Subgroups — C₁₃₁×#13363 #45699247

Order 33536.

450 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#357×C₁₃₁	C₂	1	2	yes	maximal
3	#1970×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1380×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1374×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Q₈×C₁₃₁	—	2	1	no	—
8	C₂²×C₁₆×C₁₃₁	C₂²	1	2	yes	—
9	#500×C₁₃₁	—	2	4	no	—
10	C₂×#107×C₁₃₁	—	2	1	no	—
11	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
13	C₂×#102×C₁₃₁	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	#390×C₁₃₁	—	2	1	no	—
16	#357×C₁₃₁	—	2	4	no	—
17	C₂²×Q₈×C₁₃₁	—	4	1	no	—
18	C₂²×D₄×C₁₃₁	—	2	1	no	—
19	C₂×D₈×C₁₃₁	—	2	2	no	—
20	C₂×D₈×C₁₃₁	—	2	2	no	—
21	C₂×D₈×C₁₃₁	—	2	1	no	—
22	C₂×D₈×C₁₃₁	Q₈	1	2	yes	complemented
23	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
25	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
26	C₄⋊Q₈×C₁₃₁	—	4	2	no	—
27	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₃₁	—	4	1	no	—
30	C₂×C₂²⋊C₄×C₁₃₁	—	4	1	no	—
31	C₂×C₂²⋊C₄×C₁₃₁	—	4	1	no	—
32	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₁₃₁	—	2	2	no	—
34	C₂×C₁₆×C₁₃₁	—	2	2	no	—
35	C₂×C₁₆×C₁₃₁	—	2	2	no	—
36	#107×C₁₃₁	—	2	4	no	—
37	#107×C₁₃₁	—	2	2	no	—
38	#107×C₁₃₁	—	2	2	no	—
39	#107×C₁₃₁	D₄	1	4	yes	—
40	#102×C₁₃₁	—	4	2	no	—
41	C₄×C₈×C₁₃₁	—	2	2	no	—
42	C₂⁴×C₁₃₁	—	4	1	no	—
43	C₂×Q₈×C₁₃₁	—	4	4	no	—
44	C₂×Q₈×C₁₃₁	—	8	2	no	—
45	C₂×Q₈×C₁₃₁	—	8	2	no	—
46	C₂×D₄×C₁₃₁	—	2	2	no	—
47	C₂×D₄×C₁₃₁	—	4	1	no	—
48	C₂×D₄×C₁₃₁	—	4	1	no	—
49	C₂×D₄×C₁₃₁	—	4	1	no	—
50	C₂×D₄×C₁₃₁	—	4	1	no	—
51	C₂×D₄×C₁₃₁	—	4	1	no	—
52	C₂²×C₄×C₁₃₁	—	4	2	no	—
53	C₂²×C₄×C₁₃₁	—	4	1	no	—
54	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
57	D₈×C₁₃₁	—	2	4	no	—
58	D₈×C₁₃₁	—	4	2	no	—
59	C₂×C₈×C₁₃₁	—	2	1	no	—
60	C₂×C₈×C₁₃₁	—	2	1	no	—
61	C₂×C₈×C₁₃₁	SD₁₆	1	2	yes	—
62	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₁₃₁	#41	1	2	yes	—
65	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
66	C₄⋊C₄×C₁₃₁	—	2	4	no	—
67	C₄⋊C₄×C₁₃₁	—	4	4	no	—
68	C₄⋊C₄×C₁₃₁	—	4	2	no	—
69	C₄⋊C₄×C₁₃₁	—	4	1	no	—
70	C₄⋊C₄×C₁₃₁	—	4	1	no	—
71	C₂²⋊C₄×C₁₃₁	—	8	2	no	—
72	C₂²⋊C₄×C₁₃₁	—	8	2	no	—
73	C₄²×C₁₃₁	—	2	1	no	—
74	C₄²×C₁₃₁	—	2	1	no	—
75	C₁₆×C₁₃₁	—	4	2	no	—
76	C₂³×C₁₃₁	—	4	2	no	—
77	C₂³×C₁₃₁	—	8	1	no	—
78	C₂³×C₁₃₁	—	8	1	no	—
79	C₂³×C₁₃₁	—	8	1	no	—
80	C₂³×C₁₃₁	—	8	1	no	—
81	C₂³×C₁₃₁	—	8	1	no	—
82	C₂³×C₁₃₁	—	8	1	no	—
83	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
84	Q₈×C₁₃₁	—	8	8	no	—
85	D₄×C₁₃₁	—	4	2	no	—
86	D₄×C₁₃₁	—	4	2	no	—
87	D₄×C₁₃₁	—	8	1	no	—
88	D₄×C₁₃₁	—	8	1	no	—
89	C₂×C₄×C₁₃₁	—	2	1	no	—
90	C₂×C₄×C₁₃₁	—	2	1	no	—
91	C₂×C₄×C₁₃₁	—	2	1	no	—
92	C₂×C₄×C₁₃₁	—	2	1	no	—
93	C₂×C₄×C₁₃₁	—	4	4	no	—
94	C₂×C₄×C₁₃₁	—	8	2	no	—
95	C₂×C₄×C₁₃₁	—	8	2	no	—
96	C₂×C₄×C₁₃₁	—	8	1	no	—
97	C₂×C₄×C₁₃₁	—	8	1	no	—
98	C₂×C₄×C₁₃₁	—	8	1	no	—
99	C₂×C₄×C₁₃₁	D₁₆	1	2	yes	—
100	C₂×C₄×C₁₃₁	SD₃₂	1	2	yes	—
101	C₂×C₄×C₁₃₁	#120	1	1	yes	characteristic
102	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃₁	#124	1	1	yes	characteristic
104	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₂×C₄×C₁₃₁	#137	1	1	yes	characteristic
107	C₈×C₁₃₁	—	2	2	no	—
108	C₈×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
109	C₈×C₁₃₁	#124	1	2	yes	—
110	C₂²×C₁₃₁	—	8	2	no	—
111	C₂²×C₁₃₁	—	8	2	no	—
112	C₂²×C₁₃₁	—	8	2	no	—
113	C₂²×C₁₃₁	—	16	1	no	—
114	C₂²×C₁₃₁	—	16	1	no	—
115	C₂²×C₁₃₁	—	16	1	no	—
116	C₂²×C₁₃₁	—	16	1	no	—
117	C₂²×C₁₃₁	#447	1	1	yes	characteristic
118	C₂²×C₁₃₁	#474	1	1	yes	characteristic
119	C₂²×C₁₃₁	#486	1	1	yes	characteristic
120	C₂²×C₁₃₁	C₂×D₁₆	1	1	yes	characteristic
121	C₂²×C₁₃₁	C₂×SD₃₂	1	1	yes	characteristic
122	C₂²×C₁₃₁	#509	1	1	yes	characteristic
123	C₂²×C₁₃₁	#510	1	1	yes	characteristic
124	C₄×C₁₃₁	—	2	2	no	—
125	C₄×C₁₃₁	—	2	2	no	—
126	C₄×C₁₃₁	—	8	4	no	—
127	C₄×C₁₃₁	—	16	1	no	—
128	C₄×C₁₃₁	#393	1	1	yes	characteristic
129	C₄×C₁₃₁	#448	1	1	yes	characteristic
130	C₄×C₁₃₁	#476	1	1	yes	characteristic
131	C₄×C₁₃₁	#489	1	1	yes	characteristic
132	C₂×C₁₃₁	—	16	2	no	—
133	C₂×C₁₃₁	#2023	1	1	yes	characteristic
134	C₂×C₁₃₁	#2184	1	1	yes	characteristic
135	C₂×C₁₃₁	#2185	1	1	yes	characteristic
136	C₂×C₁₃₁	#2224	1	1	yes	characteristic
137	C₂×C₁₃₁	#2226	1	1	yes	characteristic
138	C₂×C₁₃₁	#2240	1	1	yes	characteristic
139	C₂×C₁₃₁	#2243	1	1	yes	characteristic
140	#13363	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₁₃₁	#13363	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×#500	C₂×C₁₃₁	1	1	yes	characteristic, complemented
143	C₂×#357	C₂×C₁₃₁	1	2	yes	—
144	#1970	C₂×C₁₃₁	1	1	yes	characteristic
145	#1380	C₂×C₁₃₁	1	2	yes	complemented
146	#1374	C₂×C₁₃₁	1	1	yes	characteristic, complemented
147	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
148	C₂×C₄⋊Q₈	—	2	1	no	—
149	C₂²×C₁₆	C₂²×C₁₃₁	1	2	yes	—
150	#500	—	2	4	no	—
151	C₂×#107	—	2	1	no	—
152	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
153	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
154	C₂×#102	—	2	1	no	—
155	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
156	#390	—	2	1	no	—
157	#357	—	2	4	no	—
158	C₂²×Q₈	—	4	1	no	—
159	C₂²×D₄	—	2	1	no	—
160	C₂×D₈	—	2	2	no	—
161	C₂×D₈	—	2	2	no	—
162	C₂×D₈	—	2	1	no	—
163	C₂×D₈	Q₈×C₁₃₁	1	2	yes	complemented
164	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
166	C₄⋊Q₈	—	2	4	no	—
167	C₄⋊Q₈	—	4	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	4	1	no	—
171	C₂×C₂²⋊C₄	—	4	1	no	—
172	C₂×C₂²⋊C₄	—	4	1	no	—
173	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	2	2	no	—
177	#107	—	2	4	no	—
178	#107	—	2	2	no	—
179	#107	—	2	2	no	—
180	#107	D₄×C₁₃₁	1	4	yes	—
181	#102	—	4	2	no	—
182	C₄×C₈	—	2	2	no	—
183	C₂⁴	—	4	1	no	—
184	C₂×Q₈	—	4	4	no	—
185	C₂×Q₈	—	8	2	no	—
186	C₂×Q₈	—	8	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	1	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
196	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
198	D₈	—	2	4	no	—
199	D₈	—	4	2	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	SD₁₆×C₁₃₁	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
204	C₂×C₈	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
205	C₂×C₈	#41×C₁₃₁	1	2	yes	—
206	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₄⋊C₄	—	4	1	no	—
212	C₂²⋊C₄	—	8	2	no	—
213	C₂²⋊C₄	—	8	2	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	2	1	no	—
216	C₁₆	—	4	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	8	1	no	—
219	C₂³	—	8	1	no	—
220	C₂³	—	8	1	no	—
221	C₂³	—	8	1	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	8	1	no	—
224	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
225	Q₈	—	8	8	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	2	no	—
228	D₄	—	8	1	no	—
229	D₄	—	8	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	1	no	—
238	C₂×C₄	—	8	1	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	D₁₆×C₁₃₁	1	2	yes	—
241	C₂×C₄	SD₃₂×C₁₃₁	1	2	yes	—
242	C₂×C₄	#120×C₁₃₁	1	1	yes	characteristic
243	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
244	C₂×C₄	#124×C₁₃₁	1	1	yes	characteristic
245	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
246	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
247	C₂×C₄	#137×C₁₃₁	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
250	C₈	#124×C₁₃₁	1	2	yes	—
251	C₂²	—	8	2	no	—
252	C₂²	—	8	2	no	—
253	C₂²	—	8	2	no	—
254	C₂²	—	16	1	no	—
255	C₂²	—	16	1	no	—
256	C₂²	—	16	1	no	—
257	C₂²	—	16	1	no	—
258	C₂²	#447×C₁₃₁	1	1	yes	characteristic
259	C₂²	#474×C₁₃₁	1	1	yes	characteristic
260	C₂²	#486×C₁₃₁	1	1	yes	characteristic
261	C₂²	C₂×D₁₆×C₁₃₁	1	1	yes	characteristic
262	C₂²	C₂×SD₃₂×C₁₃₁	1	1	yes	characteristic
263	C₂²	#509×C₁₃₁	1	1	yes	characteristic
264	C₂²	#510×C₁₃₁	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	8	4	no	—
268	C₄	—	16	1	no	—
269	C₄	#393×C₁₃₁	1	1	yes	characteristic
270	C₄	#448×C₁₃₁	1	1	yes	characteristic
271	C₄	#476×C₁₃₁	1	1	yes	characteristic
272	C₄	#489×C₁₃₁	1	1	yes	characteristic
273	C₂	—	16	2	no	—
274	C₂	#2023×C₁₃₁	1	1	yes	characteristic
275	C₂	#2184×C₁₃₁	1	1	yes	characteristic
276	C₂	#2185×C₁₃₁	1	1	yes	characteristic
277	C₂	#2224×C₁₃₁	1	1	yes	characteristic
278	C₂	#2226×C₁₃₁	1	1	yes	characteristic
279	C₂	#2240×C₁₃₁	1	1	yes	characteristic
280	C₂	#2243×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
prime radical	C₂²×D₈×C₁₃₁	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₁₃₁	C₂	semidirect
4. #1380×C₁₃₁	C₂	semidirect
5. #1374×C₁₃₁	C₂	semidirect
22. C₂×D₈×C₁₃₁	Q₈	semidirect
140. #13363	C₁₃₁	direct (N × H)
141. C₁₃₁	#13363	direct (N × H)
142. C₂×#500	C₂×C₁₃₁	semidirect
145. #1380	C₂×C₁₃₁	semidirect
146. #1374	C₂×C₁₃₁	semidirect
163. C₂×D₈	Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₃₁	2
C₂×#357×C₁₃₁	2
#1970×C₁₃₁	2
#1380×C₁₃₁	2
#1374×C₁₃₁	2
#13363	131

Subgroups — C131×#13363 #45699247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13363 #45699247