Subgroups — C₁₃₁×#23173 #45709057

Order 33536.

1228 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₃₁	C₂	1	4	yes	maximal, complemented
2	C₄×#127×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1954×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	#1869×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	C₄×C₄⋊C₈×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₁₃₁	C₂²	1	4	yes	—
7	C₂×#127×C₁₃₁	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₃₁	C₂²	1	4	yes	—
9	C₂×C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
10	C₈×D₄×C₁₃₁	—	2	16	no	—
11	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	8	yes	—
14	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₃₁	C₂²	1	4	yes	—
16	#390×C₁₃₁	C₄	1	4	yes	—
17	#390×C₁₃₁	C₄	1	2	yes	—
18	#384×C₁₃₁	C₄	1	1	yes	characteristic
19	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
20	#336×C₁₃₁	C₂²	1	4	yes	—
21	C₂²×D₄×C₁₃₁	C₂×C₄	1	4	yes	—
22	C₂²×D₄×C₁₃₁	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₃₁	C₂³	1	4	yes	—
24	C₂²×C₈×C₁₃₁	—	2	16	no	—
25	C₂²×C₈×C₁₃₁	—	4	4	no	—
26	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
27	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
28	#127×C₁₃₁	C₈	1	8	yes	complemented
29	C₄×D₄×C₁₃₁	—	2	8	no	—
30	C₄×D₄×C₁₃₁	—	2	8	no	—
31	C₄×D₄×C₁₃₁	—	4	2	no	—
32	C₄×D₄×C₁₃₁	—	4	2	no	—
33	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₁₃₁	C₂×C₄	1	8	yes	—
36	C₂×C₂²⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
42	C₄⋊C₈×C₁₃₁	—	2	4	no	—
43	C₄⋊C₈×C₁₃₁	D₄	1	8	yes	complemented
44	C₂²⋊C₈×C₁₃₁	—	2	16	no	—
45	C₂²⋊C₈×C₁₃₁	—	4	4	no	—
46	C₄×C₈×C₁₃₁	—	2	8	no	—
47	C₂⁴×C₁₃₁	C₂²×C₄	1	4	yes	—
48	C₂×D₄×C₁₃₁	—	2	8	no	—
49	C₂×D₄×C₁₃₁	—	2	8	no	—
50	C₂×D₄×C₁₃₁	—	4	2	no	—
51	C₂×D₄×C₁₃₁	—	4	2	no	—
52	C₂×D₄×C₁₃₁	C₂×C₈	1	16	yes	complemented
53	C₂×D₄×C₁₃₁	C₂×C₈	1	8	yes	complemented
54	C₂²×C₄×C₁₃₁	—	2	8	no	—
55	C₂²×C₄×C₁₃₁	—	2	8	no	—
56	C₂²×C₄×C₁₃₁	—	2	4	no	—
57	C₂²×C₄×C₁₃₁	—	2	4	no	—
58	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₁₃₁	—	2	4	no	—
65	C₂×C₈×C₁₃₁	—	2	4	no	—
66	C₂×C₈×C₁₃₁	—	2	4	no	—
67	C₂×C₈×C₁₃₁	—	2	4	no	—
68	C₂×C₈×C₁₃₁	—	2	4	no	—
69	C₂×C₈×C₁₃₁	—	4	16	no	—
70	C₂×C₈×C₁₃₁	C₂×D₄	1	8	yes	—
71	C₄⋊C₄×C₁₃₁	—	2	8	no	—
72	C₄⋊C₄×C₁₃₁	—	4	2	no	—
73	C₂²⋊C₄×C₁₃₁	—	2	16	no	—
74	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
75	C₄²×C₁₃₁	—	2	4	no	—
76	C₄²×C₁₃₁	—	2	1	no	—
77	C₄²×C₁₃₁	—	2	1	no	—
78	C₄²×C₁₃₁	—	4	1	no	—
79	C₄²×C₁₃₁	C₂×C₈	1	4	yes	complemented
80	C₄²×C₁₃₁	C₂×D₄	1	4	yes	—
81	C₄²×C₁₃₁	#41	1	4	yes	—
82	C₂³×C₁₃₁	—	2	8	no	—
83	C₂³×C₁₃₁	—	2	4	no	—
84	C₂³×C₁₃₁	—	2	8	no	—
85	C₂³×C₁₃₁	—	2	4	no	—
86	C₂³×C₁₃₁	C₂²×C₈	1	8	yes	—
87	C₂³×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
88	D₄×C₁₃₁	—	2	32	no	—
89	D₄×C₁₃₁	—	4	8	no	—
90	C₂×C₄×C₁₃₁	—	2	1	no	—
91	C₂×C₄×C₁₃₁	—	2	2	no	—
92	C₂×C₄×C₁₃₁	—	2	2	no	—
93	C₂×C₄×C₁₃₁	—	2	1	no	—
94	C₂×C₄×C₁₃₁	—	2	8	no	—
95	C₂×C₄×C₁₃₁	—	2	1	no	—
96	C₂×C₄×C₁₃₁	—	2	4	no	—
97	C₂×C₄×C₁₃₁	—	2	2	no	—
98	C₂×C₄×C₁₃₁	—	4	4	no	—
99	C₂×C₄×C₁₃₁	—	4	8	no	—
100	C₂×C₄×C₁₃₁	—	4	4	no	—
101	C₂×C₄×C₁₃₁	C₄×D₄	1	4	yes	—
102	C₂×C₄×C₁₃₁	C₄×D₄	1	4	yes	—
103	C₂×C₄×C₁₃₁	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₁₃₁	C₂²×C₈	1	4	yes	—
105	C₂×C₄×C₁₃₁	#131	1	4	yes	—
106	C₂×C₄×C₁₃₁	C₂²×D₄	1	2	yes	—
107	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
108	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
109	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
110	C₈×C₁₃₁	—	2	8	no	—
111	C₈×C₁₃₁	—	4	4	no	—
112	C₂²×C₁₃₁	—	2	16	no	—
113	C₂²×C₁₃₁	—	2	8	no	—
114	C₂²×C₁₃₁	—	4	4	no	—
115	C₂²×C₁₃₁	—	4	8	no	—
116	C₂²×C₁₃₁	—	4	4	no	—
117	C₂²×C₁₃₁	C₂×C₄×D₄	1	2	yes	—
118	C₂²×C₁₃₁	#518	1	1	yes	characteristic
119	C₂²×C₁₃₁	C₂³×C₈	1	1	yes	characteristic
120	C₂²×C₁₃₁	C₂×#131	1	2	yes	—
121	C₂²×C₁₃₁	#568	1	1	yes	characteristic
122	C₄×C₁₃₁	—	2	4	no	—
123	C₄×C₁₃₁	—	2	2	no	—
124	C₄×C₁₃₁	—	4	4	no	—
125	C₄×C₁₃₁	—	4	1	no	—
126	C₄×C₁₃₁	C₈×D₄	1	8	yes	complemented
127	C₄×C₁₃₁	D₄²	1	1	yes	characteristic
128	C₄×C₁₃₁	#550	1	2	yes	—
129	C₄×C₁₃₁	#559	1	1	yes	characteristic
130	C₂×C₁₃₁	—	4	8	no	—
131	C₂×C₁₃₁	#2366	1	1	yes	characteristic
132	C₂×C₁₃₁	C₂×C₈×D₄	1	2	yes	—
133	C₂×C₁₃₁	#2935	1	2	yes	—
134	C₂×C₁₃₁	#2972	1	1	yes	characteristic
135	C₂×C₁₃₁	#2979	1	1	yes	characteristic
136	#23173	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₁₃₁	#23173	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×C₈×D₄	C₂×C₁₃₁	1	4	yes	complemented
139	C₄×#127	C₂×C₁₃₁	1	1	yes	characteristic
140	#1954	C₂×C₁₃₁	1	4	yes	complemented
141	#1869	C₂×C₁₃₁	1	4	yes	complemented
142	C₄×C₄⋊C₈	C₂×C₁₃₁	1	2	yes	complemented
143	C₂³×C₈	C₂²×C₁₃₁	1	4	yes	—
144	C₂×#127	C₄×C₁₃₁	1	1	yes	characteristic
145	C₂×C₄×D₄	C₂²×C₁₃₁	1	4	yes	—
146	C₂×C₄×D₄	C₂²×C₁₃₁	1	2	yes	—
147	C₈×D₄	—	2	16	no	—
148	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
149	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
150	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	8	yes	—
151	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	4	yes	—
152	C₂×C₄×C₈	C₂²×C₁₃₁	1	4	yes	—
153	#390	C₄×C₁₃₁	1	4	yes	—
154	#390	C₄×C₁₃₁	1	2	yes	—
155	#384	C₄×C₁₃₁	1	1	yes	characteristic
156	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
157	#336	C₂²×C₁₃₁	1	4	yes	—
158	C₂²×D₄	C₂×C₄×C₁₃₁	1	4	yes	—
159	C₂²×D₄	C₂×C₄×C₁₃₁	1	2	yes	—
160	C₂³×C₄	C₂³×C₁₃₁	1	4	yes	—
161	C₂²×C₈	—	2	16	no	—
162	C₂²×C₈	—	4	4	no	—
163	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
164	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
165	#127	C₈×C₁₃₁	1	8	yes	complemented
166	C₄×D₄	—	2	8	no	—
167	C₄×D₄	—	2	8	no	—
168	C₄×D₄	—	4	2	no	—
169	C₄×D₄	—	4	2	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
171	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
172	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₁	1	8	yes	—
173	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
174	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
175	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
176	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
177	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
179	C₄⋊C₈	—	2	4	no	—
180	C₄⋊C₈	D₄×C₁₃₁	1	8	yes	complemented
181	C₂²⋊C₈	—	2	16	no	—
182	C₂²⋊C₈	—	4	4	no	—
183	C₄×C₈	—	2	8	no	—
184	C₂⁴	C₂²×C₄×C₁₃₁	1	4	yes	—
185	C₂×D₄	—	2	8	no	—
186	C₂×D₄	—	2	8	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	C₂×C₈×C₁₃₁	1	16	yes	complemented
190	C₂×D₄	C₂×C₈×C₁₃₁	1	8	yes	complemented
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
196	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
197	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
199	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
200	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	4	16	no	—
207	C₂×C₈	C₂×D₄×C₁₃₁	1	8	yes	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	2	16	no	—
211	C₂²⋊C₄	—	4	4	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	4	1	no	—
216	C₄²	C₂×C₈×C₁₃₁	1	4	yes	complemented
217	C₄²	C₂×D₄×C₁₃₁	1	4	yes	—
218	C₄²	#41×C₁₃₁	1	4	yes	—
219	C₂³	—	2	8	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	2	8	no	—
222	C₂³	—	2	4	no	—
223	C₂³	C₂²×C₈×C₁₃₁	1	8	yes	—
224	C₂³	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
225	D₄	—	2	32	no	—
226	D₄	—	4	8	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	8	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	C₄×D₄×C₁₃₁	1	4	yes	—
239	C₂×C₄	C₄×D₄×C₁₃₁	1	4	yes	—
240	C₂×C₄	C₂²×C₈×C₁₃₁	1	2	yes	—
241	C₂×C₄	C₂²×C₈×C₁₃₁	1	4	yes	—
242	C₂×C₄	#131×C₁₃₁	1	4	yes	—
243	C₂×C₄	C₂²×D₄×C₁₃₁	1	2	yes	—
244	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
245	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
246	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
247	C₈	—	2	8	no	—
248	C₈	—	4	4	no	—
249	C₂²	—	2	16	no	—
250	C₂²	—	2	8	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	8	no	—
253	C₂²	—	4	4	no	—
254	C₂²	C₂×C₄×D₄×C₁₃₁	1	2	yes	—
255	C₂²	#518×C₁₃₁	1	1	yes	characteristic
256	C₂²	C₂³×C₈×C₁₃₁	1	1	yes	characteristic
257	C₂²	C₂×#131×C₁₃₁	1	2	yes	—
258	C₂²	#568×C₁₃₁	1	1	yes	characteristic
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	4	4	no	—
262	C₄	—	4	1	no	—
263	C₄	C₈×D₄×C₁₃₁	1	8	yes	complemented
264	C₄	D₄²×C₁₃₁	1	1	yes	characteristic
265	C₄	#550×C₁₃₁	1	2	yes	—
266	C₄	#559×C₁₃₁	1	1	yes	characteristic
267	C₂	—	4	8	no	—
268	C₂	#2366×C₁₃₁	1	1	yes	characteristic
269	C₂	C₂×C₈×D₄×C₁₃₁	1	2	yes	—
270	C₂	#2935×C₁₃₁	1	2	yes	—
271	C₂	#2972×C₁₃₁	1	1	yes	characteristic
272	C₂	#2979×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
socle	C₂³×C₁₃₁	C₂³×C₄
prime radical	C₂×#127×C₁₃₁	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×D₄×C₁₃₁	C₂	semidirect
3. #1954×C₁₃₁	C₂	semidirect
4. #1869×C₁₃₁	C₂	semidirect
5. C₄×C₄⋊C₈×C₁₃₁	C₂	semidirect
28. #127×C₁₃₁	C₈	semidirect
43. C₄⋊C₈×C₁₃₁	D₄	semidirect
52. C₂×D₄×C₁₃₁	C₂×C₈	semidirect
53. C₂×D₄×C₁₃₁	C₂×C₈	semidirect
79. C₄²×C₁₃₁	C₂×C₈	semidirect
126. C₄×C₁₃₁	C₈×D₄	semidirect
136. #23173	C₁₃₁	direct (N × H)
137. C₁₃₁	#23173	direct (N × H)
138. C₂×C₈×D₄	C₂×C₁₃₁	semidirect
140. #1954	C₂×C₁₃₁	semidirect
141. #1869	C₂×C₁₃₁	semidirect
142. C₄×C₄⋊C₈	C₂×C₁₃₁	semidirect
165. #127	C₈×C₁₃₁	semidirect
180. C₄⋊C₈	D₄×C₁₃₁	semidirect
189. C₂×D₄	C₂×C₈×C₁₃₁	semidirect
190. C₂×D₄	C₂×C₈×C₁₃₁	semidirect
216. C₄²	C₂×C₈×C₁₃₁	semidirect
263. C₄	C₈×D₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₁₃₁	2
C₄×#127×C₁₃₁	2
#1954×C₁₃₁	2
#1869×C₁₃₁	2
C₄×C₄⋊C₈×C₁₃₁	2
#23173	131

Subgroups — C131×#23173 #45709057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#23173 #45709057