Subgroups — C₁₃₁×#32808 #45718692

Order 33536.

732 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2862×C₁₃₁	C₂	1	2	yes	maximal
2	#2056×C₁₃₁	C₂	1	4	yes	maximal
3	#2024×C₁₃₁	C₂	1	4	yes	maximal
4	#2016×C₁₃₁	C₂	1	4	yes	maximal
5	#1831×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₁₃₁	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
10	C₂×#103×C₁₃₁	C₂²	1	8	yes	—
11	#391×C₁₃₁	—	2	2	no	—
12	#391×C₁₃₁	—	2	2	no	—
13	#391×C₁₃₁	C₂²	1	4	yes	—
14	#391×C₁₃₁	C₂²	1	4	yes	—
15	#382×C₁₃₁	—	2	2	no	—
16	#382×C₁₃₁	—	2	2	no	—
17	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
18	#340×C₁₃₁	C₂²	1	4	yes	—
19	#336×C₁₃₁	C₂²	1	4	yes	—
20	C₂²×Q₈×C₁₃₁	—	2	2	no	—
21	C₂²×Q₈×C₁₃₁	C₂³	1	4	yes	—
22	C₂×Q₁₆×C₁₃₁	—	4	8	no	—
23	C₂×Q₁₆×C₁₃₁	—	4	4	no	—
24	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
25	C₄⋊Q₈×C₁₃₁	—	4	4	no	—
26	C₄⋊Q₈×C₁₃₁	—	4	4	no	—
27	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
31	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
32	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
36	C₄⋊C₈×C₁₃₁	—	4	2	no	—
37	#103×C₁₃₁	—	4	8	no	—
38	#95×C₁₃₁	—	2	4	no	—
39	#95×C₁₃₁	—	2	4	no	—
40	#95×C₁₃₁	—	2	4	no	—
41	#95×C₁₃₁	—	2	4	no	—
42	C₂×Q₈×C₁₃₁	—	2	8	no	—
43	C₂×Q₈×C₁₃₁	—	2	8	no	—
44	C₂×Q₈×C₁₃₁	—	2	4	no	—
45	C₂×Q₈×C₁₃₁	—	2	4	no	—
46	C₂×Q₈×C₁₃₁	—	4	4	no	—
47	C₂×Q₈×C₁₃₁	—	4	4	no	—
48	C₂×Q₈×C₁₃₁	—	4	4	no	—
49	C₂²×C₄×C₁₃₁	—	2	4	no	—
50	C₂²×C₄×C₁₃₁	—	2	4	no	—
51	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
56	Q₁₆×C₁₃₁	—	4	16	no	—
57	C₂×C₈×C₁₃₁	—	2	8	no	—
58	C₂×C₈×C₁₃₁	—	2	4	no	—
59	C₄⋊C₄×C₁₃₁	—	2	8	no	—
60	C₄⋊C₄×C₁₃₁	—	4	4	no	—
61	C₄⋊C₄×C₁₃₁	—	4	4	no	—
62	C₄⋊C₄×C₁₃₁	—	4	4	no	—
63	C₄²×C₁₃₁	—	2	4	no	—
64	C₄²×C₁₃₁	—	2	1	no	—
65	C₄²×C₁₃₁	—	2	4	no	—
66	C₄²×C₁₃₁	—	2	1	no	—
67	C₄²×C₁₃₁	—	4	1	no	—
68	C₄²×C₁₃₁	Q₁₆	1	4	yes	complemented
69	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
70	Q₈×C₁₃₁	—	4	8	no	—
71	Q₈×C₁₃₁	—	4	8	no	—
72	Q₈×C₁₃₁	—	8	8	no	—
73	C₂×C₄×C₁₃₁	—	2	2	no	—
74	C₂×C₄×C₁₃₁	—	2	2	no	—
75	C₂×C₄×C₁₃₁	—	2	1	no	—
76	C₂×C₄×C₁₃₁	—	2	2	no	—
77	C₂×C₄×C₁₃₁	—	2	1	no	—
78	C₂×C₄×C₁₃₁	—	2	2	no	—
79	C₂×C₄×C₁₃₁	—	2	1	no	—
80	C₂×C₄×C₁₃₁	—	2	2	no	—
81	C₂×C₄×C₁₃₁	—	4	4	no	—
82	C₂×C₄×C₁₃₁	—	4	4	no	—
83	C₂×C₄×C₁₃₁	—	4	4	no	—
84	C₂×C₄×C₁₃₁	—	4	4	no	—
85	C₂×C₄×C₁₃₁	—	4	4	no	—
86	C₂×C₄×C₁₃₁	—	4	4	no	—
87	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	4	yes	—
88	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	2	yes	—
89	C₂×C₄×C₁₃₁	#137	1	4	yes	—
90	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
91	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	4	yes	—
92	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
93	C₈×C₁₃₁	—	4	4	no	—
94	C₂²×C₁₃₁	#535	1	2	yes	—
95	C₂²×C₁₃₁	#537	1	1	yes	characteristic
96	C₂²×C₁₃₁	C₂²×Q₁₆	1	1	yes	characteristic
97	C₂²×C₁₃₁	C₂×#137	1	2	yes	—
98	C₂²×C₁₃₁	#575	1	1	yes	characteristic
99	C₄×C₁₃₁	—	2	4	no	—
100	C₄×C₁₃₁	—	2	4	no	—
101	C₄×C₁₃₁	—	2	2	no	—
102	C₄×C₁₃₁	—	4	1	no	—
103	C₄×C₁₃₁	—	8	4	no	—
104	C₄×C₁₃₁	—	8	4	no	—
105	C₄×C₁₃₁	H(3,𝔽₄)	1	2	yes	—
106	C₄×C₁₃₁	#562	1	2	yes	—
107	C₂×C₁₃₁	#2819	1	1	yes	characteristic
108	C₂×C₁₃₁	#3209	1	2	yes	—
109	C₂×C₁₃₁	#3215	1	2	yes	—
110	C₂×C₁₃₁	#3249	1	1	yes	characteristic
111	C₂×C₁₃₁	#3255	1	1	yes	characteristic
112	#32808	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₁₃₁	#32808	1	1	yes	characteristic, direct product factor, Sylow
114	#2862	C₂×C₁₃₁	1	2	yes	—
115	#2056	C₂×C₁₃₁	1	4	yes	—
116	#2024	C₂×C₁₃₁	1	4	yes	—
117	#2016	C₂×C₁₃₁	1	4	yes	—
118	#1831	C₂×C₁₃₁	1	1	yes	characteristic
119	C₂²×Q₁₆	C₂²×C₁₃₁	1	4	yes	—
120	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
121	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
122	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
123	C₂×#103	C₂²×C₁₃₁	1	8	yes	—
124	#391	—	2	2	no	—
125	#391	—	2	2	no	—
126	#391	C₂²×C₁₃₁	1	4	yes	—
127	#391	C₂²×C₁₃₁	1	4	yes	—
128	#382	—	2	2	no	—
129	#382	—	2	2	no	—
130	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
131	#340	C₂²×C₁₃₁	1	4	yes	—
132	#336	C₂²×C₁₃₁	1	4	yes	—
133	C₂²×Q₈	—	2	2	no	—
134	C₂²×Q₈	C₂³×C₁₃₁	1	4	yes	—
135	C₂×Q₁₆	—	4	8	no	—
136	C₂×Q₁₆	—	4	4	no	—
137	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
138	C₄⋊Q₈	—	4	4	no	—
139	C₄⋊Q₈	—	4	4	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
144	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
145	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
146	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
147	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
148	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
149	C₄⋊C₈	—	4	2	no	—
150	#103	—	4	8	no	—
151	#95	—	2	4	no	—
152	#95	—	2	4	no	—
153	#95	—	2	4	no	—
154	#95	—	2	4	no	—
155	C₂×Q₈	—	2	8	no	—
156	C₂×Q₈	—	2	8	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	—	4	4	no	—
160	C₂×Q₈	—	4	4	no	—
161	C₂×Q₈	—	4	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
167	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
168	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
169	Q₁₆	—	4	16	no	—
170	C₂×C₈	—	2	8	no	—
171	C₂×C₈	—	2	4	no	—
172	C₄⋊C₄	—	2	8	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄²	—	2	4	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	4	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	4	1	no	—
181	C₄²	Q₁₆×C₁₃₁	1	4	yes	complemented
182	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
183	Q₈	—	4	8	no	—
184	Q₈	—	4	8	no	—
185	Q₈	—	8	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	2	yes	—
201	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	4	yes	—
202	C₂×C₄	#137×C₁₃₁	1	4	yes	—
203	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
204	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	4	yes	—
205	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
206	C₈	—	4	4	no	—
207	C₂²	#535×C₁₃₁	1	2	yes	—
208	C₂²	#537×C₁₃₁	1	1	yes	characteristic
209	C₂²	C₂²×Q₁₆×C₁₃₁	1	1	yes	characteristic
210	C₂²	C₂×#137×C₁₃₁	1	2	yes	—
211	C₂²	#575×C₁₃₁	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	4	no	—
214	C₄	—	2	4	no	—
215	C₄	—	4	1	no	—
216	C₄	—	8	4	no	—
217	C₄	—	8	4	no	—
218	C₄	H(3,𝔽₄)×C₁₃₁	1	2	yes	—
219	C₄	#562×C₁₃₁	1	2	yes	—
220	C₂	#2819×C₁₃₁	1	1	yes	characteristic
221	C₂	#3209×C₁₃₁	1	2	yes	—
222	C₂	#3215×C₁₃₁	1	2	yes	—
223	C₂	#3249×C₁₃₁	1	1	yes	characteristic
224	C₂	#3255×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
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
68. C₄²×C₁₃₁	Q₁₆	semidirect
112. #32808	C₁₃₁	direct (N × H)
113. C₁₃₁	#32808	direct (N × H)
181. C₄²	Q₁₆×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2862×C₁₃₁	2
#2056×C₁₃₁	2
#2024×C₁₃₁	2
#2016×C₁₃₁	2
#1831×C₁₃₁	2
#32808	131

Subgroups — C131×#32808 #45718692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#32808 #45718692