Subgroups — C₁₃₁×#20364 #45706248

Order 33536.

1286 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₁₃₁	C₂	1	2	yes	maximal
2	C₂×C₄³×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1745×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	#1744×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	#1728×C₁₃₁	C₂	1	4	yes	maximal
6	C₂²×M₁₆×C₁₃₁	C₂²	1	4	yes	—
7	C₂²×C₄²×C₁₃₁	C₄	1	4	yes	—
8	C₂²×C₄²×C₁₃₁	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₄×M₁₆×C₁₃₁	C₄	1	16	yes	complemented
11	C₂×#97×C₁₃₁	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₁₃₁	C₂²	1	4	yes	—
13	C₄³×C₁₃₁	C₄	1	4	yes	—
14	C₄³×C₁₃₁	C₂²	1	2	yes	—
15	C₄³×C₁₃₁	C₂²	1	2	yes	—
16	#336×C₁₃₁	C₂²	1	16	yes	—
17	C₂³×C₄×C₁₃₁	C₂×C₄	1	4	yes	—
18	C₂³×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₃₁	—	2	16	no	—
21	C₂×M₁₆×C₁₃₁	C₂×C₄	1	16	yes	—
22	C₂²×C₈×C₁₃₁	C₂³	1	8	yes	—
23	C₂×C₄²×C₁₃₁	—	2	8	no	—
24	C₂×C₄²×C₁₃₁	—	2	16	no	—
25	C₂×C₄²×C₁₃₁	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₁₃₁	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₃₁	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃₁	C₂×C₄	1	8	yes	—
30	C₂×C₄²×C₁₃₁	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
32	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
34	#97×C₁₃₁	C₂×C₄	1	16	yes	complemented
35	C₄×C₈×C₁₃₁	C₂×C₄	1	16	yes	complemented
36	C₂⁴×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₁	—	2	8	no	—
38	C₂²×C₄×C₁₃₁	—	2	16	no	—
39	C₂²×C₄×C₁₃₁	—	2	8	no	—
40	C₂²×C₄×C₁₃₁	C₄²	1	8	yes	—
41	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
43	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₁₃₁	#41	1	8	yes	—
47	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
48	M₁₆×C₁₃₁	—	2	32	no	—
49	C₂×C₈×C₁₃₁	—	2	16	no	—
50	C₂×C₈×C₁₃₁	C₂²×C₄	1	16	yes	—
51	C₄²×C₁₃₁	—	2	8	no	—
52	C₄²×C₁₃₁	—	2	16	no	—
53	C₄²×C₁₃₁	—	2	8	no	—
54	C₄²×C₁₃₁	—	2	8	no	—
55	C₄²×C₁₃₁	C₄²	1	8	yes	—
56	C₄²×C₁₃₁	M₁₆	1	16	yes	complemented
57	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
58	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
59	C₂³×C₁₃₁	—	2	4	no	—
60	C₂³×C₁₃₁	C₂×C₄²	1	2	yes	—
61	C₂³×C₁₃₁	#117	1	4	yes	—
62	C₂³×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃₁	—	2	32	no	—
64	C₂×C₄×C₁₃₁	—	2	8	no	—
65	C₂×C₄×C₁₃₁	—	2	4	no	—
66	C₂×C₄×C₁₃₁	—	2	8	no	—
67	C₂×C₄×C₁₃₁	—	2	16	no	—
68	C₂×C₄×C₁₃₁	C₂×C₄²	1	4	yes	—
69	C₂×C₄×C₁₃₁	C₂×C₄²	1	2	yes	—
70	C₂×C₄×C₁₃₁	C₂×C₄²	1	4	yes	—
71	C₂×C₄×C₁₃₁	#117	1	16	yes	—
72	C₂×C₄×C₁₃₁	#117	1	4	yes	—
73	C₂×C₄×C₁₃₁	C₂×M₁₆	1	16	yes	—
74	C₂×C₄×C₁₃₁	C₂×M₁₆	1	8	yes	—
75	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
76	C₂×C₄×C₁₃₁	C₂×#41	1	4	yes	—
77	C₈×C₁₃₁	—	2	16	no	—
78	C₂²×C₁₃₁	—	2	4	no	—
79	C₂²×C₁₃₁	—	2	8	no	—
80	C₂²×C₁₃₁	#376	1	4	yes	—
81	C₂²×C₁₃₁	C₂²×C₄²	1	1	yes	characteristic
82	C₂²×C₁₃₁	C₂×#117	1	2	yes	—
83	C₂²×C₁₃₁	C₂²×M₁₆	1	4	yes	—
84	C₄×C₁₃₁	—	2	4	no	—
85	C₄×C₁₃₁	—	2	16	no	—
86	C₄×C₁₃₁	#376	1	4	yes	—
87	C₄×C₁₃₁	C₄×M₁₆	1	8	yes	—
88	C₄×C₁₃₁	C₂×#117	1	4	yes	—
89	C₂×C₁₃₁	—	2	4	no	—
90	C₂×C₁₃₁	C₂×#376	1	1	yes	characteristic
91	C₂×C₁₃₁	C₂×C₄×M₁₆	1	2	yes	—
92	C₂×C₁₃₁	#2920	1	4	yes	—
93	#20364	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₁₃₁	#20364	1	1	yes	characteristic, direct product factor, Sylow
95	C₂×C₄×M₁₆	C₂×C₁₃₁	1	2	yes	—
96	C₂×C₄³	C₂×C₁₃₁	1	1	yes	characteristic
97	#1745	C₂×C₁₃₁	1	4	yes	complemented
98	#1744	C₂×C₁₃₁	1	4	yes	complemented
99	#1728	C₂×C₁₃₁	1	4	yes	—
100	C₂²×M₁₆	C₂²×C₁₃₁	1	4	yes	—
101	C₂²×C₄²	C₄×C₁₃₁	1	4	yes	—
102	C₂²×C₄²	C₂²×C₁₃₁	1	2	yes	—
103	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
104	C₄×M₁₆	C₄×C₁₃₁	1	16	yes	complemented
105	C₂×#97	C₂²×C₁₃₁	1	4	yes	—
106	C₂×C₄×C₈	C₂²×C₁₃₁	1	4	yes	—
107	C₄³	C₄×C₁₃₁	1	4	yes	—
108	C₄³	C₂²×C₁₃₁	1	2	yes	—
109	C₄³	C₂²×C₁₃₁	1	2	yes	—
110	#336	C₂²×C₁₃₁	1	16	yes	—
111	C₂³×C₄	C₂×C₄×C₁₃₁	1	4	yes	—
112	C₂³×C₄	C₂×C₄×C₁₃₁	1	2	yes	—
113	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
114	C₂×M₁₆	—	2	16	no	—
115	C₂×M₁₆	C₂×C₄×C₁₃₁	1	16	yes	—
116	C₂²×C₈	C₂³×C₁₃₁	1	8	yes	—
117	C₂×C₄²	—	2	8	no	—
118	C₂×C₄²	—	2	16	no	—
119	C₂×C₄²	C₂×C₄×C₁₃₁	1	4	yes	—
120	C₂×C₄²	C₂×C₄×C₁₃₁	1	8	yes	—
121	C₂×C₄²	C₂×C₄×C₁₃₁	1	4	yes	—
122	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
123	C₂×C₄²	C₂×C₄×C₁₃₁	1	8	yes	—
124	C₂×C₄²	C₂×C₄×C₁₃₁	1	4	yes	—
125	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
126	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
127	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
128	#97	C₂×C₄×C₁₃₁	1	16	yes	complemented
129	C₄×C₈	C₂×C₄×C₁₃₁	1	16	yes	complemented
130	C₂⁴	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
131	C₂²×C₄	—	2	8	no	—
132	C₂²×C₄	—	2	16	no	—
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	C₄²×C₁₃₁	1	8	yes	—
135	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
136	C₂²×C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
137	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
138	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
139	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
140	C₂²×C₄	#41×C₁₃₁	1	8	yes	—
141	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
142	M₁₆	—	2	32	no	—
143	C₂×C₈	—	2	16	no	—
144	C₂×C₈	C₂²×C₄×C₁₃₁	1	16	yes	—
145	C₄²	—	2	8	no	—
146	C₄²	—	2	16	no	—
147	C₄²	—	2	8	no	—
148	C₄²	—	2	8	no	—
149	C₄²	C₄²×C₁₃₁	1	8	yes	—
150	C₄²	M₁₆×C₁₃₁	1	16	yes	complemented
151	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
152	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
153	C₂³	—	2	4	no	—
154	C₂³	C₂×C₄²×C₁₃₁	1	2	yes	—
155	C₂³	#117×C₁₃₁	1	4	yes	—
156	C₂³	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
157	C₂×C₄	—	2	32	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	16	no	—
162	C₂×C₄	C₂×C₄²×C₁₃₁	1	4	yes	—
163	C₂×C₄	C₂×C₄²×C₁₃₁	1	4	yes	—
164	C₂×C₄	C₂×C₄²×C₁₃₁	1	2	yes	—
165	C₂×C₄	#117×C₁₃₁	1	16	yes	—
166	C₂×C₄	#117×C₁₃₁	1	4	yes	—
167	C₂×C₄	C₂×M₁₆×C₁₃₁	1	16	yes	—
168	C₂×C₄	C₂×M₁₆×C₁₃₁	1	8	yes	—
169	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
170	C₂×C₄	C₂×#41×C₁₃₁	1	4	yes	—
171	C₈	—	2	16	no	—
172	C₂²	—	2	4	no	—
173	C₂²	—	2	8	no	—
174	C₂²	#376×C₁₃₁	1	4	yes	—
175	C₂²	C₂²×C₄²×C₁₃₁	1	1	yes	characteristic
176	C₂²	C₂×#117×C₁₃₁	1	2	yes	—
177	C₂²	C₂²×M₁₆×C₁₃₁	1	4	yes	—
178	C₄	—	2	4	no	—
179	C₄	—	2	16	no	—
180	C₄	#376×C₁₃₁	1	4	yes	—
181	C₄	C₄×M₁₆×C₁₃₁	1	8	yes	—
182	C₄	C₂×#117×C₁₃₁	1	4	yes	—
183	C₂	—	2	4	no	—
184	C₂	C₂×#376×C₁₃₁	1	1	yes	characteristic
185	C₂	C₂×C₄×M₁₆×C₁₃₁	1	2	yes	—
186	C₂	#2920×C₁₃₁	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₁₃₁	C₂³
commutator	C₂²	C₂²×C₄²×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
socle	C₂³×C₁₃₁	C₂³×C₄
prime radical	C₂⁴×C₁₃₁	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1745×C₁₃₁	C₂	semidirect
4. #1744×C₁₃₁	C₂	semidirect
10. C₄×M₁₆×C₁₃₁	C₄	semidirect
34. #97×C₁₃₁	C₂×C₄	semidirect
35. C₄×C₈×C₁₃₁	C₂×C₄	semidirect
56. C₄²×C₁₃₁	M₁₆	semidirect
93. #20364	C₁₃₁	direct (N × H)
94. C₁₃₁	#20364	direct (N × H)
97. #1745	C₂×C₁₃₁	semidirect
98. #1744	C₂×C₁₃₁	semidirect
104. C₄×M₁₆	C₄×C₁₃₁	semidirect
128. #97	C₂×C₄×C₁₃₁	semidirect
129. C₄×C₈	C₂×C₄×C₁₃₁	semidirect
150. C₄²	M₁₆×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×M₁₆×C₁₃₁	2
C₂×C₄³×C₁₃₁	2
#1745×C₁₃₁	2
#1744×C₁₃₁	2
#1728×C₁₃₁	2
#20364	131

Subgroups — C131×#20364 #45706248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#20364 #45706248