Subgroups — C₁₃₁×#13243 #45699127

Order 33536.

410 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2158×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2145×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#2108×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1976×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₁₃₁	C₂²	1	2	yes	—
7	#508×C₁₃₁	—	2	1	no	—
8	#508×C₁₃₁	—	2	1	no	—
9	#508×C₁₃₁	—	2	1	no	—
10	#508×C₁₃₁	—	2	1	no	—
11	C₂×Q₃₂×C₁₃₁	C₄	1	1	yes	characteristic, complemented
12	C₂×SD₃₂×C₁₃₁	C₄	1	2	yes	complemented
13	C₂×D₁₆×C₁₃₁	C₄	1	1	yes	characteristic, complemented
14	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
16	#427×C₁₃₁	C₂²	1	2	yes	—
17	#421×C₁₃₁	—	2	2	no	—
18	#410×C₁₃₁	—	2	2	no	—
19	#405×C₁₃₁	C₂²	1	1	yes	characteristic
20	#367×C₁₃₁	—	2	1	no	—
21	#366×C₁₃₁	—	2	1	no	—
22	#360×C₁₃₁	—	2	4	no	—
23	#346×C₁₃₁	—	2	1	no	—
24	C₄×C₁₆×C₁₃₁	—	2	1	no	—
25	C₂×#41×C₁₃₁	—	2	2	no	—
26	#135×C₁₃₁	—	2	2	no	—
27	#135×C₁₃₁	—	2	2	no	—
28	#135×C₁₃₁	—	2	2	no	—
29	#135×C₁₃₁	—	2	2	no	—
30	C₂×Q₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
31	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
32	C₂×D₈×C₁₃₁	C₂×C₄	1	2	yes	—
33	C₂×M₁₆×C₁₃₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
35	#117×C₁₃₁	—	2	2	no	—
36	#117×C₁₃₁	D₄	1	1	yes	characteristic
37	Q₃₂×C₁₃₁	—	2	1	no	—
38	Q₃₂×C₁₃₁	—	2	1	no	—
39	SD₃₂×C₁₃₁	—	2	2	no	—
40	SD₃₂×C₁₃₁	—	2	2	no	—
41	D₁₆×C₁₃₁	—	2	1	no	—
42	D₁₆×C₁₃₁	—	2	1	no	—
43	M₃₂×C₁₃₁	—	2	1	no	—
44	M₃₂×C₁₃₁	D₄	1	2	yes	—
45	C₂×C₁₆×C₁₃₁	—	2	1	no	—
46	C₂×C₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₁₆×C₁₃₁	D₄	1	1	yes	characteristic
49	C₂×C₁₆×C₁₃₁	D₄	1	1	yes	characteristic
50	C₂×C₁₆×C₁₃₁	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₁₃₁	D₄	1	1	yes	characteristic
52	#107×C₁₃₁	—	2	2	no	—
53	#106×C₁₃₁	—	2	2	no	—
54	#104×C₁₃₁	—	4	4	no	—
55	#99×C₁₃₁	—	4	4	no	—
56	#97×C₁₃₁	—	2	1	no	—
57	C₄×C₈×C₁₃₁	—	2	1	no	—
58	#41×C₁₃₁	—	4	2	no	—
59	#41×C₁₃₁	—	4	2	no	—
60	#41×C₁₃₁	—	4	2	no	—
61	#41×C₁₃₁	—	4	2	no	—
62	C₂×Q₈×C₁₃₁	—	2	2	no	—
63	C₂×D₄×C₁₃₁	—	2	2	no	—
64	C₂×D₄×C₁₃₁	—	4	2	no	—
65	C₂²×C₄×C₁₃₁	—	4	2	no	—
66	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₁₃₁	—	2	2	no	—
68	Q₁₆×C₁₃₁	—	2	2	no	—
69	SD₁₆×C₁₃₁	—	4	2	no	—
70	SD₁₆×C₁₃₁	—	4	2	no	—
71	D₈×C₁₃₁	—	2	2	no	—
72	D₈×C₁₃₁	—	2	2	no	—
73	M₁₆×C₁₃₁	—	2	1	no	—
74	M₁₆×C₁₃₁	D₈	1	2	yes	—
75	C₂×C₈×C₁₃₁	—	2	1	no	—
76	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
81	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
82	C₄⋊C₄×C₁₃₁	—	2	1	no	—
83	C₄⋊C₄×C₁₃₁	—	4	2	no	—
84	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
85	C₂²⋊C₄×C₁₃₁	D₈	1	2	yes	—
86	C₄²×C₁₃₁	—	2	1	no	—
87	C₄²×C₁₃₁	—	4	2	no	—
88	C₁₆×C₁₃₁	—	2	2	no	—
89	C₁₆×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
90	C₁₆×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
91	C₁₆×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
92	C₁₆×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
93	C₂³×C₁₃₁	—	4	2	no	—
94	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
95	Q₈×C₁₃₁	—	4	2	no	—
96	Q₈×C₁₃₁	—	4	2	no	—
97	D₄×C₁₃₁	—	4	2	no	—
98	D₄×C₁₃₁	—	4	2	no	—
99	D₄×C₁₃₁	—	8	2	no	—
100	D₄×C₁₃₁	—	8	2	no	—
101	C₂×C₄×C₁₃₁	—	2	2	no	—
102	C₂×C₄×C₁₃₁	—	4	2	no	—
103	C₂×C₄×C₁₃₁	—	8	2	no	—
104	C₂×C₄×C₁₃₁	—	8	2	no	—
105	C₂×C₄×C₁₃₁	—	8	2	no	—
106	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
108	C₂×C₄×C₁₃₁	#127	1	1	yes	characteristic
109	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃₁	#135	1	1	yes	characteristic
111	C₂×C₄×C₁₃₁	#135	1	1	yes	characteristic
112	C₈×C₁₃₁	—	2	2	no	—
113	C₈×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₈×C₁₃₁	C₄×D₄	1	1	yes	characteristic
115	C₈×C₁₃₁	#121	1	1	yes	characteristic
116	C₈×C₁₃₁	#124	1	1	yes	characteristic
117	C₂²×C₁₃₁	—	4	1	no	—
118	C₂²×C₁₃₁	—	8	2	no	—
119	C₂²×C₁₃₁	—	16	2	no	—
120	C₂²×C₁₃₁	C₄×D₈	1	1	yes	characteristic
121	C₂²×C₁₃₁	#466	1	1	yes	characteristic
122	C₂²×C₁₃₁	#493	1	1	yes	characteristic
123	C₄×C₁₃₁	—	4	2	no	—
124	C₄×C₁₃₁	—	8	2	no	—
125	C₄×C₁₃₁	—	16	2	no	—
126	C₄×C₁₃₁	#390	1	1	yes	characteristic
127	C₄×C₁₃₁	C₄×D₈	1	1	yes	characteristic
128	C₄×C₁₃₁	#466	1	1	yes	characteristic
129	C₄×C₁₃₁	#495	1	1	yes	characteristic
130	C₂×C₁₃₁	—	2	1	no	—
131	C₂×C₁₃₁	—	2	1	no	—
132	C₂×C₁₃₁	—	2	1	no	—
133	C₂×C₁₃₁	—	16	2	no	—
134	C₂×C₁₃₁	#1970	1	1	yes	characteristic
135	#13243	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₁₃₁	#13243	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×#508	C₂×C₁₃₁	1	1	yes	characteristic
138	#2158	C₂×C₁₃₁	1	1	yes	characteristic, complemented
139	#2145	C₂×C₁₃₁	1	2	yes	complemented
140	#2108	C₂×C₁₃₁	1	1	yes	characteristic, complemented
141	#1976	C₂×C₁₃₁	1	2	yes	complemented
142	C₂×#135	C₂²×C₁₃₁	1	2	yes	—
143	#508	—	2	1	no	—
144	#508	—	2	1	no	—
145	#508	—	2	1	no	—
146	#508	—	2	1	no	—
147	C₂×Q₃₂	C₄×C₁₃₁	1	1	yes	characteristic, complemented
148	C₂×SD₃₂	C₄×C₁₃₁	1	2	yes	complemented
149	C₂×D₁₆	C₄×C₁₃₁	1	1	yes	characteristic, complemented
150	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
151	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
152	#427	C₂²×C₁₃₁	1	2	yes	—
153	#421	—	2	2	no	—
154	#410	—	2	2	no	—
155	#405	C₂²×C₁₃₁	1	1	yes	characteristic
156	#367	—	2	1	no	—
157	#366	—	2	1	no	—
158	#360	—	2	4	no	—
159	#346	—	2	1	no	—
160	C₄×C₁₆	—	2	1	no	—
161	C₂×#41	—	2	2	no	—
162	#135	—	2	2	no	—
163	#135	—	2	2	no	—
164	#135	—	2	2	no	—
165	#135	—	2	2	no	—
166	C₂×Q₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
167	C₂×SD₁₆	—	2	2	no	—
168	C₂×D₈	C₂×C₄×C₁₃₁	1	2	yes	—
169	C₂×M₁₆	D₄×C₁₃₁	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
171	#117	—	2	2	no	—
172	#117	D₄×C₁₃₁	1	1	yes	characteristic
173	Q₃₂	—	2	1	no	—
174	Q₃₂	—	2	1	no	—
175	SD₃₂	—	2	2	no	—
176	SD₃₂	—	2	2	no	—
177	D₁₆	—	2	1	no	—
178	D₁₆	—	2	1	no	—
179	M₃₂	—	2	1	no	—
180	M₃₂	D₄×C₁₃₁	1	2	yes	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	C₂×C₄×C₁₃₁	1	1	yes	characteristic
183	C₂×C₁₆	C₂×C₄×C₁₃₁	1	1	yes	characteristic
184	C₂×C₁₆	D₄×C₁₃₁	1	1	yes	characteristic
185	C₂×C₁₆	D₄×C₁₃₁	1	1	yes	characteristic
186	C₂×C₁₆	D₄×C₁₃₁	1	1	yes	characteristic
187	C₂×C₁₆	D₄×C₁₃₁	1	1	yes	characteristic
188	#107	—	2	2	no	—
189	#106	—	2	2	no	—
190	#104	—	4	4	no	—
191	#99	—	4	4	no	—
192	#97	—	2	1	no	—
193	C₄×C₈	—	2	1	no	—
194	#41	—	4	2	no	—
195	#41	—	4	2	no	—
196	#41	—	4	2	no	—
197	#41	—	4	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
203	Q₁₆	—	2	2	no	—
204	Q₁₆	—	2	2	no	—
205	SD₁₆	—	4	2	no	—
206	SD₁₆	—	4	2	no	—
207	D₈	—	2	2	no	—
208	D₈	—	2	2	no	—
209	M₁₆	—	2	1	no	—
210	M₁₆	D₈×C₁₃₁	1	2	yes	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
213	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
215	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
216	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
217	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
218	C₄⋊C₄	—	2	1	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	D₈×C₁₃₁	1	2	yes	—
222	C₄²	—	2	1	no	—
223	C₄²	—	4	2	no	—
224	C₁₆	—	2	2	no	—
225	C₁₆	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
226	C₁₆	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
227	C₁₆	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
228	C₁₆	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
229	C₂³	—	4	2	no	—
230	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
231	Q₈	—	4	2	no	—
232	Q₈	—	4	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	8	2	no	—
236	D₄	—	8	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
243	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
244	C₂×C₄	#127×C₁₃₁	1	1	yes	characteristic
245	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
246	C₂×C₄	#135×C₁₃₁	1	1	yes	characteristic
247	C₂×C₄	#135×C₁₃₁	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
250	C₈	C₄×D₄×C₁₃₁	1	1	yes	characteristic
251	C₈	#121×C₁₃₁	1	1	yes	characteristic
252	C₈	#124×C₁₃₁	1	1	yes	characteristic
253	C₂²	—	4	1	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	16	2	no	—
256	C₂²	C₄×D₈×C₁₃₁	1	1	yes	characteristic
257	C₂²	#466×C₁₃₁	1	1	yes	characteristic
258	C₂²	#493×C₁₃₁	1	1	yes	characteristic
259	C₄	—	4	2	no	—
260	C₄	—	8	2	no	—
261	C₄	—	16	2	no	—
262	C₄	#390×C₁₃₁	1	1	yes	characteristic
263	C₄	C₄×D₈×C₁₃₁	1	1	yes	characteristic
264	C₄	#466×C₁₃₁	1	1	yes	characteristic
265	C₄	#495×C₁₃₁	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	16	2	no	—
270	C₂	#1970×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₁	#466
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
socle	C₂×C₁₃₁	#1970
prime radical	C₂×#508×C₁₃₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ #405×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2158×C₁₃₁	C₂	semidirect
3. #2145×C₁₃₁	C₂	semidirect
4. #2108×C₁₃₁	C₂	semidirect
5. #1976×C₁₃₁	C₂	semidirect
11. C₂×Q₃₂×C₁₃₁	C₄	semidirect
12. C₂×SD₃₂×C₁₃₁	C₄	semidirect
13. C₂×D₁₆×C₁₃₁	C₄	semidirect
135. #13243	C₁₃₁	direct (N × H)
136. C₁₃₁	#13243	direct (N × H)
138. #2158	C₂×C₁₃₁	semidirect
139. #2145	C₂×C₁₃₁	semidirect
140. #2108	C₂×C₁₃₁	semidirect
141. #1976	C₂×C₁₃₁	semidirect
147. C₂×Q₃₂	C₄×C₁₃₁	semidirect
148. C₂×SD₃₂	C₄×C₁₃₁	semidirect
149. C₂×D₁₆	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508×C₁₃₁	2
#2158×C₁₃₁	2
#2145×C₁₃₁	2
#2108×C₁₃₁	2
#1976×C₁₃₁	2
#13243	131

Subgroups — C131×#13243 #45699127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13243 #45699127