Subgroups — C₁₃₁×#23366 #45709250

Order 33536.

616 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1985×C₁₃₁	C₂	1	4	yes	maximal
3	#1917×C₁₃₁	C₂	1	8	yes	maximal
4	#1839×C₁₃₁	C₂	1	2	yes	maximal
5	C₂×C₄⋊Q₈×C₁₃₁	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃₁	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	4	yes	—
8	C₂×#97×C₁₃₁	C₂²	1	4	yes	—
9	#391×C₁₃₁	C₄	1	2	yes	—
10	#389×C₁₃₁	C₄	1	4	yes	—
11	#384×C₁₃₁	C₄	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	—
13	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
14	#336×C₁₃₁	C₂²	1	4	yes	—
15	#336×C₁₃₁	C₂²	1	8	yes	—
16	#336×C₁₃₁	C₂²	1	8	yes	—
17	C₂²×Q₈×C₁₃₁	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₃₁	C₂³	1	8	yes	—
19	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
20	C₄×Q₈×C₁₃₁	—	4	2	no	—
21	C₄×Q₈×C₁₃₁	—	4	2	no	—
22	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
30	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
32	C₄⋊C₈×C₁₃₁	—	4	4	no	—
33	#97×C₁₃₁	—	4	4	no	—
34	#95×C₁₃₁	C₂×C₄	1	8	yes	—
35	C₂×Q₈×C₁₃₁	—	2	4	no	—
36	C₂×Q₈×C₁₃₁	—	4	2	no	—
37	C₂×Q₈×C₁₃₁	—	4	2	no	—
38	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
39	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
40	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
41	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
45	C₂×C₈×C₁₃₁	—	2	8	no	—
46	C₂×C₈×C₁₃₁	—	2	16	no	—
47	C₄⋊C₄×C₁₃₁	—	2	8	no	—
48	C₄⋊C₄×C₁₃₁	—	4	4	no	—
49	C₄⋊C₄×C₁₃₁	—	4	2	no	—
50	C₄⋊C₄×C₁₃₁	—	4	4	no	—
51	C₄²×C₁₃₁	—	2	4	no	—
52	C₄²×C₁₃₁	—	2	4	no	—
53	C₄²×C₁₃₁	—	2	2	no	—
54	C₄²×C₁₃₁	—	2	4	no	—
55	C₄²×C₁₃₁	—	4	2	no	—
56	C₄²×C₁₃₁	—	4	1	no	—
57	C₄²×C₁₃₁	M₁₆	1	4	yes	—
58	C₂³×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
59	Q₈×C₁₃₁	—	4	8	no	—
60	C₂×C₄×C₁₃₁	—	2	1	no	—
61	C₂×C₄×C₁₃₁	—	2	2	no	—
62	C₂×C₄×C₁₃₁	—	2	2	no	—
63	C₂×C₄×C₁₃₁	—	2	1	no	—
64	C₂×C₄×C₁₃₁	—	2	1	no	—
65	C₂×C₄×C₁₃₁	—	2	4	no	—
66	C₂×C₄×C₁₃₁	—	2	4	no	—
67	C₂×C₄×C₁₃₁	—	2	2	no	—
68	C₂×C₄×C₁₃₁	—	2	4	no	—
69	C₂×C₄×C₁₃₁	—	2	2	no	—
70	C₂×C₄×C₁₃₁	—	2	2	no	—
71	C₂×C₄×C₁₃₁	—	2	2	no	—
72	C₂×C₄×C₁₃₁	—	2	8	no	—
73	C₂×C₄×C₁₃₁	—	2	4	no	—
74	C₂×C₄×C₁₃₁	C₂×M₁₆	1	4	yes	—
75	C₂×C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
77	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	4	yes	—
78	C₈×C₁₃₁	—	4	8	no	—
79	C₂²×C₁₃₁	#518	1	1	yes	characteristic
80	C₂²×C₁₃₁	#519	1	2	yes	—
81	C₂²×C₁₃₁	C₂²×M₁₆	1	1	yes	characteristic
82	C₂²×C₁₃₁	#568	1	1	yes	characteristic
83	C₂²×C₁₃₁	#568	1	2	yes	—
84	C₄×C₁₃₁	—	2	2	no	—
85	C₄×C₁₃₁	—	2	4	no	—
86	C₄×C₁₃₁	—	4	4	no	—
87	C₄×C₁₃₁	—	4	1	no	—
88	C₄×C₁₃₁	—	4	4	no	—
89	C₄×C₁₃₁	—	4	2	no	—
90	C₄×C₁₃₁	#563	1	4	yes	—
91	C₂×C₁₃₁	#2381	1	1	yes	characteristic
92	C₂×C₁₃₁	#2975	1	1	yes	characteristic
93	C₂×C₁₃₁	#2980	1	1	yes	characteristic
94	C₂×C₁₃₁	#2983	1	2	yes	—
95	C₂×C₁₃₁	#2987	1	2	yes	—
96	#23366	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₁₃₁	#23366	1	1	yes	characteristic, direct product factor, Sylow
98	C₄×C₄⋊Q₈	C₂×C₁₃₁	1	1	yes	characteristic
99	#1985	C₂×C₁₃₁	1	4	yes	—
100	#1917	C₂×C₁₃₁	1	8	yes	—
101	#1839	C₂×C₁₃₁	1	2	yes	—
102	C₂×C₄⋊Q₈	C₄×C₁₃₁	1	1	yes	characteristic
103	C₂×C₄×Q₈	C₂²×C₁₃₁	1	2	yes	—
104	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	4	yes	—
105	C₂×#97	C₂²×C₁₃₁	1	4	yes	—
106	#391	C₄×C₁₃₁	1	2	yes	—
107	#389	C₄×C₁₃₁	1	4	yes	—
108	#384	C₄×C₁₃₁	1	1	yes	characteristic
109	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	—
110	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
111	#336	C₂²×C₁₃₁	1	8	yes	—
112	#336	C₂²×C₁₃₁	1	4	yes	—
113	#336	C₂²×C₁₃₁	1	8	yes	—
114	C₂²×Q₈	C₂×C₄×C₁₃₁	1	2	yes	—
115	C₂²×C₈	C₂³×C₁₃₁	1	8	yes	—
116	C₄⋊Q₈	—	2	4	no	—
117	C₄×Q₈	—	4	2	no	—
118	C₄×Q₈	—	4	2	no	—
119	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
120	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
121	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
122	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
123	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
124	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
125	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
126	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
127	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
128	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
129	C₄⋊C₈	—	4	4	no	—
130	#97	—	4	4	no	—
131	#95	C₂×C₄×C₁₃₁	1	8	yes	—
132	C₂×Q₈	—	2	4	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
136	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
137	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
138	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
139	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
140	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
141	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
142	C₂×C₈	—	2	8	no	—
143	C₂×C₈	—	2	16	no	—
144	C₄⋊C₄	—	2	8	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	2	4	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	4	no	—
152	C₄²	—	4	2	no	—
153	C₄²	—	4	1	no	—
154	C₄²	M₁₆×C₁₃₁	1	4	yes	—
155	C₂³	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
156	Q₈	—	4	8	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	C₂×M₁₆×C₁₃₁	1	4	yes	—
172	C₂×C₄	C₂×M₁₆×C₁₃₁	1	2	yes	—
173	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	2	yes	—
174	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	4	yes	—
175	C₈	—	4	8	no	—
176	C₂²	#518×C₁₃₁	1	1	yes	characteristic
177	C₂²	#519×C₁₃₁	1	2	yes	—
178	C₂²	C₂²×M₁₆×C₁₃₁	1	1	yes	characteristic
179	C₂²	#568×C₁₃₁	1	1	yes	characteristic
180	C₂²	#568×C₁₃₁	1	2	yes	—
181	C₄	—	2	2	no	—
182	C₄	—	2	4	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	4	no	—
185	C₄	—	4	1	no	—
186	C₄	—	4	4	no	—
187	C₄	#563×C₁₃₁	1	4	yes	—
188	C₂	#2381×C₁₃₁	1	1	yes	characteristic
189	C₂	#2975×C₁₃₁	1	1	yes	characteristic
190	C₂	#2980×C₁₃₁	1	1	yes	characteristic
191	C₂	#2983×C₁₃₁	1	2	yes	—
192	C₂	#2987×C₁₃₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
96. #23366	C₁₃₁	direct (N × H)
97. C₁₃₁	#23366	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₁₃₁	2
#1985×C₁₃₁	2
#1917×C₁₃₁	2
#1839×C₁₃₁	2
#23366	131

Subgroups — C131×#23366 #45709250

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#23366 #45709250