Subgroups — C₁₃₁×#11658 #45697542

Order 33536.

366 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2162×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1567×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1332×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₁₆×C₁₃₁	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃₁	—	2	1	no	—
7	C₄×Q₁₆×C₁₃₁	—	2	1	no	—
8	C₄×Q₁₆×C₁₃₁	—	2	1	no	—
9	C₄×Q₁₆×C₁₃₁	C₄	1	2	yes	—
10	C₄×Q₁₆×C₁₃₁	C₂²	1	2	yes	—
11	C₂×#107×C₁₃₁	C₄	1	1	yes	characteristic
12	#423×C₁₃₁	—	2	1	no	—
13	C₂×#103×C₁₃₁	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₁₃₁	—	2	1	no	—
16	C₂²⋊C₁₆×C₁₃₁	—	2	1	no	—
17	C₄×C₁₆×C₁₃₁	C₂²	1	2	yes	—
18	#335×C₁₃₁	—	2	1	no	—
19	#335×C₁₃₁	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₃₁	—	2	1	no	—
21	C₂×Q₁₆×C₁₃₁	—	2	4	no	—
22	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
23	C₂×Q₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
24	C₂×M₁₆×C₁₃₁	—	4	1	no	—
25	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
27	C₄×Q₈×C₁₃₁	—	2	2	no	—
28	C₄×Q₈×C₁₃₁	—	2	2	no	—
29	C₄×Q₈×C₁₃₁	—	4	1	no	—
30	C₄×Q₈×C₁₃₁	—	4	1	no	—
31	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₃₁	—	4	1	no	—
33	C₂×C₄²×C₁₃₁	—	4	1	no	—
34	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₁₃₁	—	2	2	no	—
36	#107×C₁₃₁	—	2	1	no	—
37	#107×C₁₃₁	C₂×C₄	1	2	yes	—
38	C₄⋊C₈×C₁₃₁	—	2	2	no	—
39	C₄⋊C₈×C₁₃₁	—	4	1	no	—
40	#103×C₁₃₁	—	2	2	no	—
41	#103×C₁₃₁	—	2	2	no	—
42	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₁₃₁	—	2	4	no	—
47	C₂×Q₈×C₁₃₁	—	2	2	no	—
48	C₂×Q₈×C₁₃₁	—	4	2	no	—
49	C₂×Q₈×C₁₃₁	—	4	1	no	—
50	C₂²×C₄×C₁₃₁	—	4	1	no	—
51	C₂²×C₄×C₁₃₁	—	4	1	no	—
52	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₁	SD₁₆	1	1	yes	characteristic
55	Q₁₆×C₁₃₁	—	2	4	no	—
56	Q₁₆×C₁₃₁	—	2	2	no	—
57	Q₁₆×C₁₃₁	M₁₆	1	4	yes	complemented
58	M₁₆×C₁₃₁	—	8	2	no	—
59	C₂×C₈×C₁₃₁	—	2	2	no	—
60	C₂×C₈×C₁₃₁	—	4	1	no	—
61	C₂×C₈×C₁₃₁	—	4	2	no	—
62	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₃₁	—	2	2	no	—
67	C₄⋊C₄×C₁₃₁	—	4	1	no	—
68	C₄⋊C₄×C₁₃₁	—	4	2	no	—
69	C₄⋊C₄×C₁₃₁	—	4	2	no	—
70	C₄²×C₁₃₁	—	4	2	no	—
71	C₄²×C₁₃₁	—	4	2	no	—
72	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₁₃₁	D₈	1	1	yes	characteristic
74	C₄²×C₁₃₁	SD₁₆	1	1	yes	characteristic
75	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
76	C₁₆×C₁₃₁	—	4	2	no	—
77	C₂³×C₁₃₁	#102	1	1	yes	characteristic
78	Q₈×C₁₃₁	—	2	4	no	—
79	Q₈×C₁₃₁	—	4	4	no	—
80	Q₈×C₁₃₁	—	4	2	no	—
81	C₂×C₄×C₁₃₁	—	2	1	no	—
82	C₂×C₄×C₁₃₁	—	2	1	no	—
83	C₂×C₄×C₁₃₁	—	4	1	no	—
84	C₂×C₄×C₁₃₁	—	4	4	no	—
85	C₂×C₄×C₁₃₁	—	4	2	no	—
86	C₂×C₄×C₁₃₁	—	4	2	no	—
87	C₂×C₄×C₁₃₁	—	8	2	no	—
88	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
89	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
90	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
91	C₂×C₄×C₁₃₁	SD₃₂	1	2	yes	—
92	C₂×C₄×C₁₃₁	Q₃₂	1	2	yes	—
93	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
96	C₈×C₁₃₁	—	2	1	no	—
97	C₈×C₁₃₁	—	2	1	no	—
98	C₈×C₁₃₁	—	2	1	no	—
99	C₈×C₁₃₁	—	8	2	no	—
100	C₈×C₁₃₁	C₂×M₁₆	1	2	yes	—
101	C₂²×C₁₃₁	—	2	1	no	—
102	C₂²×C₁₃₁	—	2	1	no	—
103	C₂²×C₁₃₁	#358	1	2	yes	—
104	C₂²×C₁₃₁	C₂×#102	1	1	yes	characteristic
105	C₄×C₁₃₁	—	2	1	no	—
106	C₄×C₁₃₁	—	2	1	no	—
107	C₄×C₁₃₁	—	2	1	no	—
108	C₄×C₁₃₁	—	4	4	no	—
109	C₄×C₁₃₁	—	8	2	no	—
110	C₄×C₁₃₁	#358	1	2	yes	—
111	C₄×C₁₃₁	#407	1	1	yes	characteristic
112	C₄×C₁₃₁	#421	1	1	yes	characteristic
113	C₄×C₁₃₁	C₂×SD₃₂	1	1	yes	characteristic
114	C₄×C₁₃₁	C₂×Q₃₂	1	1	yes	characteristic
115	C₂×C₁₃₁	—	2	2	no	—
116	C₂×C₁₃₁	#1486	1	1	yes	characteristic
117	C₂×C₁₃₁	C₂×#358	1	1	yes	characteristic
118	C₂×C₁₃₁	#2143	1	1	yes	characteristic
119	#11658	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₁₃₁	#11658	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×C₄×Q₁₆	C₂×C₁₃₁	1	1	yes	characteristic
122	#2162	C₂×C₁₃₁	1	1	yes	characteristic
123	#1567	C₂×C₁₃₁	1	1	yes	characteristic
124	#1332	C₂×C₁₃₁	1	4	yes	complemented
125	C₂²×Q₁₆	C₄×C₁₃₁	1	1	yes	characteristic
126	C₂×C₄×Q₈	—	2	1	no	—
127	C₄×Q₁₆	—	2	1	no	—
128	C₄×Q₁₆	—	2	1	no	—
129	C₄×Q₁₆	C₄×C₁₃₁	1	2	yes	—
130	C₄×Q₁₆	C₂²×C₁₃₁	1	2	yes	—
131	C₂×#107	C₄×C₁₃₁	1	1	yes	characteristic
132	#423	—	2	1	no	—
133	C₂×#103	—	2	1	no	—
134	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
135	C₄⋊C₁₆	—	2	1	no	—
136	C₂²⋊C₁₆	—	2	1	no	—
137	C₄×C₁₆	C₂²×C₁₃₁	1	2	yes	—
138	#335	—	2	1	no	—
139	#335	C₂²×C₁₃₁	1	2	yes	—
140	C₂²×Q₈	—	2	1	no	—
141	C₂×Q₁₆	—	2	4	no	—
142	C₂×Q₁₆	—	2	1	no	—
143	C₂×Q₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
144	C₂×M₁₆	—	4	1	no	—
145	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
146	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
147	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₄×Q₈	—	4	1	no	—
150	C₄×Q₈	—	4	1	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	—	4	1	no	—
153	C₂×C₄²	—	4	1	no	—
154	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
155	C₂×C₁₆	—	2	2	no	—
156	#107	—	2	1	no	—
157	#107	C₂×C₄×C₁₃₁	1	2	yes	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	4	1	no	—
160	#103	—	2	2	no	—
161	#103	—	2	2	no	—
162	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
163	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
164	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
165	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
166	C₂×Q₈	—	2	4	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	—	4	1	no	—
170	C₂²×C₄	—	4	1	no	—
171	C₂²×C₄	—	4	1	no	—
172	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
173	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
174	C₂²×C₄	SD₁₆×C₁₃₁	1	1	yes	characteristic
175	Q₁₆	—	2	4	no	—
176	Q₁₆	—	2	2	no	—
177	Q₁₆	M₁₆×C₁₃₁	1	4	yes	complemented
178	M₁₆	—	8	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
183	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
184	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
185	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄²	—	4	2	no	—
191	C₄²	—	4	2	no	—
192	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
193	C₄²	D₈×C₁₃₁	1	1	yes	characteristic
194	C₄²	SD₁₆×C₁₃₁	1	1	yes	characteristic
195	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
196	C₁₆	—	4	2	no	—
197	C₂³	#102×C₁₃₁	1	1	yes	characteristic
198	Q₈	—	2	4	no	—
199	Q₈	—	4	4	no	—
200	Q₈	—	4	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	8	2	no	—
208	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
209	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
210	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
211	C₂×C₄	SD₃₂×C₁₃₁	1	2	yes	—
212	C₂×C₄	Q₃₂×C₁₃₁	1	2	yes	—
213	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
215	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	8	2	no	—
220	C₈	C₂×M₁₆×C₁₃₁	1	2	yes	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	#358×C₁₃₁	1	2	yes	—
224	C₂²	C₂×#102×C₁₃₁	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	4	no	—
229	C₄	—	8	2	no	—
230	C₄	#358×C₁₃₁	1	2	yes	—
231	C₄	#407×C₁₃₁	1	1	yes	characteristic
232	C₄	#421×C₁₃₁	1	1	yes	characteristic
233	C₄	C₂×SD₃₂×C₁₃₁	1	1	yes	characteristic
234	C₄	C₂×Q₃₂×C₁₃₁	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	#1486×C₁₃₁	1	1	yes	characteristic
237	C₂	C₂×#358×C₁₃₁	1	1	yes	characteristic
238	C₂	#2143×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₁₃₁
socle	C₂²×C₁₃₁	C₂×#102
prime radical	C₂³×C₁₃₁	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. #1332×C₁₃₁	C₂	semidirect
57. Q₁₆×C₁₃₁	M₁₆	semidirect
119. #11658	C₁₃₁	direct (N × H)
120. C₁₃₁	#11658	direct (N × H)
124. #1332	C₂×C₁₃₁	semidirect
177. Q₁₆	M₁₆×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×Q₁₆×C₁₃₁	2
#2162×C₁₃₁	2
#1567×C₁₃₁	2
#1332×C₁₃₁	2
#11658	131

Subgroups — C131×#11658 #45697542

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#11658 #45697542