Subgroups — C₁₃₁×#33726 #45719610

Order 33536.

490 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3160×C₁₃₁	C₂	1	2	yes	maximal
2	#2254×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#2253×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	#2252×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2158×C₁₃₁	C₂	1	4	yes	maximal
6	C₂×C₄×C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
7	#533×C₁₃₁	—	2	2	no	—
8	C₂²×C₁₆×C₁₃₁	C₂²	1	2	yes	—
9	#500×C₁₃₁	C₂²	1	4	yes	—
10	#499×C₁₃₁	—	2	2	no	—
11	#499×C₁₃₁	C₂²	1	4	yes	—
12	#498×C₁₃₁	—	2	2	no	—
13	#427×C₁₃₁	—	2	2	no	—
14	#427×C₁₃₁	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	#367×C₁₃₁	C₂²	1	8	yes	—
17	#366×C₁₃₁	C₂²	1	8	yes	—
18	C₄×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₄×C₁₆×C₁₃₁	C₂²	1	2	yes	—
20	C₄×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
24	#125×C₁₃₁	—	2	4	no	—
25	C₂²⋊Q₈×C₁₃₁	—	4	4	no	—
26	C₄×Q₈×C₁₃₁	—	4	4	no	—
27	#117×C₁₃₁	—	2	4	no	—
28	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₁₃₁	Q₈	1	8	yes	complemented
30	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
31	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
32	#107×C₁₃₁	—	2	4	no	—
33	#107×C₁₃₁	C₂³	1	8	yes	—
34	#106×C₁₃₁	—	2	4	no	—
35	#106×C₁₃₁	—	2	4	no	—
36	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
37	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
38	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
40	C₂×Q₈×C₁₃₁	—	4	4	no	—
41	C₂²×C₄×C₁₃₁	D₈	1	2	yes	—
42	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃₁	Q₁₆	1	4	yes	—
44	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃₁	C₂×Q₈	1	4	yes	—
48	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄×C₁₃₁	—	2	8	no	—
50	C₄⋊C₄×C₁₃₁	—	4	8	no	—
51	C₄⋊C₄×C₁₃₁	—	4	4	no	—
52	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
53	C₄²×C₁₃₁	—	4	4	no	—
54	C₄²×C₁₃₁	D₈	1	2	yes	—
55	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
57	C₁₆×C₁₃₁	C₂×Q₈	1	4	yes	—
58	C₁₆×C₁₃₁	C₂×Q₈	1	4	yes	—
59	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
60	Q₈×C₁₃₁	—	8	8	no	—
61	C₂×C₄×C₁₃₁	—	2	4	no	—
62	C₂×C₄×C₁₃₁	—	4	8	no	—
63	C₂×C₄×C₁₃₁	C₄⋊Q₈	1	2	yes	—
64	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
65	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
66	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	2	yes	—
68	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
69	C₈×C₁₃₁	C₄⋊Q₈	1	2	yes	—
70	C₈×C₁₃₁	C₂×Q₁₆	1	2	yes	—
71	C₈×C₁₃₁	C₂×Q₁₆	1	2	yes	—
72	C₈×C₁₃₁	C₂²×Q₈	1	2	yes	—
73	C₂²×C₁₃₁	—	2	2	no	—
74	C₂²×C₁₃₁	#500	1	2	yes	—
75	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
76	C₄×C₁₃₁	—	2	2	no	—
77	C₄×C₁₃₁	—	8	8	no	—
78	C₄×C₁₃₁	#500	1	2	yes	—
79	C₄×C₁₃₁	#508	1	4	yes	—
80	C₄×C₁₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
81	C₄×C₁₃₁	C₂²×Q₁₆	1	1	yes	characteristic
82	C₂×C₁₃₁	—	2	2	no	—
83	C₂×C₁₃₁	C₂×#500	1	1	yes	characteristic
84	C₂×C₁₃₁	C₂×#508	1	2	yes	—
85	#33726	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₁₃₁	#33726	1	1	yes	characteristic, direct product factor, Sylow
87	#3160	C₂×C₁₃₁	1	2	yes	—
88	#2254	C₂×C₁₃₁	1	2	yes	complemented
89	#2253	C₂×C₁₃₁	1	4	yes	complemented
90	#2252	C₂×C₁₃₁	1	2	yes	complemented
91	#2158	C₂×C₁₃₁	1	4	yes	—
92	C₂×C₄×C₁₆	C₂×C₁₃₁	1	1	yes	characteristic
93	#533	—	2	2	no	—
94	C₂²×C₁₆	C₂²×C₁₃₁	1	2	yes	—
95	#500	C₂²×C₁₃₁	1	4	yes	—
96	#499	—	2	2	no	—
97	#499	C₂²×C₁₃₁	1	4	yes	—
98	#498	—	2	2	no	—
99	#427	—	2	2	no	—
100	#427	C₂²×C₁₃₁	1	4	yes	—
101	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
102	#367	C₂²×C₁₃₁	1	8	yes	—
103	#366	C₂²×C₁₃₁	1	8	yes	—
104	C₄×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
105	C₄×C₁₆	C₂²×C₁₃₁	1	2	yes	—
106	C₄×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
107	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
108	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
109	C₄⋊Q₈	—	2	4	no	—
110	#125	—	2	4	no	—
111	C₂²⋊Q₈	—	4	4	no	—
112	C₄×Q₈	—	4	4	no	—
113	#117	—	2	4	no	—
114	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
115	C₂×C₁₆	Q₈×C₁₃₁	1	8	yes	complemented
116	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
117	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
118	#107	—	2	4	no	—
119	#107	C₂³×C₁₃₁	1	8	yes	—
120	#106	—	2	4	no	—
121	#106	—	2	4	no	—
122	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
123	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
124	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
125	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
126	C₂×Q₈	—	4	4	no	—
127	C₂²×C₄	D₈×C₁₃₁	1	2	yes	—
128	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
129	C₂×C₈	Q₁₆×C₁₃₁	1	4	yes	—
130	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
131	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
132	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
133	C₂×C₈	C₂×Q₈×C₁₃₁	1	4	yes	—
134	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
135	C₄⋊C₄	—	2	8	no	—
136	C₄⋊C₄	—	4	8	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₂²⋊C₄	—	4	4	no	—
139	C₄²	—	4	4	no	—
140	C₄²	D₈×C₁₃₁	1	2	yes	—
141	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
142	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
143	C₁₆	C₂×Q₈×C₁₃₁	1	4	yes	—
144	C₁₆	C₂×Q₈×C₁₃₁	1	4	yes	—
145	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
146	Q₈	—	8	8	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	4	8	no	—
149	C₂×C₄	C₄⋊Q₈×C₁₃₁	1	2	yes	—
150	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
151	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
152	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
153	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	2	yes	—
154	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
155	C₈	C₄⋊Q₈×C₁₃₁	1	2	yes	—
156	C₈	C₂×Q₁₆×C₁₃₁	1	2	yes	—
157	C₈	C₂×Q₁₆×C₁₃₁	1	2	yes	—
158	C₈	C₂²×Q₈×C₁₃₁	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	#500×C₁₃₁	1	2	yes	—
161	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	8	8	no	—
164	C₄	#500×C₁₃₁	1	2	yes	—
165	C₄	#508×C₁₃₁	1	4	yes	—
166	C₄	C₂×C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
167	C₄	C₂²×Q₁₆×C₁₃₁	1	1	yes	characteristic
168	C₂	—	2	2	no	—
169	C₂	C₂×#500×C₁₃₁	1	1	yes	characteristic
170	C₂	C₂×#508×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
socle	C₂²×C₁₃₁	C₂²×D₈
prime radical	C₂³×C₁₃₁	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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
2. #2254×C₁₃₁	C₂	semidirect
3. #2253×C₁₃₁	C₂	semidirect
4. #2252×C₁₃₁	C₂	semidirect
29. C₂×C₁₆×C₁₃₁	Q₈	semidirect
85. #33726	C₁₃₁	direct (N × H)
86. C₁₃₁	#33726	direct (N × H)
88. #2254	C₂×C₁₃₁	semidirect
89. #2253	C₂×C₁₃₁	semidirect
90. #2252	C₂×C₁₃₁	semidirect
115. C₂×C₁₆	Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3160×C₁₃₁	2
#2254×C₁₃₁	2
#2253×C₁₃₁	2
#2252×C₁₃₁	2
#2158×C₁₃₁	2
C₂×C₄×C₁₆×C₁₃₁	2
#33726	131

Subgroups — C131×#33726 #45719610

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33726 #45719610