Subgroups — C₁₃₁×#29232 #45715116

Order 33536.

552 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2102×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#2050×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2031×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#1994×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1932×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#1906×C₁₃₁	C₂	1	2	yes	maximal
7	#1841×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	#1827×C₁₃₁	C₂	1	2	yes	maximal, complemented
9	C₄×#104×C₁₃₁	C₂	1	2	yes	maximal, complemented
10	#575×C₁₃₁	C₂²	1	1	yes	characteristic
11	#533×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₄×#41×C₁₃₁	C₂²	1	2	yes	—
13	#430×C₁₃₁	C₂²	1	1	yes	characteristic
14	#423×C₁₃₁	C₂²	1	1	yes	characteristic
15	#421×C₁₃₁	C₂²	1	2	yes	—
16	C₂×#104×C₁₃₁	C₂²	1	2	yes	—
17	C₂×#104×C₁₃₁	C₂²	1	2	yes	—
18	#413×C₁₃₁	C₂²	1	2	yes	—
19	#410×C₁₃₁	C₂²	1	2	yes	—
20	C₄×M₁₆×C₁₃₁	C₂²	1	2	yes	—
21	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
22	#344×C₁₃₁	C₂²	1	2	yes	—
23	#344×C₁₃₁	C₂²	1	2	yes	—
24	#339×C₁₃₁	C₂²	1	2	yes	—
25	#339×C₁₃₁	C₂²	1	2	yes	—
26	#339×C₁₃₁	C₂²	1	2	yes	—
27	#339×C₁₃₁	C₂²	1	2	yes	—
28	#338×C₁₃₁	C₂²	1	1	yes	characteristic
29	H(3,ℤ/4)×C₁₃₁	C₂²	1	2	yes	—
30	H(3,ℤ/4)×C₁₃₁	C₂²	1	1	yes	characteristic
31	C₂×#41×C₁₃₁	C₂³	1	2	yes	—
32	#137×C₁₃₁	—	2	2	no	—
33	AGL(1,ℤ/8)×C₁₃₁	—	2	2	no	—
34	#135×C₁₃₁	—	4	1	no	—
35	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
36	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
37	C₂×M₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
40	#125×C₁₃₁	—	2	1	no	—
41	C₂²⋊Q₈×C₁₃₁	—	4	1	no	—
42	C₄×Q₈×C₁₃₁	—	2	2	no	—
43	C₄×Q₈×C₁₃₁	—	4	1	no	—
44	C₄×D₄×C₁₃₁	—	2	2	no	—
45	C₄×D₄×C₁₃₁	—	4	2	no	—
46	#117×C₁₃₁	—	2	2	no	—
47	#117×C₁₃₁	C₂³	1	2	yes	—
48	#117×C₁₃₁	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃₁	—	2	2	no	—
50	C₂×C₄²×C₁₃₁	—	4	1	no	—
51	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
52	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
53	#108×C₁₃₁	—	4	1	no	—
54	C₄⋊C₈×C₁₃₁	—	4	1	no	—
55	#104×C₁₃₁	—	2	2	no	—
56	#104×C₁₃₁	—	2	2	no	—
57	#104×C₁₃₁	—	2	2	no	—
58	#104×C₁₃₁	—	2	2	no	—
59	#104×C₁₃₁	—	4	2	no	—
60	#101×C₁₃₁	—	2	2	no	—
61	#100×C₁₃₁	—	2	2	no	—
62	#99×C₁₃₁	—	4	2	no	—
63	#97×C₁₃₁	—	2	2	no	—
64	C₄×C₈×C₁₃₁	—	2	2	no	—
65	#41×C₁₃₁	—	2	2	no	—
66	#41×C₁₃₁	—	2	2	no	—
67	#41×C₁₃₁	—	2	2	no	—
68	#41×C₁₃₁	—	2	2	no	—
69	C₂×Q₈×C₁₃₁	—	4	1	no	—
70	C₂×Q₈×C₁₃₁	#41	1	2	yes	—
71	C₂×D₄×C₁₃₁	—	2	2	no	—
72	C₂×D₄×C₁₃₁	#41	1	2	yes	—
73	C₂²×C₄×C₁₃₁	—	2	2	no	—
74	C₂²×C₄×C₁₃₁	—	2	1	no	—
75	C₂²×C₄×C₁₃₁	—	2	2	no	—
76	C₂²×C₄×C₁₃₁	—	4	2	no	—
77	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
78	Q₁₆×C₁₃₁	—	4	1	no	—
79	SD₁₆×C₁₃₁	—	4	2	no	—
80	D₈×C₁₃₁	—	4	1	no	—
81	M₁₆×C₁₃₁	—	2	2	no	—
82	M₁₆×C₁₃₁	—	2	2	no	—
83	M₁₆×C₁₃₁	—	2	2	no	—
84	M₁₆×C₁₃₁	—	2	2	no	—
85	M₁₆×C₁₃₁	—	2	2	no	—
86	M₁₆×C₁₃₁	—	2	2	no	—
87	C₂×C₈×C₁₃₁	—	2	2	no	—
88	C₂×C₈×C₁₃₁	—	2	2	no	—
89	C₂×C₈×C₁₃₁	—	2	1	no	—
90	C₂×C₈×C₁₃₁	—	2	1	no	—
91	C₄⋊C₄×C₁₃₁	—	2	2	no	—
92	C₄⋊C₄×C₁₃₁	—	2	2	no	—
93	C₄⋊C₄×C₁₃₁	—	4	2	no	—
94	C₄⋊C₄×C₁₃₁	—	4	2	no	—
95	C₄⋊C₄×C₁₃₁	—	4	1	no	—
96	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
97	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
98	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
99	C₄²×C₁₃₁	—	2	2	no	—
100	C₄²×C₁₃₁	—	2	1	no	—
101	C₄²×C₁₃₁	—	2	2	no	—
102	C₄²×C₁₃₁	—	2	2	no	—
103	C₄²×C₁₃₁	—	4	2	no	—
104	C₄²×C₁₃₁	—	4	4	no	—
105	C₄²×C₁₃₁	#41	1	2	yes	—
106	C₄²×C₁₃₁	#41	1	2	yes	—
107	C₄²×C₁₃₁	#41	1	1	yes	characteristic
108	C₄²×C₁₃₁	#41	1	1	yes	characteristic
109	C₂³×C₁₃₁	—	2	2	no	—
110	C₂³×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
111	Q₈×C₁₃₁	—	2	2	no	—
112	Q₈×C₁₃₁	—	2	2	no	—
113	Q₈×C₁₃₁	—	8	2	no	—
114	D₄×C₁₃₁	—	2	2	no	—
115	D₄×C₁₃₁	—	2	2	no	—
116	D₄×C₁₃₁	—	4	2	no	—
117	D₄×C₁₃₁	—	4	2	no	—
118	C₂×C₄×C₁₃₁	—	2	1	no	—
119	C₂×C₄×C₁₃₁	—	2	2	no	—
120	C₂×C₄×C₁₃₁	—	2	1	no	—
121	C₂×C₄×C₁₃₁	—	2	2	no	—
122	C₂×C₄×C₁₃₁	—	2	2	no	—
123	C₂×C₄×C₁₃₁	—	4	4	no	—
124	C₂×C₄×C₁₃₁	—	4	2	no	—
125	C₂×C₄×C₁₃₁	—	4	2	no	—
126	C₂×C₄×C₁₃₁	—	4	2	no	—
127	C₂×C₄×C₁₃₁	—	4	2	no	—
128	C₂×C₄×C₁₃₁	—	4	2	no	—
129	C₂×C₄×C₁₃₁	—	8	4	no	—
130	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
132	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
133	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
134	C₈×C₁₃₁	—	4	2	no	—
135	C₈×C₁₃₁	—	4	1	no	—
136	C₈×C₁₃₁	—	4	1	no	—
137	C₈×C₁₃₁	—	4	2	no	—
138	C₂²×C₁₃₁	—	4	1	no	—
139	C₂²×C₁₃₁	—	4	2	no	—
140	C₂²×C₁₃₁	—	8	2	no	—
141	C₂²×C₁₃₁	#538	1	1	yes	characteristic
142	C₂²×C₁₃₁	#551	1	2	yes	—
143	C₄×C₁₃₁	—	2	4	no	—
144	C₄×C₁₃₁	—	2	2	no	—
145	C₄×C₁₃₁	—	4	1	no	—
146	C₄×C₁₃₁	—	4	2	no	—
147	C₄×C₁₃₁	—	4	2	no	—
148	C₄×C₁₃₁	—	8	2	no	—
149	C₄×C₁₃₁	—	8	2	no	—
150	C₄×C₁₃₁	#529	1	2	yes	—
151	C₄×C₁₃₁	#551	1	1	yes	characteristic
152	C₄×C₁₃₁	#555	1	1	yes	characteristic
153	C₂×C₁₃₁	—	2	1	no	—
154	C₂×C₁₃₁	—	2	2	no	—
155	C₂×C₁₃₁	—	8	2	no	—
156	C₂×C₁₃₁	#2572	1	1	yes	characteristic
157	#29232	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₁₃₁	#29232	1	1	yes	characteristic, direct product factor, Sylow
159	#2102	C₂×C₁₃₁	1	1	yes	characteristic, complemented
160	#2050	C₂×C₁₃₁	1	1	yes	characteristic
161	#2031	C₂×C₁₃₁	1	2	yes	complemented
162	#1994	C₂×C₁₃₁	1	1	yes	characteristic, complemented
163	#1932	C₂×C₁₃₁	1	2	yes	complemented
164	#1906	C₂×C₁₃₁	1	2	yes	—
165	#1841	C₂×C₁₃₁	1	2	yes	complemented
166	#1827	C₂×C₁₃₁	1	2	yes	complemented
167	C₄×#104	C₂×C₁₃₁	1	2	yes	complemented
168	#575	C₂²×C₁₃₁	1	1	yes	characteristic
169	#533	C₂²×C₁₃₁	1	1	yes	characteristic
170	C₄×#41	C₂²×C₁₃₁	1	2	yes	—
171	#430	C₂²×C₁₃₁	1	1	yes	characteristic
172	#423	C₂²×C₁₃₁	1	1	yes	characteristic
173	#421	C₂²×C₁₃₁	1	2	yes	—
174	C₂×#104	C₂²×C₁₃₁	1	2	yes	—
175	C₂×#104	C₂²×C₁₃₁	1	2	yes	—
176	#413	C₂²×C₁₃₁	1	2	yes	—
177	#410	C₂²×C₁₃₁	1	2	yes	—
178	C₄×M₁₆	C₂²×C₁₃₁	1	2	yes	—
179	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
180	#344	C₂²×C₁₃₁	1	2	yes	—
181	#344	C₂²×C₁₃₁	1	2	yes	—
182	#339	C₂²×C₁₃₁	1	2	yes	—
183	#339	C₂²×C₁₃₁	1	2	yes	—
184	#339	C₂²×C₁₃₁	1	2	yes	—
185	#339	C₂²×C₁₃₁	1	2	yes	—
186	#338	C₂²×C₁₃₁	1	1	yes	characteristic
187	H(3,ℤ/4)	C₂²×C₁₃₁	1	1	yes	characteristic
188	H(3,ℤ/4)	C₂²×C₁₃₁	1	2	yes	—
189	C₂×#41	C₂³×C₁₃₁	1	2	yes	—
190	#137	—	2	2	no	—
191	AGL(1,ℤ/8)	—	2	2	no	—
192	#135	—	4	1	no	—
193	C₂×M₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
194	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
195	C₂×M₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
196	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
197	C₄⋊Q₈	—	2	1	no	—
198	#125	—	2	1	no	—
199	C₂²⋊Q₈	—	4	1	no	—
200	C₄×Q₈	—	2	2	no	—
201	C₄×Q₈	—	4	1	no	—
202	C₄×D₄	—	2	2	no	—
203	C₄×D₄	—	4	2	no	—
204	#117	—	2	2	no	—
205	#117	C₂³×C₁₃₁	1	2	yes	—
206	#117	C₂³×C₁₃₁	1	2	yes	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	4	1	no	—
209	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
210	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
211	#108	—	4	1	no	—
212	C₄⋊C₈	—	4	1	no	—
213	#104	—	2	2	no	—
214	#104	—	2	2	no	—
215	#104	—	2	2	no	—
216	#104	—	2	2	no	—
217	#104	—	4	2	no	—
218	#101	—	2	2	no	—
219	#100	—	2	2	no	—
220	#99	—	4	2	no	—
221	#97	—	2	2	no	—
222	C₄×C₈	—	2	2	no	—
223	#41	—	2	2	no	—
224	#41	—	2	2	no	—
225	#41	—	2	2	no	—
226	#41	—	2	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	#41×C₁₃₁	1	2	yes	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	#41×C₁₃₁	1	2	yes	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
236	Q₁₆	—	4	1	no	—
237	SD₁₆	—	4	2	no	—
238	D₈	—	4	1	no	—
239	M₁₆	—	2	2	no	—
240	M₁₆	—	2	2	no	—
241	M₁₆	—	2	2	no	—
242	M₁₆	—	2	2	no	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	4	2	no	—
262	C₄²	—	4	4	no	—
263	C₄²	#41×C₁₃₁	1	2	yes	—
264	C₄²	#41×C₁₃₁	1	2	yes	—
265	C₄²	#41×C₁₃₁	1	1	yes	characteristic
266	C₄²	#41×C₁₃₁	1	1	yes	characteristic
267	C₂³	—	2	2	no	—
268	C₂³	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
269	Q₈	—	2	2	no	—
270	Q₈	—	2	2	no	—
271	Q₈	—	8	2	no	—
272	D₄	—	2	2	no	—
273	D₄	—	2	2	no	—
274	D₄	—	4	2	no	—
275	D₄	—	4	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	8	4	no	—
288	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
289	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
290	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
291	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
292	C₈	—	4	2	no	—
293	C₈	—	4	1	no	—
294	C₈	—	4	1	no	—
295	C₈	—	4	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	#538×C₁₃₁	1	1	yes	characteristic
300	C₂²	#551×C₁₃₁	1	2	yes	—
301	C₄	—	2	4	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	#529×C₁₃₁	1	2	yes	—
309	C₄	#551×C₁₃₁	1	1	yes	characteristic
310	C₄	#555×C₁₃₁	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	—	8	2	no	—
314	C₂	#2572×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₁	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
socle	C₂×C₁₃₁	#2572
prime radical	#575×C₁₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #2102×C₁₃₁	C₂	semidirect
3. #2031×C₁₃₁	C₂	semidirect
4. #1994×C₁₃₁	C₂	semidirect
5. #1932×C₁₃₁	C₂	semidirect
7. #1841×C₁₃₁	C₂	semidirect
8. #1827×C₁₃₁	C₂	semidirect
9. C₄×#104×C₁₃₁	C₂	semidirect
157. #29232	C₁₃₁	direct (N × H)
158. C₁₃₁	#29232	direct (N × H)
159. #2102	C₂×C₁₃₁	semidirect
161. #2031	C₂×C₁₃₁	semidirect
162. #1994	C₂×C₁₃₁	semidirect
163. #1932	C₂×C₁₃₁	semidirect
165. #1841	C₂×C₁₃₁	semidirect
166. #1827	C₂×C₁₃₁	semidirect
167. C₄×#104	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2102×C₁₃₁	2
#2050×C₁₃₁	2
#2031×C₁₃₁	2
#1994×C₁₃₁	2
#1932×C₁₃₁	2
#1906×C₁₃₁	2
#1841×C₁₃₁	2
#1827×C₁₃₁	2
C₄×#104×C₁₃₁	2
#29232	131

Subgroups — C131×#29232 #45715116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#29232 #45715116