Subgroups — C₁₃₁×#22281 #45708165

Order 33536.

818 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3150×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2931×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#1963×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#1876×C₁₃₁	C₂	1	8	yes	maximal, complemented
5	#1838×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#1746×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#135×C₁₃₁	C₂²	1	4	yes	—
8	C₂×#131×C₁₃₁	C₂²	1	4	yes	—
9	#532×C₁₃₁	C₂²	1	2	yes	—
10	#497×C₁₃₁	C₄	1	4	yes	complemented
11	#496×C₁₃₁	C₄	1	4	yes	complemented
12	#436×C₁₃₁	—	2	2	no	—
13	#436×C₁₃₁	—	2	2	no	—
14	C₈×D₄×C₁₃₁	—	2	2	no	—
15	C₈×D₄×C₁₃₁	—	2	2	no	—
16	#423×C₁₃₁	C₂²	1	2	yes	—
17	C₂×#104×C₁₃₁	C₂²	1	8	yes	—
18	#408×C₁₃₁	C₂²	1	4	yes	—
19	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
21	#376×C₁₃₁	C₂²	1	1	yes	characteristic
22	#341×C₁₃₁	C₂²	1	4	yes	—
23	#336×C₁₃₁	C₂²	1	4	yes	—
24	C₂×#41×C₁₃₁	C₂³	1	4	yes	—
25	#135×C₁₃₁	—	4	8	no	—
26	C₂×Q₁₆×C₁₃₁	C₂×C₄	1	4	yes	—
27	C₂×SD₁₆×C₁₃₁	C₂×C₄	1	8	yes	—
28	C₂×D₈×C₁₃₁	C₂×C₄	1	4	yes	—
29	#131×C₁₃₁	—	4	8	no	—
30	C₂×M₁₆×C₁₃₁	—	2	4	no	—
31	C₂×M₁₆×C₁₃₁	C₂³	1	4	yes	—
32	C₂²×C₈×C₁₃₁	—	2	4	no	—
33	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
35	C₄⋊Q₈×C₁₃₁	C₂×C₄	1	2	yes	—
36	#127×C₁₃₁	C₂×C₄	1	2	yes	—
37	#124×C₁₃₁	C₂×C₄	1	4	yes	—
38	#121×C₁₃₁	—	2	2	no	—
39	#121×C₁₃₁	—	2	2	no	—
40	C₄×D₄×C₁₃₁	—	2	2	no	—
41	C₄×D₄×C₁₃₁	—	2	2	no	—
42	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₁₃₁	—	2	2	no	—
45	C₄⋊C₈×C₁₃₁	—	2	2	no	—
46	#104×C₁₃₁	—	4	8	no	—
47	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
48	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
49	#97×C₁₃₁	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
52	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
53	#95×C₁₃₁	—	2	2	no	—
54	#41×C₁₃₁	—	2	8	no	—
55	#41×C₁₃₁	—	2	8	no	—
56	C₂×Q₈×C₁₃₁	C₂²×C₄	1	4	yes	—
57	C₂×D₄×C₁₃₁	—	2	4	no	—
58	C₂×D₄×C₁₃₁	—	2	2	no	—
59	C₂×D₄×C₁₃₁	—	2	2	no	—
60	C₂×D₄×C₁₃₁	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₁₃₁	—	2	4	no	—
62	C₂²×C₄×C₁₃₁	—	2	2	no	—
63	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
64	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
65	Q₁₆×C₁₃₁	—	4	4	no	—
66	SD₁₆×C₁₃₁	—	4	8	no	—
67	D₈×C₁₃₁	—	4	4	no	—
68	M₁₆×C₁₃₁	—	2	8	no	—
69	M₁₆×C₁₃₁	—	4	8	no	—
70	C₂×C₈×C₁₃₁	—	2	4	no	—
71	C₂×C₈×C₁₃₁	—	2	4	no	—
72	C₂×C₈×C₁₃₁	—	2	4	no	—
73	C₂×C₈×C₁₃₁	—	4	8	no	—
74	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
77	C₄⋊C₄×C₁₃₁	—	2	2	no	—
78	C₄⋊C₄×C₁₃₁	—	2	2	no	—
79	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
80	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
81	C₄²×C₁₃₁	—	2	4	no	—
82	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
83	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂³×C₁₃₁	—	2	4	no	—
86	C₂³×C₁₃₁	C₂×#41	1	1	yes	characteristic
87	Q₈×C₁₃₁	—	2	8	no	—
88	D₄×C₁₃₁	—	2	8	no	—
89	D₄×C₁₃₁	—	4	4	no	—
90	D₄×C₁₃₁	—	4	4	no	—
91	D₄×C₁₃₁	—	4	8	no	—
92	C₂×C₄×C₁₃₁	—	2	4	no	—
93	C₂×C₄×C₁₃₁	—	4	2	no	—
94	C₂×C₄×C₁₃₁	—	4	4	no	—
95	C₂×C₄×C₁₃₁	—	4	4	no	—
96	C₂×C₄×C₁₃₁	—	4	4	no	—
97	C₂×C₄×C₁₃₁	—	4	2	no	—
98	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	4	yes	—
103	C₈×C₁₃₁	—	2	4	no	—
104	C₈×C₁₃₁	—	2	2	no	—
105	C₈×C₁₃₁	—	2	2	no	—
106	C₈×C₁₃₁	—	4	4	no	—
107	C₂²×C₁₃₁	—	2	2	no	—
108	C₂²×C₁₃₁	—	4	4	no	—
109	C₂²×C₁₃₁	—	4	4	no	—
110	C₂²×C₁₃₁	—	4	4	no	—
111	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂²×C₁₃₁	#540	1	2	yes	—
113	C₄×C₁₃₁	—	2	2	no	—
114	C₄×C₁₃₁	—	2	2	no	—
115	C₄×C₁₃₁	—	4	4	no	—
116	C₄×C₁₃₁	—	8	2	no	—
117	C₄×C₁₃₁	#518	1	2	yes	—
118	C₄×C₁₃₁	#535	1	2	yes	—
119	C₂×C₁₃₁	—	2	2	no	—
120	C₂×C₁₃₁	—	8	4	no	—
121	C₂×C₁₃₁	#2328	1	1	yes	characteristic
122	C₂×C₁₃₁	#2955	1	2	yes	—
123	#22281	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₁₃₁	#22281	1	1	yes	characteristic, direct product factor, Sylow
125	#3150	C₂×C₁₃₁	1	1	yes	characteristic
126	#2931	C₂×C₁₃₁	1	2	yes	complemented
127	#1963	C₂×C₁₃₁	1	2	yes	complemented
128	#1876	C₂×C₁₃₁	1	8	yes	complemented
129	#1838	C₂×C₁₃₁	1	1	yes	characteristic, complemented
130	#1746	C₂×C₁₃₁	1	1	yes	characteristic, complemented
131	C₂×#135	C₂²×C₁₃₁	1	4	yes	—
132	C₂×#131	C₂²×C₁₃₁	1	4	yes	—
133	#532	C₂²×C₁₃₁	1	2	yes	—
134	#497	C₄×C₁₃₁	1	4	yes	complemented
135	#496	C₄×C₁₃₁	1	4	yes	complemented
136	#436	—	2	2	no	—
137	#436	—	2	2	no	—
138	C₈×D₄	—	2	2	no	—
139	C₈×D₄	—	2	2	no	—
140	#423	C₂²×C₁₃₁	1	2	yes	—
141	C₂×#104	C₂²×C₁₃₁	1	8	yes	—
142	#408	C₂²×C₁₃₁	1	4	yes	—
143	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
145	#376	C₂²×C₁₃₁	1	1	yes	characteristic
146	#341	C₂²×C₁₃₁	1	4	yes	—
147	#336	C₂²×C₁₃₁	1	4	yes	—
148	C₂×#41	C₂³×C₁₃₁	1	4	yes	—
149	#135	—	4	8	no	—
150	C₂×Q₁₆	C₂×C₄×C₁₃₁	1	4	yes	—
151	C₂×SD₁₆	C₂×C₄×C₁₃₁	1	8	yes	—
152	C₂×D₈	C₂×C₄×C₁₃₁	1	4	yes	—
153	#131	—	4	8	no	—
154	C₂×M₁₆	—	2	4	no	—
155	C₂×M₁₆	C₂³×C₁₃₁	1	4	yes	—
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
158	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
159	C₄⋊Q₈	C₂×C₄×C₁₃₁	1	2	yes	—
160	#127	C₂×C₄×C₁₃₁	1	2	yes	—
161	#124	C₂×C₄×C₁₃₁	1	4	yes	—
162	#121	—	2	2	no	—
163	#121	—	2	2	no	—
164	C₄×D₄	—	2	2	no	—
165	C₄×D₄	—	2	2	no	—
166	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
167	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	#104	—	4	8	no	—
171	C₂²⋊C₈	—	2	4	no	—
172	C₂²⋊C₈	—	2	4	no	—
173	#97	C₂×C₄×C₁₃₁	1	4	yes	—
174	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
175	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
176	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
177	#95	—	2	2	no	—
178	#41	—	2	8	no	—
179	#41	—	2	8	no	—
180	C₂×Q₈	C₂²×C₄×C₁₃₁	1	4	yes	—
181	C₂×D₄	—	2	4	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	C₂²×C₄×C₁₃₁	1	4	yes	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
188	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
189	Q₁₆	—	4	4	no	—
190	SD₁₆	—	4	8	no	—
191	D₈	—	4	4	no	—
192	M₁₆	—	2	8	no	—
193	M₁₆	—	4	8	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	4	8	no	—
198	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
199	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
200	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₄²	—	2	4	no	—
206	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
207	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
208	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
209	C₂³	—	2	4	no	—
210	C₂³	C₂×#41×C₁₃₁	1	1	yes	characteristic
211	Q₈	—	2	8	no	—
212	D₄	—	2	8	no	—
213	D₄	—	4	4	no	—
214	D₄	—	4	4	no	—
215	D₄	—	4	8	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
223	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
224	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
226	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	4	yes	—
227	C₈	—	2	4	no	—
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	4	4	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	4	no	—
233	C₂²	—	4	4	no	—
234	C₂²	—	4	4	no	—
235	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
236	C₂²	#540×C₁₃₁	1	2	yes	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	4	4	no	—
240	C₄	—	8	2	no	—
241	C₄	#518×C₁₃₁	1	2	yes	—
242	C₄	#535×C₁₃₁	1	2	yes	—
243	C₂	—	2	2	no	—
244	C₂	—	8	4	no	—
245	C₂	#2328×C₁₃₁	1	1	yes	characteristic
246	C₂	#2955×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
socle	C₂²×C₁₃₁	C₂×C₄×D₄
prime radical	#3150×C₁₃₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2931×C₁₃₁	C₂	semidirect
3. #1963×C₁₃₁	C₂	semidirect
4. #1876×C₁₃₁	C₂	semidirect
5. #1838×C₁₃₁	C₂	semidirect
6. #1746×C₁₃₁	C₂	semidirect
10. #497×C₁₃₁	C₄	semidirect
11. #496×C₁₃₁	C₄	semidirect
123. #22281	C₁₃₁	direct (N × H)
124. C₁₃₁	#22281	direct (N × H)
126. #2931	C₂×C₁₃₁	semidirect
127. #1963	C₂×C₁₃₁	semidirect
128. #1876	C₂×C₁₃₁	semidirect
129. #1838	C₂×C₁₃₁	semidirect
130. #1746	C₂×C₁₃₁	semidirect
134. #497	C₄×C₁₃₁	semidirect
135. #496	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3150×C₁₃₁	2
#2931×C₁₃₁	2
#1963×C₁₃₁	2
#1876×C₁₃₁	2
#1838×C₁₃₁	2
#1746×C₁₃₁	2
#22281	131

Subgroups — C131×#22281 #45708165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#22281 #45708165