Subgroups — C₁₃₁×#11485 #45697369

Order 33536.

372 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2931×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2115×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#1327×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#1321×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	C₂×#131×C₁₃₁	C₄	1	1	yes	characteristic
6	C₂×#131×C₁₃₁	C₂²	1	1	yes	characteristic
7	#532×C₁₃₁	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₁₃₁	—	2	2	no	—
9	#436×C₁₃₁	—	2	2	no	—
10	C₈×D₄×C₁₃₁	—	2	2	no	—
11	#423×C₁₃₁	C₂²	1	1	yes	characteristic
12	#408×C₁₃₁	C₄	1	1	yes	characteristic
13	#408×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃₁	C₄	1	1	yes	characteristic
15	#349×C₁₃₁	—	2	2	no	—
16	#349×C₁₃₁	C₂²	1	4	yes	—
17	C₂×#41×C₁₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂×#41×C₁₃₁	C₂×C₄	1	1	yes	characteristic
19	#131×C₁₃₁	—	4	1	no	—
20	#131×C₁₃₁	—	4	1	no	—
21	#131×C₁₃₁	—	4	1	no	—
22	#131×C₁₃₁	—	4	1	no	—
23	C₂×M₁₆×C₁₃₁	—	2	1	no	—
24	C₂×M₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂×M₁₆×C₁₃₁	D₄	1	2	yes	—
26	C₂×M₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₁₃₁	—	2	1	no	—
28	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
31	C₄⋊Q₈×C₁₃₁	C₈	1	1	yes	characteristic
32	#127×C₁₃₁	C₈	1	1	yes	characteristic
33	#124×C₁₃₁	C₈	1	1	yes	characteristic
34	#124×C₁₃₁	C₈	1	1	yes	characteristic
35	#121×C₁₃₁	—	2	2	no	—
36	C₄×D₄×C₁₃₁	—	2	2	no	—
37	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
38	M₃₂×C₁₃₁	—	2	4	no	—
39	M₃₂×C₁₃₁	—	4	2	no	—
40	C₂×C₁₆×C₁₃₁	—	4	2	no	—
41	C₄⋊C₈×C₁₃₁	—	2	2	no	—
42	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
43	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
44	C₄×C₈×C₁₃₁	—	2	1	no	—
45	C₄×C₈×C₁₃₁	—	2	1	no	—
46	#41×C₁₃₁	—	4	1	no	—
47	#41×C₁₃₁	—	4	1	no	—
48	#41×C₁₃₁	—	4	1	no	—
49	#41×C₁₃₁	—	4	1	no	—
50	C₂×Q₈×C₁₃₁	C₂×C₈	1	1	yes	characteristic
51	C₂×Q₈×C₁₃₁	C₂×C₈	1	1	yes	characteristic
52	C₂×D₄×C₁₃₁	—	2	2	no	—
53	C₂×D₄×C₁₃₁	—	2	1	no	—
54	C₂×D₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
55	C₂×D₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
56	C₂×D₄×C₁₃₁	M₁₆	1	2	yes	—
57	C₂²×C₄×C₁₃₁	—	2	1	no	—
58	C₂²×C₄×C₁₃₁	—	2	1	no	—
59	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₁₃₁	—	4	2	no	—
62	M₁₆×C₁₃₁	—	4	1	no	—
63	M₁₆×C₁₃₁	—	4	2	no	—
64	M₁₆×C₁₃₁	—	4	1	no	—
65	C₂×C₈×C₁₃₁	—	2	2	no	—
66	C₂×C₈×C₁₃₁	—	2	1	no	—
67	C₂×C₈×C₁₃₁	—	2	2	no	—
68	C₂×C₈×C₁₃₁	—	2	1	no	—
69	C₂×C₈×C₁₃₁	—	2	1	no	—
70	C₂×C₈×C₁₃₁	—	2	1	no	—
71	C₂×C₈×C₁₃₁	—	4	1	no	—
72	C₂×C₈×C₁₃₁	—	4	1	no	—
73	C₂×C₈×C₁₃₁	—	4	1	no	—
74	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
77	C₄⋊C₄×C₁₃₁	—	2	2	no	—
78	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
79	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
80	C₄²×C₁₃₁	—	2	1	no	—
81	C₄²×C₁₃₁	C₂×C₈	1	1	yes	characteristic
82	C₄²×C₁₃₁	C₂×C₈	1	1	yes	characteristic
83	C₁₆×C₁₃₁	—	4	4	no	—
84	C₂³×C₁₃₁	—	2	1	no	—
85	C₂³×C₁₃₁	C₂²⋊C₈	1	2	yes	—
86	C₂³×C₁₃₁	C₂×M₁₆	1	1	yes	characteristic
87	Q₈×C₁₃₁	—	4	1	no	—
88	Q₈×C₁₃₁	—	4	1	no	—
89	D₄×C₁₃₁	—	4	1	no	—
90	D₄×C₁₃₁	—	4	2	no	—
91	D₄×C₁₃₁	—	4	1	no	—
92	D₄×C₁₃₁	—	4	4	no	—
93	D₄×C₁₃₁	—	4	2	no	—
94	C₂×C₄×C₁₃₁	—	2	1	no	—
95	C₂×C₄×C₁₃₁	—	2	1	no	—
96	C₂×C₄×C₁₃₁	—	2	2	no	—
97	C₂×C₄×C₁₃₁	—	2	1	no	—
98	C₂×C₄×C₁₃₁	—	2	2	no	—
99	C₂×C₄×C₁₃₁	—	2	1	no	—
100	C₂×C₄×C₁₃₁	—	2	1	no	—
101	C₂×C₄×C₁₃₁	—	4	2	no	—
102	C₂×C₄×C₁₃₁	—	4	1	no	—
103	C₂×C₄×C₁₃₁	—	4	1	no	—
104	C₂×C₄×C₁₃₁	C₂²⋊C₈	1	2	yes	—
105	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
107	C₈×C₁₃₁	—	2	2	no	—
108	C₈×C₁₃₁	—	2	1	no	—
109	C₈×C₁₃₁	—	4	1	no	—
110	C₈×C₁₃₁	—	4	1	no	—
111	C₈×C₁₃₁	#99	1	1	yes	characteristic
112	C₈×C₁₃₁	#99	1	1	yes	characteristic
113	C₂²×C₁₃₁	—	2	1	no	—
114	C₂²×C₁₃₁	—	2	2	no	—
115	C₂²×C₁₃₁	—	4	1	no	—
116	C₂²×C₁₃₁	—	4	2	no	—
117	C₂²×C₁₃₁	—	4	2	no	—
118	C₂²×C₁₃₁	—	4	1	no	—
119	C₂²×C₁₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄×C₁₃₁	—	2	2	no	—
121	C₄×C₁₃₁	—	2	2	no	—
122	C₄×C₁₃₁	—	2	1	no	—
123	C₄×C₁₃₁	—	4	1	no	—
124	C₄×C₁₃₁	—	4	1	no	—
125	C₄×C₁₃₁	C₂×#99	1	1	yes	characteristic
126	C₄×C₁₃₁	#413	1	1	yes	characteristic
127	C₂×C₁₃₁	—	2	1	no	—
128	C₂×C₁₃₁	—	4	1	no	—
129	C₂×C₁₃₁	—	4	2	no	—
130	C₂×C₁₃₁	—	8	1	no	—
131	C₂×C₁₃₁	#1459	1	1	yes	characteristic
132	#11485	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₁₃₁	#11485	1	1	yes	characteristic, direct product factor, Sylow
134	#2931	C₂×C₁₃₁	1	1	yes	characteristic
135	#2115	C₂×C₁₃₁	1	2	yes	complemented
136	#1327	C₂×C₁₃₁	1	2	yes	complemented
137	#1321	C₂×C₁₃₁	1	2	yes	complemented
138	C₂×#131	C₄×C₁₃₁	1	1	yes	characteristic
139	C₂×#131	C₂²×C₁₃₁	1	1	yes	characteristic
140	#532	C₄×C₁₃₁	1	1	yes	characteristic
141	C₂×M₃₂	—	2	2	no	—
142	#436	—	2	2	no	—
143	C₈×D₄	—	2	2	no	—
144	#423	C₂²×C₁₃₁	1	1	yes	characteristic
145	#408	C₄×C₁₃₁	1	1	yes	characteristic
146	#408	C₂²×C₁₃₁	1	1	yes	characteristic
147	C₂×C₄×C₈	C₄×C₁₃₁	1	1	yes	characteristic
148	#349	—	2	2	no	—
149	#349	C₂²×C₁₃₁	1	4	yes	—
150	C₂×#41	C₂×C₄×C₁₃₁	1	1	yes	characteristic
151	C₂×#41	C₂×C₄×C₁₃₁	1	1	yes	characteristic
152	#131	—	4	1	no	—
153	#131	—	4	1	no	—
154	#131	—	4	1	no	—
155	#131	—	4	1	no	—
156	C₂×M₁₆	—	2	1	no	—
157	C₂×M₁₆	C₂×C₄×C₁₃₁	1	1	yes	characteristic
158	C₂×M₁₆	D₄×C₁₃₁	1	2	yes	—
159	C₂×M₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
162	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
163	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
164	C₄⋊Q₈	C₈×C₁₃₁	1	1	yes	characteristic
165	#127	C₈×C₁₃₁	1	1	yes	characteristic
166	#124	C₈×C₁₃₁	1	1	yes	characteristic
167	#124	C₈×C₁₃₁	1	1	yes	characteristic
168	#121	—	2	2	no	—
169	C₄×D₄	—	2	2	no	—
170	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
171	M₃₂	—	2	4	no	—
172	M₃₂	—	4	2	no	—
173	C₂×C₁₆	—	4	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₄×C₈	—	2	1	no	—
178	C₄×C₈	—	2	1	no	—
179	#41	—	4	1	no	—
180	#41	—	4	1	no	—
181	#41	—	4	1	no	—
182	#41	—	4	1	no	—
183	C₂×Q₈	C₂×C₈×C₁₃₁	1	1	yes	characteristic
184	C₂×Q₈	C₂×C₈×C₁₃₁	1	1	yes	characteristic
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
188	C₂×D₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
189	C₂×D₄	M₁₆×C₁₃₁	1	2	yes	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	2	yes	—
193	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
194	M₁₆	—	4	2	no	—
195	M₁₆	—	4	1	no	—
196	M₁₆	—	4	2	no	—
197	M₁₆	—	4	1	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
210	C₄⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₄²	—	2	1	no	—
214	C₄²	C₂×C₈×C₁₃₁	1	1	yes	characteristic
215	C₄²	C₂×C₈×C₁₃₁	1	1	yes	characteristic
216	C₁₆	—	4	4	no	—
217	C₂³	—	2	1	no	—
218	C₂³	C₂²⋊C₈×C₁₃₁	1	2	yes	—
219	C₂³	C₂×M₁₆×C₁₃₁	1	1	yes	characteristic
220	Q₈	—	4	1	no	—
221	Q₈	—	4	1	no	—
222	D₄	—	4	1	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	1	no	—
225	D₄	—	4	4	no	—
226	D₄	—	4	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	C₂²⋊C₈×C₁₃₁	1	2	yes	—
238	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	1	no	—
242	C₈	—	4	1	no	—
243	C₈	—	4	1	no	—
244	C₈	#99×C₁₃₁	1	1	yes	characteristic
245	C₈	#99×C₁₃₁	1	1	yes	characteristic
246	C₂²	—	2	1	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	1	no	—
252	C₂²	C₂×C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	1	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	C₂×#99×C₁₃₁	1	1	yes	characteristic
259	C₄	#413×C₁₃₁	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	4	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	8	1	no	—
264	C₂	#1459×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₃₁	#99
commutator	C₂×C₄	C₂²×C₈×C₁₃₁
Frattini, cosocle	C₂×M₁₆	C₂³×C₁₃₁
socle	C₂×C₁₃₁	#1459
prime radical	#532×C₁₃₁	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃₁ ◁ C₂×C₈×C₁₃₁ ◁ C₂×#131×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2115×C₁₃₁	C₂	semidirect
3. #1327×C₁₃₁	C₂	semidirect
4. #1321×C₁₃₁	C₂	semidirect
132. #11485	C₁₃₁	direct (N × H)
133. C₁₃₁	#11485	direct (N × H)
135. #2115	C₂×C₁₃₁	semidirect
136. #1327	C₂×C₁₃₁	semidirect
137. #1321	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2931×C₁₃₁	2
#2115×C₁₃₁	2
#1327×C₁₃₁	2
#1321×C₁₃₁	2
#11485	131

Subgroups — C131×#11485 #45697369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#11485 #45697369