Subgroups — C₁₃₁×#13288 #45699172

Order 33536.

354 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3160×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₄×M₃₂×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1846×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1386×C₁₃₁	C₂	1	4	yes	maximal
5	#533×C₁₃₁	—	2	1	no	—
6	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
7	#500×C₁₃₁	C₄	1	2	yes	complemented
8	#499×C₁₃₁	—	2	1	no	—
9	#499×C₁₃₁	C₄	1	2	yes	complemented
10	#498×C₁₃₁	—	2	1	no	—
11	#427×C₁₃₁	—	2	1	no	—
12	#427×C₁₃₁	C₂²	1	2	yes	—
13	C₂×#107×C₁₃₁	—	2	1	no	—
14	C₂×#106×C₁₃₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	#382×C₁₃₁	—	2	1	no	—
17	#346×C₁₃₁	—	2	1	no	—
18	C₄×C₁₆×C₁₃₁	—	2	1	no	—
19	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
21	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
22	#125×C₁₃₁	—	2	2	no	—
23	C₂²⋊Q₈×C₁₃₁	—	4	2	no	—
24	C₄×Q₈×C₁₃₁	—	4	2	no	—
25	#117×C₁₃₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
27	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
28	M₃₂×C₁₃₁	—	2	2	no	—
29	M₃₂×C₁₃₁	Q₈	1	4	yes	complemented
30	C₂×C₁₆×C₁₃₁	—	2	2	no	—
31	#107×C₁₃₁	—	2	2	no	—
32	#107×C₁₃₁	—	4	2	no	—
33	#107×C₁₃₁	C₂×C₄	1	4	yes	—
34	#106×C₁₃₁	—	2	2	no	—
35	#106×C₁₃₁	—	2	2	no	—
36	#106×C₁₃₁	—	2	4	no	—
37	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
41	#95×C₁₃₁	—	4	2	no	—
42	C₂×Q₈×C₁₃₁	—	4	2	no	—
43	C₂²×C₄×C₁₃₁	—	4	2	no	—
44	C₂²×C₄×C₁₃₁	D₈	1	2	yes	—
45	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃₁	SD₁₆	1	2	yes	—
49	C₂×C₈×C₁₃₁	Q₁₆	1	2	yes	—
50	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₁₃₁	#41	1	2	yes	—
54	C₄⋊C₄×C₁₃₁	—	2	4	no	—
55	C₄⋊C₄×C₁₃₁	—	4	4	no	—
56	C₄⋊C₄×C₁₃₁	—	4	2	no	—
57	C₄⋊C₄×C₁₃₁	—	4	2	no	—
58	C₄⋊C₄×C₁₃₁	—	4	2	no	—
59	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
60	C₄²×C₁₃₁	—	4	2	no	—
61	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₁₃₁	SD₁₆	1	2	yes	—
64	C₁₆×C₁₃₁	—	2	4	no	—
65	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
66	Q₈×C₁₃₁	—	8	4	no	—
67	C₂×C₄×C₁₃₁	—	4	2	no	—
68	C₂×C₄×C₁₃₁	—	4	4	no	—
69	C₂×C₄×C₁₃₁	—	8	4	no	—
70	C₂×C₄×C₁₃₁	—	8	2	no	—
71	C₂×C₄×C₁₃₁	#102	1	2	yes	—
72	C₂×C₄×C₁₃₁	#102	1	2	yes	—
73	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃₁	#124	1	1	yes	characteristic
75	C₂×C₄×C₁₃₁	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	1	yes	characteristic
79	C₈×C₁₃₁	#103	1	2	yes	—
80	C₈×C₁₃₁	#103	1	2	yes	—
81	C₈×C₁₃₁	C₄×Q₈	1	2	yes	—
82	C₈×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
83	C₂²×C₁₃₁	—	2	2	no	—
84	C₂²×C₁₃₁	C₂×#102	1	1	yes	characteristic
85	C₂²×C₁₃₁	#486	1	1	yes	characteristic
86	C₂²×C₁₃₁	#500	1	1	yes	characteristic
87	C₄×C₁₃₁	—	2	2	no	—
88	C₄×C₁₃₁	—	2	2	no	—
89	C₄×C₁₃₁	—	8	4	no	—
90	C₄×C₁₃₁	—	16	2	no	—
91	C₄×C₁₃₁	#391	1	1	yes	characteristic
92	C₄×C₁₃₁	C₂×#103	1	1	yes	characteristic
93	C₄×C₁₃₁	#487	1	1	yes	characteristic
94	C₄×C₁₃₁	#498	1	1	yes	characteristic
95	C₂×C₁₃₁	—	2	1	no	—
96	C₂×C₁₃₁	—	2	1	no	—
97	C₂×C₁₃₁	#1990	1	1	yes	characteristic
98	C₂×C₁₃₁	#2145	1	2	yes	—
99	#13288	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₁₃₁	#13288	1	1	yes	characteristic, direct product factor, Sylow
101	#3160	C₂×C₁₃₁	1	1	yes	characteristic
102	C₄×M₃₂	C₂×C₁₃₁	1	1	yes	characteristic
103	#1846	C₂×C₁₃₁	1	1	yes	characteristic
104	#1386	C₂×C₁₃₁	1	4	yes	—
105	#533	—	2	1	no	—
106	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
107	#500	C₄×C₁₃₁	1	2	yes	complemented
108	#499	—	2	1	no	—
109	#499	C₄×C₁₃₁	1	2	yes	complemented
110	#498	—	2	1	no	—
111	#427	—	2	1	no	—
112	#427	C₂²×C₁₃₁	1	2	yes	—
113	C₂×#107	—	2	1	no	—
114	C₂×#106	C₂²×C₁₃₁	1	2	yes	—
115	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
116	#382	—	2	1	no	—
117	#346	—	2	1	no	—
118	C₄×C₁₆	—	2	1	no	—
119	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
120	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
121	C₄⋊Q₈	—	2	2	no	—
122	#125	—	2	2	no	—
123	C₂²⋊Q₈	—	4	2	no	—
124	C₄×Q₈	—	4	2	no	—
125	#117	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
128	M₃₂	—	2	2	no	—
129	M₃₂	Q₈×C₁₃₁	1	4	yes	complemented
130	C₂×C₁₆	—	2	2	no	—
131	#107	—	2	2	no	—
132	#107	—	4	2	no	—
133	#107	C₂×C₄×C₁₃₁	1	4	yes	—
134	#106	—	2	2	no	—
135	#106	—	2	2	no	—
136	#106	—	2	4	no	—
137	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
138	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
139	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
140	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
141	#95	—	4	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	D₈×C₁₃₁	1	2	yes	—
145	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
146	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
147	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
148	C₂×C₈	SD₁₆×C₁₃₁	1	2	yes	—
149	C₂×C₈	Q₁₆×C₁₃₁	1	2	yes	—
150	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
151	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
152	C₂×C₈	C₂×Q₈×C₁₃₁	1	2	yes	—
153	C₂×C₈	#41×C₁₃₁	1	2	yes	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₂²⋊C₄	—	4	2	no	—
160	C₄²	—	4	2	no	—
161	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
162	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
163	C₄²	SD₁₆×C₁₃₁	1	2	yes	—
164	C₁₆	—	2	4	no	—
165	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
166	Q₈	—	8	4	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	8	4	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	#102×C₁₃₁	1	2	yes	—
172	C₂×C₄	#102×C₁₃₁	1	2	yes	—
173	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
174	C₂×C₄	#124×C₁₃₁	1	1	yes	characteristic
175	C₂×C₄	C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
176	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
177	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
178	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	1	yes	characteristic
179	C₈	#103×C₁₃₁	1	2	yes	—
180	C₈	#103×C₁₃₁	1	2	yes	—
181	C₈	C₄×Q₈×C₁₃₁	1	2	yes	—
182	C₈	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	C₂×#102×C₁₃₁	1	1	yes	characteristic
185	C₂²	#486×C₁₃₁	1	1	yes	characteristic
186	C₂²	#500×C₁₃₁	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	8	4	no	—
190	C₄	—	16	2	no	—
191	C₄	#391×C₁₃₁	1	1	yes	characteristic
192	C₄	C₂×#103×C₁₃₁	1	1	yes	characteristic
193	C₄	#487×C₁₃₁	1	1	yes	characteristic
194	C₄	#498×C₁₃₁	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	#1990×C₁₃₁	1	1	yes	characteristic
198	C₂	#2145×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
prime radical	C₂³×C₁₃₁	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
7. #500×C₁₃₁	C₄	semidirect
9. #499×C₁₃₁	C₄	semidirect
29. M₃₂×C₁₃₁	Q₈	semidirect
99. #13288	C₁₃₁	direct (N × H)
100. C₁₃₁	#13288	direct (N × H)
107. #500	C₄×C₁₃₁	semidirect
109. #499	C₄×C₁₃₁	semidirect
129. M₃₂	Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3160×C₁₃₁	2
C₄×M₃₂×C₁₃₁	2
#1846×C₁₃₁	2
#1386×C₁₃₁	2
#13288	131

Subgroups — C131×#13288 #45699172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13288 #45699172