Subgroups — C₁₃₁×#13166 #45699050

Order 33536.

326 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#367×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#367×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1942×C₁₃₁	C₂	1	2	yes	maximal
5	#1380×C₁₃₁	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
8	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
9	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
11	#389×C₁₃₁	—	2	2	no	—
12	#367×C₁₃₁	—	2	2	no	—
13	#367×C₁₃₁	C₄	1	4	yes	complemented
14	C₄×C₁₆×C₁₃₁	—	2	2	no	—
15	#340×C₁₃₁	—	2	2	no	—
16	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
19	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₃₁	—	4	2	no	—
21	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₃₁	—	2	1	no	—
23	C₂×C₁₆×C₁₃₁	—	2	1	no	—
24	C₂×C₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₁₃₁	D₄	1	2	yes	—
26	C₂×C₁₆×C₁₃₁	Q₈	1	2	yes	—
27	C₂×C₁₆×C₁₃₁	Q₈	1	2	yes	—
28	#107×C₁₃₁	—	2	2	no	—
29	#107×C₁₃₁	—	2	2	no	—
30	#107×C₁₃₁	—	2	2	no	—
31	#107×C₁₃₁	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₁₃₁	—	2	1	no	—
33	C₄×C₈×C₁₃₁	—	2	1	no	—
34	#95×C₁₃₁	—	4	2	no	—
35	C₂²×C₄×C₁₃₁	—	4	2	no	—
36	C₂²×C₄×C₁₃₁	—	4	2	no	—
37	C₂²×C₄×C₁₃₁	D₈	1	2	yes	—
38	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃₁	—	2	1	no	—
40	C₂×C₈×C₁₃₁	—	2	1	no	—
41	C₂×C₈×C₁₃₁	Q₁₆	1	2	yes	—
42	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
47	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
48	C₄⋊C₄×C₁₃₁	—	2	4	no	—
49	C₄⋊C₄×C₁₃₁	—	4	2	no	—
50	C₄⋊C₄×C₁₃₁	—	4	2	no	—
51	C₄⋊C₄×C₁₃₁	—	4	2	no	—
52	C₄⋊C₄×C₁₃₁	—	8	4	no	—
53	C₄²×C₁₃₁	—	2	1	no	—
54	C₄²×C₁₃₁	—	2	1	no	—
55	C₁₆×C₁₃₁	—	2	2	no	—
56	C₁₆×C₁₃₁	C₄⋊C₄	1	4	yes	complemented
57	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃₁	—	2	2	no	—
59	C₂×C₄×C₁₃₁	—	2	2	no	—
60	C₂×C₄×C₁₃₁	—	4	4	no	—
61	C₂×C₄×C₁₃₁	—	8	2	no	—
62	C₂×C₄×C₁₃₁	—	8	2	no	—
63	C₂×C₄×C₁₃₁	—	8	2	no	—
64	C₂×C₄×C₁₃₁	—	8	2	no	—
65	C₂×C₄×C₁₃₁	—	8	2	no	—
66	C₂×C₄×C₁₃₁	SD₃₂	1	4	yes	—
67	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
69	C₂×C₄×C₁₃₁	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₃₁	#135	1	1	yes	characteristic
72	C₂×C₄×C₁₃₁	#135	1	1	yes	characteristic
73	C₈×C₁₃₁	—	2	2	no	—
74	C₈×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₁₃₁	C₄×Q₈	1	1	yes	characteristic
76	C₈×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
77	C₈×C₁₃₁	#125	1	1	yes	characteristic
78	C₂²×C₁₃₁	C₄×D₈	1	1	yes	characteristic
79	C₂²×C₁₃₁	#466	1	1	yes	characteristic
80	C₂²×C₁₃₁	#500	1	1	yes	characteristic
81	C₂²×C₁₃₁	C₂×SD₃₂	1	2	yes	—
82	C₂²×C₁₃₁	#508	1	2	yes	—
83	C₄×C₁₃₁	—	2	4	no	—
84	C₄×C₁₃₁	—	8	4	no	—
85	C₄×C₁₃₁	—	16	2	no	—
86	C₄×C₁₃₁	#389	1	1	yes	characteristic
87	C₄×C₁₃₁	C₄×Q₁₆	1	1	yes	characteristic
88	C₄×C₁₃₁	#467	1	1	yes	characteristic
89	C₄×C₁₃₁	#499	1	1	yes	characteristic
90	C₂×C₁₃₁	#1942	1	1	yes	characteristic
91	C₂×C₁₃₁	C₄×SD₃₂	1	2	yes	—
92	C₂×C₁₃₁	#2217	1	2	yes	—
93	C₂×C₁₃₁	#2253	1	1	yes	characteristic
94	C₂×C₁₃₁	#2254	1	1	yes	characteristic
95	#13166	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₁₃₁	#13166	1	1	yes	characteristic, direct product factor, Sylow
97	C₂×#367	C₂×C₁₃₁	1	1	yes	characteristic
98	C₂×#367	C₂×C₁₃₁	1	1	yes	characteristic
99	C₂×C₄×C₁₆	C₂×C₁₃₁	1	1	yes	characteristic
100	#1942	C₂×C₁₃₁	1	2	yes	—
101	#1380	C₂×C₁₃₁	1	2	yes	—
102	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
103	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
104	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
105	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
106	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
107	#389	—	2	2	no	—
108	#367	—	2	2	no	—
109	#367	C₄×C₁₃₁	1	4	yes	complemented
110	C₄×C₁₆	—	2	2	no	—
111	#340	—	2	2	no	—
112	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
113	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	4	2	no	—
117	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
121	C₂×C₁₆	D₄×C₁₃₁	1	2	yes	—
122	C₂×C₁₆	Q₈×C₁₃₁	1	2	yes	—
123	C₂×C₁₆	Q₈×C₁₃₁	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	—	2	2	no	—
127	#107	C₂×C₄×C₁₃₁	1	4	yes	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	—	2	1	no	—
130	#95	—	4	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	D₈×C₁₃₁	1	2	yes	—
134	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	Q₁₆×C₁₃₁	1	2	yes	—
138	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
139	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
140	C₂×C₈	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
141	C₂×C₈	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
142	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
143	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	8	4	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₁₆	—	2	2	no	—
152	C₁₆	C₄⋊C₄×C₁₃₁	1	4	yes	complemented
153	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	SD₃₂×C₁₃₁	1	4	yes	—
163	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
164	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
165	C₂×C₄	C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
166	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	1	yes	characteristic
167	C₂×C₄	#135×C₁₃₁	1	1	yes	characteristic
168	C₂×C₄	#135×C₁₃₁	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
171	C₈	C₄×Q₈×C₁₃₁	1	1	yes	characteristic
172	C₈	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
173	C₈	#125×C₁₃₁	1	1	yes	characteristic
174	C₂²	C₄×D₈×C₁₃₁	1	1	yes	characteristic
175	C₂²	#466×C₁₃₁	1	1	yes	characteristic
176	C₂²	#500×C₁₃₁	1	1	yes	characteristic
177	C₂²	C₂×SD₃₂×C₁₃₁	1	2	yes	—
178	C₂²	#508×C₁₃₁	1	2	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	8	4	no	—
181	C₄	—	16	2	no	—
182	C₄	#389×C₁₃₁	1	1	yes	characteristic
183	C₄	C₄×Q₁₆×C₁₃₁	1	1	yes	characteristic
184	C₄	#467×C₁₃₁	1	1	yes	characteristic
185	C₄	#499×C₁₃₁	1	1	yes	characteristic
186	C₂	#1942×C₁₃₁	1	1	yes	characteristic
187	C₂	C₄×SD₃₂×C₁₃₁	1	2	yes	—
188	C₂	#2217×C₁₃₁	1	2	yes	—
189	C₂	#2253×C₁₃₁	1	1	yes	characteristic
190	C₂	#2254×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
13. #367×C₁₃₁	C₄	semidirect
56. C₁₆×C₁₃₁	C₄⋊C₄	semidirect
95. #13166	C₁₃₁	direct (N × H)
96. C₁₃₁	#13166	direct (N × H)
109. #367	C₄×C₁₃₁	semidirect
152. C₁₆	C₄⋊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×#367×C₁₃₁	2
C₂×C₄×C₁₆×C₁₃₁	2
#1942×C₁₃₁	2
#1380×C₁₃₁	2
#13166	131

Subgroups — C131×#13166 #45699050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13166 #45699050