Subgroups — C₁₃₁×#8279 #45694163

Order 33536.

394 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1767×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1744×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#335×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#334×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	#1277×C₁₃₁	C₂	1	1	yes	maximal, characteristic
7	#1277×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×#97×C₁₃₁	—	2	1	no	—
13	C₂×C₄×C₈×C₁₃₁	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
17	#336×C₁₃₁	—	2	1	no	—
18	#336×C₁₃₁	—	2	1	no	—
19	#336×C₁₃₁	C₄	1	4	yes	—
20	#335×C₁₃₁	C₄	1	4	yes	complemented
21	#334×C₁₃₁	C₄	1	4	yes	complemented
22	C₂²×C₈×C₁₃₁	—	2	1	no	—
23	C₂²×C₈×C₁₃₁	—	2	1	no	—
24	C₂²×C₈×C₁₃₁	—	2	1	no	—
25	C₂²×C₈×C₁₃₁	—	2	1	no	—
26	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃₁	Q₈	1	2	yes	—
32	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₁₃₁	—	2	1	no	—
34	C₄⋊C₈×C₁₃₁	—	2	1	no	—
35	C₄⋊C₈×C₁₃₁	—	2	1	no	—
36	C₄⋊C₈×C₁₃₁	—	2	1	no	—
37	C₄⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
38	C₄⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
39	C₄⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
40	C₄⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
41	#97×C₁₃₁	—	4	2	no	—
42	C₄×C₈×C₁₃₁	—	2	1	no	—
43	C₄×C₈×C₁₃₁	—	2	1	no	—
44	C₄×C₈×C₁₃₁	—	4	2	no	—
45	C₄×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
47	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₃₁	—	2	4	no	—
53	C₂×C₈×C₁₃₁	—	2	2	no	—
54	C₂×C₈×C₁₃₁	—	2	2	no	—
55	C₂×C₈×C₁₃₁	—	2	2	no	—
56	C₂×C₈×C₁₃₁	—	2	2	no	—
57	C₂×C₈×C₁₃₁	—	2	2	no	—
58	C₂×C₈×C₁₃₁	—	2	4	no	—
59	C₂×C₈×C₁₃₁	—	4	1	no	—
60	C₂×C₈×C₁₃₁	—	4	1	no	—
61	C₂×C₈×C₁₃₁	—	4	1	no	—
62	C₂×C₈×C₁₃₁	—	4	1	no	—
63	C₂×C₈×C₁₃₁	—	4	1	no	—
64	C₂×C₈×C₁₃₁	—	4	1	no	—
65	C₂×C₈×C₁₃₁	—	4	1	no	—
66	C₂×C₈×C₁₃₁	—	4	1	no	—
67	C₂×C₈×C₁₃₁	C₄²	1	4	yes	—
68	C₄²×C₁₃₁	—	2	4	no	—
69	C₄²×C₁₃₁	—	2	4	no	—
70	C₄²×C₁₃₁	—	2	1	no	—
71	C₄²×C₁₃₁	—	2	1	no	—
72	C₄²×C₁₃₁	—	2	1	no	—
73	C₄²×C₁₃₁	—	2	1	no	—
74	C₄²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₁₃₁	#41	1	1	yes	characteristic
77	C₄²×C₁₃₁	#41	1	1	yes	characteristic
78	C₂³×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃₁	—	2	2	no	—
80	C₂×C₄×C₁₃₁	—	2	2	no	—
81	C₂×C₄×C₁₃₁	—	2	2	no	—
82	C₂×C₄×C₁₃₁	—	2	2	no	—
83	C₂×C₄×C₁₃₁	—	2	2	no	—
84	C₂×C₄×C₁₃₁	—	2	2	no	—
85	C₂×C₄×C₁₃₁	#108	1	4	yes	—
86	C₂×C₄×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
88	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃₁	C₄×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃₁	C₄×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃₁	#131	1	4	yes	—
93	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₁₃₁	#137	1	1	yes	characteristic
96	C₂×C₄×C₁₃₁	#137	1	1	yes	characteristic
97	C₈×C₁₃₁	—	2	4	no	—
98	C₈×C₁₃₁	—	4	2	no	—
99	C₈×C₁₃₁	—	4	2	no	—
100	C₈×C₁₃₁	—	4	2	no	—
101	C₈×C₁₃₁	—	4	2	no	—
102	C₈×C₁₃₁	—	4	2	no	—
103	C₂²×C₁₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₁₃₁	#424	1	2	yes	—
105	C₂²×C₁₃₁	#428	1	1	yes	characteristic
106	C₂²×C₁₃₁	#428	1	1	yes	characteristic
107	C₂²×C₁₃₁	C₂×#108	1	2	yes	—
108	C₄×C₁₃₁	—	2	4	no	—
109	C₄×C₁₃₁	—	2	1	no	—
110	C₄×C₁₃₁	—	2	1	no	—
111	C₄×C₁₃₁	—	4	2	no	—
112	C₄×C₁₃₁	#405	1	2	yes	—
113	C₄×C₁₃₁	#433	1	2	yes	—
114	C₄×C₁₃₁	#440	1	1	yes	characteristic
115	C₄×C₁₃₁	#440	1	1	yes	characteristic
116	C₄×C₁₃₁	#441	1	1	yes	characteristic
117	C₄×C₁₃₁	#442	1	1	yes	characteristic
118	C₂×C₁₃₁	#1573	1	2	yes	—
119	C₂×C₁₃₁	#1574	1	2	yes	—
120	C₂×C₁₃₁	#1768	1	1	yes	characteristic
121	C₂×C₁₃₁	#1776	1	1	yes	characteristic
122	C₂×C₁₃₁	C₄×#108	1	1	yes	characteristic
123	#8279	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₁₃₁	#8279	1	1	yes	characteristic, direct product factor, Sylow
125	#1767	C₂×C₁₃₁	1	1	yes	characteristic
126	C₄×C₄⋊C₈	C₂×C₁₃₁	1	1	yes	characteristic
127	#1744	C₂×C₁₃₁	1	1	yes	characteristic
128	C₂×#335	C₂×C₁₃₁	1	1	yes	characteristic
129	C₂×#334	C₂×C₁₃₁	1	1	yes	characteristic
130	#1277	C₂×C₁₃₁	1	1	yes	characteristic
131	#1277	C₂×C₁₃₁	1	1	yes	characteristic
132	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
133	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
134	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
135	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
136	C₂×#97	—	2	1	no	—
137	C₂×C₄×C₈	—	2	1	no	—
138	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
140	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
141	#336	—	2	1	no	—
142	#336	—	2	1	no	—
143	#336	C₄×C₁₃₁	1	4	yes	—
144	#335	C₄×C₁₃₁	1	4	yes	complemented
145	#334	C₄×C₁₃₁	1	4	yes	complemented
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
151	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
152	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
153	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
154	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
155	C₂×C₄²	Q₈×C₁₃₁	1	2	yes	—
156	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
157	C₄⋊C₈	—	2	1	no	—
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
162	C₄⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
163	C₄⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
164	C₄⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
165	#97	—	4	2	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	—	4	2	no	—
169	C₄×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
170	C₄×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
171	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
172	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
173	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
174	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
175	C₂²×C₄	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	—	4	1	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	C₄²×C₁₃₁	1	4	yes	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
199	C₄²	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
200	C₄²	#41×C₁₃₁	1	1	yes	characteristic
201	C₄²	#41×C₁₃₁	1	1	yes	characteristic
202	C₂³	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	#108×C₁₃₁	1	4	yes	—
210	C₂×C₄	C₂×C₄²×C₁₃₁	1	1	yes	characteristic
211	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
212	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
213	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
214	C₂×C₄	C₄×Q₈×C₁₃₁	1	1	yes	characteristic
215	C₂×C₄	C₄×Q₈×C₁₃₁	1	1	yes	characteristic
216	C₂×C₄	#131×C₁₃₁	1	4	yes	—
217	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
218	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
219	C₂×C₄	#137×C₁₃₁	1	1	yes	characteristic
220	C₂×C₄	#137×C₁₃₁	1	1	yes	characteristic
221	C₈	—	2	4	no	—
222	C₈	—	4	2	no	—
223	C₈	—	4	2	no	—
224	C₈	—	4	2	no	—
225	C₈	—	4	2	no	—
226	C₈	—	4	2	no	—
227	C₂²	C₄×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
228	C₂²	#424×C₁₃₁	1	2	yes	—
229	C₂²	#428×C₁₃₁	1	1	yes	characteristic
230	C₂²	#428×C₁₃₁	1	1	yes	characteristic
231	C₂²	C₂×#108×C₁₃₁	1	2	yes	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	#405×C₁₃₁	1	2	yes	—
237	C₄	#433×C₁₃₁	1	2	yes	—
238	C₄	#440×C₁₃₁	1	1	yes	characteristic
239	C₄	#440×C₁₃₁	1	1	yes	characteristic
240	C₄	#441×C₁₃₁	1	1	yes	characteristic
241	C₄	#442×C₁₃₁	1	1	yes	characteristic
242	C₂	#1573×C₁₃₁	1	2	yes	—
243	C₂	#1574×C₁₃₁	1	2	yes	—
244	C₂	#1768×C₁₃₁	1	1	yes	characteristic
245	C₂	#1776×C₁₃₁	1	1	yes	characteristic
246	C₂	C₄×#108×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, prime radical	C₂³×C₁₃₁	C₂×C₄⋊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
20. #335×C₁₃₁	C₄	semidirect
21. #334×C₁₃₁	C₄	semidirect
123. #8279	C₁₃₁	direct (N × H)
124. C₁₃₁	#8279	direct (N × H)
144. #335	C₄×C₁₃₁	semidirect
145. #334	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#1767×C₁₃₁	2
C₄×C₄⋊C₈×C₁₃₁	2
#1744×C₁₃₁	2
C₂×#335×C₁₃₁	2
C₂×#334×C₁₃₁	2
#1277×C₁₃₁	2
#8279	131

Subgroups — C131×#8279 #45694163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#8279 #45694163