Subgroups — C₁₃₁×#8165 #45694049

Order 33536.

430 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₄×#101×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1461×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1451×C₁₃₁	C₂	1	2	yes	maximal
5	#1283×C₁₃₁	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₁₃₁	—	2	2	no	—
8	C₈×Q₈×C₁₃₁	—	2	2	no	—
9	#431×C₁₃₁	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₁₃₁	C₄	1	2	yes	—
11	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×#101×C₁₃₁	C₄	1	2	yes	—
13	C₄×M₁₆×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃₁	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	#343×C₁₃₁	C₄	1	2	yes	—
17	#324×C₁₃₁	C₄	1	4	yes	—
18	#322×C₁₃₁	—	2	2	no	—
19	#322×C₁₃₁	—	2	2	no	—
20	C₂²×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₁₃₁	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₃₁	—	2	2	no	—
26	C₄×Q₈×C₁₃₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₁₃₁	—	2	4	no	—
32	C₄⋊C₈×C₁₃₁	—	2	2	no	—
33	C₄⋊C₈×C₁₃₁	—	2	2	no	—
34	#101×C₁₃₁	—	2	4	no	—
35	C₂²⋊C₈×C₁₃₁	C₂×C₄	1	4	yes	—
36	#97×C₁₃₁	—	2	2	no	—
37	C₄×C₈×C₁₃₁	—	2	2	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	D₄	1	2	yes	—
42	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
43	C₂×Q₈×C₁₃₁	—	2	2	no	—
44	C₂×Q₈×C₁₃₁	—	2	2	no	—
45	C₂×Q₈×C₁₃₁	M₁₆	1	4	yes	—
46	C₂²×C₄×C₁₃₁	C₄²	1	2	yes	—
47	C₂²×C₄×C₁₃₁	C₄²	1	2	yes	—
48	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
51	M₁₆×C₁₃₁	—	2	4	no	—
52	M₁₆×C₁₃₁	—	2	4	no	—
53	C₂×C₈×C₁₃₁	—	2	2	no	—
54	C₂×C₈×C₁₃₁	—	2	2	no	—
55	C₂×C₈×C₁₃₁	—	2	4	no	—
56	C₂×C₈×C₁₃₁	—	2	4	no	—
57	C₂×C₈×C₁₃₁	—	4	1	no	—
58	C₂×C₈×C₁₃₁	—	4	2	no	—
59	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₄⋊C₄×C₁₃₁	—	2	4	no	—
63	C₄⋊C₄×C₁₃₁	—	2	2	no	—
64	C₄²×C₁₃₁	—	2	2	no	—
65	C₄²×C₁₃₁	—	2	2	no	—
66	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
67	C₄²×C₁₃₁	#41	1	2	yes	—
68	C₂³×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
69	Q₈×C₁₃₁	—	2	8	no	—
70	C₂×C₄×C₁₃₁	—	2	2	no	—
71	C₂×C₄×C₁₃₁	—	2	2	no	—
72	C₂×C₄×C₁₃₁	—	2	1	no	—
73	C₂×C₄×C₁₃₁	—	2	1	no	—
74	C₂×C₄×C₁₃₁	—	2	2	no	—
75	C₂×C₄×C₁₃₁	—	4	1	no	—
76	C₂×C₄×C₁₃₁	—	4	2	no	—
77	C₂×C₄×C₁₃₁	#97	1	4	yes	—
78	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
80	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₁₃₁	#131	1	2	yes	—
84	C₈×C₁₃₁	—	2	1	no	—
85	C₈×C₁₃₁	—	2	2	no	—
86	C₈×C₁₃₁	—	4	4	no	—
87	C₈×C₁₃₁	—	4	2	no	—
88	C₈×C₁₃₁	—	4	4	no	—
89	C₈×C₁₃₁	#101	1	2	yes	complemented
90	C₂²×C₁₃₁	—	2	1	no	—
91	C₂²×C₁₃₁	—	2	1	no	—
92	C₂²×C₁₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₁₃₁	C₂×#97	1	1	yes	characteristic
94	C₂²×C₁₃₁	#405	1	1	yes	characteristic
95	C₄×C₁₃₁	—	2	4	no	—
96	C₄×C₁₃₁	—	2	1	no	—
97	C₄×C₁₃₁	—	2	2	no	—
98	C₄×C₁₃₁	—	4	2	no	—
99	C₄×C₁₃₁	C₂×#101	1	1	yes	characteristic
100	C₄×C₁₃₁	#413	1	1	yes	characteristic
101	C₄×C₁₃₁	#435	1	4	yes	—
102	C₂×C₁₃₁	—	2	1	no	—
103	C₂×C₁₃₁	—	2	1	no	—
104	C₂×C₁₃₁	#1752	1	1	yes	characteristic
105	C₂×C₁₃₁	C₄×#101	1	1	yes	characteristic
106	C₂×C₁₃₁	#1757	1	1	yes	characteristic
107	#8165	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₁₃₁	#8165	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×C₈×Q₈	C₂×C₁₃₁	1	1	yes	characteristic
110	C₄×#101	C₂×C₁₃₁	1	1	yes	characteristic
111	#1461	C₂×C₁₃₁	1	1	yes	characteristic
112	#1451	C₂×C₁₃₁	1	2	yes	—
113	#1283	C₂×C₁₃₁	1	2	yes	—
114	C₂×C₄×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
115	C₈×Q₈	—	2	2	no	—
116	C₈×Q₈	—	2	2	no	—
117	#431	C₂²×C₁₃₁	1	2	yes	—
118	C₂×C₄⋊C₈	C₄×C₁₃₁	1	2	yes	—
119	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
120	C₂×#101	C₄×C₁₃₁	1	2	yes	—
121	C₄×M₁₆	C₂²×C₁₃₁	1	2	yes	—
122	C₂×C₄×C₈	C₄×C₁₃₁	1	2	yes	—
123	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
124	#343	C₄×C₁₃₁	1	2	yes	—
125	#324	C₄×C₁₃₁	1	4	yes	—
126	#322	—	2	2	no	—
127	#322	—	2	2	no	—
128	C₂²×Q₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
129	C₂×M₁₆	C₂×C₄×C₁₃₁	1	4	yes	—
130	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
131	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
132	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
133	C₄×Q₈	—	2	2	no	—
134	C₄×Q₈	—	2	2	no	—
135	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
136	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
137	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
138	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
139	C₄⋊C₈	—	2	4	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	2	2	no	—
142	#101	—	2	4	no	—
143	C₂²⋊C₈	C₂×C₄×C₁₃₁	1	4	yes	—
144	#97	—	2	2	no	—
145	C₄×C₈	—	2	2	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
150	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
151	C₂×Q₈	—	2	2	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	M₁₆×C₁₃₁	1	4	yes	—
154	C₂²×C₄	C₄²×C₁₃₁	1	2	yes	—
155	C₂²×C₄	C₄²×C₁₃₁	1	2	yes	—
156	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
157	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
158	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
159	M₁₆	—	2	4	no	—
160	M₁₆	—	2	4	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
168	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	2	yes	—
169	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
175	C₄²	#41×C₁₃₁	1	2	yes	—
176	C₂³	C₂×C₄²×C₁₃₁	1	1	yes	characteristic
177	Q₈	—	2	8	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	#97×C₁₃₁	1	4	yes	—
186	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
187	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
188	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
189	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
190	C₂×C₄	C₂×M₁₆×C₁₃₁	1	2	yes	—
191	C₂×C₄	#131×C₁₃₁	1	2	yes	—
192	C₈	—	2	1	no	—
193	C₈	—	2	2	no	—
194	C₈	—	4	4	no	—
195	C₈	—	4	2	no	—
196	C₈	—	4	4	no	—
197	C₈	#101×C₁₃₁	1	2	yes	complemented
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	C₄×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
201	C₂²	C₂×#97×C₁₃₁	1	1	yes	characteristic
202	C₂²	#405×C₁₃₁	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	2	no	—
206	C₄	—	4	2	no	—
207	C₄	C₂×#101×C₁₃₁	1	1	yes	characteristic
208	C₄	#413×C₁₃₁	1	1	yes	characteristic
209	C₄	#435×C₁₃₁	1	4	yes	—
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#1752×C₁₃₁	1	1	yes	characteristic
213	C₂	C₄×#101×C₁₃₁	1	1	yes	characteristic
214	C₂	#1757×C₁₃₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
89. C₈×C₁₃₁	#101	semidirect
107. #8165	C₁₃₁	direct (N × H)
108. C₁₃₁	#8165	direct (N × H)
197. C₈	#101×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×Q₈×C₁₃₁	2
C₄×#101×C₁₃₁	2
#1461×C₁₃₁	2
#1451×C₁₃₁	2
#1283×C₁₃₁	2
#8165	131

Subgroups — C131×#8165 #45694049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#8165 #45694049