	8	no	—
195	D₈×C₈₃	—	4	8	no	—
196	C₂×C₈×C₈₃	—	2	4	no	—
197	C₂×C₈×C₈₃	—	2	4	no	—
198	C₂×C₈×C₈₃	C₂⁴	1	4	yes	—
199	C₂×C₈×C₈₃	C₂⁴	1	4	yes	—
200	C₄⋊C₄×C₈₃	—	2	4	no	—
201	C₄⋊C₄×C₈₃	—	2	4	no	—
202	C₄⋊C₄×C₈₃	—	2	4	no	—
203	C₄⋊C₄×C₈₃	—	2	4	no	—
204	C₄⋊C₄×C₈₃	—	2	1	no	—
205	C₄⋊C₄×C₈₃	—	2	1	no	—
206	C₄⋊C₄×C₈₃	C₂⁴	1	2	yes	—
207	C₄⋊C₄×C₈₃	C₂⁴	1	2	yes	—
208	C₂²⋊C₄×C₈₃	—	2	4	no	—
209	C₂²⋊C₄×C₈₃	—	2	8	no	—
210	C₂²⋊C₄×C₈₃	—	2	8	no	—
211	C₂²⋊C₄×C₈₃	—	2	8	no	—
212	C₂²⋊C₄×C₈₃	—	2	8	no	—
213	C₂²⋊C₄×C₈₃	—	2	4	no	—
214	C₂²⋊C₄×C₈₃	—	2	4	no	—
215	C₂²⋊C₄×C₈₃	—	2	4	no	—
216	C₂²⋊C₄×C₈₃	—	2	4	no	—
217	C₄²×C₈₃	—	2	4	no	—
218	C₄²×C₈₃	C₂×D₄	1	4	yes	—
219	C₄²×C₈₃	C₂×D₄	1	2	yes	—
220	C₄²×C₈₃	C₂×D₄	1	2	yes	—
221	C₄²×C₈₃	C₂×D₄	1	4	yes	—
222	C₄²×C₈₃	C₂⁴	1	2	yes	—
223	C₄²×C₈₃	C₂⁴	1	2	yes	—
224	C₂³×C₈₃	—	2	2	no	—
225	C₂³×C₈₃	—	2	2	no	—
226	C₂³×C₈₃	—	2	2	no	—
227	C₂³×C₈₃	—	2	4	no	—
228	C₂³×C₈₃	—	4	8	no	—
229	C₂³×C₈₃	—	4	8	no	—
230	C₂³×C₈₃	—	4	4	no	—
231	C₂³×C₈₃	—	4	2	no	—
232	C₂³×C₈₃	—	4	2	no	—
233	C₂³×C₈₃	—	4	8	no	—
234	C₂³×C₈₃	—	4	4	no	—
235	C₂³×C₈₃	—	4	2	no	—
236	C₂³×C₈₃	—	4	4	no	—
237	C₂³×C₈₃	—	8	16	no	—
238	C₂³×C₈₃	#135	1	4	yes	—
239	C₂³×C₈₃	C₂²×D₄	1	2	yes	—
240	C₂³×C₈₃	C₂²×D₄	1	1	yes	characteristic
241	Q₈×C₈₃	—	2	4	no	—
242	Q₈×C₈₃	—	2	4	no	—
243	Q₈×C₈₃	—	2	4	no	—
244	Q₈×C₈₃	—	4	8	no	—
245	Q₈×C₈₃	—	4	2	no	—
246	Q₈×C₈₃	—	4	4	no	—
247	D₄×C₈₃	—	2	4	no	—
248	D₄×C₈₃	—	2	4	no	—
249	D₄×C₈₃	—	2	4	no	—
250	D₄×C₈₃	—	4	8	no	—
251	D₄×C₈₃	—	4	8	no	—
252	D₄×C₈₃	—	4	2	no	—
253	D₄×C₈₃	—	4	4	no	—
254	C₂×C₄×C₈₃	—	2	2	no	—
255	C₂×C₄×C₈₃	—	2	8	no	—
256	C₂×C₄×C₈₃	—	2	2	no	—
257	C₂×C₄×C₈₃	—	2	2	no	—
258	C₂×C₄×C₈₃	—	2	2	no	—
259	C₂×C₄×C₈₃	—	2	2	no	—
260	C₂×C₄×C₈₃	—	2	4	no	—
261	C₂×C₄×C₈₃	—	2	4	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	—	4	4	no	—
267	C₂×C₄×C₈₃	—	4	8	no	—
268	C₂×C₄×C₈₃	—	4	8	no	—
269	C₂×C₄×C₈₃	—	4	4	no	—
270	C₂×C₄×C₈₃	—	4	2	no	—
271	C₂×C₄×C₈₃	—	4	2	no	—
272	C₂×C₄×C₈₃	—	4	2	no	—
273	C₂×C₄×C₈₃	C₂²×D₄	1	2	yes	—
274	C₂×C₄×C₈₃	C₂²×D₄	1	2	yes	—
275	C₂×C₄×C₈₃	C₂²×D₄	1	2	yes	—
276	C₂×C₄×C₈₃	C₂²×D₄	1	2	yes	—
277	C₂×C₄×C₈₃	C₂²×D₄	1	2	yes	—
278	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
279	C₂×C₄×C₈₃	H(5,𝔽₂)	1	4	yes	—
280	C₂×C₄×C₈₃	C₂⁵	1	1	yes	characteristic
281	C₈×C₈₃	—	2	4	no	—
282	C₈×C₈₃	—	2	4	no	—
283	C₂²×C₈₃	—	2	2	no	—
284	C₂²×C₈₃	—	2	8	no	—
285	C₂²×C₈₃	—	2	2	no	—
286	C₂²×C₈₃	—	4	4	no	—
287	C₂²×C₈₃	—	4	2	no	—
288	C₂²×C₈₃	—	4	2	no	—
289	C₂²×C₈₃	—	4	4	no	—
290	C₂²×C₈₃	—	4	2	no	—
291	C₂²×C₈₃	—	4	4	no	—
292	C₂²×C₈₃	—	8	16	no	—
293	C₂²×C₈₃	—	8	4	no	—
294	C₂²×C₈₃	—	8	8	no	—
295	C₂²×C₈₃	#535	1	2	yes	—
296	C₂²×C₈₃	C₂×#135	1	4	yes	—
297	C₂²×C₈₃	C₂×#135	1	2	yes	—
298	C₂²×C₈₃	#575	1	2	yes	—
299	C₂²×C₈₃	C₂³×D₄	1	1	yes	characteristic
300	C₄×C₈₃	—	2	2	no	—
301	C₄×C₈₃	—	2	2	no	—
302	C₄×C₈₃	—	2	2	no	—
303	C₄×C₈₃	—	2	2	no	—
304	C₄×C₈₃	—	2	2	no	—
305	C₄×C₈₃	—	4	4	no	—
306	C₄×C₈₃	—	4	2	no	—
307	C₄×C₈₃	—	4	2	no	—
308	C₄×C₈₃	—	4	2	no	—
309	C₄×C₈₃	#535	1	2	yes	—
310	C₄×C₈₃	C₂×H(5,𝔽₂)	1	2	yes	—
311	C₂×#3211	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
312	C₂×C₈₃	—	2	4	no	—
313	C₂×C₈₃	—	8	2	no	—
314	C₂×C₈₃	—	8	4	no	—
315	C₂×C₈₃	#3211	1	4	yes	direct product factor
316	C₂×C₈₃	C₂×#535	1	1	yes	characteristic
317	C₂×C₈₃	C₂²×#135	1	1	yes	characteristic
318	C₂×C₈₃	C₂×#575	1	1	yes	characteristic
319	C₂×#529	C₂×C₈₃	1	1	yes	characteristic, complemented
320	C₂²×#124	C₂×C₈₃	1	1	yes	characteristic, complemented
321	#3211	C₂×C₈₃	1	16	yes	direct product factor
322	C₂×#495	C₂×C₈₃	1	2	yes	complemented
323	C₂×#488	C₂×C₈₃	1	2	yes	complemented
324	C₂×#464	C₂×C₈₃	1	2	yes	complemented
325	C₂×#463	C₂×C₈₃	1	2	yes	complemented
326	C₂×#451	C₂×C₈₃	1	1	yes	characteristic, complemented
327	C₂×#448	C₂×C₈₃	1	1	yes	characteristic, complemented
328	C₂×#450	C₂×C₈₃	1	1	yes	characteristic, complemented
329	C₂×#447	C₂×C₈₃	1	1	yes	characteristic
330	C₂×#431	C₂×C₈₃	1	1	yes	characteristic, complemented
331	C₈₃	C₂×#3211	1	1	yes	characteristic, direct product factor, Sylow
332	C₂³×Q₈	C₂²×C₈₃	1	1	yes	characteristic
333	C₂³×D₄	C₂²×C₈₃	1	1	yes	characteristic
334	C₂²×Q₁₆	C₂²×C₈₃	1	2	yes	complemented
335	C₂²×SD₁₆	C₂²×C₈₃	1	2	yes	—
336	C₂²×SD₁₆	C₂²×C₈₃	1	2	yes	complemented
337	C₂²×D₈	C₂²×C₈₃	1	2	yes	—
338	#529	—	2	4	no	—
339	#529	C₂²×C₈₃	1	8	yes	complemented
340	C₂×#126	—	2	1	no	—
341	C₂×#125	C₂²×C₈₃	1	1	yes	characteristic, complemented
342	C₂×#124	—	2	1	no	—
343	C₂×#124	—	2	1	no	—
344	C₂×#124	—	2	4	no	—
345	C₂×#124	C₂²×C₈₃	1	2	yes	—
346	C₂×#124	C₂²×C₈₃	1	2	yes	—
347	C₂×#124	C₂²×C₈₃	1	8	yes	complemented
348	C₂×#124	C₂²×C₈₃	1	1	yes	characteristic, complemented
349	C₂×#123	—	2	1	no	—
350	C₂×C₂²⋊Q₈	C₂²×C₈₃	1	1	yes	characteristic
351	C₂×#121	C₂²×C₈₃	1	1	yes	characteristic
352	C₂×C₄×Q₈	C₂²×C₈₃	1	1	yes	characteristic, complemented
353	C₂×C₄×D₄	—	2	1	no	—
354	C₂×C₄×D₄	C₂²×C₈₃	1	1	yes	characteristic, complemented
355	C₂×#117	—	2	1	no	—
356	C₂²×C₂²⋊C₄	—	2	1	no	—
357	C₂²×C₂²⋊C₄	—	2	1	no	—
358	C₂²×C₄²	C₂²×C₈₃	1	1	yes	characteristic
359	#495	C₂²×C₈₃	1	16	yes	complemented
360	#488	C₂²×C₈₃	1	16	yes	complemented
361	#464	C₂²×C₈₃	1	16	yes	complemented
362	#463	C₂²×C₈₃	1	16	yes	complemented
363	#451	C₂²×C₈₃	1	8	yes	complemented
364	#450	C₂²×C₈₃	1	8	yes	complemented
365	#448	C₂²×C₈₃	1	8	yes	complemented
366	#447	C₂²×C₈₃	1	8	yes	—
367	#431	C₂²×C₈₃	1	8	yes	complemented
368	C₂×C₄⋊C₈	C₂²×C₈₃	1	2	yes	complemented
369	C₂×#103	C₂²×C₈₃	1	2	yes	complemented
370	C₂×#103	C₂²×C₈₃	1	2	yes	complemented
371	C₂×#102	C₂²×C₈₃	1	2	yes	—
372	C₂×#102	C₂²×C₈₃	1	2	yes	—
373	C₂×C₂²⋊C₈	C₂²×C₈₃	1	1	yes	characteristic
374	C₂×C₂²⋊C₈	C₂²×C₈₃	1	1	yes	characteristic
375	C₂×C₄×C₈	C₂²×C₈₃	1	2	yes	complemented
376	C₂⁵	—	2	1	no	—
377	C₂²×Q₈	—	2	1	no	—
378	C₂²×Q₈	—	2	2	no	—
379	C₂²×Q₈	—	4	4	no	—
380	C₂²×Q₈	C₂³×C₈₃	1	1	yes	characteristic
381	C₂²×Q₈	C₂³×C₈₃	1	2	yes	—
382	C₂²×Q₈	C₂³×C₈₃	1	4	yes	—
383	C₂²×D₄	—	2	1	no	—
384	C₂²×D₄	—	2	1	no	—
385	C₂²×D₄	—	2	2	no	—
386	C₂²×D₄	—	4	4	no	—
387	C₂²×D₄	C₂³×C₈₃	1	1	yes	characteristic
388	C₂²×D₄	C₂³×C₈₃	1	2	yes	—
389	C₂²×D₄	C₂³×C₈₃	1	4	yes	—
390	C₂³×C₄	—	2	1	no	—
391	C₂³×C₄	—	2	1	no	—
392	C₂³×C₄	D₄×C₈₃	1	1	yes	characteristic
393	C₂³×C₄	D₄×C₈₃	1	1	yes	characteristic
394	C₂³×C₄	C₂³×C₈₃	1	1	yes	characteristic
395	C₂×Q₁₆	—	4	4	no	—
396	C₂×Q₁₆	C₂³×C₈₃	1	8	yes	complemented
397	C₂×SD₁₆	—	4	4	no	—
398	C₂×SD₁₆	—	4	4	no	—
399	C₂×SD₁₆	C₂³×C₈₃	1	8	yes	—
400	C₂×SD₁₆	C₂³×C₈₃	1	8	yes	complemented
401	C₂×D₈	—	4	4	no	—
402	C₂×D₈	C₂³×C₈₃	1	8	yes	—
403	C₂²×C₈	C₂³×C₈₃	1	2	yes	—
404	C₂²×C₈	C₂³×C₈₃	1	2	yes	—
405	#126	—	2	8	no	—
406	#125	—	2	2	no	—
407	#125	C₂³×C₈₃	1	4	yes	complemented
408	#124	—	2	4	no	—
409	#124	—	2	4	no	—
410	#124	—	2	8	no	—
411	#124	—	2	8	no	—
412	#124	—	2	2	no	—
413	#124	C₂³×C₈₃	1	8	yes	—
414	#124	C₂³×C₈₃	1	8	yes	—
415	#124	C₂³×C₈₃	1	4	yes	complemented
416	#123	—	2	8	no	—
417	C₂²⋊Q₈	—	2	2	no	—
418	C₂²⋊Q₈	C₂³×C₈₃	1	4	yes	—
419	#121	—	2	2	no	—
420	#121	C₂³×C₈₃	1	4	yes	—
421	C₄×Q₈	—	2	2	no	—
422	C₄×Q₈	C₂³×C₈₃	1	4	yes	complemented
423	C₄×D₄	—	2	8	no	—
424	C₄×D₄	—	2	2	no	—
425	C₄×D₄	C₂³×C₈₃	1	4	yes	complemented
426	#117	—	2	8	no	—
427	C₂×C₄⋊C₄	—	2	1	no	—
428	C₂×C₄⋊C₄	—	2	1	no	—
429	C₂×C₄⋊C₄	—	2	1	no	—
430	C₂×C₄⋊C₄	—	2	1	no	—
431	C₂×C₄⋊C₄	C₂³×C₈₃	1	1	yes	characteristic
432	C₂×C₄⋊C₄	C₂³×C₈₃	1	1	yes	characteristic
433	C₂×C₂²⋊C₄	—	2	1	no	—
434	C₂×C₂²⋊C₄	—	2	2	no	—
435	C₂×C₂²⋊C₄	—	2	2	no	—
436	C₂×C₂²⋊C₄	—	2	2	no	—
437	C₂×C₂²⋊C₄	—	2	2	no	—
438	C₂×C₂²⋊C₄	—	2	8	no	—
439	C₂×C₂²⋊C₄	—	2	8	no	—
440	C₂×C₂²⋊C₄	—	2	1	no	—
441	C₂×C₂²⋊C₄	—	2	1	no	—
442	C₂×C₂²⋊C₄	—	2	1	no	—
443	C₂×C₂²⋊C₄	—	2	1	no	—
444	C₂×C₄²	—	2	1	no	—
445	C₂×C₄²	D₄×C₈₃	1	1	yes	characteristic
446	C₂×C₄²	D₄×C₈₃	1	1	yes	characteristic
447	C₂×C₄²	D₄×C₈₃	1	4	yes	—
448	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
449	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
450	C₂×C₄²	C₂³×C₈₃	1	4	yes	—
451	C₄⋊C₈	C₂³×C₈₃	1	8	yes	complemented
452	#103	C₂³×C₈₃	1	8	yes	complemented
453	#103	C₂³×C₈₃	1	8	yes	complemented
454	#102	C₂³×C₈₃	1	8	yes	—
455	#102	C₂³×C₈₃	1	8	yes	—
456	C₂²⋊C₈	C₂³×C₈₃	1	4	yes	—
457	C₂²⋊C₈	C₂³×C₈₃	1	4	yes	—
458	C₄×C₈	C₂³×C₈₃	1	8	yes	complemented
459	C₂⁴	—	2	1	no	—
460	C₂⁴	—	2	4	no	—
461	C₂⁴	—	2	2	no	—
462	C₂⁴	—	4	4	no	—
463	C₂⁴	—	4	4	no	—
464	C₂⁴	—	4	4	no	—
465	C₂⁴	C₂×D₄×C₈₃	1	1	yes	characteristic
466	C₂×Q₈	—	2	1	no	—
467	C₂×Q₈	—	2	4	no	—
468	C₂×Q₈	—	2	2	no	—
469	C₂×Q₈	—	2	4	no	—
470	C₂×Q₈	—	2	4	no	—
471	C₂×Q₈	—	2	4	no	—
472	C₂×Q₈	—	4	8	no	—
473	C₂×Q₈	—	4	4	no	—
474	C₂×Q₈	—	4	8	no	—
475	C₂×Q₈	C₂⁴×C₈₃	1	4	yes	—
476	C₂×Q₈	C₂⁴×C₈₃	1	2	yes	—
477	C₂×D₄	—	2	4	no	—
478	C₂×D₄	—	2	1	no	—
479	C₂×D₄	—	2	4	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	2	4	no	—
482	C₂×D₄	—	2	4	no	—
483	C₂×D₄	—	2	4	no	—
484	C₂×D₄	—	4	8	no	—
485	C₂×D₄	—	4	4	no	—
486	C₂×D₄	—	4	8	no	—
487	C₂×D₄	—	4	4	no	—
488	C₂×D₄	C₂⁴×C₈₃	1	4	yes	—
489	C₂×D₄	C₂⁴×C₈₃	1	2	yes	—
490	C₂²×C₄	—	2	4	no	—
491	C₂²×C₄	—	2	4	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	4	no	—
495	C₂²×C₄	—	2	2	no	—
496	C₂²×C₄	—	2	1	no	—
497	C₂²×C₄	—	2	4	no	—
498	C₂²×C₄	—	4	2	no	—
499	C₂²×C₄	—	4	4	no	—
500	C₂²×C₄	—	4	4	no	—
501	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
502	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
503	C₂²×C₄	C₂×D₄×C₈₃	1	2	yes	—
504	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
505	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
506	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
507	C₂²×C₄	C₂×D₄×C₈₃	1	4	yes	—
508	C₂²×C₄	C₂×D₄×C₈₃	1	4	yes	—
509	C₂²×C₄	C₂⁴×C₈₃	1	2	yes	—
510	C₂²×C₄	C₂⁴×C₈₃	1	1	yes	characteristic
511	Q₁₆	—	4	8	no	—
512	SD₁₆	—	4	8	no	—
513	SD₁₆	—	4	8	no	—
514	D₈	—	4	8	no	—
515	C₂×C₈	—	2	4	no	—
516	C₂×C₈	—	2	4	no	—
517	C₂×C₈	C₂⁴×C₈₃	1	4	yes	—
518	C₂×C₈	C₂⁴×C₈₃	1	4	yes	—
519	C₄⋊C₄	—	2	4	no	—
520	C₄⋊C₄	—	2	4	no	—
521	C₄⋊C₄	—	2	4	no	—
522	C₄⋊C₄	—	2	4	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	C₂⁴×C₈₃	1	2	yes	—
526	C₄⋊C₄	C₂⁴×C₈₃	1	2	yes	—
527	C₂²⋊C₄	—	2	4	no	—
528	C₂²⋊C₄	—	2	8	no	—
529	C₂²⋊C₄	—	2	8	no	—
530	C₂²⋊C₄	—	2	8	no	—
531	C₂²⋊C₄	—	2	8	no	—
532	C₂²⋊C₄	—	2	4	no	—
533	C₂²⋊C₄	—	2	4	no	—
534	C₂²⋊C₄	—	2	4	no	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₄²	—	2	4	no	—
537	C₄²	C₂×D₄×C₈₃	1	4	yes	—
538	C₄²	C₂×D₄×C₈₃	1	2	yes	—
539	C₄²	C₂×D₄×C₈₃	1	2	yes	—
540	C₄²	C₂×D₄×C₈₃	1	4	yes	—
541	C₄²	C₂⁴×C₈₃	1	2	yes	—
542	C₄²	C₂⁴×C₈₃	1	2	yes	—
543	C₂³	—	2	2	no	—
544	C₂³	—	2	2	no	—
545	C₂³	—	2	2	no	—
546	C₂³	—	2	4	no	—
547	C₂³	—	4	8	no	—
548	C₂³	—	4	8	no	—
549	C₂³	—	4	4	no	—
550	C₂³	—	4	2	no	—
551	C₂³	—	4	2	no	—
552	C₂³	—	4	8	no	—
553	C₂³	—	4	4	no	—
554	C₂³	—	4	2	no	—
555	C₂³	—	4	4	no	—
556	C₂³	—	8	16	no	—
557	C₂³	#135×C₈₃	1	4	yes	—
558	C₂³	C₂²×D₄×C₈₃	1	2	yes	—
559	C₂³	C₂²×D₄×C₈₃	1	1	yes	characteristic
560	Q₈	—	2	4	no	—
561	Q₈	—	2	4	no	—
562	Q₈	—	2	4	no	—
563	Q₈	—	4	8	no	—
564	Q₈	—	4	4	no	—
565	Q₈	—	4	2	no	—
566	D₄	—	2	4	no	—
567	D₄	—	2	4	no	—
568	D₄	—	2	4	no	—
569	D₄	—	4	8	no	—
570	D₄	—	4	8	no	—
571	D₄	—	4	4	no	—
572	D₄	—	4	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	8	no	—
575	C₂×C₄	—	2	2	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	4	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	4	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	4	4	no	—
586	C₂×C₄	—	4	8	no	—
587	C₂×C₄	—	4	8	no	—
588	C₂×C₄	—	4	4	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	2	no	—
591	C₂×C₄	—	4	2	no	—
592	C₂×C₄	C₂²×D₄×C₈₃	1	2	yes	—
593	C₂×C₄	C₂²×D₄×C₈₃	1	2	yes	—
594	C₂×C₄	C₂²×D₄×C₈₃	1	2	yes	—
595	C₂×C₄	C₂²×D₄×C₈₃	1	2	yes	—
596	C₂×C₄	C₂²×D₄×C₈₃	1	2	yes	—
597	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
598	C₂×C₄	H(5,𝔽₂)×C₈₃	1	4	yes	—
599	C₂×C₄	C₂⁵×C₈₃	1	1	yes	characteristic
600	C₈	—	2	4	no	—
601	C₈	—	2	4	no	—
602	C₂²	—	2	2	no	—
603	C₂²	—	2	2	no	—
604	C₂²	—	2	8	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	4	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	4	no	—
609	C₂²	—	4	4	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	8	16	no	—
612	C₂²	—	8	8	no	—
613	C₂²	—	8	4	no	—
614	C₂²	#535×C₈₃	1	2	yes	—
615	C₂²	C₂×#135×C₈₃	1	4	yes	—
616	C₂²	C₂×#135×C₈₃	1	2	yes	—
617	C₂²	#575×C₈₃	1	2	yes	—
618	C₂²	C₂³×D₄×C₈₃	1	1	yes	characteristic
619	C₄	—	2	2	no	—
620	C₄	—	2	2	no	—
621	C₄	—	2	2	no	—
622	C₄	—	2	2	no	—
623	C₄	—	2	2	no	—
624	C₄	—	4	2	no	—
625	C₄	—	4	2	no	—
626	C₄	—	4	4	no	—
627	C₄	—	4	2	no	—
628	C₄	#535×C₈₃	1	2	yes	—
629	C₄	C₂×H(5,𝔽₂)×C₈₃	1	2	yes	—
630	C₂	—	2	4	no	—
631	C₂	—	8	4	no	—
632	C₂	—	8	2	no	—
633	C₂	C₈₃×#3211	1	4	yes	direct product factor
634	C₂	C₂×#535×C₈₃	1	1	yes	characteristic
635	C₂	C₂²×#135×C₈₃	1	1	yes	characteristic
636	C₂	C₂×#575×C₈₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₃
prime radical	C₂×#447×C₈₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₃ ◁ C₂²×C₄²×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#529×C₈₃	C₂	semidirect
2. C₂²×#124×C₈₃	C₂	semidirect
3. C₈₃×#3211	C₂	direct (N × H)
4. C₂×#495×C₈₃	C₂	semidirect
5. C₂×#488×C₈₃	C₂	semidirect
6. C₂×#464×C₈₃	C₂	semidirect
7. C₂×#463×C₈₃	C₂	semidirect
8. C₂×#451×C₈₃	C₂	semidirect
9. C₂×#448×C₈₃	C₂	semidirect
10. C₂×#450×C₈₃	C₂	semidirect
12. C₂×#431×C₈₃	C₂	semidirect
15. C₂²×Q₁₆×C₈₃	C₂²	semidirect
17. C₂²×SD₁₆×C₈₃	C₂²	semidirect
20. #529×C₈₃	C₂²	semidirect
22. C₂×#125×C₈₃	C₂²	semidirect
28. C₂×#124×C₈₃	C₂²	semidirect
29. C₂×#124×C₈₃	C₂²	semidirect
33. C₂×C₄×Q₈×C₈₃	C₂²	semidirect
35. C₂×C₄×D₄×C₈₃	C₂²	semidirect
40. #495×C₈₃	C₂²	semidirect
41. #488×C₈₃	C₂²	semidirect
42. #464×C₈₃	C₂²	semidirect
43. #463×C₈₃	C₂²	semidirect
44. #451×C₈₃	C₂²	semidirect
45. #450×C₈₃	C₂²	semidirect
46. #448×C₈₃	C₂²	semidirect
48. #431×C₈₃	C₂²	semidirect
49. C₂×C₄⋊C₈×C₈₃	C₂²	semidirect
50. C₂×#103×C₈₃	C₂²	semidirect
51. C₂×#103×C₈₃	C₂²	semidirect
56. C₂×C₄×C₈×C₈₃	C₂²	semidirect
77. C₂×Q₁₆×C₈₃	C₂³	semidirect
81. C₂×SD₁₆×C₈₃	C₂³	semidirect
88. #125×C₈₃	C₂³	semidirect
96. #124×C₈₃	C₂³	semidirect
103. C₄×Q₈×C₈₃	C₂³	semidirect
106. C₄×D₄×C₈₃	C₂³	semidirect
132. C₄⋊C₈×C₈₃	C₂³	semidirect
133. #103×C₈₃	C₂³	semidirect
134. #103×C₈₃	C₂³	semidirect
139. C₄×C₈×C₈₃	C₂³	semidirect
311. C₂×#3211	C₈₃	direct (N × H)
315. C₂×C₈₃	#3211	direct (N × H)
319. C₂×#529	C₂×C₈₃	semidirect
320. C₂²×#124	C₂×C₈₃	semidirect
321. #3211	C₂×C₈₃	direct (N × H)
322. C₂×#495	C₂×C₈₃	semidirect
323. C₂×#488	C₂×C₈₃	semidirect
324. C₂×#464	C₂×C₈₃	semidirect
325. C₂×#463	C₂×C₈₃	semidirect
326. C₂×#451	C₂×C₈₃	semidirect
327. C₂×#448	C₂×C₈₃	semidirect
328. C₂×#450	C₂×C₈₃	semidirect
330. C₂×#431	C₂×C₈₃	semidirect
331. C₈₃	C₂×#3211	direct (N × H)
334. C₂²×Q₁₆	C₂²×C₈₃	semidirect
336. C₂²×SD₁₆	C₂²×C₈₃	semidirect
339. #529	C₂²×C₈₃	semidirect
341. C₂×#125	C₂²×C₈₃	semidirect
347. C₂×#124	C₂²×C₈₃	semidirect
348. C₂×#124	C₂²×C₈₃	semidirect
352. C₂×C₄×Q₈	C₂²×C₈₃	semidirect
354. C₂×C₄×D₄	C₂²×C₈₃	semidirect
359. #495	C₂²×C₈₃	semidirect
360. #488	C₂²×C₈₃	semidirect
361. #464	C₂²×C₈₃	semidirect
362. #463	C₂²×C₈₃	semidirect
363. #451	C₂²×C₈₃	semidirect
364. #450	C₂²×C₈₃	semidirect
365. #448	C₂²×C₈₃	semidirect
367. #431	C₂²×C₈₃	semidirect
368. C₂×C₄⋊C₈	C₂²×C₈₃	semidirect
369. C₂×#103	C₂²×C₈₃	semidirect
370. C₂×#103	C₂²×C₈₃	semidirect
375. C₂×C₄×C₈	C₂²×C₈₃	semidirect
396. C₂×Q₁₆	C₂³×C₈₃	semidirect
400. C₂×SD₁₆	C₂³×C₈₃	semidirect
407. #125	C₂³×C₈₃	semidirect
415. #124	C₂³×C₈₃	semidirect
422. C₄×Q₈	C₂³×C₈₃	semidirect
425. C₄×D₄	C₂³×C₈₃	semidirect
451. C₄⋊C₈	C₂³×C₈₃	semidirect
452. #103	C₂³×C₈₃	semidirect
453. #103	C₂³×C₈₃	semidirect
458. C₄×C₈	C₂³×C₈₃	semidirect
633. C₂	C₈₃×#3211	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#529×C₈₃	2
C₂²×#124×C₈₃	2
C₈₃×#3211	2
C₂×#495×C₈₃	2
C₂×#488×C₈₃	2
C₂×#464×C₈₃	2
C₂×#463×C₈₃	2
C₂×#451×C₈₃	2
C₂×#448×C₈₃	2
C₂×#450×C₈₃	2
C₂×#447×C₈₃	2
C₂×#431×C₈₃	2
C₂×#3211	83

Subgroups — C2×C83×#3211 #35288600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₃×#3211 #35288600