Subgroups — C₁₃₁×#13271 #45699155

Order 33536.

332 conjugacy classes of subgroups.

The subgroup lattice has 234 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	C₂×#364×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1772×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1389×C₁₃₁	C₂	1	2	yes	maximal
5	#1384×C₁₃₁	C₂	1	2	yes	maximal
6	#533×C₁₃₁	—	2	1	no	—
7	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
8	#500×C₁₃₁	—	2	1	no	—
9	#499×C₁₃₁	—	2	1	no	—
10	#499×C₁₃₁	C₄	1	2	yes	complemented
11	#498×C₁₃₁	C₄	1	2	yes	complemented
12	C₂×#108×C₁₃₁	C₂²	1	1	yes	characteristic
13	#427×C₁₃₁	—	2	1	no	—
14	#427×C₁₃₁	C₂²	1	2	yes	—
15	#427×C₁₃₁	C₂²	1	1	yes	characteristic
16	#405×C₁₃₁	—	2	1	no	—
17	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
18	#364×C₁₃₁	—	2	1	no	—
19	#364×C₁₃₁	—	2	1	no	—
20	#339×C₁₃₁	—	2	1	no	—
21	C₂×M₁₆×C₁₃₁	—	2	1	no	—
22	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
24	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
25	#125×C₁₃₁	—	2	2	no	—
26	C₂²⋊Q₈×C₁₃₁	—	4	2	no	—
27	C₄×Q₈×C₁₃₁	—	4	2	no	—
28	#117×C₁₃₁	—	2	1	no	—
29	#117×C₁₃₁	—	2	2	no	—
30	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
31	M₃₂×C₁₃₁	—	2	2	no	—
32	M₃₂×C₁₃₁	—	2	2	no	—
33	C₂×C₁₆×C₁₃₁	—	2	2	no	—
34	#108×C₁₃₁	—	2	1	no	—
35	#108×C₁₃₁	Q₈	1	2	yes	complemented
36	#107×C₁₃₁	—	2	2	no	—
37	#107×C₁₃₁	—	2	2	no	—
38	#107×C₁₃₁	—	2	1	no	—
39	#106×C₁₃₁	—	2	2	no	—
40	#106×C₁₃₁	C₂×C₄	1	4	yes	—
41	#106×C₁₃₁	Q₈	1	2	yes	complemented
42	#97×C₁₃₁	—	4	1	no	—
43	C₄×C₈×C₁₃₁	—	4	1	no	—
44	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
48	C₂×Q₈×C₁₃₁	—	4	2	no	—
49	C₂²×C₄×C₁₃₁	—	2	1	no	—
50	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₁₃₁	—	2	2	no	—
52	M₁₆×C₁₃₁	—	4	1	no	—
53	C₂×C₈×C₁₃₁	—	4	1	no	—
54	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
57	C₂×C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
58	C₂×C₈×C₁₃₁	Q₁₆	1	1	yes	characteristic
59	C₂×C₈×C₁₃₁	Q₁₆	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₁₃₁	C₂×Q₈	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	—	2	4	no	—
67	C₄⋊C₄×C₁₃₁	—	2	2	no	—
68	C₄⋊C₄×C₁₃₁	—	4	4	no	—
69	C₄⋊C₄×C₁₃₁	—	4	2	no	—
70	C₂²⋊C₄×C₁₃₁	—	4	1	no	—
71	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
72	C₄²×C₁₃₁	—	2	1	no	—
73	C₄²×C₁₃₁	—	4	1	no	—
74	C₄²×C₁₃₁	—	4	2	no	—
75	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₁₆×C₁₃₁	—	4	2	no	—
78	C₂³×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
79	Q₈×C₁₃₁	—	8	4	no	—
80	C₂×C₄×C₁₃₁	—	2	1	no	—
81	C₂×C₄×C₁₃₁	—	2	1	no	—
82	C₂×C₄×C₁₃₁	—	4	2	no	—
83	C₂×C₄×C₁₃₁	—	4	4	no	—
84	C₂×C₄×C₁₃₁	—	4	2	no	—
85	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	1	yes	characteristic
90	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₈×C₁₃₁	—	4	2	no	—
92	C₈×C₁₃₁	#103	1	1	yes	characteristic
93	C₈×C₁₃₁	#103	1	1	yes	characteristic
94	C₈×C₁₃₁	#103	1	1	yes	characteristic
95	C₈×C₁₃₁	#103	1	1	yes	characteristic
96	C₈×C₁₃₁	C₄×Q₈	1	1	yes	characteristic
97	C₈×C₁₃₁	C₄×Q₈	1	1	yes	characteristic
98	C₈×C₁₃₁	#124	1	1	yes	characteristic
99	C₈×C₁₃₁	C₄⋊Q₈	1	1	yes	characteristic
100	C₂²×C₁₃₁	—	4	1	no	—
101	C₂²×C₁₃₁	#418	1	1	yes	characteristic
102	C₂²×C₁₃₁	#476	1	1	yes	characteristic
103	C₂²×C₁₃₁	#477	1	1	yes	characteristic
104	C₄×C₁₃₁	—	2	2	no	—
105	C₄×C₁₃₁	—	4	1	no	—
106	C₄×C₁₃₁	—	8	2	no	—
107	C₄×C₁₃₁	—	8	4	no	—
108	C₄×C₁₃₁	#391	1	1	yes	characteristic
109	C₄×C₁₃₁	C₂×#103	1	1	yes	characteristic
110	C₄×C₁₃₁	#476	1	1	yes	characteristic
111	C₄×C₁₃₁	#477	1	1	yes	characteristic
112	C₂×C₁₃₁	—	2	1	no	—
113	C₂×C₁₃₁	—	2	1	no	—
114	C₂×C₁₃₁	—	2	1	no	—
115	C₂×C₁₃₁	#1987	1	1	yes	characteristic
116	#13271	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₁₃₁	#13271	1	1	yes	characteristic, direct product factor, Sylow
118	#3160	C₂×C₁₃₁	1	1	yes	characteristic
119	C₂×#364	C₂×C₁₃₁	1	1	yes	characteristic
120	#1772	C₂×C₁₃₁	1	1	yes	characteristic
121	#1389	C₂×C₁₃₁	1	2	yes	—
122	#1384	C₂×C₁₃₁	1	2	yes	—
123	#533	—	2	1	no	—
124	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
125	#500	—	2	1	no	—
126	#499	—	2	1	no	—
127	#499	C₄×C₁₃₁	1	2	yes	complemented
128	#498	C₄×C₁₃₁	1	2	yes	complemented
129	C₂×#108	C₂²×C₁₃₁	1	1	yes	characteristic
130	#427	—	2	1	no	—
131	#427	C₂²×C₁₃₁	1	2	yes	—
132	#427	C₂²×C₁₃₁	1	1	yes	characteristic
133	#405	—	2	1	no	—
134	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
135	#364	—	2	1	no	—
136	#364	—	2	1	no	—
137	#339	—	2	1	no	—
138	C₂×M₁₆	—	2	1	no	—
139	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
140	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	2	no	—
142	#125	—	2	2	no	—
143	C₂²⋊Q₈	—	4	2	no	—
144	C₄×Q₈	—	4	2	no	—
145	#117	—	2	2	no	—
146	#117	—	2	1	no	—
147	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
148	M₃₂	—	2	2	no	—
149	M₃₂	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	#108	—	2	1	no	—
152	#108	Q₈×C₁₃₁	1	2	yes	complemented
153	#107	—	2	2	no	—
154	#107	—	2	2	no	—
155	#107	—	2	1	no	—
156	#106	—	2	2	no	—
157	#106	C₂×C₄×C₁₃₁	1	4	yes	—
158	#106	Q₈×C₁₃₁	1	2	yes	complemented
159	#97	—	4	1	no	—
160	C₄×C₈	—	4	1	no	—
161	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
162	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
163	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
164	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
165	C₂×Q₈	—	4	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
168	M₁₆	—	2	2	no	—
169	M₁₆	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
173	C₂×C₈	SD₁₆×C₁₃₁	1	1	yes	characteristic
174	C₂×C₈	SD₁₆×C₁₃₁	1	1	yes	characteristic
175	C₂×C₈	Q₁₆×C₁₃₁	1	1	yes	characteristic
176	C₂×C₈	Q₁₆×C₁₃₁	1	1	yes	characteristic
177	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
178	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
179	C₂×C₈	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
180	C₂×C₈	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
181	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
182	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₂²⋊C₄	—	4	1	no	—
188	C₂²⋊C₄	—	4	2	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	4	1	no	—
191	C₄²	—	4	2	no	—
192	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
193	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
194	C₁₆	—	4	2	no	—
195	C₂³	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
196	Q₈	—	8	4	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
203	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
204	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
205	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
206	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	1	yes	characteristic
207	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
208	C₈	—	4	2	no	—
209	C₈	#103×C₁₃₁	1	1	yes	characteristic
210	C₈	#103×C₁₃₁	1	1	yes	characteristic
211	C₈	#103×C₁₃₁	1	1	yes	characteristic
212	C₈	#103×C₁₃₁	1	1	yes	characteristic
213	C₈	C₄×Q₈×C₁₃₁	1	1	yes	characteristic
214	C₈	C₄×Q₈×C₁₃₁	1	1	yes	characteristic
215	C₈	#124×C₁₃₁	1	1	yes	characteristic
216	C₈	C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
217	C₂²	—	4	1	no	—
218	C₂²	#418×C₁₃₁	1	1	yes	characteristic
219	C₂²	#476×C₁₃₁	1	1	yes	characteristic
220	C₂²	#477×C₁₃₁	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	8	4	no	—
224	C₄	—	8	2	no	—
225	C₄	#391×C₁₃₁	1	1	yes	characteristic
226	C₄	C₂×#103×C₁₃₁	1	1	yes	characteristic
227	C₄	#476×C₁₃₁	1	1	yes	characteristic
228	C₄	#477×C₁₃₁	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	#1987×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₁	#476
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
socle	C₂×C₁₃₁	#1987
prime radical	C₂³×C₁₃₁	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. #499×C₁₃₁	C₄	semidirect
11. #498×C₁₃₁	C₄	semidirect
35. #108×C₁₃₁	Q₈	semidirect
41. #106×C₁₃₁	Q₈	semidirect
116. #13271	C₁₃₁	direct (N × H)
117. C₁₃₁	#13271	direct (N × H)
127. #499	C₄×C₁₃₁	semidirect
128. #498	C₄×C₁₃₁	semidirect
152. #108	Q₈×C₁₃₁	semidirect
158. #106	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
C₂×#364×C₁₃₁	2
#1772×C₁₃₁	2
#1389×C₁₃₁	2
#1384×C₁₃₁	2
#13271	131

Subgroups — C131×#13271 #45699155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13271 #45699155