Subgroups — C₁₃₁×#13279 #45699163

Order 33536.

366 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2150×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₄×#108×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1387×C₁₃₁	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₁₃₁	—	2	1	no	—
6	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
7	#500×C₁₃₁	—	2	1	no	—
8	#500×C₁₃₁	—	2	1	no	—
9	#500×C₁₃₁	C₄	1	2	yes	—
10	#500×C₁₃₁	C₄	1	2	yes	—
11	C₂×#108×C₁₃₁	C₂²	1	2	yes	—
12	C₂×#107×C₁₃₁	—	2	1	no	—
13	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
14	C₄×M₁₆×C₁₃₁	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₁₃₁	—	2	1	no	—
17	C₄⋊C₁₆×C₁₃₁	—	2	1	no	—
18	#341×C₁₃₁	—	2	1	no	—
19	C₂²×Q₈×C₁₃₁	—	4	1	no	—
20	C₂×M₁₆×C₁₃₁	—	2	2	no	—
21	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
24	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
25	C₄⋊Q₈×C₁₃₁	—	4	1	no	—
26	C₄⋊Q₈×C₁₃₁	—	4	1	no	—
27	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₃₁	—	4	1	no	—
29	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
30	M₃₂×C₁₃₁	—	2	4	no	—
31	C₂×C₁₆×C₁₃₁	—	2	2	no	—
32	#108×C₁₃₁	—	2	2	no	—
33	#108×C₁₃₁	Q₈	1	4	yes	complemented
34	#107×C₁₃₁	—	2	2	no	—
35	#107×C₁₃₁	—	2	2	no	—
36	#107×C₁₃₁	—	2	2	no	—
37	#107×C₁₃₁	C₂×C₄	1	4	yes	—
38	#97×C₁₃₁	—	4	1	no	—
39	C₄×C₈×C₁₃₁	—	4	1	no	—
40	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₁₃₁	—	4	2	no	—
45	C₂×Q₈×C₁₃₁	—	4	2	no	—
46	C₂×Q₈×C₁₃₁	—	8	4	no	—
47	C₂²×C₄×C₁₃₁	—	4	2	no	—
48	C₂²×C₄×C₁₃₁	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₁	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₁₃₁	—	2	4	no	—
52	M₁₆×C₁₃₁	—	4	2	no	—
53	C₂×C₈×C₁₃₁	—	2	2	no	—
54	C₂×C₈×C₁₃₁	—	4	2	no	—
55	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₁₃₁	#41	1	2	yes	—
61	C₄⋊C₄×C₁₃₁	—	2	4	no	—
62	C₄⋊C₄×C₁₃₁	—	4	2	no	—
63	C₄⋊C₄×C₁₃₁	—	4	2	no	—
64	C₄⋊C₄×C₁₃₁	—	4	2	no	—
65	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₁₃₁	D₈	1	1	yes	characteristic
68	C₄²×C₁₃₁	D₈	1	1	yes	characteristic
69	C₁₆×C₁₃₁	—	4	2	no	—
70	C₂³×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
71	Q₈×C₁₃₁	—	8	4	no	—
72	Q₈×C₁₃₁	—	8	4	no	—
73	C₂×C₄×C₁₃₁	—	2	2	no	—
74	C₂×C₄×C₁₃₁	—	2	2	no	—
75	C₂×C₄×C₁₃₁	—	4	4	no	—
76	C₂×C₄×C₁₃₁	—	8	4	no	—
77	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
78	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
79	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
80	C₂×C₄×C₁₃₁	#102	1	1	yes	characteristic
81	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
83	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃₁	#137	1	2	yes	—
85	C₈×C₁₃₁	—	2	1	no	—
86	C₈×C₁₃₁	—	2	1	no	—
87	C₈×C₁₃₁	—	4	4	no	—
88	C₈×C₁₃₁	C₄×Q₈	1	2	yes	—
89	C₈×C₁₃₁	#124	1	1	yes	characteristic
90	C₈×C₁₃₁	C₄⋊Q₈	1	1	yes	characteristic
91	C₂²×C₁₃₁	—	2	1	no	—
92	C₂²×C₁₃₁	—	2	1	no	—
93	C₂²×C₁₃₁	C₂×#102	1	1	yes	characteristic
94	C₂²×C₁₃₁	#475	1	2	yes	—
95	C₄×C₁₃₁	—	2	1	no	—
96	C₄×C₁₃₁	—	2	1	no	—
97	C₄×C₁₃₁	—	8	4	no	—
98	C₄×C₁₃₁	#361	1	2	yes	—
99	C₄×C₁₃₁	#362	1	2	yes	—
100	C₄×C₁₃₁	#391	1	1	yes	characteristic
101	C₄×C₁₃₁	#419	1	1	yes	characteristic
102	C₄×C₁₃₁	#474	1	2	yes	—
103	C₂×C₁₃₁	—	2	2	no	—
104	C₂×C₁₃₁	#1988	1	1	yes	characteristic
105	C₂×C₁₃₁	C₂×#361	1	1	yes	characteristic
106	C₂×C₁₃₁	C₂×#362	1	1	yes	characteristic
107	#13279	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₁₃₁	#13279	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×#500	C₂×C₁₃₁	1	1	yes	characteristic
110	#2150	C₂×C₁₃₁	1	1	yes	characteristic
111	C₄×#108	C₂×C₁₃₁	1	1	yes	characteristic
112	#1387	C₂×C₁₃₁	1	4	yes	—
113	C₂×C₄⋊Q₈	—	2	1	no	—
114	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
115	#500	—	2	1	no	—
116	#500	—	2	1	no	—
117	#500	C₄×C₁₃₁	1	2	yes	—
118	#500	C₄×C₁₃₁	1	2	yes	—
119	C₂×#108	C₂²×C₁₃₁	1	2	yes	—
120	C₂×#107	—	2	1	no	—
121	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
122	C₄×M₁₆	—	2	1	no	—
123	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
124	C₄⋊C₁₆	—	2	1	no	—
125	C₄⋊C₁₆	—	2	1	no	—
126	#341	—	2	1	no	—
127	C₂²×Q₈	—	4	1	no	—
128	C₂×M₁₆	—	2	2	no	—
129	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
130	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	4	1	no	—
134	C₄⋊Q₈	—	4	1	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
138	M₃₂	—	2	4	no	—
139	C₂×C₁₆	—	2	2	no	—
140	#108	—	2	2	no	—
141	#108	Q₈×C₁₃₁	1	4	yes	complemented
142	#107	—	2	2	no	—
143	#107	—	2	2	no	—
144	#107	—	2	2	no	—
145	#107	C₂×C₄×C₁₃₁	1	4	yes	—
146	#97	—	4	1	no	—
147	C₄×C₈	—	4	1	no	—
148	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
149	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
150	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
151	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	8	4	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	SD₁₆×C₁₃₁	1	1	yes	characteristic
157	C₂²×C₄	SD₁₆×C₁₃₁	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
159	M₁₆	—	2	4	no	—
160	M₁₆	—	4	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
164	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
165	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
166	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
167	C₂×C₈	C₂×Q₈×C₁₃₁	1	2	yes	—
168	C₂×C₈	#41×C₁₃₁	1	2	yes	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
174	C₄²	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
175	C₄²	D₈×C₁₃₁	1	1	yes	characteristic
176	C₄²	D₈×C₁₃₁	1	1	yes	characteristic
177	C₁₆	—	4	2	no	—
178	C₂³	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
179	Q₈	—	8	4	no	—
180	Q₈	—	8	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	8	4	no	—
185	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
186	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
187	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
188	C₂×C₄	#102×C₁₃₁	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
190	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
191	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
192	C₂×C₄	#137×C₁₃₁	1	2	yes	—
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	4	4	no	—
196	C₈	C₄×Q₈×C₁₃₁	1	2	yes	—
197	C₈	#124×C₁₃₁	1	1	yes	characteristic
198	C₈	C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	C₂×#102×C₁₃₁	1	1	yes	characteristic
202	C₂²	#475×C₁₃₁	1	2	yes	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	8	4	no	—
206	C₄	#361×C₁₃₁	1	2	yes	—
207	C₄	#362×C₁₃₁	1	2	yes	—
208	C₄	#391×C₁₃₁	1	1	yes	characteristic
209	C₄	#419×C₁₃₁	1	1	yes	characteristic
210	C₄	#474×C₁₃₁	1	2	yes	—
211	C₂	—	2	2	no	—
212	C₂	#1988×C₁₃₁	1	1	yes	characteristic
213	C₂	C₂×#361×C₁₃₁	1	1	yes	characteristic
214	C₂	C₂×#362×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
prime radical	C₂³×C₁₃₁	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
33. #108×C₁₃₁	Q₈	semidirect
107. #13279	C₁₃₁	direct (N × H)
108. C₁₃₁	#13279	direct (N × H)
141. #108	Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₃₁	2
#2150×C₁₃₁	2
C₄×#108×C₁₃₁	2
#1387×C₁₃₁	2
#13279	131

Subgroups — C131×#13279 #45699163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13279 #45699163