Subgroups — C₁₃₁×#23369 #45709253

Order 33536.

632 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	C₄×#125×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1939×C₁₃₁	C₂	1	4	yes	maximal
3	#1916×C₁₃₁	C₂	1	4	yes	maximal
4	#1916×C₁₃₁	C₂	1	4	yes	maximal
5	#1831×C₁₃₁	C₂	1	2	yes	maximal
6	C₂×#125×C₁₃₁	C₄	1	1	yes	characteristic
7	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
8	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	4	yes	—
10	C₂×C₄×C₈×C₁₃₁	C₂²	1	4	yes	—
11	#389×C₁₃₁	C₄	1	2	yes	—
12	#387×C₁₃₁	C₄	1	4	yes	—
13	#382×C₁₃₁	C₄	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
15	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	—
16	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
17	#336×C₁₃₁	C₂²	1	4	yes	—
18	#336×C₁₃₁	C₂²	1	4	yes	—
19	#336×C₁₃₁	C₂²	1	4	yes	—
20	#336×C₁₃₁	C₂²	1	4	yes	—
21	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
22	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
23	#125×C₁₃₁	C₈	1	8	yes	complemented
24	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
32	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
33	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₁₃₁	—	4	4	no	—
35	C₄⋊C₈×C₁₃₁	—	4	2	no	—
36	C₄⋊C₈×C₁₃₁	—	4	2	no	—
37	C₄×C₈×C₁₃₁	—	4	4	no	—
38	#95×C₁₃₁	C₂×C₄	1	4	yes	—
39	#95×C₁₃₁	C₂×C₄	1	4	yes	—
40	#95×C₁₃₁	C₂×C₄	1	4	yes	—
41	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
43	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
48	C₂×C₈×C₁₃₁	—	2	4	no	—
49	C₂×C₈×C₁₃₁	—	2	4	no	—
50	C₂×C₈×C₁₃₁	—	2	4	no	—
51	C₂×C₈×C₁₃₁	—	2	4	no	—
52	C₂×C₈×C₁₃₁	—	2	4	no	—
53	C₂×C₈×C₁₃₁	—	2	4	no	—
54	C₄⋊C₄×C₁₃₁	—	4	2	no	—
55	C₄⋊C₄×C₁₃₁	—	4	4	no	—
56	C₄⋊C₄×C₁₃₁	C₂×C₈	1	8	yes	—
57	C₄⋊C₄×C₁₃₁	C₂×C₈	1	16	yes	—
58	C₄²×C₁₃₁	—	2	4	no	—
59	C₄²×C₁₃₁	—	2	2	no	—
60	C₄²×C₁₃₁	—	2	1	no	—
61	C₄²×C₁₃₁	—	2	1	no	—
62	C₄²×C₁₃₁	—	2	4	no	—
63	C₄²×C₁₃₁	—	2	2	no	—
64	C₄²×C₁₃₁	—	4	1	no	—
65	C₄²×C₁₃₁	—	4	2	no	—
66	C₄²×C₁₃₁	C₂×C₈	1	4	yes	—
67	C₂³×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃₁	—	2	1	no	—
69	C₂×C₄×C₁₃₁	—	2	4	no	—
70	C₂×C₄×C₁₃₁	—	2	2	no	—
71	C₂×C₄×C₁₃₁	—	2	1	no	—
72	C₂×C₄×C₁₃₁	—	2	2	no	—
73	C₂×C₄×C₁₃₁	—	2	4	no	—
74	C₂×C₄×C₁₃₁	—	2	1	no	—
75	C₂×C₄×C₁₃₁	—	2	4	no	—
76	C₂×C₄×C₁₃₁	—	2	2	no	—
77	C₂×C₄×C₁₃₁	—	2	2	no	—
78	C₂×C₄×C₁₃₁	—	2	2	no	—
79	C₂×C₄×C₁₃₁	—	2	2	no	—
80	C₂×C₄×C₁₃₁	—	2	4	no	—
81	C₂×C₄×C₁₃₁	—	2	4	no	—
82	C₂×C₄×C₁₃₁	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₁₃₁	C₂²×C₈	1	4	yes	—
84	C₂×C₄×C₁₃₁	C₂²×C₈	1	8	yes	—
85	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
86	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
87	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
88	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
89	C₈×C₁₃₁	—	4	4	no	—
90	C₈×C₁₃₁	—	4	4	no	—
91	C₂²×C₁₃₁	#519	1	1	yes	characteristic
92	C₂²×C₁₃₁	#520	1	2	yes	—
93	C₂²×C₁₃₁	C₂³×C₈	1	1	yes	characteristic
94	C₂²×C₁₃₁	#568	1	1	yes	characteristic
95	C₂²×C₁₃₁	#568	1	2	yes	—
96	C₄×C₁₃₁	—	2	2	no	—
97	C₄×C₁₃₁	—	2	4	no	—
98	C₄×C₁₃₁	—	2	8	no	—
99	C₄×C₁₃₁	—	4	1	no	—
100	C₄×C₁₃₁	—	4	2	no	—
101	C₄×C₁₃₁	—	4	4	no	—
102	C₄×C₁₃₁	#562	1	1	yes	characteristic
103	C₄×C₁₃₁	#563	1	2	yes	—
104	C₄×C₁₃₁	#564	1	1	yes	characteristic
105	C₂×C₁₃₁	#2382	1	1	yes	characteristic
106	C₂×C₁₃₁	#2982	1	1	yes	characteristic
107	C₂×C₁₃₁	#2987	1	1	yes	characteristic
108	C₂×C₁₃₁	#2988	1	2	yes	—
109	C₂×C₁₃₁	#2995	1	2	yes	—
110	#23369	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₁₃₁	#23369	1	1	yes	characteristic, direct product factor, Sylow
112	C₄×#125	C₂×C₁₃₁	1	1	yes	characteristic
113	#1939	C₂×C₁₃₁	1	4	yes	—
114	#1916	C₂×C₁₃₁	1	4	yes	—
115	#1916	C₂×C₁₃₁	1	4	yes	—
116	#1831	C₂×C₁₃₁	1	2	yes	—
117	C₂×#125	C₄×C₁₃₁	1	1	yes	characteristic
118	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
119	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
120	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	4	yes	—
121	C₂×C₄×C₈	C₂²×C₁₃₁	1	4	yes	—
122	#389	C₄×C₁₃₁	1	2	yes	—
123	#387	C₄×C₁₃₁	1	4	yes	—
124	#382	C₄×C₁₃₁	1	1	yes	characteristic
125	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
126	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	—
127	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
128	#336	C₂²×C₁₃₁	1	4	yes	—
129	#336	C₂²×C₁₃₁	1	4	yes	—
130	#336	C₂²×C₁₃₁	1	4	yes	—
131	#336	C₂²×C₁₃₁	1	4	yes	—
132	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
133	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
134	#125	C₈×C₁₃₁	1	8	yes	complemented
135	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
136	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
137	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
138	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
139	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
140	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
141	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
142	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
143	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
144	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
145	C₄⋊C₈	—	4	4	no	—
146	C₄⋊C₈	—	4	2	no	—
147	C₄⋊C₈	—	4	2	no	—
148	C₄×C₈	—	4	4	no	—
149	#95	C₂×C₄×C₁₃₁	1	4	yes	—
150	#95	C₂×C₄×C₁₃₁	1	4	yes	—
151	#95	C₂×C₄×C₁₃₁	1	4	yes	—
152	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
153	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
154	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
155	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
156	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
157	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
158	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	C₂×C₈×C₁₃₁	1	16	yes	—
168	C₄⋊C₄	C₂×C₈×C₁₃₁	1	8	yes	—
169	C₄²	—	2	4	no	—
170	C₄²	—	2	2	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	1	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	4	no	—
175	C₄²	—	4	1	no	—
176	C₄²	—	4	2	no	—
177	C₄²	C₂×C₈×C₁₃₁	1	4	yes	—
178	C₂³	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	C₂²×C₈×C₁₃₁	1	4	yes	—
194	C₂×C₄	C₂²×C₈×C₁₃₁	1	2	yes	—
195	C₂×C₄	C₂²×C₈×C₁₃₁	1	8	yes	—
196	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	2	yes	—
197	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
198	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
199	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	2	yes	—
200	C₈	—	4	4	no	—
201	C₈	—	4	4	no	—
202	C₂²	#519×C₁₃₁	1	1	yes	characteristic
203	C₂²	#520×C₁₃₁	1	2	yes	—
204	C₂²	C₂³×C₈×C₁₃₁	1	1	yes	characteristic
205	C₂²	#568×C₁₃₁	1	2	yes	—
206	C₂²	#568×C₁₃₁	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	2	8	no	—
209	C₄	—	2	2	no	—
210	C₄	—	4	1	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	4	no	—
213	C₄	#562×C₁₃₁	1	1	yes	characteristic
214	C₄	#563×C₁₃₁	1	2	yes	—
215	C₄	#564×C₁₃₁	1	1	yes	characteristic
216	C₂	#2382×C₁₃₁	1	1	yes	characteristic
217	C₂	#2982×C₁₃₁	1	1	yes	characteristic
218	C₂	#2987×C₁₃₁	1	1	yes	characteristic
219	C₂	#2988×C₁₃₁	1	2	yes	—
220	C₂	#2995×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
socle, prime radical	C₂³×C₁₃₁	C₂³×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
23. #125×C₁₃₁	C₈	semidirect
110. #23369	C₁₃₁	direct (N × H)
111. C₁₃₁	#23369	direct (N × H)
134. #125	C₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#125×C₁₃₁	2
#1939×C₁₃₁	2
#1916×C₁₃₁	2
#1831×C₁₃₁	2
#23369	131

Subgroups — C131×#23369 #45709253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#23369 #45709253