Subgroups — C₁₃₁×#20421 #45706305

Order 33536.

1490 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#104×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2896×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#2873×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×M₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#1765×C₁₃₁	C₂	1	4	yes	maximal, complemented
7	#1764×C₁₃₁	C₂	1	4	yes	maximal, complemented
8	#1738×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#339×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₁₃₁	C₂²	1	1	yes	characteristic
11	#568×C₁₃₁	—	2	4	no	—
12	#568×C₁₃₁	C₄	1	8	yes	complemented
13	C₂×#131×C₁₃₁	—	2	2	no	—
14	C₂×#131×C₁₃₁	C₂²	1	4	yes	—
15	C₂²×M₁₆×C₁₃₁	—	2	2	no	—
16	C₂²×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
22	C₂×#104×C₁₃₁	C₄	1	8	yes	complemented
23	C₂×#104×C₁₃₁	C₂²	1	2	yes	—
24	C₂×#104×C₁₃₁	C₂²	1	2	yes	—
25	#419×C₁₃₁	C₄	1	2	yes	complemented
26	#418×C₁₃₁	C₄	1	2	yes	complemented
27	#417×C₁₃₁	C₄	1	4	yes	complemented
28	#416×C₁₃₁	C₂²	1	4	yes	—
29	#405×C₁₃₁	—	2	2	no	—
30	#405×C₁₃₁	C₂²	1	4	yes	complemented
31	C₄×M₁₆×C₁₃₁	—	2	4	no	—
32	C₄×M₁₆×C₁₃₁	—	2	2	no	—
33	C₂×#97×C₁₃₁	C₂²	1	2	yes	complemented
34	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	complemented
35	#341×C₁₃₁	C₂²	1	4	yes	complemented
36	#339×C₁₃₁	C₂²	1	2	yes	complemented
37	#339×C₁₃₁	C₂²	1	2	yes	complemented
38	C₂×#41×C₁₃₁	—	2	1	no	—
39	C₂×#41×C₁₃₁	—	2	2	no	—
40	C₂×#41×C₁₃₁	—	2	1	no	—
41	C₂×#41×C₁₃₁	—	2	2	no	—
42	C₂×#41×C₁₃₁	C₂×C₄	1	4	yes	—
43	C₂×#41×C₁₃₁	C₂×C₄	1	4	yes	—
44	C₂×#41×C₁₃₁	C₂×C₄	1	2	yes	—
45	C₂×#41×C₁₃₁	C₂³	1	2	yes	—
46	C₂²×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×D₄×C₁₃₁	—	2	1	no	—
48	C₂²×D₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₁₃₁	—	2	1	no	—
50	C₂³×C₄×C₁₃₁	—	2	1	no	—
51	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
52	#131×C₁₃₁	—	2	16	no	—
53	#131×C₁₃₁	—	2	8	no	—
54	#131×C₁₃₁	C₂×C₄	1	16	yes	complemented
55	C₂×M₁₆×C₁₃₁	—	2	2	no	—
56	C₂×M₁₆×C₁₃₁	—	2	1	no	—
57	C₂×M₁₆×C₁₃₁	—	2	1	no	—
58	C₂×M₁₆×C₁₃₁	—	2	2	no	—
59	C₂×M₁₆×C₁₃₁	—	2	2	no	—
60	C₂×M₁₆×C₁₃₁	—	2	2	no	—
61	C₂×M₁₆×C₁₃₁	—	2	16	no	—
62	C₂×M₁₆×C₁₃₁	—	2	4	no	—
63	C₂×M₁₆×C₁₃₁	C₂×C₄	1	4	yes	—
64	C₂×M₁₆×C₁₃₁	C₂×C₄	1	4	yes	—
65	C₂×M₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
66	C₂×M₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
67	C₂×M₁₆×C₁₃₁	C₂×C₄	1	4	yes	—
68	C₂×M₁₆×C₁₃₁	D₄	1	2	yes	—
69	C₂×M₁₆×C₁₃₁	D₄	1	4	yes	—
70	C₂×M₁₆×C₁₃₁	D₄	1	2	yes	—
71	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
72	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
73	C₂²×C₈×C₁₃₁	—	2	1	no	—
74	C₂²×C₈×C₁₃₁	—	2	1	no	—
75	C₂²×C₈×C₁₃₁	—	2	4	no	—
76	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
77	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
78	#117×C₁₃₁	—	2	1	no	—
79	#117×C₁₃₁	—	2	1	no	—
80	#117×C₁₃₁	C₂×C₄	1	2	yes	—
81	#117×C₁₃₁	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
83	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
84	C₂×C₄²×C₁₃₁	—	2	2	no	—
85	C₂×C₄²×C₁₃₁	—	2	1	no	—
86	C₂×C₄²×C₁₃₁	—	2	1	no	—
87	C₂×C₄²×C₁₃₁	C₂×C₄	1	4	yes	—
88	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
90	#104×C₁₃₁	C₂×C₄	1	8	yes	complemented
91	#104×C₁₃₁	C₂×C₄	1	8	yes	complemented
92	#103×C₁₃₁	C₂×C₄	1	8	yes	complemented
93	#102×C₁₃₁	C₂×C₄	1	8	yes	complemented
94	#97×C₁₃₁	—	2	4	no	—
95	#97×C₁₃₁	—	2	4	no	—
96	C₄×C₈×C₁₃₁	—	2	4	no	—
97	C₄×C₈×C₁₃₁	—	2	4	no	—
98	C₂⁴×C₁₃₁	—	2	1	no	—
99	C₂⁴×C₁₃₁	#41	1	1	yes	characteristic
100	#41×C₁₃₁	—	2	4	no	—
101	#41×C₁₃₁	—	2	4	no	—
102	#41×C₁₃₁	—	2	4	no	—
103	#41×C₁₃₁	—	2	4	no	—
104	#41×C₁₃₁	—	2	4	no	—
105	#41×C₁₃₁	—	2	2	no	—
106	#41×C₁₃₁	—	2	2	no	—
107	#41×C₁₃₁	C₄²	1	8	yes	—
108	#41×C₁₃₁	C₂²×C₄	1	4	yes	—
109	#41×C₁₃₁	C₂²×C₄	1	4	yes	—
110	C₂×Q₈×C₁₃₁	—	2	1	no	—
111	C₂×Q₈×C₁₃₁	—	2	2	no	—
112	C₂×Q₈×C₁₃₁	C₄²	1	4	yes	—
113	C₂×Q₈×C₁₃₁	C₂²×C₄	1	2	yes	—
114	C₂×D₄×C₁₃₁	—	2	4	no	—
115	C₂×D₄×C₁₃₁	—	2	2	no	—
116	C₂×D₄×C₁₃₁	—	2	4	no	—
117	C₂×D₄×C₁₃₁	—	2	2	no	—
118	C₂×D₄×C₁₃₁	—	2	2	no	—
119	C₂×D₄×C₁₃₁	—	2	1	no	—
120	C₂×D₄×C₁₃₁	C₄²	1	4	yes	—
121	C₂×D₄×C₁₃₁	C₂²×C₄	1	2	yes	—
122	C₂²×C₄×C₁₃₁	—	2	1	no	—
123	C₂²×C₄×C₁₃₁	—	2	1	no	—
124	C₂²×C₄×C₁₃₁	—	2	4	no	—
125	C₂²×C₄×C₁₃₁	—	2	2	no	—
126	C₂²×C₄×C₁₃₁	—	2	4	no	—
127	C₂²×C₄×C₁₃₁	—	2	2	no	—
128	C₂²×C₄×C₁₃₁	—	2	4	no	—
129	C₂²×C₄×C₁₃₁	—	2	1	no	—
130	C₂²×C₄×C₁₃₁	—	2	2	no	—
131	C₂²×C₄×C₁₃₁	—	4	4	no	—
132	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
133	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
134	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
140	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
141	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
142	M₁₆×C₁₃₁	—	2	4	no	—
143	M₁₆×C₁₃₁	—	2	4	no	—
144	M₁₆×C₁₃₁	—	2	4	no	—
145	M₁₆×C₁₃₁	—	2	4	no	—
146	M₁₆×C₁₃₁	—	2	2	no	—
147	M₁₆×C₁₃₁	—	2	2	no	—
148	M₁₆×C₁₃₁	—	2	16	no	—
149	M₁₆×C₁₃₁	C₂²⋊C₄	1	8	yes	—
150	M₁₆×C₁₃₁	C₂²⋊C₄	1	8	yes	—
151	M₁₆×C₁₃₁	C₂²×C₄	1	4	yes	—
152	M₁₆×C₁₃₁	C₂²×C₄	1	4	yes	—
153	C₂×C₈×C₁₃₁	—	2	4	no	—
154	C₂×C₈×C₁₃₁	—	2	4	no	—
155	C₂×C₈×C₁₃₁	—	2	2	no	—
156	C₂×C₈×C₁₃₁	—	2	2	no	—
157	C₂×C₈×C₁₃₁	—	2	2	no	—
158	C₂×C₈×C₁₃₁	—	2	2	no	—
159	C₂×C₈×C₁₃₁	—	2	16	no	—
160	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
161	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
162	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
163	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
164	C₄⋊C₄×C₁₃₁	—	2	2	no	—
165	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
166	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
167	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
168	C₄²×C₁₃₁	—	2	4	no	—
169	C₄²×C₁₃₁	—	2	2	no	—
170	C₄²×C₁₃₁	—	2	1	no	—
171	C₄²×C₁₃₁	—	2	1	no	—
172	C₄²×C₁₃₁	—	2	2	no	—
173	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
174	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
175	C₂³×C₁₃₁	—	2	2	no	—
176	C₂³×C₁₃₁	—	2	4	no	—
177	C₂³×C₁₃₁	—	4	2	no	—
178	C₂³×C₁₃₁	—	8	2	no	—
179	C₂³×C₁₃₁	#117	1	2	yes	—
180	C₂³×C₁₃₁	C₄×D₄	1	1	yes	characteristic
181	C₂³×C₁₃₁	C₄×D₄	1	1	yes	characteristic
182	C₂³×C₁₃₁	C₄×D₄	1	2	yes	—
183	C₂³×C₁₃₁	C₂×#41	1	1	yes	characteristic
184	Q₈×C₁₃₁	—	2	2	no	—
185	Q₈×C₁₃₁	—	2	4	no	—
186	Q₈×C₁₃₁	C₂×C₄²	1	4	yes	—
187	D₄×C₁₃₁	—	2	4	no	—
188	D₄×C₁₃₁	—	2	4	no	—
189	D₄×C₁₃₁	—	2	8	no	—
190	D₄×C₁₃₁	—	2	2	no	—
191	D₄×C₁₃₁	—	2	4	no	—
192	D₄×C₁₃₁	C₂×C₄²	1	4	yes	—
193	C₂×C₄×C₁₃₁	—	2	4	no	—
194	C₂×C₄×C₁₃₁	—	2	2	no	—
195	C₂×C₄×C₁₃₁	—	2	2	no	—
196	C₂×C₄×C₁₃₁	—	2	4	no	—
197	C₂×C₄×C₁₃₁	—	2	4	no	—
198	C₂×C₄×C₁₃₁	—	2	4	no	—
199	C₂×C₄×C₁₃₁	—	2	8	no	—
200	C₂×C₄×C₁₃₁	—	2	2	no	—
201	C₂×C₄×C₁₃₁	—	2	4	no	—
202	C₂×C₄×C₁₃₁	—	4	8	no	—
203	C₂×C₄×C₁₃₁	—	4	2	no	—
204	C₂×C₄×C₁₃₁	—	8	2	no	—
205	C₂×C₄×C₁₃₁	C₂×C₄²	1	2	yes	—
206	C₂×C₄×C₁₃₁	C₂×C₄²	1	2	yes	—
207	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	2	yes	—
208	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	2	yes	—
209	C₂×C₄×C₁₃₁	#117	1	2	yes	—
210	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
211	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
212	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
213	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
214	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
215	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
216	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
217	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
218	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
219	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
220	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
223	C₈×C₁₃₁	—	2	4	no	—
224	C₈×C₁₃₁	—	2	4	no	—
225	C₈×C₁₃₁	—	2	4	no	—
226	C₈×C₁₃₁	C₂×C₂²⋊C₄	1	8	yes	—
227	C₂²×C₁₃₁	—	2	2	no	—
228	C₂²×C₁₃₁	—	2	2	no	—
229	C₂²×C₁₃₁	—	2	2	no	—
230	C₂²×C₁₃₁	—	2	4	no	—
231	C₂²×C₁₃₁	—	4	2	no	—
232	C₂²×C₁₃₁	—	4	2	no	—
233	C₂²×C₁₃₁	—	4	4	no	—
234	C₂²×C₁₃₁	—	8	4	no	—
235	C₂²×C₁₃₁	C₄×C₂²⋊C₄	1	2	yes	—
236	C₂²×C₁₃₁	C₄×C₂²⋊C₄	1	2	yes	—
237	C₂²×C₁₃₁	C₂×#117	1	1	yes	characteristic
238	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
239	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
240	C₄×C₁₃₁	—	2	4	no	—
241	C₄×C₁₃₁	—	2	4	no	—
242	C₄×C₁₃₁	—	4	2	no	—
243	C₄×C₁₃₁	—	4	4	no	—
244	C₄×C₁₃₁	C₄×C₂²⋊C₄	1	2	yes	—
245	C₄×C₁₃₁	C₄×C₂²⋊C₄	1	2	yes	—
246	C₄×C₁₃₁	C₂²×C₄²	1	1	yes	characteristic
247	C₄×C₁₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
248	C₄×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
249	C₄×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
250	C₂×C₁₃₁	—	2	1	no	—
251	C₂×C₁₃₁	—	2	2	no	—
252	C₂×C₁₃₁	—	2	1	no	—
253	C₂×C₁₃₁	—	2	2	no	—
254	C₂×C₁₃₁	—	4	4	no	—
255	C₂×C₁₃₁	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₁₃₁	#2956	1	2	yes	—
257	#20421	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
258	C₁₃₁	#20421	1	1	yes	characteristic, direct product factor, Sylow
259	C₂×#568	C₂×C₁₃₁	1	1	yes	characteristic
260	C₂²×#104	C₂×C₁₃₁	1	1	yes	characteristic
261	#2896	C₂×C₁₃₁	1	1	yes	characteristic
262	#2873	C₂×C₁₃₁	1	1	yes	characteristic, complemented
263	C₂×C₄×M₁₆	C₂×C₁₃₁	1	1	yes	characteristic, complemented
264	#1765	C₂×C₁₃₁	1	4	yes	complemented
265	#1764	C₂×C₁₃₁	1	4	yes	complemented
266	#1738	C₂×C₁₃₁	1	1	yes	characteristic, complemented
267	C₂×#339	C₂×C₁₃₁	1	1	yes	characteristic, complemented
268	C₂²×#41	C₂²×C₁₃₁	1	1	yes	characteristic
269	#568	—	2	4	no	—
270	#568	C₄×C₁₃₁	1	8	yes	complemented
271	C₂×#131	—	2	2	no	—
272	C₂×#131	C₂²×C₁₃₁	1	4	yes	—
273	C₂²×M₁₆	—	2	2	no	—
274	C₂²×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
275	C₂²×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
276	C₂²×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
277	C₂²×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
278	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
279	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
280	C₂×#104	C₄×C₁₃₁	1	8	yes	complemented
281	C₂×#104	C₂²×C₁₃₁	1	2	yes	—
282	C₂×#104	C₂²×C₁₃₁	1	2	yes	—
283	#419	C₄×C₁₃₁	1	2	yes	complemented
284	#418	C₄×C₁₃₁	1	2	yes	complemented
285	#417	C₄×C₁₃₁	1	4	yes	complemented
286	#416	C₂²×C₁₃₁	1	4	yes	—
287	#405	—	2	2	no	—
288	#405	C₂²×C₁₃₁	1	4	yes	complemented
289	C₄×M₁₆	—	2	4	no	—
290	C₄×M₁₆	—	2	2	no	—
291	C₂×#97	C₂²×C₁₃₁	1	2	yes	complemented
292	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	complemented
293	#341	C₂²×C₁₃₁	1	4	yes	complemented
294	#339	C₂²×C₁₃₁	1	2	yes	complemented
295	#339	C₂²×C₁₃₁	1	2	yes	complemented
296	C₂×#41	—	2	1	no	—
297	C₂×#41	—	2	2	no	—
298	C₂×#41	—	2	1	no	—
299	C₂×#41	—	2	2	no	—
300	C₂×#41	C₂×C₄×C₁₃₁	1	4	yes	—
301	C₂×#41	C₂×C₄×C₁₃₁	1	4	yes	—
302	C₂×#41	C₂×C₄×C₁₃₁	1	2	yes	—
303	C₂×#41	C₂³×C₁₃₁	1	2	yes	—
304	C₂²×Q₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
305	C₂²×D₄	—	2	1	no	—
306	C₂²×D₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
307	C₂³×C₄	—	2	1	no	—
308	C₂³×C₄	—	2	1	no	—
309	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
310	#131	—	2	16	no	—
311	#131	—	2	8	no	—
312	#131	C₂×C₄×C₁₃₁	1	16	yes	complemented
313	C₂×M₁₆	—	2	1	no	—
314	C₂×M₁₆	—	2	2	no	—
315	C₂×M₁₆	—	2	1	no	—
316	C₂×M₁₆	—	2	2	no	—
317	C₂×M₁₆	—	2	2	no	—
318	C₂×M₁₆	—	2	2	no	—
319	C₂×M₁₆	—	2	16	no	—
320	C₂×M₁₆	—	2	4	no	—
321	C₂×M₁₆	C₂×C₄×C₁₃₁	1	4	yes	—
322	C₂×M₁₆	C₂×C₄×C₁₃₁	1	4	yes	—
323	C₂×M₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
324	C₂×M₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
325	C₂×M₁₆	C₂×C₄×C₁₃₁	1	4	yes	—
326	C₂×M₁₆	D₄×C₁₃₁	1	2	yes	—
327	C₂×M₁₆	D₄×C₁₃₁	1	2	yes	—
328	C₂×M₁₆	D₄×C₁₃₁	1	4	yes	—
329	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
330	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
331	C₂²×C₈	—	2	1	no	—
332	C₂²×C₈	—	2	1	no	—
333	C₂²×C₈	—	2	4	no	—
334	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
335	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
336	#117	—	2	1	no	—
337	#117	—	2	1	no	—
338	#117	C₂×C₄×C₁₃₁	1	2	yes	—
339	#117	C₂³×C₁₃₁	1	2	yes	—
340	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
341	C₂×C₂²⋊C₄	—	2	1	no	—
342	C₂×C₄²	—	2	1	no	—
343	C₂×C₄²	—	2	2	no	—
344	C₂×C₄²	—	2	1	no	—
345	C₂×C₄²	C₂×C₄×C₁₃₁	1	4	yes	—
346	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
347	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
348	#104	C₂×C₄×C₁₃₁	1	8	yes	complemented
349	#104	C₂×C₄×C₁₃₁	1	8	yes	complemented
350	#103	C₂×C₄×C₁₃₁	1	8	yes	complemented
351	#102	C₂×C₄×C₁₃₁	1	8	yes	complemented
352	#97	—	2	4	no	—
353	#97	—	2	4	no	—
354	C₄×C₈	—	2	4	no	—
355	C₄×C₈	—	2	4	no	—
356	C₂⁴	—	2	1	no	—
357	C₂⁴	#41×C₁₃₁	1	1	yes	characteristic
358	#41	—	2	4	no	—
359	#41	—	2	4	no	—
360	#41	—	2	4	no	—
361	#41	—	2	4	no	—
362	#41	—	2	4	no	—
363	#41	—	2	2	no	—
364	#41	—	2	2	no	—
365	#41	C₄²×C₁₃₁	1	8	yes	—
366	#41	C₂²×C₄×C₁₃₁	1	4	yes	—
367	#41	C₂²×C₄×C₁₃₁	1	4	yes	—
368	C₂×Q₈	—	2	1	no	—
369	C₂×Q₈	—	2	2	no	—
370	C₂×Q₈	C₄²×C₁₃₁	1	4	yes	—
371	C₂×Q₈	C₂²×C₄×C₁₃₁	1	2	yes	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	4	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	C₄²×C₁₃₁	1	4	yes	—
379	C₂×D₄	C₂²×C₄×C₁₃₁	1	2	yes	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	4	4	no	—
390	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
391	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
392	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
393	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
395	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
396	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
397	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
398	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
399	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
400	M₁₆	—	2	4	no	—
401	M₁₆	—	2	4	no	—
402	M₁₆	—	2	4	no	—
403	M₁₆	—	2	4	no	—
404	M₁₆	—	2	2	no	—
405	M₁₆	—	2	2	no	—
406	M₁₆	—	2	16	no	—
407	M₁₆	C₂²⋊C₄×C₁₃₁	1	8	yes	—
408	M₁₆	C₂²⋊C₄×C₁₃₁	1	8	yes	—
409	M₁₆	C₂²×C₄×C₁₃₁	1	4	yes	—
410	M₁₆	C₂²×C₄×C₁₃₁	1	4	yes	—
411	C₂×C₈	—	2	4	no	—
412	C₂×C₈	—	2	4	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	16	no	—
418	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
419	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
420	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
421	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
424	C₂²⋊C₄	—	2	2	no	—
425	C₂²⋊C₄	—	2	2	no	—
426	C₄²	—	2	4	no	—
427	C₄²	—	2	2	no	—
428	C₄²	—	2	1	no	—
429	C₄²	—	2	2	no	—
430	C₄²	—	2	1	no	—
431	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
432	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
433	C₂³	—	2	2	no	—
434	C₂³	—	2	4	no	—
435	C₂³	—	4	2	no	—
436	C₂³	—	8	2	no	—
437	C₂³	#117×C₁₃₁	1	2	yes	—
438	C₂³	C₄×D₄×C₁₃₁	1	1	yes	characteristic
439	C₂³	C₄×D₄×C₁₃₁	1	1	yes	characteristic
440	C₂³	C₄×D₄×C₁₃₁	1	2	yes	—
441	C₂³	C₂×#41×C₁₃₁	1	1	yes	characteristic
442	Q₈	—	2	2	no	—
443	Q₈	—	2	4	no	—
444	Q₈	C₂×C₄²×C₁₃₁	1	4	yes	—
445	D₄	—	2	4	no	—
446	D₄	—	2	4	no	—
447	D₄	—	2	8	no	—
448	D₄	—	2	2	no	—
449	D₄	—	2	4	no	—
450	D₄	C₂×C₄²×C₁₃₁	1	4	yes	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	8	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	4	8	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	8	2	no	—
463	C₂×C₄	C₂×C₄²×C₁₃₁	1	2	yes	—
464	C₂×C₄	C₂×C₄²×C₁₃₁	1	2	yes	—
465	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	2	yes	—
466	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	2	yes	—
467	C₂×C₄	#117×C₁₃₁	1	2	yes	—
468	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
469	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
470	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
471	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
472	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
473	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
474	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
475	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
476	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
477	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
478	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
479	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
480	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
481	C₈	—	2	4	no	—
482	C₈	—	2	4	no	—
483	C₈	—	2	4	no	—
484	C₈	C₂×C₂²⋊C₄×C₁₃₁	1	8	yes	—
485	C₂²	—	2	2	no	—
486	C₂²	—	2	2	no	—
487	C₂²	—	2	2	no	—
488	C₂²	—	2	4	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	4	no	—
492	C₂²	—	8	4	no	—
493	C₂²	C₄×C₂²⋊C₄×C₁₃₁	1	2	yes	—
494	C₂²	C₄×C₂²⋊C₄×C₁₃₁	1	2	yes	—
495	C₂²	C₂×#117×C₁₃₁	1	1	yes	characteristic
496	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
497	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
498	C₄	—	2	4	no	—
499	C₄	—	2	4	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	4	no	—
502	C₄	C₄×C₂²⋊C₄×C₁₃₁	1	2	yes	—
503	C₄	C₄×C₂²⋊C₄×C₁₃₁	1	2	yes	—
504	C₄	C₂²×C₄²×C₁₃₁	1	1	yes	characteristic
505	C₄	C₂²×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
506	C₄	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
507	C₄	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
508	C₂	—	2	1	no	—
509	C₂	—	2	2	no	—
510	C₂	—	2	1	no	—
511	C₂	—	2	2	no	—
512	C₂	—	4	4	no	—
513	C₂	C₂×C₄×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
514	C₂	#2956×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₁	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
socle	C₂²×C₁₃₁	C₂×C₄×D₄
prime radical	C₂²×#41×C₁₃₁	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂²×M₁₆×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. #2873×C₁₃₁	C₂	semidirect
5. C₂×C₄×M₁₆×C₁₃₁	C₂	semidirect
6. #1765×C₁₃₁	C₂	semidirect
7. #1764×C₁₃₁	C₂	semidirect
8. #1738×C₁₃₁	C₂	semidirect
9. C₂×#339×C₁₃₁	C₂	semidirect
12. #568×C₁₃₁	C₄	semidirect
22. C₂×#104×C₁₃₁	C₄	semidirect
25. #419×C₁₃₁	C₄	semidirect
26. #418×C₁₃₁	C₄	semidirect
27. #417×C₁₃₁	C₄	semidirect
30. #405×C₁₃₁	C₂²	semidirect
33. C₂×#97×C₁₃₁	C₂²	semidirect
34. C₂×C₄×C₈×C₁₃₁	C₂²	semidirect
35. #341×C₁₃₁	C₂²	semidirect
36. #339×C₁₃₁	C₂²	semidirect
37. #339×C₁₃₁	C₂²	semidirect
54. #131×C₁₃₁	C₂×C₄	semidirect
90. #104×C₁₃₁	C₂×C₄	semidirect
91. #104×C₁₃₁	C₂×C₄	semidirect
92. #103×C₁₃₁	C₂×C₄	semidirect
93. #102×C₁₃₁	C₂×C₄	semidirect
257. #20421	C₁₃₁	direct (N × H)
258. C₁₃₁	#20421	direct (N × H)
262. #2873	C₂×C₁₃₁	semidirect
263. C₂×C₄×M₁₆	C₂×C₁₃₁	semidirect
264. #1765	C₂×C₁₃₁	semidirect
265. #1764	C₂×C₁₃₁	semidirect
266. #1738	C₂×C₁₃₁	semidirect
267. C₂×#339	C₂×C₁₃₁	semidirect
270. #568	C₄×C₁₃₁	semidirect
280. C₂×#104	C₄×C₁₃₁	semidirect
283. #419	C₄×C₁₃₁	semidirect
284. #418	C₄×C₁₃₁	semidirect
285. #417	C₄×C₁₃₁	semidirect
288. #405	C₂²×C₁₃₁	semidirect
291. C₂×#97	C₂²×C₁₃₁	semidirect
292. C₂×C₄×C₈	C₂²×C₁₃₁	semidirect
293. #341	C₂²×C₁₃₁	semidirect
294. #339	C₂²×C₁₃₁	semidirect
295. #339	C₂²×C₁₃₁	semidirect
312. #131	C₂×C₄×C₁₃₁	semidirect
348. #104	C₂×C₄×C₁₃₁	semidirect
349. #104	C₂×C₄×C₁₃₁	semidirect
350. #103	C₂×C₄×C₁₃₁	semidirect
351. #102	C₂×C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#568×C₁₃₁	2
C₂²×#104×C₁₃₁	2
#2896×C₁₃₁	2
#2873×C₁₃₁	2
C₂×C₄×M₁₆×C₁₃₁	2
#1765×C₁₃₁	2
#1764×C₁₃₁	2
#1738×C₁₃₁	2
C₂×#339×C₁₃₁	2
#20421	131

Subgroups — C131×#20421 #45706305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#20421 #45706305