Subgroups — C₁₃₁×#8179 #45694063

Order 33536.

562 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2293×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1767×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#1744×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄⋊C₈×C₁₃₁	C₂	1	2	yes	maximal
5	#1279×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₂×#117×C₁₃₁	—	2	1	no	—
8	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
9	C₂×#97×C₁₃₁	—	2	1	no	—
10	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
11	#390×C₁₃₁	C₄	1	2	yes	—
12	#389×C₁₃₁	C₄	1	2	yes	—
13	#388×C₁₃₁	—	2	2	no	—
14	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
16	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
17	#336×C₁₃₁	—	2	2	no	—
18	#336×C₁₃₁	—	2	1	no	—
19	D₄⋊C₈×C₁₃₁	C₄	1	8	yes	complemented
20	C₂²×D₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃₁	—	2	1	no	—
22	C₂²×C₈×C₁₃₁	—	2	2	no	—
23	C₂²×C₈×C₁₃₁	—	2	2	no	—
24	C₄×D₄×C₁₃₁	—	2	2	no	—
25	C₄×D₄×C₁₃₁	C₂×C₄	1	4	yes	—
26	#117×C₁₃₁	—	4	4	no	—
27	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
31	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
32	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
33	C₂×C₄²×C₁₃₁	—	2	1	no	—
34	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
38	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₁₃₁	—	2	2	no	—
40	C₄⋊C₈×C₁₃₁	C₂×C₄	1	4	yes	—
41	#97×C₁₃₁	—	4	2	no	—
42	C₄×C₈×C₁₃₁	—	2	2	no	—
43	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
44	#95×C₁₃₁	—	2	2	no	—
45	#95×C₁₃₁	—	2	1	no	—
46	C₂⁴×C₁₃₁	—	2	1	no	—
47	C₂×D₄×C₁₃₁	—	2	1	no	—
48	C₂×D₄×C₁₃₁	—	2	4	no	—
49	C₂×D₄×C₁₃₁	C₄²	1	2	yes	—
50	C₂²×C₄×C₁₃₁	—	2	1	no	—
51	C₂²×C₄×C₁₃₁	—	2	4	no	—
52	C₂²×C₄×C₁₃₁	—	2	2	no	—
53	C₂²×C₄×C₁₃₁	—	4	2	no	—
54	C₂²×C₄×C₁₃₁	—	4	1	no	—
55	C₂²×C₄×C₁₃₁	—	4	1	no	—
56	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃₁	—	2	4	no	—
62	C₂×C₈×C₁₃₁	—	2	4	no	—
63	C₂×C₈×C₁₃₁	—	4	4	no	—
64	C₂×C₈×C₁₃₁	—	4	2	no	—
65	C₂×C₈×C₁₃₁	—	4	2	no	—
66	C₄⋊C₄×C₁₃₁	—	2	1	no	—
67	C₄⋊C₄×C₁₃₁	—	2	4	no	—
68	C₄⋊C₄×C₁₃₁	—	4	4	no	—
69	C₄⋊C₄×C₁₃₁	C₄²	1	2	yes	—
70	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
71	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
72	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
73	C₄²×C₁₃₁	—	2	4	no	—
74	C₄²×C₁₃₁	—	2	1	no	—
75	C₄²×C₁₃₁	—	2	2	no	—
76	C₄²×C₁₃₁	—	2	1	no	—
77	C₄²×C₁₃₁	—	4	2	no	—
78	C₄²×C₁₃₁	—	4	2	no	—
79	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
80	C₄²×C₁₃₁	#41	1	2	yes	—
81	C₂³×C₁₃₁	—	2	2	no	—
82	C₂³×C₁₃₁	—	4	2	no	—
83	C₂³×C₁₃₁	—	4	1	no	—
84	C₂³×C₁₃₁	—	4	1	no	—
85	C₂³×C₁₃₁	—	4	2	no	—
86	C₂³×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	D₄×C₁₃₁	—	2	4	no	—
88	D₄×C₁₃₁	—	4	2	no	—
89	C₂×C₄×C₁₃₁	—	2	2	no	—
90	C₂×C₄×C₁₃₁	—	2	2	no	—
91	C₂×C₄×C₁₃₁	—	2	2	no	—
92	C₂×C₄×C₁₃₁	—	2	2	no	—
93	C₂×C₄×C₁₃₁	—	2	2	no	—
94	C₂×C₄×C₁₃₁	—	2	2	no	—
95	C₂×C₄×C₁₃₁	—	2	2	no	—
96	C₂×C₄×C₁₃₁	—	2	2	no	—
97	C₂×C₄×C₁₃₁	—	4	1	no	—
98	C₂×C₄×C₁₃₁	—	4	1	no	—
99	C₂×C₄×C₁₃₁	—	4	4	no	—
100	C₂×C₄×C₁₃₁	—	4	4	no	—
101	C₂×C₄×C₁₃₁	—	4	2	no	—
102	C₂×C₄×C₁₃₁	—	4	2	no	—
103	C₂×C₄×C₁₃₁	—	4	2	no	—
104	C₂×C₄×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
106	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₁₃₁	#131	1	4	yes	—
109	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₁₃₁	#137	1	1	yes	characteristic
112	C₈×C₁₃₁	—	4	4	no	—
113	C₈×C₁₃₁	—	4	4	no	—
114	C₂²×C₁₃₁	—	4	2	no	—
115	C₂²×C₁₃₁	—	4	2	no	—
116	C₂²×C₁₃₁	—	4	2	no	—
117	C₂²×C₁₃₁	—	8	2	no	—
118	C₂²×C₁₃₁	—	8	2	no	—
119	C₂²×C₁₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×C₁₃₁	#408	1	2	yes	—
121	C₂²×C₁₃₁	#417	1	1	yes	characteristic
122	C₂²×C₁₃₁	#418	1	1	yes	characteristic
123	C₂²×C₁₃₁	#421	1	2	yes	—
124	C₄×C₁₃₁	—	2	2	no	—
125	C₄×C₁₃₁	—	4	2	no	—
126	C₄×C₁₃₁	—	4	2	no	—
127	C₄×C₁₃₁	—	4	2	no	—
128	C₄×C₁₃₁	—	8	2	no	—
129	C₄×C₁₃₁	#405	1	2	yes	—
130	C₄×C₁₃₁	#433	1	2	yes	—
131	C₄×C₁₃₁	#440	1	2	yes	—
132	C₄×C₁₃₁	#442	1	2	yes	—
133	C₂×C₁₃₁	—	8	2	no	—
134	C₂×C₁₃₁	#1495	1	2	yes	—
135	C₂×C₁₃₁	#1496	1	2	yes	—
136	C₂×C₁₃₁	#1750	1	1	yes	characteristic
137	C₂×C₁₃₁	#1759	1	1	yes	characteristic
138	C₂×C₁₃₁	#1762	1	1	yes	characteristic
139	#8179	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₁₃₁	#8179	1	1	yes	characteristic, direct product factor, Sylow
141	#2293	C₂×C₁₃₁	1	1	yes	characteristic
142	#1767	C₂×C₁₃₁	1	1	yes	characteristic, complemented
143	#1744	C₂×C₁₃₁	1	1	yes	characteristic, complemented
144	C₂×D₄⋊C₈	C₂×C₁₃₁	1	2	yes	—
145	#1279	C₂×C₁₃₁	1	2	yes	complemented
146	C₂×C₄×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
147	C₂×#117	—	2	1	no	—
148	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
149	C₂×#97	—	2	1	no	—
150	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
151	#390	C₄×C₁₃₁	1	2	yes	—
152	#389	C₄×C₁₃₁	1	2	yes	—
153	#388	—	2	2	no	—
154	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
155	C₄×C₂²⋊C₄	—	2	1	no	—
156	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
157	#336	—	2	2	no	—
158	#336	—	2	1	no	—
159	D₄⋊C₈	C₄×C₁₃₁	1	8	yes	complemented
160	C₂²×D₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
161	C₂³×C₄	—	2	1	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₄×D₄	—	2	2	no	—
165	C₄×D₄	C₂×C₄×C₁₃₁	1	4	yes	—
166	#117	—	4	4	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
169	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	2	1	no	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₄²	—	2	1	no	—
174	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
175	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
176	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
177	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
178	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	C₂×C₄×C₁₃₁	1	4	yes	—
181	#97	—	4	2	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
184	#95	—	2	2	no	—
185	#95	—	2	1	no	—
186	C₂⁴	—	2	1	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	C₄²×C₁₃₁	1	2	yes	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	2	yes	—
197	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	4	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₄⋊C₄	C₄²×C₁₃₁	1	2	yes	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	4	4	no	—
212	C₂²⋊C₄	—	4	4	no	—
213	C₄²	—	2	4	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	4	2	no	—
218	C₄²	—	4	2	no	—
219	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
220	C₄²	#41×C₁₃₁	1	2	yes	—
221	C₂³	—	2	2	no	—
222	C₂³	—	4	2	no	—
223	C₂³	—	4	1	no	—
224	C₂³	—	4	1	no	—
225	C₂³	—	4	2	no	—
226	C₂³	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
227	D₄	—	2	4	no	—
228	D₄	—	4	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	C₂×C₄²×C₁₃₁	1	1	yes	characteristic
245	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
246	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
247	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
248	C₂×C₄	#131×C₁₃₁	1	4	yes	—
249	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
250	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
251	C₂×C₄	#137×C₁₃₁	1	1	yes	characteristic
252	C₈	—	4	4	no	—
253	C₈	—	4	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	C₄×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
260	C₂²	#408×C₁₃₁	1	2	yes	—
261	C₂²	#417×C₁₃₁	1	1	yes	characteristic
262	C₂²	#418×C₁₃₁	1	1	yes	characteristic
263	C₂²	#421×C₁₃₁	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	8	2	no	—
269	C₄	#405×C₁₃₁	1	2	yes	—
270	C₄	#433×C₁₃₁	1	2	yes	—
271	C₄	#440×C₁₃₁	1	2	yes	—
272	C₄	#442×C₁₃₁	1	2	yes	—
273	C₂	—	8	2	no	—
274	C₂	#1495×C₁₃₁	1	2	yes	—
275	C₂	#1496×C₁₃₁	1	2	yes	—
276	C₂	#1750×C₁₃₁	1	1	yes	characteristic
277	C₂	#1759×C₁₃₁	1	1	yes	characteristic
278	C₂	#1762×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
socle	C₂³×C₁₃₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₃₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1767×C₁₃₁	C₂	semidirect
3. #1744×C₁₃₁	C₂	semidirect
5. #1279×C₁₃₁	C₂	semidirect
19. D₄⋊C₈×C₁₃₁	C₄	semidirect
139. #8179	C₁₃₁	direct (N × H)
140. C₁₃₁	#8179	direct (N × H)
142. #1767	C₂×C₁₃₁	semidirect
143. #1744	C₂×C₁₃₁	semidirect
145. #1279	C₂×C₁₃₁	semidirect
159. D₄⋊C₈	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2293×C₁₃₁	2
#1767×C₁₃₁	2
#1744×C₁₃₁	2
C₂×D₄⋊C₈×C₁₃₁	2
#1279×C₁₃₁	2
#8179	131

Subgroups — C131×#8179 #45694063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#8179 #45694063