Subgroups — C₁₃₁×#9898 #45695782

Order 33536.

538 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	#2868×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1842×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1296×C₁₃₁	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₁₃₁	—	2	1	no	—
6	C₂×C₄⋊Q₈×C₁₃₁	—	2	1	no	—
7	C₂×C₄⋊Q₈×C₁₃₁	C₄	1	2	yes	—
8	C₂×C₄⋊C₈×C₁₃₁	—	2	1	no	—
9	C₂×C₄⋊C₈×C₁₃₁	—	2	1	no	—
10	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃₁	—	2	1	no	—
13	#384×C₁₃₁	—	2	1	no	—
14	#384×C₁₃₁	—	2	1	no	—
15	#384×C₁₃₁	C₄	1	2	yes	—
16	#384×C₁₃₁	C₂²	1	2	yes	—
17	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
18	#336×C₁₃₁	—	2	1	no	—
19	C₂²×Q₈×C₁₃₁	—	2	2	no	—
20	C₂²×C₈×C₁₃₁	—	2	2	no	—
21	C₂²×C₈×C₁₃₁	—	2	2	no	—
22	C₄⋊Q₈×C₁₃₁	—	4	4	no	—
23	C₄⋊Q₈×C₁₃₁	—	4	2	no	—
24	C₄⋊Q₈×C₁₃₁	—	4	2	no	—
25	C₄⋊Q₈×C₁₃₁	—	4	2	no	—
26	C₄⋊Q₈×C₁₃₁	—	4	2	no	—
27	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₁₃₁	—	2	2	no	—
41	C₄⋊C₈×C₁₃₁	—	2	4	no	—
42	C₄⋊C₈×C₁₃₁	—	4	2	no	—
43	C₄⋊C₈×C₁₃₁	—	4	2	no	—
44	C₄⋊C₈×C₁₃₁	Q₈	1	4	yes	complemented
45	C₄×C₈×C₁₃₁	—	4	2	no	—
46	C₂×Q₈×C₁₃₁	—	4	4	no	—
47	C₂×Q₈×C₁₃₁	—	4	4	no	—
48	C₂×Q₈×C₁₃₁	—	4	4	no	—
49	C₂²×C₄×C₁₃₁	—	2	4	no	—
50	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃₁	—	2	4	no	—
58	C₂×C₈×C₁₃₁	—	4	2	no	—
59	C₂×C₈×C₁₃₁	—	4	2	no	—
60	C₂×C₈×C₁₃₁	—	4	2	no	—
61	C₂×C₈×C₁₃₁	—	4	2	no	—
62	C₂×C₈×C₁₃₁	—	4	2	no	—
63	C₄⋊C₄×C₁₃₁	—	4	4	no	—
64	C₄⋊C₄×C₁₃₁	—	4	4	no	—
65	C₄⋊C₄×C₁₃₁	—	4	4	no	—
66	C₄⋊C₄×C₁₃₁	—	4	4	no	—
67	C₄⋊C₄×C₁₃₁	—	4	2	no	—
68	C₄⋊C₄×C₁₃₁	—	4	2	no	—
69	C₄⋊C₄×C₁₃₁	—	4	2	no	—
70	C₄⋊C₄×C₁₃₁	—	4	2	no	—
71	C₄²×C₁₃₁	—	2	2	no	—
72	C₄²×C₁₃₁	—	2	2	no	—
73	C₄²×C₁₃₁	—	2	2	no	—
74	C₄²×C₁₃₁	D₈	1	2	yes	—
75	C₄²×C₁₃₁	D₈	1	2	yes	—
76	C₄²×C₁₃₁	SD₁₆	1	2	yes	—
77	C₄²×C₁₃₁	SD₁₆	1	2	yes	—
78	C₄²×C₁₃₁	SD₁₆	1	2	yes	—
79	C₄²×C₁₃₁	Q₁₆	1	2	yes	—
80	C₄²×C₁₃₁	C₂×Q₈	1	2	yes	—
81	C₄²×C₁₃₁	#41	1	2	yes	—
82	C₂³×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	Q₈×C₁₃₁	—	8	8	no	—
84	C₂×C₄×C₁₃₁	—	2	1	no	—
85	C₂×C₄×C₁₃₁	—	2	1	no	—
86	C₂×C₄×C₁₃₁	—	2	1	no	—
87	C₂×C₄×C₁₃₁	—	2	1	no	—
88	C₂×C₄×C₁₃₁	—	2	1	no	—
89	C₂×C₄×C₁₃₁	—	2	1	no	—
90	C₂×C₄×C₁₃₁	—	4	4	no	—
91	C₂×C₄×C₁₃₁	—	4	4	no	—
92	C₂×C₄×C₁₃₁	—	4	4	no	—
93	C₂×C₄×C₁₃₁	#100	1	2	yes	—
94	C₂×C₄×C₁₃₁	#101	1	2	yes	—
95	C₂×C₄×C₁₃₁	#102	1	2	yes	—
96	C₂×C₄×C₁₃₁	#102	1	2	yes	—
97	C₂×C₄×C₁₃₁	#102	1	2	yes	—
98	C₂×C₄×C₁₃₁	#102	1	2	yes	—
99	C₂×C₄×C₁₃₁	#103	1	2	yes	—
100	C₂×C₄×C₁₃₁	#103	1	2	yes	—
101	C₂×C₄×C₁₃₁	C₄×Q₈	1	2	yes	—
102	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₁₃₁	#124	1	1	yes	characteristic
104	C₂×C₄×C₁₃₁	C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×C₄×C₁₃₁	#137	1	2	yes	—
112	C₈×C₁₃₁	—	4	4	no	—
113	C₈×C₁₃₁	—	8	2	no	—
114	C₂²×C₁₃₁	#391	1	1	yes	characteristic
115	C₂²×C₁₃₁	C₂×#100	1	1	yes	characteristic
116	C₂²×C₁₃₁	C₂×#101	1	1	yes	characteristic
117	C₂²×C₁₃₁	C₂×#102	1	1	yes	characteristic
118	C₂²×C₁₃₁	C₂×#102	1	1	yes	characteristic
119	C₂²×C₁₃₁	C₂×#103	1	1	yes	characteristic
120	C₂²×C₁₃₁	#419	1	1	yes	characteristic
121	C₄×C₁₃₁	—	2	2	no	—
122	C₄×C₁₃₁	—	2	2	no	—
123	C₄×C₁₃₁	—	2	2	no	—
124	C₄×C₁₃₁	—	8	4	no	—
125	C₄×C₁₃₁	#331	1	4	yes	—
126	C₄×C₁₃₁	#332	1	4	yes	—
127	C₄×C₁₃₁	#474	1	2	yes	—
128	C₄×C₁₃₁	#475	1	2	yes	—
129	C₄×C₁₃₁	#486	1	1	yes	characteristic
130	C₄×C₁₃₁	#487	1	1	yes	characteristic
131	C₄×C₁₃₁	#498	1	1	yes	characteristic
132	C₄×C₁₃₁	#500	1	1	yes	characteristic
133	C₂×C₁₃₁	C₂×#331	1	1	yes	characteristic
134	C₂×C₁₃₁	C₂×#332	1	1	yes	characteristic
135	C₂×C₁₃₁	#1546	1	1	yes	characteristic
136	C₂×C₁₃₁	#1548	1	1	yes	characteristic
137	C₂×C₁₃₁	#1988	1	1	yes	characteristic
138	C₂×C₁₃₁	#1990	1	1	yes	characteristic
139	C₂×C₁₃₁	#1997	1	1	yes	characteristic
140	#9898	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₁₃₁	#9898	1	1	yes	characteristic, direct product factor, Sylow
142	#2868	C₂×C₁₃₁	1	1	yes	characteristic
143	#1842	C₂×C₁₃₁	1	1	yes	characteristic
144	C₄×C₄⋊C₈	C₂×C₁₃₁	1	1	yes	characteristic
145	#1296	C₂×C₁₃₁	1	4	yes	—
146	C₂×C₄⋊Q₈	—	2	1	no	—
147	C₂×C₄⋊Q₈	—	2	1	no	—
148	C₂×C₄⋊Q₈	C₄×C₁₃₁	1	2	yes	—
149	C₂×C₄⋊C₈	—	2	1	no	—
150	C₂×C₄⋊C₈	—	2	1	no	—
151	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
152	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
153	C₂×C₄×C₈	—	2	1	no	—
154	#384	—	2	1	no	—
155	#384	—	2	1	no	—
156	#384	C₄×C₁₃₁	1	2	yes	—
157	#384	C₂²×C₁₃₁	1	2	yes	—
158	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
159	#336	—	2	1	no	—
160	C₂²×Q₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₄⋊Q₈	—	4	4	no	—
164	C₄⋊Q₈	—	4	2	no	—
165	C₄⋊Q₈	—	4	2	no	—
166	C₄⋊Q₈	—	4	2	no	—
167	C₄⋊Q₈	—	4	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
174	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
176	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
177	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
178	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
179	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	2	4	no	—
183	C₄⋊C₈	—	4	2	no	—
184	C₄⋊C₈	—	4	2	no	—
185	C₄⋊C₈	Q₈×C₁₃₁	1	4	yes	complemented
186	C₄×C₈	—	4	2	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂×Q₈	—	4	4	no	—
189	C₂×Q₈	—	4	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	D₈×C₁₃₁	1	2	yes	—
216	C₄²	D₈×C₁₃₁	1	2	yes	—
217	C₄²	SD₁₆×C₁₃₁	1	2	yes	—
218	C₄²	SD₁₆×C₁₃₁	1	2	yes	—
219	C₄²	SD₁₆×C₁₃₁	1	2	yes	—
220	C₄²	Q₁₆×C₁₃₁	1	2	yes	—
221	C₄²	C₂×Q₈×C₁₃₁	1	2	yes	—
222	C₄²	#41×C₁₃₁	1	2	yes	—
223	C₂³	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
224	Q₈	—	8	8	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	#100×C₁₃₁	1	2	yes	—
235	C₂×C₄	#101×C₁₃₁	1	2	yes	—
236	C₂×C₄	#102×C₁₃₁	1	2	yes	—
237	C₂×C₄	#102×C₁₃₁	1	2	yes	—
238	C₂×C₄	#102×C₁₃₁	1	2	yes	—
239	C₂×C₄	#102×C₁₃₁	1	2	yes	—
240	C₂×C₄	#103×C₁₃₁	1	2	yes	—
241	C₂×C₄	#103×C₁₃₁	1	2	yes	—
242	C₂×C₄	C₄×Q₈×C₁₃₁	1	2	yes	—
243	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
244	C₂×C₄	#124×C₁₃₁	1	1	yes	characteristic
245	C₂×C₄	C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
246	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
247	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
249	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
250	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
251	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	1	yes	characteristic
252	C₂×C₄	#137×C₁₃₁	1	2	yes	—
253	C₈	—	4	4	no	—
254	C₈	—	8	2	no	—
255	C₂²	#391×C₁₃₁	1	1	yes	characteristic
256	C₂²	C₂×#100×C₁₃₁	1	1	yes	characteristic
257	C₂²	C₂×#101×C₁₃₁	1	1	yes	characteristic
258	C₂²	C₂×#102×C₁₃₁	1	1	yes	characteristic
259	C₂²	C₂×#102×C₁₃₁	1	1	yes	characteristic
260	C₂²	C₂×#103×C₁₃₁	1	1	yes	characteristic
261	C₂²	#419×C₁₃₁	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	8	4	no	—
266	C₄	#331×C₁₃₁	1	4	yes	—
267	C₄	#332×C₁₃₁	1	4	yes	—
268	C₄	#474×C₁₃₁	1	2	yes	—
269	C₄	#475×C₁₃₁	1	2	yes	—
270	C₄	#486×C₁₃₁	1	1	yes	characteristic
271	C₄	#487×C₁₃₁	1	1	yes	characteristic
272	C₄	#498×C₁₃₁	1	1	yes	characteristic
273	C₄	#500×C₁₃₁	1	1	yes	characteristic
274	C₂	C₂×#331×C₁₃₁	1	1	yes	characteristic
275	C₂	C₂×#332×C₁₃₁	1	1	yes	characteristic
276	C₂	#1546×C₁₃₁	1	1	yes	characteristic
277	C₂	#1548×C₁₃₁	1	1	yes	characteristic
278	C₂	#1988×C₁₃₁	1	1	yes	characteristic
279	C₂	#1990×C₁₃₁	1	1	yes	characteristic
280	C₂	#1997×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
44. C₄⋊C₈×C₁₃₁	Q₈	semidirect
140. #9898	C₁₃₁	direct (N × H)
141. C₁₃₁	#9898	direct (N × H)
185. C₄⋊C₈	Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2868×C₁₃₁	2
#1842×C₁₃₁	2
C₄×C₄⋊C₈×C₁₃₁	2
#1296×C₁₃₁	2
#9898	131

Subgroups — C131×#9898 #45695782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#9898 #45695782