Subgroups — C₁₃₁×#33265 #45719149

Order 33536.

732 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2863×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#2860×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1968×C₁₃₁	C₂	1	3	yes	maximal
4	#1941×C₁₃₁	C₂	1	3	yes	maximal, complemented
5	#1934×C₁₃₁	C₂	1	3	yes	maximal
6	#1933×C₁₃₁	C₂	1	3	yes	maximal, complemented
7	C₄²×C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂×#126×C₁₃₁	—	2	1	no	—
10	C₂×#106×C₁₃₁	C₂²	1	3	yes	—
11	C₂×#103×C₁₃₁	C₂²	1	3	yes	—
12	C₂×#102×C₁₃₁	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₁₃₁	C₂²	1	3	yes	—
14	C₂×C₄×C₈×C₁₃₁	C₂²	1	3	yes	—
15	#391×C₁₃₁	C₂²	1	3	yes	—
16	#390×C₁₃₁	C₂²	1	3	yes	—
17	#389×C₁₃₁	C₂²	1	3	yes	—
18	#389×C₁₃₁	C₂²	1	3	yes	—
19	#388×C₁₃₁	—	2	3	no	—
20	#387×C₁₃₁	—	2	3	no	—
21	#383×C₁₃₁	—	2	1	no	—
22	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
23	#340×C₁₃₁	C₂²	1	6	yes	—
24	C₂²×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
25	C₂²×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
26	C₂×SD₁₆×C₁₃₁	—	2	6	no	—
27	C₂²×C₈×C₁₃₁	C₂³	1	3	yes	—
28	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
29	#126×C₁₃₁	—	4	4	no	—
30	C₂×C₄⋊C₄×C₁₃₁	—	2	3	no	—
31	C₂×C₄⋊C₄×C₁₃₁	—	2	3	no	—
32	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	3	yes	—
33	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	3	yes	—
34	C₂×C₂²⋊C₄×C₁₃₁	—	2	3	no	—
35	C₂×C₂²⋊C₄×C₁₃₁	—	2	3	no	—
36	C₂×C₄²×C₁₃₁	D₄	1	3	yes	—
37	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃₁	C₂³	1	3	yes	—
39	#106×C₁₃₁	—	2	6	no	—
40	#103×C₁₃₁	—	2	6	no	—
41	#102×C₁₃₁	—	2	6	no	—
42	C₄×C₈×C₁₃₁	—	2	6	no	—
43	C₄×C₈×C₁₃₁	—	2	6	no	—
44	#95×C₁₃₁	—	2	3	no	—
45	#95×C₁₃₁	—	2	3	no	—
46	#95×C₁₃₁	—	2	3	no	—
47	#95×C₁₃₁	—	2	3	no	—
48	C₂⁴×C₁₃₁	—	2	1	no	—
49	C₂×Q₈×C₁₃₁	—	2	3	no	—
50	C₂×Q₈×C₁₃₁	—	2	3	no	—
51	C₂×D₄×C₁₃₁	—	2	3	no	—
52	C₂×D₄×C₁₃₁	—	2	3	no	—
53	C₂²×C₄×C₁₃₁	—	2	1	no	—
54	C₂²×C₄×C₁₃₁	—	2	3	no	—
55	C₂²×C₄×C₁₃₁	—	2	3	no	—
56	C₂²×C₄×C₁₃₁	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₁₃₁	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
59	SD₁₆×C₁₃₁	—	4	4	no	—
60	C₂×C₈×C₁₃₁	—	2	3	no	—
61	C₂×C₈×C₁₃₁	—	2	3	no	—
62	C₂×C₈×C₁₃₁	#41	1	6	yes	—
63	C₂×C₈×C₁₃₁	#41	1	6	yes	—
64	C₄⋊C₄×C₁₃₁	—	2	3	no	—
65	C₄⋊C₄×C₁₃₁	—	2	3	no	—
66	C₄⋊C₄×C₁₃₁	—	2	3	no	—
67	C₄⋊C₄×C₁₃₁	—	2	3	no	—
68	C₄⋊C₄×C₁₃₁	—	4	6	no	—
69	C₄⋊C₄×C₁₃₁	—	4	6	no	—
70	C₂²⋊C₄×C₁₃₁	—	4	6	no	—
71	C₂²⋊C₄×C₁₃₁	—	4	6	no	—
72	C₄²×C₁₃₁	—	2	6	no	—
73	C₄²×C₁₃₁	—	2	3	no	—
74	C₄²×C₁₃₁	—	2	3	no	—
75	C₄²×C₁₃₁	SD₁₆	1	4	yes	complemented
76	C₂³×C₁₃₁	—	4	3	no	—
77	C₂³×C₁₃₁	—	4	1	no	—
78	C₂³×C₁₃₁	—	4	3	no	—
79	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
80	Q₈×C₁₃₁	—	4	1	no	—
81	Q₈×C₁₃₁	—	4	3	no	—
82	D₄×C₁₃₁	—	4	1	no	—
83	D₄×C₁₃₁	—	4	3	no	—
84	C₂×C₄×C₁₃₁	—	2	3	no	—
85	C₂×C₄×C₁₃₁	—	2	3	no	—
86	C₂×C₄×C₁₃₁	—	2	3	no	—
87	C₂×C₄×C₁₃₁	—	2	3	no	—
88	C₂×C₄×C₁₃₁	—	4	3	no	—
89	C₂×C₄×C₁₃₁	—	4	3	no	—
90	C₂×C₄×C₁₃₁	—	4	3	no	—
91	C₂×C₄×C₁₃₁	—	4	3	no	—
92	C₂×C₄×C₁₃₁	—	4	3	no	—
93	C₂×C₄×C₁₃₁	—	4	6	no	—
94	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	6	yes	—
95	C₂×C₄×C₁₃₁	#135	1	6	yes	—
96	C₂×C₄×C₁₃₁	C₂×#41	1	3	yes	—
97	C₂×C₄×C₁₃₁	C₂×#41	1	3	yes	—
98	C₈×C₁₃₁	—	2	6	no	—
99	C₈×C₁₃₁	#126	1	4	yes	complemented
100	C₂²×C₁₃₁	—	8	1	no	—
101	C₂²×C₁₃₁	—	8	3	no	—
102	C₂²×C₁₃₁	—	8	3	no	—
103	C₂²×C₁₃₁	#532	1	3	yes	—
104	C₂²×C₁₃₁	C₂²×SD₁₆	1	1	yes	characteristic
105	C₂²×C₁₃₁	C₂×#135	1	3	yes	—
106	C₄×C₁₃₁	—	2	6	no	—
107	C₄×C₁₃₁	—	2	6	no	—
108	C₄×C₁₃₁	—	8	1	no	—
109	C₄×C₁₃₁	—	8	3	no	—
110	C₄×C₁₃₁	—	8	3	no	—
111	C₄×C₁₃₁	C₂×#126	1	1	yes	characteristic
112	C₄×C₁₃₁	#529	1	3	yes	—
113	C₂×C₁₃₁	—	16	1	no	—
114	C₂×C₁₃₁	#2860	1	1	yes	characteristic
115	C₂×C₁₃₁	#3167	1	3	yes	—
116	C₂×C₁₃₁	#3168	1	3	yes	—
117	#33265	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₁₃₁	#33265	1	1	yes	characteristic, direct product factor, Sylow
119	#2863	C₂×C₁₃₁	1	1	yes	characteristic, complemented
120	#2860	C₂×C₁₃₁	1	1	yes	characteristic
121	#1968	C₂×C₁₃₁	1	3	yes	—
122	#1941	C₂×C₁₃₁	1	3	yes	complemented
123	#1934	C₂×C₁₃₁	1	3	yes	—
124	#1933	C₂×C₁₃₁	1	3	yes	complemented
125	C₄²×C₈	C₂×C₁₃₁	1	1	yes	characteristic, complemented
126	C₂²×SD₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
127	C₂×#126	—	2	1	no	—
128	C₂×#106	C₂²×C₁₃₁	1	3	yes	—
129	C₂×#103	C₂²×C₁₃₁	1	3	yes	—
130	C₂×#102	C₂²×C₁₃₁	1	3	yes	—
131	C₂×C₄×C₈	C₂²×C₁₃₁	1	3	yes	—
132	C₂×C₄×C₈	C₂²×C₁₃₁	1	3	yes	—
133	#391	C₂²×C₁₃₁	1	3	yes	—
134	#390	C₂²×C₁₃₁	1	3	yes	—
135	#389	C₂²×C₁₃₁	1	3	yes	—
136	#389	C₂²×C₁₃₁	1	3	yes	—
137	#388	—	2	3	no	—
138	#387	—	2	3	no	—
139	#383	—	2	1	no	—
140	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
141	#340	C₂²×C₁₃₁	1	6	yes	—
142	C₂²×Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
143	C₂²×D₄	C₂³×C₁₃₁	1	1	yes	characteristic
144	C₂×SD₁₆	—	2	6	no	—
145	C₂²×C₈	C₂³×C₁₃₁	1	3	yes	—
146	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
147	#126	—	4	4	no	—
148	C₂×C₄⋊C₄	—	2	3	no	—
149	C₂×C₄⋊C₄	—	2	3	no	—
150	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	3	yes	—
151	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	3	yes	—
152	C₂×C₂²⋊C₄	—	2	3	no	—
153	C₂×C₂²⋊C₄	—	2	3	no	—
154	C₂×C₄²	D₄×C₁₃₁	1	3	yes	—
155	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
156	C₂×C₄²	C₂³×C₁₃₁	1	3	yes	—
157	#106	—	2	6	no	—
158	#103	—	2	6	no	—
159	#102	—	2	6	no	—
160	C₄×C₈	—	2	6	no	—
161	C₄×C₈	—	2	6	no	—
162	#95	—	2	3	no	—
163	#95	—	2	3	no	—
164	#95	—	2	3	no	—
165	#95	—	2	3	no	—
166	C₂⁴	—	2	1	no	—
167	C₂×Q₈	—	2	3	no	—
168	C₂×Q₈	—	2	3	no	—
169	C₂×D₄	—	2	3	no	—
170	C₂×D₄	—	2	3	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	3	no	—
173	C₂²×C₄	—	2	3	no	—
174	C₂²×C₄	C₂×D₄×C₁₃₁	1	3	yes	—
175	C₂²×C₄	C₂×D₄×C₁₃₁	1	3	yes	—
176	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
177	SD₁₆	—	4	4	no	—
178	C₂×C₈	—	2	3	no	—
179	C₂×C₈	—	2	3	no	—
180	C₂×C₈	#41×C₁₃₁	1	6	yes	—
181	C₂×C₈	#41×C₁₃₁	1	6	yes	—
182	C₄⋊C₄	—	2	3	no	—
183	C₄⋊C₄	—	2	3	no	—
184	C₄⋊C₄	—	2	3	no	—
185	C₄⋊C₄	—	2	3	no	—
186	C₄⋊C₄	—	4	6	no	—
187	C₄⋊C₄	—	4	6	no	—
188	C₂²⋊C₄	—	4	6	no	—
189	C₂²⋊C₄	—	4	6	no	—
190	C₄²	—	2	6	no	—
191	C₄²	—	2	3	no	—
192	C₄²	—	2	3	no	—
193	C₄²	SD₁₆×C₁₃₁	1	4	yes	complemented
194	C₂³	—	4	3	no	—
195	C₂³	—	4	1	no	—
196	C₂³	—	4	3	no	—
197	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
198	Q₈	—	4	1	no	—
199	Q₈	—	4	3	no	—
200	D₄	—	4	1	no	—
201	D₄	—	4	3	no	—
202	C₂×C₄	—	2	3	no	—
203	C₂×C₄	—	2	3	no	—
204	C₂×C₄	—	2	3	no	—
205	C₂×C₄	—	2	3	no	—
206	C₂×C₄	—	4	3	no	—
207	C₂×C₄	—	4	3	no	—
208	C₂×C₄	—	4	3	no	—
209	C₂×C₄	—	4	3	no	—
210	C₂×C₄	—	4	3	no	—
211	C₂×C₄	—	4	6	no	—
212	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	6	yes	—
213	C₂×C₄	#135×C₁₃₁	1	6	yes	—
214	C₂×C₄	C₂×#41×C₁₃₁	1	3	yes	—
215	C₂×C₄	C₂×#41×C₁₃₁	1	3	yes	—
216	C₈	—	2	6	no	—
217	C₈	#126×C₁₃₁	1	4	yes	complemented
218	C₂²	—	8	1	no	—
219	C₂²	—	8	3	no	—
220	C₂²	—	8	3	no	—
221	C₂²	#532×C₁₃₁	1	3	yes	—
222	C₂²	C₂²×SD₁₆×C₁₃₁	1	1	yes	characteristic
223	C₂²	C₂×#135×C₁₃₁	1	3	yes	—
224	C₄	—	2	6	no	—
225	C₄	—	2	6	no	—
226	C₄	—	8	1	no	—
227	C₄	—	8	3	no	—
228	C₄	—	8	3	no	—
229	C₄	C₂×#126×C₁₃₁	1	1	yes	characteristic
230	C₄	#529×C₁₃₁	1	3	yes	—
231	C₂	—	16	1	no	—
232	C₂	#2860×C₁₃₁	1	1	yes	characteristic
233	C₂	#3167×C₁₃₁	1	3	yes	—
234	C₂	#3168×C₁₃₁	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂²×D₄×C₁₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #2863×C₁₃₁	C₂	semidirect
4. #1941×C₁₃₁	C₂	semidirect
6. #1933×C₁₃₁	C₂	semidirect
7. C₄²×C₈×C₁₃₁	C₂	semidirect
75. C₄²×C₁₃₁	SD₁₆	semidirect
99. C₈×C₁₃₁	#126	semidirect
117. #33265	C₁₃₁	direct (N × H)
118. C₁₃₁	#33265	direct (N × H)
119. #2863	C₂×C₁₃₁	semidirect
122. #1941	C₂×C₁₃₁	semidirect
124. #1933	C₂×C₁₃₁	semidirect
125. C₄²×C₈	C₂×C₁₃₁	semidirect
193. C₄²	SD₁₆×C₁₃₁	semidirect
217. C₈	#126×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2863×C₁₃₁	2
#2860×C₁₃₁	2
#1968×C₁₃₁	2
#1941×C₁₃₁	2
#1934×C₁₃₁	2
#1933×C₁₃₁	2
C₄²×C₈×C₁₃₁	2
#33265	131

Subgroups — C131×#33265 #45719149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33265 #45719149