Subgroups — C₁₃₁×#19297 #45705181

Order 33536.

616 conjugacy classes of subgroups.

The subgroup lattice has 316 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, complemented
2	#3079×C₁₃₁	C₂	1	2	yes	maximal
3	#2965×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1696×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	#1695×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#1566×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#1548×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
9	C₄×#41×C₁₃₁	C₂²	1	1	yes	characteristic
10	#500×C₁₃₁	—	2	1	no	—
11	#500×C₁₃₁	—	2	1	no	—
12	#500×C₁₃₁	C₂²	1	4	yes	—
13	#477×C₁₃₁	C₂²	1	4	yes	—
14	#476×C₁₃₁	C₂²	1	4	yes	—
15	#475×C₁₃₁	—	2	2	no	—
16	#474×C₁₃₁	—	2	2	no	—
17	#446×C₁₃₁	C₂²	1	2	yes	—
18	#436×C₁₃₁	C₂²	1	2	yes	—
19	#435×C₁₃₁	—	2	2	no	—
20	#431×C₁₃₁	—	2	1	no	—
21	#431×C₁₃₁	C₂²	1	1	yes	characteristic
22	#428×C₁₃₁	—	2	2	no	—
23	C₂×#107×C₁₃₁	—	2	2	no	—
24	#423×C₁₃₁	C₂²	1	2	yes	—
25	#417×C₁₃₁	—	2	2	no	—
26	C₄×M₁₆×C₁₃₁	—	2	1	no	—
27	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
28	#334×C₁₃₁	C₂²	1	4	yes	—
29	#333×C₁₃₁	C₂²	1	8	yes	—
30	C₂×#41×C₁₃₁	D₄	1	1	yes	characteristic
31	C₂²×Q₈×C₁₃₁	—	2	2	no	—
32	C₂×M₁₆×C₁₃₁	—	2	2	no	—
33	C₂×M₁₆×C₁₃₁	—	2	2	no	—
34	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
35	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
36	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
37	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
38	C₄⋊Q₈×C₁₃₁	C₂³	1	4	yes	—
39	C₄×Q₈×C₁₃₁	D₄	1	1	yes	characteristic
40	C₄×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
41	C₄×D₄×C₁₃₁	—	2	1	no	—
42	C₄×D₄×C₁₃₁	—	2	1	no	—
43	C₄×D₄×C₁₃₁	D₄	1	1	yes	characteristic
44	C₄×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
45	#117×C₁₃₁	—	2	1	no	—
46	#117×C₁₃₁	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
49	C₂×C₄²×C₁₃₁	—	2	1	no	—
50	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
51	#107×C₁₃₁	—	2	4	no	—
52	#107×C₁₃₁	—	2	4	no	—
53	#107×C₁₃₁	—	2	4	no	—
54	#106×C₁₃₁	—	2	4	no	—
55	C₄⋊C₈×C₁₃₁	—	2	2	no	—
56	C₄⋊C₈×C₁₃₁	—	2	2	no	—
57	C₄⋊C₈×C₁₃₁	C₂³	1	4	yes	—
58	C₄⋊C₈×C₁₃₁	C₂³	1	2	yes	—
59	#103×C₁₃₁	—	2	4	no	—
60	#102×C₁₃₁	—	2	4	no	—
61	C₂²⋊C₈×C₁₃₁	—	2	1	no	—
62	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
63	#97×C₁₃₁	—	2	2	no	—
64	C₄×C₈×C₁₃₁	—	2	1	no	—
65	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
66	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
67	C₄×C₈×C₁₃₁	C₂³	1	2	yes	—
68	#41×C₁₃₁	—	2	4	no	—
69	C₂×Q₈×C₁₃₁	—	2	4	no	—
70	C₂×Q₈×C₁₃₁	—	4	2	no	—
71	C₂×Q₈×C₁₃₁	—	8	4	no	—
72	C₂×Q₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂×D₄×C₁₃₁	—	2	1	no	—
74	C₂×D₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₁	—	2	1	no	—
76	C₂²×C₄×C₁₃₁	—	2	1	no	—
77	C₂²×C₄×C₁₃₁	—	2	1	no	—
78	C₂²×C₄×C₁₃₁	—	4	2	no	—
79	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
82	M₁₆×C₁₃₁	—	4	4	no	—
83	M₁₆×C₁₃₁	—	4	4	no	—
84	C₂×C₈×C₁₃₁	—	2	4	no	—
85	C₂×C₈×C₁₃₁	—	2	2	no	—
86	C₂×C₈×C₁₃₁	—	2	4	no	—
87	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
89	C₄⋊C₄×C₁₃₁	—	2	1	no	—
90	C₄⋊C₄×C₁₃₁	—	2	1	no	—
91	C₄⋊C₄×C₁₃₁	—	2	4	no	—
92	C₄⋊C₄×C₁₃₁	—	2	4	no	—
93	C₄⋊C₄×C₁₃₁	—	4	2	no	—
94	C₄⋊C₄×C₁₃₁	—	4	2	no	—
95	C₄⋊C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
96	C₄⋊C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
98	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
99	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
100	C₄²×C₁₃₁	—	2	1	no	—
101	C₄²×C₁₃₁	—	2	1	no	—
102	C₄²×C₁₃₁	—	2	1	no	—
103	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₃₁	#41	1	1	yes	characteristic
106	C₄²×C₁₃₁	C₂⁴	1	1	yes	characteristic
107	C₂³×C₁₃₁	—	2	1	no	—
108	C₂³×C₁₃₁	#121	1	1	yes	characteristic
109	Q₈×C₁₃₁	—	2	2	no	—
110	Q₈×C₁₃₁	—	8	4	no	—
111	Q₈×C₁₃₁	—	8	4	no	—
112	D₄×C₁₃₁	—	2	2	no	—
113	D₄×C₁₃₁	—	4	2	no	—
114	C₂×C₄×C₁₃₁	—	2	2	no	—
115	C₂×C₄×C₁₃₁	—	2	2	no	—
116	C₂×C₄×C₁₃₁	—	2	1	no	—
117	C₂×C₄×C₁₃₁	—	2	1	no	—
118	C₂×C₄×C₁₃₁	—	2	1	no	—
119	C₂×C₄×C₁₃₁	—	4	2	no	—
120	C₂×C₄×C₁₃₁	—	4	2	no	—
121	C₂×C₄×C₁₃₁	—	4	2	no	—
122	C₂×C₄×C₁₃₁	—	4	4	no	—
123	C₂×C₄×C₁₃₁	—	8	4	no	—
124	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
125	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
126	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
127	C₂×C₄×C₁₃₁	#137	1	4	yes	—
128	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
131	C₈×C₁₃₁	—	2	2	no	—
132	C₈×C₁₃₁	—	4	2	no	—
133	C₈×C₁₃₁	—	4	4	no	—
134	C₈×C₁₃₁	#137	1	4	yes	—
135	C₂²×C₁₃₁	—	2	1	no	—
136	C₂²×C₁₃₁	—	2	1	no	—
137	C₂²×C₁₃₁	—	4	1	no	—
138	C₂²×C₁₃₁	—	4	1	no	—
139	C₂²×C₁₃₁	—	4	1	no	—
140	C₂²×C₁₃₁	#470	1	2	yes	—
141	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
142	C₄×C₁₃₁	—	2	1	no	—
143	C₄×C₁₃₁	—	2	1	no	—
144	C₄×C₁₃₁	—	4	1	no	—
145	C₄×C₁₃₁	—	4	1	no	—
146	C₄×C₁₃₁	—	4	1	no	—
147	C₄×C₁₃₁	—	8	4	no	—
148	C₄×C₁₃₁	#470	1	2	yes	—
149	C₄×C₁₃₁	#472	1	2	yes	—
150	C₄×C₁₃₁	C₂×#137	1	2	yes	—
151	C₄×C₁₃₁	C₂×#137	1	2	yes	—
152	C₂×C₁₃₁	—	2	2	no	—
153	C₂×C₁₃₁	—	8	1	no	—
154	C₂×C₁₃₁	#3039	1	1	yes	characteristic
155	C₂×C₁₃₁	C₂×#470	1	1	yes	characteristic
156	C₂×C₁₃₁	C₂×#472	1	1	yes	characteristic
157	#19297	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₁₃₁	#19297	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×#500	C₂×C₁₃₁	1	1	yes	characteristic, complemented
160	#3079	C₂×C₁₃₁	1	2	yes	—
161	#2965	C₂×C₁₃₁	1	1	yes	characteristic
162	#1696	C₂×C₁₃₁	1	4	yes	complemented
163	#1695	C₂×C₁₃₁	1	4	yes	complemented
164	#1566	C₂×C₁₃₁	1	1	yes	characteristic, complemented
165	#1548	C₂×C₁₃₁	1	2	yes	complemented
166	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
167	C₄×#41	C₂²×C₁₃₁	1	1	yes	characteristic
168	#500	—	2	1	no	—
169	#500	—	2	1	no	—
170	#500	C₂²×C₁₃₁	1	4	yes	—
171	#477	C₂²×C₁₃₁	1	4	yes	—
172	#476	C₂²×C₁₃₁	1	4	yes	—
173	#475	—	2	2	no	—
174	#474	—	2	2	no	—
175	#446	C₂²×C₁₃₁	1	2	yes	—
176	#436	C₂²×C₁₃₁	1	2	yes	—
177	#435	—	2	2	no	—
178	#431	—	2	1	no	—
179	#431	C₂²×C₁₃₁	1	1	yes	characteristic
180	#428	—	2	2	no	—
181	C₂×#107	—	2	2	no	—
182	#423	C₂²×C₁₃₁	1	2	yes	—
183	#417	—	2	2	no	—
184	C₄×M₁₆	—	2	1	no	—
185	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
186	#334	C₂²×C₁₃₁	1	4	yes	—
187	#333	C₂²×C₁₃₁	1	8	yes	—
188	C₂×#41	D₄×C₁₃₁	1	1	yes	characteristic
189	C₂²×Q₈	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	C₂³×C₁₃₁	1	4	yes	—
197	C₄×Q₈	D₄×C₁₃₁	1	1	yes	characteristic
198	C₄×Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	—	2	1	no	—
201	C₄×D₄	D₄×C₁₃₁	1	1	yes	characteristic
202	C₄×D₄	C₂³×C₁₃₁	1	1	yes	characteristic
203	#117	—	2	1	no	—
204	#117	D₄×C₁₃₁	1	1	yes	characteristic
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
209	#107	—	2	4	no	—
210	#107	—	2	4	no	—
211	#107	—	2	4	no	—
212	#106	—	2	4	no	—
213	C₄⋊C₈	—	2	2	no	—
214	C₄⋊C₈	—	2	2	no	—
215	C₄⋊C₈	C₂³×C₁₃₁	1	2	yes	—
216	C₄⋊C₈	C₂³×C₁₃₁	1	4	yes	—
217	#103	—	2	4	no	—
218	#102	—	2	4	no	—
219	C₂²⋊C₈	—	2	1	no	—
220	C₂²⋊C₈	—	2	2	no	—
221	#97	—	2	2	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
224	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
225	C₄×C₈	C₂³×C₁₃₁	1	2	yes	—
226	#41	—	2	4	no	—
227	C₂×Q₈	—	2	4	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×Q₈	—	8	4	no	—
230	C₂×Q₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
239	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
240	M₁₆	—	4	4	no	—
241	M₁₆	—	4	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
246	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
254	C₄⋊C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
255	C₂²⋊C₄	—	2	1	no	—
256	C₂²⋊C₄	—	2	1	no	—
257	C₂²⋊C₄	—	2	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
263	C₄²	#41×C₁₃₁	1	1	yes	characteristic
264	C₄²	C₂⁴×C₁₃₁	1	1	yes	characteristic
265	C₂³	—	2	1	no	—
266	C₂³	#121×C₁₃₁	1	1	yes	characteristic
267	Q₈	—	2	2	no	—
268	Q₈	—	8	4	no	—
269	Q₈	—	8	4	no	—
270	D₄	—	2	2	no	—
271	D₄	—	4	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	8	4	no	—
282	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
283	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
284	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
285	C₂×C₄	#137×C₁₃₁	1	4	yes	—
286	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
288	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	4	2	no	—
291	C₈	—	4	4	no	—
292	C₈	#137×C₁₃₁	1	4	yes	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	#470×C₁₃₁	1	2	yes	—
299	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	8	4	no	—
306	C₄	#470×C₁₃₁	1	2	yes	—
307	C₄	#472×C₁₃₁	1	2	yes	—
308	C₄	C₂×#137×C₁₃₁	1	2	yes	—
309	C₄	C₂×#137×C₁₃₁	1	2	yes	—
310	C₂	—	2	2	no	—
311	C₂	—	8	1	no	—
312	C₂	#3039×C₁₃₁	1	1	yes	characteristic
313	C₂	C₂×#470×C₁₃₁	1	1	yes	characteristic
314	C₂	C₂×#472×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₁
prime radical	C₂×#41×C₁₃₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₁₃₁	C₂	semidirect
4. #1696×C₁₃₁	C₂	semidirect
5. #1695×C₁₃₁	C₂	semidirect
6. #1566×C₁₃₁	C₂	semidirect
7. #1548×C₁₃₁	C₂	semidirect
157. #19297	C₁₃₁	direct (N × H)
158. C₁₃₁	#19297	direct (N × H)
159. C₂×#500	C₂×C₁₃₁	semidirect
162. #1696	C₂×C₁₃₁	semidirect
163. #1695	C₂×C₁₃₁	semidirect
164. #1566	C₂×C₁₃₁	semidirect
165. #1548	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×#500×C₁₃₁	2
#3079×C₁₃₁	2
#2965×C₁₃₁	2
#1696×C₁₃₁	2
#1695×C₁₃₁	2
#1566×C₁₃₁	2
#1548×C₁₃₁	2
#19297	131

Subgroups — C131×#19297 #45705181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#19297 #45705181