Subgroups — C₁₃₁×#7878 #45693762

Order 33536.

574 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#431×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1277×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
6	#432×C₁₃₁	—	2	2	no	—
7	#432×C₁₃₁	C₄	1	4	yes	—
8	#431×C₁₃₁	—	2	2	no	—
9	#431×C₁₃₁	—	2	2	no	—
10	#431×C₁₃₁	—	2	2	no	—
11	#431×C₁₃₁	C₄	1	4	yes	—
12	C₂×C₄⋊C₈×C₁₃₁	C₄	1	2	yes	—
13	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₁₃₁	—	2	1	no	—
16	C₂×C₂²⋊C₈×C₁₃₁	—	2	1	no	—
17	C₂×C₂²⋊C₈×C₁₃₁	C₄	1	2	yes	—
18	C₂×#97×C₁₃₁	—	2	1	no	—
19	C₂×C₄×C₈×C₁₃₁	—	2	1	no	—
20	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
21	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃₁	Q₈	1	1	yes	characteristic
24	C₂²×C₈×C₁₃₁	—	2	2	no	—
25	C₂²×C₈×C₁₃₁	—	2	2	no	—
26	C₂²×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
32	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
33	C₂×C₄²×C₁₃₁	Q₈	1	1	yes	characteristic
34	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₁₃₁	—	2	4	no	—
36	C₄⋊C₈×C₁₃₁	—	2	2	no	—
37	C₄⋊C₈×C₁₃₁	—	2	2	no	—
38	C₄⋊C₈×C₁₃₁	C₂×C₄	1	4	yes	—
39	C₄⋊C₈×C₁₃₁	C₂×C₄	1	4	yes	—
40	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
41	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
42	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
43	#97×C₁₃₁	—	2	4	no	—
44	C₄×C₈×C₁₃₁	—	2	4	no	—
45	C₄×C₈×C₁₃₁	—	2	2	no	—
46	C₄×C₈×C₁₃₁	—	2	2	no	—
47	C₂⁴×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₁	—	2	4	no	—
49	C₂²×C₄×C₁₃₁	—	2	4	no	—
50	C₂²×C₄×C₁₃₁	—	4	2	no	—
51	C₂²×C₄×C₁₃₁	C₄²	1	2	yes	—
52	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₁₃₁	—	2	4	no	—
64	C₂×C₈×C₁₃₁	—	2	4	no	—
65	C₂×C₈×C₁₃₁	—	2	4	no	—
66	C₂×C₈×C₁₃₁	—	2	8	no	—
67	C₂×C₈×C₁₃₁	—	4	4	no	—
68	C₂×C₈×C₁₃₁	—	4	4	no	—
69	C₄²×C₁₃₁	C₄²	1	2	yes	—
70	C₄²×C₁₃₁	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₁₃₁	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₁₃₁	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₁₃₁	C₄⋊C₄	1	2	yes	—
74	C₄²×C₁₃₁	C₄⋊C₄	1	2	yes	—
75	C₄²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂³×C₁₃₁	—	2	2	no	—
80	C₂³×C₁₃₁	—	2	2	no	—
81	C₂³×C₁₃₁	—	2	2	no	—
82	C₂³×C₁₃₁	#95	1	2	yes	—
83	C₂³×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃₁	—	2	2	no	—
85	C₂×C₄×C₁₃₁	—	2	2	no	—
86	C₂×C₄×C₁₃₁	—	2	2	no	—
87	C₂×C₄×C₁₃₁	—	2	2	no	—
88	C₂×C₄×C₁₃₁	—	2	4	no	—
89	C₂×C₄×C₁₃₁	—	2	4	no	—
90	C₂×C₄×C₁₃₁	—	2	4	no	—
91	C₂×C₄×C₁₃₁	—	2	4	no	—
92	C₂×C₄×C₁₃₁	—	4	2	no	—
93	C₂×C₄×C₁₃₁	—	4	2	no	—
94	C₂×C₄×C₁₃₁	#95	1	2	yes	—
95	C₂×C₄×C₁₃₁	#95	1	2	yes	—
96	C₂×C₄×C₁₃₁	#95	1	2	yes	—
97	C₂×C₄×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃₁	#131	1	4	yes	—
104	C₂×C₄×C₁₃₁	#135	1	4	yes	—
105	C₈×C₁₃₁	—	4	4	no	—
106	C₈×C₁₃₁	—	4	4	no	—
107	C₈×C₁₃₁	—	4	4	no	—
108	C₂²×C₁₃₁	—	2	2	no	—
109	C₂²×C₁₃₁	—	2	2	no	—
110	C₂²×C₁₃₁	—	2	2	no	—
111	C₂²×C₁₃₁	—	4	4	no	—
112	C₂²×C₁₃₁	C₂×#95	1	1	yes	characteristic
113	C₂²×C₁₃₁	#424	1	2	yes	—
114	C₂²×C₁₃₁	#427	1	2	yes	—
115	C₂²×C₁₃₁	#430	1	2	yes	—
116	C₄×C₁₃₁	—	2	2	no	—
117	C₄×C₁₃₁	—	2	2	no	—
118	C₄×C₁₃₁	—	2	2	no	—
119	C₄×C₁₃₁	—	2	1	no	—
120	C₄×C₁₃₁	—	2	1	no	—
121	C₄×C₁₃₁	#405	1	2	yes	—
122	C₄×C₁₃₁	#408	1	2	yes	—
123	C₄×C₁₃₁	#416	1	2	yes	—
124	C₄×C₁₃₁	#416	1	2	yes	—
125	C₂×C₁₃₁	—	4	2	no	—
126	C₂×C₁₃₁	#1572	1	4	yes	—
127	C₂×C₁₃₁	#1729	1	1	yes	characteristic
128	C₂×C₁₃₁	#1735	1	1	yes	characteristic
129	C₂×C₁₃₁	#1738	1	1	yes	characteristic
130	#7878	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₁₃₁	#7878	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×#432	C₂×C₁₃₁	1	1	yes	characteristic
133	C₂×#431	C₂×C₁₃₁	1	1	yes	characteristic
134	C₂×#431	C₂×C₁₃₁	1	1	yes	characteristic
135	#1277	C₂×C₁₃₁	1	4	yes	complemented
136	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
137	#432	—	2	2	no	—
138	#432	C₄×C₁₃₁	1	4	yes	—
139	#431	—	2	2	no	—
140	#431	—	2	2	no	—
141	#431	—	2	2	no	—
142	#431	C₄×C₁₃₁	1	4	yes	—
143	C₂×C₄⋊C₈	C₄×C₁₃₁	1	2	yes	—
144	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
145	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
146	C₂×C₂²⋊C₈	—	2	1	no	—
147	C₂×C₂²⋊C₈	—	2	1	no	—
148	C₂×C₂²⋊C₈	C₄×C₁₃₁	1	2	yes	—
149	C₂×#97	—	2	1	no	—
150	C₂×C₄×C₈	—	2	1	no	—
151	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
152	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
153	C₂³×C₄	D₄×C₁₃₁	1	1	yes	characteristic
154	C₂³×C₄	Q₈×C₁₃₁	1	1	yes	characteristic
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
158	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
159	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
160	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
161	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
162	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
163	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
164	C₂×C₄²	Q₈×C₁₃₁	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	C₂×C₄×C₁₃₁	1	4	yes	—
170	C₄⋊C₈	C₂×C₄×C₁₃₁	1	4	yes	—
171	C₂²⋊C₈	—	2	4	no	—
172	C₂²⋊C₈	—	2	4	no	—
173	C₂²⋊C₈	—	2	4	no	—
174	#97	—	2	4	no	—
175	C₄×C₈	—	2	4	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	2	2	no	—
178	C₂⁴	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₄²×C₁₃₁	1	2	yes	—
183	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	2	yes	—
184	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	2	yes	—
185	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	2	yes	—
186	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
187	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
189	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
190	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
191	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
193	C₂²×C₄	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	—	4	4	no	—
199	C₂×C₈	—	4	4	no	—
200	C₄²	C₄²×C₁₃₁	1	2	yes	—
201	C₄²	C₂²⋊C₄×C₁₃₁	1	2	yes	—
202	C₄²	C₂²⋊C₄×C₁₃₁	1	2	yes	—
203	C₄²	C₂²⋊C₄×C₁₃₁	1	2	yes	—
204	C₄²	C₄⋊C₄×C₁₃₁	1	2	yes	—
205	C₄²	C₄⋊C₄×C₁₃₁	1	2	yes	—
206	C₄²	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
207	C₄²	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
208	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
209	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	#95×C₁₃₁	1	2	yes	—
214	C₂³	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	#95×C₁₃₁	1	2	yes	—
226	C₂×C₄	#95×C₁₃₁	1	2	yes	—
227	C₂×C₄	#95×C₁₃₁	1	2	yes	—
228	C₂×C₄	C₂×C₄²×C₁₃₁	1	1	yes	characteristic
229	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
230	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
231	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
234	C₂×C₄	#131×C₁₃₁	1	4	yes	—
235	C₂×C₄	#135×C₁₃₁	1	4	yes	—
236	C₈	—	4	4	no	—
237	C₈	—	4	4	no	—
238	C₈	—	4	4	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	4	no	—
243	C₂²	C₂×#95×C₁₃₁	1	1	yes	characteristic
244	C₂²	#424×C₁₃₁	1	2	yes	—
245	C₂²	#427×C₁₃₁	1	2	yes	—
246	C₂²	#430×C₁₃₁	1	2	yes	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	#405×C₁₃₁	1	2	yes	—
253	C₄	#408×C₁₃₁	1	2	yes	—
254	C₄	#416×C₁₃₁	1	2	yes	—
255	C₄	#416×C₁₃₁	1	2	yes	—
256	C₂	—	4	2	no	—
257	C₂	#1572×C₁₃₁	1	4	yes	—
258	C₂	#1729×C₁₃₁	1	1	yes	characteristic
259	C₂	#1735×C₁₃₁	1	1	yes	characteristic
260	C₂	#1738×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×C₄⋊C₄
prime radical	C₂⁴×C₁₃₁	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂²×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. #1277×C₁₃₁	C₂	semidirect
130. #7878	C₁₃₁	direct (N × H)
131. C₁₃₁	#7878	direct (N × H)
135. #1277	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×#432×C₁₃₁	2
C₂×#431×C₁₃₁	2
#1277×C₁₃₁	2
#7878	131

Subgroups — C131×#7878 #45693762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#7878 #45693762