Subgroups — C₁₃₁×#8156 #45694040

Order 33536.

506 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#405×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1458×C₁₃₁	C₂	1	2	yes	maximal
3	#1450×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#1281×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₁₃₁	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃₁	C₄	1	1	yes	characteristic
7	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
8	#431×C₁₃₁	C₂²	1	4	yes	—
9	#405×C₁₃₁	—	2	2	no	—
10	#405×C₁₃₁	—	2	2	no	—
11	C₂×#97×C₁₃₁	C₄	1	1	yes	characteristic
12	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃₁	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	#324×C₁₃₁	C₄	1	4	yes	—
16	#323×C₁₃₁	C₄	1	4	yes	—
17	#322×C₁₃₁	—	2	2	no	—
18	C₈²×C₁₃₁	—	2	2	no	—
19	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₁₃₁	—	2	2	no	—
21	C₂×M₁₆×C₁₃₁	—	2	2	no	—
22	C₂×M₁₆×C₁₃₁	C₈	1	4	yes	complemented
23	C₂²×C₈×C₁₃₁	—	2	1	no	—
24	C₂²×C₈×C₁₃₁	—	2	1	no	—
25	C₂²×C₈×C₁₃₁	—	2	2	no	—
26	C₂²×C₈×C₁₃₁	C₈	1	4	yes	complemented
27	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
31	#117×C₁₃₁	—	2	2	no	—
32	#117×C₁₃₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₁₃₁	—	2	4	no	—
38	C₂²⋊C₈×C₁₃₁	C₂×C₄	1	8	yes	—
39	#97×C₁₃₁	—	2	2	no	—
40	#97×C₁₃₁	—	2	2	no	—
41	C₄×C₈×C₁₃₁	—	2	1	no	—
42	C₄×C₈×C₁₃₁	—	2	1	no	—
43	C₄×C₈×C₁₃₁	—	2	4	no	—
44	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
45	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
46	C₂⁴×C₁₃₁	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₁₃₁	—	2	1	no	—
48	C₂²×C₄×C₁₃₁	—	2	1	no	—
49	C₂²×C₄×C₁₃₁	—	2	1	no	—
50	C₂²×C₄×C₁₃₁	—	2	1	no	—
51	C₂²×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₁	C₄²	1	2	yes	—
53	C₂²×C₄×C₁₃₁	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₁₃₁	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
57	M₁₆×C₁₃₁	—	2	8	no	—
58	C₂×C₈×C₁₃₁	—	2	1	no	—
59	C₂×C₈×C₁₃₁	—	2	1	no	—
60	C₂×C₈×C₁₃₁	—	2	1	no	—
61	C₂×C₈×C₁₃₁	—	2	1	no	—
62	C₂×C₈×C₁₃₁	—	2	4	no	—
63	C₂×C₈×C₁₃₁	—	2	4	no	—
64	C₂×C₈×C₁₃₁	—	2	8	no	—
65	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃₁	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₁₃₁	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₁₃₁	C₂×C₈	1	2	yes	—
71	C₂×C₈×C₁₃₁	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₁₃₁	M₁₆	1	2	yes	—
73	C₂×C₈×C₁₃₁	M₁₆	1	2	yes	—
74	C₄⋊C₄×C₁₃₁	—	2	4	no	—
75	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
76	C₄²×C₁₃₁	—	2	1	no	—
77	C₄²×C₁₃₁	—	2	1	no	—
78	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
79	C₄²×C₁₃₁	#41	1	2	yes	—
80	C₂³×C₁₃₁	—	2	1	no	—
81	C₂³×C₁₃₁	—	2	1	no	—
82	C₂³×C₁₃₁	—	4	2	no	—
83	C₂³×C₁₃₁	C₄×C₈	1	2	yes	—
84	C₂³×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₁₃₁	—	2	1	no	—
86	C₂×C₄×C₁₃₁	—	2	1	no	—
87	C₂×C₄×C₁₃₁	—	2	2	no	—
88	C₂×C₄×C₁₃₁	—	2	2	no	—
89	C₂×C₄×C₁₃₁	—	2	2	no	—
90	C₂×C₄×C₁₃₁	—	2	2	no	—
91	C₂×C₄×C₁₃₁	—	2	2	no	—
92	C₂×C₄×C₁₃₁	—	4	2	no	—
93	C₂×C₄×C₁₃₁	—	4	2	no	—
94	C₂×C₄×C₁₃₁	C₄×C₈	1	2	yes	—
95	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
97	C₂×C₄×C₁₃₁	C₄×D₄	1	4	yes	—
98	C₂×C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃₁	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₃₁	C₂×M₁₆	1	1	yes	characteristic
102	C₈×C₁₃₁	—	2	2	no	—
103	C₈×C₁₃₁	—	2	2	no	—
104	C₈×C₁₃₁	—	4	8	no	—
105	C₈×C₁₃₁	C₂²⋊C₈	1	2	yes	—
106	C₈×C₁₃₁	C₂²⋊C₈	1	2	yes	—
107	C₈×C₁₃₁	C₂²⋊C₈	1	4	yes	—
108	C₂²×C₁₃₁	—	2	2	no	—
109	C₂²×C₁₃₁	—	2	2	no	—
110	C₂²×C₁₃₁	—	4	2	no	—
111	C₂²×C₁₃₁	—	4	2	no	—
112	C₂²×C₁₃₁	—	4	2	no	—
113	C₂²×C₁₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₁₃₁	C₂×C₄×C₈	1	1	yes	characteristic
115	C₂²×C₁₃₁	C₄×M₁₆	1	1	yes	characteristic
116	C₄×C₁₃₁	—	2	2	no	—
117	C₄×C₁₃₁	—	2	4	no	—
118	C₄×C₁₃₁	—	4	2	no	—
119	C₄×C₁₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄×C₁₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₄×C₁₃₁	#431	1	1	yes	characteristic
122	C₄×C₁₃₁	#431	1	1	yes	characteristic
123	C₂×C₁₃₁	—	2	1	no	—
124	C₂×C₁₃₁	—	2	1	no	—
125	C₂×C₁₃₁	—	4	2	no	—
126	C₂×C₁₃₁	C₄×C₂²⋊C₈	1	1	yes	characteristic
127	C₂×C₁₃₁	#1757	1	2	yes	—
128	#8156	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₁₃₁	#8156	1	1	yes	characteristic, direct product factor, Sylow
130	C₂×#405	C₂×C₁₃₁	1	1	yes	characteristic
131	#1458	C₂×C₁₃₁	1	2	yes	—
132	#1450	C₂×C₁₃₁	1	2	yes	complemented
133	#1281	C₂×C₁₃₁	1	2	yes	complemented
134	C₂²×M₁₆	C₄×C₁₃₁	1	1	yes	characteristic
135	C₂³×C₈	C₄×C₁₃₁	1	1	yes	characteristic
136	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
137	#431	C₂²×C₁₃₁	1	4	yes	—
138	#405	—	2	2	no	—
139	#405	—	2	2	no	—
140	C₂×#97	C₄×C₁₃₁	1	1	yes	characteristic
141	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
142	C₂×C₄×C₈	C₄×C₁₃₁	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
144	#324	C₄×C₁₃₁	1	4	yes	—
145	#323	C₄×C₁₃₁	1	4	yes	—
146	#322	—	2	2	no	—
147	C₈²	—	2	2	no	—
148	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	C₈×C₁₃₁	1	4	yes	complemented
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	C₈×C₁₃₁	1	4	yes	complemented
156	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
157	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
158	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
159	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
160	#117	—	2	2	no	—
161	#117	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
163	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
164	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
166	C₄⋊C₈	—	2	4	no	—
167	C₂²⋊C₈	C₂×C₄×C₁₃₁	1	8	yes	—
168	#97	—	2	2	no	—
169	#97	—	2	2	no	—
170	C₄×C₈	—	2	1	no	—
171	C₄×C₈	—	2	1	no	—
172	C₄×C₈	—	2	4	no	—
173	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
174	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
175	C₂⁴	C₄²×C₁₃₁	1	1	yes	characteristic
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	C₄²×C₁₃₁	1	1	yes	characteristic
181	C₂²×C₄	C₄²×C₁₃₁	1	2	yes	—
182	C₂²×C₄	C₂×C₈×C₁₃₁	1	2	yes	—
183	C₂²×C₄	C₂×C₈×C₁₃₁	1	2	yes	—
184	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
186	M₁₆	—	2	8	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	8	no	—
194	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
195	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	2	yes	—
196	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
197	C₂×C₈	C₂×C₈×C₁₃₁	1	2	yes	—
198	C₂×C₈	C₂×C₈×C₁₃₁	1	2	yes	—
199	C₂×C₈	C₂×C₈×C₁₃₁	1	2	yes	—
200	C₂×C₈	C₂×C₈×C₁₃₁	1	2	yes	—
201	C₂×C₈	M₁₆×C₁₃₁	1	2	yes	—
202	C₂×C₈	M₁₆×C₁₃₁	1	2	yes	—
203	C₄⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
208	C₄²	#41×C₁₃₁	1	2	yes	—
209	C₂³	—	2	1	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	4	2	no	—
212	C₂³	C₄×C₈×C₁₃₁	1	2	yes	—
213	C₂³	C₂×C₄²×C₁₃₁	1	1	yes	characteristic
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	C₄×C₈×C₁₃₁	1	2	yes	—
224	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
225	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
226	C₂×C₄	C₄×D₄×C₁₃₁	1	4	yes	—
227	C₂×C₄	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
229	C₂×C₄	C₂×M₁₆×C₁₃₁	1	1	yes	characteristic
230	C₂×C₄	C₂×M₁₆×C₁₃₁	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	2	2	no	—
233	C₈	—	4	8	no	—
234	C₈	C₂²⋊C₈×C₁₃₁	1	2	yes	—
235	C₈	C₂²⋊C₈×C₁₃₁	1	2	yes	—
236	C₈	C₂²⋊C₈×C₁₃₁	1	4	yes	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	C₄×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
243	C₂²	C₂×C₄×C₈×C₁₃₁	1	1	yes	characteristic
244	C₂²	C₄×M₁₆×C₁₃₁	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	C₂×C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
249	C₄	C₂×C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
250	C₄	#431×C₁₃₁	1	1	yes	characteristic
251	C₄	#431×C₁₃₁	1	1	yes	characteristic
252	C₂	—	2	1	no	—
253	C₂	—	2	1	no	—
254	C₂	—	4	2	no	—
255	C₂	C₄×C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
256	C₂	#1757×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₃₁	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₈×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
socle	C₂²×C₁₃₁	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₁₃₁	C₄²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1450×C₁₃₁	C₂	semidirect
4. #1281×C₁₃₁	C₂	semidirect
22. C₂×M₁₆×C₁₃₁	C₈	semidirect
26. C₂²×C₈×C₁₃₁	C₈	semidirect
128. #8156	C₁₃₁	direct (N × H)
129. C₁₃₁	#8156	direct (N × H)
132. #1450	C₂×C₁₃₁	semidirect
133. #1281	C₂×C₁₃₁	semidirect
151. C₂×M₁₆	C₈×C₁₃₁	semidirect
155. C₂²×C₈	C₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#405×C₁₃₁	2
#1458×C₁₃₁	2
#1450×C₁₃₁	2
#1281×C₁₃₁	2
#8156	131

Subgroups — C131×#8156 #45694040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#8156 #45694040