Subgroups — C₁₃₁×#8146 #45694030

Order 33536.

642 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊C₈×C₁₃₁	C₂	1	2	yes	maximal
3	C₂×#322×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈²×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1273×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₁₃₁	C₄	1	2	yes	—
7	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊C₈×C₁₃₁	C₄	1	4	yes	—
9	C₄×M₁₆×C₁₃₁	—	2	2	no	—
10	C₄×M₁₆×C₁₃₁	—	2	2	no	—
11	C₂×#97×C₁₃₁	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	#336×C₁₃₁	C₄	1	4	yes	—
17	#322×C₁₃₁	—	2	2	no	—
18	C₈²×C₁₃₁	—	2	2	no	—
19	C₂³×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₁₃₁	—	2	4	no	—
22	C₂×M₁₆×C₁₃₁	—	2	4	no	—
23	C₂×M₁₆×C₁₃₁	C₈	1	8	yes	complemented
24	C₂²×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₃₁	—	2	1	no	—
29	C₂×C₄²×C₁₃₁	—	2	1	no	—
30	C₂×C₄²×C₁₃₁	—	2	2	no	—
31	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
34	C₂²⋊C₈×C₁₃₁	—	2	8	no	—
35	#97×C₁₃₁	—	2	4	no	—
36	C₄×C₈×C₁₃₁	—	2	1	no	—
37	C₄×C₈×C₁₃₁	—	2	1	no	—
38	C₄×C₈×C₁₃₁	—	2	2	no	—
39	C₄×C₈×C₁₃₁	—	2	2	no	—
40	C₄×C₈×C₁₃₁	—	2	2	no	—
41	C₄×C₈×C₁₃₁	—	2	2	no	—
42	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
43	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
44	C₂⁴×C₁₃₁	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₁	—	2	1	no	—
46	C₂²×C₄×C₁₃₁	—	2	1	no	—
47	C₂²×C₄×C₁₃₁	—	2	2	no	—
48	C₂²×C₄×C₁₃₁	—	2	2	no	—
49	C₂²×C₄×C₁₃₁	—	2	2	no	—
50	C₂²×C₄×C₁₃₁	—	2	4	no	—
51	C₂²×C₄×C₁₃₁	—	2	4	no	—
52	C₂²×C₄×C₁₃₁	C₄²	1	2	yes	—
53	C₂²×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₁	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
57	M₁₆×C₁₃₁	—	2	16	no	—
58	C₂×C₈×C₁₃₁	—	2	2	no	—
59	C₂×C₈×C₁₃₁	—	2	4	no	—
60	C₂×C₈×C₁₃₁	—	2	2	no	—
61	C₂×C₈×C₁₃₁	—	2	4	no	—
62	C₂×C₈×C₁₃₁	—	2	4	no	—
63	C₂×C₈×C₁₃₁	—	2	4	no	—
64	C₂×C₈×C₁₃₁	—	2	4	no	—
65	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	4	yes	—
66	C₂×C₈×C₁₃₁	C₂×C₈	1	4	yes	—
67	C₂×C₈×C₁₃₁	C₂×C₈	1	4	yes	—
68	C₂×C₈×C₁₃₁	M₁₆	1	4	yes	—
69	C₂×C₈×C₁₃₁	M₁₆	1	8	yes	complemented
70	C₄²×C₁₃₁	—	2	1	no	—
71	C₄²×C₁₃₁	—	2	1	no	—
72	C₄²×C₁₃₁	—	2	2	no	—
73	C₄²×C₁₃₁	—	2	2	no	—
74	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₁₃₁	#41	1	2	yes	—
77	C₂³×C₁₃₁	—	2	1	no	—
78	C₂³×C₁₃₁	—	2	2	no	—
79	C₂³×C₁₃₁	—	2	1	no	—
80	C₂³×C₁₃₁	—	2	2	no	—
81	C₂³×C₁₃₁	C₄×C₈	1	2	yes	—
82	C₂³×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₁₃₁	—	2	1	no	—
84	C₂×C₄×C₁₃₁	—	2	1	no	—
85	C₂×C₄×C₁₃₁	—	2	4	no	—
86	C₂×C₄×C₁₃₁	—	2	4	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	—	4	4	no	—
91	C₂×C₄×C₁₃₁	—	4	4	no	—
92	C₂×C₄×C₁₃₁	C₄×C₈	1	2	yes	—
93	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
95	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₁₃₁	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₁₃₁	C₂×M₁₆	1	4	yes	—
100	C₂×C₄×C₁₃₁	#131	1	4	yes	—
101	C₈×C₁₃₁	—	2	4	no	—
102	C₈×C₁₃₁	—	2	8	no	—
103	C₈×C₁₃₁	—	4	4	no	—
104	C₈×C₁₃₁	C₂²⋊C₈	1	8	yes	complemented
105	C₂²×C₁₃₁	—	2	4	no	—
106	C₂²×C₁₃₁	—	2	2	no	—
107	C₂²×C₁₃₁	—	4	1	no	—
108	C₂²×C₁₃₁	—	4	1	no	—
109	C₂²×C₁₃₁	—	4	2	no	—
110	C₂²×C₁₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
111	C₂²×C₁₃₁	C₂×C₄×C₈	1	1	yes	characteristic
112	C₂²×C₁₃₁	C₄×M₁₆	1	2	yes	—
113	C₂²×C₁₃₁	C₄×M₁₆	1	1	yes	characteristic
114	C₂²×C₁₃₁	#405	1	2	yes	—
115	C₄×C₁₃₁	—	2	2	no	—
116	C₄×C₁₃₁	—	4	2	no	—
117	C₄×C₁₃₁	C₂×C₂²⋊C₈	1	2	yes	—
118	C₄×C₁₃₁	#431	1	2	yes	—
119	C₄×C₁₃₁	#435	1	4	yes	—
120	C₄×C₁₃₁	#436	1	4	yes	—
121	C₂×C₁₃₁	—	4	2	no	—
122	C₂×C₁₃₁	C₈×M₁₆	1	2	yes	—
123	C₂×C₁₃₁	#1453	1	1	yes	characteristic
124	C₂×C₁₃₁	#1455	1	1	yes	characteristic
125	C₂×C₁₃₁	C₄×C₂²⋊C₈	1	1	yes	characteristic
126	C₂×C₁₃₁	#1753	1	2	yes	—
127	#8146	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₁₃₁	#8146	1	1	yes	characteristic, direct product factor, Sylow
129	C₂×C₄×M₁₆	C₂×C₁₃₁	1	1	yes	characteristic
130	C₄×C₂²⋊C₈	C₂×C₁₃₁	1	2	yes	—
131	C₂×#322	C₂×C₁₃₁	1	1	yes	characteristic, complemented
132	C₂×C₈²	C₂×C₁₃₁	1	1	yes	characteristic, complemented
133	#1273	C₂×C₁₃₁	1	2	yes	complemented
134	C₂²×M₁₆	C₄×C₁₃₁	1	2	yes	—
135	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
136	C₂×C₂²⋊C₈	C₄×C₁₃₁	1	4	yes	—
137	C₄×M₁₆	—	2	2	no	—
138	C₄×M₁₆	—	2	2	no	—
139	C₂×#97	C₄×C₁₃₁	1	2	yes	—
140	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
142	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
143	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
144	#336	C₄×C₁₃₁	1	4	yes	—
145	#322	—	2	2	no	—
146	C₈²	—	2	2	no	—
147	C₂³×C₄	C₂×C₄×C₁₃₁	1	2	yes	—
148	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
149	C₂×M₁₆	—	2	4	no	—
150	C₂×M₁₆	—	2	4	no	—
151	C₂×M₁₆	C₈×C₁₃₁	1	8	yes	complemented
152	C₂²×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
153	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
154	C₂²×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
155	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
156	C₂×C₄²	—	2	1	no	—
157	C₂×C₄²	—	2	1	no	—
158	C₂×C₄²	—	2	2	no	—
159	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
160	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
162	C₂²⋊C₈	—	2	8	no	—
163	#97	—	2	4	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	—	2	2	no	—
170	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
171	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
172	C₂⁴	C₄²×C₁₃₁	1	1	yes	characteristic
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	C₄²×C₁₃₁	1	2	yes	—
181	C₂²×C₄	C₄²×C₁₃₁	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₈×C₁₃₁	1	4	yes	—
183	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
184	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
185	M₁₆	—	2	16	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	4	yes	—
194	C₂×C₈	C₂×C₈×C₁₃₁	1	4	yes	—
195	C₂×C₈	C₂×C₈×C₁₃₁	1	4	yes	—
196	C₂×C₈	M₁₆×C₁₃₁	1	4	yes	—
197	C₂×C₈	M₁₆×C₁₃₁	1	8	yes	complemented
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
203	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
204	C₄²	#41×C₁₃₁	1	2	yes	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	1	no	—
209	C₂³	C₄×C₈×C₁₃₁	1	2	yes	—
210	C₂³	C₂×C₄²×C₁₃₁	1	1	yes	characteristic
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	C₄×C₈×C₁₃₁	1	2	yes	—
221	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
222	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
223	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
224	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
225	C₂×C₄	C₂²×C₈×C₁₃₁	1	2	yes	—
226	C₂×C₄	C₂×M₁₆×C₁₃₁	1	2	yes	—
227	C₂×C₄	C₂×M₁₆×C₁₃₁	1	4	yes	—
228	C₂×C₄	#131×C₁₃₁	1	4	yes	—
229	C₈	—	2	4	no	—
230	C₈	—	2	8	no	—
231	C₈	—	4	4	no	—
232	C₈	C₂²⋊C₈×C₁₃₁	1	8	yes	complemented
233	C₂²	—	2	4	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	2	no	—
238	C₂²	C₄×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
239	C₂²	C₂×C₄×C₈×C₁₃₁	1	1	yes	characteristic
240	C₂²	C₄×M₁₆×C₁₃₁	1	2	yes	—
241	C₂²	C₄×M₁₆×C₁₃₁	1	1	yes	characteristic
242	C₂²	#405×C₁₃₁	1	2	yes	—
243	C₄	—	2	2	no	—
244	C₄	—	4	2	no	—
245	C₄	C₂×C₂²⋊C₈×C₁₃₁	1	2	yes	—
246	C₄	#431×C₁₃₁	1	2	yes	—
247	C₄	#435×C₁₃₁	1	4	yes	—
248	C₄	#436×C₁₃₁	1	4	yes	—
249	C₂	—	4	2	no	—
250	C₂	C₈×M₁₆×C₁₃₁	1	2	yes	—
251	C₂	#1453×C₁₃₁	1	1	yes	characteristic
252	C₂	#1455×C₁₃₁	1	1	yes	characteristic
253	C₂	C₄×C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
254	C₂	#1753×C₁₃₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂×#322×C₁₃₁	C₂	semidirect
4. C₂×C₈²×C₁₃₁	C₂	semidirect
5. #1273×C₁₃₁	C₂	semidirect
23. C₂×M₁₆×C₁₃₁	C₈	semidirect
69. C₂×C₈×C₁₃₁	M₁₆	semidirect
104. C₈×C₁₃₁	C₂²⋊C₈	semidirect
127. #8146	C₁₃₁	direct (N × H)
128. C₁₃₁	#8146	direct (N × H)
131. C₂×#322	C₂×C₁₃₁	semidirect
132. C₂×C₈²	C₂×C₁₃₁	semidirect
133. #1273	C₂×C₁₃₁	semidirect
151. C₂×M₁₆	C₈×C₁₃₁	semidirect
197. C₂×C₈	M₁₆×C₁₃₁	semidirect
232. C₈	C₂²⋊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×C₄×M₁₆×C₁₃₁	2
C₄×C₂²⋊C₈×C₁₃₁	2
C₂×#322×C₁₃₁	2
C₂×C₈²×C₁₃₁	2
#1273×C₁₃₁	2
#8146	131

Subgroups — C131×#8146 #45694030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#8146 #45694030