Subgroups — C₁₃₁×#7171 #45693055

Order 33536.

282 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₄×#107×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#335×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₁₃₁	—	2	1	no	—
5	C₂×#107×C₁₃₁	C₄	1	2	yes	—
6	C₂×C₄⋊C₈×C₁₃₁	—	2	1	no	—
7	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
8	C₄×C₄⋊C₄×C₁₃₁	—	2	1	no	—
9	C₄×C₁₆×C₁₃₁	—	2	1	no	—
10	C₄×C₁₆×C₁₃₁	C₄	1	2	yes	complemented
11	#340×C₁₃₁	—	2	1	no	—
12	#335×C₁₃₁	—	2	1	no	—
13	#335×C₁₃₁	C₄	1	2	yes	complemented
14	C₂²×C₈×C₁₃₁	—	4	1	no	—
15	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
18	C₂×C₄²×C₁₃₁	—	4	1	no	—
19	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₁₃₁	—	2	4	no	—
21	C₂×C₁₆×C₁₃₁	—	2	2	no	—
22	#107×C₁₃₁	—	2	2	no	—
23	#107×C₁₃₁	C₈	1	4	yes	complemented
24	C₄⋊C₈×C₁₃₁	—	2	2	no	—
25	C₄⋊C₈×C₁₃₁	—	4	1	no	—
26	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃₁	Q₈	1	1	yes	characteristic
30	#95×C₁₃₁	—	4	1	no	—
31	C₂²×C₄×C₁₃₁	—	4	2	no	—
32	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₃₁	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₁₃₁	—	2	2	no	—
36	C₂×C₈×C₁₃₁	—	4	2	no	—
37	C₂×C₈×C₁₃₁	—	8	1	no	—
38	C₂×C₈×C₁₃₁	—	8	1	no	—
39	C₂×C₈×C₁₃₁	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃₁	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₁₃₁	M₁₆	1	2	yes	—
45	C₄⋊C₄×C₁₃₁	—	2	4	no	—
46	C₄⋊C₄×C₁₃₁	—	4	2	no	—
47	C₄²×C₁₃₁	—	8	1	no	—
48	C₄²×C₁₃₁	—	8	1	no	—
49	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₄²×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₁₃₁	SD₁₆	1	1	yes	characteristic
52	C₄²×C₁₃₁	Q₁₆	1	1	yes	characteristic
53	C₁₆×C₁₃₁	—	2	4	no	—
54	C₂³×C₁₃₁	#102	1	1	yes	characteristic
55	C₂×C₄×C₁₃₁	—	2	2	no	—
56	C₂×C₄×C₁₃₁	—	4	4	no	—
57	C₂×C₄×C₁₃₁	—	8	2	no	—
58	C₂×C₄×C₁₃₁	—	8	2	no	—
59	C₂×C₄×C₁₃₁	#95	1	1	yes	characteristic
60	C₂×C₄×C₁₃₁	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
62	C₂×C₄×C₁₃₁	#103	1	1	yes	characteristic
63	C₂×C₄×C₁₃₁	#103	1	1	yes	characteristic
64	C₂×C₄×C₁₃₁	#104	1	2	yes	—
65	C₂×C₄×C₁₃₁	#106	1	1	yes	characteristic
66	C₂×C₄×C₁₃₁	#107	1	1	yes	characteristic
67	C₂×C₄×C₁₃₁	D₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₃₁	SD₃₂	1	1	yes	characteristic
69	C₂×C₄×C₁₃₁	SD₃₂	1	1	yes	characteristic
70	C₂×C₄×C₁₃₁	Q₃₂	1	1	yes	characteristic
71	C₈×C₁₃₁	—	2	1	no	—
72	C₈×C₁₃₁	—	2	1	no	—
73	C₈×C₁₃₁	—	8	2	no	—
74	C₈×C₁₃₁	C₄×C₈	1	1	yes	characteristic
75	C₈×C₁₃₁	#97	1	1	yes	characteristic
76	C₈×C₁₃₁	C₄⋊C₈	1	2	yes	—
77	C₂²×C₁₃₁	D₄⋊C₈	1	2	yes	—
78	C₂²×C₁₃₁	#340	1	1	yes	characteristic
79	C₂²×C₁₃₁	#357	1	1	yes	characteristic
80	C₂²×C₁₃₁	#358	1	1	yes	characteristic
81	C₂²×C₁₃₁	#359	1	2	yes	—
82	C₄×C₁₃₁	—	2	1	no	—
83	C₄×C₁₃₁	—	2	1	no	—
84	C₄×C₁₃₁	—	8	4	no	—
85	C₄×C₁₃₁	Q₈⋊C₈	1	2	yes	—
86	C₄×C₁₃₁	#336	1	1	yes	characteristic
87	C₄×C₁₃₁	#339	1	1	yes	characteristic
88	C₄×C₁₃₁	#357	1	1	yes	characteristic
89	C₄×C₁₃₁	#358	1	1	yes	characteristic
90	C₄×C₁₃₁	#366	1	1	yes	characteristic
91	C₄×C₁₃₁	#367	1	1	yes	characteristic
92	C₂×C₁₃₁	#1279	1	1	yes	characteristic
93	C₂×C₁₃₁	#1331	1	2	yes	—
94	C₂×C₁₃₁	#1332	1	2	yes	—
95	C₂×C₁₃₁	#1380	1	1	yes	characteristic
96	C₂×C₁₃₁	#1382	1	1	yes	characteristic
97	#7171	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₁₃₁	#7171	1	1	yes	characteristic, direct product factor, Sylow
99	C₂×C₄×C₁₆	C₂×C₁₃₁	1	1	yes	characteristic
100	C₄×#107	C₂×C₁₃₁	1	1	yes	characteristic
101	C₂×#335	C₂×C₁₃₁	1	1	yes	characteristic
102	C₂²×C₁₆	—	2	1	no	—
103	C₂×#107	C₄×C₁₃₁	1	2	yes	—
104	C₂×C₄⋊C₈	—	2	1	no	—
105	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
106	C₄×C₄⋊C₄	—	2	1	no	—
107	C₄×C₁₆	—	2	1	no	—
108	C₄×C₁₆	C₄×C₁₃₁	1	2	yes	complemented
109	#340	—	2	1	no	—
110	#335	—	2	1	no	—
111	#335	C₄×C₁₃₁	1	2	yes	complemented
112	C₂²×C₈	—	4	1	no	—
113	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
114	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄²	—	4	1	no	—
117	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
118	C₂×C₁₆	—	2	4	no	—
119	C₂×C₁₆	—	2	2	no	—
120	#107	—	2	2	no	—
121	#107	C₈×C₁₃₁	1	4	yes	complemented
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	4	1	no	—
124	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
125	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
126	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
127	C₄×C₈	Q₈×C₁₃₁	1	1	yes	characteristic
128	#95	—	4	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
131	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
132	C₂²×C₄	SD₁₆×C₁₃₁	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	8	1	no	—
136	C₂×C₈	—	8	1	no	—
137	C₂×C₈	C₄²×C₁₃₁	1	1	yes	characteristic
138	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
139	C₂×C₈	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
140	C₂×C₈	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
141	C₂×C₈	C₂×C₈×C₁₃₁	1	2	yes	—
142	C₂×C₈	M₁₆×C₁₃₁	1	2	yes	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄²	—	8	1	no	—
146	C₄²	—	8	1	no	—
147	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
148	C₄²	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
149	C₄²	SD₁₆×C₁₃₁	1	1	yes	characteristic
150	C₄²	Q₁₆×C₁₃₁	1	1	yes	characteristic
151	C₁₆	—	2	4	no	—
152	C₂³	#102×C₁₃₁	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	#95×C₁₃₁	1	1	yes	characteristic
158	C₂×C₄	C₂²⋊C₈×C₁₃₁	1	2	yes	—
159	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
160	C₂×C₄	#103×C₁₃₁	1	1	yes	characteristic
161	C₂×C₄	#103×C₁₃₁	1	1	yes	characteristic
162	C₂×C₄	#104×C₁₃₁	1	2	yes	—
163	C₂×C₄	#106×C₁₃₁	1	1	yes	characteristic
164	C₂×C₄	#107×C₁₃₁	1	1	yes	characteristic
165	C₂×C₄	D₁₆×C₁₃₁	1	1	yes	characteristic
166	C₂×C₄	SD₃₂×C₁₃₁	1	1	yes	characteristic
167	C₂×C₄	SD₃₂×C₁₃₁	1	1	yes	characteristic
168	C₂×C₄	Q₃₂×C₁₃₁	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	8	2	no	—
172	C₈	C₄×C₈×C₁₃₁	1	1	yes	characteristic
173	C₈	#97×C₁₃₁	1	1	yes	characteristic
174	C₈	C₄⋊C₈×C₁₃₁	1	2	yes	—
175	C₂²	D₄⋊C₈×C₁₃₁	1	2	yes	—
176	C₂²	#340×C₁₃₁	1	1	yes	characteristic
177	C₂²	#357×C₁₃₁	1	1	yes	characteristic
178	C₂²	#358×C₁₃₁	1	1	yes	characteristic
179	C₂²	#359×C₁₃₁	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	8	4	no	—
183	C₄	Q₈⋊C₈×C₁₃₁	1	2	yes	—
184	C₄	#336×C₁₃₁	1	1	yes	characteristic
185	C₄	#339×C₁₃₁	1	1	yes	characteristic
186	C₄	#357×C₁₃₁	1	1	yes	characteristic
187	C₄	#358×C₁₃₁	1	1	yes	characteristic
188	C₄	#366×C₁₃₁	1	1	yes	characteristic
189	C₄	#367×C₁₃₁	1	1	yes	characteristic
190	C₂	#1279×C₁₃₁	1	1	yes	characteristic
191	C₂	#1331×C₁₃₁	1	2	yes	—
192	C₂	#1332×C₁₃₁	1	2	yes	—
193	C₂	#1380×C₁₃₁	1	1	yes	characteristic
194	C₂	#1382×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	D₈
commutator	C₈	C₄×C₈×C₁₃₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₃₁
socle, prime radical	C₂³×C₁₃₁	#102
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₄×C₁₆×C₁₃₁	C₄	semidirect
13. #335×C₁₃₁	C₄	semidirect
23. #107×C₁₃₁	C₈	semidirect
97. #7171	C₁₃₁	direct (N × H)
98. C₁₃₁	#7171	direct (N × H)
108. C₄×C₁₆	C₄×C₁₃₁	semidirect
111. #335	C₄×C₁₃₁	semidirect
121. #107	C₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₆×C₁₃₁	2
C₄×#107×C₁₃₁	2
C₂×#335×C₁₃₁	2
#7171	131

Subgroups — C131×#7171 #45693055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#7171 #45693055