Subgroups — C₁₃₁×#8618 #45694502

Order 33536.

454 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#436×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1800×C₁₃₁	C₂	1	2	yes	maximal
3	#1567×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#1566×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1292×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₁₃₁	C₄	1	2	yes	—
7	C₂×C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
8	#436×C₁₃₁	—	2	1	no	—
9	#436×C₁₃₁	—	2	1	no	—
10	#436×C₁₃₁	—	2	1	no	—
11	#436×C₁₃₁	—	2	1	no	—
12	#423×C₁₃₁	C₂²	1	2	yes	—
13	#423×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#100×C₁₃₁	—	2	2	no	—
16	C₂×C₂²⋊C₈×C₁₃₁	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
18	#343×C₁₃₁	—	2	2	no	—
19	#335×C₁₃₁	—	2	1	no	—
20	#335×C₁₃₁	—	2	1	no	—
21	#334×C₁₃₁	—	2	1	no	—
22	#334×C₁₃₁	—	2	1	no	—
23	C₂²×D₄×C₁₃₁	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₁₃₁	—	2	2	no	—
26	C₂×M₁₆×C₁₃₁	—	2	2	no	—
27	C₂×M₁₆×C₁₃₁	—	2	2	no	—
28	C₂×M₁₆×C₁₃₁	—	2	2	no	—
29	C₂×M₁₆×C₁₃₁	—	4	4	no	—
30	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
33	C₄×D₄×C₁₃₁	—	2	1	no	—
34	C₄×D₄×C₁₃₁	—	2	1	no	—
35	C₄×D₄×C₁₃₁	—	2	1	no	—
36	C₄×D₄×C₁₃₁	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₁₃₁	Q₈	1	2	yes	—
39	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₁₃₁	—	2	1	no	—
41	C₄⋊C₈×C₁₃₁	—	2	1	no	—
42	C₄⋊C₈×C₁₃₁	—	2	2	no	—
43	C₄⋊C₈×C₁₃₁	—	2	2	no	—
44	C₄⋊C₈×C₁₃₁	—	2	2	no	—
45	C₄⋊C₈×C₁₃₁	—	2	2	no	—
46	#100×C₁₃₁	—	4	4	no	—
47	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
48	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
49	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
53	C₂⁴×C₁₃₁	C₄⋊C₄	1	2	yes	—
54	C₂×D₄×C₁₃₁	—	2	1	no	—
55	C₂×D₄×C₁₃₁	—	2	1	no	—
56	C₂×D₄×C₁₃₁	—	2	2	no	—
57	C₂×D₄×C₁₃₁	—	2	2	no	—
58	C₂²×C₄×C₁₃₁	—	2	2	no	—
59	C₂²×C₄×C₁₃₁	—	2	2	no	—
60	C₂²×C₄×C₁₃₁	—	4	2	no	—
61	C₂²×C₄×C₁₃₁	—	4	2	no	—
62	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
66	M₁₆×C₁₃₁	—	4	4	no	—
67	M₁₆×C₁₃₁	—	4	4	no	—
68	M₁₆×C₁₃₁	—	4	2	no	—
69	M₁₆×C₁₃₁	—	4	2	no	—
70	M₁₆×C₁₃₁	—	4	2	no	—
71	M₁₆×C₁₃₁	—	4	2	no	—
72	C₂×C₈×C₁₃₁	—	2	2	no	—
73	C₂×C₈×C₁₃₁	—	2	2	no	—
74	C₂×C₈×C₁₃₁	—	4	1	no	—
75	C₂×C₈×C₁₃₁	—	4	2	no	—
76	C₂×C₈×C₁₃₁	—	4	2	no	—
77	C₂×C₈×C₁₃₁	—	4	2	no	—
78	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₁₃₁	—	2	1	no	—
83	C₄⋊C₄×C₁₃₁	—	2	1	no	—
84	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
85	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
86	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₃₁	#41	1	1	yes	characteristic
89	C₄²×C₁₃₁	#41	1	1	yes	characteristic
90	C₂³×C₁₃₁	—	2	2	no	—
91	C₂³×C₁₃₁	—	2	4	no	—
92	C₂³×C₁₃₁	—	4	2	no	—
93	C₂³×C₁₃₁	—	4	2	no	—
94	C₂³×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
95	D₄×C₁₃₁	—	4	2	no	—
96	D₄×C₁₃₁	—	4	1	no	—
97	D₄×C₁₃₁	—	4	1	no	—
98	C₂×C₄×C₁₃₁	—	2	1	no	—
99	C₂×C₄×C₁₃₁	—	2	1	no	—
100	C₂×C₄×C₁₃₁	—	2	2	no	—
101	C₂×C₄×C₁₃₁	—	4	1	no	—
102	C₂×C₄×C₁₃₁	—	4	4	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
108	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
109	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
110	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×C₄×C₁₃₁	#137	1	2	yes	—
114	C₈×C₁₃₁	—	2	1	no	—
115	C₈×C₁₃₁	—	2	1	no	—
116	C₈×C₁₃₁	—	2	1	no	—
117	C₈×C₁₃₁	—	4	2	no	—
118	C₈×C₁₃₁	—	8	2	no	—
119	C₈×C₁₃₁	—	8	2	no	—
120	C₈×C₁₃₁	#100	1	2	yes	complemented
121	C₂²×C₁₃₁	—	2	1	no	—
122	C₂²×C₁₃₁	—	2	1	no	—
123	C₂²×C₁₃₁	—	4	2	no	—
124	C₂²×C₁₃₁	—	4	2	no	—
125	C₂²×C₁₃₁	—	4	2	no	—
126	C₂²×C₁₃₁	—	8	4	no	—
127	C₂²×C₁₃₁	#380	1	1	yes	characteristic
128	C₂²×C₁₃₁	#428	1	2	yes	—
129	C₄×C₁₃₁	—	2	1	no	—
130	C₄×C₁₃₁	—	2	1	no	—
131	C₄×C₁₃₁	—	2	1	no	—
132	C₄×C₁₃₁	—	4	2	no	—
133	C₄×C₁₃₁	C₂×#100	1	1	yes	characteristic
134	C₄×C₁₃₁	#413	1	1	yes	characteristic
135	C₄×C₁₃₁	#468	1	2	yes	—
136	C₄×C₁₃₁	#469	1	1	yes	characteristic
137	C₄×C₁₃₁	#470	1	1	yes	characteristic
138	C₂×C₁₃₁	—	2	2	no	—
139	C₂×C₁₃₁	—	8	2	no	—
140	C₂×C₁₃₁	#1800	1	1	yes	characteristic
141	C₂×C₁₃₁	#1809	1	1	yes	characteristic
142	C₂×C₁₃₁	#1813	1	1	yes	characteristic
143	#8618	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₁₃₁	#8618	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×#436	C₂×C₁₃₁	1	1	yes	characteristic
146	#1800	C₂×C₁₃₁	1	2	yes	—
147	#1567	C₂×C₁₃₁	1	1	yes	characteristic, complemented
148	#1566	C₂×C₁₃₁	1	1	yes	characteristic, complemented
149	#1292	C₂×C₁₃₁	1	2	yes	complemented
150	C₂²×M₁₆	C₄×C₁₃₁	1	2	yes	—
151	C₂×C₄×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
152	#436	—	2	1	no	—
153	#436	—	2	1	no	—
154	#436	—	2	1	no	—
155	#436	—	2	1	no	—
156	#423	C₂²×C₁₃₁	1	2	yes	—
157	#423	C₂²×C₁₃₁	1	2	yes	—
158	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
159	C₂×#100	—	2	2	no	—
160	C₂×C₂²⋊C₈	C₄×C₁₃₁	1	2	yes	—
161	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
162	#343	—	2	2	no	—
163	#335	—	2	1	no	—
164	#335	—	2	1	no	—
165	#334	—	2	1	no	—
166	#334	—	2	1	no	—
167	C₂²×D₄	D₄×C₁₃₁	1	1	yes	characteristic
168	C₂³×C₄	C₂×C₄×C₁₃₁	1	2	yes	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	4	4	no	—
174	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
175	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
176	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	2	1	no	—
180	C₄×D₄	—	2	1	no	—
181	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	Q₈×C₁₃₁	1	2	yes	—
183	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	2	2	no	—
189	C₄⋊C₈	—	2	2	no	—
190	#100	—	4	4	no	—
191	C₂²⋊C₈	—	2	2	no	—
192	C₂²⋊C₈	—	2	2	no	—
193	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
194	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
195	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
196	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
197	C₂⁴	C₄⋊C₄×C₁₃₁	1	2	yes	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
207	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
209	C₂²×C₄	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
210	M₁₆	—	4	4	no	—
211	M₁₆	—	4	4	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
223	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
225	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
231	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
232	C₄²	#41×C₁₃₁	1	1	yes	characteristic
233	C₄²	#41×C₁₃₁	1	1	yes	characteristic
234	C₂³	—	2	2	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	4	2	no	—
237	C₂³	—	4	2	no	—
238	C₂³	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
239	D₄	—	4	2	no	—
240	D₄	—	4	1	no	—
241	D₄	—	4	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
251	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
252	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
253	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
254	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
255	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
256	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
257	C₂×C₄	#137×C₁₃₁	1	2	yes	—
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	4	2	no	—
262	C₈	—	8	2	no	—
263	C₈	—	8	2	no	—
264	C₈	#100×C₁₃₁	1	2	yes	complemented
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	8	4	no	—
271	C₂²	#380×C₁₃₁	1	1	yes	characteristic
272	C₂²	#428×C₁₃₁	1	2	yes	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	2	no	—
277	C₄	C₂×#100×C₁₃₁	1	1	yes	characteristic
278	C₄	#413×C₁₃₁	1	1	yes	characteristic
279	C₄	#468×C₁₃₁	1	2	yes	—
280	C₄	#469×C₁₃₁	1	1	yes	characteristic
281	C₄	#470×C₁₃₁	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	8	2	no	—
284	C₂	#1800×C₁₃₁	1	1	yes	characteristic
285	C₂	#1809×C₁₃₁	1	1	yes	characteristic
286	C₂	#1813×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	#380
commutator	C₂²×C₄	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
prime radical	C₂²×D₄×C₁₃₁	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1567×C₁₃₁	C₂	semidirect
4. #1566×C₁₃₁	C₂	semidirect
5. #1292×C₁₃₁	C₂	semidirect
120. C₈×C₁₃₁	#100	semidirect
143. #8618	C₁₃₁	direct (N × H)
144. C₁₃₁	#8618	direct (N × H)
147. #1567	C₂×C₁₃₁	semidirect
148. #1566	C₂×C₁₃₁	semidirect
149. #1292	C₂×C₁₃₁	semidirect
264. C₈	#100×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#436×C₁₃₁	2
#1800×C₁₃₁	2
#1567×C₁₃₁	2
#1566×C₁₃₁	2
#1292×C₁₃₁	2
#8618	131

Subgroups — C131×#8618 #45694502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#8618 #45694502