Subgroups — C₁₃₁×#22259 #45708143

Order 33536.

594 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3162×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2961×C₁₃₁	C₂	1	2	yes	maximal
3	#1994×C₁₃₁	C₂	1	2	yes	maximal
4	#1922×C₁₃₁	C₂	1	8	yes	maximal
5	#1838×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	#1745×C₁₃₁	C₂	1	1	yes	maximal, characteristic
7	#533×C₁₃₁	C₂²	1	2	yes	—
8	#501×C₁₃₁	C₄	1	8	yes	complemented
9	#446×C₁₃₁	—	2	8	no	—
10	#427×C₁₃₁	C₂²	1	4	yes	—
11	#424×C₁₃₁	C₂²	1	4	yes	—
12	#423×C₁₃₁	C₂²	1	2	yes	—
13	#405×C₁₃₁	C₂²	1	4	yes	—
14	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
17	#341×C₁₃₁	C₂²	1	4	yes	—
18	#339×C₁₃₁	C₂²	1	8	yes	—
19	#336×C₁₃₁	C₂²	1	4	yes	—
20	C₂×M₁₆×C₁₃₁	C₂³	1	4	yes	—
21	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
22	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
23	C₄⋊Q₈×C₁₃₁	C₂×C₄	1	4	yes	—
24	#125×C₁₃₁	C₂×C₄	1	4	yes	—
25	C₂²⋊Q₈×C₁₃₁	—	2	4	no	—
26	C₄×Q₈×C₁₃₁	—	2	4	no	—
27	#117×C₁₃₁	C₂³	1	4	yes	—
28	C₂×C₄²×C₁₃₁	—	2	2	no	—
29	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
30	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
31	#107×C₁₃₁	C₂×C₄	1	8	yes	—
32	#106×C₁₃₁	C₂×C₄	1	8	yes	—
33	C₄⋊C₈×C₁₃₁	—	2	8	no	—
34	C₄⋊C₈×C₁₃₁	—	2	4	no	—
35	#97×C₁₃₁	—	2	4	no	—
36	#97×C₁₃₁	C₂×C₄	1	4	yes	—
37	#97×C₁₃₁	D₄	1	4	yes	—
38	C₄×C₈×C₁₃₁	—	2	4	no	—
39	C₂×Q₈×C₁₃₁	—	2	4	no	—
40	C₂²×C₄×C₁₃₁	—	2	2	no	—
41	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
42	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
43	M₁₆×C₁₃₁	—	2	8	no	—
44	C₂×C₈×C₁₃₁	—	2	4	no	—
45	C₂×C₈×C₁₃₁	—	2	4	no	—
46	C₂×C₈×C₁₃₁	—	2	4	no	—
47	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
48	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
49	C₄⋊C₄×C₁₃₁	—	2	8	no	—
50	C₄⋊C₄×C₁₃₁	—	2	4	no	—
51	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
52	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
53	C₄²×C₁₃₁	—	2	4	no	—
54	C₄²×C₁₃₁	—	2	4	no	—
55	C₄²×C₁₃₁	—	4	4	no	—
56	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
57	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
58	C₂³×C₁₃₁	C₂×#41	1	1	yes	characteristic
59	Q₈×C₁₃₁	—	8	4	no	—
60	C₂×C₄×C₁₃₁	—	2	4	no	—
61	C₂×C₄×C₁₃₁	—	2	8	no	—
62	C₂×C₄×C₁₃₁	—	4	2	no	—
63	C₂×C₄×C₁₃₁	—	4	2	no	—
64	C₂×C₄×C₁₃₁	—	4	2	no	—
65	C₂×C₄×C₁₃₁	C₄×D₄	1	4	yes	—
66	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	4	yes	—
69	C₈×C₁₃₁	—	2	4	no	—
70	C₈×C₁₃₁	—	2	4	no	—
71	C₈×C₁₃₁	—	4	4	no	—
72	C₂²×C₁₃₁	—	2	2	no	—
73	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
74	C₂²×C₁₃₁	#541	1	2	yes	—
75	C₄×C₁₃₁	—	2	4	no	—
76	C₄×C₁₃₁	—	4	4	no	—
77	C₄×C₁₃₁	—	4	8	no	—
78	C₄×C₁₃₁	#519	1	2	yes	—
79	C₄×C₁₃₁	#536	1	2	yes	—
80	C₂×C₁₃₁	—	2	2	no	—
81	C₂×C₁₃₁	#2327	1	1	yes	characteristic
82	C₂×C₁₃₁	#2956	1	2	yes	—
83	#22259	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₁₃₁	#22259	1	1	yes	characteristic, direct product factor, Sylow
85	#3162	C₂×C₁₃₁	1	1	yes	characteristic
86	#2961	C₂×C₁₃₁	1	2	yes	—
87	#1994	C₂×C₁₃₁	1	2	yes	—
88	#1922	C₂×C₁₃₁	1	8	yes	—
89	#1838	C₂×C₁₃₁	1	1	yes	characteristic
90	#1745	C₂×C₁₃₁	1	1	yes	characteristic
91	#533	C₂²×C₁₃₁	1	2	yes	—
92	#501	C₄×C₁₃₁	1	8	yes	complemented
93	#446	—	2	8	no	—
94	#427	C₂²×C₁₃₁	1	4	yes	—
95	#424	C₂²×C₁₃₁	1	4	yes	—
96	#423	C₂²×C₁₃₁	1	2	yes	—
97	#405	C₂²×C₁₃₁	1	4	yes	—
98	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
99	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
100	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
101	#341	C₂²×C₁₃₁	1	4	yes	—
102	#339	C₂²×C₁₃₁	1	8	yes	—
103	#336	C₂²×C₁₃₁	1	4	yes	—
104	C₂×M₁₆	C₂³×C₁₃₁	1	4	yes	—
105	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
106	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
107	C₄⋊Q₈	C₂×C₄×C₁₃₁	1	4	yes	—
108	#125	C₂×C₄×C₁₃₁	1	4	yes	—
109	C₂²⋊Q₈	—	2	4	no	—
110	C₄×Q₈	—	2	4	no	—
111	#117	C₂³×C₁₃₁	1	4	yes	—
112	C₂×C₄²	—	2	2	no	—
113	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
114	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
115	#107	C₂×C₄×C₁₃₁	1	8	yes	—
116	#106	C₂×C₄×C₁₃₁	1	8	yes	—
117	C₄⋊C₈	—	2	8	no	—
118	C₄⋊C₈	—	2	4	no	—
119	#97	—	2	4	no	—
120	#97	C₂×C₄×C₁₃₁	1	4	yes	—
121	#97	D₄×C₁₃₁	1	4	yes	—
122	C₄×C₈	—	2	4	no	—
123	C₂×Q₈	—	2	4	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
126	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
127	M₁₆	—	2	8	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
132	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
133	C₄⋊C₄	—	2	8	no	—
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
136	C₂²⋊C₄	—	2	4	no	—
137	C₄²	—	2	4	no	—
138	C₄²	—	2	4	no	—
139	C₄²	—	4	4	no	—
140	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
141	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
142	C₂³	C₂×#41×C₁₃₁	1	1	yes	characteristic
143	Q₈	—	8	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	C₄×D₄×C₁₃₁	1	4	yes	—
150	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
151	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
152	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	4	yes	—
153	C₈	—	2	4	no	—
154	C₈	—	2	4	no	—
155	C₈	—	4	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
158	C₂²	#541×C₁₃₁	1	2	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	4	4	no	—
161	C₄	—	4	8	no	—
162	C₄	#519×C₁₃₁	1	2	yes	—
163	C₄	#536×C₁₃₁	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	#2327×C₁₃₁	1	1	yes	characteristic
166	C₂	#2956×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	C₂×C₄×D₄
prime radical	C₂³×C₁₃₁	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂×#97×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
8. #501×C₁₃₁	C₄	semidirect
83. #22259	C₁₃₁	direct (N × H)
84. C₁₃₁	#22259	direct (N × H)
92. #501	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3162×C₁₃₁	2
#2961×C₁₃₁	2
#1994×C₁₃₁	2
#1922×C₁₃₁	2
#1838×C₁₃₁	2
#1745×C₁₃₁	2
#22259	131

Subgroups — C131×#22259 #45708143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#22259 #45708143