Subgroups — C₁₃₁×#29145 #45715029

Order 33536.

894 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2862×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#127×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2023×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	#1961×C₁₃₁	C₂	1	2	yes	maximal
5	#1836×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#1804×C₁₃₁	C₂	1	4	yes	maximal, complemented
7	#1767×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
9	C₂×#127×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃₁	—	2	2	no	—
11	C₂×C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
13	C₂×#102×C₁₃₁	C₂²	1	8	yes	—
14	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
15	#391×C₁₃₁	—	2	2	no	—
16	#391×C₁₃₁	—	2	2	no	—
17	#390×C₁₃₁	—	2	2	no	—
18	#390×C₁₃₁	C₂²	1	2	yes	—
19	#384×C₁₃₁	C₂²	1	1	yes	characteristic
20	#382×C₁₃₁	C₂²	1	2	yes	—
21	#382×C₁₃₁	C₂²	1	2	yes	—
22	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
23	#340×C₁₃₁	C₂²	1	8	yes	—
24	#336×C₁₃₁	C₂²	1	2	yes	—
25	C₂²×Q₈×C₁₃₁	—	2	2	no	—
26	C₂²×D₄×C₁₃₁	—	2	2	no	—
27	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
28	C₂³×C₄×C₁₃₁	—	2	2	no	—
29	C₂²×C₈×C₁₃₁	C₂³	1	4	yes	—
30	C₄⋊Q₈×C₁₃₁	—	4	4	no	—
31	#127×C₁₃₁	—	2	2	no	—
32	#127×C₁₃₁	Q₈	1	4	yes	complemented
33	C₄×D₄×C₁₃₁	—	2	8	no	—
34	C₄×D₄×C₁₃₁	—	4	8	no	—
35	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
38	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
39	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
40	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
41	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
42	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
43	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₁₃₁	—	2	4	no	—
47	#102×C₁₃₁	—	4	8	no	—
48	#97×C₁₃₁	—	2	4	no	—
49	#95×C₁₃₁	—	2	4	no	—
50	#95×C₁₃₁	—	2	2	no	—
51	#95×C₁₃₁	—	2	2	no	—
52	C₂⁴×C₁₃₁	—	2	2	no	—
53	C₂×Q₈×C₁₃₁	—	4	4	no	—
54	C₂×Q₈×C₁₃₁	—	4	4	no	—
55	C₂×Q₈×C₁₃₁	—	4	4	no	—
56	C₂×D₄×C₁₃₁	—	2	8	no	—
57	C₂×D₄×C₁₃₁	—	2	4	no	—
58	C₂×D₄×C₁₃₁	—	2	2	no	—
59	C₂×D₄×C₁₃₁	—	2	4	no	—
60	C₂×D₄×C₁₃₁	—	4	4	no	—
61	C₂×D₄×C₁₃₁	—	4	4	no	—
62	C₂×D₄×C₁₃₁	C₂×Q₈	1	4	yes	—
63	C₂²×C₄×C₁₃₁	—	2	4	no	—
64	C₂²×C₄×C₁₃₁	—	2	2	no	—
65	C₂²×C₄×C₁₃₁	—	4	8	no	—
66	C₂²×C₄×C₁₃₁	—	4	4	no	—
67	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₁₃₁	—	2	4	no	—
73	C₂×C₈×C₁₃₁	—	2	4	no	—
74	C₂×C₈×C₁₃₁	—	2	4	no	—
75	C₄⋊C₄×C₁₃₁	—	2	4	no	—
76	C₄⋊C₄×C₁₃₁	—	2	8	no	—
77	C₄⋊C₄×C₁₃₁	—	4	4	no	—
78	C₄⋊C₄×C₁₃₁	—	4	4	no	—
79	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
80	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
81	C₄²×C₁₃₁	—	2	4	no	—
82	C₄²×C₁₃₁	—	2	2	no	—
83	C₄²×C₁₃₁	—	4	2	no	—
84	C₄²×C₁₃₁	C₂×Q₈	1	2	yes	—
85	C₄²×C₁₃₁	#41	1	2	yes	—
86	C₄²×C₁₃₁	#41	1	4	yes	—
87	C₂³×C₁₃₁	—	2	4	no	—
88	C₂³×C₁₃₁	—	4	2	no	—
89	C₂³×C₁₃₁	—	4	2	no	—
90	C₂³×C₁₃₁	—	4	4	no	—
91	C₂³×C₁₃₁	—	4	4	no	—
92	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
93	Q₈×C₁₃₁	—	8	8	no	—
94	D₄×C₁₃₁	—	2	8	no	—
95	D₄×C₁₃₁	—	4	16	no	—
96	D₄×C₁₃₁	—	4	4	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	—	2	2	no	—
103	C₂×C₄×C₁₃₁	—	4	2	no	—
104	C₂×C₄×C₁₃₁	—	4	4	no	—
105	C₂×C₄×C₁₃₁	—	4	4	no	—
106	C₂×C₄×C₁₃₁	—	4	4	no	—
107	C₂×C₄×C₁₃₁	—	4	4	no	—
108	C₂×C₄×C₁₃₁	—	8	8	no	—
109	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	4	yes	—
110	C₂×C₄×C₁₃₁	#124	1	2	yes	—
111	C₂×C₄×C₁₃₁	#124	1	2	yes	—
112	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	4	yes	—
113	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
115	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
117	C₈×C₁₃₁	—	4	4	no	—
118	C₂²×C₁₃₁	—	4	4	no	—
119	C₂²×C₁₃₁	—	4	8	no	—
120	C₂²×C₁₃₁	—	8	2	no	—
121	C₂²×C₁₃₁	—	8	2	no	—
122	C₂²×C₁₃₁	—	8	4	no	—
123	C₂²×C₁₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
124	C₂²×C₁₃₁	C₂×#124	1	1	yes	characteristic
125	C₂²×C₁₃₁	#535	1	1	yes	characteristic
126	C₂²×C₁₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
127	C₂²×C₁₃₁	#575	1	2	yes	—
128	C₄×C₁₃₁	—	2	4	no	—
129	C₄×C₁₃₁	—	2	2	no	—
130	C₄×C₁₃₁	—	2	2	no	—
131	C₄×C₁₃₁	—	2	4	no	—
132	C₄×C₁₃₁	—	8	2	no	—
133	C₄×C₁₃₁	—	8	4	no	—
134	C₄×C₁₃₁	#554	1	2	yes	—
135	C₄×C₁₃₁	#555	1	2	yes	—
136	C₂×C₁₃₁	—	8	4	no	—
137	C₂×C₁₃₁	#2563	1	1	yes	characteristic
138	C₂×C₁₃₁	#3077	1	2	yes	—
139	C₂×C₁₃₁	#3135	1	1	yes	characteristic
140	C₂×C₁₃₁	#3138	1	1	yes	characteristic
141	C₂×C₁₃₁	#3213	1	2	yes	—
142	#29145	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₁₃₁	#29145	1	1	yes	characteristic, direct product factor, Sylow
144	#2862	C₂×C₁₃₁	1	1	yes	characteristic, complemented
145	C₄×#127	C₂×C₁₃₁	1	1	yes	characteristic
146	#2023	C₂×C₁₃₁	1	4	yes	complemented
147	#1961	C₂×C₁₃₁	1	2	yes	—
148	#1836	C₂×C₁₃₁	1	2	yes	complemented
149	#1804	C₂×C₁₃₁	1	4	yes	complemented
150	#1767	C₂×C₁₃₁	1	1	yes	characteristic, complemented
151	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
152	C₂×#127	C₂²×C₁₃₁	1	1	yes	characteristic
153	C₂×C₄×D₄	—	2	2	no	—
154	C₂×C₄×D₄	C₂²×C₁₃₁	1	2	yes	—
155	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
156	C₂×#102	C₂²×C₁₃₁	1	8	yes	—
157	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
158	#391	—	2	2	no	—
159	#391	—	2	2	no	—
160	#390	—	2	2	no	—
161	#390	C₂²×C₁₃₁	1	2	yes	—
162	#384	C₂²×C₁₃₁	1	1	yes	characteristic
163	#382	C₂²×C₁₃₁	1	2	yes	—
164	#382	C₂²×C₁₃₁	1	2	yes	—
165	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
166	#340	C₂²×C₁₃₁	1	8	yes	—
167	#336	C₂²×C₁₃₁	1	2	yes	—
168	C₂²×Q₈	—	2	2	no	—
169	C₂²×D₄	—	2	2	no	—
170	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
171	C₂³×C₄	—	2	2	no	—
172	C₂²×C₈	C₂³×C₁₃₁	1	4	yes	—
173	C₄⋊Q₈	—	4	4	no	—
174	#127	—	2	2	no	—
175	#127	Q₈×C₁₃₁	1	4	yes	complemented
176	C₄×D₄	—	2	8	no	—
177	C₄×D₄	—	4	8	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
181	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
182	C₂×C₂²⋊C₄	—	2	4	no	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
185	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
186	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
187	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
188	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
189	C₄⋊C₈	—	2	4	no	—
190	#102	—	4	8	no	—
191	#97	—	2	4	no	—
192	#95	—	2	4	no	—
193	#95	—	2	2	no	—
194	#95	—	2	2	no	—
195	C₂⁴	—	2	2	no	—
196	C₂×Q₈	—	4	4	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂×Q₈	—	4	4	no	—
199	C₂×D₄	—	2	8	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	4	4	no	—
204	C₂×D₄	—	4	4	no	—
205	C₂×D₄	C₂×Q₈×C₁₃₁	1	4	yes	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	4	8	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
211	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
214	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	8	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₂²⋊C₄	—	4	4	no	—
223	C₂²⋊C₄	—	4	8	no	—
224	C₄²	—	2	4	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	4	2	no	—
227	C₄²	C₂×Q₈×C₁₃₁	1	2	yes	—
228	C₄²	#41×C₁₃₁	1	2	yes	—
229	C₄²	#41×C₁₃₁	1	4	yes	—
230	C₂³	—	2	4	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	4	no	—
235	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
236	Q₈	—	8	8	no	—
237	D₄	—	2	8	no	—
238	D₄	—	4	16	no	—
239	D₄	—	4	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	8	8	no	—
252	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	4	yes	—
253	C₂×C₄	#124×C₁₃₁	1	2	yes	—
254	C₂×C₄	#124×C₁₃₁	1	2	yes	—
255	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	4	yes	—
256	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
257	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
258	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
259	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	2	yes	—
260	C₈	—	4	4	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	4	8	no	—
263	C₂²	—	8	2	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	4	no	—
266	C₂²	C₂×C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
267	C₂²	C₂×#124×C₁₃₁	1	1	yes	characteristic
268	C₂²	#535×C₁₃₁	1	1	yes	characteristic
269	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
270	C₂²	#575×C₁₃₁	1	2	yes	—
271	C₄	—	2	4	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	4	no	—
275	C₄	—	8	2	no	—
276	C₄	—	8	4	no	—
277	C₄	#554×C₁₃₁	1	2	yes	—
278	C₄	#555×C₁₃₁	1	2	yes	—
279	C₂	—	8	4	no	—
280	C₂	#2563×C₁₃₁	1	1	yes	characteristic
281	C₂	#3077×C₁₃₁	1	2	yes	—
282	C₂	#3135×C₁₃₁	1	1	yes	characteristic
283	C₂	#3138×C₁₃₁	1	1	yes	characteristic
284	C₂	#3213×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂×#127×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. #2862×C₁₃₁	C₂	semidirect
3. #2023×C₁₃₁	C₂	semidirect
5. #1836×C₁₃₁	C₂	semidirect
6. #1804×C₁₃₁	C₂	semidirect
7. #1767×C₁₃₁	C₂	semidirect
32. #127×C₁₃₁	Q₈	semidirect
142. #29145	C₁₃₁	direct (N × H)
143. C₁₃₁	#29145	direct (N × H)
144. #2862	C₂×C₁₃₁	semidirect
146. #2023	C₂×C₁₃₁	semidirect
148. #1836	C₂×C₁₃₁	semidirect
149. #1804	C₂×C₁₃₁	semidirect
150. #1767	C₂×C₁₃₁	semidirect
175. #127	Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2862×C₁₃₁	2
C₄×#127×C₁₃₁	2
#2023×C₁₃₁	2
#1961×C₁₃₁	2
#1836×C₁₃₁	2
#1804×C₁₃₁	2
#1767×C₁₃₁	2
#29145	131

Subgroups — C131×#29145 #45715029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#29145 #45715029