Subgroups — C₁₃₁×#33721 #45719605

Order 33536.

794 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#510×C₁₃₁	C₂	1	2	yes	maximal, complemented
2	C₂×#508×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×Q₃₂×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#3156×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2251×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#2249×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#2247×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	#2108×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#137×C₁₃₁	—	2	2	no	—
10	C₂×#135×C₁₃₁	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₁₃₁	C₂²	1	2	yes	—
12	#536×C₁₃₁	—	2	2	no	—
13	#510×C₁₃₁	—	2	8	no	—
14	#508×C₁₃₁	—	2	4	no	—
15	C₂×Q₃₂×C₁₃₁	—	2	4	no	—
16	C₂×Q₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×Q₃₂×C₁₃₁	C₂²	1	2	yes	—
18	C₂×Q₃₂×C₁₃₁	C₂²	1	4	yes	—
19	C₂×Q₃₂×C₁₃₁	C₂²	1	2	yes	—
20	C₂×SD₃₂×C₁₃₁	C₂²	1	4	yes	—
21	C₂×SD₃₂×C₁₃₁	C₂²	1	2	yes	—
22	C₂×D₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
25	#497×C₁₃₁	—	2	4	no	—
26	#495×C₁₃₁	C₂²	1	4	yes	—
27	#494×C₁₃₁	C₂²	1	4	yes	—
28	#405×C₁₃₁	C₂²	1	1	yes	characteristic
29	#346×C₁₃₁	C₂²	1	2	yes	—
30	C₄×C₁₆×C₁₃₁	C₂²	1	2	yes	—
31	C₂×#41×C₁₃₁	—	2	2	no	—
32	C₂²×Q₈×C₁₃₁	—	2	2	no	—
33	#137×C₁₃₁	—	4	8	no	—
34	#135×C₁₃₁	—	2	4	no	—
35	#135×C₁₃₁	—	2	4	no	—
36	C₂×Q₁₆×C₁₃₁	—	2	4	no	—
37	C₂×Q₁₆×C₁₃₁	—	2	4	no	—
38	C₂×Q₁₆×C₁₃₁	—	2	4	no	—
39	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
40	C₂×Q₁₆×C₁₃₁	C₂³	1	2	yes	—
41	C₂×Q₁₆×C₁₃₁	C₂³	1	4	yes	—
42	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
43	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
44	C₂×D₈×C₁₃₁	C₂³	1	2	yes	—
45	C₂×M₁₆×C₁₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
48	#124×C₁₃₁	—	2	4	no	—
49	#123×C₁₃₁	—	4	4	no	—
50	C₂²⋊Q₈×C₁₃₁	—	4	4	no	—
51	#117×C₁₃₁	D₄	1	1	yes	characteristic
52	Q₃₂×C₁₃₁	—	2	4	no	—
53	Q₃₂×C₁₃₁	—	2	4	no	—
54	Q₃₂×C₁₃₁	—	2	2	no	—
55	Q₃₂×C₁₃₁	—	2	8	no	—
56	SD₃₂×C₁₃₁	—	2	4	no	—
57	SD₃₂×C₁₃₁	—	2	8	no	—
58	D₁₆×C₁₃₁	—	2	2	no	—
59	M₃₂×C₁₃₁	D₄	1	4	yes	—
60	C₂×C₁₆×C₁₃₁	D₄	1	4	yes	—
61	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
64	#97×C₁₃₁	D₄	1	2	yes	—
65	C₄×C₈×C₁₃₁	C₂³	1	2	yes	—
66	#41×C₁₃₁	—	4	4	no	—
67	#41×C₁₃₁	—	4	4	no	—
68	C₂×Q₈×C₁₃₁	—	2	4	no	—
69	C₂×Q₈×C₁₃₁	—	2	2	no	—
70	C₂×Q₈×C₁₃₁	—	4	2	no	—
71	C₂×Q₈×C₁₃₁	—	4	4	no	—
72	C₂×Q₈×C₁₃₁	—	4	4	no	—
73	C₂×D₄×C₁₃₁	—	2	2	no	—
74	C₂×D₄×C₁₃₁	—	4	2	no	—
75	C₂²×C₄×C₁₃₁	—	4	2	no	—
76	C₂²×C₄×C₁₃₁	—	4	2	no	—
77	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₁₃₁	—	2	4	no	—
79	Q₁₆×C₁₃₁	—	2	2	no	—
80	Q₁₆×C₁₃₁	—	2	4	no	—
81	Q₁₆×C₁₃₁	—	2	2	no	—
82	Q₁₆×C₁₃₁	—	4	4	no	—
83	Q₁₆×C₁₃₁	—	4	8	no	—
84	SD₁₆×C₁₃₁	—	4	4	no	—
85	SD₁₆×C₁₃₁	—	4	8	no	—
86	D₈×C₁₃₁	—	2	2	no	—
87	D₈×C₁₃₁	—	2	2	no	—
88	M₁₆×C₁₃₁	D₈	1	4	yes	—
89	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₁₃₁	—	4	8	no	—
95	C₄⋊C₄×C₁₃₁	D₈	1	2	yes	—
96	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
97	C₂²⋊C₄×C₁₃₁	D₈	1	2	yes	—
98	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
99	C₁₆×C₁₃₁	C₂×D₄	1	2	yes	—
100	C₁₆×C₁₃₁	C₂×D₄	1	2	yes	—
101	C₁₆×C₁₃₁	C₂×D₄	1	4	yes	—
102	C₂³×C₁₃₁	—	4	2	no	—
103	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
104	Q₈×C₁₃₁	—	4	2	no	—
105	Q₈×C₁₃₁	—	4	4	no	—
106	Q₈×C₁₃₁	—	4	4	no	—
107	Q₈×C₁₃₁	—	4	2	no	—
108	Q₈×C₁₃₁	—	8	4	no	—
109	D₄×C₁₃₁	—	4	2	no	—
110	D₄×C₁₃₁	—	4	2	no	—
111	D₄×C₁₃₁	—	8	4	no	—
112	C₂×C₄×C₁₃₁	—	4	4	no	—
113	C₂×C₄×C₁₃₁	—	4	2	no	—
114	C₂×C₄×C₁₃₁	—	8	4	no	—
115	C₂×C₄×C₁₃₁	—	8	4	no	—
116	C₂×C₄×C₁₃₁	#127	1	2	yes	—
117	C₂×C₄×C₁₃₁	C₂×D₈	1	4	yes	—
118	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₁₃₁	#127	1	2	yes	—
122	C₈×C₁₃₁	C₂×D₈	1	4	yes	—
123	C₈×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
125	C₂²×C₁₃₁	—	2	2	no	—
126	C₂²×C₁₃₁	—	8	2	no	—
127	C₂²×C₁₃₁	—	8	4	no	—
128	C₂²×C₁₃₁	#493	1	2	yes	—
129	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
130	C₄×C₁₃₁	—	2	4	no	—
131	C₄×C₁₃₁	—	8	2	no	—
132	C₄×C₁₃₁	—	8	4	no	—
133	C₄×C₁₃₁	#493	1	2	yes	—
134	C₄×C₁₃₁	C₂×#127	1	1	yes	characteristic
135	C₄×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
136	C₂×C₁₃₁	—	2	2	no	—
137	C₂×C₁₃₁	—	16	2	no	—
138	C₂×C₁₃₁	C₂×#493	1	1	yes	characteristic
139	C₂×C₁₃₁	#3417	1	2	yes	—
140	#33721	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₁₃₁	#33721	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×#510	C₂×C₁₃₁	1	2	yes	complemented
143	C₂×#508	C₂×C₁₃₁	1	1	yes	characteristic
144	C₂²×Q₃₂	C₂×C₁₃₁	1	1	yes	characteristic, complemented
145	#3156	C₂×C₁₃₁	1	2	yes	complemented
146	#2251	C₂×C₁₃₁	1	4	yes	complemented
147	#2249	C₂×C₁₃₁	1	2	yes	complemented
148	#2247	C₂×C₁₃₁	1	2	yes	complemented
149	#2108	C₂×C₁₃₁	1	1	yes	characteristic, complemented
150	C₂×#137	—	2	2	no	—
151	C₂×#135	C₂²×C₁₃₁	1	2	yes	—
152	C₂²×Q₁₆	C₂²×C₁₃₁	1	2	yes	—
153	#536	—	2	2	no	—
154	#510	—	2	8	no	—
155	#508	—	2	4	no	—
156	C₂×Q₃₂	—	2	4	no	—
157	C₂×Q₃₂	C₂²×C₁₃₁	1	2	yes	—
158	C₂×Q₃₂	C₂²×C₁₃₁	1	2	yes	—
159	C₂×Q₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
160	C₂×Q₃₂	C₂²×C₁₃₁	1	4	yes	—
161	C₂×SD₃₂	C₂²×C₁₃₁	1	2	yes	—
162	C₂×SD₃₂	C₂²×C₁₃₁	1	4	yes	—
163	C₂×D₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
164	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
165	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
166	#497	—	2	4	no	—
167	#495	C₂²×C₁₃₁	1	4	yes	—
168	#494	C₂²×C₁₃₁	1	4	yes	—
169	#405	C₂²×C₁₃₁	1	1	yes	characteristic
170	#346	C₂²×C₁₃₁	1	2	yes	—
171	C₄×C₁₆	C₂²×C₁₃₁	1	2	yes	—
172	C₂×#41	—	2	2	no	—
173	C₂²×Q₈	—	2	2	no	—
174	#137	—	4	8	no	—
175	#135	—	2	4	no	—
176	#135	—	2	4	no	—
177	C₂×Q₁₆	—	2	4	no	—
178	C₂×Q₁₆	—	2	4	no	—
179	C₂×Q₁₆	—	2	2	no	—
180	C₂×Q₁₆	—	2	4	no	—
181	C₂×Q₁₆	C₂³×C₁₃₁	1	4	yes	—
182	C₂×Q₁₆	C₂³×C₁₃₁	1	2	yes	—
183	C₂×SD₁₆	—	2	2	no	—
184	C₂×SD₁₆	—	2	4	no	—
185	C₂×D₈	C₂³×C₁₃₁	1	2	yes	—
186	C₂×M₁₆	D₄×C₁₃₁	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
188	C₄⋊Q₈	—	2	4	no	—
189	#124	—	2	4	no	—
190	#123	—	4	4	no	—
191	C₂²⋊Q₈	—	4	4	no	—
192	#117	D₄×C₁₃₁	1	1	yes	characteristic
193	Q₃₂	—	2	4	no	—
194	Q₃₂	—	2	4	no	—
195	Q₃₂	—	2	2	no	—
196	Q₃₂	—	2	8	no	—
197	SD₃₂	—	2	4	no	—
198	SD₃₂	—	2	8	no	—
199	D₁₆	—	2	2	no	—
200	M₃₂	D₄×C₁₃₁	1	4	yes	—
201	C₂×C₁₆	D₄×C₁₃₁	1	4	yes	—
202	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
203	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
204	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
205	#97	D₄×C₁₃₁	1	2	yes	—
206	C₄×C₈	C₂³×C₁₃₁	1	2	yes	—
207	#41	—	4	4	no	—
208	#41	—	4	4	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂×Q₈	—	4	4	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
219	Q₁₆	—	2	4	no	—
220	Q₁₆	—	2	2	no	—
221	Q₁₆	—	2	4	no	—
222	Q₁₆	—	2	2	no	—
223	Q₁₆	—	4	4	no	—
224	Q₁₆	—	4	8	no	—
225	SD₁₆	—	4	4	no	—
226	SD₁₆	—	4	8	no	—
227	D₈	—	2	2	no	—
228	D₈	—	2	2	no	—
229	M₁₆	D₈×C₁₃₁	1	4	yes	—
230	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
232	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
233	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
234	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
235	C₄⋊C₄	—	4	8	no	—
236	C₄⋊C₄	D₈×C₁₃₁	1	2	yes	—
237	C₂²⋊C₄	—	4	8	no	—
238	C₂²⋊C₄	D₈×C₁₃₁	1	2	yes	—
239	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
240	C₁₆	C₂×D₄×C₁₃₁	1	2	yes	—
241	C₁₆	C₂×D₄×C₁₃₁	1	2	yes	—
242	C₁₆	C₂×D₄×C₁₃₁	1	4	yes	—
243	C₂³	—	4	2	no	—
244	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
245	Q₈	—	4	2	no	—
246	Q₈	—	4	4	no	—
247	Q₈	—	4	4	no	—
248	Q₈	—	4	2	no	—
249	Q₈	—	8	4	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	8	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	#127×C₁₃₁	1	2	yes	—
258	C₂×C₄	C₂×D₈×C₁₃₁	1	4	yes	—
259	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
260	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
261	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
262	C₈	#127×C₁₃₁	1	2	yes	—
263	C₈	C₂×D₈×C₁₃₁	1	4	yes	—
264	C₈	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
265	C₈	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	8	2	no	—
268	C₂²	—	8	4	no	—
269	C₂²	#493×C₁₃₁	1	2	yes	—
270	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
271	C₄	—	2	4	no	—
272	C₄	—	8	2	no	—
273	C₄	—	8	4	no	—
274	C₄	#493×C₁₃₁	1	2	yes	—
275	C₄	C₂×#127×C₁₃₁	1	1	yes	characteristic
276	C₄	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	16	2	no	—
279	C₂	C₂×#493×C₁₃₁	1	1	yes	characteristic
280	C₂	#3417×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
prime radical	C₂×#508×C₁₃₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ #117×C₁₃₁ ◁ #405×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#510×C₁₃₁	C₂	semidirect
3. C₂²×Q₃₂×C₁₃₁	C₂	semidirect
4. #3156×C₁₃₁	C₂	semidirect
5. #2251×C₁₃₁	C₂	semidirect
6. #2249×C₁₃₁	C₂	semidirect
7. #2247×C₁₃₁	C₂	semidirect
8. #2108×C₁₃₁	C₂	semidirect
140. #33721	C₁₃₁	direct (N × H)
141. C₁₃₁	#33721	direct (N × H)
142. C₂×#510	C₂×C₁₃₁	semidirect
144. C₂²×Q₃₂	C₂×C₁₃₁	semidirect
145. #3156	C₂×C₁₃₁	semidirect
146. #2251	C₂×C₁₃₁	semidirect
147. #2249	C₂×C₁₃₁	semidirect
148. #2247	C₂×C₁₃₁	semidirect
149. #2108	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×#510×C₁₃₁	2
C₂×#508×C₁₃₁	2
C₂²×Q₃₂×C₁₃₁	2
#3156×C₁₃₁	2
#2251×C₁₃₁	2
#2249×C₁₃₁	2
#2247×C₁₃₁	2
#2108×C₁₃₁	2
#33721	131

Subgroups — C131×#33721 #45719605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33721 #45719605