Subgroups — C₁₃₁×#12913 #45698797

Order 33536.

438 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₈×C₂²⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#1374×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₁₃₁	—	2	2	no	—
8	C₂×M₃₂×C₁₃₁	—	2	4	no	—
9	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₃₁	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₈×C₁₃₁	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
14	C₄×C₂²⋊C₄×C₁₃₁	C₄	1	1	yes	characteristic
15	C₄⋊C₁₆×C₁₃₁	—	2	4	no	—
16	C₂²⋊C₁₆×C₁₃₁	—	2	2	no	—
17	#336×C₁₃₁	C₄	1	2	yes	—
18	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃₁	—	2	2	no	—
20	C₂²×C₈×C₁₃₁	—	2	2	no	—
21	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
24	C₂²×C₈×C₁₃₁	Q₈	1	2	yes	—
25	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₁₃₁	C₈	1	2	yes	—
27	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
28	M₃₂×C₁₃₁	—	2	8	no	—
29	C₂×C₁₆×C₁₃₁	—	2	2	no	—
30	C₂×C₁₆×C₁₃₁	—	2	2	no	—
31	C₂×C₁₆×C₁₃₁	—	2	1	no	—
32	C₂×C₁₆×C₁₃₁	—	2	1	no	—
33	C₂×C₁₆×C₁₃₁	—	2	2	no	—
34	C₂×C₁₆×C₁₃₁	—	2	2	no	—
35	C₂×C₁₆×C₁₃₁	D₄	1	4	yes	—
36	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
37	C₄×C₈×C₁₃₁	—	2	2	no	—
38	C₄×C₈×C₁₃₁	—	2	2	no	—
39	#95×C₁₃₁	C₈	1	1	yes	characteristic
40	#95×C₁₃₁	C₈	1	1	yes	characteristic
41	C₂⁴×C₁₃₁	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₁	—	2	2	no	—
43	C₂²×C₄×C₁₃₁	—	2	2	no	—
44	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₃₁	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₃₁	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₁₃₁	M₁₆	1	2	yes	—
49	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃₁	—	2	2	no	—
51	C₂×C₈×C₁₃₁	—	2	2	no	—
52	C₂×C₈×C₁₃₁	—	2	4	no	—
53	C₂×C₈×C₁₃₁	—	2	2	no	—
54	C₂×C₈×C₁₃₁	—	2	2	no	—
55	C₂×C₈×C₁₃₁	—	2	2	no	—
56	C₂×C₈×C₁₃₁	—	2	2	no	—
57	C₂×C₈×C₁₃₁	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₁₃₁	#41	1	2	yes	—
62	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
63	C₄²×C₁₃₁	—	2	2	no	—
64	C₄²×C₁₃₁	—	2	2	no	—
65	C₁₆×C₁₃₁	—	2	4	no	—
66	C₁₆×C₁₃₁	—	4	1	no	—
67	C₁₆×C₁₃₁	—	4	1	no	—
68	C₂³×C₁₃₁	—	2	2	no	—
69	C₂³×C₁₃₁	—	2	2	no	—
70	C₂³×C₁₃₁	—	2	1	no	—
71	C₂³×C₁₃₁	—	2	1	no	—
72	C₂³×C₁₃₁	C₄⋊C₈	1	2	yes	—
73	C₂³×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃₁	—	2	2	no	—
75	C₂×C₄×C₁₃₁	—	2	2	no	—
76	C₂×C₄×C₁₃₁	—	2	2	no	—
77	C₂×C₄×C₁₃₁	—	2	2	no	—
78	C₂×C₄×C₁₃₁	—	2	2	no	—
79	C₂×C₄×C₁₃₁	—	2	4	no	—
80	C₂×C₄×C₁₃₁	—	2	2	no	—
81	C₂×C₄×C₁₃₁	—	2	2	no	—
82	C₂×C₄×C₁₃₁	—	2	2	no	—
83	C₂×C₄×C₁₃₁	C₄⋊C₈	1	2	yes	—
84	C₂×C₄×C₁₃₁	M₃₂	1	4	yes	complemented
85	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₁₃₁	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₃₁	#131	1	2	yes	—
89	C₈×C₁₃₁	—	2	2	no	—
90	C₈×C₁₃₁	—	4	2	no	—
91	C₈×C₁₃₁	#120	1	1	yes	characteristic
92	C₈×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
93	C₈×C₁₃₁	#123	1	1	yes	characteristic
94	C₂²×C₁₃₁	—	2	2	no	—
95	C₂²×C₁₃₁	—	2	2	no	—
96	C₂²×C₁₃₁	—	2	2	no	—
97	C₂²×C₁₃₁	—	4	2	no	—
98	C₂²×C₁₃₁	—	4	2	no	—
99	C₂²×C₁₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₁₃₁	C₈×D₄	1	2	yes	—
101	C₂²×C₁₃₁	C₂×M₃₂	1	2	yes	—
102	C₂²×C₁₃₁	#504	1	1	yes	characteristic
103	C₂²×C₁₃₁	#504	1	1	yes	characteristic
104	C₄×C₁₃₁	—	2	2	no	—
105	C₄×C₁₃₁	—	2	4	no	—
106	C₄×C₁₃₁	—	4	2	no	—
107	C₄×C₁₃₁	#385	1	1	yes	characteristic
108	C₄×C₁₃₁	#424	1	1	yes	characteristic
109	C₄×C₁₃₁	#435	1	2	yes	—
110	C₂×C₁₃₁	—	4	2	no	—
111	C₂×C₁₃₁	#1850	1	1	yes	characteristic
112	C₂×C₁₃₁	#2150	1	1	yes	characteristic
113	C₂×C₁₃₁	#2151	1	1	yes	characteristic
114	C₂×C₁₃₁	#2168	1	2	yes	—
115	C₂×C₁₃₁	#2169	1	2	yes	—
116	#12913	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₁₃₁	#12913	1	1	yes	characteristic, direct product factor, Sylow
118	C₂²×M₃₂	C₂×C₁₃₁	1	1	yes	characteristic
119	C₂×C₄⋊C₁₆	C₂×C₁₃₁	1	2	yes	complemented
120	C₂×C₂²⋊C₁₆	C₂×C₁₃₁	1	1	yes	characteristic
121	C₈×C₂²⋊C₄	C₂×C₁₃₁	1	1	yes	characteristic
122	#1374	C₂×C₁₃₁	1	2	yes	complemented
123	C₂³×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
124	C₂×M₃₂	—	2	2	no	—
125	C₂×M₃₂	—	2	4	no	—
126	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
127	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
128	C₂²×C₁₆	C₂²×C₁₃₁	1	2	yes	—
129	C₂×C₂²⋊C₈	C₄×C₁₃₁	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
131	C₄×C₂²⋊C₄	C₄×C₁₃₁	1	1	yes	characteristic
132	C₄⋊C₁₆	—	2	4	no	—
133	C₂²⋊C₁₆	—	2	2	no	—
134	#336	C₄×C₁₃₁	1	2	yes	—
135	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
139	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
140	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
141	C₂²×C₈	Q₈×C₁₃₁	1	2	yes	—
142	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	C₈×C₁₃₁	1	2	yes	—
144	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
145	M₃₂	—	2	8	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	2	1	no	—
149	C₂×C₁₆	—	2	1	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	D₄×C₁₃₁	1	4	yes	—
153	C₂²⋊C₈	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	#95	C₈×C₁₃₁	1	1	yes	characteristic
157	#95	C₈×C₁₃₁	1	1	yes	characteristic
158	C₂⁴	C₂×C₈×C₁₃₁	1	1	yes	characteristic
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
162	C₂²×C₄	C₂×C₈×C₁₃₁	1	2	yes	—
163	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₈×C₁₃₁	1	2	yes	—
165	C₂²×C₄	M₁₆×C₁₃₁	1	2	yes	—
166	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₄⋊C₄×C₁₃₁	1	2	yes	—
175	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
176	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
177	C₂×C₈	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
178	C₂×C₈	#41×C₁₃₁	1	2	yes	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₁₆	—	2	4	no	—
183	C₁₆	—	4	1	no	—
184	C₁₆	—	4	1	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	1	no	—
188	C₂³	—	2	1	no	—
189	C₂³	C₄⋊C₈×C₁₃₁	1	2	yes	—
190	C₂³	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	C₄⋊C₈×C₁₃₁	1	2	yes	—
201	C₂×C₄	M₃₂×C₁₃₁	1	4	yes	complemented
202	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
203	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
204	C₂×C₄	C₂×M₁₆×C₁₃₁	1	1	yes	characteristic
205	C₂×C₄	#131×C₁₃₁	1	2	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	4	2	no	—
208	C₈	#120×C₁₃₁	1	1	yes	characteristic
209	C₈	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
210	C₈	#123×C₁₃₁	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	C₂×C₄⋊C₈×C₁₃₁	1	1	yes	characteristic
217	C₂²	C₈×D₄×C₁₃₁	1	2	yes	—
218	C₂²	C₂×M₃₂×C₁₃₁	1	2	yes	—
219	C₂²	#504×C₁₃₁	1	1	yes	characteristic
220	C₂²	#504×C₁₃₁	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	4	no	—
223	C₄	—	4	2	no	—
224	C₄	#385×C₁₃₁	1	1	yes	characteristic
225	C₄	#424×C₁₃₁	1	1	yes	characteristic
226	C₄	#435×C₁₃₁	1	2	yes	—
227	C₂	—	4	2	no	—
228	C₂	#1850×C₁₃₁	1	1	yes	characteristic
229	C₂	#2150×C₁₃₁	1	1	yes	characteristic
230	C₂	#2151×C₁₃₁	1	1	yes	characteristic
231	C₂	#2168×C₁₃₁	1	2	yes	—
232	C₂	#2169×C₁₃₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊C₁₆×C₁₃₁	C₂	semidirect
5. #1374×C₁₃₁	C₂	semidirect
84. C₂×C₄×C₁₃₁	M₃₂	semidirect
116. #12913	C₁₃₁	direct (N × H)
117. C₁₃₁	#12913	direct (N × H)
119. C₂×C₄⋊C₁₆	C₂×C₁₃₁	semidirect
122. #1374	C₂×C₁₃₁	semidirect
201. C₂×C₄	M₃₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₁₃₁	2
C₂×C₄⋊C₁₆×C₁₃₁	2
C₂×C₂²⋊C₁₆×C₁₃₁	2
C₈×C₂²⋊C₄×C₁₃₁	2
#1374×C₁₃₁	2
#12913	131

Subgroups — C131×#12913 #45698797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#12913 #45698797