Subgroups — C₁₃₁×#33253 #45719137

Order 33536.

922 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#495×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#488×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#2861×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1973×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#1972×C₁₃₁	C₂	1	2	yes	maximal
7	#1971×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	C₄×#97×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂²×SD₁₆×C₁₃₁	C₂²	1	2	yes	—
11	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×#125×C₁₃₁	C₂²	1	2	yes	—
13	C₂×#124×C₁₃₁	C₂²	1	2	yes	—
14	#499×C₁₃₁	—	2	4	no	—
15	#495×C₁₃₁	—	2	4	no	—
16	#488×C₁₃₁	—	2	8	no	—
17	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
18	C₂×#106×C₁₃₁	C₂²	1	2	yes	—
19	C₂×#103×C₁₃₁	C₂²	1	4	yes	—
20	C₂×#102×C₁₃₁	C₂²	1	4	yes	—
21	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
22	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
25	#391×C₁₃₁	C₂²	1	4	yes	—
26	#390×C₁₃₁	C₂²	1	4	yes	—
27	#388×C₁₃₁	—	2	8	no	—
28	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
29	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
30	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
31	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
32	C₂×Q₁₆×C₁₃₁	—	2	4	no	—
33	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
34	C₂×SD₁₆×C₁₃₁	—	2	8	no	—
35	C₂×D₈×C₁₃₁	—	2	2	no	—
36	C₂×D₈×C₁₃₁	—	2	4	no	—
37	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
40	#125×C₁₃₁	—	2	4	no	—
41	#125×C₁₃₁	—	2	4	no	—
42	#124×C₁₃₁	—	2	4	no	—
43	#124×C₁₃₁	—	2	4	no	—
44	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
45	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
46	C₂×C₂²⋊C₄×C₁₃₁	—	2	8	no	—
47	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
48	C₂×C₄²×C₁₃₁	D₄	1	4	yes	—
49	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
51	#107×C₁₃₁	—	2	4	no	—
52	#106×C₁₃₁	—	2	4	no	—
53	#103×C₁₃₁	—	2	8	no	—
54	#102×C₁₃₁	—	2	8	no	—
55	#97×C₁₃₁	—	2	4	no	—
56	#97×C₁₃₁	—	2	4	no	—
57	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
58	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
59	#95×C₁₃₁	—	2	8	no	—
60	C₂⁴×C₁₃₁	—	2	2	no	—
61	C₂×Q₈×C₁₃₁	—	2	2	no	—
62	C₂×Q₈×C₁₃₁	—	2	2	no	—
63	C₂×Q₈×C₁₃₁	—	2	4	no	—
64	C₂×Q₈×C₁₃₁	—	2	4	no	—
65	C₂×D₄×C₁₃₁	—	2	2	no	—
66	C₂×D₄×C₁₃₁	—	2	2	no	—
67	C₂×D₄×C₁₃₁	—	2	4	no	—
68	C₂×D₄×C₁₃₁	—	2	4	no	—
69	C₂²×C₄×C₁₃₁	—	2	2	no	—
70	C₂²×C₄×C₁₃₁	—	2	4	no	—
71	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
74	Q₁₆×C₁₃₁	—	4	4	no	—
75	SD₁₆×C₁₃₁	—	4	8	no	—
76	D₈×C₁₃₁	—	4	4	no	—
77	C₂×C₈×C₁₃₁	—	2	2	no	—
78	C₂×C₈×C₁₃₁	—	2	2	no	—
79	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₃₁	#41	1	4	yes	—
83	C₂×C₈×C₁₃₁	#41	1	4	yes	—
84	C₄⋊C₄×C₁₃₁	—	2	4	no	—
85	C₄⋊C₄×C₁₃₁	—	2	4	no	—
86	C₄⋊C₄×C₁₃₁	—	4	8	no	—
87	C₂²⋊C₄×C₁₃₁	—	4	16	no	—
88	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
89	C₄²×C₁₃₁	—	2	2	no	—
90	C₄²×C₁₃₁	—	2	2	no	—
91	C₄²×C₁₃₁	—	4	4	no	—
92	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
93	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
94	C₂³×C₁₃₁	—	4	2	no	—
95	C₂³×C₁₃₁	—	4	4	no	—
96	C₂³×C₁₃₁	—	4	8	no	—
97	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₁₃₁	—	4	2	no	—
99	Q₈×C₁₃₁	—	4	2	no	—
100	Q₈×C₁₃₁	—	4	4	no	—
101	D₄×C₁₃₁	—	4	2	no	—
102	D₄×C₁₃₁	—	4	2	no	—
103	D₄×C₁₃₁	—	4	4	no	—
104	C₂×C₄×C₁₃₁	—	2	4	no	—
105	C₂×C₄×C₁₃₁	—	2	4	no	—
106	C₂×C₄×C₁₃₁	—	2	8	no	—
107	C₂×C₄×C₁₃₁	—	4	2	no	—
108	C₂×C₄×C₁₃₁	—	4	4	no	—
109	C₂×C₄×C₁₃₁	—	4	4	no	—
110	C₂×C₄×C₁₃₁	—	4	8	no	—
111	C₂×C₄×C₁₃₁	#127	1	4	yes	—
112	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
115	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
116	C₈×C₁₃₁	—	2	4	no	—
117	C₈×C₁₃₁	#124	1	4	yes	—
118	C₈×C₁₃₁	#124	1	4	yes	—
119	C₂²×C₁₃₁	—	8	2	no	—
120	C₂²×C₁₃₁	—	8	4	no	—
121	C₂²×C₁₃₁	—	8	8	no	—
122	C₂²×C₁₃₁	C₂×#127	1	1	yes	characteristic
123	C₂²×C₁₃₁	#532	1	2	yes	—
124	C₂²×C₁₃₁	#575	1	4	yes	—
125	C₄×C₁₃₁	—	2	4	no	—
126	C₄×C₁₃₁	—	4	4	no	—
127	C₄×C₁₃₁	—	8	2	no	—
128	C₄×C₁₃₁	—	8	4	no	—
129	C₄×C₁₃₁	C₂×#124	1	1	yes	characteristic
130	C₄×C₁₃₁	C₂×#124	1	1	yes	characteristic
131	C₄×C₁₃₁	#532	1	2	yes	—
132	C₂×C₁₃₁	—	16	2	no	—
133	C₂×C₁₃₁	#2858	1	1	yes	characteristic
134	C₂×C₁₃₁	#3150	1	2	yes	—
135	C₂×C₁₃₁	#3183	1	4	yes	—
136	#33253	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₁₃₁	#33253	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×#499	C₂×C₁₃₁	1	1	yes	characteristic, complemented
139	C₂×#495	C₂×C₁₃₁	1	1	yes	characteristic
140	C₂×#488	C₂×C₁₃₁	1	2	yes	complemented
141	#2861	C₂×C₁₃₁	1	2	yes	complemented
142	#1973	C₂×C₁₃₁	1	4	yes	complemented
143	#1972	C₂×C₁₃₁	1	2	yes	—
144	#1971	C₂×C₁₃₁	1	2	yes	complemented
145	C₄×#97	C₂×C₁₃₁	1	1	yes	characteristic, complemented
146	C₂²×Q₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
147	C₂²×SD₁₆	C₂²×C₁₃₁	1	2	yes	—
148	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
149	C₂×#125	C₂²×C₁₃₁	1	2	yes	—
150	C₂×#124	C₂²×C₁₃₁	1	2	yes	—
151	#499	—	2	4	no	—
152	#495	—	2	4	no	—
153	#488	—	2	8	no	—
154	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
155	C₂×#106	C₂²×C₁₃₁	1	2	yes	—
156	C₂×#103	C₂²×C₁₃₁	1	4	yes	—
157	C₂×#102	C₂²×C₁₃₁	1	4	yes	—
158	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
159	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
160	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
162	#391	C₂²×C₁₃₁	1	4	yes	—
163	#390	C₂²×C₁₃₁	1	4	yes	—
164	#388	—	2	8	no	—
165	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
166	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
167	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
168	C₂×Q₁₆	—	2	2	no	—
169	C₂×Q₁₆	—	2	4	no	—
170	C₂×SD₁₆	—	2	4	no	—
171	C₂×SD₁₆	—	2	8	no	—
172	C₂×D₈	—	2	2	no	—
173	C₂×D₈	—	2	4	no	—
174	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
175	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
177	#125	—	2	4	no	—
178	#125	—	2	4	no	—
179	#124	—	2	4	no	—
180	#124	—	2	4	no	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
183	C₂×C₂²⋊C₄	—	2	4	no	—
184	C₂×C₂²⋊C₄	—	2	8	no	—
185	C₂×C₄²	D₄×C₁₃₁	1	4	yes	—
186	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
187	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
188	#107	—	2	4	no	—
189	#106	—	2	4	no	—
190	#103	—	2	8	no	—
191	#102	—	2	8	no	—
192	#97	—	2	4	no	—
193	#97	—	2	4	no	—
194	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
195	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
196	#95	—	2	8	no	—
197	C₂⁴	—	2	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
209	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
210	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
211	Q₁₆	—	4	4	no	—
212	SD₁₆	—	4	8	no	—
213	D₈	—	4	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
217	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
219	C₂×C₈	#41×C₁₃₁	1	4	yes	—
220	C₂×C₈	#41×C₁₃₁	1	4	yes	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	8	no	—
224	C₂²⋊C₄	—	4	16	no	—
225	C₂²⋊C₄	—	4	8	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	4	4	no	—
229	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
230	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	4	no	—
233	C₂³	—	4	8	no	—
234	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
235	Q₈	—	4	2	no	—
236	Q₈	—	4	2	no	—
237	Q₈	—	4	4	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	D₄	—	4	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	#127×C₁₃₁	1	4	yes	—
249	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
250	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
251	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
252	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
253	C₈	—	2	4	no	—
254	C₈	#124×C₁₃₁	1	4	yes	—
255	C₈	#124×C₁₃₁	1	4	yes	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	4	no	—
258	C₂²	—	8	8	no	—
259	C₂²	C₂×#127×C₁₃₁	1	1	yes	characteristic
260	C₂²	#532×C₁₃₁	1	2	yes	—
261	C₂²	#575×C₁₃₁	1	4	yes	—
262	C₄	—	2	4	no	—
263	C₄	—	4	4	no	—
264	C₄	—	8	2	no	—
265	C₄	—	8	4	no	—
266	C₄	C₂×#124×C₁₃₁	1	1	yes	characteristic
267	C₄	C₂×#124×C₁₃₁	1	1	yes	characteristic
268	C₄	#532×C₁₃₁	1	2	yes	—
269	C₂	—	16	2	no	—
270	C₂	#2858×C₁₃₁	1	1	yes	characteristic
271	C₂	#3150×C₁₃₁	1	2	yes	—
272	C₂	#3183×C₁₃₁	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂²×D₈×C₁₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#499×C₁₃₁	C₂	semidirect
3. C₂×#488×C₁₃₁	C₂	semidirect
4. #2861×C₁₃₁	C₂	semidirect
5. #1973×C₁₃₁	C₂	semidirect
7. #1971×C₁₃₁	C₂	semidirect
8. C₄×#97×C₁₃₁	C₂	semidirect
136. #33253	C₁₃₁	direct (N × H)
137. C₁₃₁	#33253	direct (N × H)
138. C₂×#499	C₂×C₁₃₁	semidirect
140. C₂×#488	C₂×C₁₃₁	semidirect
141. #2861	C₂×C₁₃₁	semidirect
142. #1973	C₂×C₁₃₁	semidirect
144. #1971	C₂×C₁₃₁	semidirect
145. C₄×#97	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#499×C₁₃₁	2
C₂×#495×C₁₃₁	2
C₂×#488×C₁₃₁	2
#2861×C₁₃₁	2
#1973×C₁₃₁	2
#1972×C₁₃₁	2
#1971×C₁₃₁	2
C₄×#97×C₁₃₁	2
#33253	131

Subgroups — C131×#33253 #45719137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33253 #45719137