Subgroups — #35426472

Order 21248.

720 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24917247	C₂	1	1	yes	maximal, characteristic
2	#24916407	C₂	1	1	yes	maximal, characteristic, complemented
3	#24916029	C₂	1	1	yes	maximal, characteristic
4	#24908438	C₂	1	1	yes	maximal, characteristic, complemented
5	#24908237	C₂	1	1	yes	maximal, characteristic, complemented
6	#24908211	C₂	1	1	yes	maximal, characteristic, complemented
7	#24908210	C₂	1	1	yes	maximal, characteristic, complemented
8	#24908062	C₂	1	1	yes	maximal, characteristic, complemented
9	#24908061	C₂	1	1	yes	maximal, characteristic, complemented
10	#24907740	C₂	1	1	yes	maximal, characteristic
11	#24907738	C₂	1	1	yes	maximal, characteristic
12	C₄×Dic₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#24907301	C₂	1	1	yes	maximal, characteristic
14	#104×Dic₈₃	C₂	1	1	yes	maximal, characteristic
15	C₈₃×#1764	C₂	1	1	yes	maximal, characteristic
16	C₂×#14581799	C₂²	1	1	yes	characteristic
17	#19128254	C₂²	1	1	yes	characteristic
18	#41×Dic₈₃	C₂²	1	1	yes	characteristic
19	#19128010	C₄	1	1	yes	characteristic, complemented
20	#19128006	C₄	1	1	yes	characteristic, complemented
21	#19128003	C₄	1	1	yes	characteristic, complemented
22	#19127999	C₄	1	1	yes	characteristic, complemented
23	#19127994	C₄	1	1	yes	characteristic, complemented
24	#19127989	C₄	1	1	yes	characteristic, complemented
25	#19127984	C₄	1	1	yes	characteristic, complemented
26	#19127979	C₄	1	1	yes	characteristic, complemented
27	#19127831	—	2	1	no	—
28	#19127829	—	2	1	no	—
29	#19127829	—	2	1	no	—
30	C₈×Dic₁₆₆	—	2	1	no	—
31	C₂×#14581728	C₂²	1	1	yes	characteristic
32	C₂×#14581727	C₂²	1	1	yes	characteristic
33	#19127739	C₂²	1	1	yes	characteristic
34	#19127737	C₂²	1	1	yes	characteristic
35	#19127729	—	2	1	no	—
36	#19127729	C₂²	1	1	yes	characteristic
37	#19127728	C₂²	1	1	yes	characteristic
38	#19127727	—	2	1	no	—
39	#19127727	C₂²	1	1	yes	characteristic
40	#19127727	C₂²	1	1	yes	characteristic
41	#19127724	—	2	1	no	—
42	#19127724	C₂²	1	1	yes	characteristic
43	C₂×Dic₈₃⋊C₈	—	2	1	no	—
44	C₂×Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
45	C₂×Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
46	C₂×#14581719	C₂²	1	1	yes	characteristic
47	#19127697	C₂²	1	1	yes	characteristic
48	#19127659	C₂²	1	1	yes	characteristic
49	M₁₆×Dic₈₃	C₂²	1	1	yes	characteristic
50	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
51	C₄²×Dic₈₃	C₂²	1	1	yes	characteristic
52	#19127465	C₂²	1	1	yes	characteristic
53	#19127462	C₂²	1	1	yes	characteristic
54	#19127462	C₂²	1	1	yes	characteristic
55	#19127461	C₂²	1	1	yes	characteristic
56	#19127461	C₂²	1	1	yes	characteristic
57	#19127460	C₂²	1	1	yes	characteristic
58	#19127455	C₂²	1	1	yes	characteristic
59	#19127455	C₂²	1	1	yes	characteristic
60	C₂×#131×C₈₃	C₂²	1	1	yes	characteristic
61	C₂×#104×C₈₃	C₂²	1	1	yes	characteristic
62	#416×C₈₃	C₂²	1	1	yes	characteristic
63	#405×C₈₃	C₂²	1	1	yes	characteristic
64	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
65	#341×C₈₃	C₂²	1	1	yes	characteristic
66	#339×C₈₃	C₂²	1	1	yes	characteristic
67	#14581799	—	2	2	no	—
68	#14581799	—	4	1	no	—
69	C₂×#13050574	—	2	1	no	—
70	C₂×#13050574	C₂³	1	1	yes	characteristic
71	C₂×#13050574	C₂³	1	1	yes	characteristic
72	C₂²×C₈₃⋊C₈	—	2	1	no	—
73	C₂²×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₂²×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
75	#14581791	C₂×C₄	1	1	yes	characteristic
76	#14581791	C₂×C₄	1	1	yes	characteristic
77	#14581783	C₂×C₄	1	1	yes	characteristic
78	#14581783	C₂×C₄	1	1	yes	characteristic
79	#14581769	—	2	1	no	—
80	#14581769	—	2	1	no	—
81	Q₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
82	Q₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
83	C₄×Dic₁₆₆	—	2	1	no	—
84	C₄×Dic₁₆₆	—	2	1	no	—
85	D₄×Dic₈₃	—	2	1	no	—
86	D₄×Dic₈₃	—	2	1	no	—
87	D₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
88	D₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
89	#14581751	—	2	1	no	—
90	#14581751	C₂³	1	1	yes	characteristic
91	#14581751	C₂³	1	1	yes	characteristic
92	C₂×C₄×Dic₈₃	—	2	1	no	—
93	C₂×C₄×Dic₈₃	—	2	1	no	—
94	C₂×C₄×Dic₈₃	—	2	1	no	—
95	C₂×C₄×Dic₈₃	C₂³	1	1	yes	characteristic
96	C₂×C₄×Dic₈₃	C₂³	1	1	yes	characteristic
97	#14581728	—	4	1	no	—
98	#14581727	—	4	1	no	—
99	#14581726	C₂×C₄	1	1	yes	characteristic
100	#14581726	C₂×C₄	1	1	yes	characteristic
101	#14581725	C₂×C₄	1	1	yes	characteristic
102	#14581725	C₂×C₄	1	1	yes	characteristic
103	#14581724	C₂×C₄	1	1	yes	characteristic
104	#14581724	C₂×C₄	1	1	yes	characteristic
105	#14581723	C₂×C₄	1	1	yes	characteristic
106	#14581723	C₂×C₄	1	1	yes	characteristic
107	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
108	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
109	Dic₈₃⋊C₈	—	2	1	no	—
110	Dic₈₃⋊C₈	—	2	1	no	—
111	Dic₈₃⋊C₈	—	2	1	no	—
112	Dic₈₃⋊C₈	—	2	1	no	—
113	Dic₈₃⋊C₈	—	2	1	no	—
114	Dic₈₃⋊C₈	—	2	1	no	—
115	Dic₈₃⋊C₈	—	2	1	no	—
116	Dic₈₃⋊C₈	—	2	1	no	—
117	Dic₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
118	Dic₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
119	Dic₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
120	Dic₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
121	Dic₈₃⋊C₈	D₄	1	1	yes	characteristic
122	Dic₈₃⋊C₈	D₄	1	1	yes	characteristic
123	Dic₈₃⋊C₈	D₄	1	1	yes	characteristic
124	Dic₈₃⋊C₈	D₄	1	1	yes	characteristic
125	#14581719	—	2	2	no	—
126	#14581716	C₂×C₄	1	1	yes	characteristic
127	#14581716	C₂×C₄	1	1	yes	characteristic
128	#14581713	C₂×C₄	1	1	yes	characteristic
129	#14581713	C₂×C₄	1	1	yes	characteristic
130	#14581702	—	2	1	no	—
131	#14581702	—	2	1	no	—
132	C₄×C₈₃⋊C₈	—	4	1	no	—
133	C₈×Dic₈₃	—	2	1	no	—
134	C₈×Dic₈₃	—	2	1	no	—
135	C₂×#41×C₈₃	C₂³	1	1	yes	characteristic
136	#131×C₈₃	—	2	2	no	—
137	#131×C₈₃	—	4	1	no	—
138	C₂×M₁₆×C₈₃	—	2	1	no	—
139	C₂×M₁₆×C₈₃	C₂³	1	1	yes	characteristic
140	C₂×M₁₆×C₈₃	C₂³	1	1	yes	characteristic
141	C₂²×C₈×C₈₃	—	2	1	no	—
142	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
143	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
144	#117×C₈₃	C₂³	1	1	yes	characteristic
145	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
146	#104×C₈₃	Q₈	1	4	yes	complemented
147	#103×C₈₃	C₂×C₄	1	1	yes	characteristic
148	#103×C₈₃	C₂×C₄	1	1	yes	characteristic
149	#102×C₈₃	C₂×C₄	1	1	yes	characteristic
150	#102×C₈₃	C₂×C₄	1	1	yes	characteristic
151	#97×C₈₃	—	2	1	no	—
152	C₄×C₈×C₈₃	—	2	1	no	—
153	C₄×C₈×C₈₃	—	4	1	no	—
154	C₂×Dic₁₆₆	—	2	1	no	—
155	C₂×Dic₁₆₆	—	2	1	no	—
156	C₂²×Dic₈₃	—	2	1	no	—
157	C₂²×Dic₈₃	—	2	1	no	—
158	C₂²×Dic₈₃	—	2	1	no	—
159	C₂²×Dic₈₃	#41	1	1	yes	characteristic
160	C₂²×Dic₈₃	#41	1	1	yes	characteristic
161	#13050574	—	2	1	no	—
162	#13050574	—	2	1	no	—
163	#13050574	—	2	1	no	—
164	#13050574	—	2	1	no	—
165	#13050574	—	4	2	no	—
166	C₂×C₈₃⋊C₈	—	2	1	no	—
167	C₂×C₈₃⋊C₈	—	2	1	no	—
168	C₂×C₈₃⋊C₈	—	2	1	no	—
169	C₂×C₈₃⋊C₈	—	2	1	no	—
170	C₂×C₈₃⋊C₈	—	2	1	no	—
171	C₂×C₈₃⋊C₈	—	2	1	no	—
172	C₂×C₈₃⋊C₈	—	4	2	no	—
173	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
174	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
176	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
177	C₄⋊Dic₈₃	—	2	1	no	—
178	C₄⋊Dic₈₃	—	2	1	no	—
179	C₄⋊Dic₈₃	—	2	1	no	—
180	C₄⋊Dic₈₃	—	2	1	no	—
181	C₄⋊Dic₈₃	C₂²×C₄	1	1	yes	characteristic
182	C₄⋊Dic₈₃	C₂²×C₄	1	1	yes	characteristic
183	C₄⋊Dic₈₃	C₂²×C₄	1	1	yes	characteristic
184	C₄⋊Dic₈₃	C₂²×C₄	1	1	yes	characteristic
185	Dic₈₃⋊C₄	—	2	1	no	—
186	Dic₈₃⋊C₄	—	2	1	no	—
187	Dic₈₃⋊C₄	—	2	1	no	—
188	Dic₈₃⋊C₄	—	2	1	no	—
189	C₂²⋊Dic₈₃	—	2	1	no	—
190	C₂²⋊Dic₈₃	—	2	1	no	—
191	C₂²⋊Dic₈₃	—	2	1	no	—
192	C₂²⋊Dic₈₃	—	2	1	no	—
193	C₄×Dic₈₃	—	2	4	no	—
194	C₄×Dic₈₃	—	2	1	no	—
195	C₄×Dic₈₃	—	2	1	no	—
196	C₄×Dic₈₃	—	2	1	no	—
197	C₄×Dic₈₃	—	2	1	no	—
198	C₄×Dic₈₃	—	2	1	no	—
199	C₄×Dic₈₃	—	2	1	no	—
200	C₄×Dic₈₃	—	4	2	no	—
201	C₄×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
202	C₄×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
203	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
204	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
205	#41×C₈₃	—	2	1	no	—
206	#41×C₈₃	—	2	1	no	—
207	#41×C₈₃	C₂×Q₈	1	2	yes	—
208	#41×C₈₃	#41	1	2	yes	—
209	C₂×Q₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
210	C₂×D₄×C₈₃	—	2	1	no	—
211	C₂×D₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
212	C₂²×C₄×C₈₃	—	2	1	no	—
213	C₂²×C₄×C₈₃	—	2	1	no	—
214	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
215	M₁₆×C₈₃	—	2	1	no	—
216	M₁₆×C₈₃	—	2	1	no	—
217	M₁₆×C₈₃	—	2	1	no	—
218	M₁₆×C₈₃	—	4	2	no	—
219	M₁₆×C₈₃	C₂×Q₈	1	2	yes	—
220	C₂×C₈×C₈₃	—	2	1	no	—
221	C₂×C₈×C₈₃	—	2	1	no	—
222	C₂×C₈×C₈₃	—	2	1	no	—
223	C₂×C₈×C₈₃	—	2	1	no	—
224	C₂×C₈×C₈₃	—	2	1	no	—
225	C₂×C₈×C₈₃	—	2	1	no	—
226	C₂×C₈×C₈₃	—	4	2	no	—
227	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
228	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
229	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
230	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
231	C₄⋊C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
232	C₄⋊C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
233	C₂²⋊C₄×C₈₃	—	2	1	no	—
234	C₄²×C₈₃	—	2	1	no	—
235	C₄²×C₈₃	—	2	1	no	—
236	C₄²×C₈₃	C₂×Q₈	1	2	yes	—
237	C₂³×C₈₃	—	2	1	no	—
238	C₂³×C₈₃	C₂×#41	1	1	yes	characteristic
239	Q₈×C₈₃	—	2	1	no	—
240	Q₈×C₈₃	C₄×Q₈	1	2	yes	—
241	D₄×C₈₃	—	2	2	no	—
242	D₄×C₈₃	—	2	2	no	—
243	D₄×C₈₃	—	2	1	no	—
244	D₄×C₈₃	C₄×Q₈	1	2	yes	—
245	C₂×C₄×C₈₃	—	2	2	no	—
246	C₂×C₄×C₈₃	—	2	2	no	—
247	C₂×C₄×C₈₃	—	2	1	no	—
248	C₂×C₄×C₈₃	—	2	2	no	—
249	C₂×C₄×C₈₃	—	2	2	no	—
250	C₂×C₄×C₈₃	—	2	2	no	—
251	C₂×C₄×C₈₃	—	2	1	no	—
252	C₂×C₄×C₈₃	—	2	1	no	—
253	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
254	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
256	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
257	C₂×C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
258	C₂×C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
259	C₂×Dic₈₃	—	2	2	no	—
260	C₂×Dic₈₃	—	2	2	no	—
261	C₂×Dic₈₃	—	2	1	no	—
262	C₂×Dic₈₃	—	2	1	no	—
263	C₂×Dic₈₃	—	2	1	no	—
264	C₂×Dic₈₃	—	2	1	no	—
265	C₂×Dic₈₃	—	4	1	no	—
266	C₂×Dic₈₃	—	4	1	no	—
267	C₂×Dic₈₃	—	4	1	no	—
268	C₂×Dic₈₃	—	4	2	no	—
269	C₂×Dic₈₃	—	8	1	no	—
270	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
271	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
272	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
273	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
274	Dic₁₆₆	—	4	2	no	—
275	Dic₁₆₆	—	8	1	no	—
276	C₈×C₈₃	—	2	1	no	—
277	C₈×C₈₃	—	2	1	no	—
278	C₈×C₈₃	—	2	1	no	—
279	C₈×C₈₃	—	2	1	no	—
280	C₈×C₈₃	—	2	2	no	—
281	C₈×C₈₃	—	4	1	no	—
282	C₈₃⋊C₈	—	2	1	no	—
283	C₈₃⋊C₈	—	2	1	no	—
284	C₈₃⋊C₈	—	2	1	no	—
285	C₈₃⋊C₈	—	2	1	no	—
286	C₈₃⋊C₈	—	4	1	no	—
287	C₈₃⋊C₈	—	4	1	no	—
288	C₂²×C₈₃	—	2	1	no	—
289	C₂²×C₈₃	—	2	1	no	—
290	C₂²×C₈₃	—	2	2	no	—
291	C₂²×C₈₃	—	4	1	no	—
292	C₂²×C₈₃	—	4	1	no	—
293	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
294	C₂²×C₈₃	#552	1	1	yes	characteristic
295	C₂²×C₈₃	#554	1	1	yes	characteristic
296	C₄×C₈₃	—	2	4	no	—
297	C₄×C₈₃	—	2	2	no	—
298	C₄×C₈₃	—	4	1	no	—
299	C₄×C₈₃	—	4	1	no	—
300	C₄×C₈₃	C₂×C₄×Q₈	1	1	yes	characteristic
301	C₄×C₈₃	#520	1	1	yes	characteristic
302	C₄×C₈₃	#546	1	1	yes	characteristic
303	C₄×C₈₃	D₄×Q₈	1	1	yes	characteristic
304	Dic₈₃	—	2	1	no	—
305	Dic₈₃	—	2	1	no	—
306	Dic₈₃	—	2	1	no	—
307	Dic₈₃	—	4	2	no	—
308	Dic₈₃	—	4	1	no	—
309	Dic₈₃	—	4	1	no	—
310	Dic₈₃	—	4	1	no	—
311	Dic₈₃	—	4	1	no	—
312	Dic₈₃	#443	1	2	yes	—
313	#22982	—	83	1	no	maximal, Sylow
314	C₂×C₈₃	—	2	1	no	—
315	C₂×C₈₃	—	2	1	no	—
316	C₂×C₈₃	—	4	2	no	—
317	C₂×C₈₃	#2360	1	1	yes	characteristic
318	C₂×C₈₃	C₂×#443	1	1	yes	characteristic
319	C₂×C₈₃	#2956	1	1	yes	characteristic
320	#3076	—	83	1	no	—
321	#2962	—	83	1	no	—
322	#2906	—	83	1	no	—
323	#1994	—	83	1	no	—
324	#1950	—	83	1	no	—
325	#1945	—	83	1	no	—
326	#1945	—	83	1	no	—
327	#1922	—	83	1	no	—
328	#1922	—	83	1	no	—
329	#1866	—	83	1	no	—
330	#1866	—	83	1	no	—
331	C₄×C₄⋊C₈	—	83	1	no	—
332	#1764	—	83	1	no	—
333	#1764	D₈₃	1	1	yes	characteristic
334	C₄×#104	—	83	1	no	—
335	C₈₃	#22982	1	1	yes	characteristic, complemented, Sylow
336	C₂×#131	—	83	1	no	—
337	C₂×#131	C₂×D₈₃	1	1	yes	characteristic
338	#533	—	83	1	no	—
339	C₄×#41	—	83	1	no	—
340	#479	—	83	1	no	—
341	#479	—	83	1	no	—
342	#478	—	83	1	no	—
343	#478	—	83	1	no	—
344	#477	—	83	1	no	—
345	#476	—	83	1	no	—
346	#475	—	83	1	no	—
347	#474	—	83	1	no	—
348	#446	—	166	1	no	—
349	#446	—	166	1	no	—
350	#446	—	166	1	no	—
351	C₈×Q₈	—	166	1	no	—
352	C₂×#108	—	83	1	no	—
353	C₂×#108	—	83	1	no	—
354	#427	—	83	1	no	—
355	#427	—	83	1	no	—
356	#424	—	83	1	no	—
357	#424	—	83	1	no	—
358	#424	—	83	1	no	—
359	#424	—	83	1	no	—
360	#424	—	166	1	no	—
361	#424	—	166	1	no	—
362	#423	—	83	1	no	—
363	#423	—	166	1	no	—
364	C₂×C₄⋊C₈	—	83	1	no	—
365	C₂×C₄⋊C₈	—	83	1	no	—
366	C₂×C₄⋊C₈	—	166	1	no	—
367	C₂×#104	—	83	1	no	—
368	C₂×#104	C₂×D₈₃	1	1	yes	characteristic
369	#416	—	83	1	no	—
370	#416	C₂×D₈₃	1	1	yes	characteristic
371	#405	—	83	1	no	—
372	#405	C₂×D₈₃	1	1	yes	characteristic
373	C₄×M₁₆	—	83	1	no	—
374	C₂×C₄×C₈	—	83	1	no	—
375	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
376	C₄³	—	83	1	no	—
377	#341	—	83	1	no	—
378	#341	C₂×D₈₃	1	1	yes	characteristic
379	#339	—	83	1	no	—
380	#339	—	83	1	no	—
381	#339	—	83	1	no	—
382	#339	—	83	1	no	—
383	#339	—	83	1	no	—
384	#339	C₂×D₈₃	1	1	yes	characteristic
385	#336	—	83	1	no	—
386	#336	—	83	1	no	—
387	C₂×#41	C₂²×D₈₃	1	1	yes	characteristic
388	#131	—	2	2	no	—
389	#131	—	4	1	no	—
390	#131	—	332	1	no	—
391	#131	—	166	2	no	—
392	C₂×M₁₆	—	2	1	no	—
393	C₂×M₁₆	—	83	1	no	—
394	C₂×M₁₆	—	83	1	no	—
395	C₂×M₁₆	—	166	1	no	—
396	C₂×M₁₆	C₂²×D₈₃	1	1	yes	characteristic
397	C₂×M₁₆	C₂²×D₈₃	1	1	yes	characteristic
398	C₂²×C₈	—	2	1	no	—
399	C₂²×C₈	—	83	1	no	—
400	C₂²×C₈	—	83	1	no	—
401	C₂²×C₈	—	166	1	no	—
402	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
403	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
404	C₄⋊Q₈	—	83	1	no	—
405	C₄⋊Q₈	—	83	1	no	—
406	#125	—	83	1	no	—
407	#125	—	83	1	no	—
408	C₂²⋊Q₈	—	166	1	no	—
409	C₂²⋊Q₈	—	166	1	no	—
410	C₄×Q₈	—	83	1	no	—
411	C₄×Q₈	—	83	1	no	—
412	C₄×Q₈	—	166	1	no	—
413	C₄×Q₈	—	166	1	no	—
414	C₄×D₄	—	83	1	no	—
415	C₄×D₄	—	83	1	no	—
416	C₄×D₄	—	166	1	no	—
417	C₄×D₄	—	166	1	no	—
418	#117	—	83	1	no	—
419	#117	—	83	1	no	—
420	#117	—	166	1	no	—
421	#117	C₂²×D₈₃	1	1	yes	characteristic
422	C₂×C₄²	—	83	1	no	—
423	C₂×C₄²	—	83	1	no	—
424	C₂×C₄²	—	166	1	no	—
425	C₂×C₄²	—	166	1	no	—
426	C₂×C₄²	—	166	1	no	—
427	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
428	#108	—	332	1	no	—
429	#108	—	332	1	no	—
430	#107	—	83	1	no	—
431	#107	—	83	1	no	—
432	#107	—	83	1	no	—
433	#107	—	83	1	no	—
434	#106	—	83	1	no	—
435	#106	—	83	1	no	—
436	#106	—	83	1	no	—
437	#106	—	83	1	no	—
438	C₄⋊C₈	—	83	1	no	—
439	C₄⋊C₈	—	83	1	no	—
440	C₄⋊C₈	—	83	1	no	—
441	C₄⋊C₈	—	83	1	no	—
442	C₄⋊C₈	—	83	1	no	—
443	C₄⋊C₈	—	83	1	no	—
444	C₄⋊C₈	—	83	1	no	—
445	C₄⋊C₈	—	83	1	no	—
446	C₄⋊C₈	—	166	1	no	—
447	C₄⋊C₈	—	166	1	no	—
448	C₄⋊C₈	—	166	1	no	—
449	C₄⋊C₈	—	166	1	no	—
450	C₄⋊C₈	—	166	1	no	—
451	C₄⋊C₈	—	166	1	no	—
452	C₄⋊C₈	—	166	1	no	—
453	C₄⋊C₈	—	166	1	no	—
454	C₄⋊C₈	—	166	1	no	—
455	C₄⋊C₈	—	166	1	no	—
456	#104	—	166	2	no	—
457	#104	Dic₁₆₆	1	4	yes	complemented
458	#103	—	83	1	no	—
459	#103	—	83	1	no	—
460	#103	C₄×D₈₃	1	1	yes	characteristic
461	#103	C₄×D₈₃	1	1	yes	characteristic
462	#102	—	83	1	no	—
463	#102	—	83	1	no	—
464	#102	C₄×D₈₃	1	1	yes	characteristic
465	#102	C₄×D₈₃	1	1	yes	characteristic
466	#97	—	2	1	no	—
467	#97	—	166	1	no	—
468	#97	—	166	1	no	—
469	C₄×C₈	—	2	1	no	—
470	C₄×C₈	—	4	1	no	—
471	C₄×C₈	—	332	1	no	—
472	C₄×C₈	—	166	1	no	—
473	C₄×C₈	—	166	1	no	—
474	#41	—	2	1	no	—
475	#41	—	2	1	no	—
476	#41	C₂×Dic₁₆₆	1	2	yes	—
477	#41	#13050590	1	2	yes	—
478	C₂×Q₈	—	166	1	no	—
479	C₂×Q₈	—	166	1	no	—
480	C₂×Q₈	C₂×C₄×D₈₃	1	1	yes	characteristic
481	C₂×D₄	—	2	1	no	—
482	C₂×D₄	C₂×C₄×D₈₃	1	1	yes	characteristic
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	83	1	no	—
486	C₂²×C₄	—	83	1	no	—
487	C₂²×C₄	—	166	1	no	—
488	C₂²×C₄	—	166	1	no	—
489	C₂²×C₄	—	166	1	no	—
490	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
491	M₁₆	—	2	1	no	—
492	M₁₆	—	2	1	no	—
493	M₁₆	—	2	1	no	—
494	M₁₆	—	4	2	no	—
495	M₁₆	—	332	2	no	—
496	M₁₆	—	166	1	no	—
497	M₁₆	—	166	1	no	—
498	M₁₆	—	166	1	no	—
499	M₁₆	—	166	1	no	—
500	M₁₆	C₂×Dic₁₆₆	1	2	yes	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	4	2	no	—
508	C₂×C₈	—	332	2	no	—
509	C₂×C₈	—	83	1	no	—
510	C₂×C₈	—	83	1	no	—
511	C₂×C₈	—	83	1	no	—
512	C₂×C₈	—	83	1	no	—
513	C₂×C₈	—	166	1	no	—
514	C₂×C₈	—	166	1	no	—
515	C₂×C₈	—	166	1	no	—
516	C₂×C₈	—	166	1	no	—
517	C₂×C₈	—	166	1	no	—
518	C₂×C₈	—	166	1	no	—
519	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
520	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
521	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
522	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
523	C₄⋊C₄	—	83	1	no	—
524	C₄⋊C₄	—	83	1	no	—
525	C₄⋊C₄	—	83	1	no	—
526	C₄⋊C₄	—	83	1	no	—
527	C₄⋊C₄	—	166	1	no	—
528	C₄⋊C₄	—	166	1	no	—
529	C₄⋊C₄	—	166	1	no	—
530	C₄⋊C₄	—	166	1	no	—
531	C₄⋊C₄	—	166	1	no	—
532	C₄⋊C₄	—	166	1	no	—
533	C₄⋊C₄	—	166	1	no	—
534	C₄⋊C₄	—	166	1	no	—
535	C₄⋊C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
536	C₄⋊C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
537	C₂²⋊C₄	—	2	1	no	—
538	C₂²⋊C₄	—	166	1	no	—
539	C₂²⋊C₄	—	166	1	no	—
540	C₂²⋊C₄	—	166	1	no	—
541	C₂²⋊C₄	—	166	1	no	—
542	C₄²	—	2	1	no	—
543	C₄²	—	2	1	no	—
544	C₄²	—	332	2	no	—
545	C₄²	—	83	1	no	—
546	C₄²	—	83	1	no	—
547	C₄²	—	83	1	no	—
548	C₄²	—	83	1	no	—
549	C₄²	—	166	1	no	—
550	C₄²	—	166	1	no	—
551	C₄²	—	166	4	no	—
552	C₄²	—	166	1	no	—
553	C₄²	—	166	1	no	—
554	C₄²	—	166	1	no	—
555	C₄²	—	166	1	no	—
556	C₄²	C₂×Dic₁₆₆	1	2	yes	—
557	C₂³	—	2	1	no	—
558	C₂³	#41×D₈₃	1	1	yes	characteristic
559	Q₈	—	2	1	no	—
560	Q₈	—	332	2	no	—
561	Q₈	—	664	1	no	—
562	Q₈	C₄×Dic₁₆₆	1	2	yes	—
563	D₄	—	2	2	no	—
564	D₄	—	2	2	no	—
565	D₄	—	2	1	no	—
566	D₄	C₄×Dic₁₆₆	1	2	yes	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	332	1	no	—
576	C₂×C₄	—	332	1	no	—
577	C₂×C₄	—	332	2	no	—
578	C₂×C₄	—	332	1	no	—
579	C₂×C₄	—	83	1	no	—
580	C₂×C₄	—	83	1	no	—
581	C₂×C₄	—	83	1	no	—
582	C₂×C₄	—	83	1	no	—
583	C₂×C₄	—	664	1	no	—
584	C₂×C₄	—	166	2	no	—
585	C₂×C₄	—	166	1	no	—
586	C₂×C₄	—	166	1	no	—
587	C₂×C₄	—	166	1	no	—
588	C₂×C₄	—	166	1	no	—
589	C₂×C₄	—	166	2	no	—
590	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
591	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
592	C₂×C₄	C₂²×Dic₁₆₆	1	1	yes	characteristic
593	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
594	C₂×C₄	#14581837	1	1	yes	characteristic
595	C₂×C₄	#14581839	1	1	yes	characteristic
596	C₈	—	2	1	no	—
597	C₈	—	2	1	no	—
598	C₈	—	2	2	no	—
599	C₈	—	2	1	no	—
600	C₈	—	2	1	no	—
601	C₈	—	4	1	no	—
602	C₈	—	332	1	no	—
603	C₈	—	332	1	no	—
604	C₈	—	166	1	no	—
605	C₈	—	166	1	no	—
606	C₈	—	166	1	no	—
607	C₈	—	166	1	no	—
608	C₂²	—	2	1	no	—
609	C₂²	—	2	1	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	4	1	no	—
612	C₂²	—	4	1	no	—
613	C₂²	C₄×D₄×D₈₃	1	1	yes	characteristic
614	C₂²	#19128384	1	1	yes	characteristic
615	C₂²	#19128410	1	1	yes	characteristic
616	C₄	—	2	4	no	—
617	C₄	—	2	2	no	—
618	C₄	—	4	1	no	—
619	C₄	—	4	1	no	—
620	C₄	—	332	2	no	—
621	C₄	—	332	1	no	—
622	C₄	—	332	1	no	—
623	C₄	—	332	1	no	—
624	C₄	—	332	1	no	—
625	C₄	—	83	2	no	—
626	C₄	—	166	1	no	—
627	C₄	—	166	1	no	—
628	C₄	—	166	1	no	—
629	C₄	C₂×C₄×Dic₁₆₆	1	1	yes	characteristic
630	C₄	#19128180	1	1	yes	characteristic
631	C₄	#19128341	1	1	yes	characteristic
632	C₄	D₄×Dic₁₆₆	1	1	yes	characteristic
633	C₂	—	2	1	no	—
634	C₂	—	2	1	no	—
635	C₂	—	4	2	no	—
636	C₂	#24910469	1	1	yes	characteristic
637	C₂	#443×D₈₃	1	1	yes	characteristic
638	C₂	#24916368	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	C₈₃×#1764	C₂
socle	C₂²×C₈₃	C₂×C₄×D₄
prime radical	C₂×#41×C₈₃	C₂³
cosocle	C₂²×C₄×C₈₃	C₂⁴
hypercenter	#1764	D₈₃
hypocenter	C₈₃	#22982
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1764

Complements

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

N	H ≅ G/N	type
2. #24916407	C₂	semidirect
4. #24908438	C₂	semidirect
5. #24908237	C₂	semidirect
6. #24908211	C₂	semidirect
7. #24908210	C₂	semidirect
8. #24908062	C₂	semidirect
9. #24908061	C₂	semidirect
12. C₄×Dic₈₃⋊C₈	C₂	semidirect
19. #19128010	C₄	semidirect
20. #19128006	C₄	semidirect
21. #19128003	C₄	semidirect
22. #19127999	C₄	semidirect
23. #19127994	C₄	semidirect
24. #19127989	C₄	semidirect
25. #19127984	C₄	semidirect
26. #19127979	C₄	semidirect
146. #104×C₈₃	Q₈	semidirect
335. C₈₃	#22982	semidirect
457. #104	Dic₁₆₆	semidirect

Maximal subgroups

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

subgroup	index
#24917247	2
#24916407	2
#24916029	2
#24908438	2
#24908237	2
#24908211	2
#24908210	2
#24908062	2
#24908061	2
#24907740	2
#24907738	2
C₄×Dic₈₃⋊C₈	2
#24907301	2
#104×Dic₈₃	2
C₈₃×#1764	2
#22982	83