Subgroups — C₁₃₁×#20543 #45706427

Order 33536.

1090 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₈×C₄⋊C₄×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	C₄×C₄⋊C₈×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	C₈×C₂²⋊C₄×C₁₃₁	C₂	1	4	yes	maximal
7	C₂³×C₈×C₁₃₁	C₂²	1	2	yes	—
8	C₂×#117×C₁₃₁	C₄	1	2	yes	—
9	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
10	#431×C₁₃₁	C₄	1	8	yes	complemented
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	2	yes	—
14	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃₁	C₂²	1	8	yes	—
17	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₁₃₁	C₄	1	2	yes	—
19	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
20	C₄×C₂²⋊C₄×C₁₃₁	C₄	1	2	yes	—
21	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
22	#376×C₁₃₁	C₄	1	2	yes	—
23	C₄³×C₁₃₁	C₂²	1	2	yes	—
24	#336×C₁₃₁	C₂²	1	8	yes	—
25	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₁₃₁	—	4	4	no	—
29	C₂²×C₈×C₁₃₁	C₂×C₄	1	8	yes	—
30	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
31	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
32	#117×C₁₃₁	C₈	1	16	yes	complemented
33	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₁₃₁	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃₁	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₁₃₁	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₁₃₁	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
42	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₁₃₁	C₂×C₄	1	8	yes	—
45	C₄⋊C₈×C₁₃₁	C₂×C₄	1	8	yes	—
46	C₄⋊C₈×C₁₃₁	C₂×C₄	1	8	yes	—
47	C₂²⋊C₈×C₁₃₁	C₂×C₄	1	8	yes	—
48	C₄×C₈×C₁₃₁	—	4	8	no	—
49	C₄×C₈×C₁₃₁	C₂×C₄	1	8	yes	—
50	#95×C₁₃₁	C₂×C₄	1	4	yes	—
51	#95×C₁₃₁	C₂×C₄	1	4	yes	—
52	C₂⁴×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₁	—	2	2	no	—
54	C₂²×C₄×C₁₃₁	—	2	2	no	—
55	C₂²×C₄×C₁₃₁	—	4	2	no	—
56	C₂²×C₄×C₁₃₁	C₄²	1	4	yes	—
57	C₂²×C₄×C₁₃₁	C₄²	1	4	yes	—
58	C₂²×C₄×C₁₃₁	C₂×C₈	1	8	yes	—
59	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
65	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₁₃₁	—	2	4	no	—
69	C₂×C₈×C₁₃₁	—	2	8	no	—
70	C₂×C₈×C₁₃₁	—	2	4	no	—
71	C₂×C₈×C₁₃₁	—	4	8	no	—
72	C₂×C₈×C₁₃₁	C₂²×C₄	1	8	yes	—
73	C₂×C₈×C₁₃₁	C₂²×C₄	1	8	yes	—
74	C₄⋊C₄×C₁₃₁	C₂×C₈	1	16	yes	complemented
75	C₂²⋊C₄×C₁₃₁	C₂×C₈	1	16	yes	—
76	C₄²×C₁₃₁	—	2	4	no	—
77	C₄²×C₁₃₁	—	2	4	no	—
78	C₄²×C₁₃₁	—	4	4	no	—
79	C₄²×C₁₃₁	C₄²	1	4	yes	—
80	C₄²×C₁₃₁	C₄²	1	4	yes	—
81	C₄²×C₁₃₁	C₂×C₈	1	16	yes	complemented
82	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
83	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
84	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
85	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
86	C₂³×C₁₃₁	—	2	2	no	—
87	C₂³×C₁₃₁	—	2	2	no	—
88	C₂³×C₁₃₁	C₂×C₄²	1	2	yes	—
89	C₂³×C₁₃₁	C₂²×C₈	1	4	yes	—
90	C₂³×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃₁	—	2	8	no	—
92	C₂×C₄×C₁₃₁	—	2	2	no	—
93	C₂×C₄×C₁₃₁	—	2	2	no	—
94	C₂×C₄×C₁₃₁	—	2	8	no	—
95	C₂×C₄×C₁₃₁	—	2	2	no	—
96	C₂×C₄×C₁₃₁	—	2	2	no	—
97	C₂×C₄×C₁₃₁	—	2	2	no	—
98	C₂×C₄×C₁₃₁	—	2	4	no	—
99	C₂×C₄×C₁₃₁	—	4	4	no	—
100	C₂×C₄×C₁₃₁	C₄×C₈	1	16	yes	complemented
101	C₂×C₄×C₁₃₁	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₁₃₁	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₁₃₁	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₁₃₁	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₁₃₁	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₁₃₁	C₂²×C₈	1	16	yes	—
107	C₂×C₄×C₁₃₁	C₂²×C₈	1	4	yes	—
108	C₂×C₄×C₁₃₁	C₂²×C₈	1	4	yes	—
109	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
112	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
113	C₈×C₁₃₁	—	2	8	no	—
114	C₈×C₁₃₁	—	2	8	no	—
115	C₂²×C₁₃₁	—	2	2	no	—
116	C₂²×C₁₃₁	—	2	8	no	—
117	C₂²×C₁₃₁	—	2	2	no	—
118	C₂²×C₁₃₁	C₂×C₄×C₈	1	4	yes	—
119	C₂²×C₁₃₁	C₂²×C₄²	1	1	yes	characteristic
120	C₂²×C₁₃₁	#520	1	2	yes	—
121	C₂²×C₁₃₁	C₂³×C₈	1	2	yes	—
122	C₂²×C₁₃₁	#568	1	2	yes	—
123	C₄×C₁₃₁	—	2	2	no	—
124	C₄×C₁₃₁	—	2	2	no	—
125	C₄×C₁₃₁	—	2	16	no	—
126	C₄×C₁₃₁	—	2	2	no	—
127	C₄×C₁₃₁	C₂×C₄×C₈	1	4	yes	—
128	C₄×C₁₃₁	C₂×C₄×C₈	1	4	yes	—
129	C₄×C₁₃₁	#518	1	1	yes	characteristic
130	C₄×C₁₃₁	#519	1	1	yes	characteristic
131	C₄×C₁₃₁	#520	1	2	yes	—
132	C₂×C₁₃₁	—	2	4	no	—
133	C₂×C₁₃₁	#2274	1	1	yes	characteristic
134	C₂×C₁₃₁	C₂²×C₄×C₈	1	1	yes	characteristic
135	C₂×C₁₃₁	#2873	1	1	yes	characteristic
136	C₂×C₁₃₁	#2988	1	4	yes	—
137	#20543	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₁₃₁	#20543	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×#431	C₂×C₁₃₁	1	1	yes	characteristic
140	C₂²×C₄⋊C₈	C₂×C₁₃₁	1	1	yes	characteristic
141	C₄×#117	C₂×C₁₃₁	1	1	yes	characteristic
142	C₈×C₄⋊C₄	C₂×C₁₃₁	1	4	yes	complemented
143	C₄×C₄⋊C₈	C₂×C₁₃₁	1	4	yes	complemented
144	C₈×C₂²⋊C₄	C₂×C₁₃₁	1	4	yes	—
145	C₂³×C₈	C₂²×C₁₃₁	1	2	yes	—
146	C₂×#117	C₄×C₁₃₁	1	2	yes	—
147	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
148	#431	C₄×C₁₃₁	1	8	yes	complemented
149	C₂×C₄⋊C₈	C₄×C₁₃₁	1	8	yes	complemented
150	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
151	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
152	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
153	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	2	yes	—
154	C₂×C₄×C₈	C₂²×C₁₃₁	1	8	yes	—
155	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
156	C₄×C₄⋊C₄	C₄×C₁₃₁	1	2	yes	—
157	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
158	C₄×C₂²⋊C₄	C₄×C₁₃₁	1	2	yes	—
159	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	2	yes	—
160	#376	C₄×C₁₃₁	1	2	yes	—
161	C₄³	C₂²×C₁₃₁	1	2	yes	—
162	#336	C₂²×C₁₃₁	1	8	yes	—
163	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
164	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
165	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
166	C₂²×C₈	—	4	4	no	—
167	C₂²×C₈	C₂×C₄×C₁₃₁	1	8	yes	—
168	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
169	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
170	#117	C₈×C₁₃₁	1	16	yes	complemented
171	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
172	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
173	C₂×C₄²	C₂×C₄×C₁₃₁	1	4	yes	—
174	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
176	C₂×C₄²	C₂×C₄×C₁₃₁	1	4	yes	—
177	C₂×C₄²	C₂×C₄×C₁₃₁	1	4	yes	—
178	C₂×C₄²	C₂×C₄×C₁₃₁	1	4	yes	—
179	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
180	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
181	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
182	C₄⋊C₈	C₂×C₄×C₁₃₁	1	8	yes	—
183	C₄⋊C₈	C₂×C₄×C₁₃₁	1	8	yes	—
184	C₄⋊C₈	C₂×C₄×C₁₃₁	1	8	yes	—
185	C₂²⋊C₈	C₂×C₄×C₁₃₁	1	8	yes	—
186	C₄×C₈	—	4	8	no	—
187	C₄×C₈	C₂×C₄×C₁₃₁	1	8	yes	—
188	#95	C₂×C₄×C₁₃₁	1	4	yes	—
189	#95	C₂×C₄×C₁₃₁	1	4	yes	—
190	C₂⁴	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₄²×C₁₃₁	1	4	yes	—
195	C₂²×C₄	C₄²×C₁₃₁	1	4	yes	—
196	C₂²×C₄	C₂×C₈×C₁₃₁	1	8	yes	—
197	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
199	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
201	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
202	C₂²×C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
203	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
204	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
205	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	8	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	8	no	—
210	C₂×C₈	C₂²×C₄×C₁₃₁	1	8	yes	—
211	C₂×C₈	C₂²×C₄×C₁₃₁	1	8	yes	—
212	C₄⋊C₄	C₂×C₈×C₁₃₁	1	16	yes	complemented
213	C₂²⋊C₄	C₂×C₈×C₁₃₁	1	16	yes	—
214	C₄²	—	2	4	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	4	4	no	—
217	C₄²	C₄²×C₁₃₁	1	4	yes	—
218	C₄²	C₄²×C₁₃₁	1	4	yes	—
219	C₄²	C₂×C₈×C₁₃₁	1	16	yes	complemented
220	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
221	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
222	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
223	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₂×C₄²×C₁₃₁	1	2	yes	—
227	C₂³	C₂²×C₈×C₁₃₁	1	4	yes	—
228	C₂³	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	C₄×C₈×C₁₃₁	1	16	yes	complemented
239	C₂×C₄	C₂×C₄²×C₁₃₁	1	2	yes	—
240	C₂×C₄	C₂×C₄²×C₁₃₁	1	2	yes	—
241	C₂×C₄	C₂×C₄²×C₁₃₁	1	2	yes	—
242	C₂×C₄	C₂×C₄²×C₁₃₁	1	2	yes	—
243	C₂×C₄	C₂×C₄²×C₁₃₁	1	2	yes	—
244	C₂×C₄	C₂²×C₈×C₁₃₁	1	16	yes	—
245	C₂×C₄	C₂²×C₈×C₁₃₁	1	4	yes	—
246	C₂×C₄	C₂²×C₈×C₁₃₁	1	4	yes	—
247	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
249	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	2	yes	—
250	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	2	yes	—
251	C₈	—	2	8	no	—
252	C₈	—	2	8	no	—
253	C₂²	—	2	8	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	C₂×C₄×C₈×C₁₃₁	1	4	yes	—
257	C₂²	C₂²×C₄²×C₁₃₁	1	1	yes	characteristic
258	C₂²	#520×C₁₃₁	1	2	yes	—
259	C₂²	C₂³×C₈×C₁₃₁	1	2	yes	—
260	C₂²	#568×C₁₃₁	1	2	yes	—
261	C₄	—	2	16	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	C₂×C₄×C₈×C₁₃₁	1	4	yes	—
266	C₄	C₂×C₄×C₈×C₁₃₁	1	4	yes	—
267	C₄	#518×C₁₃₁	1	1	yes	characteristic
268	C₄	#519×C₁₃₁	1	1	yes	characteristic
269	C₄	#520×C₁₃₁	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	#2274×C₁₃₁	1	1	yes	characteristic
272	C₂	C₂²×C₄×C₈×C₁₃₁	1	1	yes	characteristic
273	C₂	#2873×C₁₃₁	1	1	yes	characteristic
274	C₂	#2988×C₁₃₁	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₈×C₄⋊C₄×C₁₃₁	C₂	semidirect
5. C₄×C₄⋊C₈×C₁₃₁	C₂	semidirect
10. #431×C₁₃₁	C₄	semidirect
11. C₂×C₄⋊C₈×C₁₃₁	C₄	semidirect
32. #117×C₁₃₁	C₈	semidirect
74. C₄⋊C₄×C₁₃₁	C₂×C₈	semidirect
81. C₄²×C₁₃₁	C₂×C₈	semidirect
100. C₂×C₄×C₁₃₁	C₄×C₈	semidirect
137. #20543	C₁₃₁	direct (N × H)
138. C₁₃₁	#20543	direct (N × H)
142. C₈×C₄⋊C₄	C₂×C₁₃₁	semidirect
143. C₄×C₄⋊C₈	C₂×C₁₃₁	semidirect
148. #431	C₄×C₁₃₁	semidirect
149. C₂×C₄⋊C₈	C₄×C₁₃₁	semidirect
170. #117	C₈×C₁₃₁	semidirect
212. C₄⋊C₄	C₂×C₈×C₁₃₁	semidirect
219. C₄²	C₂×C₈×C₁₃₁	semidirect
238. C₂×C₄	C₄×C₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431×C₁₃₁	2
C₂²×C₄⋊C₈×C₁₃₁	2
C₄×#117×C₁₃₁	2
C₈×C₄⋊C₄×C₁₃₁	2
C₄×C₄⋊C₈×C₁₃₁	2
C₈×C₂²⋊C₄×C₁₃₁	2
#20543	131

Subgroups — C131×#20543 #45706427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#20543 #45706427