Subgroups — C₁₃₁×#8664 #45694548

Order 33536.

434 conjugacy classes of subgroups.

The subgroup lattice has 272 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	#1837×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#1564×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#1486×C₁₃₁	C₂	1	2	yes	maximal
5	#1301×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₁₃₁	—	2	1	no	—
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	C₄	1	2	yes	—
11	#436×C₁₃₁	C₄	1	2	yes	—
12	#431×C₁₃₁	C₂²	1	2	yes	—
13	#423×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₈×C₁₃₁	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
17	#341×C₁₃₁	—	2	1	no	—
18	#341×C₁₃₁	—	2	1	no	—
19	#335×C₁₃₁	—	2	1	no	—
20	#334×C₁₃₁	—	2	1	no	—
21	D₄⋊C₈×C₁₃₁	—	2	4	no	—
22	C₂²×D₄×C₁₃₁	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₁₃₁	—	2	1	no	—
24	C₂×M₁₆×C₁₃₁	—	2	2	no	—
25	C₂×M₁₆×C₁₃₁	—	2	2	no	—
26	C₂×M₁₆×C₁₃₁	—	2	2	no	—
27	C₂×M₁₆×C₁₃₁	—	4	4	no	—
28	C₂²×C₈×C₁₃₁	—	2	1	no	—
29	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
31	C₄×D₄×C₁₃₁	—	2	1	no	—
32	C₄×D₄×C₁₃₁	—	2	1	no	—
33	C₄×D₄×C₁₃₁	C₂×C₄	1	2	yes	—
34	C₄×D₄×C₁₃₁	D₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₃₁	Q₈	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
37	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₁₃₁	—	2	1	no	—
39	C₄⋊C₈×C₁₃₁	—	2	2	no	—
40	C₄⋊C₈×C₁₃₁	—	2	2	no	—
41	C₄⋊C₈×C₁₃₁	—	2	2	no	—
42	C₄⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
43	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
44	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
45	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
46	C₄×C₈×C₁₃₁	—	2	2	no	—
47	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
51	C₂⁴×C₁₃₁	—	2	1	no	—
52	C₂×D₄×C₁₃₁	—	2	1	no	—
53	C₂×D₄×C₁₃₁	—	4	2	no	—
54	C₂×D₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₃₁	—	2	1	no	—
56	C₂²×C₄×C₁₃₁	—	2	2	no	—
57	C₂²×C₄×C₁₃₁	—	2	2	no	—
58	C₂²×C₄×C₁₃₁	—	4	2	no	—
59	C₂²×C₄×C₁₃₁	—	4	2	no	—
60	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
63	M₁₆×C₁₃₁	—	4	4	no	—
64	M₁₆×C₁₃₁	—	4	4	no	—
65	M₁₆×C₁₃₁	—	4	2	no	—
66	M₁₆×C₁₃₁	—	4	2	no	—
67	C₂×C₈×C₁₃₁	—	2	2	no	—
68	C₂×C₈×C₁₃₁	—	2	2	no	—
69	C₂×C₈×C₁₃₁	—	2	2	no	—
70	C₂×C₈×C₁₃₁	—	4	2	no	—
71	C₂×C₈×C₁₃₁	—	4	2	no	—
72	C₂×C₈×C₁₃₁	—	4	2	no	—
73	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₃₁	—	2	1	no	—
78	C₄⋊C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
79	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
80	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
81	C₄²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₃₁	#41	1	1	yes	characteristic
85	C₂³×C₁₃₁	—	2	2	no	—
86	C₂³×C₁₃₁	—	4	2	no	—
87	C₂³×C₁₃₁	—	4	2	no	—
88	C₂³×C₁₃₁	—	4	2	no	—
89	C₂³×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
90	D₄×C₁₃₁	—	2	4	no	—
91	D₄×C₁₃₁	—	4	2	no	—
92	C₂×C₄×C₁₃₁	—	2	2	no	—
93	C₂×C₄×C₁₃₁	—	2	2	no	—
94	C₂×C₄×C₁₃₁	—	4	1	no	—
95	C₂×C₄×C₁₃₁	—	4	2	no	—
96	C₂×C₄×C₁₃₁	—	4	2	no	—
97	C₂×C₄×C₁₃₁	—	4	2	no	—
98	C₂×C₄×C₁₃₁	—	4	4	no	—
99	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
102	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
103	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
104	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃₁	#135	1	2	yes	—
106	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
107	C₈×C₁₃₁	—	2	1	no	—
108	C₈×C₁₃₁	—	2	1	no	—
109	C₈×C₁₃₁	—	2	1	no	—
110	C₈×C₁₃₁	—	2	1	no	—
111	C₈×C₁₃₁	—	4	2	no	—
112	C₈×C₁₃₁	—	8	2	no	—
113	C₈×C₁₃₁	—	8	2	no	—
114	C₂²×C₁₃₁	—	2	1	no	—
115	C₂²×C₁₃₁	—	2	1	no	—
116	C₂²×C₁₃₁	—	4	2	no	—
117	C₂²×C₁₃₁	—	4	2	no	—
118	C₂²×C₁₃₁	—	4	2	no	—
119	C₂²×C₁₃₁	—	8	4	no	—
120	C₂²×C₁₃₁	#380	1	1	yes	characteristic
121	C₂²×C₁₃₁	#478	1	2	yes	—
122	C₄×C₁₃₁	—	2	1	no	—
123	C₄×C₁₃₁	—	2	1	no	—
124	C₄×C₁₃₁	—	2	1	no	—
125	C₄×C₁₃₁	—	4	2	no	—
126	C₄×C₁₃₁	#416	1	1	yes	characteristic
127	C₄×C₁₃₁	#418	1	1	yes	characteristic
128	C₄×C₁₃₁	#430	1	2	yes	—
129	C₄×C₁₃₁	#463	1	2	yes	—
130	C₂×C₁₃₁	—	2	2	no	—
131	C₂×C₁₃₁	—	8	2	no	—
132	C₂×C₁₃₁	#1804	1	1	yes	characteristic
133	C₂×C₁₃₁	#1811	1	1	yes	characteristic
134	C₂×C₁₃₁	#1814	1	1	yes	characteristic
135	#8664	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₁₃₁	#8664	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×#436	C₂×C₁₃₁	1	1	yes	characteristic
138	#1837	C₂×C₁₃₁	1	1	yes	characteristic, complemented
139	#1564	C₂×C₁₃₁	1	1	yes	characteristic, complemented
140	#1486	C₂×C₁₃₁	1	2	yes	—
141	#1301	C₂×C₁₃₁	1	2	yes	complemented
142	C₂²×M₁₆	—	2	1	no	—
143	C₂×C₄×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
144	#436	—	2	1	no	—
145	#436	—	2	1	no	—
146	#436	C₄×C₁₃₁	1	2	yes	—
147	#436	C₄×C₁₃₁	1	2	yes	—
148	#431	C₂²×C₁₃₁	1	2	yes	—
149	#423	C₂²×C₁₃₁	1	2	yes	—
150	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
151	C₂×C₂²⋊C₈	—	2	1	no	—
152	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
153	#341	—	2	1	no	—
154	#341	—	2	1	no	—
155	#335	—	2	1	no	—
156	#334	—	2	1	no	—
157	D₄⋊C₈	—	2	4	no	—
158	C₂²×D₄	Q₈×C₁₃₁	1	1	yes	characteristic
159	C₂³×C₄	—	2	1	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	4	4	no	—
164	C₂²×C₈	—	2	1	no	—
165	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
166	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
167	C₄×D₄	—	2	1	no	—
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	C₂×C₄×C₁₃₁	1	2	yes	—
170	C₄×D₄	D₄×C₁₃₁	1	2	yes	—
171	C₂×C₄⋊C₄	Q₈×C₁₃₁	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	—	2	1	no	—
173	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
179	C₂²⋊C₈	—	2	2	no	—
180	C₂²⋊C₈	—	2	2	no	—
181	C₂²⋊C₈	—	2	2	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
184	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
185	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
186	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
187	C₂⁴	—	2	1	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
197	C₂²×C₄	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
198	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
199	M₁₆	—	4	4	no	—
200	M₁₆	—	4	4	no	—
201	M₁₆	—	4	2	no	—
202	M₁₆	—	4	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₄²	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
219	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
220	C₄²	#41×C₁₃₁	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	C₂³	—	4	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
226	D₄	—	2	4	no	—
227	D₄	—	4	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
236	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
237	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
238	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
239	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
240	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
241	C₂×C₄	#135×C₁₃₁	1	2	yes	—
242	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	4	2	no	—
248	C₈	—	8	2	no	—
249	C₈	—	8	2	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	8	4	no	—
256	C₂²	#380×C₁₃₁	1	1	yes	characteristic
257	C₂²	#478×C₁₃₁	1	2	yes	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	2	no	—
262	C₄	#416×C₁₃₁	1	1	yes	characteristic
263	C₄	#418×C₁₃₁	1	1	yes	characteristic
264	C₄	#430×C₁₃₁	1	2	yes	—
265	C₄	#463×C₁₃₁	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	8	2	no	—
268	C₂	#1804×C₁₃₁	1	1	yes	characteristic
269	C₂	#1811×C₁₃₁	1	1	yes	characteristic
270	C₂	#1814×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₁	#121
commutator	C₄²	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
socle	C₂²×C₁₃₁	#380
prime radical	C₂²×D₄×C₁₃₁	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1837×C₁₃₁	C₂	semidirect
3. #1564×C₁₃₁	C₂	semidirect
5. #1301×C₁₃₁	C₂	semidirect
135. #8664	C₁₃₁	direct (N × H)
136. C₁₃₁	#8664	direct (N × H)
138. #1837	C₂×C₁₃₁	semidirect
139. #1564	C₂×C₁₃₁	semidirect
141. #1301	C₂×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
#1837×C₁₃₁	2
#1564×C₁₃₁	2
#1486×C₁₃₁	2
#1301×C₁₃₁	2
#8664	131

Subgroups — C131×#8664 #45694548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#8664 #45694548