Subgroups — C₁₃₁×#34123 #45720007

Order 33536.

2598 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2356×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#2353×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	#2349×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2346×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#2343×C₁₃₁	C₂	1	4	yes	maximal, complemented
7	#2339×C₁₃₁	C₂	1	2	yes	maximal
8	#2295×C₁₃₁	C₂	1	2	yes	maximal, complemented
9	#2290×C₁₃₁	C₂	1	4	yes	maximal, complemented
10	#2283×C₁₃₁	C₂	1	2	yes	maximal, complemented
11	#2283×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄²×D₄×C₁₃₁	C₂	1	2	yes	maximal, complemented
13	C₄×#117×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#376×C₁₃₁	C₂	1	2	yes	maximal, complemented
15	C₂²×#41×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₄×#41×C₁₃₁	C₄	1	16	yes	complemented
17	C₂×C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₁₃₁	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
24	C₂×#117×C₁₃₁	C₂²	1	2	yes	—
25	C₂×#117×C₁₃₁	C₂²	1	2	yes	—
26	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₁₃₁	C₂²	1	4	yes	—
28	C₂²×C₄²×C₁₃₁	C₂²	1	2	yes	—
29	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
30	#391×C₁₃₁	C₂²	1	4	yes	—
31	#389×C₁₃₁	C₂²	1	4	yes	—
32	#388×C₁₃₁	C₂²	1	8	yes	—
33	#387×C₁₃₁	C₂²	1	8	yes	—
34	#387×C₁₃₁	C₂²	1	8	yes	—
35	#386×C₁₃₁	C₂²	1	8	yes	—
36	#385×C₁₃₁	C₂²	1	8	yes	—
37	#383×C₁₃₁	C₂²	1	2	yes	—
38	#383×C₁₃₁	C₂²	1	4	yes	—
39	#382×C₁₃₁	C₂²	1	4	yes	—
40	#382×C₁₃₁	C₂²	1	2	yes	—
41	#381×C₁₃₁	C₂²	1	4	yes	—
42	#381×C₁₃₁	C₂²	1	2	yes	—
43	#380×C₁₃₁	C₂²	1	2	yes	—
44	#380×C₁₃₁	C₂²	1	4	yes	—
45	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
46	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
47	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
48	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	—
49	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	—
50	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
51	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
52	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	4	yes	—
53	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	8	yes	—
54	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
55	#376×C₁₃₁	C₂²	1	1	yes	characteristic
56	#376×C₁₃₁	C₂²	1	2	yes	—
57	#376×C₁₃₁	C₂²	1	4	yes	—
58	#376×C₁₃₁	C₂²	1	4	yes	—
59	#376×C₁₃₁	C₂²	1	4	yes	—
60	#376×C₁₃₁	C₂²	1	1	yes	characteristic
61	C₂×#95×C₁₃₁	C₂²	1	4	yes	—
62	C₂×#95×C₁₃₁	C₂²	1	4	yes	—
63	C₄³×C₁₃₁	C₂²	1	2	yes	—
64	C₂×#41×C₁₃₁	—	2	8	no	—
65	C₂×#41×C₁₃₁	C₂×C₄	1	8	yes	—
66	C₂²×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₁₃₁	C₂³	1	8	yes	—
70	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
71	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
73	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
76	C₄×Q₈×C₁₃₁	C₂×C₄	1	8	yes	complemented
77	C₄×Q₈×C₁₃₁	C₂×C₄	1	8	yes	complemented
78	C₄×D₄×C₁₃₁	—	2	16	no	—
79	C₄×D₄×C₁₃₁	C₂×C₄	1	8	yes	—
80	C₄×D₄×C₁₃₁	C₂×C₄	1	8	yes	—
81	C₄×D₄×C₁₃₁	C₂×C₄	1	16	yes	complemented
82	C₄×D₄×C₁₃₁	C₂×C₄	1	16	yes	complemented
83	#117×C₁₃₁	—	2	8	no	—
84	#117×C₁₃₁	C₂×C₄	1	16	yes	complemented
85	#117×C₁₃₁	C₂×C₄	1	8	yes	complemented
86	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	8	yes	—
89	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
93	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	8	yes	—
95	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
97	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
98	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
99	C₂×C₄²×C₁₃₁	—	2	16	no	—
100	C₂×C₄²×C₁₃₁	C₂×C₄	1	8	yes	complemented
101	C₂×C₄²×C₁₃₁	C₂×C₄	1	16	yes	complemented
102	C₂×C₄²×C₁₃₁	C₂³	1	8	yes	—
103	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
104	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
105	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
106	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
107	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
108	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
110	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
114	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
115	#95×C₁₃₁	C₂³	1	8	yes	—
116	#95×C₁₃₁	C₂³	1	8	yes	—
117	#95×C₁₃₁	C₂³	1	8	yes	—
118	#95×C₁₃₁	C₂³	1	8	yes	—
119	#95×C₁₃₁	C₂³	1	2	yes	—
120	#95×C₁₃₁	C₂³	1	2	yes	—
121	#95×C₁₃₁	C₂³	1	4	yes	—
122	#95×C₁₃₁	C₂³	1	4	yes	—
123	#95×C₁₃₁	C₂³	1	2	yes	—
124	#95×C₁₃₁	C₂³	1	2	yes	—
125	#95×C₁₃₁	C₂³	1	4	yes	—
126	#95×C₁₃₁	C₂³	1	4	yes	—
127	C₂⁴×C₁₃₁	C₂⁴	1	1	yes	characteristic
128	C₂⁴×C₁₃₁	C₂⁴	1	2	yes	—
129	#41×C₁₃₁	—	2	32	no	—
130	C₂×Q₈×C₁₃₁	—	2	4	no	—
131	C₂×Q₈×C₁₃₁	C₂²×C₄	1	4	yes	—
132	C₂×D₄×C₁₃₁	—	2	8	no	—
133	C₂×D₄×C₁₃₁	C₂²×C₄	1	8	yes	—
134	C₂×D₄×C₁₃₁	C₂²×C₄	1	4	yes	—
135	C₂×D₄×C₁₃₁	#41	1	8	yes	—
136	C₂²×C₄×C₁₃₁	—	2	8	no	—
137	C₂²×C₄×C₁₃₁	—	2	8	no	—
138	C₂²×C₄×C₁₃₁	—	2	4	no	—
139	C₂²×C₄×C₁₃₁	—	2	4	no	—
140	C₂²×C₄×C₁₃₁	—	2	16	no	—
141	C₂²×C₄×C₁₃₁	—	2	16	no	—
142	C₂²×C₄×C₁₃₁	—	4	2	no	—
143	C₂²×C₄×C₁₃₁	—	4	8	no	—
144	C₂²×C₄×C₁₃₁	—	4	4	no	—
145	C₂²×C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
146	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
147	C₂²×C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
148	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
149	C₂²×C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
150	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
151	C₂²×C₄×C₁₃₁	C₂⁴	1	8	yes	—
152	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
154	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
155	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
159	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
162	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
163	C₄⋊C₄×C₁₃₁	—	2	8	no	—
164	C₄⋊C₄×C₁₃₁	—	2	8	no	—
165	C₄⋊C₄×C₁₃₁	—	4	8	no	—
166	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
167	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
168	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
169	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
170	C₂²⋊C₄×C₁₃₁	—	2	16	no	—
171	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
172	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
173	C₂²⋊C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
174	C₂²⋊C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
175	C₂²⋊C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
176	C₄²×C₁₃₁	—	2	16	no	—
177	C₄²×C₁₃₁	—	2	8	no	—
178	C₄²×C₁₃₁	—	2	4	no	—
179	C₄²×C₁₃₁	—	2	8	no	—
180	C₄²×C₁₃₁	—	4	4	no	—
181	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
182	C₄²×C₁₃₁	C₂²×C₄	1	8	yes	—
183	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
184	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
185	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
186	C₄²×C₁₃₁	C₂²×C₄	1	8	yes	—
187	C₄²×C₁₃₁	C₂²×C₄	1	8	yes	—
188	C₄²×C₁₃₁	#41	1	8	yes	complemented
189	C₂³×C₁₃₁	—	2	4	no	—
190	C₂³×C₁₃₁	—	2	4	no	—
191	C₂³×C₁₃₁	—	2	2	no	—
192	C₂³×C₁₃₁	—	2	2	no	—
193	C₂³×C₁₃₁	—	2	2	no	—
194	C₂³×C₁₃₁	C₂³×C₄	1	2	yes	—
195	C₂³×C₁₃₁	C₂³×C₄	1	4	yes	—
196	C₂³×C₁₃₁	C₂×#41	1	8	yes	—
197	C₂³×C₁₃₁	C₂⁵	1	1	yes	characteristic
198	Q₈×C₁₃₁	—	2	8	no	—
199	D₄×C₁₃₁	—	2	16	no	—
200	D₄×C₁₃₁	#117	1	16	yes	complemented
201	C₂×C₄×C₁₃₁	—	2	16	no	—
202	C₂×C₄×C₁₃₁	—	2	4	no	—
203	C₂×C₄×C₁₃₁	—	2	4	no	—
204	C₂×C₄×C₁₃₁	—	2	2	no	—
205	C₂×C₄×C₁₃₁	—	2	2	no	—
206	C₂×C₄×C₁₃₁	—	2	4	no	—
207	C₂×C₄×C₁₃₁	—	2	2	no	—
208	C₂×C₄×C₁₃₁	—	2	4	no	—
209	C₂×C₄×C₁₃₁	—	2	4	no	—
210	C₂×C₄×C₁₃₁	—	2	2	no	—
211	C₂×C₄×C₁₃₁	—	2	8	no	—
212	C₂×C₄×C₁₃₁	—	2	16	no	—
213	C₂×C₄×C₁₃₁	—	2	8	no	—
214	C₂×C₄×C₁₃₁	—	2	8	no	—
215	C₂×C₄×C₁₃₁	—	2	4	no	—
216	C₂×C₄×C₁₃₁	—	2	4	no	—
217	C₂×C₄×C₁₃₁	—	2	8	no	—
218	C₂×C₄×C₁₃₁	—	2	4	no	—
219	C₂×C₄×C₁₃₁	—	2	8	no	—
220	C₂×C₄×C₁₃₁	—	4	8	no	—
221	C₂×C₄×C₁₃₁	—	4	4	no	—
222	C₂×C₄×C₁₃₁	—	4	32	no	—
223	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
224	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
225	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
226	C₂×C₄×C₁₃₁	C₂³×C₄	1	4	yes	—
227	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
228	C₂×C₄×C₁₃₁	C₂³×C₄	1	4	yes	—
229	C₂×C₄×C₁₃₁	C₂³×C₄	1	4	yes	—
230	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
231	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
232	C₂×C₄×C₁₃₁	C₂×#41	1	8	yes	—
233	C₂×C₄×C₁₃₁	C₂×#41	1	4	yes	—
234	C₂×C₄×C₁₃₁	C₂×#41	1	4	yes	—
235	C₂²×C₁₃₁	—	2	4	no	—
236	C₂²×C₁₃₁	—	2	4	no	—
237	C₂²×C₁₃₁	—	2	16	no	—
238	C₂²×C₁₃₁	—	2	2	no	—
239	C₂²×C₁₃₁	—	2	2	no	—
240	C₂²×C₁₃₁	—	2	2	no	—
241	C₂²×C₁₃₁	—	4	4	no	—
242	C₂²×C₁₃₁	C₂×#117	1	8	yes	—
243	C₂²×C₁₃₁	C₂⁴×C₄	1	1	yes	characteristic
244	C₂²×C₁₃₁	C₂²×#41	1	2	yes	—
245	C₂²×C₁₃₁	C₂²×#41	1	2	yes	—
246	C₂²×C₁₃₁	#585	1	2	yes	—
247	C₄×C₁₃₁	—	2	2	no	—
248	C₄×C₁₃₁	—	2	4	no	—
249	C₄×C₁₃₁	—	2	4	no	—
250	C₄×C₁₃₁	—	2	2	no	—
251	C₄×C₁₃₁	—	2	2	no	—
252	C₄×C₁₃₁	—	2	2	no	—
253	C₄×C₁₃₁	—	2	4	no	—
254	C₄×C₁₃₁	—	2	8	no	—
255	C₄×C₁₃₁	—	4	4	no	—
256	C₄×C₁₃₁	—	4	8	no	—
257	C₄×C₁₃₁	C₂×#117	1	4	yes	—
258	C₄×C₁₃₁	C₄×#41	1	4	yes	—
259	C₂×C₁₃₁	—	2	8	no	—
260	C₂×C₁₃₁	—	4	2	no	—
261	C₂×C₁₃₁	C₂²×#117	1	1	yes	characteristic
262	C₂×C₁₃₁	C₂×C₄×#41	1	1	yes	characteristic
263	C₂×C₁₃₁	#3428	1	1	yes	characteristic
264	C₂×C₁₃₁	#3475	1	4	yes	—
265	#34123	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₁₃₁	#34123	1	1	yes	characteristic, direct product factor, Sylow
267	C₂×C₄×#41	C₂×C₁₃₁	1	1	yes	characteristic
268	#2356	C₂×C₁₃₁	1	2	yes	complemented
269	#2353	C₂×C₁₃₁	1	4	yes	complemented
270	#2349	C₂×C₁₃₁	1	2	yes	complemented
271	#2346	C₂×C₁₃₁	1	2	yes	complemented
272	#2343	C₂×C₁₃₁	1	4	yes	complemented
273	#2339	C₂×C₁₃₁	1	2	yes	—
274	#2295	C₂×C₁₃₁	1	2	yes	complemented
275	#2290	C₂×C₁₃₁	1	4	yes	complemented
276	#2283	C₂×C₁₃₁	1	1	yes	characteristic, complemented
277	#2283	C₂×C₁₃₁	1	2	yes	complemented
278	C₄²×D₄	C₂×C₁₃₁	1	2	yes	complemented
279	C₄×#117	C₂×C₁₃₁	1	1	yes	characteristic, complemented
280	C₂×#376	C₂×C₁₃₁	1	2	yes	complemented
281	C₂²×#41	C₂²×C₁₃₁	1	1	yes	characteristic
282	C₄×#41	C₄×C₁₃₁	1	16	yes	complemented
283	C₂×C₄×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
284	C₂×C₄×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
285	C₂×C₄×D₄	C₂²×C₁₃₁	1	4	yes	—
286	C₂×C₄×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
287	C₂×C₄×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
288	C₂×C₄×D₄	C₂²×C₁₃₁	1	2	yes	—
289	C₂×C₄×D₄	C₂²×C₁₃₁	1	2	yes	—
290	C₂×#117	C₂²×C₁₃₁	1	2	yes	—
291	C₂×#117	C₂²×C₁₃₁	1	2	yes	—
292	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
293	C₂²×C₄²	C₂²×C₁₃₁	1	4	yes	—
294	C₂²×C₄²	C₂²×C₁₃₁	1	2	yes	—
295	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
296	#391	C₂²×C₁₃₁	1	4	yes	—
297	#389	C₂²×C₁₃₁	1	4	yes	—
298	#388	C₂²×C₁₃₁	1	8	yes	—
299	#387	C₂²×C₁₃₁	1	8	yes	—
300	#387	C₂²×C₁₃₁	1	8	yes	—
301	#386	C₂²×C₁₃₁	1	8	yes	—
302	#385	C₂²×C₁₃₁	1	8	yes	—
303	#383	C₂²×C₁₃₁	1	2	yes	—
304	#383	C₂²×C₁₃₁	1	4	yes	—
305	#382	C₂²×C₁₃₁	1	4	yes	—
306	#382	C₂²×C₁₃₁	1	2	yes	—
307	#381	C₂²×C₁₃₁	1	4	yes	—
308	#381	C₂²×C₁₃₁	1	2	yes	—
309	#380	C₂²×C₁₃₁	1	2	yes	—
310	#380	C₂²×C₁₃₁	1	4	yes	—
311	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
312	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	—
313	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
314	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
315	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	—
316	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	2	yes	—
317	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	8	yes	—
318	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	2	yes	—
319	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	4	yes	—
320	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	2	yes	—
321	#376	C₂²×C₁₃₁	1	4	yes	—
322	#376	C₂²×C₁₃₁	1	4	yes	—
323	#376	C₂²×C₁₃₁	1	1	yes	characteristic
324	#376	C₂²×C₁₃₁	1	2	yes	—
325	#376	C₂²×C₁₃₁	1	4	yes	—
326	#376	C₂²×C₁₃₁	1	1	yes	characteristic
327	C₂×#95	C₂²×C₁₃₁	1	4	yes	—
328	C₂×#95	C₂²×C₁₃₁	1	4	yes	—
329	C₄³	C₂²×C₁₃₁	1	2	yes	—
330	C₂×#41	—	2	8	no	—
331	C₂×#41	C₂×C₄×C₁₃₁	1	8	yes	—
332	C₂²×Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
333	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
334	C₂²×D₄	C₂³×C₁₃₁	1	1	yes	characteristic
335	C₂³×C₄	C₂³×C₁₃₁	1	8	yes	—
336	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
337	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
338	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
339	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
340	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
341	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
342	C₄×Q₈	C₂×C₄×C₁₃₁	1	8	yes	complemented
343	C₄×Q₈	C₂×C₄×C₁₃₁	1	8	yes	complemented
344	C₄×D₄	—	2	16	no	—
345	C₄×D₄	C₂×C₄×C₁₃₁	1	8	yes	—
346	C₄×D₄	C₂×C₄×C₁₃₁	1	8	yes	—
347	C₄×D₄	C₂×C₄×C₁₃₁	1	16	yes	complemented
348	C₄×D₄	C₂×C₄×C₁₃₁	1	16	yes	complemented
349	#117	—	2	8	no	—
350	#117	C₂×C₄×C₁₃₁	1	16	yes	complemented
351	#117	C₂×C₄×C₁₃₁	1	8	yes	complemented
352	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
353	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
354	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	8	yes	—
355	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
356	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
357	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
358	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
359	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	4	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	8	yes	—
361	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	4	yes	—
362	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	2	yes	—
363	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	2	yes	—
364	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	2	yes	—
365	C₂×C₄²	—	2	16	no	—
366	C₂×C₄²	C₂×C₄×C₁₃₁	1	8	yes	complemented
367	C₂×C₄²	C₂×C₄×C₁₃₁	1	16	yes	complemented
368	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
369	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
370	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
371	C₂×C₄²	C₂³×C₁₃₁	1	8	yes	—
372	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
373	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
374	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
375	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
376	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
377	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
378	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
379	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
380	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
381	#95	C₂³×C₁₃₁	1	2	yes	—
382	#95	C₂³×C₁₃₁	1	2	yes	—
383	#95	C₂³×C₁₃₁	1	4	yes	—
384	#95	C₂³×C₁₃₁	1	4	yes	—
385	#95	C₂³×C₁₃₁	1	8	yes	—
386	#95	C₂³×C₁₃₁	1	8	yes	—
387	#95	C₂³×C₁₃₁	1	2	yes	—
388	#95	C₂³×C₁₃₁	1	2	yes	—
389	#95	C₂³×C₁₃₁	1	4	yes	—
390	#95	C₂³×C₁₃₁	1	8	yes	—
391	#95	C₂³×C₁₃₁	1	4	yes	—
392	#95	C₂³×C₁₃₁	1	8	yes	—
393	C₂⁴	C₂⁴×C₁₃₁	1	1	yes	characteristic
394	C₂⁴	C₂⁴×C₁₃₁	1	2	yes	—
395	#41	—	2	32	no	—
396	C₂×Q₈	—	2	4	no	—
397	C₂×Q₈	C₂²×C₄×C₁₃₁	1	4	yes	—
398	C₂×D₄	—	2	8	no	—
399	C₂×D₄	C₂²×C₄×C₁₃₁	1	8	yes	—
400	C₂×D₄	C₂²×C₄×C₁₃₁	1	4	yes	—
401	C₂×D₄	#41×C₁₃₁	1	8	yes	—
402	C₂²×C₄	—	2	8	no	—
403	C₂²×C₄	—	2	8	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	4	no	—
406	C₂²×C₄	—	2	16	no	—
407	C₂²×C₄	—	2	16	no	—
408	C₂²×C₄	—	4	2	no	—
409	C₂²×C₄	—	4	8	no	—
410	C₂²×C₄	—	4	4	no	—
411	C₂²×C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
412	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
413	C₂²×C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
414	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
415	C₂²×C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
416	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
417	C₂²×C₄	C₂⁴×C₁₃₁	1	8	yes	—
418	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
419	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
420	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
421	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
422	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
423	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
424	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
425	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
426	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
427	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
428	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
429	C₄⋊C₄	—	2	8	no	—
430	C₄⋊C₄	—	2	8	no	—
431	C₄⋊C₄	—	4	8	no	—
432	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
433	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
434	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
435	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
436	C₂²⋊C₄	—	2	16	no	—
437	C₂²⋊C₄	—	2	8	no	—
438	C₂²⋊C₄	—	4	4	no	—
439	C₂²⋊C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
440	C₂²⋊C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
441	C₂²⋊C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
442	C₄²	—	2	16	no	—
443	C₄²	—	2	8	no	—
444	C₄²	—	2	4	no	—
445	C₄²	—	2	8	no	—
446	C₄²	—	4	4	no	—
447	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
448	C₄²	C₂²×C₄×C₁₃₁	1	8	yes	—
449	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
450	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
451	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
452	C₄²	C₂²×C₄×C₁₃₁	1	8	yes	—
453	C₄²	C₂²×C₄×C₁₃₁	1	8	yes	—
454	C₄²	#41×C₁₃₁	1	8	yes	complemented
455	C₂³	—	2	4	no	—
456	C₂³	—	2	4	no	—
457	C₂³	—	2	2	no	—
458	C₂³	—	2	2	no	—
459	C₂³	—	2	2	no	—
460	C₂³	C₂³×C₄×C₁₃₁	1	2	yes	—
461	C₂³	C₂³×C₄×C₁₃₁	1	4	yes	—
462	C₂³	C₂×#41×C₁₃₁	1	8	yes	—
463	C₂³	C₂⁵×C₁₃₁	1	1	yes	characteristic
464	Q₈	—	2	8	no	—
465	D₄	—	2	16	no	—
466	D₄	#117×C₁₃₁	1	16	yes	complemented
467	C₂×C₄	—	2	16	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	8	no	—
478	C₂×C₄	—	2	16	no	—
479	C₂×C₄	—	2	8	no	—
480	C₂×C₄	—	2	8	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	8	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	8	no	—
486	C₂×C₄	—	4	8	no	—
487	C₂×C₄	—	4	4	no	—
488	C₂×C₄	—	4	32	no	—
489	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
490	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
491	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
492	C₂×C₄	C₂³×C₄×C₁₃₁	1	4	yes	—
493	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
494	C₂×C₄	C₂³×C₄×C₁₃₁	1	4	yes	—
495	C₂×C₄	C₂³×C₄×C₁₃₁	1	4	yes	—
496	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
497	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
498	C₂×C₄	C₂×#41×C₁₃₁	1	8	yes	—
499	C₂×C₄	C₂×#41×C₁₃₁	1	4	yes	—
500	C₂×C₄	C₂×#41×C₁₃₁	1	4	yes	—
501	C₂²	—	2	4	no	—
502	C₂²	—	2	4	no	—
503	C₂²	—	2	16	no	—
504	C₂²	—	2	2	no	—
505	C₂²	—	2	2	no	—
506	C₂²	—	2	2	no	—
507	C₂²	—	4	4	no	—
508	C₂²	C₂×#117×C₁₃₁	1	8	yes	—
509	C₂²	C₂⁴×C₄×C₁₃₁	1	1	yes	characteristic
510	C₂²	C₂²×#41×C₁₃₁	1	2	yes	—
511	C₂²	C₂²×#41×C₁₃₁	1	2	yes	—
512	C₂²	#585×C₁₃₁	1	2	yes	—
513	C₄	—	2	2	no	—
514	C₄	—	2	4	no	—
515	C₄	—	2	4	no	—
516	C₄	—	2	2	no	—
517	C₄	—	2	2	no	—
518	C₄	—	2	2	no	—
519	C₄	—	2	4	no	—
520	C₄	—	2	8	no	—
521	C₄	—	4	8	no	—
522	C₄	—	4	4	no	—
523	C₄	C₂×#117×C₁₃₁	1	4	yes	—
524	C₄	C₄×#41×C₁₃₁	1	4	yes	—
525	C₂	—	2	8	no	—
526	C₂	—	4	2	no	—
527	C₂	C₂²×#117×C₁₃₁	1	1	yes	characteristic
528	C₂	C₂×C₄×#41×C₁₃₁	1	1	yes	characteristic
529	C₂	#3428×C₁₃₁	1	1	yes	characteristic
530	C₂	#3475×C₁₃₁	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₁₃₁
Frattini, cosocle	C₂³	C₂⁵×C₁₃₁
socle	C₂³×C₁₃₁	C₂⁵
prime radical	C₂²×#41×C₁₃₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2356×C₁₃₁	C₂	semidirect
3. #2353×C₁₃₁	C₂	semidirect
4. #2349×C₁₃₁	C₂	semidirect
5. #2346×C₁₃₁	C₂	semidirect
6. #2343×C₁₃₁	C₂	semidirect
8. #2295×C₁₃₁	C₂	semidirect
9. #2290×C₁₃₁	C₂	semidirect
10. #2283×C₁₃₁	C₂	semidirect
11. #2283×C₁₃₁	C₂	semidirect
12. C₄²×D₄×C₁₃₁	C₂	semidirect
13. C₄×#117×C₁₃₁	C₂	semidirect
14. C₂×#376×C₁₃₁	C₂	semidirect
16. C₄×#41×C₁₃₁	C₄	semidirect
76. C₄×Q₈×C₁₃₁	C₂×C₄	semidirect
77. C₄×Q₈×C₁₃₁	C₂×C₄	semidirect
81. C₄×D₄×C₁₃₁	C₂×C₄	semidirect
82. C₄×D₄×C₁₃₁	C₂×C₄	semidirect
84. #117×C₁₃₁	C₂×C₄	semidirect
85. #117×C₁₃₁	C₂×C₄	semidirect
100. C₂×C₄²×C₁₃₁	C₂×C₄	semidirect
101. C₂×C₄²×C₁₃₁	C₂×C₄	semidirect
188. C₄²×C₁₃₁	#41	semidirect
200. D₄×C₁₃₁	#117	semidirect
265. #34123	C₁₃₁	direct (N × H)
266. C₁₃₁	#34123	direct (N × H)
268. #2356	C₂×C₁₃₁	semidirect
269. #2353	C₂×C₁₃₁	semidirect
270. #2349	C₂×C₁₃₁	semidirect
271. #2346	C₂×C₁₃₁	semidirect
272. #2343	C₂×C₁₃₁	semidirect
274. #2295	C₂×C₁₃₁	semidirect
275. #2290	C₂×C₁₃₁	semidirect
276. #2283	C₂×C₁₃₁	semidirect
277. #2283	C₂×C₁₃₁	semidirect
278. C₄²×D₄	C₂×C₁₃₁	semidirect
279. C₄×#117	C₂×C₁₃₁	semidirect
280. C₂×#376	C₂×C₁₃₁	semidirect
282. C₄×#41	C₄×C₁₃₁	semidirect
342. C₄×Q₈	C₂×C₄×C₁₃₁	semidirect
343. C₄×Q₈	C₂×C₄×C₁₃₁	semidirect
347. C₄×D₄	C₂×C₄×C₁₃₁	semidirect
348. C₄×D₄	C₂×C₄×C₁₃₁	semidirect
350. #117	C₂×C₄×C₁₃₁	semidirect
351. #117	C₂×C₄×C₁₃₁	semidirect
366. C₂×C₄²	C₂×C₄×C₁₃₁	semidirect
367. C₂×C₄²	C₂×C₄×C₁₃₁	semidirect
454. C₄²	#41×C₁₃₁	semidirect
466. D₄	#117×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₁₃₁	2
#2356×C₁₃₁	2
#2353×C₁₃₁	2
#2349×C₁₃₁	2
#2346×C₁₃₁	2
#2343×C₁₃₁	2
#2339×C₁₃₁	2
#2295×C₁₃₁	2
#2290×C₁₃₁	2
#2283×C₁₃₁	2
C₄²×D₄×C₁₃₁	2
C₄×#117×C₁₃₁	2
C₂×#376×C₁₃₁	2
#34123	131

Subgroups — C131×#34123 #45720007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#34123 #45720007