Subgroups — C₁₃₁×#7170 #45693054

Order 33536.

262 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#346×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1770×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₁₃₁	—	2	1	no	—
5	C₂²×C₁₆×C₁₃₁	—	2	1	no	—
6	C₂×C₄⋊C₈×C₁₃₁	C₄	1	1	yes	characteristic
7	C₂×#97×C₁₃₁	—	2	1	no	—
8	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊C₄×C₁₃₁	C₄	1	1	yes	characteristic
10	C₄⋊C₁₆×C₁₃₁	—	2	1	no	—
11	C₄⋊C₁₆×C₁₃₁	C₄	1	2	yes	complemented
12	#346×C₁₃₁	—	2	1	no	—
13	#346×C₁₃₁	C₄	1	2	yes	complemented
14	#336×C₁₃₁	—	2	1	no	—
15	C₂²×C₈×C₁₃₁	—	2	1	no	—
16	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₃₁	C₈	1	2	yes	—
19	C₂×C₄²×C₁₃₁	—	2	1	no	—
20	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₁₃₁	—	2	2	no	—
22	C₂×C₁₆×C₁₃₁	—	2	2	no	—
23	C₂×C₁₆×C₁₃₁	—	2	2	no	—
24	C₂×C₁₆×C₁₃₁	—	2	2	no	—
25	C₂×C₁₆×C₁₃₁	—	4	1	no	—
26	C₂×C₁₆×C₁₃₁	—	4	1	no	—
27	C₄⋊C₈×C₁₃₁	—	2	1	no	—
28	C₄⋊C₈×C₁₃₁	C₈	1	2	yes	—
29	#97×C₁₃₁	—	4	1	no	—
30	#97×C₁₃₁	—	4	1	no	—
31	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₃₁	Q₈	1	1	yes	characteristic
35	#95×C₁₃₁	—	2	1	no	—
36	C₂²×C₄×C₁₃₁	—	2	2	no	—
37	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₃₁	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₁₃₁	—	2	1	no	—
41	C₂×C₈×C₁₃₁	—	2	1	no	—
42	C₂×C₈×C₁₃₁	—	2	2	no	—
43	C₂×C₈×C₁₃₁	—	4	1	no	—
44	C₂×C₈×C₁₃₁	—	4	1	no	—
45	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃₁	D₈	1	1	yes	characteristic
50	C₂×C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₁₃₁	Q₁₆	1	1	yes	characteristic
53	C₄⋊C₄×C₁₃₁	—	4	2	no	—
54	C₄²×C₁₃₁	—	4	1	no	—
55	C₄²×C₁₃₁	—	4	1	no	—
56	C₄²×C₁₃₁	C₄²	1	1	yes	characteristic
57	C₄²×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₁₃₁	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₁₃₁	M₁₆	1	1	yes	characteristic
60	C₁₆×C₁₃₁	—	2	4	no	—
61	C₁₆×C₁₃₁	—	4	2	no	—
62	C₂³×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₃₁	—	2	2	no	—
64	C₂×C₄×C₁₃₁	—	2	2	no	—
65	C₂×C₄×C₁₃₁	—	4	2	no	—
66	C₂×C₄×C₁₃₁	—	4	2	no	—
67	C₂×C₄×C₁₃₁	—	4	2	no	—
68	C₂×C₄×C₁₃₁	#95	1	1	yes	characteristic
69	C₂×C₄×C₁₃₁	C₄×C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₃₁	#97	1	1	yes	characteristic
71	C₂×C₄×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
73	C₂×C₄×C₁₃₁	#103	1	1	yes	characteristic
74	C₂×C₄×C₁₃₁	#104	1	1	yes	characteristic
75	C₂×C₄×C₁₃₁	#104	1	1	yes	characteristic
76	C₂×C₄×C₁₃₁	C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃₁	C₄⋊C₈	1	1	yes	characteristic
78	C₈×C₁₃₁	—	2	1	no	—
79	C₈×C₁₃₁	—	2	1	no	—
80	C₈×C₁₃₁	—	4	2	no	—
81	C₈×C₁₃₁	#102	1	1	yes	characteristic
82	C₈×C₁₃₁	#103	1	1	yes	characteristic
83	C₈×C₁₃₁	#106	1	1	yes	characteristic
84	C₈×C₁₃₁	#107	1	1	yes	characteristic
85	C₂²×C₁₃₁	D₄⋊C₈	1	1	yes	characteristic
86	C₂²×C₁₃₁	Q₈⋊C₈	1	1	yes	characteristic
87	C₂²×C₁₃₁	#336	1	1	yes	characteristic
88	C₂²×C₁₃₁	#349	1	2	yes	—
89	C₂²×C₁₃₁	#350	1	2	yes	—
90	C₄×C₁₃₁	—	2	1	no	—
91	C₄×C₁₃₁	—	2	1	no	—
92	C₄×C₁₃₁	—	2	1	no	—
93	C₄×C₁₃₁	—	2	1	no	—
94	C₄×C₁₃₁	—	8	2	no	—
95	C₄×C₁₃₁	D₄⋊C₈	1	1	yes	characteristic
96	C₄×C₁₃₁	Q₈⋊C₈	1	1	yes	characteristic
97	C₄×C₁₃₁	#339	1	1	yes	characteristic
98	C₄×C₁₃₁	#340	1	1	yes	characteristic
99	C₂×C₁₃₁	#1279	1	1	yes	characteristic
100	C₂×C₁₃₁	#1330	1	2	yes	—
101	C₂×C₁₃₁	#1338	1	2	yes	—
102	C₂×C₁₃₁	#1377	1	1	yes	characteristic
103	C₂×C₁₃₁	#1379	1	1	yes	characteristic
104	#7170	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₁₃₁	#7170	1	1	yes	characteristic, direct product factor, Sylow
106	C₂×C₄⋊C₁₆	C₂×C₁₃₁	1	1	yes	characteristic
107	C₂×#346	C₂×C₁₃₁	1	1	yes	characteristic
108	#1770	C₂×C₁₃₁	1	1	yes	characteristic
109	C₂²×C₁₆	—	2	1	no	—
110	C₂²×C₁₆	—	2	1	no	—
111	C₂×C₄⋊C₈	C₄×C₁₃₁	1	1	yes	characteristic
112	C₂×#97	—	2	1	no	—
113	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
114	C₄×C₄⋊C₄	C₄×C₁₃₁	1	1	yes	characteristic
115	C₄⋊C₁₆	—	2	1	no	—
116	C₄⋊C₁₆	C₄×C₁₃₁	1	2	yes	complemented
117	#346	—	2	1	no	—
118	#346	C₄×C₁₃₁	1	2	yes	complemented
119	#336	—	2	1	no	—
120	C₂²×C₈	—	2	1	no	—
121	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
122	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
123	C₂×C₄⋊C₄	C₈×C₁₃₁	1	2	yes	—
124	C₂×C₄²	—	2	1	no	—
125	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	4	1	no	—
131	C₂×C₁₆	—	4	1	no	—
132	C₄⋊C₈	—	2	1	no	—
133	C₄⋊C₈	C₈×C₁₃₁	1	2	yes	—
134	#97	—	4	1	no	—
135	#97	—	4	1	no	—
136	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
137	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
138	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
139	C₄×C₈	Q₈×C₁₃₁	1	1	yes	characteristic
140	#95	—	2	1	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
144	C₂²×C₄	M₁₆×C₁₃₁	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
152	C₂×C₈	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
153	C₂×C₈	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
154	C₂×C₈	D₈×C₁₃₁	1	1	yes	characteristic
155	C₂×C₈	SD₁₆×C₁₃₁	1	1	yes	characteristic
156	C₂×C₈	SD₁₆×C₁₃₁	1	1	yes	characteristic
157	C₂×C₈	Q₁₆×C₁₃₁	1	1	yes	characteristic
158	C₄⋊C₄	—	4	2	no	—
159	C₄²	—	4	1	no	—
160	C₄²	—	4	1	no	—
161	C₄²	C₄²×C₁₃₁	1	1	yes	characteristic
162	C₄²	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
163	C₄²	C₂×C₈×C₁₃₁	1	1	yes	characteristic
164	C₄²	M₁₆×C₁₃₁	1	1	yes	characteristic
165	C₁₆	—	2	4	no	—
166	C₁₆	—	4	2	no	—
167	C₂³	C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	#95×C₁₃₁	1	1	yes	characteristic
174	C₂×C₄	C₄×C₈×C₁₃₁	1	1	yes	characteristic
175	C₂×C₄	#97×C₁₃₁	1	1	yes	characteristic
176	C₂×C₄	C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
177	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
178	C₂×C₄	#103×C₁₃₁	1	1	yes	characteristic
179	C₂×C₄	#104×C₁₃₁	1	1	yes	characteristic
180	C₂×C₄	#104×C₁₃₁	1	1	yes	characteristic
181	C₂×C₄	C₄⋊C₈×C₁₃₁	1	1	yes	characteristic
182	C₂×C₄	C₄⋊C₈×C₁₃₁	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	4	2	no	—
186	C₈	#102×C₁₃₁	1	1	yes	characteristic
187	C₈	#103×C₁₃₁	1	1	yes	characteristic
188	C₈	#106×C₁₃₁	1	1	yes	characteristic
189	C₈	#107×C₁₃₁	1	1	yes	characteristic
190	C₂²	D₄⋊C₈×C₁₃₁	1	1	yes	characteristic
191	C₂²	Q₈⋊C₈×C₁₃₁	1	1	yes	characteristic
192	C₂²	#336×C₁₃₁	1	1	yes	characteristic
193	C₂²	#349×C₁₃₁	1	2	yes	—
194	C₂²	#350×C₁₃₁	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	8	2	no	—
200	C₄	D₄⋊C₈×C₁₃₁	1	1	yes	characteristic
201	C₄	Q₈⋊C₈×C₁₃₁	1	1	yes	characteristic
202	C₄	#339×C₁₃₁	1	1	yes	characteristic
203	C₄	#340×C₁₃₁	1	1	yes	characteristic
204	C₂	#1279×C₁₃₁	1	1	yes	characteristic
205	C₂	#1330×C₁₃₁	1	2	yes	—
206	C₂	#1338×C₁₃₁	1	2	yes	—
207	C₂	#1377×C₁₃₁	1	1	yes	characteristic
208	C₂	#1379×C₁₃₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
11. C₄⋊C₁₆×C₁₃₁	C₄	semidirect
13. #346×C₁₃₁	C₄	semidirect
104. #7170	C₁₃₁	direct (N × H)
105. C₁₃₁	#7170	direct (N × H)
116. C₄⋊C₁₆	C₄×C₁₃₁	semidirect
118. #346	C₄×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⋊C₁₆×C₁₃₁	2
C₂×#346×C₁₃₁	2
#1770×C₁₃₁	2
#7170	131

Subgroups — C131×#7170 #45693054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#7170 #45693054