Subgroups — C₁₃₁×#13262 #45699146

Order 33536.

462 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#364×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#362×C₁₃₁	C₂	1	2	yes	maximal
4	#2141×C₁₃₁	C₂	1	2	yes	maximal
5	#1772×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₁₃₁	—	2	1	no	—
7	C₂²×Q₁₆×C₁₃₁	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈×C₁₃₁	—	2	1	no	—
9	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
10	#494×C₁₃₁	—	2	2	no	—
11	#494×C₁₃₁	C₄	1	4	yes	complemented
12	C₂×#108×C₁₃₁	C₂²	1	1	yes	characteristic
13	#427×C₁₃₁	C₂²	1	1	yes	characteristic
14	#405×C₁₃₁	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	#364×C₁₃₁	—	2	2	no	—
17	#362×C₁₃₁	—	2	4	no	—
18	#358×C₁₃₁	—	2	4	no	—
19	#339×C₁₃₁	—	2	1	no	—
20	C₂²×Q₈×C₁₃₁	—	2	2	no	—
21	C₂×Q₁₆×C₁₃₁	—	2	4	no	—
22	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
23	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
24	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
25	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
26	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
27	C₂×Q₁₆×C₁₃₁	—	4	1	no	—
28	C₂×Q₁₆×C₁₃₁	—	4	1	no	—
29	C₂×Q₁₆×C₁₃₁	—	4	2	no	—
30	C₂×Q₁₆×C₁₃₁	C₂×C₄	1	4	yes	—
31	C₂×M₁₆×C₁₃₁	—	2	1	no	—
32	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
35	C₄⋊Q₈×C₁₃₁	—	4	2	no	—
36	#117×C₁₃₁	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₃₁	—	4	2	no	—
38	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
39	M₃₂×C₁₃₁	—	2	2	no	—
40	M₃₂×C₁₃₁	—	2	2	no	—
41	C₂×C₁₆×C₁₃₁	—	2	2	no	—
42	#108×C₁₃₁	—	2	1	no	—
43	#108×C₁₃₁	D₄	1	2	yes	—
44	#107×C₁₃₁	D₄	1	2	yes	—
45	#106×C₁₃₁	—	2	1	no	—
46	#97×C₁₃₁	—	4	1	no	—
47	C₄×C₈×C₁₃₁	—	4	1	no	—
48	C₄×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
50	C₂×Q₈×C₁₃₁	—	2	4	no	—
51	C₂×Q₈×C₁₃₁	—	4	2	no	—
52	C₂×Q₈×C₁₃₁	—	4	2	no	—
53	C₂×Q₈×C₁₃₁	—	4	2	no	—
54	C₂×Q₈×C₁₃₁	—	4	2	no	—
55	C₂×Q₈×C₁₃₁	—	4	2	no	—
56	C₂²×C₄×C₁₃₁	—	2	1	no	—
57	C₂²×C₄×C₁₃₁	—	4	2	no	—
58	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
59	Q₁₆×C₁₃₁	—	2	4	no	—
60	Q₁₆×C₁₃₁	—	2	4	no	—
61	Q₁₆×C₁₃₁	—	4	4	no	—
62	Q₁₆×C₁₃₁	—	4	4	no	—
63	Q₁₆×C₁₃₁	—	4	2	no	—
64	Q₁₆×C₁₃₁	—	4	2	no	—
65	M₁₆×C₁₃₁	—	2	2	no	—
66	M₁₆×C₁₃₁	—	4	1	no	—
67	C₂×C₈×C₁₃₁	—	4	1	no	—
68	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₁₃₁	D₈	1	1	yes	characteristic
70	C₂×C₈×C₁₃₁	D₈	1	1	yes	characteristic
71	C₂×C₈×C₁₃₁	SD₁₆	1	2	yes	—
72	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
77	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
78	C₄⋊C₄×C₁₃₁	—	2	2	no	—
79	C₄⋊C₄×C₁₃₁	—	4	8	no	—
80	C₂²⋊C₄×C₁₃₁	—	4	1	no	—
81	C₄²×C₁₃₁	—	2	1	no	—
82	C₄²×C₁₃₁	—	4	1	no	—
83	C₄²×C₁₃₁	C₂²⋊C₄	1	2	yes	—
84	C₁₆×C₁₃₁	—	4	2	no	—
85	C₂³×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
86	Q₈×C₁₃₁	—	4	4	no	—
87	Q₈×C₁₃₁	—	4	4	no	—
88	Q₈×C₁₃₁	—	8	2	no	—
89	Q₈×C₁₃₁	—	8	2	no	—
90	C₂×C₄×C₁₃₁	—	2	4	no	—
91	C₂×C₄×C₁₃₁	—	2	2	no	—
92	C₂×C₄×C₁₃₁	—	4	4	no	—
93	C₂×C₄×C₁₃₁	—	4	2	no	—
94	C₂×C₄×C₁₃₁	—	8	2	no	—
95	C₂×C₄×C₁₃₁	—	8	2	no	—
96	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
98	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
99	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₈×C₁₃₁	—	4	2	no	—
103	C₈×C₁₃₁	#102	1	2	yes	—
104	C₈×C₁₃₁	#102	1	2	yes	—
105	C₈×C₁₃₁	C₄×D₄	1	1	yes	characteristic
106	C₈×C₁₃₁	C₄×D₄	1	1	yes	characteristic
107	C₈×C₁₃₁	#124	1	1	yes	characteristic
108	C₈×C₁₃₁	#127	1	1	yes	characteristic
109	C₂²×C₁₃₁	—	2	2	no	—
110	C₂²×C₁₃₁	#418	1	1	yes	characteristic
111	C₂²×C₁₃₁	#459	1	1	yes	characteristic
112	C₂²×C₁₃₁	#460	1	1	yes	characteristic
113	C₄×C₁₃₁	—	2	4	no	—
114	C₄×C₁₃₁	—	8	2	no	—
115	C₄×C₁₃₁	—	8	4	no	—
116	C₄×C₁₃₁	#390	1	1	yes	characteristic
117	C₄×C₁₃₁	C₂×#102	1	1	yes	characteristic
118	C₄×C₁₃₁	#459	1	1	yes	characteristic
119	C₄×C₁₃₁	#460	1	1	yes	characteristic
120	C₂×C₁₃₁	—	2	1	no	—
121	C₂×C₁₃₁	—	2	1	no	—
122	C₂×C₁₃₁	#1982	1	1	yes	characteristic
123	C₂×C₁₃₁	#2214	1	2	yes	—
124	#13262	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₁₃₁	#13262	1	1	yes	characteristic, direct product factor, Sylow
126	C₂×#494	C₂×C₁₃₁	1	1	yes	characteristic
127	C₂×#364	C₂×C₁₃₁	1	1	yes	characteristic
128	C₂×#362	C₂×C₁₃₁	1	2	yes	—
129	#2141	C₂×C₁₃₁	1	2	yes	—
130	#1772	C₂×C₁₃₁	1	1	yes	characteristic
131	C₂²×Q₁₆	—	2	1	no	—
132	C₂²×Q₁₆	C₂²×C₁₃₁	1	2	yes	—
133	C₂×C₄⋊Q₈	—	2	1	no	—
134	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
135	#494	—	2	2	no	—
136	#494	C₄×C₁₃₁	1	4	yes	complemented
137	C₂×#108	C₂²×C₁₃₁	1	1	yes	characteristic
138	#427	C₂²×C₁₃₁	1	1	yes	characteristic
139	#405	—	2	1	no	—
140	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
141	#364	—	2	2	no	—
142	#362	—	2	4	no	—
143	#358	—	2	4	no	—
144	#339	—	2	1	no	—
145	C₂²×Q₈	—	2	2	no	—
146	C₂×Q₁₆	—	2	4	no	—
147	C₂×Q₁₆	—	2	2	no	—
148	C₂×Q₁₆	—	2	2	no	—
149	C₂×Q₁₆	—	2	2	no	—
150	C₂×Q₁₆	—	2	2	no	—
151	C₂×Q₁₆	—	2	2	no	—
152	C₂×Q₁₆	—	4	1	no	—
153	C₂×Q₁₆	—	4	1	no	—
154	C₂×Q₁₆	—	4	2	no	—
155	C₂×Q₁₆	C₂×C₄×C₁₃₁	1	4	yes	—
156	C₂×M₁₆	—	2	1	no	—
157	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
158	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
159	C₄⋊Q₈	—	2	4	no	—
160	C₄⋊Q₈	—	4	2	no	—
161	#117	—	2	1	no	—
162	C₂×C₄⋊C₄	—	4	2	no	—
163	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
164	M₃₂	—	2	2	no	—
165	M₃₂	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	#108	—	2	1	no	—
168	#108	D₄×C₁₃₁	1	2	yes	—
169	#107	D₄×C₁₃₁	1	2	yes	—
170	#106	—	2	1	no	—
171	#97	—	4	1	no	—
172	C₄×C₈	—	4	1	no	—
173	C₄×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
174	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
184	Q₁₆	—	2	4	no	—
185	Q₁₆	—	2	4	no	—
186	Q₁₆	—	4	4	no	—
187	Q₁₆	—	4	4	no	—
188	Q₁₆	—	4	2	no	—
189	Q₁₆	—	4	2	no	—
190	M₁₆	—	2	2	no	—
191	M₁₆	—	4	1	no	—
192	C₂×C₈	—	4	1	no	—
193	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	2	yes	—
194	C₂×C₈	D₈×C₁₃₁	1	1	yes	characteristic
195	C₂×C₈	D₈×C₁₃₁	1	1	yes	characteristic
196	C₂×C₈	SD₁₆×C₁₃₁	1	2	yes	—
197	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
198	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
199	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
200	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
201	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
202	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	8	no	—
205	C₂²⋊C₄	—	4	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	4	1	no	—
208	C₄²	C₂²⋊C₄×C₁₃₁	1	2	yes	—
209	C₁₆	—	4	2	no	—
210	C₂³	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
211	Q₈	—	4	4	no	—
212	Q₈	—	4	4	no	—
213	Q₈	—	8	2	no	—
214	Q₈	—	8	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
222	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
223	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
224	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
225	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
226	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
227	C₈	—	4	2	no	—
228	C₈	#102×C₁₃₁	1	2	yes	—
229	C₈	#102×C₁₃₁	1	2	yes	—
230	C₈	C₄×D₄×C₁₃₁	1	1	yes	characteristic
231	C₈	C₄×D₄×C₁₃₁	1	1	yes	characteristic
232	C₈	#124×C₁₃₁	1	1	yes	characteristic
233	C₈	#127×C₁₃₁	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	#418×C₁₃₁	1	1	yes	characteristic
236	C₂²	#459×C₁₃₁	1	1	yes	characteristic
237	C₂²	#460×C₁₃₁	1	1	yes	characteristic
238	C₄	—	2	4	no	—
239	C₄	—	8	4	no	—
240	C₄	—	8	2	no	—
241	C₄	#390×C₁₃₁	1	1	yes	characteristic
242	C₄	C₂×#102×C₁₃₁	1	1	yes	characteristic
243	C₄	#459×C₁₃₁	1	1	yes	characteristic
244	C₄	#460×C₁₃₁	1	1	yes	characteristic
245	C₂	—	2	1	no	—
246	C₂	—	2	1	no	—
247	C₂	#1982×C₁₃₁	1	1	yes	characteristic
248	C₂	#2214×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	#459
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
prime radical	C₂³×C₁₃₁	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
11. #494×C₁₃₁	C₄	semidirect
124. #13262	C₁₃₁	direct (N × H)
125. C₁₃₁	#13262	direct (N × H)
136. #494	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494×C₁₃₁	2
C₂×#364×C₁₃₁	2
C₂×#362×C₁₃₁	2
#2141×C₁₃₁	2
#1772×C₁₃₁	2
#13262	131

Subgroups — C131×#13262 #45699146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13262 #45699146