Subgroups — C₁₃₁×#23161 #45709045

Order 33536.

794 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2365×C₁₃₁	C₂	1	2	yes	maximal
3	#1988×C₁₃₁	C₂	1	2	yes	maximal
4	#1943×C₁₃₁	C₂	1	2	yes	maximal
5	#1921×C₁₃₁	C₂	1	4	yes	maximal
6	#1776×C₁₃₁	C₂	1	2	yes	maximal
7	C₄×#106×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	#1770×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₁₃₁	—	2	2	no	—
11	#500×C₁₃₁	C₄	1	8	yes	complemented
12	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
13	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
14	C₂×#106×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#106×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	#391×C₁₃₁	—	2	2	no	—
20	#389×C₁₃₁	C₂²	1	4	yes	—
21	#387×C₁₃₁	—	2	4	no	—
22	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	—
24	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
25	#340×C₁₃₁	C₂²	1	4	yes	—
26	#340×C₁₃₁	C₂²	1	4	yes	—
27	#340×C₁₃₁	C₂²	1	2	yes	—
28	#336×C₁₃₁	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₃₁	—	2	2	no	—
30	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
34	C₄⋊Q₈×C₁₃₁	C₂×C₄	1	8	yes	—
35	C₄×Q₈×C₁₃₁	—	4	8	no	—
36	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₃₁	D₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₃₁	—	2	2	no	—
43	C₂×C₄²×C₁₃₁	—	2	4	no	—
44	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
45	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
46	#107×C₁₃₁	C₂×C₄	1	8	yes	—
47	#107×C₁₃₁	C₂×C₄	1	8	yes	—
48	#106×C₁₃₁	—	2	2	no	—
49	#106×C₁₃₁	Q₈	1	4	yes	complemented
50	C₄⋊C₈×C₁₃₁	—	2	2	no	—
51	#97×C₁₃₁	—	2	4	no	—
52	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
53	#95×C₁₃₁	—	2	4	no	—
54	#95×C₁₃₁	—	2	4	no	—
55	#95×C₁₃₁	—	2	2	no	—
56	#95×C₁₃₁	D₄	1	2	yes	—
57	C₂×Q₈×C₁₃₁	—	2	8	no	—
58	C₂×Q₈×C₁₃₁	—	4	4	no	—
59	C₂×Q₈×C₁₃₁	—	4	4	no	—
60	C₂²×C₄×C₁₃₁	—	2	2	no	—
61	C₂²×C₄×C₁₃₁	—	2	4	no	—
62	C₂²×C₄×C₁₃₁	—	2	2	no	—
63	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₁₃₁	—	2	1	no	—
68	C₂×C₈×C₁₃₁	—	2	1	no	—
69	C₂×C₈×C₁₃₁	—	2	2	no	—
70	C₂×C₈×C₁₃₁	—	2	2	no	—
71	C₂×C₈×C₁₃₁	—	2	2	no	—
72	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₁₃₁	#41	1	2	yes	—
76	C₄⋊C₄×C₁₃₁	—	2	8	no	—
77	C₄⋊C₄×C₁₃₁	—	2	4	no	—
78	C₄⋊C₄×C₁₃₁	—	2	4	no	—
79	C₄⋊C₄×C₁₃₁	—	2	2	no	—
80	C₄⋊C₄×C₁₃₁	—	2	2	no	—
81	C₄⋊C₄×C₁₃₁	—	2	2	no	—
82	C₄⋊C₄×C₁₃₁	—	4	8	no	—
83	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
84	C₄⋊C₄×C₁₃₁	C₂×Q₈	1	4	yes	—
85	C₄²×C₁₃₁	—	2	2	no	—
86	C₄²×C₁₃₁	—	2	2	no	—
87	C₄²×C₁₃₁	—	4	8	no	—
88	C₄²×C₁₃₁	—	4	4	no	—
89	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
90	C₄²×C₁₃₁	#41	1	2	yes	—
91	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
92	Q₈×C₁₃₁	—	4	16	no	—
93	C₂×C₄×C₁₃₁	—	2	2	no	—
94	C₂×C₄×C₁₃₁	—	2	8	no	—
95	C₂×C₄×C₁₃₁	—	2	4	no	—
96	C₂×C₄×C₁₃₁	—	2	4	no	—
97	C₂×C₄×C₁₃₁	—	2	4	no	—
98	C₂×C₄×C₁₃₁	—	2	4	no	—
99	C₂×C₄×C₁₃₁	—	4	2	no	—
100	C₂×C₄×C₁₃₁	—	4	4	no	—
101	C₂×C₄×C₁₃₁	—	4	4	no	—
102	C₂×C₄×C₁₃₁	—	4	2	no	—
103	C₂×C₄×C₁₃₁	—	4	2	no	—
104	C₂×C₄×C₁₃₁	—	4	2	no	—
105	C₂×C₄×C₁₃₁	—	4	2	no	—
106	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	4	yes	—
109	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
114	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
115	C₈×C₁₃₁	—	2	2	no	—
116	C₈×C₁₃₁	—	4	2	no	—
117	C₈×C₁₃₁	C₄×Q₈	1	4	yes	complemented
118	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂²×C₁₃₁	#547	1	1	yes	characteristic
120	C₂²×C₁₃₁	D₄×Q₈	1	1	yes	characteristic
121	C₂²×C₁₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
122	C₂²×C₁₃₁	#578	1	2	yes	—
123	C₄×C₁₃₁	—	2	2	no	—
124	C₄×C₁₃₁	—	4	4	no	—
125	C₄×C₁₃₁	—	4	4	no	—
126	C₄×C₁₃₁	—	4	8	no	—
127	C₄×C₁₃₁	—	8	2	no	—
128	C₄×C₁₃₁	#442	1	4	yes	—
129	C₄×C₁₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
130	C₄×C₁₃₁	#519	1	1	yes	characteristic
131	C₄×C₁₃₁	#556	1	1	yes	characteristic
132	C₄×C₁₃₁	#557	1	1	yes	characteristic
133	C₂×C₁₃₁	#2365	1	1	yes	characteristic
134	C₂×C₁₃₁	C₂×#442	1	1	yes	characteristic
135	C₂×C₁₃₁	#2950	1	1	yes	characteristic
136	C₂×C₁₃₁	#3353	1	2	yes	—
137	C₂×C₁₃₁	#3386	1	2	yes	—
138	#23161	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₁₃₁	#23161	1	1	yes	characteristic, direct product factor, Sylow
140	C₂×#500	C₂×C₁₃₁	1	1	yes	characteristic
141	#2365	C₂×C₁₃₁	1	2	yes	—
142	#1988	C₂×C₁₃₁	1	2	yes	—
143	#1943	C₂×C₁₃₁	1	2	yes	—
144	#1921	C₂×C₁₃₁	1	4	yes	—
145	#1776	C₂×C₁₃₁	1	2	yes	—
146	C₄×#106	C₂×C₁₃₁	1	1	yes	characteristic
147	#1770	C₂×C₁₃₁	1	1	yes	characteristic
148	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
149	C₂×C₄×Q₈	—	2	2	no	—
150	#500	C₄×C₁₃₁	1	8	yes	complemented
151	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
152	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
153	C₂×#106	C₂²×C₁₃₁	1	1	yes	characteristic
154	C₂×#106	C₂²×C₁₃₁	1	1	yes	characteristic
155	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
156	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
158	#391	—	2	2	no	—
159	#389	C₂²×C₁₃₁	1	4	yes	—
160	#387	—	2	4	no	—
161	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	—
162	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
163	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
164	#340	C₂²×C₁₃₁	1	2	yes	—
165	#340	C₂²×C₁₃₁	1	4	yes	—
166	#340	C₂²×C₁₃₁	1	4	yes	—
167	#336	C₂²×C₁₃₁	1	2	yes	—
168	C₂²×Q₈	—	2	2	no	—
169	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
170	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
172	C₄⋊Q₈	—	2	4	no	—
173	C₄⋊Q₈	C₂×C₄×C₁₃₁	1	8	yes	—
174	C₄×Q₈	—	4	8	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	D₄×C₁₃₁	1	2	yes	—
178	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
179	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
180	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
181	C₂×C₄²	—	2	4	no	—
182	C₂×C₄²	—	2	2	no	—
183	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
184	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
185	#107	C₂×C₄×C₁₃₁	1	8	yes	—
186	#107	C₂×C₄×C₁₃₁	1	8	yes	—
187	#106	—	2	2	no	—
188	#106	Q₈×C₁₃₁	1	4	yes	complemented
189	C₄⋊C₈	—	2	2	no	—
190	#97	—	2	4	no	—
191	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
192	#95	—	2	2	no	—
193	#95	—	2	4	no	—
194	#95	—	2	4	no	—
195	#95	D₄×C₁₃₁	1	2	yes	—
196	C₂×Q₈	—	2	8	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂×Q₈	—	4	4	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
203	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
205	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
212	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
213	C₂×C₈	C₂×Q₈×C₁₃₁	1	2	yes	—
214	C₂×C₈	#41×C₁₃₁	1	2	yes	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	4	8	no	—
222	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
223	C₄⋊C₄	C₂×Q₈×C₁₃₁	1	4	yes	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	4	8	no	—
227	C₄²	—	4	4	no	—
228	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
229	C₄²	#41×C₁₃₁	1	2	yes	—
230	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
231	Q₈	—	4	16	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
246	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
247	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	4	yes	—
248	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
249	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
250	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
251	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
252	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
253	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
254	C₈	—	2	2	no	—
255	C₈	—	4	2	no	—
256	C₈	C₄×Q₈×C₁₃₁	1	4	yes	complemented
257	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
258	C₂²	#547×C₁₃₁	1	1	yes	characteristic
259	C₂²	D₄×Q₈×C₁₃₁	1	1	yes	characteristic
260	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
261	C₂²	#578×C₁₃₁	1	2	yes	—
262	C₄	—	2	2	no	—
263	C₄	—	4	4	no	—
264	C₄	—	4	4	no	—
265	C₄	—	4	8	no	—
266	C₄	—	8	2	no	—
267	C₄	#442×C₁₃₁	1	4	yes	—
268	C₄	C₂×C₄×Q₈×C₁₃₁	1	1	yes	characteristic
269	C₄	#519×C₁₃₁	1	1	yes	characteristic
270	C₄	#556×C₁₃₁	1	1	yes	characteristic
271	C₄	#557×C₁₃₁	1	1	yes	characteristic
272	C₂	#2365×C₁₃₁	1	1	yes	characteristic
273	C₂	C₂×#442×C₁₃₁	1	1	yes	characteristic
274	C₂	#2950×C₁₃₁	1	1	yes	characteristic
275	C₂	#3353×C₁₃₁	1	2	yes	—
276	C₂	#3386×C₁₃₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. #500×C₁₃₁	C₄	semidirect
49. #106×C₁₃₁	Q₈	semidirect
117. C₈×C₁₃₁	C₄×Q₈	semidirect
138. #23161	C₁₃₁	direct (N × H)
139. C₁₃₁	#23161	direct (N × H)
150. #500	C₄×C₁₃₁	semidirect
188. #106	Q₈×C₁₃₁	semidirect
256. C₈	C₄×Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₃₁	2
#2365×C₁₃₁	2
#1988×C₁₃₁	2
#1943×C₁₃₁	2
#1921×C₁₃₁	2
#1776×C₁₃₁	2
C₄×#106×C₁₃₁	2
#1770×C₁₃₁	2
#23161	131

Subgroups — C131×#23161 #45709045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#23161 #45709045