Subgroups — #35319441

Order 21248.

514 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127656	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₈₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127476	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#24905540	C₂	1	1	yes	maximal, characteristic, complemented
6	#24905540	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#13050574	C₄	1	1	yes	characteristic
9	C₂×#14581751	C₄	1	1	yes	characteristic
10	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	#19127656	—	2	2	no	—
13	#19127656	—	2	2	no	—
14	C₂×#14581702	C₄	1	1	yes	characteristic
15	C₂×#14581702	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
18	C₂²⋊(C₈₃⋊C₁₆)	C₄	1	4	yes	complemented
19	#19127476	—	2	2	no	—
20	C₁₆×Dic₈₃	—	2	2	no	—
21	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
24	C₂²⋊C₁₆×C₈₃	C₄	1	4	yes	complemented
25	C₂×#13050574	—	2	2	no	—
26	C₂×#13050574	—	2	2	no	—
27	C₂×#13050574	C₈	1	4	yes	—
28	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
30	#14581751	—	2	2	no	—
31	#14581751	—	2	2	no	—
32	C₂×C₄⋊Dic₈₃	C₈	1	2	yes	—
33	C₂×C₂²⋊Dic₈₃	C₈	1	2	yes	—
34	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈₃⋊C₁₆	—	2	1	no	—
37	C₂×C₈₃⋊C₁₆	—	2	1	no	—
38	C₂×C₈₃⋊C₁₆	—	2	1	no	—
39	C₂×C₈₃⋊C₁₆	—	2	1	no	—
40	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
41	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
42	#14581702	—	2	2	no	—
43	#14581702	—	2	2	no	—
44	C₈×Dic₈₃	—	2	1	no	—
45	C₈×Dic₈₃	—	2	1	no	—
46	C₈×Dic₈₃	D₄	1	2	yes	—
47	C₈×Dic₈₃	D₄	1	2	yes	—
48	C₂³×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₃	—	2	1	no	—
50	C₂²×C₈×C₈₃	—	2	1	no	—
51	C₂²×C₈×C₈₃	—	2	1	no	—
52	C₂²×C₈×C₈₃	—	2	1	no	—
53	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
56	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₈₃	—	2	1	no	—
58	C₂×C₁₆×C₈₃	—	2	1	no	—
59	C₂×C₁₆×C₈₃	—	2	1	no	—
60	C₂×C₁₆×C₈₃	—	2	1	no	—
61	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
62	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
63	C₂²×Dic₈₃	C₂×C₈	1	2	yes	—
64	C₂²×Dic₈₃	C₂×C₈	1	2	yes	—
65	#13050574	—	2	8	no	—
66	C₂×C₈₃⋊C₈	—	2	1	no	—
67	C₂×C₈₃⋊C₈	—	2	1	no	—
68	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈₃⋊C₈	C₂×C₈	1	2	yes	—
70	C₂×C₈₃⋊C₈	C₂×C₈	1	2	yes	—
71	C₂×C₈₃⋊C₈	M₁₆	1	2	yes	—
72	C₄⋊Dic₈₃	—	2	4	no	—
73	C₂²⋊Dic₈₃	—	2	4	no	—
74	C₄×Dic₈₃	—	2	1	no	—
75	C₄×Dic₈₃	—	2	1	no	—
76	C₄×Dic₈₃	C₂²⋊C₄	1	2	yes	—
77	C₄×Dic₈₃	M₁₆	1	2	yes	—
78	C₈₃⋊C₁₆	—	2	2	no	—
79	C₈₃⋊C₁₆	—	2	2	no	—
80	C₂⁴×C₈₃	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₈₃	—	2	1	no	—
82	C₂²×C₄×C₈₃	—	2	1	no	—
83	C₂²×C₄×C₈₃	—	2	1	no	—
84	C₂²×C₄×C₈₃	—	2	1	no	—
85	C₂²×C₄×C₈₃	C₄²	1	2	yes	—
86	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
87	C₂²×C₄×C₈₃	C₂×C₈	1	2	yes	—
88	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₈₃	—	2	1	no	—
90	C₂×C₈×C₈₃	—	2	1	no	—
91	C₂×C₈×C₈₃	—	2	2	no	—
92	C₂×C₈×C₈₃	—	2	2	no	—
93	C₂×C₈×C₈₃	—	2	2	no	—
94	C₂×C₈×C₈₃	—	2	2	no	—
95	C₂×C₈×C₈₃	C₄²	1	2	yes	—
96	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
101	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
102	C₁₆×C₈₃	—	2	2	no	—
103	C₁₆×C₈₃	—	2	2	no	—
104	C₂³×C₈₃	—	2	1	no	—
105	C₂³×C₈₃	—	2	1	no	—
106	C₂³×C₈₃	—	2	2	no	—
107	C₂³×C₈₃	—	2	2	no	—
108	C₂³×C₈₃	C₄×C₈	1	2	yes	—
109	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₈₃	—	2	1	no	—
111	C₂×C₄×C₈₃	—	2	1	no	—
112	C₂×C₄×C₈₃	—	2	2	no	—
113	C₂×C₄×C₈₃	—	2	2	no	—
114	C₂×C₄×C₈₃	—	2	2	no	—
115	C₂×C₄×C₈₃	—	2	2	no	—
116	C₂×C₄×C₈₃	C₄×C₈	1	2	yes	—
117	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
123	C₂×Dic₈₃	—	2	2	no	—
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⋊C₈	1	4	yes	—
129	C₈×C₈₃	—	2	2	no	—
130	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
131	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
132	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
133	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
134	C₈₃⋊C₈	—	2	2	no	—
135	C₈₃⋊C₈	C₂²⋊C₈	1	4	yes	—
136	C₂²×C₈₃	—	2	2	no	—
137	C₂²×C₈₃	—	2	2	no	—
138	C₂²×C₈₃	—	2	2	no	—
139	C₂²×C₈₃	—	4	2	no	—
140	C₂²×C₈₃	—	4	2	no	—
141	C₂²×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
142	C₂²×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
143	C₂²×C₈₃	#431	1	1	yes	characteristic
144	C₂²×C₈₃	#504	1	2	yes	—
145	C₂²×C₈₃	#504	1	2	yes	—
146	C₄×C₈₃	—	2	2	no	—
147	C₄×C₈₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
148	C₄×C₈₃	C₄×M₁₆	1	1	yes	characteristic
149	C₄×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₄×C₈₃	#431	1	1	yes	characteristic
151	Dic₈₃	—	2	4	no	—
152	Dic₈₃	—	4	2	no	—
153	#12600	—	83	1	no	maximal, Sylow
154	C₂×C₈₃	—	4	2	no	—
155	C₂×C₈₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
156	C₂×C₈₃	#2108	1	1	yes	characteristic
157	C₂×C₈₃	#2108	1	1	yes	characteristic
158	C₂×C₈₃	#2113	1	2	yes	—
159	C₂×C₈₃	#2164	1	2	yes	—
160	C₂×#405	—	83	1	no	—
161	C₂×C₂²⋊C₁₆	—	83	1	no	—
162	C₂×C₂²⋊C₁₆	D₈₃	1	1	yes	characteristic
163	C₂×#346	—	83	1	no	—
164	C₂×C₄×C₁₆	—	83	1	no	—
165	#1374	—	83	1	no	—
166	#1374	—	83	1	no	—
167	C₈₃	#12600	1	1	yes	characteristic, complemented, Sylow
168	C₂²×M₁₆	—	83	1	no	—
169	C₂³×C₈	C₂×D₈₃	1	1	yes	characteristic
170	C₂×#117	—	83	1	no	—
171	C₂²×C₁₆	—	83	1	no	—
172	C₂²×C₁₆	—	83	1	no	—
173	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
174	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
175	#405	—	166	2	no	—
176	#405	—	166	2	no	—
177	C₂×#97	—	83	1	no	—
178	C₂×#97	—	83	1	no	—
179	C₂×C₄×C₈	—	83	1	no	—
180	C₂×C₄×C₈	—	83	1	no	—
181	C₂²⋊C₁₆	—	83	4	no	—
182	C₂²⋊C₁₆	Dic₈₃	1	4	yes	complemented
183	#346	—	166	2	no	—
184	C₄×C₁₆	—	166	2	no	—
185	C₂³×C₄	C₄×D₈₃	1	1	yes	characteristic
186	C₂×M₁₆	—	83	4	no	—
187	C₂×M₁₆	—	166	2	no	—
188	C₂×M₁₆	—	166	2	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	83	1	no	—
194	C₂²×C₈	—	83	1	no	—
195	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
196	C₂²×C₈	C₄×D₈₃	1	2	yes	—
197	C₂²×C₈	C₂×Dic₈₃	1	2	yes	—
198	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
199	#117	—	166	2	no	—
200	#117	—	166	2	no	—
201	C₂×C₄⋊C₄	—	83	2	no	—
202	C₂×C₂²⋊C₄	—	83	2	no	—
203	C₂×C₄²	—	83	1	no	—
204	C₂×C₄²	—	83	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	2	1	no	—
209	C₂×C₁₆	—	83	2	no	—
210	C₂×C₁₆	—	83	2	no	—
211	C₂×C₁₆	—	166	1	no	—
212	C₂×C₁₆	—	166	1	no	—
213	C₂×C₁₆	—	166	1	no	—
214	C₂×C₁₆	—	166	1	no	—
215	C₂×C₁₆	C₂×Dic₈₃	1	2	yes	—
216	C₂×C₁₆	C₂×Dic₈₃	1	2	yes	—
217	#97	—	166	2	no	—
218	#97	—	166	2	no	—
219	C₄×C₈	—	83	2	no	—
220	C₄×C₈	—	83	2	no	—
221	C₄×C₈	—	166	1	no	—
222	C₄×C₈	—	166	1	no	—
223	C₂⁴	C₈×D₈₃	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	83	2	no	—
229	C₂²×C₄	—	83	2	no	—
230	C₂²×C₄	C₄×Dic₈₃	1	2	yes	—
231	C₂²×C₄	C₈×D₈₃	1	2	yes	—
232	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
234	M₁₆	—	166	8	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	83	2	no	—
242	C₂×C₈	—	83	2	no	—
243	C₂×C₈	—	83	2	no	—
244	C₂×C₈	—	83	2	no	—
245	C₂×C₈	—	166	1	no	—
246	C₂×C₈	—	166	1	no	—
247	C₂×C₈	C₄×Dic₈₃	1	2	yes	—
248	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
249	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
250	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
251	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
252	C₂×C₈	#13050589	1	1	yes	characteristic
253	C₂×C₈	#13050589	1	1	yes	characteristic
254	C₄⋊C₄	—	166	4	no	—
255	C₂²⋊C₄	—	166	4	no	—
256	C₄²	—	83	2	no	—
257	C₄²	—	83	2	no	—
258	C₄²	—	166	1	no	—
259	C₄²	—	166	1	no	—
260	C₁₆	—	2	2	no	—
261	C₁₆	—	2	2	no	—
262	C₁₆	—	166	2	no	—
263	C₁₆	—	166	2	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	C₈×Dic₈₃	1	2	yes	—
269	C₂³	C₂×C₈×D₈₃	1	1	yes	characteristic
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	83	4	no	—
277	C₂×C₄	—	166	2	no	—
278	C₂×C₄	—	166	2	no	—
279	C₂×C₄	—	166	2	no	—
280	C₂×C₄	—	166	2	no	—
281	C₂×C₄	—	166	2	no	—
282	C₂×C₄	C₈×Dic₈₃	1	2	yes	—
283	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
284	C₂×C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
285	C₂×C₄	#14581748	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
287	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	83	4	no	—
291	C₈	—	166	2	no	—
292	C₈	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
293	C₈	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
294	C₈	D₄×Dic₈₃	1	1	yes	characteristic
295	C₈	D₄×Dic₈₃	1	1	yes	characteristic
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	C₂×C₈×Dic₈₃	1	1	yes	characteristic
302	C₂²	C₂²⋊C₈×D₈₃	1	1	yes	characteristic
303	C₂²	#19127751	1	1	yes	characteristic
304	C₂²	#19128117	1	2	yes	—
305	C₂²	#19128119	1	2	yes	—
306	C₄	—	2	2	no	—
307	C₄	—	332	2	no	—
308	C₄	—	166	4	no	—
309	C₄	C₂²⋊C₄×Dic₈₃	1	1	yes	characteristic
310	C₄	M₁₆×Dic₈₃	1	1	yes	characteristic
311	C₄	C₂²⋊C₈×D₈₃	1	1	yes	characteristic
312	C₄	#19127751	1	1	yes	characteristic
313	C₂	—	4	2	no	—
314	C₂	C₂²⋊C₈×Dic₈₃	1	1	yes	characteristic
315	C₂	#24909059	1	1	yes	characteristic
316	C₂	#24909062	1	1	yes	characteristic
317	C₂	#24909075	1	2	yes	—
318	C₂	#24909267	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂×C₄×C₈
Fitting	C₂×C₂²⋊C₁₆×C₈₃	C₂
socle	C₂³×C₈₃	C₂²×C₈
prime radical	C₂⁴×C₈₃	C₂×C₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₈₃
hypocenter	C₈₃	#12600
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#19127476	C₂	semidirect
4. C₂×C₁₆×Dic₈₃	C₂	semidirect
5. #24905540	C₂	semidirect
6. #24905540	C₂	semidirect
18. C₂²⋊(C₈₃⋊C₁₆)	C₄	semidirect
24. C₂²⋊C₁₆×C₈₃	C₄	semidirect
167. C₈₃	#12600	semidirect
182. C₂²⋊C₁₆	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127656	2
C₂×C₂²⋊(C₈₃⋊C₁₆)	2
C₂×#19127476	2
C₂×C₁₆×Dic₈₃	2
#24905540	2
C₂×C₂²⋊C₁₆×C₈₃	2
#12600	83