Subgroups — C₁₃₁×#7275 #45693159

Order 33536.

246 conjugacy classes of subgroups.

The subgroup lattice has 198 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	#1846×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#334×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₁₃₁	—	2	1	no	—
5	C₂×#107×C₁₃₁	C₄	1	2	yes	—
6	C₂×#106×C₁₃₁	—	2	1	no	—
7	C₂×C₄⋊C₈×C₁₃₁	—	2	1	no	—
8	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
9	#382×C₁₃₁	—	2	1	no	—
10	C₄⋊C₁₆×C₁₃₁	—	2	1	no	—
11	C₄⋊C₁₆×C₁₃₁	C₄	1	2	yes	complemented
12	#334×C₁₃₁	—	2	1	no	—
13	#334×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	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₁₃₁	—	2	2	no	—
20	C₂×C₁₆×C₁₃₁	—	4	1	no	—
21	C₂×C₁₆×C₁₃₁	—	4	1	no	—
22	#107×C₁₃₁	—	4	2	no	—
23	#106×C₁₃₁	—	4	1	no	—
24	#106×C₁₃₁	—	4	1	no	—
25	C₄⋊C₈×C₁₃₁	—	2	2	no	—
26	C₄⋊C₈×C₁₃₁	—	4	1	no	—
27	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃₁	Q₈	1	1	yes	characteristic
31	#95×C₁₃₁	—	4	1	no	—
32	#95×C₁₃₁	—	4	1	no	—
33	C₂²×C₄×C₁₃₁	—	4	2	no	—
34	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₃₁	SD₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₃₁	—	2	1	no	—
38	C₂×C₈×C₁₃₁	—	2	1	no	—
39	C₂×C₈×C₁₃₁	—	4	2	no	—
40	C₂×C₈×C₁₃₁	—	8	2	no	—
41	C₂×C₈×C₁₃₁	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃₁	D₈	1	1	yes	characteristic
46	C₂×C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₁₃₁	Q₁₆	1	1	yes	characteristic
49	C₄⋊C₄×C₁₃₁	—	4	2	no	—
50	C₄⋊C₄×C₁₃₁	—	4	2	no	—
51	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₄²×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₁₃₁	SD₁₆	1	1	yes	characteristic
54	C₄²×C₁₃₁	Q₁₆	1	1	yes	characteristic
55	C₁₆×C₁₃₁	—	4	2	no	—
56	C₂³×C₁₃₁	#102	1	1	yes	characteristic
57	C₂×C₄×C₁₃₁	—	2	2	no	—
58	C₂×C₄×C₁₃₁	—	2	2	no	—
59	C₂×C₄×C₁₃₁	—	8	2	no	—
60	C₂×C₄×C₁₃₁	—	8	2	no	—
61	C₂×C₄×C₁₃₁	—	8	2	no	—
62	C₂×C₄×C₁₃₁	#95	1	1	yes	characteristic
63	C₂×C₄×C₁₃₁	#100	1	1	yes	characteristic
64	C₂×C₄×C₁₃₁	#101	1	1	yes	characteristic
65	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
66	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
67	C₂×C₄×C₁₃₁	#103	1	1	yes	characteristic
68	C₂×C₄×C₁₃₁	#103	1	1	yes	characteristic
69	C₂×C₄×C₁₃₁	#103	1	1	yes	characteristic
70	C₂×C₄×C₁₃₁	#106	1	1	yes	characteristic
71	C₂×C₄×C₁₃₁	#107	1	1	yes	characteristic
72	C₈×C₁₃₁	—	2	1	no	—
73	C₈×C₁₃₁	—	2	1	no	—
74	C₈×C₁₃₁	—	8	2	no	—
75	C₈×C₁₃₁	#102	1	1	yes	characteristic
76	C₈×C₁₃₁	#103	1	1	yes	characteristic
77	C₈×C₁₃₁	#106	1	1	yes	characteristic
78	C₈×C₁₃₁	#107	1	1	yes	characteristic
79	C₂²×C₁₃₁	#331	1	1	yes	characteristic
80	C₂²×C₁₃₁	#332	1	1	yes	characteristic
81	C₂²×C₁₃₁	#340	1	1	yes	characteristic
82	C₂²×C₁₃₁	#359	1	2	yes	—
83	C₂²×C₁₃₁	#360	1	2	yes	—
84	C₄×C₁₃₁	—	2	1	no	—
85	C₄×C₁₃₁	—	2	1	no	—
86	C₄×C₁₃₁	—	2	1	no	—
87	C₄×C₁₃₁	—	2	1	no	—
88	C₄×C₁₃₁	—	16	2	no	—
89	C₄×C₁₃₁	#332	1	1	yes	characteristic
90	C₄×C₁₃₁	#333	1	1	yes	characteristic
91	C₄×C₁₃₁	#340	1	1	yes	characteristic
92	C₄×C₁₃₁	#343	1	1	yes	characteristic
93	C₂×C₁₃₁	#1296	1	1	yes	characteristic
94	C₂×C₁₃₁	#1360	1	2	yes	—
95	C₂×C₁₃₁	#1363	1	2	yes	—
96	C₂×C₁₃₁	#1382	1	1	yes	characteristic
97	C₂×C₁₃₁	#1386	1	1	yes	characteristic
98	#7275	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₁₃₁	#7275	1	1	yes	characteristic, direct product factor, Sylow
100	C₂×C₄⋊C₁₆	C₂×C₁₃₁	1	1	yes	characteristic
101	#1846	C₂×C₁₃₁	1	1	yes	characteristic
102	C₂×#334	C₂×C₁₃₁	1	1	yes	characteristic
103	C₂²×C₁₆	—	2	1	no	—
104	C₂×#107	C₄×C₁₃₁	1	2	yes	—
105	C₂×#106	—	2	1	no	—
106	C₂×C₄⋊C₈	—	2	1	no	—
107	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
108	#382	—	2	1	no	—
109	C₄⋊C₁₆	—	2	1	no	—
110	C₄⋊C₁₆	C₄×C₁₃₁	1	2	yes	complemented
111	#334	—	2	1	no	—
112	#334	C₄×C₁₃₁	1	2	yes	complemented
113	C₂²×C₈	—	4	1	no	—
114	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
115	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	—	4	1	no	—
120	C₂×C₁₆	—	4	1	no	—
121	#107	—	4	2	no	—
122	#106	—	4	1	no	—
123	#106	—	4	1	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	4	1	no	—
126	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
127	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
128	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
129	C₄×C₈	Q₈×C₁₃₁	1	1	yes	characteristic
130	#95	—	4	1	no	—
131	#95	—	4	1	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
134	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
135	C₂²×C₄	SD₁₆×C₁₃₁	1	1	yes	characteristic
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	8	2	no	—
140	C₂×C₈	C₄²×C₁₃₁	1	1	yes	characteristic
141	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
142	C₂×C₈	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
143	C₂×C₈	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
144	C₂×C₈	D₈×C₁₃₁	1	1	yes	characteristic
145	C₂×C₈	SD₁₆×C₁₃₁	1	1	yes	characteristic
146	C₂×C₈	SD₁₆×C₁₃₁	1	1	yes	characteristic
147	C₂×C₈	Q₁₆×C₁₃₁	1	1	yes	characteristic
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
151	C₄²	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
152	C₄²	SD₁₆×C₁₃₁	1	1	yes	characteristic
153	C₄²	Q₁₆×C₁₃₁	1	1	yes	characteristic
154	C₁₆	—	4	2	no	—
155	C₂³	#102×C₁₃₁	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	#95×C₁₃₁	1	1	yes	characteristic
162	C₂×C₄	#100×C₁₃₁	1	1	yes	characteristic
163	C₂×C₄	#101×C₁₃₁	1	1	yes	characteristic
164	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
165	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
166	C₂×C₄	#103×C₁₃₁	1	1	yes	characteristic
167	C₂×C₄	#103×C₁₃₁	1	1	yes	characteristic
168	C₂×C₄	#103×C₁₃₁	1	1	yes	characteristic
169	C₂×C₄	#106×C₁₃₁	1	1	yes	characteristic
170	C₂×C₄	#107×C₁₃₁	1	1	yes	characteristic
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	8	2	no	—
174	C₈	#102×C₁₃₁	1	1	yes	characteristic
175	C₈	#103×C₁₃₁	1	1	yes	characteristic
176	C₈	#106×C₁₃₁	1	1	yes	characteristic
177	C₈	#107×C₁₃₁	1	1	yes	characteristic
178	C₂²	#331×C₁₃₁	1	1	yes	characteristic
179	C₂²	#332×C₁₃₁	1	1	yes	characteristic
180	C₂²	#340×C₁₃₁	1	1	yes	characteristic
181	C₂²	#359×C₁₃₁	1	2	yes	—
182	C₂²	#360×C₁₃₁	1	2	yes	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	16	2	no	—
188	C₄	#332×C₁₃₁	1	1	yes	characteristic
189	C₄	#333×C₁₃₁	1	1	yes	characteristic
190	C₄	#340×C₁₃₁	1	1	yes	characteristic
191	C₄	#343×C₁₃₁	1	1	yes	characteristic
192	C₂	#1296×C₁₃₁	1	1	yes	characteristic
193	C₂	#1360×C₁₃₁	1	2	yes	—
194	C₂	#1363×C₁₃₁	1	2	yes	—
195	C₂	#1382×C₁₃₁	1	1	yes	characteristic
196	C₂	#1386×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	#102
commutator	C₂×C₈	C₄²×C₁₃₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₃₁
LCS	G ▷ C₂×C₈ ▷ C₂×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
11. C₄⋊C₁₆×C₁₃₁	C₄	semidirect
13. #334×C₁₃₁	C₄	semidirect
98. #7275	C₁₃₁	direct (N × H)
99. C₁₃₁	#7275	direct (N × H)
110. C₄⋊C₁₆	C₄×C₁₃₁	semidirect
112. #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
C₂×C₄⋊C₁₆×C₁₃₁	2
#1846×C₁₃₁	2
C₂×#334×C₁₃₁	2
#7275	131

Subgroups — C131×#7275 #45693159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#7275 #45693159