Subgroups — C₁₃₁×#33465 #45719349

Order 33536.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2957×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#2179×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	#2178×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	C₂×#368×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#2143×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#2108×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₁₃₁	C₂²	1	2	yes	—
9	#568×C₁₃₁	—	2	2	no	—
10	#508×C₁₃₁	C₄	1	2	yes	complemented
11	#508×C₁₃₁	C₄	1	2	yes	complemented
12	#508×C₁₃₁	C₂²	1	2	yes	—
13	#508×C₁₃₁	C₂²	1	2	yes	—
14	C₂×Q₃₂×C₁₃₁	C₄	1	1	yes	characteristic, complemented
15	C₂×SD₃₂×C₁₃₁	C₄	1	2	yes	complemented
16	C₂×D₁₆×C₁₃₁	C₄	1	1	yes	characteristic, complemented
17	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
19	#444×C₁₃₁	—	2	4	no	—
20	#443×C₁₃₁	C₂²	1	8	yes	complemented
21	C₂×#108×C₁₃₁	C₂²	1	2	yes	—
22	#421×C₁₃₁	—	2	2	no	—
23	#405×C₁₃₁	C₂²	1	1	yes	characteristic
24	#368×C₁₃₁	C₂²	1	2	yes	complemented
25	#368×C₁₃₁	C₂²	1	2	yes	complemented
26	#359×C₁₃₁	C₂²	1	8	yes	complemented
27	#346×C₁₃₁	C₂²	1	2	yes	complemented
28	C₄×C₁₆×C₁₃₁	C₂²	1	2	yes	complemented
29	C₂×#41×C₁₃₁	—	2	2	no	—
30	#135×C₁₃₁	—	2	2	no	—
31	#135×C₁₃₁	—	2	2	no	—
32	#135×C₁₃₁	C₂×C₄	1	4	yes	—
33	#135×C₁₃₁	C₂³	1	4	yes	—
34	C₂×Q₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
35	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
36	C₂×D₈×C₁₃₁	C₂×C₄	1	2	yes	—
37	#131×C₁₃₁	—	2	8	no	—
38	#131×C₁₃₁	—	4	4	no	—
39	C₂×M₁₆×C₁₃₁	—	2	2	no	—
40	C₂×M₁₆×C₁₃₁	—	4	4	no	—
41	C₂×M₁₆×C₁₃₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
43	#117×C₁₃₁	D₄	1	1	yes	characteristic
44	Q₃₂×C₁₃₁	C₂×C₄	1	2	yes	—
45	Q₃₂×C₁₃₁	C₂×C₄	1	2	yes	—
46	SD₃₂×C₁₃₁	C₂×C₄	1	4	yes	—
47	SD₃₂×C₁₃₁	C₂×C₄	1	4	yes	—
48	D₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
49	D₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
50	M₃₂×C₁₃₁	—	2	1	no	—
51	M₃₂×C₁₃₁	—	2	1	no	—
52	C₂×C₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
57	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
59	#108×C₁₃₁	—	2	2	no	—
60	#108×C₁₃₁	C₂³	1	4	yes	—
61	#104×C₁₃₁	—	2	8	no	—
62	#97×C₁₃₁	D₄	1	2	yes	—
63	C₄×C₈×C₁₃₁	C₂³	1	2	yes	—
64	#41×C₁₃₁	—	2	4	no	—
65	#41×C₁₃₁	—	2	4	no	—
66	#41×C₁₃₁	—	4	2	no	—
67	#41×C₁₃₁	—	4	2	no	—
68	C₂×Q₈×C₁₃₁	—	2	2	no	—
69	C₂×D₄×C₁₃₁	—	2	2	no	—
70	C₂×D₄×C₁₃₁	—	4	2	no	—
71	C₂²×C₄×C₁₃₁	—	4	2	no	—
72	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₁₃₁	—	2	2	no	—
74	Q₁₆×C₁₃₁	C₂²×C₄	1	4	yes	—
75	SD₁₆×C₁₃₁	—	2	4	no	—
76	SD₁₆×C₁₃₁	—	2	4	no	—
77	D₈×C₁₃₁	—	2	2	no	—
78	D₈×C₁₃₁	C₂²×C₄	1	4	yes	—
79	M₁₆×C₁₃₁	—	2	4	no	—
80	M₁₆×C₁₃₁	—	4	2	no	—
81	M₁₆×C₁₃₁	—	4	8	no	—
82	M₁₆×C₁₃₁	D₈	1	2	yes	—
83	M₁₆×C₁₃₁	D₈	1	2	yes	—
84	C₂×C₈×C₁₃₁	—	4	2	no	—
85	C₂×C₈×C₁₃₁	—	4	8	no	—
86	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
91	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
92	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₁₃₁	—	2	1	no	—
94	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
95	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
96	C₁₆×C₁₃₁	—	2	2	no	—
97	C₁₆×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₁₆×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₁₆×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₁₃₁	—	4	2	no	—
102	C₂³×C₁₃₁	#135	1	1	yes	characteristic
103	Q₈×C₁₃₁	—	2	4	no	—
104	Q₈×C₁₃₁	—	4	2	no	—
105	D₄×C₁₃₁	—	2	4	no	—
106	D₄×C₁₃₁	—	4	4	no	—
107	D₄×C₁₃₁	—	4	4	no	—
108	D₄×C₁₃₁	—	4	2	no	—
109	C₂×C₄×C₁₃₁	—	2	2	no	—
110	C₂×C₄×C₁₃₁	—	4	2	no	—
111	C₂×C₄×C₁₃₁	—	4	4	no	—
112	C₂×C₄×C₁₃₁	—	4	4	no	—
113	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃₁	#135	1	1	yes	characteristic
118	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
119	C₈×C₁₃₁	—	4	4	no	—
120	C₈×C₁₃₁	C₄×D₄	1	1	yes	characteristic
121	C₈×C₁₃₁	C₄×D₄	1	1	yes	characteristic
122	C₈×C₁₃₁	C₂×D₈	1	4	yes	—
123	C₈×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
124	C₈×C₁₃₁	C₂×#41	1	1	yes	characteristic
125	C₂²×C₁₃₁	—	4	1	no	—
126	C₂²×C₁₃₁	—	4	4	no	—
127	C₂²×C₁₃₁	—	16	2	no	—
128	C₂²×C₁₃₁	C₄×D₈	1	1	yes	characteristic
129	C₂²×C₁₃₁	C₄×D₈	1	1	yes	characteristic
130	C₂²×C₁₃₁	C₂×#135	1	1	yes	characteristic
131	C₄×C₁₃₁	—	4	2	no	—
132	C₄×C₁₃₁	—	4	4	no	—
133	C₄×C₁₃₁	C₄×D₈	1	1	yes	characteristic
134	C₄×C₁₃₁	C₄×D₈	1	1	yes	characteristic
135	C₄×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
136	C₄×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
137	C₂×C₁₃₁	—	2	1	no	—
138	C₂×C₁₃₁	—	2	1	no	—
139	C₂×C₁₃₁	—	2	1	no	—
140	C₂×C₁₃₁	—	8	4	no	—
141	C₂×C₁₃₁	C₂×C₄×D₈	1	1	yes	characteristic
142	#33465	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₁₃₁	#33465	1	1	yes	characteristic, direct product factor, Sylow
144	C₂×#508	C₂×C₁₃₁	1	1	yes	characteristic
145	#2957	C₂×C₁₃₁	1	2	yes	complemented
146	#2179	C₂×C₁₃₁	1	4	yes	complemented
147	#2178	C₂×C₁₃₁	1	4	yes	complemented
148	C₂×#368	C₂×C₁₃₁	1	1	yes	characteristic, complemented
149	#2143	C₂×C₁₃₁	1	2	yes	complemented
150	#2108	C₂×C₁₃₁	1	1	yes	characteristic, complemented
151	C₂×#135	C₂²×C₁₃₁	1	2	yes	—
152	#568	—	2	2	no	—
153	#508	C₄×C₁₃₁	1	2	yes	complemented
154	#508	C₄×C₁₃₁	1	2	yes	complemented
155	#508	C₂²×C₁₃₁	1	2	yes	—
156	#508	C₂²×C₁₃₁	1	2	yes	—
157	C₂×Q₃₂	C₄×C₁₃₁	1	1	yes	characteristic, complemented
158	C₂×SD₃₂	C₄×C₁₃₁	1	2	yes	complemented
159	C₂×D₁₆	C₄×C₁₃₁	1	1	yes	characteristic, complemented
160	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
161	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
162	#444	—	2	4	no	—
163	#443	C₂²×C₁₃₁	1	8	yes	complemented
164	C₂×#108	C₂²×C₁₃₁	1	2	yes	—
165	#421	—	2	2	no	—
166	#405	C₂²×C₁₃₁	1	1	yes	characteristic
167	#368	C₂²×C₁₃₁	1	2	yes	complemented
168	#368	C₂²×C₁₃₁	1	2	yes	complemented
169	#359	C₂²×C₁₃₁	1	8	yes	complemented
170	#346	C₂²×C₁₃₁	1	2	yes	complemented
171	C₄×C₁₆	C₂²×C₁₃₁	1	2	yes	complemented
172	C₂×#41	—	2	2	no	—
173	#135	—	2	2	no	—
174	#135	—	2	2	no	—
175	#135	C₂×C₄×C₁₃₁	1	4	yes	—
176	#135	C₂³×C₁₃₁	1	4	yes	—
177	C₂×Q₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
178	C₂×SD₁₆	—	2	2	no	—
179	C₂×D₈	C₂×C₄×C₁₃₁	1	2	yes	—
180	#131	—	2	8	no	—
181	#131	—	4	4	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	4	4	no	—
184	C₂×M₁₆	D₄×C₁₃₁	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
186	#117	D₄×C₁₃₁	1	1	yes	characteristic
187	Q₃₂	C₂×C₄×C₁₃₁	1	2	yes	—
188	Q₃₂	C₂×C₄×C₁₃₁	1	2	yes	—
189	SD₃₂	C₂×C₄×C₁₃₁	1	4	yes	—
190	SD₃₂	C₂×C₄×C₁₃₁	1	4	yes	—
191	D₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
192	D₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
193	M₃₂	—	2	1	no	—
194	M₃₂	—	2	1	no	—
195	C₂×C₁₆	C₂×C₄×C₁₃₁	1	1	yes	characteristic
196	C₂×C₁₆	C₂×C₄×C₁₃₁	1	1	yes	characteristic
197	C₂×C₁₆	C₂×C₄×C₁₃₁	1	1	yes	characteristic
198	C₂×C₁₆	C₂×C₄×C₁₃₁	1	1	yes	characteristic
199	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
200	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
201	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
202	#108	—	2	2	no	—
203	#108	C₂³×C₁₃₁	1	4	yes	—
204	#104	—	2	8	no	—
205	#97	D₄×C₁₃₁	1	2	yes	—
206	C₄×C₈	C₂³×C₁₃₁	1	2	yes	—
207	#41	—	2	4	no	—
208	#41	—	2	4	no	—
209	#41	—	4	2	no	—
210	#41	—	4	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
216	Q₁₆	—	2	2	no	—
217	Q₁₆	C₂²×C₄×C₁₃₁	1	4	yes	—
218	SD₁₆	—	2	4	no	—
219	SD₁₆	—	2	4	no	—
220	D₈	—	2	2	no	—
221	D₈	C₂²×C₄×C₁₃₁	1	4	yes	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	8	no	—
225	M₁₆	D₈×C₁₃₁	1	2	yes	—
226	M₁₆	D₈×C₁₃₁	1	2	yes	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	8	no	—
229	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
230	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
232	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
233	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
234	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
235	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	2	1	no	—
238	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
239	C₁₆	—	2	2	no	—
240	C₁₆	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
241	C₁₆	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
242	C₁₆	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
243	C₁₆	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
244	C₂³	—	4	2	no	—
245	C₂³	#135×C₁₃₁	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	4	2	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
257	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
259	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
260	C₂×C₄	#135×C₁₃₁	1	1	yes	characteristic
261	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
262	C₈	—	4	4	no	—
263	C₈	C₄×D₄×C₁₃₁	1	1	yes	characteristic
264	C₈	C₄×D₄×C₁₃₁	1	1	yes	characteristic
265	C₈	C₂×D₈×C₁₃₁	1	4	yes	—
266	C₈	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
267	C₈	C₂×#41×C₁₃₁	1	1	yes	characteristic
268	C₂²	—	4	1	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	16	2	no	—
271	C₂²	C₄×D₈×C₁₃₁	1	1	yes	characteristic
272	C₂²	C₄×D₈×C₁₃₁	1	1	yes	characteristic
273	C₂²	C₂×#135×C₁₃₁	1	1	yes	characteristic
274	C₄	—	4	2	no	—
275	C₄	—	4	4	no	—
276	C₄	C₄×D₈×C₁₃₁	1	1	yes	characteristic
277	C₄	C₄×D₈×C₁₃₁	1	1	yes	characteristic
278	C₄	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
279	C₄	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	8	4	no	—
284	C₂	C₂×C₄×D₈×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₁	C₂²×D₈
commutator	C₈	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
socle	C₂×C₁₃₁	C₂×C₄×D₈
prime radical	C₂×#508×C₁₃₁	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₁ ◁ C₂×M₁₆×C₁₃₁ ◁ #405×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2957×C₁₃₁	C₂	semidirect
3. #2179×C₁₃₁	C₂	semidirect
4. #2178×C₁₃₁	C₂	semidirect
5. C₂×#368×C₁₃₁	C₂	semidirect
6. #2143×C₁₃₁	C₂	semidirect
7. #2108×C₁₃₁	C₂	semidirect
10. #508×C₁₃₁	C₄	semidirect
11. #508×C₁₃₁	C₄	semidirect
14. C₂×Q₃₂×C₁₃₁	C₄	semidirect
15. C₂×SD₃₂×C₁₃₁	C₄	semidirect
16. C₂×D₁₆×C₁₃₁	C₄	semidirect
20. #443×C₁₃₁	C₂²	semidirect
24. #368×C₁₃₁	C₂²	semidirect
25. #368×C₁₃₁	C₂²	semidirect
26. #359×C₁₃₁	C₂²	semidirect
27. #346×C₁₃₁	C₂²	semidirect
28. C₄×C₁₆×C₁₃₁	C₂²	semidirect
142. #33465	C₁₃₁	direct (N × H)
143. C₁₃₁	#33465	direct (N × H)
145. #2957	C₂×C₁₃₁	semidirect
146. #2179	C₂×C₁₃₁	semidirect
147. #2178	C₂×C₁₃₁	semidirect
148. C₂×#368	C₂×C₁₃₁	semidirect
149. #2143	C₂×C₁₃₁	semidirect
150. #2108	C₂×C₁₃₁	semidirect
153. #508	C₄×C₁₃₁	semidirect
154. #508	C₄×C₁₃₁	semidirect
157. C₂×Q₃₂	C₄×C₁₃₁	semidirect
158. C₂×SD₃₂	C₄×C₁₃₁	semidirect
159. C₂×D₁₆	C₄×C₁₃₁	semidirect
163. #443	C₂²×C₁₃₁	semidirect
167. #368	C₂²×C₁₃₁	semidirect
168. #368	C₂²×C₁₃₁	semidirect
169. #359	C₂²×C₁₃₁	semidirect
170. #346	C₂²×C₁₃₁	semidirect
171. C₄×C₁₆	C₂²×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508×C₁₃₁	2
#2957×C₁₃₁	2
#2179×C₁₃₁	2
#2178×C₁₃₁	2
C₂×#368×C₁₃₁	2
#2143×C₁₃₁	2
#2108×C₁₃₁	2
#33465	131

Subgroups — C131×#33465 #45719349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33465 #45719349