Subgroups — C₁₃₁×#20830 #45706714

Order 33536.

770 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2920×C₁₃₁	C₂	1	2	yes	maximal
2	C₂²×#97×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1849×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#1838×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1810×C₁₃₁	C₂	1	4	yes	maximal
6	#1778×C₁₃₁	C₂	1	4	yes	maximal, complemented
7	C₂³×C₈×C₁₃₁	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
9	#432×C₁₃₁	—	2	4	no	—
10	#431×C₁₃₁	—	2	4	no	—
11	C₂×#108×C₁₃₁	C₂²	1	8	yes	—
12	#423×C₁₃₁	C₂²	1	4	yes	—
13	#407×C₁₃₁	C₂²	1	4	yes	—
14	C₄×M₁₆×C₁₃₁	C₂²	1	4	yes	—
15	C₂×#97×C₁₃₁	C₄	1	8	yes	—
16	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
17	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
19	#341×C₁₃₁	C₂²	1	8	yes	—
20	C₂³×C₄×C₁₃₁	Q₈	1	2	yes	—
21	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₁₃₁	C₂³	1	8	yes	—
23	C₂²×C₈×C₁₃₁	—	2	8	no	—
24	C₂²×C₈×C₁₃₁	C₂×C₄	1	8	yes	—
25	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
26	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
27	C₂×C₄²×C₁₃₁	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₁₃₁	D₄	1	4	yes	—
29	C₂×C₄²×C₁₃₁	Q₈	1	2	yes	—
30	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
32	#108×C₁₃₁	—	2	16	no	—
33	C₄⋊C₈×C₁₃₁	—	2	8	no	—
34	C₂²⋊C₈×C₁₃₁	—	2	8	no	—
35	#97×C₁₃₁	—	2	4	no	—
36	#97×C₁₃₁	C₂×C₄	1	4	yes	—
37	#97×C₁₃₁	C₂×C₄	1	8	yes	—
38	#97×C₁₃₁	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₁₃₁	—	2	4	no	—
40	C₂⁴×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₁	—	2	4	no	—
42	C₂²×C₄×C₁₃₁	—	4	4	no	—
43	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
49	C₂²×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
51	M₁₆×C₁₃₁	—	2	16	no	—
52	C₂×C₈×C₁₃₁	—	2	4	no	—
53	C₂×C₈×C₁₃₁	—	2	16	no	—
54	C₂×C₈×C₁₃₁	—	2	8	no	—
55	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
56	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₁₃₁	#41	1	8	yes	—
58	C₄²×C₁₃₁	C₄⋊C₄	1	8	yes	—
59	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
60	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
61	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
62	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
63	C₂³×C₁₃₁	—	2	2	no	—
64	C₂³×C₁₃₁	—	2	4	no	—
65	C₂³×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
66	C₂³×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃₁	—	2	2	no	—
68	C₂×C₄×C₁₃₁	—	2	8	no	—
69	C₂×C₄×C₁₃₁	—	2	4	no	—
70	C₂×C₄×C₁₃₁	—	2	4	no	—
71	C₂×C₄×C₁₃₁	—	4	4	no	—
72	C₂×C₄×C₁₃₁	—	4	2	no	—
73	C₂×C₄×C₁₃₁	—	4	2	no	—
74	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	4	yes	—
76	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	4	yes	—
77	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃₁	C₂×#41	1	4	yes	—
80	C₈×C₁₃₁	—	2	4	no	—
81	C₈×C₁₃₁	—	4	8	no	—
82	C₈×C₁₃₁	#117	1	8	yes	—
83	C₂²×C₁₃₁	—	2	8	no	—
84	C₂²×C₁₃₁	—	2	2	no	—
85	C₂²×C₁₃₁	—	2	2	no	—
86	C₂²×C₁₃₁	—	2	2	no	—
87	C₂²×C₁₃₁	#430	1	4	yes	—
88	C₂²×C₁₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₁₃₁	#533	1	2	yes	—
90	C₄×C₁₃₁	—	2	4	no	—
91	C₄×C₁₃₁	—	2	4	no	—
92	C₄×C₁₃₁	—	2	2	no	—
93	C₄×C₁₃₁	C₂×#117	1	2	yes	—
94	C₄×C₁₃₁	#532	1	2	yes	—
95	C₂×C₁₃₁	—	2	2	no	—
96	C₂×C₁₃₁	—	2	4	no	—
97	C₂×C₁₃₁	#2285	1	1	yes	characteristic
98	C₂×C₁₃₁	C₂×#430	1	2	yes	—
99	#20830	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₁₃₁	#20830	1	1	yes	characteristic, direct product factor, Sylow
101	#2920	C₂×C₁₃₁	1	2	yes	—
102	C₂²×#97	C₂×C₁₃₁	1	1	yes	characteristic
103	#1849	C₂×C₁₃₁	1	2	yes	complemented
104	#1838	C₂×C₁₃₁	1	2	yes	complemented
105	#1810	C₂×C₁₃₁	1	4	yes	—
106	#1778	C₂×C₁₃₁	1	4	yes	complemented
107	C₂³×C₈	C₂²×C₁₃₁	1	2	yes	—
108	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
109	#432	—	2	4	no	—
110	#431	—	2	4	no	—
111	C₂×#108	C₂²×C₁₃₁	1	8	yes	—
112	#423	C₂²×C₁₃₁	1	4	yes	—
113	#407	C₂²×C₁₃₁	1	4	yes	—
114	C₄×M₁₆	C₂²×C₁₃₁	1	4	yes	—
115	C₂×#97	C₄×C₁₃₁	1	8	yes	—
116	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
117	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
118	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
119	#341	C₂²×C₁₃₁	1	8	yes	—
120	C₂³×C₄	Q₈×C₁₃₁	1	2	yes	—
121	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
122	C₂×M₁₆	C₂³×C₁₃₁	1	8	yes	—
123	C₂²×C₈	—	2	8	no	—
124	C₂²×C₈	C₂×C₄×C₁₃₁	1	8	yes	—
125	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
126	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
127	C₂×C₄²	C₂×C₄×C₁₃₁	1	4	yes	—
128	C₂×C₄²	D₄×C₁₃₁	1	4	yes	—
129	C₂×C₄²	Q₈×C₁₃₁	1	2	yes	—
130	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
131	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
132	#108	—	2	16	no	—
133	C₄⋊C₈	—	2	8	no	—
134	C₂²⋊C₈	—	2	8	no	—
135	#97	—	2	4	no	—
136	#97	C₂×C₄×C₁₃₁	1	4	yes	—
137	#97	C₂×C₄×C₁₃₁	1	8	yes	—
138	#97	C₂×C₄×C₁₃₁	1	4	yes	—
139	C₄×C₈	—	2	4	no	—
140	C₂⁴	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	4	4	no	—
143	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	4	yes	—
144	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	4	yes	—
145	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
146	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
147	C₂²×C₄	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
148	C₂²×C₄	C₂×Q₈×C₁₃₁	1	2	yes	—
149	C₂²×C₄	C₂×Q₈×C₁₃₁	1	2	yes	—
150	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
151	M₁₆	—	2	16	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	16	no	—
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
156	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
157	C₂×C₈	#41×C₁₃₁	1	8	yes	—
158	C₄²	C₄⋊C₄×C₁₃₁	1	8	yes	—
159	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
160	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
161	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
162	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
163	C₂³	—	2	4	no	—
164	C₂³	—	2	2	no	—
165	C₂³	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
166	C₂³	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
175	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	4	yes	—
176	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	4	yes	—
177	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
179	C₂×C₄	C₂×#41×C₁₃₁	1	4	yes	—
180	C₈	—	2	4	no	—
181	C₈	—	4	8	no	—
182	C₈	#117×C₁₃₁	1	8	yes	—
183	C₂²	—	2	8	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	#430×C₁₃₁	1	4	yes	—
188	C₂²	C₂²×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
189	C₂²	#533×C₁₃₁	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	4	no	—
193	C₄	C₂×#117×C₁₃₁	1	2	yes	—
194	C₄	#532×C₁₃₁	1	2	yes	—
195	C₂	—	2	4	no	—
196	C₂	—	2	2	no	—
197	C₂	#2285×C₁₃₁	1	1	yes	characteristic
198	C₂	C₂×#430×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
socle	C₂²×C₁₃₁	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃₁	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂²×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1849×C₁₃₁	C₂	semidirect
4. #1838×C₁₃₁	C₂	semidirect
6. #1778×C₁₃₁	C₂	semidirect
99. #20830	C₁₃₁	direct (N × H)
100. C₁₃₁	#20830	direct (N × H)
103. #1849	C₂×C₁₃₁	semidirect
104. #1838	C₂×C₁₃₁	semidirect
106. #1778	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2920×C₁₃₁	2
C₂²×#97×C₁₃₁	2
#1849×C₁₃₁	2
#1838×C₁₃₁	2
#1810×C₁₃₁	2
#1778×C₁₃₁	2
#20830	131

Subgroups — C131×#20830 #45706714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#20830 #45706714