Subgroups — C₁₃₁×#13275 #45699159

Order 33536.

398 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×#500×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₄×#107×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1380×C₁₃₁	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₁₃₁	—	2	1	no	—
6	C₂²×C₁₆×C₁₃₁	C₂²	1	2	yes	—
7	#500×C₁₃₁	—	2	2	no	—
8	#500×C₁₃₁	C₄	1	4	yes	complemented
9	C₂×#107×C₁₃₁	—	2	1	no	—
10	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
11	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₁₃₁	—	2	1	no	—
14	C₄⋊C₁₆×C₁₃₁	—	2	2	no	—
15	#340×C₁₃₁	—	2	1	no	—
16	C₂²×Q₈×C₁₃₁	—	4	1	no	—
17	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
20	C₄⋊Q₈×C₁₃₁	—	4	2	no	—
21	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₃₁	—	4	1	no	—
24	C₂×C₄²×C₁₃₁	—	4	1	no	—
25	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₁₃₁	—	2	2	no	—
27	C₂×C₁₆×C₁₃₁	—	2	4	no	—
28	#107×C₁₃₁	—	2	4	no	—
29	#107×C₁₃₁	—	2	2	no	—
30	#107×C₁₃₁	—	2	2	no	—
31	#107×C₁₃₁	C₂×C₄	1	4	yes	—
32	#107×C₁₃₁	Q₈	1	4	yes	complemented
33	C₄×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
35	#95×C₁₃₁	—	4	1	no	—
36	C₂×Q₈×C₁₃₁	—	4	4	no	—
37	C₂×Q₈×C₁₃₁	—	8	2	no	—
38	C₂×Q₈×C₁₃₁	—	8	2	no	—
39	C₂²×C₄×C₁₃₁	—	4	2	no	—
40	C₂²×C₄×C₁₃₁	—	4	2	no	—
41	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃₁	—	2	2	no	—
45	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₁₃₁	#41	1	2	yes	—
50	C₄⋊C₄×C₁₃₁	—	2	4	no	—
51	C₄⋊C₄×C₁₃₁	—	2	4	no	—
52	C₄⋊C₄×C₁₃₁	—	4	4	no	—
53	C₄⋊C₄×C₁₃₁	—	4	2	no	—
54	C₄⋊C₄×C₁₃₁	—	4	2	no	—
55	C₄²×C₁₃₁	—	8	2	no	—
56	C₄²×C₁₃₁	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₁₃₁	SD₁₆	1	2	yes	—
58	C₁₆×C₁₃₁	—	4	2	no	—
59	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
60	Q₈×C₁₃₁	—	8	8	no	—
61	C₂×C₄×C₁₃₁	—	2	2	no	—
62	C₂×C₄×C₁₃₁	—	4	4	no	—
63	C₂×C₄×C₁₃₁	—	4	4	no	—
64	C₂×C₄×C₁₃₁	—	8	2	no	—
65	C₂×C₄×C₁₃₁	—	8	2	no	—
66	C₂×C₄×C₁₃₁	—	8	2	no	—
67	C₂×C₄×C₁₃₁	—	8	2	no	—
68	C₂×C₄×C₁₃₁	#102	1	2	yes	—
69	C₂×C₄×C₁₃₁	#102	1	2	yes	—
70	C₂×C₄×C₁₃₁	D₁₆	1	2	yes	—
71	C₂×C₄×C₁₃₁	SD₃₂	1	2	yes	—
72	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₃₁	#137	1	2	yes	—
76	C₈×C₁₃₁	—	2	2	no	—
77	C₈×C₁₃₁	C₄×Q₈	1	2	yes	—
78	C₈×C₁₃₁	#124	1	1	yes	characteristic
79	C₈×C₁₃₁	C₄⋊Q₈	1	1	yes	characteristic
80	C₂²×C₁₃₁	C₂×#102	1	1	yes	characteristic
81	C₂²×C₁₃₁	#474	1	2	yes	—
82	C₂²×C₁₃₁	C₂×D₁₆	1	1	yes	characteristic
83	C₂²×C₁₃₁	C₂×SD₃₂	1	1	yes	characteristic
84	C₂²×C₁₃₁	#510	1	2	yes	—
85	C₄×C₁₃₁	—	2	2	no	—
86	C₄×C₁₃₁	—	8	4	no	—
87	C₄×C₁₃₁	—	8	4	no	—
88	C₄×C₁₃₁	#357	1	4	yes	—
89	C₄×C₁₃₁	#391	1	1	yes	characteristic
90	C₄×C₁₃₁	#419	1	1	yes	characteristic
91	C₄×C₁₃₁	#475	1	2	yes	—
92	C₂×C₁₃₁	#1988	1	1	yes	characteristic
93	C₂×C₁₃₁	C₂×#357	1	1	yes	characteristic
94	C₂×C₁₃₁	#2141	1	1	yes	characteristic
95	C₂×C₁₃₁	#2224	1	2	yes	—
96	C₂×C₁₃₁	#2225	1	2	yes	—
97	#13275	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₁₃₁	#13275	1	1	yes	characteristic, direct product factor, Sylow
99	C₂×#500	C₂×C₁₃₁	1	1	yes	characteristic
100	C₂×C₄⋊C₁₆	C₂×C₁₃₁	1	1	yes	characteristic
101	C₄×#107	C₂×C₁₃₁	1	1	yes	characteristic
102	#1380	C₂×C₁₃₁	1	4	yes	—
103	C₂×C₄⋊Q₈	—	2	1	no	—
104	C₂²×C₁₆	C₂²×C₁₃₁	1	2	yes	—
105	#500	—	2	2	no	—
106	#500	C₄×C₁₃₁	1	4	yes	complemented
107	C₂×#107	—	2	1	no	—
108	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
109	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
110	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
111	C₄×C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₁₆	—	2	2	no	—
113	#340	—	2	1	no	—
114	C₂²×Q₈	—	4	1	no	—
115	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
116	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
117	C₄⋊Q₈	—	2	4	no	—
118	C₄⋊Q₈	—	4	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	4	1	no	—
122	C₂×C₄²	—	4	1	no	—
123	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	2	4	no	—
126	#107	—	2	4	no	—
127	#107	—	2	2	no	—
128	#107	—	2	2	no	—
129	#107	C₂×C₄×C₁₃₁	1	4	yes	—
130	#107	Q₈×C₁₃₁	1	4	yes	complemented
131	C₄×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
132	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
133	#95	—	4	1	no	—
134	C₂×Q₈	—	4	4	no	—
135	C₂×Q₈	—	8	2	no	—
136	C₂×Q₈	—	8	2	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
140	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	2	yes	—
144	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
145	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
146	C₂×C₈	C₂×Q₈×C₁₃₁	1	2	yes	—
147	C₂×C₈	#41×C₁₃₁	1	2	yes	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄²	—	8	2	no	—
154	C₄²	C₂²⋊C₄×C₁₃₁	1	2	yes	—
155	C₄²	SD₁₆×C₁₃₁	1	2	yes	—
156	C₁₆	—	4	2	no	—
157	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
158	Q₈	—	8	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	#102×C₁₃₁	1	2	yes	—
167	C₂×C₄	#102×C₁₃₁	1	2	yes	—
168	C₂×C₄	D₁₆×C₁₃₁	1	2	yes	—
169	C₂×C₄	SD₃₂×C₁₃₁	1	2	yes	—
170	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
171	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
172	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
173	C₂×C₄	#137×C₁₃₁	1	2	yes	—
174	C₈	—	2	2	no	—
175	C₈	C₄×Q₈×C₁₃₁	1	2	yes	—
176	C₈	#124×C₁₃₁	1	1	yes	characteristic
177	C₈	C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
178	C₂²	C₂×#102×C₁₃₁	1	1	yes	characteristic
179	C₂²	#474×C₁₃₁	1	2	yes	—
180	C₂²	C₂×D₁₆×C₁₃₁	1	1	yes	characteristic
181	C₂²	C₂×SD₃₂×C₁₃₁	1	1	yes	characteristic
182	C₂²	#510×C₁₃₁	1	2	yes	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	—	8	4	no	—
186	C₄	#357×C₁₃₁	1	4	yes	—
187	C₄	#391×C₁₃₁	1	1	yes	characteristic
188	C₄	#419×C₁₃₁	1	1	yes	characteristic
189	C₄	#475×C₁₃₁	1	2	yes	—
190	C₂	#1988×C₁₃₁	1	1	yes	characteristic
191	C₂	C₂×#357×C₁₃₁	1	1	yes	characteristic
192	C₂	#2141×C₁₃₁	1	1	yes	characteristic
193	C₂	#2224×C₁₃₁	1	2	yes	—
194	C₂	#2225×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
8. #500×C₁₃₁	C₄	semidirect
32. #107×C₁₃₁	Q₈	semidirect
97. #13275	C₁₃₁	direct (N × H)
98. C₁₃₁	#13275	direct (N × H)
106. #500	C₄×C₁₃₁	semidirect
130. #107	Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₃₁	2
C₂×C₄⋊C₁₆×C₁₃₁	2
C₄×#107×C₁₃₁	2
#1380×C₁₃₁	2
#13275	131

Subgroups — C131×#13275 #45699159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13275 #45699159