Subgroups — C₁₃₁×#29227 #45715111

Order 33536.

626 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	#2859×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₄×#125×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2101×C₁₃₁	C₂	1	4	yes	maximal
4	#1992×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#1992×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	#1989×C₁₃₁	C₂	1	4	yes	maximal
7	#1836×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	#1836×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	#1767×C₁₃₁	C₂	1	1	yes	maximal, characteristic
10	C₂×#125×C₁₃₁	C₂²	1	2	yes	—
11	C₂×#125×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
13	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
15	#389×C₁₃₁	—	2	2	no	—
16	#389×C₁₃₁	—	2	2	no	—
17	#389×C₁₃₁	C₂²	1	2	yes	—
18	#387×C₁₃₁	—	2	2	no	—
19	#384×C₁₃₁	C₂²	1	1	yes	characteristic
20	#384×C₁₃₁	C₂²	1	1	yes	characteristic
21	#382×C₁₃₁	C₂²	1	1	yes	characteristic
22	#382×C₁₃₁	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₁₃₁	—	2	2	no	—
24	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
25	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
26	#340×C₁₃₁	C₂²	1	4	yes	—
27	#340×C₁₃₁	C₂²	1	4	yes	—
28	#340×C₁₃₁	C₂²	1	4	yes	—
29	#340×C₁₃₁	C₂²	1	4	yes	—
30	#336×C₁₃₁	C₂²	1	2	yes	—
31	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
33	#125×C₁₃₁	—	2	2	no	—
34	#125×C₁₃₁	—	2	8	no	—
35	#125×C₁₃₁	Q₈	1	4	yes	—
36	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₃₁	—	2	2	no	—
47	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
48	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
49	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₁₃₁	—	2	4	no	—
53	#97×C₁₃₁	—	2	2	no	—
54	#97×C₁₃₁	—	2	2	no	—
55	#95×C₁₃₁	—	2	4	no	—
56	#95×C₁₃₁	—	2	2	no	—
57	#95×C₁₃₁	—	2	2	no	—
58	#95×C₁₃₁	—	2	2	no	—
59	C₂²×C₄×C₁₃₁	—	2	2	no	—
60	C₂²×C₄×C₁₃₁	—	2	2	no	—
61	C₂²×C₄×C₁₃₁	—	2	2	no	—
62	C₂²×C₄×C₁₃₁	—	2	2	no	—
63	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₁₃₁	—	2	2	no	—
69	C₂×C₈×C₁₃₁	—	2	2	no	—
70	C₂×C₈×C₁₃₁	—	2	2	no	—
71	C₂×C₈×C₁₃₁	—	2	2	no	—
72	C₂×C₈×C₁₃₁	—	2	2	no	—
73	C₂×C₈×C₁₃₁	—	2	2	no	—
74	C₄⋊C₄×C₁₃₁	—	2	8	no	—
75	C₄⋊C₄×C₁₃₁	—	2	2	no	—
76	C₄⋊C₄×C₁₃₁	—	2	4	no	—
77	C₄⋊C₄×C₁₃₁	—	2	4	no	—
78	C₄⋊C₄×C₁₃₁	—	2	4	no	—
79	C₄⋊C₄×C₁₃₁	—	4	4	no	—
80	C₄⋊C₄×C₁₃₁	—	4	4	no	—
81	C₄⋊C₄×C₁₃₁	—	4	4	no	—
82	C₄⋊C₄×C₁₃₁	—	4	4	no	—
83	C₄⋊C₄×C₁₃₁	—	4	4	no	—
84	C₄⋊C₄×C₁₃₁	C₂×Q₈	1	4	yes	—
85	C₄²×C₁₃₁	—	2	2	no	—
86	C₄²×C₁₃₁	—	4	2	no	—
87	C₄²×C₁₃₁	—	4	2	no	—
88	C₄²×C₁₃₁	—	4	4	no	—
89	C₄²×C₁₃₁	C₂×Q₈	1	2	yes	—
90	C₄²×C₁₃₁	#41	1	2	yes	—
91	C₄²×C₁₃₁	#41	1	4	yes	—
92	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃₁	—	2	2	no	—
94	C₂×C₄×C₁₃₁	—	2	2	no	—
95	C₂×C₄×C₁₃₁	—	2	2	no	—
96	C₂×C₄×C₁₃₁	—	2	2	no	—
97	C₂×C₄×C₁₃₁	—	2	4	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	—	4	2	no	—
102	C₂×C₄×C₁₃₁	—	4	2	no	—
103	C₂×C₄×C₁₃₁	—	4	2	no	—
104	C₂×C₄×C₁₃₁	—	4	2	no	—
105	C₂×C₄×C₁₃₁	—	4	2	no	—
106	C₂×C₄×C₁₃₁	—	4	2	no	—
107	C₂×C₄×C₁₃₁	—	4	4	no	—
108	C₂×C₄×C₁₃₁	—	4	4	no	—
109	C₂×C₄×C₁₃₁	—	4	2	no	—
110	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	4	yes	—
111	C₂×C₄×C₁₃₁	#124	1	2	yes	—
112	C₂×C₄×C₁₃₁	#124	1	2	yes	—
113	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
116	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
117	C₈×C₁₃₁	—	4	2	no	—
118	C₈×C₁₃₁	—	4	2	no	—
119	C₂²×C₁₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
120	C₂²×C₁₃₁	C₂×#124	1	1	yes	characteristic
121	C₂²×C₁₃₁	#536	1	1	yes	characteristic
122	C₂²×C₁₃₁	#575	1	2	yes	—
123	C₂²×C₁₃₁	#575	1	2	yes	—
124	C₄×C₁₃₁	—	2	4	no	—
125	C₄×C₁₃₁	—	2	2	no	—
126	C₄×C₁₃₁	—	2	2	no	—
127	C₄×C₁₃₁	—	4	4	no	—
128	C₄×C₁₃₁	—	4	2	no	—
129	C₄×C₁₃₁	—	8	2	no	—
130	C₄×C₁₃₁	—	8	2	no	—
131	C₄×C₁₃₁	—	8	2	no	—
132	C₄×C₁₃₁	#556	1	2	yes	—
133	C₄×C₁₃₁	#557	1	2	yes	—
134	C₂×C₁₃₁	#2568	1	1	yes	characteristic
135	C₂×C₁₃₁	#3080	1	2	yes	—
136	C₂×C₁₃₁	#3135	1	1	yes	characteristic
137	C₂×C₁₃₁	#3142	1	1	yes	characteristic
138	C₂×C₁₃₁	#3240	1	2	yes	—
139	#29227	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₁₃₁	#29227	1	1	yes	characteristic, direct product factor, Sylow
141	#2859	C₂×C₁₃₁	1	1	yes	characteristic
142	C₄×#125	C₂×C₁₃₁	1	1	yes	characteristic
143	#2101	C₂×C₁₃₁	1	4	yes	—
144	#1992	C₂×C₁₃₁	1	1	yes	characteristic
145	#1992	C₂×C₁₃₁	1	1	yes	characteristic
146	#1989	C₂×C₁₃₁	1	4	yes	—
147	#1836	C₂×C₁₃₁	1	1	yes	characteristic
148	#1836	C₂×C₁₃₁	1	1	yes	characteristic
149	#1767	C₂×C₁₃₁	1	1	yes	characteristic
150	C₂×#125	C₂²×C₁₃₁	1	2	yes	—
151	C₂×#125	C₂²×C₁₃₁	1	1	yes	characteristic
152	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
153	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
154	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
155	#389	—	2	2	no	—
156	#389	—	2	2	no	—
157	#389	C₂²×C₁₃₁	1	2	yes	—
158	#387	—	2	2	no	—
159	#384	C₂²×C₁₃₁	1	1	yes	characteristic
160	#384	C₂²×C₁₃₁	1	1	yes	characteristic
161	#382	C₂²×C₁₃₁	1	1	yes	characteristic
162	#382	C₂²×C₁₃₁	1	2	yes	—
163	C₄×C₄⋊C₄	—	2	2	no	—
164	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
165	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
166	#340	C₂²×C₁₃₁	1	4	yes	—
167	#340	C₂²×C₁₃₁	1	4	yes	—
168	#340	C₂²×C₁₃₁	1	4	yes	—
169	#340	C₂²×C₁₃₁	1	4	yes	—
170	#336	C₂²×C₁₃₁	1	2	yes	—
171	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
172	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
173	#125	—	2	2	no	—
174	#125	—	2	8	no	—
175	#125	Q₈×C₁₃₁	1	4	yes	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
183	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
184	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
185	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
188	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
189	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
192	C₄⋊C₈	—	2	4	no	—
193	#97	—	2	2	no	—
194	#97	—	2	2	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	#95	—	2	2	no	—
198	#95	—	2	4	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
207	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	C₂×Q₈×C₁₃₁	1	4	yes	—
225	C₄²	—	2	2	no	—
226	C₄²	—	4	2	no	—
227	C₄²	—	4	2	no	—
228	C₄²	—	4	4	no	—
229	C₄²	C₂×Q₈×C₁₃₁	1	2	yes	—
230	C₄²	#41×C₁₃₁	1	2	yes	—
231	C₄²	#41×C₁₃₁	1	4	yes	—
232	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	4	yes	—
251	C₂×C₄	#124×C₁₃₁	1	2	yes	—
252	C₂×C₄	#124×C₁₃₁	1	2	yes	—
253	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
254	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
255	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
256	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	2	yes	—
257	C₈	—	4	2	no	—
258	C₈	—	4	2	no	—
259	C₂²	C₂×C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
260	C₂²	C₂×#124×C₁₃₁	1	1	yes	characteristic
261	C₂²	#536×C₁₃₁	1	1	yes	characteristic
262	C₂²	#575×C₁₃₁	1	2	yes	—
263	C₂²	#575×C₁₃₁	1	2	yes	—
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	4	no	—
268	C₄	—	4	2	no	—
269	C₄	—	8	2	no	—
270	C₄	—	8	2	no	—
271	C₄	—	8	2	no	—
272	C₄	#556×C₁₃₁	1	2	yes	—
273	C₄	#557×C₁₃₁	1	2	yes	—
274	C₂	#2568×C₁₃₁	1	1	yes	characteristic
275	C₂	#3080×C₁₃₁	1	2	yes	—
276	C₂	#3135×C₁₃₁	1	1	yes	characteristic
277	C₂	#3142×C₁₃₁	1	1	yes	characteristic
278	C₂	#3240×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. #29227	C₁₃₁	direct (N × H)
140. C₁₃₁	#29227	direct (N × H)

Maximal subgroups

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

subgroup	index
#2859×C₁₃₁	2
C₄×#125×C₁₃₁	2
#2101×C₁₃₁	2
#1992×C₁₃₁	2
#1989×C₁₃₁	2
#1836×C₁₃₁	2
#1767×C₁₃₁	2
#29227	131

Subgroups — C131×#29227 #45715111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#29227 #45715111