Subgroups — C₁₃₁×#29869 #45715753

Order 33536.

770 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#477×C₁₃₁	C₂	1	2	yes	maximal
2	#2868×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2295×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#2083×C₁₃₁	C₂	1	4	yes	maximal
5	#2072×C₁₃₁	C₂	1	4	yes	maximal
6	#1987×C₁₃₁	C₂	1	2	yes	maximal
7	#1831×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₁₃₁	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
11	#477×C₁₃₁	—	2	8	no	—
12	C₂×#107×C₁₃₁	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
14	C₂×#103×C₁₃₁	C₂²	1	4	yes	—
15	#391×C₁₃₁	C₂²	1	2	yes	—
16	#389×C₁₃₁	C₂²	1	2	yes	—
17	#387×C₁₃₁	—	2	2	no	—
18	#384×C₁₃₁	—	2	1	no	—
19	#384×C₁₃₁	—	2	1	no	—
20	#384×C₁₃₁	C₂²	1	4	yes	—
21	C₄×C₄⋊C₄×C₁₃₁	—	2	1	no	—
22	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
23	#376×C₁₃₁	—	2	1	no	—
24	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
25	#340×C₁₃₁	C₂²	1	8	yes	—
26	#336×C₁₃₁	C₂²	1	4	yes	—
27	C₂²×Q₈×C₁₃₁	—	2	2	no	—
28	C₂²×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
30	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
31	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
32	C₄⋊Q₈×C₁₃₁	—	4	8	no	—
33	C₄×Q₈×C₁₃₁	—	2	2	no	—
34	C₄×Q₈×C₁₃₁	D₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
39	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₃₁	—	2	1	no	—
44	C₂×C₄²×C₁₃₁	—	2	1	no	—
45	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
46	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
50	#107×C₁₃₁	—	2	8	no	—
51	C₄⋊C₈×C₁₃₁	—	2	4	no	—
52	#103×C₁₃₁	—	2	8	no	—
53	#95×C₁₃₁	—	2	2	no	—
54	#95×C₁₃₁	—	2	2	no	—
55	#95×C₁₃₁	—	2	2	no	—
56	#95×C₁₃₁	—	2	1	no	—
57	C₂×Q₈×C₁₃₁	—	2	1	no	—
58	C₂×Q₈×C₁₃₁	—	2	4	no	—
59	C₂×Q₈×C₁₃₁	—	4	8	no	—
60	C₂×Q₈×C₁₃₁	—	4	4	no	—
61	C₂×Q₈×C₁₃₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃₁	—	2	1	no	—
63	C₂²×C₄×C₁₃₁	—	2	1	no	—
64	C₂²×C₄×C₁₃₁	—	2	2	no	—
65	C₂²×C₄×C₁₃₁	—	2	4	no	—
66	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
71	C₂×C₈×C₁₃₁	—	2	4	no	—
72	C₂×C₈×C₁₃₁	—	2	8	no	—
73	C₄⋊C₄×C₁₃₁	—	2	4	no	—
74	C₄⋊C₄×C₁₃₁	—	2	1	no	—
75	C₄⋊C₄×C₁₃₁	—	2	4	no	—
76	C₄⋊C₄×C₁₃₁	—	2	4	no	—
77	C₄⋊C₄×C₁₃₁	—	2	4	no	—
78	C₄⋊C₄×C₁₃₁	—	4	8	no	—
79	C₄⋊C₄×C₁₃₁	—	4	4	no	—
80	C₄⋊C₄×C₁₃₁	—	4	4	no	—
81	C₄⋊C₄×C₁₃₁	—	4	8	no	—
82	C₄⋊C₄×C₁₃₁	C₂×D₄	1	2	yes	—
83	C₄²×C₁₃₁	—	2	4	no	—
84	C₄²×C₁₃₁	—	2	4	no	—
85	C₄²×C₁₃₁	—	2	2	no	—
86	C₄²×C₁₃₁	—	2	4	no	—
87	C₄²×C₁₃₁	—	4	2	no	—
88	C₄²×C₁₃₁	Q₁₆	1	4	yes	—
89	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
90	C₄²×C₁₃₁	#41	1	2	yes	—
91	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
92	Q₈×C₁₃₁	—	2	4	no	—
93	Q₈×C₁₃₁	—	4	2	no	—
94	Q₈×C₁₃₁	—	8	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	2	no	—
99	C₂×C₄×C₁₃₁	—	2	2	no	—
100	C₂×C₄×C₁₃₁	—	2	2	no	—
101	C₂×C₄×C₁₃₁	—	2	2	no	—
102	C₂×C₄×C₁₃₁	—	4	2	no	—
103	C₂×C₄×C₁₃₁	—	4	2	no	—
104	C₂×C₄×C₁₃₁	—	4	4	no	—
105	C₂×C₄×C₁₃₁	—	4	2	no	—
106	C₂×C₄×C₁₃₁	—	4	8	no	—
107	C₂×C₄×C₁₃₁	—	4	2	no	—
108	C₂×C₄×C₁₃₁	—	4	2	no	—
109	C₂×C₄×C₁₃₁	#121	1	2	yes	—
110	C₂×C₄×C₁₃₁	#121	1	2	yes	—
111	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	4	yes	—
112	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	2	yes	—
113	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	4	yes	—
114	C₂×C₄×C₁₃₁	#137	1	2	yes	—
115	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	4	yes	—
118	C₈×C₁₃₁	—	4	4	no	—
119	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
120	C₂²×C₁₃₁	#536	1	2	yes	—
121	C₂²×C₁₃₁	C₂²×Q₁₆	1	1	yes	characteristic
122	C₂²×C₁₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
123	C₂²×C₁₃₁	C₂×#137	1	1	yes	characteristic
124	C₄×C₁₃₁	—	2	4	no	—
125	C₄×C₁₃₁	—	2	4	no	—
126	C₄×C₁₃₁	—	2	2	no	—
127	C₄×C₁₃₁	—	4	2	no	—
128	C₄×C₁₃₁	—	4	2	no	—
129	C₄×C₁₃₁	—	8	2	no	—
130	C₄×C₁₃₁	—	8	4	no	—
131	C₄×C₁₃₁	#462	1	4	yes	—
132	C₄×C₁₃₁	#536	1	2	yes	—
133	C₄×C₁₃₁	#541	1	2	yes	—
134	C₂×C₁₃₁	#2621	1	1	yes	characteristic
135	C₂×C₁₃₁	C₂×#462	1	1	yes	characteristic
136	C₂×C₁₃₁	#3038	1	1	yes	characteristic
137	C₂×C₁₃₁	#3230	1	2	yes	—
138	C₂×C₁₃₁	#3238	1	2	yes	—
139	#29869	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₁₃₁	#29869	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×#477	C₂×C₁₃₁	1	2	yes	—
142	#2868	C₂×C₁₃₁	1	1	yes	characteristic
143	#2295	C₂×C₁₃₁	1	1	yes	characteristic
144	#2083	C₂×C₁₃₁	1	4	yes	—
145	#2072	C₂×C₁₃₁	1	4	yes	—
146	#1987	C₂×C₁₃₁	1	2	yes	—
147	#1831	C₂×C₁₃₁	1	1	yes	characteristic
148	C₂×C₄⋊Q₈	—	2	2	no	—
149	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
150	C₂×C₄×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
151	#477	—	2	8	no	—
152	C₂×#107	C₂²×C₁₃₁	1	4	yes	—
153	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
154	C₂×#103	C₂²×C₁₃₁	1	4	yes	—
155	#391	C₂²×C₁₃₁	1	2	yes	—
156	#389	C₂²×C₁₃₁	1	2	yes	—
157	#387	—	2	2	no	—
158	#384	—	2	1	no	—
159	#384	—	2	1	no	—
160	#384	C₂²×C₁₃₁	1	4	yes	—
161	C₄×C₄⋊C₄	—	2	1	no	—
162	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
163	#376	—	2	1	no	—
164	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
165	#340	C₂²×C₁₃₁	1	8	yes	—
166	#336	C₂²×C₁₃₁	1	4	yes	—
167	C₂²×Q₈	—	2	2	no	—
168	C₂²×Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
170	C₄⋊Q₈	—	2	4	no	—
171	C₄⋊Q₈	—	2	4	no	—
172	C₄⋊Q₈	—	4	8	no	—
173	C₄×Q₈	—	2	2	no	—
174	C₄×Q₈	D₄×C₁₃₁	1	4	yes	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	—	2	4	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
181	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
182	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
183	C₂×C₄²	—	2	1	no	—
184	C₂×C₄²	—	2	1	no	—
185	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
186	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
187	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
188	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
189	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
190	#107	—	2	8	no	—
191	C₄⋊C₈	—	2	4	no	—
192	#103	—	2	8	no	—
193	#95	—	2	2	no	—
194	#95	—	2	2	no	—
195	#95	—	2	2	no	—
196	#95	—	2	1	no	—
197	C₂×Q₈	—	2	1	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	4	8	no	—
200	C₂×Q₈	—	4	4	no	—
201	C₂×Q₈	C₂×D₄×C₁₃₁	1	2	yes	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
210	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	8	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	4	8	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	8	no	—
222	C₄⋊C₄	C₂×D₄×C₁₃₁	1	2	yes	—
223	C₄²	—	2	4	no	—
224	C₄²	—	2	4	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	4	no	—
227	C₄²	—	4	2	no	—
228	C₄²	Q₁₆×C₁₃₁	1	4	yes	—
229	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
230	C₄²	#41×C₁₃₁	1	2	yes	—
231	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
232	Q₈	—	2	4	no	—
233	Q₈	—	4	2	no	—
234	Q₈	—	8	8	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	#121×C₁₃₁	1	2	yes	—
250	C₂×C₄	#121×C₁₃₁	1	2	yes	—
251	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	4	yes	—
252	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	2	yes	—
253	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	4	yes	—
254	C₂×C₄	#137×C₁₃₁	1	2	yes	—
255	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
256	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
257	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	4	yes	—
258	C₈	—	4	4	no	—
259	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
260	C₂²	#536×C₁₃₁	1	2	yes	—
261	C₂²	C₂²×Q₁₆×C₁₃₁	1	1	yes	characteristic
262	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
263	C₂²	C₂×#137×C₁₃₁	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	4	no	—
266	C₄	—	2	4	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	8	4	no	—
270	C₄	—	8	2	no	—
271	C₄	#462×C₁₃₁	1	4	yes	—
272	C₄	#536×C₁₃₁	1	2	yes	—
273	C₄	#541×C₁₃₁	1	2	yes	—
274	C₂	#2621×C₁₃₁	1	1	yes	characteristic
275	C₂	C₂×#462×C₁₃₁	1	1	yes	characteristic
276	C₂	#3038×C₁₃₁	1	1	yes	characteristic
277	C₂	#3230×C₁₃₁	1	2	yes	—
278	C₂	#3238×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
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
139. #29869	C₁₃₁	direct (N × H)
140. C₁₃₁	#29869	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#477×C₁₃₁	2
#2868×C₁₃₁	2
#2295×C₁₃₁	2
#2083×C₁₃₁	2
#2072×C₁₃₁	2
#1987×C₁₃₁	2
#1831×C₁₃₁	2
#29869	131

Subgroups — C131×#29869 #45715753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#29869 #45715753