Subgroups — C₁₃₁×#9824 #45695708

Order 33536.

726 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1957×C₁₃₁	C₂	1	4	yes	maximal, complemented
3	#1845×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈²×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₁₃₁	—	2	2	no	—
6	C₂×#127×C₁₃₁	C₂²	1	2	yes	—
7	#493×C₁₃₁	—	2	2	no	—
8	#493×C₁₃₁	C₄	1	4	yes	complemented
9	C₂×#107×C₁₃₁	—	2	2	no	—
10	C₂×#102×C₁₃₁	—	2	8	no	—
11	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
13	#384×C₁₃₁	C₂²	1	2	yes	—
14	C₈²×C₁₃₁	—	2	2	no	—
15	C₂²×D₄×C₁₃₁	—	2	4	no	—
16	C₂²×D₄×C₁₃₁	—	2	2	no	—
17	C₂×D₈×C₁₃₁	—	2	8	no	—
18	C₂×D₈×C₁₃₁	—	4	4	no	—
19	C₂×D₈×C₁₃₁	—	4	4	no	—
20	C₂²×C₈×C₁₃₁	D₄	1	4	yes	—
21	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
22	#127×C₁₃₁	—	2	2	no	—
23	#127×C₁₃₁	—	2	4	no	—
24	#127×C₁₃₁	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
26	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
27	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
28	#107×C₁₃₁	—	4	4	no	—
29	#102×C₁₃₁	—	4	16	no	—
30	C₄×C₈×C₁₃₁	—	2	2	no	—
31	C₄×C₈×C₁₃₁	—	2	2	no	—
32	C₄×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
34	C₂⁴×C₁₃₁	—	4	2	no	—
35	C₂×D₄×C₁₃₁	—	2	4	no	—
36	C₂×D₄×C₁₃₁	—	2	8	no	—
37	C₂×D₄×C₁₃₁	—	4	4	no	—
38	C₂×D₄×C₁₃₁	—	4	2	no	—
39	C₂×D₄×C₁₃₁	—	4	8	no	—
40	C₂×D₄×C₁₃₁	—	4	4	no	—
41	C₂×D₄×C₁₃₁	—	4	4	no	—
42	C₂×D₄×C₁₃₁	—	4	8	no	—
43	C₂²×C₄×C₁₃₁	—	4	2	no	—
44	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
46	D₈×C₁₃₁	—	4	16	no	—
47	C₂×C₈×C₁₃₁	—	2	4	no	—
48	C₂×C₈×C₁₃₁	—	2	4	no	—
49	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	4	yes	—
50	C₂×C₈×C₁₃₁	D₈	1	4	yes	—
51	C₂×C₈×C₁₃₁	D₈	1	8	yes	complemented
52	C₂×C₈×C₁₃₁	SD₁₆	1	4	yes	—
53	C₄⋊C₄×C₁₃₁	—	4	8	no	—
54	C₄⋊C₄×C₁₃₁	—	4	4	no	—
55	C₄²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₁₃₁	#41	1	2	yes	—
58	C₂³×C₁₃₁	—	4	4	no	—
59	C₂³×C₁₃₁	—	8	2	no	—
60	C₂³×C₁₃₁	—	8	2	no	—
61	C₂³×C₁₃₁	—	8	4	no	—
62	C₂³×C₁₃₁	—	8	4	no	—
63	C₂³×C₁₃₁	#127	1	1	yes	characteristic
64	D₄×C₁₃₁	—	4	8	no	—
65	D₄×C₁₃₁	—	4	16	no	—
66	D₄×C₁₃₁	—	8	8	no	—
67	D₄×C₁₃₁	—	8	4	no	—
68	C₂×C₄×C₁₃₁	—	8	2	no	—
69	C₂×C₄×C₁₃₁	—	8	4	no	—
70	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₁₃₁	#121	1	2	yes	—
73	C₂×C₄×C₁₃₁	#124	1	1	yes	characteristic
74	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
75	C₂×C₄×C₁₃₁	C₂×D₈	1	4	yes	—
76	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₁₃₁	#135	1	4	yes	—
78	C₈×C₁₃₁	—	2	8	no	—
79	C₈×C₁₃₁	#102	1	8	yes	complemented
80	C₂²×C₁₃₁	—	8	4	no	—
81	C₂²×C₁₃₁	—	8	8	no	—
82	C₂²×C₁₃₁	—	16	4	no	—
83	C₂²×C₁₃₁	—	16	2	no	—
84	C₂²×C₁₃₁	—	16	2	no	—
85	C₂²×C₁₃₁	#390	1	1	yes	characteristic
86	C₂²×C₁₃₁	#492	1	1	yes	characteristic
87	C₂²×C₁₃₁	#493	1	1	yes	characteristic
88	C₂²×C₁₃₁	#493	1	2	yes	—
89	C₂²×C₁₃₁	#495	1	2	yes	—
90	C₄×C₁₃₁	—	16	2	no	—
91	C₄×C₁₃₁	C₂×#102	1	2	yes	—
92	C₄×C₁₃₁	C₄×D₈	1	4	yes	—
93	C₄×C₁₃₁	#466	1	4	yes	—
94	C₄×C₁₃₁	#486	1	2	yes	—
95	C₂×C₁₃₁	—	16	4	no	—
96	C₂×C₁₃₁	#1706	1	2	yes	—
97	C₂×C₁₃₁	#1709	1	1	yes	characteristic
98	C₂×C₁₃₁	#1712	1	1	yes	characteristic
99	C₂×C₁₃₁	#1957	1	1	yes	characteristic
100	C₂×C₁₃₁	#1970	1	2	yes	—
101	#9824	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₁₃₁	#9824	1	1	yes	characteristic, direct product factor, Sylow
103	C₂×#493	C₂×C₁₃₁	1	1	yes	characteristic
104	#1957	C₂×C₁₃₁	1	4	yes	complemented
105	#1845	C₂×C₁₃₁	1	1	yes	characteristic, complemented
106	C₂×C₈²	C₂×C₁₃₁	1	1	yes	characteristic, complemented
107	C₂²×D₈	—	2	2	no	—
108	C₂×#127	C₂²×C₁₃₁	1	2	yes	—
109	#493	—	2	2	no	—
110	#493	C₄×C₁₃₁	1	4	yes	complemented
111	C₂×#107	—	2	2	no	—
112	C₂×#102	—	2	8	no	—
113	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
114	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
115	#384	C₂²×C₁₃₁	1	2	yes	—
116	C₈²	—	2	2	no	—
117	C₂²×D₄	—	2	2	no	—
118	C₂²×D₄	—	2	4	no	—
119	C₂×D₈	—	2	8	no	—
120	C₂×D₈	—	4	4	no	—
121	C₂×D₈	—	4	4	no	—
122	C₂²×C₈	D₄×C₁₃₁	1	4	yes	—
123	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
124	#127	—	2	2	no	—
125	#127	—	2	4	no	—
126	#127	C₂×C₄×C₁₃₁	1	4	yes	—
127	C₂×C₄⋊C₄	—	2	4	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
130	#107	—	4	4	no	—
131	#102	—	4	16	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
135	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
136	C₂⁴	—	4	2	no	—
137	C₂×D₄	—	2	4	no	—
138	C₂×D₄	—	2	8	no	—
139	C₂×D₄	—	4	4	no	—
140	C₂×D₄	—	4	2	no	—
141	C₂×D₄	—	4	8	no	—
142	C₂×D₄	—	4	4	no	—
143	C₂×D₄	—	4	4	no	—
144	C₂×D₄	—	4	8	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
148	D₈	—	4	16	no	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	4	yes	—
152	C₂×C₈	D₈×C₁₃₁	1	4	yes	—
153	C₂×C₈	D₈×C₁₃₁	1	8	yes	complemented
154	C₂×C₈	SD₁₆×C₁₃₁	1	4	yes	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄²	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
158	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
159	C₄²	#41×C₁₃₁	1	2	yes	—
160	C₂³	—	4	4	no	—
161	C₂³	—	8	2	no	—
162	C₂³	—	8	4	no	—
163	C₂³	—	8	2	no	—
164	C₂³	—	8	4	no	—
165	C₂³	#127×C₁₃₁	1	1	yes	characteristic
166	D₄	—	4	8	no	—
167	D₄	—	4	16	no	—
168	D₄	—	8	8	no	—
169	D₄	—	8	4	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
173	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
174	C₂×C₄	#121×C₁₃₁	1	2	yes	—
175	C₂×C₄	#124×C₁₃₁	1	1	yes	characteristic
176	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
177	C₂×C₄	C₂×D₈×C₁₃₁	1	4	yes	—
178	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	2	yes	—
179	C₂×C₄	#135×C₁₃₁	1	4	yes	—
180	C₈	—	2	8	no	—
181	C₈	#102×C₁₃₁	1	8	yes	complemented
182	C₂²	—	8	8	no	—
183	C₂²	—	8	4	no	—
184	C₂²	—	16	4	no	—
185	C₂²	—	16	2	no	—
186	C₂²	—	16	2	no	—
187	C₂²	#390×C₁₃₁	1	1	yes	characteristic
188	C₂²	#492×C₁₃₁	1	1	yes	characteristic
189	C₂²	#493×C₁₃₁	1	2	yes	—
190	C₂²	#493×C₁₃₁	1	1	yes	characteristic
191	C₂²	#495×C₁₃₁	1	2	yes	—
192	C₄	—	16	2	no	—
193	C₄	C₂×#102×C₁₃₁	1	2	yes	—
194	C₄	C₄×D₈×C₁₃₁	1	4	yes	—
195	C₄	#466×C₁₃₁	1	4	yes	—
196	C₄	#486×C₁₃₁	1	2	yes	—
197	C₂	—	16	4	no	—
198	C₂	#1706×C₁₃₁	1	2	yes	—
199	C₂	#1709×C₁₃₁	1	1	yes	characteristic
200	C₂	#1712×C₁₃₁	1	1	yes	characteristic
201	C₂	#1957×C₁₃₁	1	1	yes	characteristic
202	C₂	#1970×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	#127
commutator	C₄²	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
prime radical	C₂×#493×C₁₃₁	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1957×C₁₃₁	C₂	semidirect
3. #1845×C₁₃₁	C₂	semidirect
4. C₂×C₈²×C₁₃₁	C₂	semidirect
8. #493×C₁₃₁	C₄	semidirect
51. C₂×C₈×C₁₃₁	D₈	semidirect
79. C₈×C₁₃₁	#102	semidirect
101. #9824	C₁₃₁	direct (N × H)
102. C₁₃₁	#9824	direct (N × H)
104. #1957	C₂×C₁₃₁	semidirect
105. #1845	C₂×C₁₃₁	semidirect
106. C₂×C₈²	C₂×C₁₃₁	semidirect
110. #493	C₄×C₁₃₁	semidirect
153. C₂×C₈	D₈×C₁₃₁	semidirect
181. C₈	#102×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493×C₁₃₁	2
#1957×C₁₃₁	2
#1845×C₁₃₁	2
C₂×C₈²×C₁₃₁	2
#9824	131

Subgroups — C131×#9824 #45695708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#9824 #45695708