Subgroups — C₁₃₁×#33240 #45719124

Order 33536.

946 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×#499×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#495×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#488×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#2861×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1970×C₁₃₁	C₂	1	2	yes	maximal
6	#1969×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#1968×C₁₃₁	C₂	1	4	yes	maximal, complemented
8	C₄²×C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂²×SD₁₆×C₁₃₁	C₂²	1	2	yes	—
11	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×#125×C₁₃₁	C₂²	1	2	yes	—
13	C₂×#124×C₁₃₁	C₂²	1	2	yes	—
14	#499×C₁₃₁	—	2	4	no	—
15	#495×C₁₃₁	—	2	4	no	—
16	#488×C₁₃₁	—	2	8	no	—
17	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
18	C₂×#106×C₁₃₁	C₂²	1	2	yes	—
19	C₂×#103×C₁₃₁	C₂²	1	4	yes	—
20	C₂×#102×C₁₃₁	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
25	#391×C₁₃₁	C₂²	1	4	yes	—
26	#390×C₁₃₁	C₂²	1	4	yes	—
27	#388×C₁₃₁	—	2	8	no	—
28	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
29	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
30	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
31	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
32	C₂×Q₁₆×C₁₃₁	—	2	4	no	—
33	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
34	C₂×SD₁₆×C₁₃₁	—	2	8	no	—
35	C₂×D₈×C₁₃₁	—	2	2	no	—
36	C₂×D₈×C₁₃₁	—	2	4	no	—
37	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
40	#125×C₁₃₁	—	2	4	no	—
41	#125×C₁₃₁	—	2	4	no	—
42	#124×C₁₃₁	—	2	4	no	—
43	#124×C₁₃₁	—	2	4	no	—
44	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
45	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
46	C₂×C₂²⋊C₄×C₁₃₁	—	2	8	no	—
47	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
48	C₂×C₄²×C₁₃₁	D₄	1	4	yes	—
49	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
51	#107×C₁₃₁	—	2	4	no	—
52	#106×C₁₃₁	—	2	4	no	—
53	#103×C₁₃₁	—	2	8	no	—
54	#102×C₁₃₁	—	2	8	no	—
55	C₄×C₈×C₁₃₁	—	2	4	no	—
56	C₄×C₈×C₁₃₁	—	2	4	no	—
57	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
58	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
59	#95×C₁₃₁	—	2	8	no	—
60	C₂⁴×C₁₃₁	—	2	2	no	—
61	C₂×Q₈×C₁₃₁	—	2	2	no	—
62	C₂×Q₈×C₁₃₁	—	2	2	no	—
63	C₂×Q₈×C₁₃₁	—	2	4	no	—
64	C₂×Q₈×C₁₃₁	—	2	4	no	—
65	C₂×D₄×C₁₃₁	—	2	2	no	—
66	C₂×D₄×C₁₃₁	—	2	2	no	—
67	C₂×D₄×C₁₃₁	—	2	4	no	—
68	C₂×D₄×C₁₃₁	—	2	4	no	—
69	C₂²×C₄×C₁₃₁	—	2	2	no	—
70	C₂²×C₄×C₁₃₁	—	2	4	no	—
71	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
74	Q₁₆×C₁₃₁	—	4	4	no	—
75	SD₁₆×C₁₃₁	—	4	8	no	—
76	D₈×C₁₃₁	—	4	4	no	—
77	C₂×C₈×C₁₃₁	—	2	2	no	—
78	C₂×C₈×C₁₃₁	—	2	2	no	—
79	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₃₁	#41	1	4	yes	—
83	C₂×C₈×C₁₃₁	#41	1	4	yes	—
84	C₄⋊C₄×C₁₃₁	—	2	4	no	—
85	C₄⋊C₄×C₁₃₁	—	2	4	no	—
86	C₄⋊C₄×C₁₃₁	—	4	8	no	—
87	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
88	C₂²⋊C₄×C₁₃₁	—	4	16	no	—
89	C₄²×C₁₃₁	—	2	8	no	—
90	C₄²×C₁₃₁	—	2	2	no	—
91	C₄²×C₁₃₁	—	2	2	no	—
92	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
93	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
94	C₂³×C₁₃₁	—	4	2	no	—
95	C₂³×C₁₃₁	—	4	4	no	—
96	C₂³×C₁₃₁	—	4	8	no	—
97	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₁₃₁	—	4	2	no	—
99	Q₈×C₁₃₁	—	4	2	no	—
100	Q₈×C₁₃₁	—	4	4	no	—
101	D₄×C₁₃₁	—	4	2	no	—
102	D₄×C₁₃₁	—	4	2	no	—
103	D₄×C₁₃₁	—	4	4	no	—
104	C₂×C₄×C₁₃₁	—	2	4	no	—
105	C₂×C₄×C₁₃₁	—	2	4	no	—
106	C₂×C₄×C₁₃₁	—	2	4	no	—
107	C₂×C₄×C₁₃₁	—	4	2	no	—
108	C₂×C₄×C₁₃₁	—	4	4	no	—
109	C₂×C₄×C₁₃₁	—	4	4	no	—
110	C₂×C₄×C₁₃₁	—	4	8	no	—
111	C₂×C₄×C₁₃₁	#127	1	4	yes	—
112	C₂×C₄×C₁₃₁	#135	1	8	yes	—
113	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
116	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
117	C₈×C₁₃₁	—	2	4	no	—
118	C₈×C₁₃₁	#124	1	4	yes	—
119	C₈×C₁₃₁	#124	1	4	yes	—
120	C₂²×C₁₃₁	—	8	2	no	—
121	C₂²×C₁₃₁	—	8	4	no	—
122	C₂²×C₁₃₁	—	8	8	no	—
123	C₂²×C₁₃₁	C₂×#127	1	1	yes	characteristic
124	C₂²×C₁₃₁	#532	1	2	yes	—
125	C₂²×C₁₃₁	C₂×#135	1	4	yes	—
126	C₄×C₁₃₁	—	2	4	no	—
127	C₄×C₁₃₁	—	2	8	no	—
128	C₄×C₁₃₁	—	8	2	no	—
129	C₄×C₁₃₁	—	8	4	no	—
130	C₄×C₁₃₁	C₂×#124	1	1	yes	characteristic
131	C₄×C₁₃₁	C₂×#124	1	1	yes	characteristic
132	C₄×C₁₃₁	#532	1	2	yes	—
133	C₂×C₁₃₁	—	16	2	no	—
134	C₂×C₁₃₁	#2858	1	1	yes	characteristic
135	C₂×C₁₃₁	#3147	1	2	yes	—
136	C₂×C₁₃₁	#3168	1	4	yes	—
137	#33240	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₁₃₁	#33240	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×#499	C₂×C₁₃₁	1	1	yes	characteristic, complemented
140	C₂×#495	C₂×C₁₃₁	1	1	yes	characteristic
141	C₂×#488	C₂×C₁₃₁	1	2	yes	complemented
142	#2861	C₂×C₁₃₁	1	2	yes	complemented
143	#1970	C₂×C₁₃₁	1	2	yes	—
144	#1969	C₂×C₁₃₁	1	2	yes	complemented
145	#1968	C₂×C₁₃₁	1	4	yes	complemented
146	C₄²×C₈	C₂×C₁₃₁	1	1	yes	characteristic, complemented
147	C₂²×Q₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
148	C₂²×SD₁₆	C₂²×C₁₃₁	1	2	yes	—
149	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
150	C₂×#125	C₂²×C₁₃₁	1	2	yes	—
151	C₂×#124	C₂²×C₁₃₁	1	2	yes	—
152	#499	—	2	4	no	—
153	#495	—	2	4	no	—
154	#488	—	2	8	no	—
155	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
156	C₂×#106	C₂²×C₁₃₁	1	2	yes	—
157	C₂×#103	C₂²×C₁₃₁	1	4	yes	—
158	C₂×#102	C₂²×C₁₃₁	1	4	yes	—
159	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
160	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
161	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
163	#391	C₂²×C₁₃₁	1	4	yes	—
164	#390	C₂²×C₁₃₁	1	4	yes	—
165	#388	—	2	8	no	—
166	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
167	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
168	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
169	C₂×Q₁₆	—	2	2	no	—
170	C₂×Q₁₆	—	2	4	no	—
171	C₂×SD₁₆	—	2	4	no	—
172	C₂×SD₁₆	—	2	8	no	—
173	C₂×D₈	—	2	2	no	—
174	C₂×D₈	—	2	4	no	—
175	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
176	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
177	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
178	#125	—	2	4	no	—
179	#125	—	2	4	no	—
180	#124	—	2	4	no	—
181	#124	—	2	4	no	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
184	C₂×C₂²⋊C₄	—	2	4	no	—
185	C₂×C₂²⋊C₄	—	2	8	no	—
186	C₂×C₄²	D₄×C₁₃₁	1	4	yes	—
187	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
188	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
189	#107	—	2	4	no	—
190	#106	—	2	4	no	—
191	#103	—	2	8	no	—
192	#102	—	2	8	no	—
193	C₄×C₈	—	2	4	no	—
194	C₄×C₈	—	2	4	no	—
195	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
196	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
197	#95	—	2	8	no	—
198	C₂⁴	—	2	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	2	4	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
211	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
212	Q₁₆	—	4	4	no	—
213	SD₁₆	—	4	8	no	—
214	D₈	—	4	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
218	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
219	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
220	C₂×C₈	#41×C₁₃₁	1	4	yes	—
221	C₂×C₈	#41×C₁₃₁	1	4	yes	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	4	8	no	—
225	C₂²⋊C₄	—	4	8	no	—
226	C₂²⋊C₄	—	4	16	no	—
227	C₄²	—	2	8	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
231	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
232	C₂³	—	4	2	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	8	no	—
235	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
236	Q₈	—	4	2	no	—
237	Q₈	—	4	2	no	—
238	Q₈	—	4	4	no	—
239	D₄	—	4	2	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	8	no	—
249	C₂×C₄	#127×C₁₃₁	1	4	yes	—
250	C₂×C₄	#135×C₁₃₁	1	8	yes	—
251	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
253	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
254	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
255	C₈	—	2	4	no	—
256	C₈	#124×C₁₃₁	1	4	yes	—
257	C₈	#124×C₁₃₁	1	4	yes	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	4	no	—
260	C₂²	—	8	8	no	—
261	C₂²	C₂×#127×C₁₃₁	1	1	yes	characteristic
262	C₂²	#532×C₁₃₁	1	2	yes	—
263	C₂²	C₂×#135×C₁₃₁	1	4	yes	—
264	C₄	—	2	4	no	—
265	C₄	—	2	8	no	—
266	C₄	—	8	2	no	—
267	C₄	—	8	4	no	—
268	C₄	C₂×#124×C₁₃₁	1	1	yes	characteristic
269	C₄	C₂×#124×C₁₃₁	1	1	yes	characteristic
270	C₄	#532×C₁₃₁	1	2	yes	—
271	C₂	—	16	2	no	—
272	C₂	#2858×C₁₃₁	1	1	yes	characteristic
273	C₂	#3147×C₁₃₁	1	2	yes	—
274	C₂	#3168×C₁₃₁	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂²×D₈×C₁₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#499×C₁₃₁	C₂	semidirect
3. C₂×#488×C₁₃₁	C₂	semidirect
4. #2861×C₁₃₁	C₂	semidirect
6. #1969×C₁₃₁	C₂	semidirect
7. #1968×C₁₃₁	C₂	semidirect
8. C₄²×C₈×C₁₃₁	C₂	semidirect
137. #33240	C₁₃₁	direct (N × H)
138. C₁₃₁	#33240	direct (N × H)
139. C₂×#499	C₂×C₁₃₁	semidirect
141. C₂×#488	C₂×C₁₃₁	semidirect
142. #2861	C₂×C₁₃₁	semidirect
144. #1969	C₂×C₁₃₁	semidirect
145. #1968	C₂×C₁₃₁	semidirect
146. C₄²×C₈	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#499×C₁₃₁	2
C₂×#495×C₁₃₁	2
C₂×#488×C₁₃₁	2
#2861×C₁₃₁	2
#1970×C₁₃₁	2
#1969×C₁₃₁	2
#1968×C₁₃₁	2
C₄²×C₈×C₁₃₁	2
#33240	131

Subgroups — C131×#33240 #45719124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33240 #45719124