Subgroups — C₈₃×#62249 #35290073

Order 21248.

1902 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₃×#3506	C₂	1	2	yes	maximal, complemented
2	C₈₃×#3342	C₂	1	2	yes	maximal, complemented
3	C₈₃×#3341	C₂	1	2	yes	maximal, complemented
4	C₈₃×#3304	C₂	1	4	yes	maximal, complemented
5	C₈₃×#3303	C₂	1	4	yes	maximal, complemented
6	C₈₃×#3287	C₂	1	2	yes	maximal, complemented
7	C₈₃×#3283	C₂	1	2	yes	maximal, complemented
8	C₈₃×#3246	C₂	1	2	yes	maximal, complemented
9	C₈₃×#3191	C₂	1	4	yes	maximal, complemented
10	C₈₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈₃×#3050	C₂	1	4	yes	maximal, complemented
13	C₂×C₈×D₄×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#135×C₈₃	C₂²	1	4	yes	complemented
15	C₂³×C₈×C₈₃	C₂²	1	2	yes	—
16	#559×C₈₃	—	2	4	no	—
17	#556×C₈₃	C₂²	1	4	yes	—
18	#555×C₈₃	C₂²	1	4	yes	—
19	#553×C₈₃	—	2	4	no	—
20	#550×C₈₃	C₂²	1	4	yes	—
21	#547×C₈₃	C₂²	1	4	yes	—
22	#538×C₈₃	—	2	4	no	—
23	#535×C₈₃	—	2	2	no	—
24	#533×C₈₃	C₂²	1	2	yes	—
25	#532×C₈₃	C₂²	1	2	yes	—
26	#525×C₈₃	C₂²	1	8	yes	—
27	#520×C₈₃	—	2	2	no	—
28	C₂×C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
29	#500×C₈₃	C₂²	1	1	yes	characteristic, complemented
30	#500×C₈₃	C₂²	1	1	yes	characteristic, complemented
31	#494×C₈₃	C₂²	1	1	yes	characteristic, complemented
32	#493×C₈₃	C₂²	1	1	yes	characteristic, complemented
33	#487×C₈₃	C₂²	1	2	yes	complemented
34	#486×C₈₃	C₂²	1	2	yes	complemented
35	#485×C₈₃	C₂²	1	8	yes	complemented
36	#482×C₈₃	C₂²	1	8	yes	complemented
37	#476×C₈₃	C₂²	1	4	yes	complemented
38	#475×C₈₃	C₂²	1	4	yes	complemented
39	#467×C₈₃	C₂²	1	2	yes	—
40	#467×C₈₃	C₂²	1	2	yes	—
41	#467×C₈₃	C₂²	1	4	yes	complemented
42	#466×C₈₃	C₂²	1	2	yes	—
43	#466×C₈₃	C₂²	1	2	yes	—
44	#466×C₈₃	C₂²	1	4	yes	complemented
45	#465×C₈₃	C₂²	1	8	yes	—
46	#465×C₈₃	C₂²	1	8	yes	complemented
47	#461×C₈₃	C₂²	1	4	yes	complemented
48	#460×C₈₃	C₂²	1	4	yes	complemented
49	#452×C₈₃	C₂²	1	8	yes	complemented
50	#449×C₈₃	C₂²	1	8	yes	complemented
51	C₄×Q₁₆×C₈₃	C₂²	1	2	yes	complemented
52	C₄×Q₁₆×C₈₃	C₂²	1	2	yes	complemented
53	C₄×D₈×C₈₃	C₂²	1	2	yes	complemented
54	C₄×D₈×C₈₃	C₂²	1	2	yes	complemented
55	C₈×D₄×C₈₃	C₂²	1	2	yes	—
56	C₈×D₄×C₈₃	C₂²	1	2	yes	—
57	C₈×D₄×C₈₃	C₂²	1	2	yes	—
58	C₈×D₄×C₈₃	C₂²	1	2	yes	—
59	#427×C₈₃	C₂²	1	4	yes	complemented
60	C₂×C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
61	#416×C₈₃	C₂²	1	8	yes	complemented
62	C₂×C₂²⋊C₈×C₈₃	C₂²	1	2	yes	—
63	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
64	C₂×#41×C₈₃	—	2	2	no	—
65	C₂×#41×C₈₃	C₂³	1	8	yes	—
66	C₂²×D₄×C₈₃	D₄	1	1	yes	characteristic
67	C₂³×C₄×C₈₃	C₂³	1	2	yes	—
68	#135×C₈₃	—	4	8	no	—
69	C₂×Q₁₆×C₈₃	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₈₃	C₂³	1	2	yes	—
71	C₂×SD₁₆×C₈₃	C₂³	1	8	yes	—
72	C₂×D₈×C₈₃	C₂³	1	2	yes	—
73	C₂×D₈×C₈₃	C₂³	1	2	yes	—
74	C₂²×C₈×C₈₃	—	4	4	no	—
75	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₈₃	C₂³	1	4	yes	—
77	C₂²×C₈×C₈₃	C₂³	1	2	yes	—
78	C₂²×C₈×C₈₃	C₂³	1	4	yes	—
79	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
80	C₄⋊Q₈×C₈₃	C₂³	1	2	yes	—
81	C₄⋊Q₈×C₈₃	C₂³	1	2	yes	—
82	C₄⋊Q₈×C₈₃	C₂³	1	2	yes	—
83	#127×C₈₃	—	2	4	no	—
84	#127×C₈₃	C₂³	1	2	yes	—
85	#126×C₈₃	—	2	8	no	—
86	#125×C₈₃	—	2	2	no	—
87	#124×C₈₃	—	2	4	no	—
88	#124×C₈₃	—	2	8	no	—
89	#124×C₈₃	—	2	2	no	—
90	#123×C₈₃	—	2	8	no	—
91	#123×C₈₃	—	2	8	no	—
92	C₂²⋊Q₈×C₈₃	—	2	2	no	—
93	C₂²⋊Q₈×C₈₃	C₂³	1	4	yes	—
94	C₂²⋊Q₈×C₈₃	C₂³	1	8	yes	—
95	#121×C₈₃	—	2	4	no	—
96	#121×C₈₃	—	2	4	no	—
97	#121×C₈₃	—	2	2	no	—
98	#121×C₈₃	—	2	8	no	—
99	#121×C₈₃	C₂³	1	4	yes	—
100	#121×C₈₃	C₂³	1	8	yes	—
101	#120×C₈₃	—	2	8	no	—
102	C₄×Q₈×C₈₃	—	2	4	no	—
103	C₄×Q₈×C₈₃	—	2	2	no	—
104	C₄×Q₈×C₈₃	C₂³	1	4	yes	—
105	C₄×D₄×C₈₃	—	2	8	no	—
106	C₄×D₄×C₈₃	—	2	8	no	—
107	C₄×D₄×C₈₃	—	2	4	no	—
108	C₄×D₄×C₈₃	—	2	2	no	—
109	C₄×D₄×C₈₃	D₄	1	2	yes	—
110	C₄×D₄×C₈₃	D₄	1	2	yes	—
111	C₄×D₄×C₈₃	C₂³	1	4	yes	—
112	C₄×D₄×C₈₃	C₂³	1	2	yes	—
113	C₄×D₄×C₈₃	C₂³	1	2	yes	—
114	#117×C₈₃	—	2	2	no	—
115	#117×C₈₃	—	2	4	no	—
116	#117×C₈₃	C₂³	1	8	yes	—
117	C₂×C₄⋊C₄×C₈₃	—	2	4	no	—
118	C₂×C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
119	C₂×C₂²⋊C₄×C₈₃	D₄	1	2	yes	—
120	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
121	#107×C₈₃	C₂³	1	2	yes	—
122	#107×C₈₃	C₂³	1	2	yes	—
123	#107×C₈₃	C₂³	1	2	yes	—
124	#107×C₈₃	C₂³	1	2	yes	—
125	#106×C₈₃	C₂³	1	8	yes	—
126	C₄⋊C₈×C₈₃	C₂³	1	2	yes	—
127	C₄⋊C₈×C₈₃	C₂³	1	2	yes	—
128	#103×C₈₃	C₂³	1	8	yes	—
129	#103×C₈₃	C₂³	1	4	yes	—
130	#103×C₈₃	C₂³	1	4	yes	—
131	#102×C₈₃	C₂³	1	8	yes	—
132	#102×C₈₃	C₂³	1	4	yes	—
133	#102×C₈₃	C₂³	1	4	yes	—
134	C₂²⋊C₈×C₈₃	C₂³	1	4	yes	—
135	C₂²⋊C₈×C₈₃	C₂³	1	4	yes	—
136	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
137	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
140	C₂⁴×C₈₃	C₂×D₄	1	2	yes	—
141	#41×C₈₃	—	4	8	no	—
142	#41×C₈₃	—	4	8	no	—
143	#41×C₈₃	—	4	8	no	—
144	C₂×Q₈×C₈₃	—	2	2	no	—
145	C₂×Q₈×C₈₃	—	2	2	no	—
146	C₂×Q₈×C₈₃	C₂⁴	1	4	yes	—
147	C₂×D₄×C₈₃	—	2	4	no	—
148	C₂×D₄×C₈₃	—	2	8	no	—
149	C₂×D₄×C₈₃	—	2	8	no	—
150	C₂×D₄×C₈₃	—	2	4	no	—
151	C₂×D₄×C₈₃	—	2	2	no	—
152	C₂×D₄×C₈₃	—	2	2	no	—
153	C₂×D₄×C₈₃	C₂×D₄	1	2	yes	—
154	C₂×D₄×C₈₃	C₂×D₄	1	2	yes	—
155	C₂×D₄×C₈₃	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₈₃	—	2	4	no	—
157	C₂²×C₄×C₈₃	—	2	8	no	—
158	C₂²×C₄×C₈₃	—	4	2	no	—
159	C₂²×C₄×C₈₃	—	4	2	no	—
160	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₈₃	C₂×D₄	1	4	yes	—
163	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₈₃	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
166	Q₁₆×C₈₃	—	2	4	no	—
167	Q₁₆×C₈₃	—	4	2	no	—
168	SD₁₆×C₈₃	—	4	8	no	—
169	D₈×C₈₃	—	2	4	no	—
170	D₈×C₈₃	—	4	2	no	—
171	C₂×C₈×C₈₃	—	2	2	no	—
172	C₂×C₈×C₈₃	—	2	8	no	—
173	C₂×C₈×C₈₃	—	4	4	no	—
174	C₂×C₈×C₈₃	—	4	2	no	—
175	C₂×C₈×C₈₃	—	4	1	no	—
176	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
177	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
178	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
179	C₂×C₈×C₈₃	C₂⁴	1	4	yes	—
180	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
181	C₄⋊C₄×C₈₃	—	2	2	no	—
182	C₄⋊C₄×C₈₃	—	2	8	no	—
183	C₄⋊C₄×C₈₃	—	2	2	no	—
184	C₄⋊C₄×C₈₃	—	2	4	no	—
185	C₄⋊C₄×C₈₃	—	2	2	no	—
186	C₄⋊C₄×C₈₃	—	2	2	no	—
187	C₄⋊C₄×C₈₃	—	2	4	no	—
188	C₄⋊C₄×C₈₃	C₂×D₄	1	2	yes	—
189	C₄⋊C₄×C₈₃	C₂×D₄	1	2	yes	—
190	C₄⋊C₄×C₈₃	C₂⁴	1	4	yes	—
191	C₄⋊C₄×C₈₃	C₂⁴	1	4	yes	—
192	C₂²⋊C₄×C₈₃	—	2	4	no	—
193	C₂²⋊C₄×C₈₃	—	2	8	no	—
194	C₂²⋊C₄×C₈₃	—	2	8	no	—
195	C₂²⋊C₄×C₈₃	—	2	8	no	—
196	C₂²⋊C₄×C₈₃	—	2	4	no	—
197	C₂²⋊C₄×C₈₃	—	2	4	no	—
198	C₂²⋊C₄×C₈₃	C₂×D₄	1	4	yes	—
199	C₂²⋊C₄×C₈₃	C₂×D₄	1	4	yes	—
200	C₄²×C₈₃	—	2	8	no	—
201	C₄²×C₈₃	—	2	4	no	—
202	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
203	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
204	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
205	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
206	C₂³×C₈₃	—	2	4	no	—
207	C₂³×C₈₃	—	2	4	no	—
208	C₂³×C₈₃	—	4	2	no	—
209	C₂³×C₈₃	—	4	2	no	—
210	C₂³×C₈₃	C₂²×D₄	1	4	yes	—
211	C₂³×C₈₃	C₂²×D₄	1	1	yes	characteristic
212	Q₈×C₈₃	—	2	4	no	—
213	Q₈×C₈₃	—	2	4	no	—
214	Q₈×C₈₃	—	4	4	no	—
215	Q₈×C₈₃	—	8	2	no	—
216	D₄×C₈₃	—	2	4	no	—
217	D₄×C₈₃	—	2	4	no	—
218	D₄×C₈₃	—	2	4	no	—
219	D₄×C₈₃	—	2	4	no	—
220	D₄×C₈₃	—	4	8	no	—
221	D₄×C₈₃	—	4	4	no	—
222	D₄×C₈₃	—	8	2	no	—
223	D₄×C₈₃	—	8	8	no	—
224	D₄×C₈₃	—	8	8	no	—
225	C₂×C₄×C₈₃	—	2	1	no	—
226	C₂×C₄×C₈₃	—	2	4	no	—
227	C₂×C₄×C₈₃	—	2	1	no	—
228	C₂×C₄×C₈₃	—	2	4	no	—
229	C₂×C₄×C₈₃	—	2	4	no	—
230	C₂×C₄×C₈₃	—	2	4	no	—
231	C₂×C₄×C₈₃	—	2	4	no	—
232	C₂×C₄×C₈₃	—	4	4	no	—
233	C₂×C₄×C₈₃	—	4	2	no	—
234	C₂×C₄×C₈₃	—	4	1	no	—
235	C₂×C₄×C₈₃	—	4	8	no	—
236	C₂×C₄×C₈₃	—	8	8	no	—
237	C₂×C₄×C₈₃	#135	1	4	yes	complemented
238	C₂×C₄×C₈₃	C₂²×D₄	1	4	yes	—
239	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₈₃	C₂⁵	1	1	yes	characteristic
245	C₈×C₈₃	—	2	1	no	—
246	C₈×C₈₃	—	2	4	no	—
247	C₈×C₈₃	—	2	1	no	—
248	C₈×C₈₃	—	2	1	no	—
249	C₈×C₈₃	H(5,𝔽₂)	1	2	yes	—
250	C₂²×C₈₃	—	2	1	no	—
251	C₂²×C₈₃	—	2	1	no	—
252	C₂²×C₈₃	—	2	4	no	—
253	C₂²×C₈₃	—	2	4	no	—
254	C₂²×C₈₃	—	2	4	no	—
255	C₂²×C₈₃	—	4	8	no	—
256	C₂²×C₈₃	—	4	4	no	—
257	C₂²×C₈₃	—	4	4	no	—
258	C₂²×C₈₃	—	4	4	no	—
259	C₂²×C₈₃	C₂×#135	1	2	yes	—
260	C₂²×C₈₃	C₂³×D₄	1	1	yes	characteristic
261	C₄×C₈₃	—	2	1	no	—
262	C₄×C₈₃	—	2	1	no	—
263	C₄×C₈₃	—	2	1	no	—
264	C₄×C₈₃	—	2	4	no	—
265	C₄×C₈₃	—	4	4	no	—
266	C₄×C₈₃	—	4	4	no	—
267	C₄×C₈₃	—	4	4	no	—
268	C₄×C₈₃	C₂×#135	1	2	yes	—
269	C₄×C₈₃	C₂×#135	1	2	yes	—
270	C₄×C₈₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
271	C₄×C₈₃	#585	1	1	yes	characteristic
272	#62249	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
273	C₂×C₈₃	—	2	2	no	—
274	C₂×C₈₃	—	4	4	no	—
275	C₂×C₈₃	—	8	4	no	—
276	C₂×C₈₃	#3496	1	1	yes	characteristic
277	C₂×C₈₃	C₂²×#135	1	1	yes	characteristic
278	C₂×C₈₃	#3584	1	1	yes	characteristic
279	#3506	C₂×C₈₃	1	2	yes	complemented
280	#3342	C₂×C₈₃	1	2	yes	complemented
281	#3341	C₂×C₈₃	1	2	yes	complemented
282	#3304	C₂×C₈₃	1	4	yes	complemented
283	#3303	C₂×C₈₃	1	4	yes	complemented
284	#3287	C₂×C₈₃	1	2	yes	complemented
285	#3283	C₂×C₈₃	1	2	yes	complemented
286	#3246	C₂×C₈₃	1	2	yes	complemented
287	#3191	C₂×C₈₃	1	4	yes	complemented
288	#3170	C₂×C₈₃	1	1	yes	characteristic, complemented
289	#3169	C₂×C₈₃	1	1	yes	characteristic, complemented
290	#3050	C₂×C₈₃	1	4	yes	complemented
291	C₂×C₈×D₄	C₂×C₈₃	1	1	yes	characteristic, complemented
292	C₈₃	#62249	1	1	yes	characteristic, direct product factor, Sylow
293	C₂×#135	C₂²×C₈₃	1	4	yes	complemented
294	C₂³×C₈	C₂²×C₈₃	1	2	yes	—
295	#559	—	2	4	no	—
296	#556	C₂²×C₈₃	1	4	yes	—
297	#555	C₂²×C₈₃	1	4	yes	—
298	#553	—	2	4	no	—
299	#550	C₂²×C₈₃	1	4	yes	—
300	#547	C₂²×C₈₃	1	4	yes	—
301	#538	—	2	4	no	—
302	#535	—	2	2	no	—
303	#533	C₂²×C₈₃	1	2	yes	—
304	#532	C₂²×C₈₃	1	2	yes	—
305	#525	C₂²×C₈₃	1	8	yes	—
306	#520	—	2	2	no	—
307	C₂×C₄×D₄	C₂²×C₈₃	1	1	yes	characteristic
308	#500	C₂²×C₈₃	1	1	yes	characteristic, complemented
309	#500	C₂²×C₈₃	1	1	yes	characteristic, complemented
310	#494	C₂²×C₈₃	1	1	yes	characteristic, complemented
311	#493	C₂²×C₈₃	1	1	yes	characteristic, complemented
312	#487	C₂²×C₈₃	1	2	yes	complemented
313	#486	C₂²×C₈₃	1	2	yes	complemented
314	#485	C₂²×C₈₃	1	8	yes	complemented
315	#482	C₂²×C₈₃	1	8	yes	complemented
316	#476	C₂²×C₈₃	1	4	yes	complemented
317	#475	C₂²×C₈₃	1	4	yes	complemented
318	#467	C₂²×C₈₃	1	2	yes	—
319	#467	C₂²×C₈₃	1	2	yes	—
320	#467	C₂²×C₈₃	1	4	yes	complemented
321	#466	C₂²×C₈₃	1	2	yes	—
322	#466	C₂²×C₈₃	1	2	yes	—
323	#466	C₂²×C₈₃	1	4	yes	complemented
324	#465	C₂²×C₈₃	1	8	yes	—
325	#465	C₂²×C₈₃	1	8	yes	complemented
326	#461	C₂²×C₈₃	1	4	yes	complemented
327	#460	C₂²×C₈₃	1	4	yes	complemented
328	#452	C₂²×C₈₃	1	8	yes	complemented
329	#449	C₂²×C₈₃	1	8	yes	complemented
330	C₄×Q₁₆	C₂²×C₈₃	1	2	yes	complemented
331	C₄×Q₁₆	C₂²×C₈₃	1	2	yes	complemented
332	C₄×D₈	C₂²×C₈₃	1	2	yes	complemented
333	C₄×D₈	C₂²×C₈₃	1	2	yes	complemented
334	C₈×D₄	C₂²×C₈₃	1	2	yes	—
335	C₈×D₄	C₂²×C₈₃	1	2	yes	—
336	C₈×D₄	C₂²×C₈₃	1	2	yes	—
337	C₈×D₄	C₂²×C₈₃	1	2	yes	—
338	#427	C₂²×C₈₃	1	4	yes	complemented
339	C₂×C₄⋊C₈	C₂²×C₈₃	1	1	yes	characteristic
340	#416	C₂²×C₈₃	1	8	yes	complemented
341	C₂×C₂²⋊C₈	C₂²×C₈₃	1	2	yes	—
342	C₂×C₄×C₈	C₂²×C₈₃	1	1	yes	characteristic
343	C₂×#41	—	2	2	no	—
344	C₂×#41	C₂³×C₈₃	1	8	yes	—
345	C₂²×D₄	D₄×C₈₃	1	1	yes	characteristic
346	C₂³×C₄	C₂³×C₈₃	1	2	yes	—
347	#135	—	4	8	no	—
348	C₂×Q₁₆	C₂³×C₈₃	1	2	yes	—
349	C₂×Q₁₆	C₂³×C₈₃	1	2	yes	—
350	C₂×SD₁₆	C₂³×C₈₃	1	8	yes	—
351	C₂×D₈	C₂³×C₈₃	1	2	yes	—
352	C₂×D₈	C₂³×C₈₃	1	2	yes	—
353	C₂²×C₈	—	4	4	no	—
354	C₂²×C₈	C₂³×C₈₃	1	1	yes	characteristic
355	C₂²×C₈	C₂³×C₈₃	1	4	yes	—
356	C₂²×C₈	C₂³×C₈₃	1	2	yes	—
357	C₂²×C₈	C₂³×C₈₃	1	4	yes	—
358	C₂²×C₈	C₂³×C₈₃	1	1	yes	characteristic
359	C₄⋊Q₈	C₂³×C₈₃	1	2	yes	—
360	C₄⋊Q₈	C₂³×C₈₃	1	2	yes	—
361	C₄⋊Q₈	C₂³×C₈₃	1	2	yes	—
362	#127	—	2	4	no	—
363	#127	C₂³×C₈₃	1	2	yes	—
364	#126	—	2	8	no	—
365	#125	—	2	2	no	—
366	#124	—	2	4	no	—
367	#124	—	2	8	no	—
368	#124	—	2	2	no	—
369	#123	—	2	8	no	—
370	#123	—	2	8	no	—
371	C₂²⋊Q₈	—	2	2	no	—
372	C₂²⋊Q₈	C₂³×C₈₃	1	4	yes	—
373	C₂²⋊Q₈	C₂³×C₈₃	1	8	yes	—
374	#121	—	2	4	no	—
375	#121	—	2	4	no	—
376	#121	—	2	2	no	—
377	#121	—	2	8	no	—
378	#121	C₂³×C₈₃	1	4	yes	—
379	#121	C₂³×C₈₃	1	8	yes	—
380	#120	—	2	8	no	—
381	C₄×Q₈	—	2	4	no	—
382	C₄×Q₈	—	2	2	no	—
383	C₄×Q₈	C₂³×C₈₃	1	4	yes	—
384	C₄×D₄	—	2	8	no	—
385	C₄×D₄	—	2	8	no	—
386	C₄×D₄	—	2	4	no	—
387	C₄×D₄	—	2	2	no	—
388	C₄×D₄	D₄×C₈₃	1	2	yes	—
389	C₄×D₄	D₄×C₈₃	1	2	yes	—
390	C₄×D₄	C₂³×C₈₃	1	4	yes	—
391	C₄×D₄	C₂³×C₈₃	1	2	yes	—
392	C₄×D₄	C₂³×C₈₃	1	2	yes	—
393	#117	—	2	2	no	—
394	#117	—	2	4	no	—
395	#117	C₂³×C₈₃	1	8	yes	—
396	C₂×C₄⋊C₄	—	2	4	no	—
397	C₂×C₄⋊C₄	D₄×C₈₃	1	1	yes	characteristic
398	C₂×C₂²⋊C₄	D₄×C₈₃	1	2	yes	—
399	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
400	#107	C₂³×C₈₃	1	2	yes	—
401	#107	C₂³×C₈₃	1	2	yes	—
402	#107	C₂³×C₈₃	1	2	yes	—
403	#107	C₂³×C₈₃	1	2	yes	—
404	#106	C₂³×C₈₃	1	8	yes	—
405	C₄⋊C₈	C₂³×C₈₃	1	2	yes	—
406	C₄⋊C₈	C₂³×C₈₃	1	2	yes	—
407	#103	C₂³×C₈₃	1	4	yes	—
408	#103	C₂³×C₈₃	1	8	yes	—
409	#103	C₂³×C₈₃	1	4	yes	—
410	#102	C₂³×C₈₃	1	4	yes	—
411	#102	C₂³×C₈₃	1	8	yes	—
412	#102	C₂³×C₈₃	1	4	yes	—
413	C₂²⋊C₈	C₂³×C₈₃	1	4	yes	—
414	C₂²⋊C₈	C₂³×C₈₃	1	4	yes	—
415	C₄×C₈	C₂³×C₈₃	1	1	yes	characteristic
416	C₄×C₈	C₂³×C₈₃	1	1	yes	characteristic
417	C₄×C₈	C₂³×C₈₃	1	1	yes	characteristic
418	C₄×C₈	C₂³×C₈₃	1	1	yes	characteristic
419	C₂⁴	C₂×D₄×C₈₃	1	2	yes	—
420	#41	—	4	8	no	—
421	#41	—	4	8	no	—
422	#41	—	4	8	no	—
423	C₂×Q₈	—	2	2	no	—
424	C₂×Q₈	—	2	2	no	—
425	C₂×Q₈	C₂⁴×C₈₃	1	4	yes	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	—	2	8	no	—
428	C₂×D₄	—	2	8	no	—
429	C₂×D₄	—	2	4	no	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	C₂×D₄×C₈₃	1	2	yes	—
433	C₂×D₄	C₂×D₄×C₈₃	1	2	yes	—
434	C₂×D₄	C₂⁴×C₈₃	1	4	yes	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	8	no	—
437	C₂²×C₄	—	4	2	no	—
438	C₂²×C₄	—	4	2	no	—
439	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₈₃	1	2	yes	—
441	C₂²×C₄	C₂×D₄×C₈₃	1	4	yes	—
442	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
443	C₂²×C₄	C₂⁴×C₈₃	1	4	yes	—
444	C₂²×C₄	C₂⁴×C₈₃	1	1	yes	characteristic
445	Q₁₆	—	2	4	no	—
446	Q₁₆	—	4	2	no	—
447	SD₁₆	—	4	8	no	—
448	D₈	—	2	4	no	—
449	D₈	—	4	2	no	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	—	2	8	no	—
452	C₂×C₈	—	4	4	no	—
453	C₂×C₈	—	4	2	no	—
454	C₂×C₈	—	4	1	no	—
455	C₂×C₈	C₂⁴×C₈₃	1	1	yes	characteristic
456	C₂×C₈	C₂⁴×C₈₃	1	1	yes	characteristic
457	C₂×C₈	C₂⁴×C₈₃	1	1	yes	characteristic
458	C₂×C₈	C₂⁴×C₈₃	1	4	yes	—
459	C₂×C₈	C₂⁴×C₈₃	1	1	yes	characteristic
460	C₄⋊C₄	—	2	2	no	—
461	C₄⋊C₄	—	2	8	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	2	no	—
465	C₄⋊C₄	—	2	2	no	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	C₂×D₄×C₈₃	1	2	yes	—
468	C₄⋊C₄	C₂×D₄×C₈₃	1	2	yes	—
469	C₄⋊C₄	C₂⁴×C₈₃	1	4	yes	—
470	C₄⋊C₄	C₂⁴×C₈₃	1	4	yes	—
471	C₂²⋊C₄	—	2	4	no	—
472	C₂²⋊C₄	—	2	8	no	—
473	C₂²⋊C₄	—	2	8	no	—
474	C₂²⋊C₄	—	2	8	no	—
475	C₂²⋊C₄	—	2	4	no	—
476	C₂²⋊C₄	—	2	4	no	—
477	C₂²⋊C₄	C₂×D₄×C₈₃	1	4	yes	—
478	C₂²⋊C₄	C₂×D₄×C₈₃	1	4	yes	—
479	C₄²	—	2	8	no	—
480	C₄²	—	2	4	no	—
481	C₄²	C₂×D₄×C₈₃	1	1	yes	characteristic
482	C₄²	C₂×D₄×C₈₃	1	1	yes	characteristic
483	C₄²	C₂⁴×C₈₃	1	1	yes	characteristic
484	C₄²	C₂⁴×C₈₃	1	1	yes	characteristic
485	C₂³	—	2	4	no	—
486	C₂³	—	2	4	no	—
487	C₂³	—	4	2	no	—
488	C₂³	—	4	2	no	—
489	C₂³	C₂²×D₄×C₈₃	1	4	yes	—
490	C₂³	C₂²×D₄×C₈₃	1	1	yes	characteristic
491	Q₈	—	2	4	no	—
492	Q₈	—	2	4	no	—
493	Q₈	—	4	4	no	—
494	Q₈	—	8	2	no	—
495	D₄	—	2	4	no	—
496	D₄	—	2	4	no	—
497	D₄	—	2	4	no	—
498	D₄	—	2	4	no	—
499	D₄	—	4	8	no	—
500	D₄	—	4	4	no	—
501	D₄	—	8	2	no	—
502	D₄	—	8	8	no	—
503	D₄	—	8	8	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	4	4	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	—	4	8	no	—
515	C₂×C₄	—	8	8	no	—
516	C₂×C₄	#135×C₈₃	1	4	yes	complemented
517	C₂×C₄	C₂²×D₄×C₈₃	1	4	yes	—
518	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
519	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
520	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
521	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
522	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
523	C₂×C₄	C₂⁵×C₈₃	1	1	yes	characteristic
524	C₈	—	2	1	no	—
525	C₈	—	2	4	no	—
526	C₈	—	2	1	no	—
527	C₈	—	2	1	no	—
528	C₈	H(5,𝔽₂)×C₈₃	1	2	yes	—
529	C₂²	—	2	1	no	—
530	C₂²	—	2	1	no	—
531	C₂²	—	2	4	no	—
532	C₂²	—	2	4	no	—
533	C₂²	—	2	4	no	—
534	C₂²	—	4	8	no	—
535	C₂²	—	4	4	no	—
536	C₂²	—	4	4	no	—
537	C₂²	—	4	4	no	—
538	C₂²	C₂×#135×C₈₃	1	2	yes	—
539	C₂²	C₂³×D₄×C₈₃	1	1	yes	characteristic
540	C₄	—	2	1	no	—
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	2	4	no	—
544	C₄	—	4	4	no	—
545	C₄	—	4	4	no	—
546	C₄	—	4	4	no	—
547	C₄	C₂×#135×C₈₃	1	2	yes	—
548	C₄	C₂×#135×C₈₃	1	2	yes	—
549	C₄	C₂×H(5,𝔽₂)×C₈₃	1	1	yes	characteristic
550	C₄	#585×C₈₃	1	1	yes	characteristic
551	C₂	—	2	2	no	—
552	C₂	—	4	4	no	—
553	C₂	—	8	4	no	—
554	C₂	C₈₃×#3496	1	1	yes	characteristic
555	C₂	C₂²×#135×C₈₃	1	1	yes	characteristic
556	C₂	C₈₃×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
1. C₈₃×#3506	C₂	semidirect
2. C₈₃×#3342	C₂	semidirect
3. C₈₃×#3341	C₂	semidirect
4. C₈₃×#3304	C₂	semidirect
5. C₈₃×#3303	C₂	semidirect
6. C₈₃×#3287	C₂	semidirect
7. C₈₃×#3283	C₂	semidirect
8. C₈₃×#3246	C₂	semidirect
9. C₈₃×#3191	C₂	semidirect
10. C₈₃×#3170	C₂	semidirect
11. C₈₃×#3169	C₂	semidirect
12. C₈₃×#3050	C₂	semidirect
13. C₂×C₈×D₄×C₈₃	C₂	semidirect
14. C₂×#135×C₈₃	C₂²	semidirect
29. #500×C₈₃	C₂²	semidirect
30. #500×C₈₃	C₂²	semidirect
31. #494×C₈₃	C₂²	semidirect
32. #493×C₈₃	C₂²	semidirect
33. #487×C₈₃	C₂²	semidirect
34. #486×C₈₃	C₂²	semidirect
35. #485×C₈₃	C₂²	semidirect
36. #482×C₈₃	C₂²	semidirect
37. #476×C₈₃	C₂²	semidirect
38. #475×C₈₃	C₂²	semidirect
41. #467×C₈₃	C₂²	semidirect
44. #466×C₈₃	C₂²	semidirect
46. #465×C₈₃	C₂²	semidirect
47. #461×C₈₃	C₂²	semidirect
48. #460×C₈₃	C₂²	semidirect
49. #452×C₈₃	C₂²	semidirect
50. #449×C₈₃	C₂²	semidirect
51. C₄×Q₁₆×C₈₃	C₂²	semidirect
52. C₄×Q₁₆×C₈₃	C₂²	semidirect
53. C₄×D₈×C₈₃	C₂²	semidirect
54. C₄×D₈×C₈₃	C₂²	semidirect
59. #427×C₈₃	C₂²	semidirect
61. #416×C₈₃	C₂²	semidirect
237. C₂×C₄×C₈₃	#135	semidirect
272. #62249	C₈₃	direct (N × H)
279. #3506	C₂×C₈₃	semidirect
280. #3342	C₂×C₈₃	semidirect
281. #3341	C₂×C₈₃	semidirect
282. #3304	C₂×C₈₃	semidirect
283. #3303	C₂×C₈₃	semidirect
284. #3287	C₂×C₈₃	semidirect
285. #3283	C₂×C₈₃	semidirect
286. #3246	C₂×C₈₃	semidirect
287. #3191	C₂×C₈₃	semidirect
288. #3170	C₂×C₈₃	semidirect
289. #3169	C₂×C₈₃	semidirect
290. #3050	C₂×C₈₃	semidirect
291. C₂×C₈×D₄	C₂×C₈₃	semidirect
292. C₈₃	#62249	direct (N × H)
293. C₂×#135	C₂²×C₈₃	semidirect
308. #500	C₂²×C₈₃	semidirect
309. #500	C₂²×C₈₃	semidirect
310. #494	C₂²×C₈₃	semidirect
311. #493	C₂²×C₈₃	semidirect
312. #487	C₂²×C₈₃	semidirect
313. #486	C₂²×C₈₃	semidirect
314. #485	C₂²×C₈₃	semidirect
315. #482	C₂²×C₈₃	semidirect
316. #476	C₂²×C₈₃	semidirect
317. #475	C₂²×C₈₃	semidirect
320. #467	C₂²×C₈₃	semidirect
323. #466	C₂²×C₈₃	semidirect
325. #465	C₂²×C₈₃	semidirect
326. #461	C₂²×C₈₃	semidirect
327. #460	C₂²×C₈₃	semidirect
328. #452	C₂²×C₈₃	semidirect
329. #449	C₂²×C₈₃	semidirect
330. C₄×Q₁₆	C₂²×C₈₃	semidirect
331. C₄×Q₁₆	C₂²×C₈₃	semidirect
332. C₄×D₈	C₂²×C₈₃	semidirect
333. C₄×D₈	C₂²×C₈₃	semidirect
338. #427	C₂²×C₈₃	semidirect
340. #416	C₂²×C₈₃	semidirect
516. C₂×C₄	#135×C₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₈₃×#3506	2
C₈₃×#3342	2
C₈₃×#3341	2
C₈₃×#3304	2
C₈₃×#3303	2
C₈₃×#3287	2
C₈₃×#3283	2
C₈₃×#3246	2
C₈₃×#3191	2
C₈₃×#3170	2
C₈₃×#3169	2
C₈₃×#3050	2
C₂×C₈×D₄×C₈₃	2
#62249	83

Subgroups — C83×#62249 #35290073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#62249 #35290073