Subgroups — C₁₃₁×#23157 #45709041

Order 33536.

806 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2365×C₁₃₁	C₂	1	2	yes	maximal
3	#1988×C₁₃₁	C₂	1	2	yes	maximal
4	#1942×C₁₃₁	C₂	1	2	yes	maximal
5	#1918×C₁₃₁	C₂	1	4	yes	maximal
6	C₄×#107×C₁₃₁	C₂	1	2	yes	maximal
7	C₄×#107×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₁₃₁	—	2	2	no	—
11	#500×C₁₃₁	C₄	1	8	yes	complemented
12	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
14	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
15	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	#391×C₁₃₁	—	2	2	no	—
20	#389×C₁₃₁	C₂²	1	4	yes	—
21	#387×C₁₃₁	—	2	4	no	—
22	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	—
24	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
25	#340×C₁₃₁	C₂²	1	4	yes	—
26	#340×C₁₃₁	C₂²	1	4	yes	—
27	#340×C₁₃₁	C₂²	1	2	yes	—
28	#336×C₁₃₁	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₃₁	—	2	2	no	—
30	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
34	C₄⋊Q₈×C₁₃₁	C₂×C₄	1	8	yes	—
35	C₄×Q₈×C₁₃₁	—	4	8	no	—
36	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₃₁	—	2	2	no	—
44	C₂×C₄²×C₁₃₁	—	2	4	no	—
45	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
47	#107×C₁₃₁	—	2	2	no	—
48	#107×C₁₃₁	C₂×C₄	1	8	yes	—
49	#107×C₁₃₁	C₂×C₄	1	8	yes	—
50	#107×C₁₃₁	Q₈	1	4	yes	complemented
51	C₄⋊C₈×C₁₃₁	—	2	2	no	—
52	C₄×C₈×C₁₃₁	—	2	4	no	—
53	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
54	#95×C₁₃₁	—	2	4	no	—
55	#95×C₁₃₁	—	2	4	no	—
56	#95×C₁₃₁	—	2	2	no	—
57	#95×C₁₃₁	D₄	1	2	yes	—
58	C₂×Q₈×C₁₃₁	—	2	8	no	—
59	C₂×Q₈×C₁₃₁	—	4	4	no	—
60	C₂×Q₈×C₁₃₁	—	4	4	no	—
61	C₂²×C₄×C₁₃₁	—	2	2	no	—
62	C₂²×C₄×C₁₃₁	—	2	4	no	—
63	C₂²×C₄×C₁₃₁	—	2	2	no	—
64	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₁₃₁	—	2	1	no	—
70	C₂×C₈×C₁₃₁	—	2	1	no	—
71	C₂×C₈×C₁₃₁	—	2	2	no	—
72	C₂×C₈×C₁₃₁	—	2	2	no	—
73	C₂×C₈×C₁₃₁	—	2	2	no	—
74	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
77	C₂×C₈×C₁₃₁	#41	1	2	yes	—
78	C₄⋊C₄×C₁₃₁	—	2	8	no	—
79	C₄⋊C₄×C₁₃₁	—	2	2	no	—
80	C₄⋊C₄×C₁₃₁	—	2	4	no	—
81	C₄⋊C₄×C₁₃₁	—	2	2	no	—
82	C₄⋊C₄×C₁₃₁	—	2	2	no	—
83	C₄⋊C₄×C₁₃₁	—	2	2	no	—
84	C₄⋊C₄×C₁₃₁	—	4	8	no	—
85	C₄⋊C₄×C₁₃₁	D₈	1	4	yes	—
86	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
87	C₄⋊C₄×C₁₃₁	C₂×Q₈	1	4	yes	—
88	C₄²×C₁₃₁	—	2	2	no	—
89	C₄²×C₁₃₁	—	2	2	no	—
90	C₄²×C₁₃₁	—	4	8	no	—
91	C₄²×C₁₃₁	—	4	4	no	—
92	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
93	C₄²×C₁₃₁	#41	1	2	yes	—
94	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
95	Q₈×C₁₃₁	—	4	16	no	—
96	C₂×C₄×C₁₃₁	—	2	2	no	—
97	C₂×C₄×C₁₃₁	—	2	2	no	—
98	C₂×C₄×C₁₃₁	—	2	1	no	—
99	C₂×C₄×C₁₃₁	—	2	1	no	—
100	C₂×C₄×C₁₃₁	—	2	8	no	—
101	C₂×C₄×C₁₃₁	—	2	2	no	—
102	C₂×C₄×C₁₃₁	—	2	4	no	—
103	C₂×C₄×C₁₃₁	—	2	2	no	—
104	C₂×C₄×C₁₃₁	—	2	2	no	—
105	C₂×C₄×C₁₃₁	—	4	2	no	—
106	C₂×C₄×C₁₃₁	—	4	4	no	—
107	C₂×C₄×C₁₃₁	—	4	4	no	—
108	C₂×C₄×C₁₃₁	—	4	2	no	—
109	C₂×C₄×C₁₃₁	—	4	2	no	—
110	C₂×C₄×C₁₃₁	—	4	2	no	—
111	C₂×C₄×C₁₃₁	—	4	2	no	—
112	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₃₁	C₂×D₈	1	4	yes	—
115	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₁₃₁	#135	1	2	yes	—
117	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
123	C₈×C₁₃₁	—	2	2	no	—
124	C₈×C₁₃₁	—	4	2	no	—
125	C₈×C₁₃₁	C₄×Q₈	1	4	yes	complemented
126	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂²×C₁₃₁	#547	1	1	yes	characteristic
128	C₂²×C₁₃₁	D₄×Q₈	1	1	yes	characteristic
129	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
130	C₂²×C₁₃₁	C₂×#135	1	1	yes	characteristic
131	C₂²×C₁₃₁	#578	1	1	yes	characteristic
132	C₂²×C₁₃₁	#578	1	1	yes	characteristic
133	C₄×C₁₃₁	—	2	4	no	—
134	C₄×C₁₃₁	—	2	2	no	—
135	C₄×C₁₃₁	—	4	2	no	—
136	C₄×C₁₃₁	—	4	4	no	—
137	C₄×C₁₃₁	—	4	8	no	—
138	C₄×C₁₃₁	—	8	2	no	—
139	C₄×C₁₃₁	C₄×D₈	1	4	yes	—
140	C₄×C₁₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
141	C₄×C₁₃₁	#519	1	1	yes	characteristic
142	C₄×C₁₃₁	#556	1	1	yes	characteristic
143	C₄×C₁₃₁	#557	1	1	yes	characteristic
144	C₂×C₁₃₁	#2365	1	1	yes	characteristic
145	C₂×C₁₃₁	C₂×C₄×D₈	1	1	yes	characteristic
146	C₂×C₁₃₁	#2950	1	1	yes	characteristic
147	C₂×C₁₃₁	#3334	1	1	yes	characteristic
148	C₂×C₁₃₁	#3342	1	1	yes	characteristic
149	C₂×C₁₃₁	Q₈×D₈	1	1	yes	characteristic
150	C₂×C₁₃₁	#3383	1	1	yes	characteristic
151	#23157	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₁₃₁	#23157	1	1	yes	characteristic, direct product factor, Sylow
153	C₂×#500	C₂×C₁₃₁	1	1	yes	characteristic
154	#2365	C₂×C₁₃₁	1	2	yes	—
155	#1988	C₂×C₁₃₁	1	2	yes	—
156	#1942	C₂×C₁₃₁	1	2	yes	—
157	#1918	C₂×C₁₃₁	1	4	yes	—
158	C₄×#107	C₂×C₁₃₁	1	1	yes	characteristic
159	C₄×#107	C₂×C₁₃₁	1	2	yes	—
160	C₈×C₄⋊C₄	C₂×C₁₃₁	1	1	yes	characteristic
161	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
162	C₂×C₄×Q₈	—	2	2	no	—
163	#500	C₄×C₁₃₁	1	8	yes	complemented
164	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
165	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
166	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
167	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
168	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
170	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
171	#391	—	2	2	no	—
172	#389	C₂²×C₁₃₁	1	4	yes	—
173	#387	—	2	4	no	—
174	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	—
175	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
176	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
177	#340	C₂²×C₁₃₁	1	2	yes	—
178	#340	C₂²×C₁₃₁	1	4	yes	—
179	#340	C₂²×C₁₃₁	1	4	yes	—
180	#336	C₂²×C₁₃₁	1	2	yes	—
181	C₂²×Q₈	—	2	2	no	—
182	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
183	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
185	C₄⋊Q₈	—	2	4	no	—
186	C₄⋊Q₈	C₂×C₄×C₁₃₁	1	8	yes	—
187	C₄×Q₈	—	4	8	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
191	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
193	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
194	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
195	C₂×C₄²	—	2	4	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
198	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
199	#107	—	2	2	no	—
200	#107	C₂×C₄×C₁₃₁	1	8	yes	—
201	#107	C₂×C₄×C₁₃₁	1	8	yes	—
202	#107	Q₈×C₁₃₁	1	4	yes	complemented
203	C₄⋊C₈	—	2	2	no	—
204	C₄×C₈	—	2	4	no	—
205	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
206	#95	—	2	2	no	—
207	#95	—	2	4	no	—
208	#95	—	2	4	no	—
209	#95	D₄×C₁₃₁	1	2	yes	—
210	C₂×Q₈	—	2	8	no	—
211	C₂×Q₈	—	4	4	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
219	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
220	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
227	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
228	C₂×C₈	C₂×Q₈×C₁₃₁	1	2	yes	—
229	C₂×C₈	#41×C₁₃₁	1	2	yes	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	8	no	—
237	C₄⋊C₄	D₈×C₁₃₁	1	4	yes	—
238	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
239	C₄⋊C₄	C₂×Q₈×C₁₃₁	1	4	yes	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	8	no	—
243	C₄²	—	4	4	no	—
244	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
245	C₄²	#41×C₁₃₁	1	2	yes	—
246	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
247	Q₈	—	4	16	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
265	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
266	C₂×C₄	C₂×D₈×C₁₃₁	1	4	yes	—
267	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
268	C₂×C₄	#135×C₁₃₁	1	2	yes	—
269	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
270	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
271	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
272	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
273	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
274	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	4	2	no	—
277	C₈	C₄×Q₈×C₁₃₁	1	4	yes	complemented
278	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
279	C₂²	#547×C₁₃₁	1	1	yes	characteristic
280	C₂²	D₄×Q₈×C₁₃₁	1	1	yes	characteristic
281	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
282	C₂²	C₂×#135×C₁₃₁	1	1	yes	characteristic
283	C₂²	#578×C₁₃₁	1	1	yes	characteristic
284	C₂²	#578×C₁₃₁	1	1	yes	characteristic
285	C₄	—	2	4	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	4	8	no	—
290	C₄	—	8	2	no	—
291	C₄	C₄×D₈×C₁₃₁	1	4	yes	—
292	C₄	C₂×C₄×Q₈×C₁₃₁	1	1	yes	characteristic
293	C₄	#519×C₁₃₁	1	1	yes	characteristic
294	C₄	#556×C₁₃₁	1	1	yes	characteristic
295	C₄	#557×C₁₃₁	1	1	yes	characteristic
296	C₂	#2365×C₁₃₁	1	1	yes	characteristic
297	C₂	C₂×C₄×D₈×C₁₃₁	1	1	yes	characteristic
298	C₂	#2950×C₁₃₁	1	1	yes	characteristic
299	C₂	#3334×C₁₃₁	1	1	yes	characteristic
300	C₂	#3342×C₁₃₁	1	1	yes	characteristic
301	C₂	Q₈×D₈×C₁₃₁	1	1	yes	characteristic
302	C₂	#3383×C₁₃₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
11. #500×C₁₃₁	C₄	semidirect
50. #107×C₁₃₁	Q₈	semidirect
125. C₈×C₁₃₁	C₄×Q₈	semidirect
151. #23157	C₁₃₁	direct (N × H)
152. C₁₃₁	#23157	direct (N × H)
163. #500	C₄×C₁₃₁	semidirect
202. #107	Q₈×C₁₃₁	semidirect
277. C₈	C₄×Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₃₁	2
#2365×C₁₃₁	2
#1988×C₁₃₁	2
#1942×C₁₃₁	2
#1918×C₁₃₁	2
C₄×#107×C₁₃₁	2
C₈×C₄⋊C₄×C₁₃₁	2
#23157	131

Subgroups — C131×#23157 #45709041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#23157 #45709041