Subgroups — C₁₃₁×#20538 #45706422

Order 33536.

922 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₁₃₁	C₂	1	2	yes	maximal
2	#2285×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1770×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	C₈×C₄⋊C₄×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	#1750×C₁₃₁	C₂	1	4	yes	maximal
6	C₂×#117×C₁₃₁	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
8	#431×C₁₃₁	C₄	1	16	yes	complemented
9	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
11	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	4	yes	—
12	C₂×#97×C₁₃₁	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₁₃₁	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₁₃₁	C₂²	1	4	yes	—
15	#384×C₁₃₁	C₄	1	2	yes	—
16	#382×C₁₃₁	C₄	1	2	yes	—
17	#380×C₁₃₁	C₄	1	4	yes	—
18	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	—
19	#336×C₁₃₁	C₂²	1	8	yes	—
20	C₂³×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
23	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
24	#117×C₁₃₁	—	4	2	no	—
25	#117×C₁₃₁	—	4	2	no	—
26	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	8	yes	—
27	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃₁	C₂×C₄	1	8	yes	—
32	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
33	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
34	C₄⋊C₈×C₁₃₁	C₂×C₄	1	8	yes	—
35	C₄⋊C₈×C₁₃₁	C₂×C₄	1	8	yes	—
36	C₂²⋊C₈×C₁₃₁	C₂×C₄	1	16	yes	—
37	#97×C₁₃₁	—	4	4	no	—
38	C₄×C₈×C₁₃₁	—	4	4	no	—
39	C₄×C₈×C₁₃₁	C₂×C₄	1	16	yes	—
40	#95×C₁₃₁	C₂×C₄	1	8	yes	—
41	C₂⁴×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₁	—	2	8	no	—
43	C₂²×C₄×C₁₃₁	—	2	1	no	—
44	C₂²×C₄×C₁₃₁	—	2	1	no	—
45	C₂²×C₄×C₁₃₁	—	2	2	no	—
46	C₂²×C₄×C₁₃₁	C₄²	1	8	yes	—
47	C₂²×C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
48	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
53	C₂×C₈×C₁₃₁	—	2	4	no	—
54	C₂×C₈×C₁₃₁	—	2	4	no	—
55	C₂×C₈×C₁₃₁	—	2	4	no	—
56	C₂×C₈×C₁₃₁	—	2	4	no	—
57	C₂×C₈×C₁₃₁	C₂²×C₄	1	8	yes	—
58	C₂×C₈×C₁₃₁	C₂²×C₄	1	8	yes	—
59	C₄⋊C₄×C₁₃₁	—	2	16	no	—
60	C₄⋊C₄×C₁₃₁	—	4	4	no	—
61	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
62	C₄²×C₁₃₁	—	2	4	no	—
63	C₄²×C₁₃₁	—	2	4	no	—
64	C₄²×C₁₃₁	C₄²	1	4	yes	—
65	C₄²×C₁₃₁	C₄²	1	4	yes	—
66	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
67	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
68	C₂³×C₁₃₁	—	2	1	no	—
69	C₂³×C₁₃₁	—	2	1	no	—
70	C₂³×C₁₃₁	—	2	4	no	—
71	C₂³×C₁₃₁	C₂×C₄²	1	2	yes	—
72	C₂³×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃₁	—	2	1	no	—
74	C₂×C₄×C₁₃₁	—	2	1	no	—
75	C₂×C₄×C₁₃₁	—	2	4	no	—
76	C₂×C₄×C₁₃₁	—	2	4	no	—
77	C₂×C₄×C₁₃₁	—	2	4	no	—
78	C₂×C₄×C₁₃₁	—	2	16	no	—
79	C₂×C₄×C₁₃₁	—	2	8	no	—
80	C₂×C₄×C₁₃₁	—	2	16	no	—
81	C₂×C₄×C₁₃₁	C₂×C₄²	1	2	yes	—
82	C₂×C₄×C₁₃₁	C₂×C₄²	1	4	yes	—
83	C₂×C₄×C₁₃₁	C₂×C₄²	1	4	yes	—
84	C₂×C₄×C₁₃₁	#131	1	8	yes	—
85	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
86	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
87	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
88	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
89	C₈×C₁₃₁	—	2	8	no	—
90	C₈×C₁₃₁	—	2	8	no	—
91	C₂²×C₁₃₁	—	2	4	no	—
92	C₂²×C₁₃₁	—	2	2	no	—
93	C₂²×C₁₃₁	—	4	2	no	—
94	C₂²×C₁₃₁	—	4	2	no	—
95	C₂²×C₁₃₁	C₂²×C₄²	1	1	yes	characteristic
96	C₂²×C₁₃₁	#518	1	1	yes	characteristic
97	C₂²×C₁₃₁	#519	1	1	yes	characteristic
98	C₂²×C₁₃₁	C₂×#131	1	4	yes	—
99	C₄×C₁₃₁	—	2	4	no	—
100	C₄×C₁₃₁	—	2	2	no	—
101	C₄×C₁₃₁	—	4	8	no	—
102	C₄×C₁₃₁	#405	1	8	yes	—
103	C₄×C₁₃₁	#520	1	2	yes	—
104	C₄×C₁₃₁	#520	1	2	yes	—
105	C₂×C₁₃₁	—	4	2	no	—
106	C₂×C₁₃₁	#2274	1	1	yes	characteristic
107	C₂×C₁₃₁	C₂×#405	1	2	yes	—
108	C₂×C₁₃₁	#2979	1	2	yes	—
109	C₂×C₁₃₁	#2986	1	2	yes	—
110	#20538	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₁₃₁	#20538	1	1	yes	characteristic, direct product factor, Sylow
112	C₂×#431	C₂×C₁₃₁	1	2	yes	—
113	#2285	C₂×C₁₃₁	1	1	yes	characteristic
114	#1770	C₂×C₁₃₁	1	4	yes	complemented
115	C₈×C₄⋊C₄	C₂×C₁₃₁	1	4	yes	complemented
116	#1750	C₂×C₁₃₁	1	4	yes	—
117	C₂×#117	C₂²×C₁₃₁	1	2	yes	—
118	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
119	#431	C₄×C₁₃₁	1	16	yes	complemented
120	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
121	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
122	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	4	yes	—
123	C₂×#97	C₂²×C₁₃₁	1	4	yes	—
124	C₂×C₄×C₈	C₂²×C₁₃₁	1	4	yes	—
125	C₂×C₄×C₈	C₂²×C₁₃₁	1	4	yes	—
126	#384	C₄×C₁₃₁	1	2	yes	—
127	#382	C₄×C₁₃₁	1	2	yes	—
128	#380	C₄×C₁₃₁	1	4	yes	—
129	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	—
130	#336	C₂²×C₁₃₁	1	8	yes	—
131	C₂³×C₄	C₂×C₄×C₁₃₁	1	2	yes	—
132	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
133	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
134	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
135	#117	—	4	2	no	—
136	#117	—	4	2	no	—
137	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	8	yes	—
138	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
139	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
140	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
141	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
142	C₂×C₄²	C₂×C₄×C₁₃₁	1	8	yes	—
143	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
144	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
145	C₄⋊C₈	C₂×C₄×C₁₃₁	1	8	yes	—
146	C₄⋊C₈	C₂×C₄×C₁₃₁	1	8	yes	—
147	C₂²⋊C₈	C₂×C₄×C₁₃₁	1	16	yes	—
148	#97	—	4	4	no	—
149	C₄×C₈	—	4	4	no	—
150	C₄×C₈	C₂×C₄×C₁₃₁	1	16	yes	—
151	#95	C₂×C₄×C₁₃₁	1	8	yes	—
152	C₂⁴	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
153	C₂²×C₄	—	2	8	no	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	C₄²×C₁₃₁	1	8	yes	—
158	C₂²×C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
159	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
160	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
161	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
162	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
163	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	C₂²×C₄×C₁₃₁	1	8	yes	—
169	C₂×C₈	C₂²×C₄×C₁₃₁	1	8	yes	—
170	C₄⋊C₄	—	2	16	no	—
171	C₄⋊C₄	—	4	4	no	—
172	C₂²⋊C₄	—	4	4	no	—
173	C₄²	—	2	4	no	—
174	C₄²	—	2	4	no	—
175	C₄²	C₄²×C₁₃₁	1	4	yes	—
176	C₄²	C₄²×C₁₃₁	1	4	yes	—
177	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
178	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
179	C₂³	—	2	1	no	—
180	C₂³	—	2	1	no	—
181	C₂³	—	2	4	no	—
182	C₂³	C₂×C₄²×C₁₃₁	1	2	yes	—
183	C₂³	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	16	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	—	2	16	no	—
192	C₂×C₄	C₂×C₄²×C₁₃₁	1	2	yes	—
193	C₂×C₄	C₂×C₄²×C₁₃₁	1	4	yes	—
194	C₂×C₄	C₂×C₄²×C₁₃₁	1	4	yes	—
195	C₂×C₄	#131×C₁₃₁	1	8	yes	—
196	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
197	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
198	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
199	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	2	yes	—
200	C₈	—	2	8	no	—
201	C₈	—	2	8	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²×C₄²×C₁₃₁	1	1	yes	characteristic
207	C₂²	#518×C₁₃₁	1	1	yes	characteristic
208	C₂²	#519×C₁₃₁	1	1	yes	characteristic
209	C₂²	C₂×#131×C₁₃₁	1	4	yes	—
210	C₄	—	2	2	no	—
211	C₄	—	2	4	no	—
212	C₄	—	4	8	no	—
213	C₄	#405×C₁₃₁	1	8	yes	—
214	C₄	#520×C₁₃₁	1	2	yes	—
215	C₄	#520×C₁₃₁	1	2	yes	—
216	C₂	—	4	2	no	—
217	C₂	#2274×C₁₃₁	1	1	yes	characteristic
218	C₂	C₂×#405×C₁₃₁	1	2	yes	—
219	C₂	#2979×C₁₃₁	1	2	yes	—
220	C₂	#2986×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	C₂⁴
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 ◁ C₂²×C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1770×C₁₃₁	C₂	semidirect
4. C₈×C₄⋊C₄×C₁₃₁	C₂	semidirect
8. #431×C₁₃₁	C₄	semidirect
110. #20538	C₁₃₁	direct (N × H)
111. C₁₃₁	#20538	direct (N × H)
114. #1770	C₂×C₁₃₁	semidirect
115. C₈×C₄⋊C₄	C₂×C₁₃₁	semidirect
119. #431	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×#431×C₁₃₁	2
#2285×C₁₃₁	2
#1770×C₁₃₁	2
C₈×C₄⋊C₄×C₁₃₁	2
#1750×C₁₃₁	2
#20538	131

Subgroups — C131×#20538 #45706422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#20538 #45706422