Subgroups — C₁₃₁×#25057 #45710941

Order 33536.

722 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2538×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2355×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2092×C₁₃₁	C₂	1	2	yes	maximal
4	#2085×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1984×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#1948×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#1925×C₁₃₁	C₂	1	2	yes	maximal
8	#1921×C₁₃₁	C₂	1	2	yes	maximal, complemented
9	#1845×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#1802×C₁₃₁	C₂	1	1	yes	maximal, characteristic
11	C₄×#102×C₁₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂²⋊Q₈×C₁₃₁	—	2	1	no	—
13	C₂×C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
16	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
17	C₂×#106×C₁₃₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×#102×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂×#102×C₁₃₁	C₂²	1	2	yes	—
21	C₂×#102×C₁₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
23	#396×C₁₃₁	—	2	2	no	—
24	#391×C₁₃₁	—	2	1	no	—
25	#390×C₁₃₁	C₂²	1	2	yes	—
26	#389×C₁₃₁	C₂²	1	2	yes	—
27	#387×C₁₃₁	—	2	2	no	—
28	#384×C₁₃₁	C₂²	1	1	yes	characteristic
29	#384×C₁₃₁	C₂²	1	1	yes	characteristic
30	#382×C₁₃₁	C₂²	1	2	yes	—
31	#380×C₁₃₁	—	2	1	no	—
32	C₄×C₄⋊C₄×C₁₃₁	—	2	1	no	—
33	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
35	#340×C₁₃₁	C₂²	1	1	yes	characteristic
36	#340×C₁₃₁	C₂²	1	1	yes	characteristic
37	#340×C₁₃₁	C₂²	1	2	yes	—
38	#340×C₁₃₁	C₂²	1	2	yes	—
39	#336×C₁₃₁	C₂²	1	2	yes	—
40	#336×C₁₃₁	C₂²	1	2	yes	—
41	C₂²×Q₈×C₁₃₁	D₄	1	1	yes	characteristic
42	C₂²×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₁₃₁	—	2	1	no	—
44	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
47	C₂²⋊Q₈×C₁₃₁	—	4	4	no	—
48	C₄×Q₈×C₁₃₁	—	2	2	no	—
49	C₄×Q₈×C₁₃₁	—	2	2	no	—
50	C₄×D₄×C₁₃₁	—	2	2	no	—
51	C₄×D₄×C₁₃₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₁₃₁	D₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
67	C₂×C₄²×C₁₃₁	—	2	1	no	—
68	C₂×C₄²×C₁₃₁	—	2	2	no	—
69	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
70	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
71	#107×C₁₃₁	—	2	4	no	—
72	#107×C₁₃₁	—	4	2	no	—
73	#106×C₁₃₁	—	4	2	no	—
74	C₄⋊C₈×C₁₃₁	—	2	2	no	—
75	#102×C₁₃₁	—	2	2	no	—
76	#102×C₁₃₁	—	4	2	no	—
77	#102×C₁₃₁	Q₈	1	4	yes	complemented
78	C₄×C₈×C₁₃₁	—	2	2	no	—
79	#95×C₁₃₁	—	2	2	no	—
80	#95×C₁₃₁	—	2	2	no	—
81	#95×C₁₃₁	—	2	2	no	—
82	#95×C₁₃₁	—	2	2	no	—
83	#95×C₁₃₁	—	2	1	no	—
84	C₂⁴×C₁₃₁	—	2	1	no	—
85	C₂×Q₈×C₁₃₁	—	2	2	no	—
86	C₂×Q₈×C₁₃₁	—	2	2	no	—
87	C₂×Q₈×C₁₃₁	—	2	2	no	—
88	C₂×D₄×C₁₃₁	—	2	1	no	—
89	C₂×D₄×C₁₃₁	—	2	1	no	—
90	C₂×D₄×C₁₃₁	—	2	1	no	—
91	C₂×D₄×C₁₃₁	—	2	1	no	—
92	C₂×D₄×C₁₃₁	—	2	1	no	—
93	C₂×D₄×C₁₃₁	C₂×Q₈	1	2	yes	—
94	C₂²×C₄×C₁₃₁	—	2	1	no	—
95	C₂²×C₄×C₁₃₁	—	2	1	no	—
96	C₂²×C₄×C₁₃₁	—	2	2	no	—
97	C₂²×C₄×C₁₃₁	—	2	1	no	—
98	C₂²×C₄×C₁₃₁	—	2	2	no	—
99	C₂²×C₄×C₁₃₁	—	4	2	no	—
100	C₂²×C₄×C₁₃₁	—	4	2	no	—
101	C₂²×C₄×C₁₃₁	—	4	2	no	—
102	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
107	C₂×C₈×C₁₃₁	—	2	1	no	—
108	C₂×C₈×C₁₃₁	—	2	1	no	—
109	C₂×C₈×C₁₃₁	—	2	1	no	—
110	C₂×C₈×C₁₃₁	—	2	1	no	—
111	C₂×C₈×C₁₃₁	—	2	2	no	—
112	C₂×C₈×C₁₃₁	—	2	2	no	—
113	C₂×C₈×C₁₃₁	—	2	2	no	—
114	C₂×C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
115	C₂×C₈×C₁₃₁	#41	1	2	yes	—
116	C₄⋊C₄×C₁₃₁	—	2	2	no	—
117	C₄⋊C₄×C₁₃₁	—	2	4	no	—
118	C₄⋊C₄×C₁₃₁	—	2	1	no	—
119	C₄⋊C₄×C₁₃₁	—	2	1	no	—
120	C₄⋊C₄×C₁₃₁	—	2	2	no	—
121	C₄⋊C₄×C₁₃₁	—	2	2	no	—
122	C₄⋊C₄×C₁₃₁	—	2	1	no	—
123	C₄⋊C₄×C₁₃₁	—	2	1	no	—
124	C₄⋊C₄×C₁₃₁	—	2	2	no	—
125	C₄⋊C₄×C₁₃₁	—	2	1	no	—
126	C₄⋊C₄×C₁₃₁	—	2	2	no	—
127	C₄⋊C₄×C₁₃₁	—	4	4	no	—
128	C₄⋊C₄×C₁₃₁	—	4	2	no	—
129	C₄⋊C₄×C₁₃₁	—	4	2	no	—
130	C₄⋊C₄×C₁₃₁	—	8	1	no	—
131	C₄⋊C₄×C₁₃₁	—	8	1	no	—
132	C₄⋊C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
133	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
134	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
135	C₂²⋊C₄×C₁₃₁	—	8	2	no	—
136	C₄²×C₁₃₁	—	2	2	no	—
137	C₄²×C₁₃₁	—	2	2	no	—
138	C₄²×C₁₃₁	—	4	2	no	—
139	C₄²×C₁₃₁	—	8	2	no	—
140	C₄²×C₁₃₁	#41	1	2	yes	—
141	C₄²×C₁₃₁	#41	1	2	yes	—
142	C₂³×C₁₃₁	—	2	2	no	—
143	C₂³×C₁₃₁	—	4	1	no	—
144	C₂³×C₁₃₁	—	4	1	no	—
145	C₂³×C₁₃₁	—	4	1	no	—
146	C₂³×C₁₃₁	—	4	1	no	—
147	C₂³×C₁₃₁	—	4	1	no	—
148	C₂³×C₁₃₁	—	4	1	no	—
149	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
150	Q₈×C₁₃₁	—	4	4	no	—
151	D₄×C₁₃₁	—	2	2	no	—
152	D₄×C₁₃₁	—	2	2	no	—
153	D₄×C₁₃₁	—	4	1	no	—
154	D₄×C₁₃₁	—	4	1	no	—
155	C₂×C₄×C₁₃₁	—	2	1	no	—
156	C₂×C₄×C₁₃₁	—	2	1	no	—
157	C₂×C₄×C₁₃₁	—	2	2	no	—
158	C₂×C₄×C₁₃₁	—	2	2	no	—
159	C₂×C₄×C₁₃₁	—	2	2	no	—
160	C₂×C₄×C₁₃₁	—	2	2	no	—
161	C₂×C₄×C₁₃₁	—	2	2	no	—
162	C₂×C₄×C₁₃₁	—	2	2	no	—
163	C₂×C₄×C₁₃₁	—	2	2	no	—
164	C₂×C₄×C₁₃₁	—	4	1	no	—
165	C₂×C₄×C₁₃₁	—	4	1	no	—
166	C₂×C₄×C₁₃₁	—	4	1	no	—
167	C₂×C₄×C₁₃₁	—	4	1	no	—
168	C₂×C₄×C₁₃₁	—	4	2	no	—
169	C₂×C₄×C₁₃₁	—	4	2	no	—
170	C₂×C₄×C₁₃₁	—	4	1	no	—
171	C₂×C₄×C₁₃₁	—	4	1	no	—
172	C₂×C₄×C₁₃₁	—	4	2	no	—
173	C₂×C₄×C₁₃₁	—	4	1	no	—
174	C₂×C₄×C₁₃₁	—	4	2	no	—
175	C₂×C₄×C₁₃₁	—	4	2	no	—
176	C₂×C₄×C₁₃₁	—	4	2	no	—
177	C₂×C₄×C₁₃₁	—	4	1	no	—
178	C₂×C₄×C₁₃₁	—	8	4	no	—
179	C₂×C₄×C₁₃₁	#123	1	2	yes	—
180	C₂×C₄×C₁₃₁	#123	1	2	yes	—
181	C₂×C₄×C₁₃₁	#135	1	2	yes	—
182	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
183	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
184	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
185	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
187	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
188	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
189	C₈×C₁₃₁	—	2	2	no	—
190	C₈×C₁₃₁	—	2	2	no	—
191	C₈×C₁₃₁	—	4	2	no	—
192	C₂²×C₁₃₁	—	4	2	no	—
193	C₂²×C₁₃₁	—	4	2	no	—
194	C₂²×C₁₃₁	—	4	2	no	—
195	C₂²×C₁₃₁	—	8	1	no	—
196	C₂²×C₁₃₁	—	8	1	no	—
197	C₂²×C₁₃₁	—	8	1	no	—
198	C₂²×C₁₃₁	—	8	1	no	—
199	C₂²×C₁₃₁	C₂×#123	1	1	yes	characteristic
200	C₂²×C₁₃₁	D₄×Q₈	1	1	yes	characteristic
201	C₂²×C₁₃₁	#550	1	1	yes	characteristic
202	C₂²×C₁₃₁	C₂×#135	1	1	yes	characteristic
203	C₂²×C₁₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
204	C₂²×C₁₃₁	#576	1	1	yes	characteristic
205	C₂²×C₁₃₁	#578	1	1	yes	characteristic
206	C₄×C₁₃₁	—	2	2	no	—
207	C₄×C₁₃₁	—	4	2	no	—
208	C₄×C₁₃₁	—	4	2	no	—
209	C₄×C₁₃₁	—	4	2	no	—
210	C₄×C₁₃₁	—	8	1	no	—
211	C₄×C₁₃₁	—	8	1	no	—
212	C₄×C₁₃₁	—	8	2	no	—
213	C₄×C₁₃₁	—	8	2	no	—
214	C₄×C₁₃₁	#482	1	4	yes	—
215	C₄×C₁₃₁	#533	1	1	yes	characteristic
216	C₄×C₁₃₁	#542	1	1	yes	characteristic
217	C₄×C₁₃₁	#552	1	1	yes	characteristic
218	C₄×C₁₃₁	#554	1	1	yes	characteristic
219	C₂×C₁₃₁	—	8	2	no	—
220	C₂×C₁₃₁	#2423	1	1	yes	characteristic
221	C₂×C₁₃₁	C₂×#482	1	1	yes	characteristic
222	C₂×C₁₃₁	#3098	1	1	yes	characteristic
223	C₂×C₁₃₁	#3380	1	1	yes	characteristic
224	C₂×C₁₃₁	#3386	1	1	yes	characteristic
225	C₂×C₁₃₁	#3393	1	1	yes	characteristic
226	C₂×C₁₃₁	#3399	1	1	yes	characteristic
227	#25057	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₁₃₁	#25057	1	1	yes	characteristic, direct product factor, Sylow
229	#2538	C₂×C₁₃₁	1	1	yes	characteristic
230	#2355	C₂×C₁₃₁	1	1	yes	characteristic, complemented
231	#2092	C₂×C₁₃₁	1	2	yes	—
232	#2085	C₂×C₁₃₁	1	2	yes	complemented
233	#1984	C₂×C₁₃₁	1	1	yes	characteristic, complemented
234	#1948	C₂×C₁₃₁	1	1	yes	characteristic, complemented
235	#1925	C₂×C₁₃₁	1	2	yes	—
236	#1921	C₂×C₁₃₁	1	2	yes	complemented
237	#1845	C₂×C₁₃₁	1	1	yes	characteristic, complemented
238	#1802	C₂×C₁₃₁	1	1	yes	characteristic
239	C₄×#102	C₂×C₁₃₁	1	1	yes	characteristic
240	C₂×C₂²⋊Q₈	—	2	1	no	—
241	C₂×C₄×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
242	C₂×C₄×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
243	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
244	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
245	C₂×#106	C₂²×C₁₃₁	1	2	yes	—
246	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
247	C₂×#102	C₂²×C₁₃₁	1	1	yes	characteristic
248	C₂×#102	C₂²×C₁₃₁	1	2	yes	—
249	C₂×#102	C₂²×C₁₃₁	1	1	yes	characteristic
250	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
251	#396	—	2	2	no	—
252	#391	—	2	1	no	—
253	#390	C₂²×C₁₃₁	1	2	yes	—
254	#389	C₂²×C₁₃₁	1	2	yes	—
255	#387	—	2	2	no	—
256	#384	C₂²×C₁₃₁	1	1	yes	characteristic
257	#384	C₂²×C₁₃₁	1	1	yes	characteristic
258	#382	C₂²×C₁₃₁	1	2	yes	—
259	#380	—	2	1	no	—
260	C₄×C₄⋊C₄	—	2	1	no	—
261	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
262	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
263	#340	C₂²×C₁₃₁	1	2	yes	—
264	#340	C₂²×C₁₃₁	1	1	yes	characteristic
265	#340	C₂²×C₁₃₁	1	1	yes	characteristic
266	#340	C₂²×C₁₃₁	1	2	yes	—
267	#336	C₂²×C₁₃₁	1	2	yes	—
268	#336	C₂²×C₁₃₁	1	2	yes	—
269	C₂²×Q₈	D₄×C₁₃₁	1	1	yes	characteristic
270	C₂²×D₄	C₂³×C₁₃₁	1	1	yes	characteristic
271	C₂³×C₄	—	2	1	no	—
272	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
273	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
274	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
275	C₂²⋊Q₈	—	4	4	no	—
276	C₄×Q₈	—	2	2	no	—
277	C₄×Q₈	—	2	2	no	—
278	C₄×D₄	—	2	2	no	—
279	C₄×D₄	—	2	2	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
285	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
286	C₂×C₄⋊C₄	D₄×C₁₃₁	1	2	yes	—
287	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
289	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
290	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
291	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
292	C₂×C₂²⋊C₄	—	2	1	no	—
293	C₂×C₂²⋊C₄	—	2	2	no	—
294	C₂×C₂²⋊C₄	—	2	2	no	—
295	C₂×C₄²	—	2	1	no	—
296	C₂×C₄²	—	2	2	no	—
297	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
298	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
299	#107	—	2	4	no	—
300	#107	—	4	2	no	—
301	#106	—	4	2	no	—
302	C₄⋊C₈	—	2	2	no	—
303	#102	—	2	2	no	—
304	#102	—	4	2	no	—
305	#102	Q₈×C₁₃₁	1	4	yes	complemented
306	C₄×C₈	—	2	2	no	—
307	#95	—	2	2	no	—
308	#95	—	2	1	no	—
309	#95	—	2	2	no	—
310	#95	—	2	2	no	—
311	#95	—	2	2	no	—
312	C₂⁴	—	2	1	no	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	2	2	no	—
316	C₂×D₄	—	2	1	no	—
317	C₂×D₄	—	2	1	no	—
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	C₂×Q₈×C₁₃₁	1	2	yes	—
322	C₂²×C₄	—	2	1	no	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	4	2	no	—
328	C₂²×C₄	—	4	2	no	—
329	C₂²×C₄	—	4	2	no	—
330	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
331	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
332	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
334	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	C₂×Q₈×C₁₃₁	1	2	yes	—
343	C₂×C₈	#41×C₁₃₁	1	2	yes	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	4	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	4	4	no	—
356	C₄⋊C₄	—	4	2	no	—
357	C₄⋊C₄	—	4	2	no	—
358	C₄⋊C₄	—	8	1	no	—
359	C₄⋊C₄	—	8	1	no	—
360	C₄⋊C₄	C₂×Q₈×C₁₃₁	1	2	yes	—
361	C₂²⋊C₄	—	4	4	no	—
362	C₂²⋊C₄	—	4	2	no	—
363	C₂²⋊C₄	—	8	2	no	—
364	C₄²	—	2	2	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	4	2	no	—
367	C₄²	—	8	2	no	—
368	C₄²	#41×C₁₃₁	1	2	yes	—
369	C₄²	#41×C₁₃₁	1	2	yes	—
370	C₂³	—	2	2	no	—
371	C₂³	—	4	1	no	—
372	C₂³	—	4	1	no	—
373	C₂³	—	4	1	no	—
374	C₂³	—	4	1	no	—
375	C₂³	—	4	1	no	—
376	C₂³	—	4	1	no	—
377	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
378	Q₈	—	4	4	no	—
379	D₄	—	2	2	no	—
380	D₄	—	2	2	no	—
381	D₄	—	4	1	no	—
382	D₄	—	4	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	8	4	no	—
407	C₂×C₄	#123×C₁₃₁	1	2	yes	—
408	C₂×C₄	#123×C₁₃₁	1	2	yes	—
409	C₂×C₄	#135×C₁₃₁	1	2	yes	—
410	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
411	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
412	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
413	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
415	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
416	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
417	C₈	—	2	2	no	—
418	C₈	—	2	2	no	—
419	C₈	—	4	2	no	—
420	C₂²	—	4	2	no	—
421	C₂²	—	4	2	no	—
422	C₂²	—	4	2	no	—
423	C₂²	—	8	1	no	—
424	C₂²	—	8	1	no	—
425	C₂²	—	8	1	no	—
426	C₂²	—	8	1	no	—
427	C₂²	C₂×#123×C₁₃₁	1	1	yes	characteristic
428	C₂²	D₄×Q₈×C₁₃₁	1	1	yes	characteristic
429	C₂²	#550×C₁₃₁	1	1	yes	characteristic
430	C₂²	C₂×#135×C₁₃₁	1	1	yes	characteristic
431	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
432	C₂²	#576×C₁₃₁	1	1	yes	characteristic
433	C₂²	#578×C₁₃₁	1	1	yes	characteristic
434	C₄	—	2	2	no	—
435	C₄	—	4	2	no	—
436	C₄	—	4	2	no	—
437	C₄	—	4	2	no	—
438	C₄	—	8	1	no	—
439	C₄	—	8	1	no	—
440	C₄	—	8	2	no	—
441	C₄	—	8	2	no	—
442	C₄	#482×C₁₃₁	1	4	yes	—
443	C₄	#533×C₁₃₁	1	1	yes	characteristic
444	C₄	#542×C₁₃₁	1	1	yes	characteristic
445	C₄	#552×C₁₃₁	1	1	yes	characteristic
446	C₄	#554×C₁₃₁	1	1	yes	characteristic
447	C₂	—	8	2	no	—
448	C₂	#2423×C₁₃₁	1	1	yes	characteristic
449	C₂	C₂×#482×C₁₃₁	1	1	yes	characteristic
450	C₂	#3098×C₁₃₁	1	1	yes	characteristic
451	C₂	#3380×C₁₃₁	1	1	yes	characteristic
452	C₂	#3386×C₁₃₁	1	1	yes	characteristic
453	C₂	#3393×C₁₃₁	1	1	yes	characteristic
454	C₂	#3399×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂²×D₄×C₁₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂×C₄×Q₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2355×C₁₃₁	C₂	semidirect
4. #2085×C₁₃₁	C₂	semidirect
5. #1984×C₁₃₁	C₂	semidirect
6. #1948×C₁₃₁	C₂	semidirect
8. #1921×C₁₃₁	C₂	semidirect
9. #1845×C₁₃₁	C₂	semidirect
77. #102×C₁₃₁	Q₈	semidirect
227. #25057	C₁₃₁	direct (N × H)
228. C₁₃₁	#25057	direct (N × H)
230. #2355	C₂×C₁₃₁	semidirect
232. #2085	C₂×C₁₃₁	semidirect
233. #1984	C₂×C₁₃₁	semidirect
234. #1948	C₂×C₁₃₁	semidirect
236. #1921	C₂×C₁₃₁	semidirect
237. #1845	C₂×C₁₃₁	semidirect
305. #102	Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2538×C₁₃₁	2
#2355×C₁₃₁	2
#2092×C₁₃₁	2
#2085×C₁₃₁	2
#1984×C₁₃₁	2
#1948×C₁₃₁	2
#1925×C₁₃₁	2
#1921×C₁₃₁	2
#1845×C₁₃₁	2
#1802×C₁₃₁	2
C₄×#102×C₁₃₁	2
#25057	131

Subgroups — C131×#25057 #45710941

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#25057 #45710941