Subgroups — #24916583

Order 10624.

398 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13050589	C₂	1	1	yes	maximal, characteristic
2	#19127830	C₂	1	1	yes	maximal, characteristic, complemented
3	#19127829	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127775	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127773	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127770	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	(C₂²⋊D₈₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127756	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127755	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127751	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127724	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×Dic₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄⋊C₈×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
15	#435×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050589	C₄	1	1	yes	characteristic
17	C₂×#13050574	C₂²	1	1	yes	characteristic
18	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
22	#14581757	C₄	1	1	yes	characteristic
23	D₄×Dic₈₃	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊D₈₃	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₈₃	C₂²	1	1	yes	characteristic
26	(C₂²⋊D₈₃)⋊C₄	C₄	1	1	yes	characteristic
27	(C₂²⋊D₈₃)⋊C₄	C₄	1	1	yes	characteristic
28	#14581750	C₄	1	1	yes	characteristic
29	#14581748	C₄	1	1	yes	characteristic
30	#14581748	C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
33	C₄²×D₈₃	C₂²	1	1	yes	characteristic
34	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
35	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
36	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
37	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
38	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
39	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
41	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
43	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
44	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
45	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
46	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
47	#14581703	C₂²	1	1	yes	characteristic
48	#14581702	C₂²	1	1	yes	characteristic
49	#14581702	C₂²	1	1	yes	characteristic
50	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
58	#97×C₈₃	C₂²	1	1	yes	characteristic
59	#13050589	—	2	4	no	—
60	C₂×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₈₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×Dic₈₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×Dic₈₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×Dic₈₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×Dic₈₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
70	#13050574	—	4	1	no	—
71	C₂×C₈₃⋊C₈	—	4	1	no	—
72	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₈×D₈₃	—	4	1	no	—
77	C₈×D₈₃	—	4	1	no	—
78	C₄⋊Dic₈₃	C₂×C₄	1	1	yes	characteristic
79	Dic₈₃⋊C₄	C₂×C₄	1	1	yes	characteristic
80	Dic₈₃⋊C₄	C₂×C₄	1	1	yes	characteristic
81	Dic₈₃⋊C₄	C₂×C₄	1	1	yes	characteristic
82	Dic₈₃⋊C₄	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊Dic₈₃	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊Dic₈₃	C₂×C₄	1	1	yes	characteristic
85	(C₂×D₈₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
86	(C₂×D₈₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
87	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
88	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
89	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
90	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
91	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
94	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
95	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₈₃	C₂×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
99	M₁₆×C₈₃	—	4	1	no	—
100	C₂×C₈×C₈₃	—	4	1	no	—
101	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
106	C₂²⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
107	C₂²⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
108	C₄²×C₈₃	C₂³	1	1	yes	characteristic
109	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
110	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
111	C₂²×D₈₃	C₂²×C₄	1	1	yes	characteristic
112	D₄×C₈₃	—	2	2	no	—
113	C₂×C₄×C₈₃	—	2	1	no	—
114	C₂×C₄×C₈₃	—	2	1	no	—
115	C₂×C₄×C₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₈₃	—	2	2	no	—
123	C₂²⋊D₈₃	M₁₆	1	4	yes	complemented
124	C₂×Dic₈₃	—	2	2	no	—
125	C₂×Dic₈₃	—	2	2	no	—
126	C₂×Dic₈₃	—	2	2	no	—
127	C₂×Dic₈₃	—	2	2	no	—
128	C₂×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
133	C₂×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
134	C₂×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
135	C₄×D₈₃	—	2	2	no	—
136	C₄×D₈₃	—	2	1	no	—
137	C₄×D₈₃	—	2	1	no	—
138	C₄×D₈₃	—	2	2	no	—
139	Dic₁₆₆	—	2	2	no	—
140	C₈×C₈₃	—	2	1	no	—
141	C₈×C₈₃	—	2	1	no	—
142	C₈×C₈₃	—	2	1	no	—
143	C₈×C₈₃	—	2	1	no	—
144	C₈₃⋊C₈	—	2	1	no	—
145	C₈₃⋊C₈	—	2	1	no	—
146	C₈₃⋊C₈	—	2	1	no	—
147	C₈₃⋊C₈	—	2	1	no	—
148	C₂²×C₈₃	—	2	1	no	—
149	C₂²×C₈₃	—	2	1	no	—
150	C₂²×C₈₃	—	2	1	no	—
151	C₂²×C₈₃	—	2	1	no	—
152	C₂²×C₈₃	—	2	1	no	—
153	C₂²×C₈₃	C₂×M₁₆	1	2	yes	—
154	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
155	C₂×D₈₃	—	2	1	no	—
156	C₂×D₈₃	—	2	1	no	—
157	C₂×D₈₃	C₂×M₁₆	1	2	yes	—
158	C₄×C₈₃	—	2	1	no	—
159	C₄×C₈₃	—	2	1	no	—
160	C₄×C₈₃	—	2	1	no	—
161	C₄×C₈₃	—	2	1	no	—
162	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
163	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
164	Dic₈₃	—	2	1	no	—
165	Dic₈₃	—	2	1	no	—
166	Dic₈₃	—	2	1	no	—
167	Dic₈₃	—	2	1	no	—
168	Dic₈₃	—	2	1	no	—
169	Dic₈₃	C₂×M₁₆	1	2	yes	—
170	Dic₈₃	#131	1	2	yes	—
171	C₂×C₈₃	—	2	2	no	—
172	C₂×C₈₃	—	4	1	no	—
173	C₂×C₈₃	#520	1	1	yes	characteristic
174	C₂×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
175	C₂×C₈₃	C₂×#131	1	1	yes	characteristic
176	D₈₃	—	2	2	no	—
177	#2990	—	83	1	no	maximal, Sylow
178	C₈₃	#2990	1	1	yes	characteristic, complemented, Sylow
179	C₄×#41	—	83	1	no	—
180	#446	—	83	1	no	—
181	#446	—	83	1	no	—
182	#435	—	83	1	no	—
183	#435	—	83	1	no	—
184	#435	—	83	1	no	—
185	#435	D₈₃	1	1	yes	characteristic, complemented
186	C₈×D₄	—	83	1	no	—
187	C₈×D₄	—	83	1	no	—
188	#432	—	83	1	no	—
189	#432	—	83	1	no	—
190	#431	—	83	1	no	—
191	#423	—	83	1	no	—
192	C₂×C₄⋊C₈	—	83	1	no	—
193	C₂×C₄⋊C₈	—	83	1	no	—
194	C₂×#41	—	83	1	no	—
195	C₂×M₁₆	—	83	1	no	—
196	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
197	C₂²×C₈	—	83	1	no	—
198	C₂²×C₈	—	83	1	no	—
199	C₂²×C₈	—	83	1	no	—
200	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
201	C₄×Q₈	—	83	1	no	—
202	C₄×Q₈	—	83	1	no	—
203	C₄×D₄	—	83	1	no	—
204	C₄×D₄	—	83	1	no	—
205	C₄×D₄	—	83	1	no	—
206	C₄×D₄	—	83	1	no	—
207	C₄×D₄	—	83	1	no	—
208	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
209	#117	—	83	1	no	—
210	#117	—	83	1	no	—
211	#117	—	83	1	no	—
212	C₂×C₄²	—	83	1	no	—
213	C₂×C₄²	—	83	1	no	—
214	C₂×C₄²	—	83	1	no	—
215	C₄⋊C₈	—	83	1	no	—
216	C₄⋊C₈	—	83	1	no	—
217	C₄⋊C₈	—	83	1	no	—
218	C₄⋊C₈	—	83	1	no	—
219	C₄⋊C₈	—	83	1	no	—
220	C₄⋊C₈	—	83	1	no	—
221	C₄⋊C₈	—	83	1	no	—
222	C₄⋊C₈	—	83	1	no	—
223	C₄⋊C₈	—	83	1	no	—
224	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
225	C₂²⋊C₈	—	83	1	no	—
226	C₂²⋊C₈	—	83	1	no	—
227	C₂²⋊C₈	—	83	1	no	—
228	C₂²⋊C₈	—	83	1	no	—
229	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
230	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
231	#97	—	83	1	no	—
232	#97	—	83	1	no	—
233	#97	—	83	1	no	—
234	#97	C₂×D₈₃	1	1	yes	characteristic
235	C₄×C₈	—	83	1	no	—
236	C₄×C₈	—	83	1	no	—
237	#41	—	166	4	no	—
238	C₂×Q₈	—	83	1	no	—
239	C₂×D₄	—	83	1	no	—
240	C₂×D₄	—	83	1	no	—
241	C₂×D₄	C₄×D₈₃	1	1	yes	characteristic
242	C₂²×C₄	—	83	1	no	—
243	C₂²×C₄	—	83	1	no	—
244	C₂²×C₄	—	83	1	no	—
245	C₂²×C₄	—	83	1	no	—
246	C₂²×C₄	—	83	1	no	—
247	C₂²×C₄	—	83	1	no	—
248	C₂²×C₄	—	83	1	no	—
249	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
251	M₁₆	—	4	1	no	—
252	M₁₆	—	332	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	332	1	no	—
255	C₂×C₈	—	332	1	no	—
256	C₂×C₈	—	332	1	no	—
257	C₂×C₈	—	83	1	no	—
258	C₂×C₈	—	83	1	no	—
259	C₂×C₈	—	83	1	no	—
260	C₂×C₈	—	83	1	no	—
261	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
265	C₄⋊C₄	—	83	1	no	—
266	C₄⋊C₄	—	83	1	no	—
267	C₄⋊C₄	—	83	1	no	—
268	C₄⋊C₄	—	83	1	no	—
269	C₄⋊C₄	—	83	1	no	—
270	C₄⋊C₄	C₄×D₈₃	1	1	yes	characteristic
271	C₂²⋊C₄	—	83	1	no	—
272	C₂²⋊C₄	—	83	1	no	—
273	C₂²⋊C₄	—	83	1	no	—
274	C₂²⋊C₄	—	83	1	no	—
275	C₂²⋊C₄	C₄×D₈₃	1	1	yes	characteristic
276	C₂²⋊C₄	C₄×D₈₃	1	1	yes	characteristic
277	C₄²	—	83	1	no	—
278	C₄²	—	83	1	no	—
279	C₄²	—	83	1	no	—
280	C₄²	—	83	1	no	—
281	C₄²	—	83	1	no	—
282	C₄²	—	83	1	no	—
283	C₄²	—	83	1	no	—
284	C₄²	—	83	1	no	—
285	C₄²	—	83	1	no	—
286	C₄²	C₂²×D₈₃	1	1	yes	characteristic
287	C₂³	—	83	1	no	—
288	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
289	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
290	Q₈	—	166	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	83	4	no	—
293	D₄	—	166	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	83	1	no	—
299	C₂×C₄	—	83	1	no	—
300	C₂×C₄	—	83	1	no	—
301	C₂×C₄	—	83	1	no	—
302	C₂×C₄	—	83	1	no	—
303	C₂×C₄	—	83	1	no	—
304	C₂×C₄	—	83	1	no	—
305	C₂×C₄	—	166	2	no	—
306	C₂×C₄	—	166	2	no	—
307	C₂×C₄	—	166	2	no	—
308	C₂×C₄	—	166	2	no	—
309	C₂×C₄	—	166	2	no	—
310	C₂×C₄	—	166	2	no	—
311	C₂×C₄	—	166	1	no	—
312	C₂×C₄	—	166	1	no	—
313	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	2	1	no	—
322	C₈	—	166	1	no	—
323	C₈	—	166	1	no	—
324	C₈	—	166	1	no	—
325	C₈	—	166	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	83	2	no	—
332	C₂²	—	166	1	no	—
333	C₂²	—	166	1	no	—
334	C₂²	M₁₆×D₈₃	1	2	yes	—
335	C₂²	C₂²×C₄×D₈₃	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	83	2	no	—
341	C₄	—	83	2	no	—
342	C₄	—	166	1	no	—
343	C₄	—	166	1	no	—
344	C₄	—	166	1	no	—
345	C₄	—	166	1	no	—
346	C₄	—	166	1	no	—
347	C₄	#14581837	1	1	yes	characteristic
348	C₄	#14581839	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	1	no	—
351	C₂	—	166	2	no	—
352	C₂	#19128180	1	1	yes	characteristic
353	C₂	C₂×M₁₆×D₈₃	1	1	yes	characteristic
354	C₂	#131×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₄
Fitting	#435×C₈₃	C₂
prime radical	C₂×#13050589	C₄
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	#435	D₈₃
hypocenter	C₈₃	#2990
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. #19127830	C₂	semidirect
3. #19127829	C₂	semidirect
4. #19127775	C₂	semidirect
5. #19127773	C₂	semidirect
6. #19127770	C₂	semidirect
7. C₈×C₂²⋊D₈₃	C₂	semidirect
8. (C₂²⋊D₈₃)⋊C₈	C₂	semidirect
9. #19127756	C₂	semidirect
10. #19127755	C₂	semidirect
11. #19127751	C₂	semidirect
12. #19127724	C₂	semidirect
13. C₂×Dic₈₃⋊C₈	C₂	semidirect
14. C₄⋊C₈×D₈₃	C₂	semidirect
15. #435×C₈₃	C₂	semidirect
123. C₂²⋊D₈₃	M₁₆	semidirect
178. C₈₃	#2990	semidirect
185. #435	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13050589	2
#19127830	2
#19127829	2
#19127775	2
#19127773	2
#19127770	2
C₈×C₂²⋊D₈₃	2
(C₂²⋊D₈₃)⋊C₈	2
#19127756	2
#19127755	2
#19127751	2
#19127724	2
C₂×Dic₈₃⋊C₈	2
C₄⋊C₈×D₈₃	2
#435×C₈₃	2
#2990	83