Subgroups — C₁₃₁×#20816 #45706700

Order 33536.

982 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2285×C₁₃₁	C₂	1	2	yes	maximal
3	#1844×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#1834×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1807×C₁₃₁	C₂	1	4	yes	maximal
6	C₄×#106×C₁₃₁	C₂	1	4	yes	maximal, complemented
7	C₂³×C₈×C₁₃₁	C₂²	1	2	yes	—
8	C₂×#117×C₁₃₁	—	2	2	no	—
9	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×#106×C₁₃₁	C₂²	1	8	yes	—
11	C₂×C₄×C₈×C₁₃₁	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	#384×C₁₃₁	C₂²	1	4	yes	—
16	#382×C₁₃₁	—	2	2	no	—
17	#380×C₁₃₁	—	2	2	no	—
18	#380×C₁₃₁	C₂²	1	4	yes	—
19	C₄×C₄⋊C₄×C₁₃₁	—	2	2	no	—
20	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	—
21	#340×C₁₃₁	C₂²	1	8	yes	—
22	C₂³×C₄×C₁₃₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃₁	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₁₃₁	—	2	8	no	—
26	C₂²×C₈×C₁₃₁	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
29	#117×C₁₃₁	—	4	8	no	—
30	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	8	yes	—
33	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
34	C₂×C₄²×C₁₃₁	—	2	4	no	—
35	C₂×C₄²×C₁₃₁	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃₁	Q₈	1	4	yes	—
39	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
41	#106×C₁₃₁	—	2	16	no	—
42	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₁₃₁	C₂×C₄	1	8	yes	—
44	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
45	#95×C₁₃₁	—	2	4	no	—
46	#95×C₁₃₁	—	2	4	no	—
47	C₂⁴×C₁₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₁	—	2	4	no	—
49	C₂²×C₄×C₁₃₁	—	2	4	no	—
50	C₂²×C₄×C₁₃₁	—	2	8	no	—
51	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₁₃₁	SD₁₆	1	8	yes	—
54	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
62	C₂×C₈×C₁₃₁	—	2	4	no	—
63	C₂×C₈×C₁₃₁	—	2	16	no	—
64	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
66	C₂×C₈×C₁₃₁	#41	1	8	yes	—
67	C₄⋊C₄×C₁₃₁	—	2	16	no	—
68	C₄⋊C₄×C₁₃₁	—	4	8	no	—
69	C₄⋊C₄×C₁₃₁	—	4	8	no	—
70	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
71	C₄²×C₁₃₁	—	4	8	no	—
72	C₄²×C₁₃₁	C₄⋊C₄	1	4	yes	—
73	C₄²×C₁₃₁	C₄⋊C₄	1	4	yes	—
74	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
75	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
76	C₄²×C₁₃₁	C₂×Q₈	1	2	yes	—
77	C₄²×C₁₃₁	C₂×Q₈	1	2	yes	—
78	C₂³×C₁₃₁	—	2	2	no	—
79	C₂³×C₁₃₁	—	2	2	no	—
80	C₂³×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
81	C₂³×C₁₃₁	C₂×SD₁₆	1	4	yes	—
82	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃₁	—	2	2	no	—
84	C₂×C₄×C₁₃₁	—	2	2	no	—
85	C₂×C₄×C₁₃₁	—	2	4	no	—
86	C₂×C₄×C₁₃₁	—	2	4	no	—
87	C₂×C₄×C₁₃₁	—	2	8	no	—
88	C₂×C₄×C₁₃₁	—	4	8	no	—
89	C₂×C₄×C₁₃₁	—	4	8	no	—
90	C₂×C₄×C₁₃₁	—	4	8	no	—
91	C₂×C₄×C₁₃₁	#106	1	16	yes	complemented
92	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	4	yes	—
98	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	4	yes	—
99	C₂×C₄×C₁₃₁	#135	1	4	yes	—
100	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃₁	C₂×#41	1	4	yes	—
103	C₈×C₁₃₁	—	2	4	no	—
104	C₈×C₁₃₁	#117	1	8	yes	complemented
105	C₂²×C₁₃₁	—	2	2	no	—
106	C₂²×C₁₃₁	—	2	8	no	—
107	C₂²×C₁₃₁	—	2	2	no	—
108	C₂²×C₁₃₁	C₂×#106	1	4	yes	—
109	C₂²×C₁₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₁₃₁	#532	1	2	yes	—
111	C₂²×C₁₃₁	C₂²×SD₁₆	1	2	yes	—
112	C₂²×C₁₃₁	C₂×#135	1	2	yes	—
113	C₄×C₁₃₁	—	2	2	no	—
114	C₄×C₁₃₁	—	2	2	no	—
115	C₄×C₁₃₁	—	8	8	no	—
116	C₄×C₁₃₁	C₂×#106	1	4	yes	—
117	C₄×C₁₃₁	C₂×#106	1	4	yes	—
118	C₄×C₁₃₁	#427	1	4	yes	—
119	C₄×C₁₃₁	C₂×#117	1	2	yes	—
120	C₄×C₁₃₁	#533	1	2	yes	—
121	C₂×C₁₃₁	—	2	4	no	—
122	C₂×C₁₃₁	#2285	1	1	yes	characteristic
123	C₂×C₁₃₁	C₂²×#106	1	1	yes	characteristic
124	C₂×C₁₃₁	C₂×#427	1	1	yes	characteristic
125	C₂×C₁₃₁	#3167	1	4	yes	—
126	#20816	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₁₃₁	#20816	1	1	yes	characteristic, direct product factor, Sylow
128	C₂²×C₄×C₈	C₂×C₁₃₁	1	1	yes	characteristic
129	#2285	C₂×C₁₃₁	1	2	yes	—
130	#1844	C₂×C₁₃₁	1	2	yes	complemented
131	#1834	C₂×C₁₃₁	1	2	yes	complemented
132	#1807	C₂×C₁₃₁	1	4	yes	—
133	C₄×#106	C₂×C₁₃₁	1	4	yes	complemented
134	C₂³×C₈	C₂²×C₁₃₁	1	2	yes	—
135	C₂×#117	—	2	2	no	—
136	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
137	C₂×#106	C₂²×C₁₃₁	1	8	yes	—
138	C₂×C₄×C₈	C₄×C₁₃₁	1	8	yes	complemented
139	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
142	#384	C₂²×C₁₃₁	1	4	yes	—
143	#382	—	2	2	no	—
144	#380	—	2	2	no	—
145	#380	C₂²×C₁₃₁	1	4	yes	—
146	C₄×C₄⋊C₄	—	2	2	no	—
147	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	—
148	#340	C₂²×C₁₃₁	1	8	yes	—
149	C₂³×C₄	D₄×C₁₃₁	1	1	yes	characteristic
150	C₂³×C₄	D₄×C₁₃₁	1	1	yes	characteristic
151	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
152	C₂²×C₈	—	2	8	no	—
153	C₂²×C₈	C₂×C₄×C₁₃₁	1	8	yes	—
154	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
155	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
156	#117	—	4	8	no	—
157	C₂×C₄⋊C₄	—	2	4	no	—
158	C₂×C₄⋊C₄	—	2	4	no	—
159	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	8	yes	—
160	C₂×C₂²⋊C₄	—	2	4	no	—
161	C₂×C₄²	—	2	4	no	—
162	C₂×C₄²	C₂×C₄×C₁₃₁	1	4	yes	—
163	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
164	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
165	C₂×C₄²	Q₈×C₁₃₁	1	4	yes	—
166	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
167	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
168	#106	—	2	16	no	—
169	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
170	C₄×C₈	C₂×C₄×C₁₃₁	1	8	yes	—
171	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
172	#95	—	2	4	no	—
173	#95	—	2	4	no	—
174	C₂⁴	C₂×D₄×C₁₃₁	1	1	yes	characteristic
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	4	yes	—
179	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	4	yes	—
180	C₂²×C₄	SD₁₆×C₁₃₁	1	8	yes	—
181	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
187	C₂²×C₄	C₂×Q₈×C₁₃₁	1	2	yes	—
188	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	16	no	—
191	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
192	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
193	C₂×C₈	#41×C₁₃₁	1	8	yes	—
194	C₄⋊C₄	—	2	16	no	—
195	C₄⋊C₄	—	4	8	no	—
196	C₄⋊C₄	—	4	8	no	—
197	C₂²⋊C₄	—	4	8	no	—
198	C₄²	—	4	8	no	—
199	C₄²	C₄⋊C₄×C₁₃₁	1	4	yes	—
200	C₄²	C₄⋊C₄×C₁₃₁	1	4	yes	—
201	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
202	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
203	C₄²	C₂×Q₈×C₁₃₁	1	2	yes	—
204	C₄²	C₂×Q₈×C₁₃₁	1	2	yes	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
208	C₂³	C₂×SD₁₆×C₁₃₁	1	4	yes	—
209	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	8	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	—	4	8	no	—
218	C₂×C₄	#106×C₁₃₁	1	16	yes	complemented
219	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
220	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
221	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
222	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
223	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
224	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	4	yes	—
225	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	4	yes	—
226	C₂×C₄	#135×C₁₃₁	1	4	yes	—
227	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
228	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
229	C₂×C₄	C₂×#41×C₁₃₁	1	4	yes	—
230	C₈	—	2	4	no	—
231	C₈	#117×C₁₃₁	1	8	yes	complemented
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	8	no	—
235	C₂²	C₂×#106×C₁₃₁	1	4	yes	—
236	C₂²	C₂²×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
237	C₂²	#532×C₁₃₁	1	2	yes	—
238	C₂²	C₂²×SD₁₆×C₁₃₁	1	2	yes	—
239	C₂²	C₂×#135×C₁₃₁	1	2	yes	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	8	8	no	—
243	C₄	C₂×#106×C₁₃₁	1	4	yes	—
244	C₄	C₂×#106×C₁₃₁	1	4	yes	—
245	C₄	#427×C₁₃₁	1	4	yes	—
246	C₄	C₂×#117×C₁₃₁	1	2	yes	—
247	C₄	#533×C₁₃₁	1	2	yes	—
248	C₂	—	2	4	no	—
249	C₂	#2285×C₁₃₁	1	1	yes	characteristic
250	C₂	C₂²×#106×C₁₃₁	1	1	yes	characteristic
251	C₂	C₂×#427×C₁₃₁	1	1	yes	characteristic
252	C₂	#3167×C₁₃₁	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
socle	C₂³×C₁₃₁	C₂²×D₄
prime radical	C₂⁴×C₁₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂²×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1844×C₁₃₁	C₂	semidirect
4. #1834×C₁₃₁	C₂	semidirect
6. C₄×#106×C₁₃₁	C₂	semidirect
11. C₂×C₄×C₈×C₁₃₁	C₄	semidirect
91. C₂×C₄×C₁₃₁	#106	semidirect
104. C₈×C₁₃₁	#117	semidirect
126. #20816	C₁₃₁	direct (N × H)
127. C₁₃₁	#20816	direct (N × H)
130. #1844	C₂×C₁₃₁	semidirect
131. #1834	C₂×C₁₃₁	semidirect
133. C₄×#106	C₂×C₁₃₁	semidirect
138. C₂×C₄×C₈	C₄×C₁₃₁	semidirect
218. C₂×C₄	#106×C₁₃₁	semidirect
231. C₈	#117×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₁₃₁	2
#2285×C₁₃₁	2
#1844×C₁₃₁	2
#1834×C₁₃₁	2
#1807×C₁₃₁	2
C₄×#106×C₁₃₁	2
#20816	131

Subgroups — C131×#20816 #45706700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#20816 #45706700