Subgroups — #472×D₁₅₇ #34109839

Order 20096.

488 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	AGL(1,ℤ/8)×D₁₅₇	C₂	1	2	yes	maximal, complemented
2	C₂×D₈×D₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#131×D₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#23757544	C₂	1	1	yes	maximal, characteristic, complemented
5	#23757543	C₂	1	1	yes	maximal, characteristic, complemented
6	#23757542	C₂	1	2	yes	maximal, complemented
7	#23757541	C₂	1	2	yes	maximal, complemented
8	#23757540	C₂	1	1	yes	maximal, characteristic, complemented
9	#108×D₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#100×D₁₅₇	C₂	1	2	yes	maximal, complemented
11	#472×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₅₇	C₂²	1	1	yes	characteristic
13	C₂×D₄×D₁₅₇	C₂²	1	2	yes	—
14	#19097799	C₂²	1	2	yes	complemented
15	#19097798	—	2	2	no	—
16	#19097797	C₂²	1	2	yes	complemented
17	#19097796	C₂²	1	2	yes	complemented
18	#19097795	—	2	2	no	—
19	SD₁₆×D₁₅₇	—	2	2	no	—
20	C₂×D₁₂₅₆	C₂²	1	1	yes	characteristic, complemented
21	C₂×#14578939	C₂²	1	2	yes	complemented
22	D₈×D₁₅₇	—	2	1	no	—
23	D₈×D₁₅₇	—	2	1	no	—
24	D₈×D₁₅₇	—	2	1	no	—
25	D₈×D₁₅₇	—	2	1	no	—
26	D₈×D₁₅₇	—	2	2	no	—
27	#19097779	—	2	1	no	—
28	#19097779	C₂²	1	1	yes	characteristic, complemented
29	#19097778	C₂²	1	1	yes	characteristic, complemented
30	#19097777	—	2	1	no	—
31	#19097777	C₂²	1	1	yes	characteristic, complemented
32	M₁₆×D₁₅₇	—	2	1	no	—
33	M₁₆×D₁₅₇	C₂²	1	2	yes	—
34	M₁₆×D₁₅₇	C₂²	1	1	yes	characteristic
35	C₂×C₈×D₁₅₇	—	2	1	no	—
36	C₂×C₈×D₁₅₇	C₂²	1	1	yes	characteristic
37	#19097707	C₂²	1	1	yes	characteristic, complemented
38	#19097706	C₂²	1	2	yes	complemented
39	#19097688	C₂²	1	2	yes	complemented
40	#19097687	C₂²	1	2	yes	complemented
41	#19097687	C₂²	1	2	yes	complemented
42	AGL(1,ℤ/8)×C₁₅₇	C₂²	1	2	yes	complemented
43	C₂×D₈×C₁₅₇	C₂²	1	1	yes	characteristic, complemented
44	#131×C₁₅₇	C₂²	1	1	yes	characteristic, complemented
45	#108×C₁₅₇	C₂²	1	1	yes	characteristic, complemented
46	#100×C₁₅₇	C₂²	1	2	yes	complemented
47	C₂³×D₁₅₇	—	2	2	no	—
48	#14578955	D₄	1	1	yes	characteristic
49	#14578954	—	2	1	no	—
50	#14578954	C₂³	1	1	yes	characteristic
51	#14578953	—	2	1	no	—
52	#14578953	D₄	1	1	yes	characteristic
53	Q₈×D₁₅₇	D₄	1	1	yes	characteristic
54	C₂×D₆₂₈	C₂³	1	2	yes	—
55	C₂×C₂²⋊D₁₅₇	—	2	2	no	—
56	D₄×D₁₅₇	—	2	1	no	—
57	D₄×D₁₅₇	—	2	2	no	—
58	D₄×D₁₅₇	—	2	2	no	—
59	D₄×D₁₅₇	—	2	2	no	—
60	D₄×D₁₅₇	—	2	2	no	—
61	D₄×D₁₅₇	D₄	1	1	yes	characteristic
62	C₂×C₄×D₁₅₇	—	2	1	no	—
63	C₂×C₄×D₁₅₇	C₂³	1	1	yes	characteristic
64	#14578943	—	2	2	no	—
65	#14578942	—	2	2	no	—
66	#14578941	—	2	2	no	—
67	D₁₂₅₆	—	2	2	no	—
68	D₁₂₅₆	—	2	1	no	—
69	D₁₂₅₆	—	2	1	no	—
70	#14578939	—	2	2	no	—
71	#14578939	—	2	2	no	—
72	#14578939	—	2	2	no	—
73	#14578939	—	2	2	no	—
74	#14578938	—	2	1	no	—
75	#14578938	C₂³	1	1	yes	characteristic
76	#14578938	C₂³	1	2	yes	—
77	#14578937	—	2	1	no	—
78	#14578937	—	2	2	no	—
79	#14578937	—	2	1	no	—
80	#14578937	—	2	1	no	—
81	C₂×#11929434	—	2	1	no	—
82	C₂×#11929434	C₂³	1	1	yes	characteristic
83	C₈×D₁₅₇	—	2	1	no	—
84	C₈×D₁₅₇	—	2	2	no	—
85	C₈×D₁₅₇	—	2	1	no	—
86	C₈×D₁₅₇	—	2	1	no	—
87	C₈×D₁₅₇	D₄	1	1	yes	characteristic
88	C₈×D₁₅₇	D₄	1	1	yes	characteristic
89	C₈×D₁₅₇	D₄	1	1	yes	characteristic
90	C₈×D₁₅₇	D₄	1	1	yes	characteristic
91	#41×C₁₅₇	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₁₅₇	C₂³	1	2	yes	—
93	SD₁₆×C₁₅₇	—	2	2	no	—
94	D₈×C₁₅₇	—	2	2	no	—
95	D₈×C₁₅₇	—	2	1	no	—
96	D₈×C₁₅₇	—	2	1	no	—
97	M₁₆×C₁₅₇	—	2	1	no	—
98	M₁₆×C₁₅₇	C₂³	1	1	yes	characteristic
99	M₁₆×C₁₅₇	C₂³	1	2	yes	—
100	C₂×C₈×C₁₅₇	—	2	1	no	—
101	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
102	C₂²×D₁₅₇	—	2	1	no	—
103	C₂²×D₁₅₇	—	2	2	no	—
104	C₂²×D₁₅₇	—	4	2	no	—
105	C₂²×D₁₅₇	—	4	2	no	—
106	C₂²×D₁₅₇	—	8	4	no	—
107	C₂²×D₁₅₇	#41	1	1	yes	characteristic
108	Dic₃₁₄	—	2	1	no	—
109	Dic₃₁₄	C₂×D₄	1	1	yes	characteristic
110	D₆₂₈	—	2	1	no	—
111	D₆₂₈	—	2	2	no	—
112	D₆₂₈	—	2	2	no	—
113	D₆₂₈	C₂×D₄	1	1	yes	characteristic
114	C₂²⋊D₁₅₇	—	2	1	no	—
115	C₂²⋊D₁₅₇	—	2	1	no	—
116	C₂²⋊D₁₅₇	—	2	1	no	—
117	C₂²⋊D₁₅₇	—	4	2	no	—
118	C₂²⋊D₁₅₇	—	4	2	no	—
119	C₂×Dic₁₅₇	—	2	1	no	—
120	C₂×Dic₁₅₇	#41	1	1	yes	characteristic
121	C₄×D₁₅₇	—	2	1	no	—
122	C₄×D₁₅₇	—	2	1	no	—
123	C₄×D₁₅₇	—	2	1	no	—
124	C₄×D₁₅₇	C₂×D₄	1	1	yes	characteristic
125	C₄×D₁₅₇	C₂×D₄	1	1	yes	characteristic
126	C₄×D₁₅₇	C₂×D₄	1	1	yes	characteristic
127	C₄×D₁₅₇	C₂×D₄	1	1	yes	characteristic
128	#11929434	—	2	1	no	—
129	#11929434	—	2	2	no	—
130	#11929434	C₂×D₄	1	1	yes	characteristic
131	#11929434	C₂×D₄	1	1	yes	characteristic
132	C₂³×C₁₅₇	—	2	2	no	—
133	Q₈×C₁₅₇	C₂×D₄	1	1	yes	characteristic
134	D₄×C₁₅₇	—	2	1	no	—
135	D₄×C₁₅₇	—	2	2	no	—
136	D₄×C₁₅₇	—	2	2	no	—
137	D₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₅₇	—	2	1	no	—
139	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
140	C₈×C₁₅₇	—	2	1	no	—
141	C₈×C₁₅₇	—	2	2	no	—
142	C₈×C₁₅₇	C₂×D₄	1	1	yes	characteristic
143	C₈×C₁₅₇	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₁₅₇	—	2	1	no	—
145	C₂²×C₁₅₇	—	4	2	no	—
146	C₂²×C₁₅₇	—	8	2	no	—
147	C₂²×C₁₅₇	C₂×#41	1	1	yes	characteristic
148	C₂×D₁₅₇	—	2	2	no	—
149	C₂×D₁₅₇	—	2	1	no	—
150	C₂×D₁₅₇	—	4	2	no	—
151	C₂×D₁₅₇	—	4	2	no	—
152	C₂×D₁₅₇	—	8	4	no	—
153	C₂×D₁₅₇	—	8	4	no	—
154	C₂×D₁₅₇	—	8	2	no	—
155	C₂×D₁₅₇	#121	1	1	yes	characteristic
156	C₂×D₁₅₇	#121	1	1	yes	characteristic
157	C₄×C₁₅₇	—	2	1	no	—
158	C₄×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
159	C₄×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
160	Dic₁₅₇	—	2	1	no	—
161	Dic₁₅₇	#121	1	1	yes	characteristic
162	Dic₁₅₇	#121	1	1	yes	characteristic
163	C₂×C₁₅₇	—	2	1	no	—
164	C₂×C₁₅₇	—	4	1	no	—
165	C₂×C₁₅₇	—	8	2	no	—
166	C₂×C₁₅₇	C₂×#121	1	1	yes	characteristic
167	D₁₅₇	—	2	1	no	—
168	D₁₅₇	—	4	1	no	—
169	D₁₅₇	—	8	2	no	—
170	D₁₅₇	#472	1	2	yes	direct product factor
171	C₁₅₇	C₂×#472	1	1	yes	characteristic, complemented, Sylow
172	C₂×#472	—	157	1	no	maximal, Sylow
173	C₂×AGL(1,ℤ/8)	—	157	2	no	—
174	C₂²×D₈	—	157	1	no	—
175	C₂×#131	—	157	1	no	—
176	#472	—	157	2	no	—
177	#472	—	157	1	no	—
178	#472	—	157	1	no	—
179	#472	—	157	1	no	—
180	#472	—	157	2	no	—
181	#472	D₁₅₇	1	1	yes	characteristic, direct product factor
182	C₂×#108	—	157	1	no	—
183	C₂×#100	—	157	2	no	—
184	C₂×#41	—	157	1	no	—
185	C₂²×D₄	—	157	2	no	—
186	AGL(1,ℤ/8)	—	314	2	no	—
187	AGL(1,ℤ/8)	—	314	2	no	—
188	AGL(1,ℤ/8)	—	157	2	no	—
189	AGL(1,ℤ/8)	—	157	2	no	—
190	AGL(1,ℤ/8)	—	157	2	no	—
191	AGL(1,ℤ/8)	C₂×D₁₅₇	1	2	yes	complemented
192	C₂×SD₁₆	—	314	2	no	—
193	C₂×D₈	—	314	2	no	—
194	C₂×D₈	—	314	1	no	—
195	C₂×D₈	—	314	1	no	—
196	C₂×D₈	—	314	1	no	—
197	C₂×D₈	—	314	1	no	—
198	C₂×D₈	—	157	2	no	—
199	C₂×D₈	—	157	1	no	—
200	C₂×D₈	C₂×D₁₅₇	1	1	yes	characteristic, complemented
201	#131	—	314	1	no	—
202	#131	—	314	1	no	—
203	#131	—	157	1	no	—
204	#131	—	157	1	no	—
205	#131	—	157	1	no	—
206	#131	C₂×D₁₅₇	1	1	yes	characteristic, complemented
207	C₂×M₁₆	—	314	1	no	—
208	C₂×M₁₆	—	157	1	no	—
209	C₂×M₁₆	—	157	2	no	—
210	C₂²×C₈	—	314	1	no	—
211	C₂²×C₈	—	157	1	no	—
212	#108	—	157	2	no	—
213	#108	—	157	1	no	—
214	#108	C₂×D₁₅₇	1	1	yes	characteristic, complemented
215	#100	—	157	2	no	—
216	#100	—	157	2	no	—
217	#100	—	157	2	no	—
218	#100	C₂×D₁₅₇	1	2	yes	complemented
219	C₂⁴	—	314	2	no	—
220	#41	—	314	1	no	—
221	#41	—	314	1	no	—
222	#41	—	157	1	no	—
223	#41	—	157	1	no	—
224	#41	—	157	1	no	—
225	#41	C₂²×D₁₅₇	1	1	yes	characteristic
226	C₂×Q₈	—	157	1	no	—
227	C₂×D₄	—	314	1	no	—
228	C₂×D₄	—	314	2	no	—
229	C₂×D₄	—	314	2	no	—
230	C₂×D₄	—	314	2	no	—
231	C₂×D₄	—	314	2	no	—
232	C₂×D₄	—	314	2	no	—
233	C₂×D₄	—	157	1	no	—
234	C₂×D₄	—	157	2	no	—
235	C₂×D₄	C₂²×D₁₅₇	1	2	yes	—
236	C₂²×C₄	—	314	1	no	—
237	C₂²×C₄	—	157	1	no	—
238	SD₁₆	—	2	2	no	—
239	SD₁₆	—	314	2	no	—
240	SD₁₆	—	314	2	no	—
241	SD₁₆	—	314	2	no	—
242	D₈	—	2	2	no	—
243	D₈	—	2	1	no	—
244	D₈	—	2	1	no	—
245	D₈	—	314	2	no	—
246	D₈	—	314	2	no	—
247	D₈	—	314	2	no	—
248	D₈	—	314	2	no	—
249	D₈	—	314	2	no	—
250	D₈	—	314	1	no	—
251	D₈	—	314	1	no	—
252	M₁₆	—	2	1	no	—
253	M₁₆	—	314	1	no	—
254	M₁₆	—	314	2	no	—
255	M₁₆	—	314	1	no	—
256	M₁₆	—	314	1	no	—
257	M₁₆	—	314	1	no	—
258	M₁₆	—	157	1	no	—
259	M₁₆	—	157	2	no	—
260	M₁₆	C₂²×D₁₅₇	1	1	yes	characteristic
261	M₁₆	C₂²×D₁₅₇	1	2	yes	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	314	1	no	—
264	C₂×C₈	—	314	2	no	—
265	C₂×C₈	—	314	1	no	—
266	C₂×C₈	—	314	1	no	—
267	C₂×C₈	—	314	1	no	—
268	C₂×C₈	—	157	1	no	—
269	C₂×C₈	—	157	1	no	—
270	C₂×C₈	—	157	1	no	—
271	C₂×C₈	—	157	1	no	—
272	C₂×C₈	—	157	1	no	—
273	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
274	C₂³	—	2	2	no	—
275	C₂³	—	314	1	no	—
276	C₂³	—	314	2	no	—
277	C₂³	—	628	2	no	—
278	C₂³	—	628	2	no	—
279	C₂³	—	157	1	no	—
280	C₂³	—	1256	4	no	—
281	Q₈	—	314	1	no	—
282	Q₈	—	157	1	no	—
283	Q₈	D₄×D₁₅₇	1	1	yes	characteristic
284	D₄	—	2	1	no	—
285	D₄	—	2	2	no	—
286	D₄	—	2	2	no	—
287	D₄	—	314	1	no	—
288	D₄	—	314	1	no	—
289	D₄	—	314	1	no	—
290	D₄	—	314	1	no	—
291	D₄	—	314	2	no	—
292	D₄	—	314	2	no	—
293	D₄	—	628	2	no	—
294	D₄	—	628	2	no	—
295	D₄	—	157	1	no	—
296	D₄	D₄×D₁₅₇	1	1	yes	characteristic
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	314	1	no	—
299	C₂×C₄	—	314	1	no	—
300	C₂×C₄	—	314	1	no	—
301	C₂×C₄	—	314	1	no	—
302	C₂×C₄	—	157	1	no	—
303	C₂×C₄	—	157	1	no	—
304	C₂×C₄	—	157	1	no	—
305	C₂×C₄	—	157	1	no	—
306	C₂×C₄	—	157	1	no	—
307	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	2	no	—
310	C₈	—	314	1	no	—
311	C₈	—	314	2	no	—
312	C₈	—	157	1	no	—
313	C₈	—	157	1	no	—
314	C₈	D₄×D₁₅₇	1	1	yes	characteristic
315	C₈	D₄×D₁₅₇	1	1	yes	characteristic
316	C₂²	—	2	1	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	314	2	no	—
320	C₂²	—	314	1	no	—
321	C₂²	—	628	2	no	—
322	C₂²	—	628	2	no	—
323	C₂²	—	157	1	no	—
324	C₂²	—	157	1	no	—
325	C₂²	—	1256	2	no	—
326	C₂²	—	1256	4	no	—
327	C₂²	—	1256	4	no	—
328	C₂²	#41×D₁₅₇	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	314	1	no	—
331	C₄	—	157	1	no	—
332	C₄	—	157	1	no	—
333	C₄	C₂×D₄×D₁₅₇	1	1	yes	characteristic
334	C₄	C₂×D₄×D₁₅₇	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	4	1	no	—
337	C₂	—	8	2	no	—
338	C₂	—	314	1	no	—
339	C₂	—	628	1	no	—
340	C₂	—	157	2	no	—
341	C₂	—	1256	2	no	—
342	C₂	#121×D₁₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#121×D₁₅₇
commutator, cosocle	C₂×C₄×C₁₅₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	#472×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₂×#121
hypercenter	#472	D₁₅₇
hypocenter	C₁₅₇	C₂×#472
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #472

Complements

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

N	H ≅ G/N	type
1. AGL(1,ℤ/8)×D₁₅₇	C₂	semidirect
2. C₂×D₈×D₁₅₇	C₂	semidirect
3. #131×D₁₅₇	C₂	semidirect
4. #23757544	C₂	semidirect
5. #23757543	C₂	semidirect
6. #23757542	C₂	semidirect
7. #23757541	C₂	semidirect
8. #23757540	C₂	semidirect
9. #108×D₁₅₇	C₂	semidirect
10. #100×D₁₅₇	C₂	semidirect
11. #472×C₁₅₇	C₂	semidirect
14. #19097799	C₂²	semidirect
16. #19097797	C₂²	semidirect
17. #19097796	C₂²	semidirect
20. C₂×D₁₂₅₆	C₂²	semidirect
21. C₂×#14578939	C₂²	semidirect
28. #19097779	C₂²	semidirect
29. #19097778	C₂²	semidirect
31. #19097777	C₂²	semidirect
37. #19097707	C₂²	semidirect
38. #19097706	C₂²	semidirect
39. #19097688	C₂²	semidirect
40. #19097687	C₂²	semidirect
41. #19097687	C₂²	semidirect
42. AGL(1,ℤ/8)×C₁₅₇	C₂²	semidirect
43. C₂×D₈×C₁₅₇	C₂²	semidirect
44. #131×C₁₅₇	C₂²	semidirect
45. #108×C₁₅₇	C₂²	semidirect
46. #100×C₁₅₇	C₂²	semidirect
170. D₁₅₇	#472	direct (N × H)
171. C₁₅₇	C₂×#472	semidirect
181. #472	D₁₅₇	direct (N × H)
191. AGL(1,ℤ/8)	C₂×D₁₅₇	semidirect
200. C₂×D₈	C₂×D₁₅₇	semidirect
206. #131	C₂×D₁₅₇	semidirect
214. #108	C₂×D₁₅₇	semidirect
218. #100	C₂×D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
AGL(1,ℤ/8)×D₁₅₇	2
C₂×D₈×D₁₅₇	2
#131×D₁₅₇	2
#23757544	2
#23757543	2
#23757542	2
#23757541	2
#23757540	2
#108×D₁₅₇	2
#100×D₁₅₇	2
#472×C₁₅₇	2
C₂×#472	157

Subgroups — #472×D157 #34109839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #472×D₁₅₇ #34109839