Subgroups — C₁₃₁×#13254 #45699138

Order 33536.

514 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#495×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#358×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	C₂×#357×C₁₃₁	C₂	1	2	yes	maximal
5	C₄×#107×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₂²×SD₁₆×C₁₃₁	—	2	1	no	—
8	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂×#124×C₁₃₁	—	2	1	no	—
10	C₂²×C₁₆×C₁₃₁	C₂²	1	2	yes	—
11	#495×C₁₃₁	—	2	2	no	—
12	#495×C₁₃₁	C₄	1	4	yes	complemented
13	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₁₃₁	—	2	1	no	—
16	C₄⋊C₁₆×C₁₃₁	—	2	2	no	—
17	#358×C₁₃₁	—	2	4	no	—
18	#357×C₁₃₁	—	2	4	no	—
19	#340×C₁₃₁	—	2	1	no	—
20	C₂²×Q₈×C₁₃₁	—	2	1	no	—
21	C₂²×D₄×C₁₃₁	—	2	1	no	—
22	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
23	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
24	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
25	C₂×Q₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
26	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
27	C₂×SD₁₆×C₁₃₁	—	4	4	no	—
28	C₂×D₈×C₁₃₁	—	2	2	no	—
29	C₂×D₈×C₁₃₁	—	2	1	no	—
30	C₂×D₈×C₁₃₁	—	2	2	no	—
31	C₂×D₈×C₁₃₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
34	#124×C₁₃₁	—	2	4	no	—
35	#124×C₁₃₁	—	4	2	no	—
36	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₁₃₁	—	4	1	no	—
38	C₂×C₂²⋊C₄×C₁₃₁	—	4	1	no	—
39	C₂×C₄²×C₁₃₁	—	4	1	no	—
40	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₁₃₁	—	2	2	no	—
42	C₂×C₁₆×C₁₃₁	—	2	2	no	—
43	C₂×C₁₆×C₁₃₁	—	2	2	no	—
44	#107×C₁₃₁	—	2	2	no	—
45	#107×C₁₃₁	D₄	1	4	yes	—
46	C₄×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
48	#95×C₁₃₁	—	4	1	no	—
49	C₂⁴×C₁₃₁	—	4	1	no	—
50	C₂×Q₈×C₁₃₁	—	2	2	no	—
51	C₂×Q₈×C₁₃₁	—	4	1	no	—
52	C₂×Q₈×C₁₃₁	—	4	2	no	—
53	C₂×Q₈×C₁₃₁	—	4	2	no	—
54	C₂×D₄×C₁₃₁	—	2	2	no	—
55	C₂×D₄×C₁₃₁	—	4	2	no	—
56	C₂×D₄×C₁₃₁	—	4	1	no	—
57	C₂×D₄×C₁₃₁	—	4	2	no	—
58	C₂²×C₄×C₁₃₁	—	4	2	no	—
59	C₂²×C₄×C₁₃₁	—	4	1	no	—
60	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₁₃₁	—	2	4	no	—
64	Q₁₆×C₁₃₁	—	4	1	no	—
65	Q₁₆×C₁₃₁	—	4	1	no	—
66	SD₁₆×C₁₃₁	—	4	8	no	—
67	D₈×C₁₃₁	—	2	4	no	—
68	D₈×C₁₃₁	—	4	1	no	—
69	D₈×C₁₃₁	—	4	1	no	—
70	C₂×C₈×C₁₃₁	—	2	2	no	—
71	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃₁	#41	1	2	yes	—
76	C₄⋊C₄×C₁₃₁	—	2	4	no	—
77	C₄⋊C₄×C₁₃₁	—	4	2	no	—
78	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
79	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
80	C₄²×C₁₃₁	—	8	2	no	—
81	C₄²×C₁₃₁	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₁₃₁	SD₁₆	1	2	yes	—
83	C₁₆×C₁₃₁	—	4	2	no	—
84	C₂³×C₁₃₁	—	4	2	no	—
85	C₂³×C₁₃₁	—	8	2	no	—
86	C₂³×C₁₃₁	—	8	2	no	—
87	C₂³×C₁₃₁	—	8	1	no	—
88	C₂³×C₁₃₁	—	8	1	no	—
89	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
90	Q₈×C₁₃₁	—	4	2	no	—
91	Q₈×C₁₃₁	—	4	2	no	—
92	Q₈×C₁₃₁	—	8	2	no	—
93	D₄×C₁₃₁	—	4	2	no	—
94	D₄×C₁₃₁	—	4	2	no	—
95	D₄×C₁₃₁	—	8	2	no	—
96	C₂×C₄×C₁₃₁	—	2	2	no	—
97	C₂×C₄×C₁₃₁	—	2	1	no	—
98	C₂×C₄×C₁₃₁	—	2	1	no	—
99	C₂×C₄×C₁₃₁	—	4	2	no	—
100	C₂×C₄×C₁₃₁	—	4	4	no	—
101	C₂×C₄×C₁₃₁	—	8	2	no	—
102	C₂×C₄×C₁₃₁	—	8	2	no	—
103	C₂×C₄×C₁₃₁	—	8	2	no	—
104	C₂×C₄×C₁₃₁	#102	1	2	yes	—
105	C₂×C₄×C₁₃₁	#102	1	2	yes	—
106	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₁	#121	1	2	yes	—
108	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
110	C₈×C₁₃₁	—	2	2	no	—
111	C₈×C₁₃₁	C₄×D₄	1	2	yes	—
112	C₈×C₁₃₁	#124	1	1	yes	characteristic
113	C₈×C₁₃₁	#127	1	1	yes	characteristic
114	C₂²×C₁₃₁	—	8	2	no	—
115	C₂²×C₁₃₁	—	8	2	no	—
116	C₂²×C₁₃₁	—	8	2	no	—
117	C₂²×C₁₃₁	—	16	2	no	—
118	C₂²×C₁₃₁	—	16	2	no	—
119	C₂²×C₁₃₁	C₂×#102	1	1	yes	characteristic
120	C₂²×C₁₃₁	#459	1	2	yes	—
121	C₂²×C₁₃₁	#508	1	2	yes	—
122	C₂²×C₁₃₁	#509	1	1	yes	characteristic
123	C₂²×C₁₃₁	#510	1	1	yes	characteristic
124	C₄×C₁₃₁	—	2	2	no	—
125	C₄×C₁₃₁	—	2	2	no	—
126	C₄×C₁₃₁	—	8	4	no	—
127	C₄×C₁₃₁	—	8	2	no	—
128	C₄×C₁₃₁	#390	1	1	yes	characteristic
129	C₄×C₁₃₁	#418	1	1	yes	characteristic
130	C₄×C₁₃₁	#460	1	2	yes	—
131	C₂×C₁₃₁	—	16	2	no	—
132	C₂×C₁₃₁	#1982	1	1	yes	characteristic
133	C₂×C₁₃₁	#2138	1	1	yes	characteristic
134	C₂×C₁₃₁	#2139	1	1	yes	characteristic
135	C₂×C₁₃₁	#2210	1	2	yes	—
136	C₂×C₁₃₁	#2211	1	2	yes	—
137	#13254	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₁₃₁	#13254	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×#495	C₂×C₁₃₁	1	1	yes	characteristic
140	C₂×C₄⋊C₁₆	C₂×C₁₃₁	1	1	yes	characteristic, complemented
141	C₂×#358	C₂×C₁₃₁	1	2	yes	complemented
142	C₂×#357	C₂×C₁₃₁	1	2	yes	—
143	C₄×#107	C₂×C₁₃₁	1	1	yes	characteristic, complemented
144	C₂²×Q₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
145	C₂²×SD₁₆	—	2	1	no	—
146	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
147	C₂×#124	—	2	1	no	—
148	C₂²×C₁₆	C₂²×C₁₃₁	1	2	yes	—
149	#495	—	2	2	no	—
150	#495	C₄×C₁₃₁	1	4	yes	complemented
151	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
152	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
153	C₄×C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₁₆	—	2	2	no	—
155	#358	—	2	4	no	—
156	#357	—	2	4	no	—
157	#340	—	2	1	no	—
158	C₂²×Q₈	—	2	1	no	—
159	C₂²×D₄	—	2	1	no	—
160	C₂×Q₁₆	—	2	2	no	—
161	C₂×Q₁₆	—	2	1	no	—
162	C₂×Q₁₆	—	2	2	no	—
163	C₂×Q₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
164	C₂×SD₁₆	—	2	4	no	—
165	C₂×SD₁₆	—	4	4	no	—
166	C₂×D₈	—	2	2	no	—
167	C₂×D₈	—	2	1	no	—
168	C₂×D₈	—	2	2	no	—
169	C₂×D₈	C₂×C₄×C₁₃₁	1	2	yes	—
170	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
172	#124	—	2	4	no	—
173	#124	—	4	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₂²⋊C₄	—	4	1	no	—
176	C₂×C₂²⋊C₄	—	4	1	no	—
177	C₂×C₄²	—	4	1	no	—
178	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	2	2	no	—
181	C₂×C₁₆	—	2	2	no	—
182	#107	—	2	2	no	—
183	#107	D₄×C₁₃₁	1	4	yes	—
184	C₄×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
185	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
186	#95	—	4	1	no	—
187	C₂⁴	—	4	1	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	4	1	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	—	4	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
199	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
201	Q₁₆	—	2	4	no	—
202	Q₁₆	—	4	1	no	—
203	Q₁₆	—	4	1	no	—
204	SD₁₆	—	4	8	no	—
205	D₈	—	2	4	no	—
206	D₈	—	4	1	no	—
207	D₈	—	4	1	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	2	yes	—
210	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
211	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
212	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
213	C₂×C₈	#41×C₁₃₁	1	2	yes	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₂²⋊C₄	—	4	4	no	—
217	C₂²⋊C₄	—	4	4	no	—
218	C₄²	—	8	2	no	—
219	C₄²	C₂²⋊C₄×C₁₃₁	1	2	yes	—
220	C₄²	SD₁₆×C₁₃₁	1	2	yes	—
221	C₁₆	—	4	2	no	—
222	C₂³	—	4	2	no	—
223	C₂³	—	8	2	no	—
224	C₂³	—	8	2	no	—
225	C₂³	—	8	1	no	—
226	C₂³	—	8	1	no	—
227	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
228	Q₈	—	4	2	no	—
229	Q₈	—	4	2	no	—
230	Q₈	—	8	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	4	2	no	—
233	D₄	—	8	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	#102×C₁₃₁	1	2	yes	—
243	C₂×C₄	#102×C₁₃₁	1	2	yes	—
244	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
245	C₂×C₄	#121×C₁₃₁	1	2	yes	—
246	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
247	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
248	C₈	—	2	2	no	—
249	C₈	C₄×D₄×C₁₃₁	1	2	yes	—
250	C₈	#124×C₁₃₁	1	1	yes	characteristic
251	C₈	#127×C₁₃₁	1	1	yes	characteristic
252	C₂²	—	8	2	no	—
253	C₂²	—	8	2	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	16	2	no	—
256	C₂²	—	16	2	no	—
257	C₂²	C₂×#102×C₁₃₁	1	1	yes	characteristic
258	C₂²	#459×C₁₃₁	1	2	yes	—
259	C₂²	#508×C₁₃₁	1	2	yes	—
260	C₂²	#509×C₁₃₁	1	1	yes	characteristic
261	C₂²	#510×C₁₃₁	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	8	2	no	—
265	C₄	—	8	4	no	—
266	C₄	#390×C₁₃₁	1	1	yes	characteristic
267	C₄	#418×C₁₃₁	1	1	yes	characteristic
268	C₄	#460×C₁₃₁	1	2	yes	—
269	C₂	—	16	2	no	—
270	C₂	#1982×C₁₃₁	1	1	yes	characteristic
271	C₂	#2138×C₁₃₁	1	1	yes	characteristic
272	C₂	#2139×C₁₃₁	1	1	yes	characteristic
273	C₂	#2210×C₁₃₁	1	2	yes	—
274	C₂	#2211×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
prime radical	C₂²×D₈×C₁₃₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
2. C₂×C₄⋊C₁₆×C₁₃₁	C₂	semidirect
3. C₂×#358×C₁₃₁	C₂	semidirect
5. C₄×#107×C₁₃₁	C₂	semidirect
12. #495×C₁₃₁	C₄	semidirect
137. #13254	C₁₃₁	direct (N × H)
138. C₁₃₁	#13254	direct (N × H)
140. C₂×C₄⋊C₁₆	C₂×C₁₃₁	semidirect
141. C₂×#358	C₂×C₁₃₁	semidirect
143. C₄×#107	C₂×C₁₃₁	semidirect
150. #495	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×#495×C₁₃₁	2
C₂×C₄⋊C₁₆×C₁₃₁	2
C₂×#358×C₁₃₁	2
C₂×#357×C₁₃₁	2
C₄×#107×C₁₃₁	2
#13254	131

Subgroups — C131×#13254 #45699138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#13254 #45699138