Subgroups — C₁₃₁×#7281 #45693165

Order 33536.

226 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2150×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1848×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#335×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₁₃₁	—	2	1	no	—
5	C₂×#108×C₁₃₁	—	2	1	no	—
6	C₂×#108×C₁₃₁	C₄	1	2	yes	—
7	#423×C₁₃₁	—	2	1	no	—
8	C₂×C₄⋊C₈×C₁₃₁	—	2	1	no	—
9	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₄⋊C₁₆×C₁₃₁	—	2	1	no	—
11	C₄⋊C₁₆×C₁₃₁	C₄	1	2	yes	—
12	#335×C₁₃₁	—	2	1	no	—
13	#335×C₁₃₁	C₄	1	2	yes	—
14	C₂×M₁₆×C₁₃₁	—	2	2	no	—
15	C₂²×C₈×C₁₃₁	—	4	1	no	—
16	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
19	M₃₂×C₁₃₁	—	4	1	no	—
20	M₃₂×C₁₃₁	—	4	1	no	—
21	C₂×C₁₆×C₁₃₁	—	2	2	no	—
22	#108×C₁₃₁	—	4	2	no	—
23	#108×C₁₃₁	—	4	1	no	—
24	#108×C₁₃₁	—	4	1	no	—
25	C₄⋊C₈×C₁₃₁	—	2	2	no	—
26	C₄⋊C₈×C₁₃₁	—	4	1	no	—
27	C₄⋊C₈×C₁₃₁	—	4	1	no	—
28	C₄⋊C₈×C₁₃₁	—	4	1	no	—
29	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃₁	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃₁	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₁₃₁	Q₁₆	1	1	yes	characteristic
36	M₁₆×C₁₃₁	—	4	2	no	—
37	M₁₆×C₁₃₁	—	4	2	no	—
38	C₂×C₈×C₁₃₁	—	2	1	no	—
39	C₂×C₈×C₁₃₁	—	2	1	no	—
40	C₂×C₈×C₁₃₁	—	4	2	no	—
41	C₂×C₈×C₁₃₁	—	4	2	no	—
42	C₂×C₈×C₁₃₁	—	8	2	no	—
43	C₂×C₈×C₁₃₁	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃₁	D₈	1	1	yes	characteristic
48	C₂×C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₁₃₁	Q₁₆	1	1	yes	characteristic
51	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₄²×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₁₃₁	D₈	1	1	yes	characteristic
54	C₄²×C₁₃₁	SD₁₆	1	1	yes	characteristic
55	C₁₆×C₁₃₁	—	4	2	no	—
56	C₂³×C₁₃₁	#103	1	1	yes	characteristic
57	C₂×C₄×C₁₃₁	—	2	1	no	—
58	C₂×C₄×C₁₃₁	—	2	1	no	—
59	C₂×C₄×C₁₃₁	—	4	1	no	—
60	C₂×C₄×C₁₃₁	#95	1	1	yes	characteristic
61	C₂×C₄×C₁₃₁	#100	1	1	yes	characteristic
62	C₂×C₄×C₁₃₁	#101	1	1	yes	characteristic
63	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
64	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
65	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
66	C₂×C₄×C₁₃₁	#103	1	1	yes	characteristic
67	C₂×C₄×C₁₃₁	#103	1	1	yes	characteristic
68	C₂×C₄×C₁₃₁	#106	1	1	yes	characteristic
69	C₂×C₄×C₁₃₁	#107	1	1	yes	characteristic
70	C₈×C₁₃₁	—	2	1	no	—
71	C₈×C₁₃₁	—	2	1	no	—
72	C₈×C₁₃₁	—	8	2	no	—
73	C₈×C₁₃₁	—	8	2	no	—
74	C₈×C₁₃₁	#102	1	1	yes	characteristic
75	C₈×C₁₃₁	#103	1	1	yes	characteristic
76	C₈×C₁₃₁	#106	1	1	yes	characteristic
77	C₈×C₁₃₁	#107	1	1	yes	characteristic
78	C₂²×C₁₃₁	—	2	1	no	—
79	C₂²×C₁₃₁	—	2	1	no	—
80	C₂²×C₁₃₁	#332	1	1	yes	characteristic
81	C₂²×C₁₃₁	#333	1	1	yes	characteristic
82	C₂²×C₁₃₁	#340	1	1	yes	characteristic
83	C₄×C₁₃₁	—	2	1	no	—
84	C₄×C₁₃₁	—	2	1	no	—
85	C₄×C₁₃₁	—	2	1	no	—
86	C₄×C₁₃₁	#331	1	1	yes	characteristic
87	C₄×C₁₃₁	#332	1	1	yes	characteristic
88	C₄×C₁₃₁	#340	1	1	yes	characteristic
89	C₄×C₁₃₁	#343	1	1	yes	characteristic
90	C₄×C₁₃₁	#361	1	1	yes	characteristic
91	C₄×C₁₃₁	#362	1	1	yes	characteristic
92	C₂×C₁₃₁	—	2	1	no	—
93	C₂×C₁₃₁	—	2	1	no	—
94	C₂×C₁₃₁	#1296	1	1	yes	characteristic
95	C₂×C₁₃₁	#1385	1	1	yes	characteristic
96	C₂×C₁₃₁	#1387	1	1	yes	characteristic
97	#7281	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₁₃₁	#7281	1	1	yes	characteristic, direct product factor, Sylow
99	#2150	C₂×C₁₃₁	1	1	yes	characteristic
100	#1848	C₂×C₁₃₁	1	1	yes	characteristic
101	C₂×#335	C₂×C₁₃₁	1	1	yes	characteristic
102	C₂×M₃₂	—	2	1	no	—
103	C₂×#108	—	2	1	no	—
104	C₂×#108	C₄×C₁₃₁	1	2	yes	—
105	#423	—	2	1	no	—
106	C₂×C₄⋊C₈	—	2	1	no	—
107	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
108	C₄⋊C₁₆	—	2	1	no	—
109	C₄⋊C₁₆	C₄×C₁₃₁	1	2	yes	—
110	#335	—	2	1	no	—
111	#335	C₄×C₁₃₁	1	2	yes	—
112	C₂×M₁₆	—	2	2	no	—
113	C₂²×C₈	—	4	1	no	—
114	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
115	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
116	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
117	M₃₂	—	4	1	no	—
118	M₃₂	—	4	1	no	—
119	C₂×C₁₆	—	2	2	no	—
120	#108	—	4	2	no	—
121	#108	—	4	1	no	—
122	#108	—	4	1	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	4	1	no	—
125	C₄⋊C₈	—	4	1	no	—
126	C₄⋊C₈	—	4	1	no	—
127	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
128	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
129	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
130	C₄×C₈	Q₈×C₁₃₁	1	1	yes	characteristic
131	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
132	C₂²×C₄	SD₁₆×C₁₃₁	1	1	yes	characteristic
133	C₂²×C₄	Q₁₆×C₁₃₁	1	1	yes	characteristic
134	M₁₆	—	4	2	no	—
135	M₁₆	—	4	2	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	8	2	no	—
141	C₂×C₈	C₄²×C₁₃₁	1	1	yes	characteristic
142	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
143	C₂×C₈	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
144	C₂×C₈	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
145	C₂×C₈	D₈×C₁₃₁	1	1	yes	characteristic
146	C₂×C₈	SD₁₆×C₁₃₁	1	1	yes	characteristic
147	C₂×C₈	SD₁₆×C₁₃₁	1	1	yes	characteristic
148	C₂×C₈	Q₁₆×C₁₃₁	1	1	yes	characteristic
149	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
150	C₄²	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
151	C₄²	D₈×C₁₃₁	1	1	yes	characteristic
152	C₄²	SD₁₆×C₁₃₁	1	1	yes	characteristic
153	C₁₆	—	4	2	no	—
154	C₂³	#103×C₁₃₁	1	1	yes	characteristic
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	#95×C₁₃₁	1	1	yes	characteristic
159	C₂×C₄	#100×C₁₃₁	1	1	yes	characteristic
160	C₂×C₄	#101×C₁₃₁	1	1	yes	characteristic
161	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
162	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
163	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
164	C₂×C₄	#103×C₁₃₁	1	1	yes	characteristic
165	C₂×C₄	#103×C₁₃₁	1	1	yes	characteristic
166	C₂×C₄	#106×C₁₃₁	1	1	yes	characteristic
167	C₂×C₄	#107×C₁₃₁	1	1	yes	characteristic
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	8	2	no	—
171	C₈	—	8	2	no	—
172	C₈	#102×C₁₃₁	1	1	yes	characteristic
173	C₈	#103×C₁₃₁	1	1	yes	characteristic
174	C₈	#106×C₁₃₁	1	1	yes	characteristic
175	C₈	#107×C₁₃₁	1	1	yes	characteristic
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	#332×C₁₃₁	1	1	yes	characteristic
179	C₂²	#333×C₁₃₁	1	1	yes	characteristic
180	C₂²	#340×C₁₃₁	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	#331×C₁₃₁	1	1	yes	characteristic
185	C₄	#332×C₁₃₁	1	1	yes	characteristic
186	C₄	#340×C₁₃₁	1	1	yes	characteristic
187	C₄	#343×C₁₃₁	1	1	yes	characteristic
188	C₄	#361×C₁₃₁	1	1	yes	characteristic
189	C₄	#362×C₁₃₁	1	1	yes	characteristic
190	C₂	—	2	1	no	—
191	C₂	—	2	1	no	—
192	C₂	#1296×C₁₃₁	1	1	yes	characteristic
193	C₂	#1385×C₁₃₁	1	1	yes	characteristic
194	C₂	#1387×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	#340
commutator	C₂×C₈	C₄²×C₁₃₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₃₁
prime radical	C₂³×C₁₃₁	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
97. #7281	C₁₃₁	direct (N × H)
98. C₁₃₁	#7281	direct (N × H)

Maximal subgroups

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

subgroup	index
#2150×C₁₃₁	2
#1848×C₁₃₁	2
C₂×#335×C₁₃₁	2
#7281	131

Subgroups — C131×#7281 #45693165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#7281 #45693165