Subgroups — C₂×C₈₃×#3246 #35288719

Order 21248.

2178 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#532×C₈₃	C₂	1	1	yes	maximal, characteristic
3	C₈₃×#3246	C₂	1	16	yes	maximal, direct product factor
4	C₂×#475×C₈₃	C₂	1	2	yes	maximal, complemented
5	C₂×#476×C₈₃	C₂	1	2	yes	maximal, complemented
6	C₂×#465×C₈₃	C₂	1	4	yes	maximal, complemented
7	C₂×#460×C₈₃	C₂	1	2	yes	maximal, complemented
8	C₂×#461×C₈₃	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄⋊C₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₈₃	—	2	1	no	—
11	C₂²×SD₁₆×C₈₃	C₂²	1	4	yes	—
12	C₂³×C₈×C₈₃	C₂²	1	2	yes	—
13	#533×C₈₃	—	2	4	no	—
14	#533×C₈₃	C₂²	1	8	yes	complemented
15	#532×C₈₃	—	2	4	no	—
16	#532×C₈₃	C₂²	1	8	yes	—
17	C₂×C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×#127×C₈₃	C₂²	1	1	yes	characteristic
21	C₂×#125×C₈₃	—	2	1	no	—
22	C₂×#124×C₈₃	—	2	1	no	—
23	C₂×C₂²⋊Q₈×C₈₃	—	2	1	no	—
24	C₂×C₂²⋊Q₈×C₈₃	C₂²	1	2	yes	—
25	C₂×#121×C₈₃	—	2	1	no	—
26	C₂×#121×C₈₃	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₈₃	—	2	1	no	—
28	C₂×C₄×Q₈×C₈₃	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₈₃	—	2	1	no	—
30	C₂×C₄×D₄×C₈₃	C₂²	1	2	yes	—
31	C₂×#117×C₈₃	—	2	1	no	—
32	C₂²×C₄²×C₈₃	C₂²	1	1	yes	characteristic
33	#476×C₈₃	C₂²	1	16	yes	complemented
34	#475×C₈₃	C₂²	1	16	yes	complemented
35	#465×C₈₃	C₂²	1	32	yes	complemented
36	#461×C₈₃	C₂²	1	16	yes	complemented
37	#460×C₈₃	C₂²	1	16	yes	complemented
38	C₂×#106×C₈₃	C₂²	1	4	yes	—
39	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
40	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
41	C₂×C₄⋊C₈×C₈₃	C₂²	1	8	yes	complemented
42	C₂×#103×C₈₃	C₂²	1	4	yes	—
43	C₂×#102×C₈₃	C₂²	1	4	yes	—
44	C₂×#41×C₈₃	—	2	8	no	—
45	C₂×#41×C₈₃	—	4	8	no	—
46	C₂²×Q₈×C₈₃	—	2	1	no	—
47	C₂²×Q₈×C₈₃	—	2	1	no	—
48	C₂²×Q₈×C₈₃	C₂³	1	2	yes	—
49	C₂²×D₄×C₈₃	—	2	2	no	—
50	C₂²×D₄×C₈₃	—	2	1	no	—
51	C₂²×D₄×C₈₃	—	2	1	no	—
52	C₂²×D₄×C₈₃	C₂³	1	2	yes	—
53	C₂³×C₄×C₈₃	—	2	2	no	—
54	C₂³×C₄×C₈₃	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₈₃	D₄	1	1	yes	characteristic
56	C₂³×C₄×C₈₃	C₂³	1	1	yes	characteristic
57	C₂×SD₁₆×C₈₃	—	4	8	no	—
58	C₂×SD₁₆×C₈₃	C₂³	1	16	yes	—
59	C₂²×C₈×C₈₃	—	4	4	no	—
60	C₂²×C₈×C₈₃	C₂³	1	4	yes	—
61	C₂²×C₈×C₈₃	C₂³	1	8	yes	—
62	C₄⋊Q₈×C₈₃	—	2	2	no	—
63	C₄⋊Q₈×C₈₃	—	2	2	no	—
64	C₄⋊Q₈×C₈₃	—	2	2	no	—
65	C₄⋊Q₈×C₈₃	C₂³	1	4	yes	—
66	C₄⋊Q₈×C₈₃	C₂³	1	4	yes	—
67	C₄⋊Q₈×C₈₃	C₂³	1	4	yes	—
68	#127×C₈₃	—	2	2	no	—
69	#127×C₈₃	C₂³	1	4	yes	—
70	#125×C₈₃	—	2	8	no	—
71	#124×C₈₃	—	2	8	no	—
72	C₂²⋊Q₈×C₈₃	—	2	8	no	—
73	C₂²⋊Q₈×C₈₃	—	2	4	no	—
74	C₂²⋊Q₈×C₈₃	C₂³	1	8	yes	—
75	#121×C₈₃	—	2	8	no	—
76	#121×C₈₃	—	2	4	no	—
77	#121×C₈₃	C₂³	1	8	yes	—
78	C₄×Q₈×C₈₃	—	2	8	no	—
79	C₄×Q₈×C₈₃	—	2	4	no	—
80	C₄×Q₈×C₈₃	C₂³	1	8	yes	—
81	C₄×D₄×C₈₃	—	2	8	no	—
82	C₄×D₄×C₈₃	—	2	4	no	—
83	C₄×D₄×C₈₃	C₂³	1	8	yes	—
84	#117×C₈₃	—	2	8	no	—
85	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
89	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
90	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
91	C₂×C₄⋊C₄×C₈₃	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₈₃	C₂³	1	2	yes	—
93	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
96	C₂×C₄²×C₈₃	—	2	2	no	—
97	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₈₃	D₄	1	4	yes	—
100	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₈₃	C₂³	1	4	yes	—
103	#106×C₈₃	C₂³	1	16	yes	—
104	C₄⋊C₈×C₈₃	C₂³	1	8	yes	—
105	C₄⋊C₈×C₈₃	C₂³	1	8	yes	—
106	#103×C₈₃	C₂³	1	16	yes	—
107	#102×C₈₃	C₂³	1	16	yes	—
108	C₂⁴×C₈₃	—	2	2	no	—
109	C₂⁴×C₈₃	C₂×D₄	1	1	yes	characteristic
110	#41×C₈₃	—	4	32	no	—
111	C₂×Q₈×C₈₃	—	2	2	no	—
112	C₂×Q₈×C₈₃	—	2	4	no	—
113	C₂×Q₈×C₈₃	—	2	4	no	—
114	C₂×Q₈×C₈₃	—	2	4	no	—
115	C₂×Q₈×C₈₃	—	2	4	no	—
116	C₂×Q₈×C₈₃	—	4	4	no	—
117	C₂×Q₈×C₈₃	—	4	4	no	—
118	C₂×Q₈×C₈₃	C₂⁴	1	4	yes	—
119	C₂×D₄×C₈₃	—	2	8	no	—
120	C₂×D₄×C₈₃	—	2	2	no	—
121	C₂×D₄×C₈₃	—	2	4	no	—
122	C₂×D₄×C₈₃	—	2	4	no	—
123	C₂×D₄×C₈₃	—	2	4	no	—
124	C₂×D₄×C₈₃	—	2	4	no	—
125	C₂×D₄×C₈₃	—	4	4	no	—
126	C₂×D₄×C₈₃	—	4	8	no	—
127	C₂×D₄×C₈₃	—	4	4	no	—
128	C₂×D₄×C₈₃	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₈₃	—	2	2	no	—
130	C₂²×C₄×C₈₃	—	2	2	no	—
131	C₂²×C₄×C₈₃	—	2	2	no	—
132	C₂²×C₄×C₈₃	—	2	2	no	—
133	C₂²×C₄×C₈₃	—	2	2	no	—
134	C₂²×C₄×C₈₃	—	2	8	no	—
135	C₂²×C₄×C₈₃	—	4	2	no	—
136	C₂²×C₄×C₈₃	—	4	8	no	—
137	C₂²×C₄×C₈₃	SD₁₆	1	8	yes	complemented
138	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
139	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₈₃	C₂×D₄	1	4	yes	—
143	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₈₃	C₂×D₄	1	4	yes	—
146	C₂²×C₄×C₈₃	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
148	SD₁₆×C₈₃	—	4	16	no	—
149	C₂×C₈×C₈₃	—	2	8	no	—
150	C₂×C₈×C₈₃	—	4	8	no	—
151	C₂×C₈×C₈₃	C₂⁴	1	8	yes	—
152	C₄⋊C₄×C₈₃	—	2	4	no	—
153	C₄⋊C₄×C₈₃	—	2	4	no	—
154	C₄⋊C₄×C₈₃	—	2	4	no	—
155	C₄⋊C₄×C₈₃	—	2	4	no	—
156	C₄⋊C₄×C₈₃	—	2	8	no	—
157	C₄⋊C₄×C₈₃	—	2	8	no	—
158	C₄⋊C₄×C₈₃	—	2	2	no	—
159	C₄⋊C₄×C₈₃	—	2	2	no	—
160	C₄⋊C₄×C₈₃	C₂⁴	1	4	yes	—
161	C₄⋊C₄×C₈₃	C₂⁴	1	4	yes	—
162	C₂²⋊C₄×C₈₃	—	2	8	no	—
163	C₂²⋊C₄×C₈₃	—	2	8	no	—
164	C₂²⋊C₄×C₈₃	—	2	8	no	—
165	C₄²×C₈₃	—	2	8	no	—
166	C₄²×C₈₃	C₂×D₄	1	4	yes	—
167	C₄²×C₈₃	C₂×D₄	1	2	yes	—
168	C₄²×C₈₃	C₂×D₄	1	4	yes	—
169	C₄²×C₈₃	C₂×D₄	1	2	yes	—
170	C₄²×C₈₃	C₂⁴	1	2	yes	—
171	C₄²×C₈₃	C₂⁴	1	2	yes	—
172	C₂³×C₈₃	—	2	2	no	—
173	C₂³×C₈₃	—	2	2	no	—
174	C₂³×C₈₃	—	2	4	no	—
175	C₂³×C₈₃	—	4	4	no	—
176	C₂³×C₈₃	—	4	4	no	—
177	C₂³×C₈₃	—	4	4	no	—
178	C₂³×C₈₃	C₂×SD₁₆	1	4	yes	—
179	C₂³×C₈₃	C₂²×D₄	1	2	yes	—
180	C₂³×C₈₃	C₂²×D₄	1	1	yes	characteristic
181	Q₈×C₈₃	—	2	4	no	—
182	Q₈×C₈₃	—	2	4	no	—
183	Q₈×C₈₃	—	4	8	no	—
184	Q₈×C₈₃	—	4	4	no	—
185	Q₈×C₈₃	—	4	8	no	—
186	D₄×C₈₃	—	2	4	no	—
187	D₄×C₈₃	—	2	4	no	—
188	D₄×C₈₃	—	4	8	no	—
189	D₄×C₈₃	—	4	16	no	—
190	D₄×C₈₃	—	4	4	no	—
191	D₄×C₈₃	—	4	8	no	—
192	C₂×C₄×C₈₃	—	2	2	no	—
193	C₂×C₄×C₈₃	—	2	2	no	—
194	C₂×C₄×C₈₃	—	2	2	no	—
195	C₂×C₄×C₈₃	—	2	4	no	—
196	C₂×C₄×C₈₃	—	2	2	no	—
197	C₂×C₄×C₈₃	—	2	2	no	—
198	C₂×C₄×C₈₃	—	2	4	no	—
199	C₂×C₄×C₈₃	—	2	4	no	—
200	C₂×C₄×C₈₃	—	2	4	no	—
201	C₂×C₄×C₈₃	—	4	4	no	—
202	C₂×C₄×C₈₃	—	4	4	no	—
203	C₂×C₄×C₈₃	—	4	4	no	—
204	C₂×C₄×C₈₃	—	4	4	no	—
205	C₂×C₄×C₈₃	—	4	16	no	—
206	C₂×C₄×C₈₃	C₂×SD₁₆	1	4	yes	—
207	C₂×C₄×C₈₃	C₂×SD₁₆	1	4	yes	—
208	C₂×C₄×C₈₃	C₂×SD₁₆	1	16	yes	complemented
209	C₂×C₄×C₈₃	C₂²×D₄	1	2	yes	—
210	C₂×C₄×C₈₃	C₂²×D₄	1	2	yes	—
211	C₂×C₄×C₈₃	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₈₃	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₈₃	C₂²×D₄	1	2	yes	—
215	C₂×C₄×C₈₃	H(5,𝔽₂)	1	2	yes	—
216	C₂×C₄×C₈₃	2₋¹⁺⁴	1	2	yes	—
217	C₂×C₄×C₈₃	C₂⁵	1	1	yes	characteristic
218	C₈×C₈₃	—	2	8	no	—
219	C₂²×C₈₃	—	2	2	no	—
220	C₂²×C₈₃	—	2	8	no	—
221	C₂²×C₈₃	—	2	2	no	—
222	C₂²×C₈₃	—	4	4	no	—
223	C₂²×C₈₃	—	4	4	no	—
224	C₂²×C₈₃	—	4	4	no	—
225	C₂²×C₈₃	—	8	8	no	—
226	C₂²×C₈₃	#537	1	2	yes	—
227	C₂²×C₈₃	C₂²×SD₁₆	1	4	yes	—
228	C₂²×C₈₃	C₂²×SD₁₆	1	2	yes	—
229	C₂²×C₈₃	#575	1	2	yes	—
230	C₂²×C₈₃	C₂³×D₄	1	1	yes	characteristic
231	C₄×C₈₃	—	2	2	no	—
232	C₄×C₈₃	—	2	2	no	—
233	C₄×C₈₃	—	2	2	no	—
234	C₄×C₈₃	—	4	4	no	—
235	C₄×C₈₃	—	4	4	no	—
236	C₄×C₈₃	—	4	4	no	—
237	C₄×C₈₃	#537	1	2	yes	—
238	C₄×C₈₃	C₂²×SD₁₆	1	4	yes	—
239	C₄×C₈₃	C₂²×SD₁₆	1	4	yes	—
240	C₄×C₈₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
241	C₄×C₈₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
242	C₂×#3246	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
243	C₂×C₈₃	—	2	4	no	—
244	C₂×C₈₃	—	8	4	no	—
245	C₂×C₈₃	#3246	1	4	yes	direct product factor
246	C₂×C₈₃	C₂×#537	1	1	yes	characteristic
247	C₂×C₈₃	C₂³×SD₁₆	1	1	yes	characteristic
248	C₂×C₈₃	C₂×#575	1	1	yes	characteristic
249	C₂×#533	C₂×C₈₃	1	1	yes	characteristic, complemented
250	C₂×#532	C₂×C₈₃	1	1	yes	characteristic
251	#3246	C₂×C₈₃	1	16	yes	direct product factor
252	C₂×#475	C₂×C₈₃	1	2	yes	complemented
253	C₂×#476	C₂×C₈₃	1	2	yes	complemented
254	C₂×#465	C₂×C₈₃	1	4	yes	complemented
255	C₂×#460	C₂×C₈₃	1	2	yes	complemented
256	C₂×#461	C₂×C₈₃	1	2	yes	complemented
257	C₂²×C₄⋊C₈	C₂×C₈₃	1	1	yes	characteristic, complemented
258	C₈₃	C₂×#3246	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²×#41	—	2	1	no	—
260	C₂²×SD₁₆	C₂²×C₈₃	1	4	yes	—
261	C₂³×C₈	C₂²×C₈₃	1	2	yes	—
262	#533	—	2	4	no	—
263	#533	C₂²×C₈₃	1	8	yes	complemented
264	#532	—	2	4	no	—
265	#532	C₂²×C₈₃	1	8	yes	—
266	C₂×C₄⋊Q₈	C₂²×C₈₃	1	1	yes	characteristic
267	C₂×C₄⋊Q₈	C₂²×C₈₃	1	1	yes	characteristic
268	C₂×C₄⋊Q₈	C₂²×C₈₃	1	1	yes	characteristic
269	C₂×#127	C₂²×C₈₃	1	1	yes	characteristic
270	C₂×#125	—	2	1	no	—
271	C₂×#124	—	2	1	no	—
272	C₂×C₂²⋊Q₈	—	2	1	no	—
273	C₂×C₂²⋊Q₈	C₂²×C₈₃	1	2	yes	—
274	C₂×#121	—	2	1	no	—
275	C₂×#121	C₂²×C₈₃	1	2	yes	—
276	C₂×C₄×Q₈	—	2	1	no	—
277	C₂×C₄×Q₈	C₂²×C₈₃	1	2	yes	—
278	C₂×C₄×D₄	—	2	1	no	—
279	C₂×C₄×D₄	C₂²×C₈₃	1	2	yes	—
280	C₂×#117	—	2	1	no	—
281	C₂²×C₄²	C₂²×C₈₃	1	1	yes	characteristic
282	#476	C₂²×C₈₃	1	16	yes	complemented
283	#475	C₂²×C₈₃	1	16	yes	complemented
284	#465	C₂²×C₈₃	1	32	yes	complemented
285	#461	C₂²×C₈₃	1	16	yes	complemented
286	#460	C₂²×C₈₃	1	16	yes	complemented
287	C₂×#106	C₂²×C₈₃	1	4	yes	—
288	C₂×C₄⋊C₈	C₂²×C₈₃	1	2	yes	—
289	C₂×C₄⋊C₈	C₂²×C₈₃	1	2	yes	—
290	C₂×C₄⋊C₈	C₂²×C₈₃	1	8	yes	complemented
291	C₂×#103	C₂²×C₈₃	1	4	yes	—
292	C₂×#102	C₂²×C₈₃	1	4	yes	—
293	C₂×#41	—	2	8	no	—
294	C₂×#41	—	4	8	no	—
295	C₂²×Q₈	—	2	1	no	—
296	C₂²×Q₈	—	2	1	no	—
297	C₂²×Q₈	C₂³×C₈₃	1	2	yes	—
298	C₂²×D₄	—	2	2	no	—
299	C₂²×D₄	—	2	1	no	—
300	C₂²×D₄	—	2	1	no	—
301	C₂²×D₄	C₂³×C₈₃	1	2	yes	—
302	C₂³×C₄	—	2	2	no	—
303	C₂³×C₄	D₄×C₈₃	1	1	yes	characteristic
304	C₂³×C₄	D₄×C₈₃	1	1	yes	characteristic
305	C₂³×C₄	C₂³×C₈₃	1	1	yes	characteristic
306	C₂×SD₁₆	—	4	8	no	—
307	C₂×SD₁₆	C₂³×C₈₃	1	16	yes	—
308	C₂²×C₈	—	4	4	no	—
309	C₂²×C₈	C₂³×C₈₃	1	4	yes	—
310	C₂²×C₈	C₂³×C₈₃	1	8	yes	—
311	C₄⋊Q₈	—	2	2	no	—
312	C₄⋊Q₈	—	2	2	no	—
313	C₄⋊Q₈	—	2	2	no	—
314	C₄⋊Q₈	C₂³×C₈₃	1	4	yes	—
315	C₄⋊Q₈	C₂³×C₈₃	1	4	yes	—
316	C₄⋊Q₈	C₂³×C₈₃	1	4	yes	—
317	#127	—	2	2	no	—
318	#127	C₂³×C₈₃	1	4	yes	—
319	#125	—	2	8	no	—
320	#124	—	2	8	no	—
321	C₂²⋊Q₈	—	2	8	no	—
322	C₂²⋊Q₈	—	2	4	no	—
323	C₂²⋊Q₈	C₂³×C₈₃	1	8	yes	—
324	#121	—	2	8	no	—
325	#121	—	2	4	no	—
326	#121	C₂³×C₈₃	1	8	yes	—
327	C₄×Q₈	—	2	8	no	—
328	C₄×Q₈	—	2	4	no	—
329	C₄×Q₈	C₂³×C₈₃	1	8	yes	—
330	C₄×D₄	—	2	8	no	—
331	C₄×D₄	—	2	4	no	—
332	C₄×D₄	C₂³×C₈₃	1	8	yes	—
333	#117	—	2	8	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	—	2	1	no	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	2	no	—
339	C₂×C₄⋊C₄	—	2	2	no	—
340	C₂×C₄⋊C₄	C₂³×C₈₃	1	2	yes	—
341	C₂×C₄⋊C₄	C₂³×C₈₃	1	2	yes	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₂²⋊C₄	—	2	2	no	—
344	C₂×C₂²⋊C₄	—	2	2	no	—
345	C₂×C₄²	—	2	2	no	—
346	C₂×C₄²	D₄×C₈₃	1	1	yes	characteristic
347	C₂×C₄²	D₄×C₈₃	1	1	yes	characteristic
348	C₂×C₄²	D₄×C₈₃	1	4	yes	—
349	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
350	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
351	C₂×C₄²	C₂³×C₈₃	1	4	yes	—
352	#106	C₂³×C₈₃	1	16	yes	—
353	C₄⋊C₈	C₂³×C₈₃	1	8	yes	—
354	C₄⋊C₈	C₂³×C₈₃	1	8	yes	—
355	#103	C₂³×C₈₃	1	16	yes	—
356	#102	C₂³×C₈₃	1	16	yes	—
357	C₂⁴	—	2	2	no	—
358	C₂⁴	C₂×D₄×C₈₃	1	1	yes	characteristic
359	#41	—	4	32	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	—	2	4	no	—
362	C₂×Q₈	—	2	4	no	—
363	C₂×Q₈	—	2	4	no	—
364	C₂×Q₈	—	2	4	no	—
365	C₂×Q₈	—	4	4	no	—
366	C₂×Q₈	—	4	4	no	—
367	C₂×Q₈	C₂⁴×C₈₃	1	4	yes	—
368	C₂×D₄	—	2	8	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	2	4	no	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	2	4	no	—
374	C₂×D₄	—	4	4	no	—
375	C₂×D₄	—	4	8	no	—
376	C₂×D₄	—	4	4	no	—
377	C₂×D₄	C₂⁴×C₈₃	1	4	yes	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	8	no	—
384	C₂²×C₄	—	4	2	no	—
385	C₂²×C₄	—	4	8	no	—
386	C₂²×C₄	SD₁₆×C₈₃	1	8	yes	complemented
387	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₈₃	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
390	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₈₃	1	4	yes	—
392	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₈₃	1	4	yes	—
395	C₂²×C₄	C₂⁴×C₈₃	1	2	yes	—
396	C₂²×C₄	C₂⁴×C₈₃	1	1	yes	characteristic
397	SD₁₆	—	4	16	no	—
398	C₂×C₈	—	2	8	no	—
399	C₂×C₈	—	4	8	no	—
400	C₂×C₈	C₂⁴×C₈₃	1	8	yes	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	4	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	8	no	—
406	C₄⋊C₄	—	2	8	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	C₂⁴×C₈₃	1	4	yes	—
410	C₄⋊C₄	C₂⁴×C₈₃	1	4	yes	—
411	C₂²⋊C₄	—	2	8	no	—
412	C₂²⋊C₄	—	2	8	no	—
413	C₂²⋊C₄	—	2	8	no	—
414	C₄²	—	2	8	no	—
415	C₄²	C₂×D₄×C₈₃	1	4	yes	—
416	C₄²	C₂×D₄×C₈₃	1	2	yes	—
417	C₄²	C₂×D₄×C₈₃	1	4	yes	—
418	C₄²	C₂×D₄×C₈₃	1	2	yes	—
419	C₄²	C₂⁴×C₈₃	1	2	yes	—
420	C₄²	C₂⁴×C₈₃	1	2	yes	—
421	C₂³	—	2	2	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	4	no	—
424	C₂³	—	4	4	no	—
425	C₂³	—	4	4	no	—
426	C₂³	—	4	4	no	—
427	C₂³	C₂×SD₁₆×C₈₃	1	4	yes	—
428	C₂³	C₂²×D₄×C₈₃	1	2	yes	—
429	C₂³	C₂²×D₄×C₈₃	1	1	yes	characteristic
430	Q₈	—	2	4	no	—
431	Q₈	—	2	4	no	—
432	Q₈	—	4	8	no	—
433	Q₈	—	4	4	no	—
434	Q₈	—	4	8	no	—
435	D₄	—	2	4	no	—
436	D₄	—	2	4	no	—
437	D₄	—	4	8	no	—
438	D₄	—	4	16	no	—
439	D₄	—	4	4	no	—
440	D₄	—	4	8	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	4	4	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	—	4	4	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	—	4	16	no	—
455	C₂×C₄	C₂×SD₁₆×C₈₃	1	4	yes	—
456	C₂×C₄	C₂×SD₁₆×C₈₃	1	4	yes	—
457	C₂×C₄	C₂×SD₁₆×C₈₃	1	16	yes	complemented
458	C₂×C₄	C₂²×D₄×C₈₃	1	2	yes	—
459	C₂×C₄	C₂²×D₄×C₈₃	1	2	yes	—
460	C₂×C₄	C₂²×D₄×C₈₃	1	2	yes	—
461	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
462	C₂×C₄	C₂²×D₄×C₈₃	1	2	yes	—
463	C₂×C₄	C₂²×D₄×C₈₃	1	2	yes	—
464	C₂×C₄	H(5,𝔽₂)×C₈₃	1	2	yes	—
465	C₂×C₄	2₋¹⁺⁴×C₈₃	1	2	yes	—
466	C₂×C₄	C₂⁵×C₈₃	1	1	yes	characteristic
467	C₈	—	2	8	no	—
468	C₂²	—	2	2	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	8	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	4	4	no	—
473	C₂²	—	4	4	no	—
474	C₂²	—	8	8	no	—
475	C₂²	#537×C₈₃	1	2	yes	—
476	C₂²	C₂²×SD₁₆×C₈₃	1	4	yes	—
477	C₂²	C₂²×SD₁₆×C₈₃	1	2	yes	—
478	C₂²	#575×C₈₃	1	2	yes	—
479	C₂²	C₂³×D₄×C₈₃	1	1	yes	characteristic
480	C₄	—	2	2	no	—
481	C₄	—	2	2	no	—
482	C₄	—	2	2	no	—
483	C₄	—	4	4	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	4	no	—
486	C₄	#537×C₈₃	1	2	yes	—
487	C₄	C₂²×SD₁₆×C₈₃	1	4	yes	—
488	C₄	C₂²×SD₁₆×C₈₃	1	4	yes	—
489	C₄	C₂×H(5,𝔽₂)×C₈₃	1	1	yes	characteristic
490	C₄	C₂×2₋¹⁺⁴×C₈₃	1	1	yes	characteristic
491	C₂	—	2	4	no	—
492	C₂	—	8	4	no	—
493	C₂	C₈₃×#3246	1	4	yes	direct product factor
494	C₂	C₂×#537×C₈₃	1	1	yes	characteristic
495	C₂	C₂³×SD₁₆×C₈₃	1	1	yes	characteristic
496	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×#532×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×#533×C₈₃	C₂	semidirect
3. C₈₃×#3246	C₂	direct (N × H)
4. C₂×#475×C₈₃	C₂	semidirect
5. C₂×#476×C₈₃	C₂	semidirect
6. C₂×#465×C₈₃	C₂	semidirect
7. C₂×#460×C₈₃	C₂	semidirect
8. C₂×#461×C₈₃	C₂	semidirect
9. C₂²×C₄⋊C₈×C₈₃	C₂	semidirect
14. #533×C₈₃	C₂²	semidirect
33. #476×C₈₃	C₂²	semidirect
34. #475×C₈₃	C₂²	semidirect
35. #465×C₈₃	C₂²	semidirect
36. #461×C₈₃	C₂²	semidirect
37. #460×C₈₃	C₂²	semidirect
41. C₂×C₄⋊C₈×C₈₃	C₂²	semidirect
137. C₂²×C₄×C₈₃	SD₁₆	semidirect
208. C₂×C₄×C₈₃	C₂×SD₁₆	semidirect
242. C₂×#3246	C₈₃	direct (N × H)
245. C₂×C₈₃	#3246	direct (N × H)
249. C₂×#533	C₂×C₈₃	semidirect
251. #3246	C₂×C₈₃	direct (N × H)
252. C₂×#475	C₂×C₈₃	semidirect
253. C₂×#476	C₂×C₈₃	semidirect
254. C₂×#465	C₂×C₈₃	semidirect
255. C₂×#460	C₂×C₈₃	semidirect
256. C₂×#461	C₂×C₈₃	semidirect
257. C₂²×C₄⋊C₈	C₂×C₈₃	semidirect
258. C₈₃	C₂×#3246	direct (N × H)
263. #533	C₂²×C₈₃	semidirect
282. #476	C₂²×C₈₃	semidirect
283. #475	C₂²×C₈₃	semidirect
284. #465	C₂²×C₈₃	semidirect
285. #461	C₂²×C₈₃	semidirect
286. #460	C₂²×C₈₃	semidirect
290. C₂×C₄⋊C₈	C₂²×C₈₃	semidirect
386. C₂²×C₄	SD₁₆×C₈₃	semidirect
457. C₂×C₄	C₂×SD₁₆×C₈₃	semidirect
493. C₂	C₈₃×#3246	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×#533×C₈₃	2
C₂×#532×C₈₃	2
C₈₃×#3246	2
C₂×#475×C₈₃	2
C₂×#476×C₈₃	2
C₂×#465×C₈₃	2
C₂×#460×C₈₃	2
C₂×#461×C₈₃	2
C₂²×C₄⋊C₈×C₈₃	2
C₂×#3246	83

Subgroups — C2×C83×#3246 #35288719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₃×#3246 #35288719