Subgroups — #344×D₂₅₇ #45664060

Order 32896.

388 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₂₅₇	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₂₅₇	C₂	1	1	yes	maximal, characteristic
3	#117×D₂₅₇	C₂	1	1	yes	maximal, characteristic
4	#30656509	C₂	1	1	yes	maximal, characteristic, complemented
5	#30656508	C₂	1	1	yes	maximal, characteristic, complemented
6	#30656507	C₂	1	1	yes	maximal, characteristic, complemented
7	#344×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₂₅₇	C₂²	1	1	yes	characteristic
9	C₂×#16861602	C₂²	1	1	yes	characteristic
10	C₂×#16861602	C₂²	1	1	yes	characteristic
11	C₂×#16861601	—	2	1	no	—
12	C₂×#16861601	C₄	1	2	yes	complemented
13	M₁₆×D₂₅₇	—	2	1	no	—
14	M₁₆×D₂₅₇	—	2	1	no	—
15	M₁₆×D₂₅₇	—	2	1	no	—
16	M₁₆×D₂₅₇	—	2	1	no	—
17	M₁₆×D₂₅₇	—	2	1	no	—
18	M₁₆×D₂₅₇	—	2	1	no	—
19	M₁₆×D₂₅₇	C₄	1	2	yes	complemented
20	M₁₆×D₂₅₇	C₄	1	2	yes	complemented
21	C₂×C₈×D₂₅₇	—	2	1	no	—
22	C₂×C₈×D₂₅₇	C₄	1	2	yes	complemented
23	#22551551	C₂²	1	1	yes	characteristic
24	#22551550	—	2	1	no	—
25	#22551549	—	2	1	no	—
26	#22551548	C₄	1	2	yes	—
27	C₄⋊C₄×D₂₅₇	—	2	1	no	—
28	C₂²⋊C₄×D₂₅₇	C₄	1	2	yes	—
29	C₄²×D₂₅₇	—	2	1	no	—
30	C₂×M₁₆×C₂₅₇	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₂₅₇	C₂²	1	1	yes	characteristic
32	#117×C₂₅₇	C₂²	1	1	yes	characteristic
33	C₂³×D₂₅₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₂₅₇	—	2	2	no	—
36	C₂×C₄×D₂₅₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₂₅₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₂₅₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₂₅₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×D₂₅₇	D₄	1	1	yes	characteristic
41	C₂×C₄×D₂₅₇	D₄	1	1	yes	characteristic
42	C₂×C₄×D₂₅₇	D₄	1	1	yes	characteristic
43	C₂×C₄×D₂₅₇	D₄	1	1	yes	characteristic
44	C₂×C₄×D₂₅₇	D₄	1	1	yes	characteristic
45	C₂×C₄×D₂₅₇	D₄	1	1	yes	characteristic
46	C₂×C₄×D₂₅₇	Q₈	1	1	yes	characteristic
47	C₂×C₄×D₂₅₇	Q₈	1	1	yes	characteristic
48	#16861602	—	2	1	no	—
49	#16861602	—	2	1	no	—
50	#16861602	—	2	1	no	—
51	#16861602	C₂×C₄	1	2	yes	complemented
52	#16861601	—	2	2	no	—
53	#16861601	—	2	2	no	—
54	#16861601	—	2	2	no	—
55	#16861601	—	2	2	no	—
56	C₂×#14523339	—	2	1	no	—
57	C₂×#14523339	C₂×C₄	1	2	yes	complemented
58	C₈×D₂₅₇	—	2	2	no	—
59	C₈×D₂₅₇	—	2	2	no	—
60	C₈×D₂₅₇	—	2	2	no	—
61	C₈×D₂₅₇	—	2	2	no	—
62	C₄⋊Dic₂₅₇	—	2	1	no	—
63	#16861597	—	2	2	no	—
64	C₂²⋊Dic₂₅₇	C₂×C₄	1	2	yes	—
65	#16861595	—	2	2	no	—
66	C₄×Dic₂₅₇	—	2	2	no	—
67	C₄×Dic₂₅₇	—	2	1	no	—
68	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
69	M₁₆×C₂₅₇	—	2	1	no	—
70	M₁₆×C₂₅₇	—	2	1	no	—
71	M₁₆×C₂₅₇	—	2	1	no	—
72	M₁₆×C₂₅₇	C₂×C₄	1	2	yes	complemented
73	C₂×C₈×C₂₅₇	—	2	1	no	—
74	C₂×C₈×C₂₅₇	C₂×C₄	1	2	yes	complemented
75	C₄⋊C₄×C₂₅₇	—	2	1	no	—
76	C₂²⋊C₄×C₂₅₇	C₂×C₄	1	2	yes	—
77	C₄²×C₂₅₇	—	2	1	no	—
78	C₂²×D₂₅₇	—	4	2	no	—
79	C₂²×D₂₅₇	C₄²	1	1	yes	characteristic
80	C₂²×D₂₅₇	C₄²	1	1	yes	characteristic
81	C₂²×D₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×D₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×D₂₅₇	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×D₂₅₇	C₄⋊C₄	1	1	yes	characteristic
85	C₂×Dic₂₅₇	—	2	2	no	—
86	C₂×Dic₂₅₇	C₄²	1	1	yes	characteristic
87	C₂×Dic₂₅₇	C₄²	1	1	yes	characteristic
88	C₂×Dic₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×Dic₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×Dic₂₅₇	C₄⋊C₄	1	1	yes	characteristic
91	C₂×Dic₂₅₇	C₄⋊C₄	1	1	yes	characteristic
92	C₄×D₂₅₇	—	4	4	no	—
93	C₄×D₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×D₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×D₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
96	C₄×D₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
97	C₄×D₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
98	C₄×D₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
99	C₄×D₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
100	C₄×D₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
101	C₄×D₂₅₇	C₄⋊C₄	1	1	yes	characteristic
102	C₄×D₂₅₇	C₄⋊C₄	1	1	yes	characteristic
103	C₄×D₂₅₇	C₄⋊C₄	1	1	yes	characteristic
104	C₄×D₂₅₇	C₄⋊C₄	1	1	yes	characteristic
105	C₄×D₂₅₇	C₄⋊C₄	1	1	yes	characteristic
106	C₄×D₂₅₇	C₄⋊C₄	1	1	yes	characteristic
107	C₄×D₂₅₇	C₄⋊C₄	1	1	yes	characteristic
108	C₄×D₂₅₇	C₄⋊C₄	1	1	yes	characteristic
109	#14523339	—	2	2	no	—
110	#14523339	—	2	2	no	—
111	C₂³×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₅₇	—	2	2	no	—
113	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
119	C₈×C₂₅₇	—	2	2	no	—
120	C₈×C₂₅₇	—	2	2	no	—
121	C₂²×C₂₅₇	—	4	1	no	—
122	C₂²×C₂₅₇	C₂×C₄²	1	1	yes	characteristic
123	C₂²×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×D₂₅₇	—	2	2	no	—
126	C₂×D₂₅₇	—	2	2	no	—
127	C₂×D₂₅₇	—	2	2	no	—
128	C₂×D₂₅₇	—	4	1	no	—
129	C₂×D₂₅₇	—	4	1	no	—
130	C₂×D₂₅₇	—	4	1	no	—
131	C₂×D₂₅₇	#95	1	1	yes	characteristic
132	C₂×D₂₅₇	#95	1	1	yes	characteristic
133	C₂×D₂₅₇	#95	1	1	yes	characteristic
134	C₂×D₂₅₇	#95	1	1	yes	characteristic
135	C₄×C₂₅₇	—	4	2	no	—
136	C₄×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
140	Dic₂₅₇	—	4	2	no	—
141	Dic₂₅₇	#95	1	1	yes	characteristic
142	Dic₂₅₇	#95	1	1	yes	characteristic
143	Dic₂₅₇	#95	1	1	yes	characteristic
144	Dic₂₅₇	#95	1	1	yes	characteristic
145	C₂×C₂₅₇	—	2	1	no	—
146	C₂×C₂₅₇	—	2	1	no	—
147	C₂×C₂₅₇	—	2	1	no	—
148	C₂×C₂₅₇	C₂×#95	1	1	yes	characteristic
149	D₂₅₇	—	2	1	no	—
150	D₂₅₇	—	2	1	no	—
151	D₂₅₇	—	2	1	no	—
152	D₂₅₇	#344	1	2	yes	direct product factor
153	C₂₅₇	C₂×#344	1	1	yes	characteristic, complemented, Sylow
154	C₂×#344	—	257	1	no	maximal, Sylow
155	C₂²×M₁₆	—	257	1	no	—
156	C₂²×M₁₆	—	257	1	no	—
157	C₂×#117	—	257	1	no	—
158	#344	—	257	1	no	—
159	#344	—	257	1	no	—
160	#344	—	257	1	no	—
161	#344	D₂₅₇	1	1	yes	characteristic, direct product factor
162	C₂³×C₄	—	257	1	no	—
163	C₂×M₁₆	—	257	1	no	—
164	C₂×M₁₆	—	257	2	no	—
165	C₂×M₁₆	—	257	1	no	—
166	C₂×M₁₆	—	257	2	no	—
167	C₂×M₁₆	—	257	2	no	—
168	C₂×M₁₆	—	514	1	no	—
169	C₂×M₁₆	—	514	1	no	—
170	C₂×M₁₆	—	514	1	no	—
171	C₂×M₁₆	—	514	1	no	—
172	C₂×M₁₆	—	514	1	no	—
173	C₂×M₁₆	—	514	1	no	—
174	C₂×M₁₆	—	514	1	no	—
175	C₂×M₁₆	C₂×D₂₅₇	1	1	yes	characteristic
176	C₂×M₁₆	C₂×D₂₅₇	1	1	yes	characteristic
177	C₂²×C₈	—	257	2	no	—
178	C₂²×C₈	—	514	1	no	—
179	#117	—	257	1	no	—
180	#117	—	257	2	no	—
181	#117	—	514	1	no	—
182	#117	—	514	1	no	—
183	#117	C₂×D₂₅₇	1	1	yes	characteristic
184	C₂×C₄⋊C₄	—	514	1	no	—
185	C₂×C₂²⋊C₄	—	257	2	no	—
186	C₂×C₄²	—	514	1	no	—
187	C₂⁴	—	257	1	no	—
188	C₂²×C₄	—	257	1	no	—
189	C₂²×C₄	—	257	1	no	—
190	C₂²×C₄	—	257	1	no	—
191	C₂²×C₄	—	257	1	no	—
192	C₂²×C₄	—	257	1	no	—
193	C₂²×C₄	—	257	1	no	—
194	C₂²×C₄	—	257	1	no	—
195	C₂²×C₄	—	257	1	no	—
196	C₂²×C₄	—	257	1	no	—
197	C₂²×C₄	—	257	1	no	—
198	C₂²×C₄	—	257	1	no	—
199	C₂²×C₄	—	257	1	no	—
200	C₂²×C₄	—	257	1	no	—
201	C₂²×C₄	—	514	2	no	—
202	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
203	M₁₆	—	257	2	no	—
204	M₁₆	—	2	1	no	—
205	M₁₆	—	2	1	no	—
206	M₁₆	—	2	1	no	—
207	M₁₆	—	514	2	no	—
208	M₁₆	—	514	2	no	—
209	M₁₆	—	514	2	no	—
210	M₁₆	—	514	2	no	—
211	M₁₆	—	514	1	no	—
212	M₁₆	—	514	1	no	—
213	M₁₆	—	514	1	no	—
214	M₁₆	C₄×D₂₅₇	1	2	yes	complemented
215	C₂×C₈	—	257	2	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	514	2	no	—
218	C₂×C₈	—	514	2	no	—
219	C₂×C₈	—	514	2	no	—
220	C₂×C₈	—	514	2	no	—
221	C₂×C₈	—	514	1	no	—
222	C₂×C₈	C₄×D₂₅₇	1	2	yes	complemented
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	514	2	no	—
225	C₄⋊C₄	—	514	1	no	—
226	C₂²⋊C₄	—	257	2	no	—
227	C₂²⋊C₄	—	514	2	no	—
228	C₂²⋊C₄	C₄×D₂₅₇	1	2	yes	—
229	C₄²	—	2	1	no	—
230	C₄²	—	514	2	no	—
231	C₄²	—	514	1	no	—
232	C₂³	—	257	1	no	—
233	C₂³	—	257	1	no	—
234	C₂³	—	257	1	no	—
235	C₂³	—	257	1	no	—
236	C₂³	—	257	1	no	—
237	C₂³	—	257	1	no	—
238	C₂³	—	1028	2	no	—
239	C₂³	C₂×C₄×D₂₅₇	1	1	yes	characteristic
240	C₂×C₄	—	257	1	no	—
241	C₂×C₄	—	257	1	no	—
242	C₂×C₄	—	257	1	no	—
243	C₂×C₄	—	257	1	no	—
244	C₂×C₄	—	257	1	no	—
245	C₂×C₄	—	257	1	no	—
246	C₂×C₄	—	257	1	no	—
247	C₂×C₄	—	257	1	no	—
248	C₂×C₄	—	257	1	no	—
249	C₂×C₄	—	257	1	no	—
250	C₂×C₄	—	257	1	no	—
251	C₂×C₄	—	257	1	no	—
252	C₂×C₄	—	257	1	no	—
253	C₂×C₄	—	257	1	no	—
254	C₂×C₄	—	257	1	no	—
255	C₂×C₄	—	257	1	no	—
256	C₂×C₄	—	257	1	no	—
257	C₂×C₄	—	257	1	no	—
258	C₂×C₄	—	257	1	no	—
259	C₂×C₄	—	257	1	no	—
260	C₂×C₄	—	257	1	no	—
261	C₂×C₄	—	257	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	514	2	no	—
264	C₂×C₄	—	1028	4	no	—
265	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
267	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
268	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
269	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
270	C₂×C₄	Q₈×D₂₅₇	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	2	2	no	—
273	C₈	—	514	2	no	—
274	C₈	—	514	2	no	—
275	C₂²	—	257	1	no	—
276	C₂²	—	257	1	no	—
277	C₂²	—	257	1	no	—
278	C₂²	—	257	1	no	—
279	C₂²	—	514	2	no	—
280	C₂²	—	514	2	no	—
281	C₂²	—	514	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	1028	1	no	—
284	C₂²	—	1028	1	no	—
285	C₂²	—	1028	1	no	—
286	C₂²	C₄²×D₂₅₇	1	1	yes	characteristic
287	C₂²	C₂²⋊C₄×D₂₅₇	1	1	yes	characteristic
288	C₂²	C₄⋊C₄×D₂₅₇	1	1	yes	characteristic
289	C₄	—	257	1	no	—
290	C₄	—	257	1	no	—
291	C₄	—	257	1	no	—
292	C₄	—	257	1	no	—
293	C₄	—	4	2	no	—
294	C₄	—	1028	2	no	—
295	C₄	C₂²⋊C₄×D₂₅₇	1	1	yes	characteristic
296	C₄	C₂²⋊C₄×D₂₅₇	1	1	yes	characteristic
297	C₄	C₄⋊C₄×D₂₅₇	1	1	yes	characteristic
298	C₄	C₄⋊C₄×D₂₅₇	1	1	yes	characteristic
299	C₂	—	257	2	no	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	514	1	no	—
304	C₂	—	514	1	no	—
305	C₂	—	514	1	no	—
306	C₂	#95×D₂₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊C₄×D₂₅₇
commutator	C₂²×C₂₅₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#344×C₂₅₇	C₂
socle	C₂×C₂₅₇	C₂×#95
prime radical	C₂³×D₂₅₇	C₂×C₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	#344	D₂₅₇
hypocenter	C₂₅₇	C₂×#344
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

Complements

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

N	H ≅ G/N	type
4. #30656509	C₂	semidirect
5. #30656508	C₂	semidirect
6. #30656507	C₂	semidirect
7. #344×C₂₅₇	C₂	semidirect
12. C₂×#16861601	C₄	semidirect
19. M₁₆×D₂₅₇	C₄	semidirect
20. M₁₆×D₂₅₇	C₄	semidirect
22. C₂×C₈×D₂₅₇	C₄	semidirect
51. #16861602	C₂×C₄	semidirect
57. C₂×#14523339	C₂×C₄	semidirect
72. M₁₆×C₂₅₇	C₂×C₄	semidirect
74. C₂×C₈×C₂₅₇	C₂×C₄	semidirect
152. D₂₅₇	#344	direct (N × H)
153. C₂₅₇	C₂×#344	semidirect
161. #344	D₂₅₇	direct (N × H)
214. M₁₆	C₄×D₂₅₇	semidirect
222. C₂×C₈	C₄×D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₂₅₇	2
#117×D₂₅₇	2
#30656509	2
#30656508	2
#30656507	2
#344×C₂₅₇	2
C₂×#344	257

Subgroups — #344×D257 #45664060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #344×D₂₅₇ #45664060