Subgroups — C₁₃₁×#13240 #45699124

Order 33536.

430 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	C₂²×Q₃₂×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#367×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#358×C₁₃₁	C₂	1	2	yes	maximal
4	C₂×#346×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#1969×C₁₃₁	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₁₃₁	C₂²	1	2	yes	—
7	C₂×Q₃₂×C₁₃₁	—	2	2	no	—
8	C₂×Q₃₂×C₁₃₁	—	2	2	no	—
9	C₂×Q₃₂×C₁₃₁	C₄	1	4	yes	complemented
10	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
13	C₂×#103×C₁₃₁	—	2	2	no	—
14	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	#391×C₁₃₁	—	2	2	no	—
16	#367×C₁₃₁	—	2	2	no	—
17	#358×C₁₃₁	—	2	4	no	—
18	#346×C₁₃₁	—	2	2	no	—
19	C₂²×Q₈×C₁₃₁	—	2	2	no	—
20	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
21	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
22	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
23	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
24	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
25	C₂×Q₁₆×C₁₃₁	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
29	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
30	Q₃₂×C₁₃₁	—	2	8	no	—
31	C₂×C₁₆×C₁₃₁	—	2	1	no	—
32	C₂×C₁₆×C₁₃₁	—	2	1	no	—
33	C₂×C₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₁₃₁	D₄	1	2	yes	—
35	C₂×C₁₆×C₁₃₁	D₄	1	2	yes	—
36	C₂×C₁₆×C₁₃₁	D₄	1	2	yes	—
37	#107×C₁₃₁	—	2	2	no	—
38	#107×C₁₃₁	—	2	2	no	—
39	#103×C₁₃₁	—	4	4	no	—
40	C₄×C₈×C₁₃₁	—	2	1	no	—
41	C₄×C₈×C₁₃₁	—	2	1	no	—
42	#95×C₁₃₁	—	4	4	no	—
43	C₂×Q₈×C₁₃₁	—	2	4	no	—
44	C₂×Q₈×C₁₃₁	—	4	2	no	—
45	C₂×Q₈×C₁₃₁	—	4	2	no	—
46	C₂×Q₈×C₁₃₁	—	4	2	no	—
47	C₂×Q₈×C₁₃₁	—	4	2	no	—
48	C₂×Q₈×C₁₃₁	—	4	2	no	—
49	C₂²×C₄×C₁₃₁	—	4	2	no	—
50	C₂²×C₄×C₁₃₁	—	4	2	no	—
51	C₂²×C₄×C₁₃₁	D₈	1	2	yes	—
52	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
53	Q₁₆×C₁₃₁	—	2	4	no	—
54	Q₁₆×C₁₃₁	—	2	4	no	—
55	Q₁₆×C₁₃₁	—	4	2	no	—
56	Q₁₆×C₁₃₁	—	4	2	no	—
57	C₂×C₈×C₁₃₁	—	2	1	no	—
58	C₂×C₈×C₁₃₁	—	2	1	no	—
59	C₂×C₈×C₁₃₁	D₈	1	2	yes	—
60	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
65	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
66	C₄⋊C₄×C₁₃₁	—	4	2	no	—
67	C₄⋊C₄×C₁₃₁	—	4	2	no	—
68	C₄²×C₁₃₁	—	2	1	no	—
69	C₄²×C₁₃₁	—	2	1	no	—
70	C₁₆×C₁₃₁	—	2	2	no	—
71	C₁₆×C₁₃₁	C₂²⋊C₄	1	4	yes	—
72	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
73	Q₈×C₁₃₁	—	4	4	no	—
74	Q₈×C₁₃₁	—	4	4	no	—
75	Q₈×C₁₃₁	—	8	2	no	—
76	Q₈×C₁₃₁	—	8	2	no	—
77	C₂×C₄×C₁₃₁	—	2	2	no	—
78	C₂×C₄×C₁₃₁	—	2	4	no	—
79	C₂×C₄×C₁₃₁	—	4	4	no	—
80	C₂×C₄×C₁₃₁	—	8	2	no	—
81	C₂×C₄×C₁₃₁	—	8	2	no	—
82	C₂×C₄×C₁₃₁	—	8	2	no	—
83	C₂×C₄×C₁₃₁	—	8	4	no	—
84	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
86	C₂×C₄×C₁₃₁	#127	1	1	yes	characteristic
87	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃₁	#135	1	1	yes	characteristic
89	C₂×C₄×C₁₃₁	#135	1	1	yes	characteristic
90	C₈×C₁₃₁	—	2	2	no	—
91	C₈×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₈×C₁₃₁	C₄×D₄	1	1	yes	characteristic
93	C₈×C₁₃₁	#121	1	1	yes	characteristic
94	C₈×C₁₃₁	#124	1	1	yes	characteristic
95	C₂²×C₁₃₁	C₄×D₈	1	1	yes	characteristic
96	C₂²×C₁₃₁	#466	1	1	yes	characteristic
97	C₂²×C₁₃₁	#493	1	1	yes	characteristic
98	C₂²×C₁₃₁	#510	1	4	yes	—
99	C₄×C₁₃₁	—	4	2	no	—
100	C₄×C₁₃₁	—	8	4	no	—
101	C₄×C₁₃₁	—	16	2	no	—
102	C₄×C₁₃₁	#390	1	1	yes	characteristic
103	C₄×C₁₃₁	C₄×D₈	1	1	yes	characteristic
104	C₄×C₁₃₁	#466	1	1	yes	characteristic
105	C₄×C₁₃₁	#495	1	1	yes	characteristic
106	C₂×C₁₃₁	#1970	1	1	yes	characteristic
107	C₂×C₁₃₁	#2176	1	2	yes	—
108	C₂×C₁₃₁	#2219	1	2	yes	—
109	C₂×C₁₃₁	#2251	1	2	yes	—
110	#13240	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₁₃₁	#13240	1	1	yes	characteristic, direct product factor, Sylow
112	C₂²×Q₃₂	C₂×C₁₃₁	1	1	yes	characteristic
113	C₂×#367	C₂×C₁₃₁	1	1	yes	characteristic
114	C₂×#358	C₂×C₁₃₁	1	2	yes	—
115	C₂×#346	C₂×C₁₃₁	1	1	yes	characteristic
116	#1969	C₂×C₁₃₁	1	2	yes	—
117	C₂²×Q₁₆	C₂²×C₁₃₁	1	2	yes	—
118	C₂×Q₃₂	—	2	2	no	—
119	C₂×Q₃₂	—	2	2	no	—
120	C₂×Q₃₂	C₄×C₁₃₁	1	4	yes	complemented
121	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
122	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
123	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
124	C₂×#103	—	2	2	no	—
125	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
126	#391	—	2	2	no	—
127	#367	—	2	2	no	—
128	#358	—	2	4	no	—
129	#346	—	2	2	no	—
130	C₂²×Q₈	—	2	2	no	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×Q₁₆	C₂×C₄×C₁₃₁	1	4	yes	—
137	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
138	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
141	Q₃₂	—	2	8	no	—
142	C₂×C₁₆	—	2	1	no	—
143	C₂×C₁₆	—	2	1	no	—
144	C₂×C₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
145	C₂×C₁₆	D₄×C₁₃₁	1	2	yes	—
146	C₂×C₁₆	D₄×C₁₃₁	1	2	yes	—
147	C₂×C₁₆	D₄×C₁₃₁	1	2	yes	—
148	#107	—	2	2	no	—
149	#107	—	2	2	no	—
150	#103	—	4	4	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	2	1	no	—
153	#95	—	4	4	no	—
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	2	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	D₈×C₁₃₁	1	2	yes	—
163	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
164	Q₁₆	—	2	4	no	—
165	Q₁₆	—	2	4	no	—
166	Q₁₆	—	4	2	no	—
167	Q₁₆	—	4	2	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	D₈×C₁₃₁	1	2	yes	—
171	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
172	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
174	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
175	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
176	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₁₆	—	2	2	no	—
182	C₁₆	C₂²⋊C₄×C₁₃₁	1	4	yes	—
183	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
184	Q₈	—	4	4	no	—
185	Q₈	—	4	4	no	—
186	Q₈	—	8	2	no	—
187	Q₈	—	8	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	8	2	no	—
194	C₂×C₄	—	8	4	no	—
195	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
196	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
197	C₂×C₄	#127×C₁₃₁	1	1	yes	characteristic
198	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
199	C₂×C₄	#135×C₁₃₁	1	1	yes	characteristic
200	C₂×C₄	#135×C₁₃₁	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₈	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
203	C₈	C₄×D₄×C₁₃₁	1	1	yes	characteristic
204	C₈	#121×C₁₃₁	1	1	yes	characteristic
205	C₈	#124×C₁₃₁	1	1	yes	characteristic
206	C₂²	C₄×D₈×C₁₃₁	1	1	yes	characteristic
207	C₂²	#466×C₁₃₁	1	1	yes	characteristic
208	C₂²	#493×C₁₃₁	1	1	yes	characteristic
209	C₂²	#510×C₁₃₁	1	4	yes	—
210	C₄	—	4	2	no	—
211	C₄	—	8	4	no	—
212	C₄	—	16	2	no	—
213	C₄	#390×C₁₃₁	1	1	yes	characteristic
214	C₄	C₄×D₈×C₁₃₁	1	1	yes	characteristic
215	C₄	#466×C₁₃₁	1	1	yes	characteristic
216	C₄	#495×C₁₃₁	1	1	yes	characteristic
217	C₂	#1970×C₁₃₁	1	1	yes	characteristic
218	C₂	#2176×C₁₃₁	1	2	yes	—
219	C₂	#2219×C₁₃₁	1	2	yes	—
220	C₂	#2251×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
9. C₂×Q₃₂×C₁₃₁	C₄	semidirect
110. #13240	C₁₃₁	direct (N × H)
111. C₁₃₁	#13240	direct (N × H)
120. C₂×Q₃₂	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₃₂×C₁₃₁	2
C₂×#367×C₁₃₁	2
C₂×#358×C₁₃₁	2
C₂×#346×C₁₃₁	2
#1969×C₁₃₁	2
#13240	131

Subgroups — C131×#13240 #45699124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13240 #45699124