Subgroups — C₁₃₁×#34985 #45720869

Order 33536.

2866 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#543×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2855×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#2848×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2627×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#2626×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#2450×C₁₃₁	C₂	1	8	yes	maximal, complemented
8	#2449×C₁₃₁	C₂	1	8	yes	maximal, complemented
9	#2315×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#2308×C₁₃₁	C₂	1	2	yes	maximal
11	C₂×#385×C₁₃₁	C₂	1	2	yes	maximal, complemented
12	C₂³×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
15	#543×C₁₃₁	—	2	8	no	—
16	C₂×C₂²⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
17	C₂×C₂²⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊Q₈×C₁₃₁	C₂²	1	4	yes	complemented
19	C₂×C₂²⋊Q₈×C₁₃₁	C₂²	1	4	yes	complemented
20	C₂×C₂²⋊Q₈×C₁₃₁	C₂²	1	4	yes	complemented
21	C₂×C₄×D₄×C₁₃₁	—	2	8	no	—
22	C₂×C₄×D₄×C₁₃₁	C₂²	1	8	yes	—
23	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	complemented
25	C₂²×C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
30	#400×C₁₃₁	C₂²	1	4	yes	complemented
31	#399×C₁₃₁	C₂²	1	4	yes	complemented
32	#398×C₁₃₁	C₂²	1	8	yes	complemented
33	#396×C₁₃₁	C₂²	1	8	yes	complemented
34	#395×C₁₃₁	C₂²	1	4	yes	complemented
35	#393×C₁₃₁	C₂²	1	4	yes	complemented
36	#391×C₁₃₁	C₂²	1	4	yes	complemented
37	#389×C₁₃₁	C₂²	1	4	yes	complemented
38	#387×C₁₃₁	C₂²	1	8	yes	complemented
39	#386×C₁₃₁	C₂²	1	4	yes	—
40	#386×C₁₃₁	C₂²	1	8	yes	—
41	#386×C₁₃₁	C₂²	1	4	yes	—
42	#385×C₁₃₁	C₂²	1	4	yes	—
43	#385×C₁₃₁	C₂²	1	4	yes	—
44	#385×C₁₃₁	C₂²	1	8	yes	complemented
45	#385×C₁₃₁	C₂²	1	8	yes	complemented
46	#385×C₁₃₁	C₂²	1	8	yes	complemented
47	#381×C₁₃₁	C₂²	1	4	yes	—
48	#380×C₁₃₁	C₂²	1	2	yes	—
49	#380×C₁₃₁	C₂²	1	2	yes	—
50	#380×C₁₃₁	C₂²	1	4	yes	complemented
51	#379×C₁₃₁	C₂²	1	4	yes	—
52	C₂×#95×C₁₃₁	C₂²	1	4	yes	complemented
53	C₂⁵×C₁₃₁	C₂³	1	1	yes	characteristic
54	C₂⁵×C₁₃₁	C₂³	1	1	yes	characteristic
55	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
56	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
57	C₂²×D₄×C₁₃₁	—	2	4	no	—
58	C₂²×D₄×C₁₃₁	—	2	4	no	—
59	C₂²×D₄×C₁₃₁	Q₈	1	8	yes	complemented
60	C₂²×D₄×C₁₃₁	C₂³	1	4	yes	—
61	C₂³×C₄×C₁₃₁	—	2	4	no	—
62	C₂³×C₄×C₁₃₁	—	2	4	no	—
63	C₂³×C₄×C₁₃₁	—	4	4	no	—
64	C₂³×C₄×C₁₃₁	—	4	4	no	—
65	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₁₃₁	C₂³	1	4	yes	—
67	C₂³×C₄×C₁₃₁	C₂³	1	4	yes	—
68	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
71	C₂³×C₄×C₁₃₁	C₂³	1	4	yes	—
72	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
73	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
75	C₂²⋊Q₈×C₁₃₁	—	2	8	no	—
76	C₂²⋊Q₈×C₁₃₁	—	2	8	no	—
77	C₂²⋊Q₈×C₁₃₁	—	2	16	no	—
78	C₂²⋊Q₈×C₁₃₁	—	2	16	no	—
79	C₂²⋊Q₈×C₁₃₁	—	4	8	no	—
80	C₄×D₄×C₁₃₁	—	2	32	no	—
81	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
82	C₂×C₄⋊C₄×C₁₃₁	—	4	8	no	—
83	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	8	yes	—
85	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
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	2	yes	—
89	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	8	yes	—
91	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
92	C₂×C₂²⋊C₄×C₁₃₁	—	2	8	no	—
93	C₂×C₂²⋊C₄×C₁₃₁	—	2	8	no	—
94	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
95	C₂×C₂²⋊C₄×C₁₃₁	D₄	1	8	yes	—
96	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	8	yes	—
102	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	8	yes	—
103	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	4	yes	—
104	C₂×C₄²×C₁₃₁	—	2	4	no	—
105	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
106	#95×C₁₃₁	C₂³	1	8	yes	—
107	#95×C₁₃₁	C₂³	1	8	yes	—
108	#95×C₁₃₁	C₂³	1	8	yes	—
109	#95×C₁₃₁	C₂³	1	8	yes	—
110	#95×C₁₃₁	C₂³	1	4	yes	—
111	#95×C₁₃₁	C₂³	1	4	yes	—
112	C₂⁴×C₁₃₁	—	2	2	no	—
113	C₂⁴×C₁₃₁	—	2	2	no	—
114	C₂⁴×C₁₃₁	—	4	2	no	—
115	C₂⁴×C₁₃₁	—	4	2	no	—
116	C₂⁴×C₁₃₁	—	4	2	no	—
117	C₂⁴×C₁₃₁	—	4	2	no	—
118	C₂⁴×C₁₃₁	C₂×D₄	1	4	yes	—
119	C₂⁴×C₁₃₁	C₂×Q₈	1	4	yes	—
120	C₂⁴×C₁₃₁	C₂×Q₈	1	4	yes	—
121	C₂⁴×C₁₃₁	#41	1	4	yes	—
122	C₂⁴×C₁₃₁	C₂⁴	1	2	yes	—
123	C₂⁴×C₁₃₁	C₂⁴	1	1	yes	characteristic
124	C₂⁴×C₁₃₁	C₂⁴	1	2	yes	—
125	C₂×Q₈×C₁₃₁	—	2	4	no	—
126	C₂×Q₈×C₁₃₁	—	2	4	no	—
127	C₂×Q₈×C₁₃₁	—	4	4	no	—
128	C₂×Q₈×C₁₃₁	—	4	4	no	—
129	C₂×D₄×C₁₃₁	—	2	32	no	—
130	C₂×D₄×C₁₃₁	—	2	8	no	—
131	C₂×D₄×C₁₃₁	—	2	8	no	—
132	C₂×D₄×C₁₃₁	C₂×Q₈	1	16	yes	—
133	C₂²×C₄×C₁₃₁	—	2	2	no	—
134	C₂²×C₄×C₁₃₁	—	2	2	no	—
135	C₂²×C₄×C₁₃₁	—	2	2	no	—
136	C₂²×C₄×C₁₃₁	—	2	4	no	—
137	C₂²×C₄×C₁₃₁	—	2	2	no	—
138	C₂²×C₄×C₁₃₁	—	2	8	no	—
139	C₂²×C₄×C₁₃₁	—	2	8	no	—
140	C₂²×C₄×C₁₃₁	—	2	4	no	—
141	C₂²×C₄×C₁₃₁	—	2	4	no	—
142	C₂²×C₄×C₁₃₁	—	2	8	no	—
143	C₂²×C₄×C₁₃₁	—	2	8	no	—
144	C₂²×C₄×C₁₃₁	—	2	8	no	—
145	C₂²×C₄×C₁₃₁	—	4	4	no	—
146	C₂²×C₄×C₁₃₁	—	4	2	no	—
147	C₂²×C₄×C₁₃₁	—	4	4	no	—
148	C₂²×C₄×C₁₃₁	—	4	4	no	—
149	C₂²×C₄×C₁₃₁	—	4	2	no	—
150	C₂²×C₄×C₁₃₁	—	4	2	no	—
151	C₂²×C₄×C₁₃₁	—	4	4	no	—
152	C₂²×C₄×C₁₃₁	—	4	4	no	—
153	C₂²×C₄×C₁₃₁	—	4	4	no	—
154	C₂²×C₄×C₁₃₁	—	4	16	no	—
155	C₂²×C₄×C₁₃₁	—	4	16	no	—
156	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
157	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
158	C₂²×C₄×C₁₃₁	C₂×Q₈	1	4	yes	—
159	C₂²×C₄×C₁₃₁	C₂⁴	1	8	yes	—
160	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
167	C₄⋊C₄×C₁₃₁	—	2	8	no	—
168	C₄⋊C₄×C₁₃₁	—	2	4	no	—
169	C₄⋊C₄×C₁₃₁	—	2	8	no	—
170	C₄⋊C₄×C₁₃₁	—	2	8	no	—
171	C₄⋊C₄×C₁₃₁	—	2	4	no	—
172	C₄⋊C₄×C₁₃₁	—	2	16	no	—
173	C₄⋊C₄×C₁₃₁	—	4	8	no	—
174	C₄⋊C₄×C₁₃₁	—	4	4	no	—
175	C₄⋊C₄×C₁₃₁	—	4	4	no	—
176	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
177	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
178	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
179	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
180	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
181	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
182	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
183	C₂²⋊C₄×C₁₃₁	C₂×D₄	1	16	yes	—
184	C₄²×C₁₃₁	—	2	8	no	—
185	C₂³×C₁₃₁	—	2	4	no	—
186	C₂³×C₁₃₁	—	2	2	no	—
187	C₂³×C₁₃₁	—	2	4	no	—
188	C₂³×C₁₃₁	—	2	2	no	—
189	C₂³×C₁₃₁	—	2	8	no	—
190	C₂³×C₁₃₁	—	2	2	no	—
191	C₂³×C₁₃₁	—	2	2	no	—
192	C₂³×C₁₃₁	—	2	4	no	—
193	C₂³×C₁₃₁	—	2	4	no	—
194	C₂³×C₁₃₁	—	4	4	no	—
195	C₂³×C₁₃₁	—	4	4	no	—
196	C₂³×C₁₃₁	—	4	8	no	—
197	C₂³×C₁₃₁	—	4	8	no	—
198	C₂³×C₁₃₁	—	4	8	no	—
199	C₂³×C₁₃₁	—	4	4	no	—
200	C₂³×C₁₃₁	—	4	8	no	—
201	C₂³×C₁₃₁	—	4	4	no	—
202	C₂³×C₁₃₁	C₂²⋊Q₈	1	16	yes	complemented
203	C₂³×C₁₃₁	C₂²×D₄	1	2	yes	—
204	C₂³×C₁₃₁	C₂²×D₄	1	4	yes	—
205	C₂³×C₁₃₁	C₂²×Q₈	1	4	yes	—
206	C₂³×C₁₃₁	C₂²×Q₈	1	2	yes	—
207	C₂³×C₁₃₁	C₂²×Q₈	1	4	yes	—
208	C₂³×C₁₃₁	C₂×#41	1	2	yes	—
209	C₂³×C₁₃₁	C₂×#41	1	4	yes	—
210	C₂³×C₁₃₁	C₂⁵	1	1	yes	characteristic
211	Q₈×C₁₃₁	—	8	4	no	—
212	Q₈×C₁₃₁	—	8	4	no	—
213	D₄×C₁₃₁	—	2	32	no	—
214	C₂×C₄×C₁₃₁	—	2	8	no	—
215	C₂×C₄×C₁₃₁	—	2	8	no	—
216	C₂×C₄×C₁₃₁	—	2	2	no	—
217	C₂×C₄×C₁₃₁	—	2	2	no	—
218	C₂×C₄×C₁₃₁	—	2	2	no	—
219	C₂×C₄×C₁₃₁	—	2	4	no	—
220	C₂×C₄×C₁₃₁	—	2	2	no	—
221	C₂×C₄×C₁₃₁	—	2	4	no	—
222	C₂×C₄×C₁₃₁	—	2	4	no	—
223	C₂×C₄×C₁₃₁	—	2	4	no	—
224	C₂×C₄×C₁₃₁	—	2	2	no	—
225	C₂×C₄×C₁₃₁	—	2	8	no	—
226	C₂×C₄×C₁₃₁	—	2	8	no	—
227	C₂×C₄×C₁₃₁	—	2	8	no	—
228	C₂×C₄×C₁₃₁	—	2	4	no	—
229	C₂×C₄×C₁₃₁	—	2	8	no	—
230	C₂×C₄×C₁₃₁	—	2	8	no	—
231	C₂×C₄×C₁₃₁	—	4	4	no	—
232	C₂×C₄×C₁₃₁	—	4	4	no	—
233	C₂×C₄×C₁₃₁	—	4	16	no	—
234	C₂×C₄×C₁₃₁	—	4	16	no	—
235	C₂×C₄×C₁₃₁	—	4	16	no	—
236	C₂×C₄×C₁₃₁	—	4	4	no	—
237	C₂×C₄×C₁₃₁	—	8	4	no	—
238	C₂×C₄×C₁₃₁	—	8	4	no	—
239	C₂×C₄×C₁₃₁	C₂²×D₄	1	4	yes	—
240	C₂×C₄×C₁₃₁	C₂²×D₄	1	4	yes	—
241	C₂×C₄×C₁₃₁	C₂²×Q₈	1	4	yes	—
242	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	4	yes	—
243	C₂²×C₁₃₁	—	2	8	no	—
244	C₂²×C₁₃₁	—	2	4	no	—
245	C₂²×C₁₃₁	—	2	2	no	—
246	C₂²×C₁₃₁	—	2	4	no	—
247	C₂²×C₁₃₁	—	2	2	no	—
248	C₂²×C₁₃₁	—	2	4	no	—
249	C₂²×C₁₃₁	—	2	4	no	—
250	C₂²×C₁₃₁	—	2	8	no	—
251	C₂²×C₁₃₁	—	2	4	no	—
252	C₂²×C₁₃₁	—	2	8	no	—
253	C₂²×C₁₃₁	—	4	16	no	—
254	C₂²×C₁₃₁	—	4	16	no	—
255	C₂²×C₁₃₁	—	4	4	no	—
256	C₂²×C₁₃₁	—	4	2	no	—
257	C₂²×C₁₃₁	—	4	2	no	—
258	C₂²×C₁₃₁	C₂×C₂²⋊Q₈	1	4	yes	—
259	C₂²×C₁₃₁	C₂×C₂²⋊Q₈	1	8	yes	—
260	C₂²×C₁₃₁	C₂³×D₄	1	1	yes	characteristic
261	C₂²×C₁₃₁	C₂³×Q₈	1	1	yes	characteristic
262	C₂²×C₁₃₁	C₂²×#41	1	1	yes	characteristic
263	C₂²×C₁₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
264	C₂²×C₁₃₁	#585	1	2	yes	—
265	C₄×C₁₃₁	—	2	4	no	—
266	C₄×C₁₃₁	—	2	4	no	—
267	C₄×C₁₃₁	—	2	4	no	—
268	C₄×C₁₃₁	—	2	4	no	—
269	C₄×C₁₃₁	—	4	8	no	—
270	C₄×C₁₃₁	—	4	2	no	—
271	C₄×C₁₃₁	—	4	4	no	—
272	C₄×C₁₃₁	—	4	4	no	—
273	C₂×C₁₃₁	—	2	4	no	—
274	C₂×C₁₃₁	—	2	8	no	—
275	C₂×C₁₃₁	—	4	4	no	—
276	C₂×C₁₃₁	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
277	C₂×C₁₃₁	#3459	1	2	yes	—
278	C₂×C₁₃₁	#3484	1	1	yes	characteristic
279	C₂×C₁₃₁	#3485	1	1	yes	characteristic
280	C₂×C₁₃₁	#3501	1	2	yes	—
281	#34985	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
282	C₁₃₁	#34985	1	1	yes	characteristic, direct product factor, Sylow
283	C₂×#543	C₂×C₁₃₁	1	1	yes	characteristic, complemented
284	C₂²×C₄×D₄	C₂×C₁₃₁	1	1	yes	characteristic
285	#2855	C₂×C₁₃₁	1	2	yes	complemented
286	#2848	C₂×C₁₃₁	1	2	yes	complemented
287	#2627	C₂×C₁₃₁	1	2	yes	complemented
288	#2626	C₂×C₁₃₁	1	2	yes	complemented
289	#2450	C₂×C₁₃₁	1	8	yes	complemented
290	#2449	C₂×C₁₃₁	1	8	yes	complemented
291	#2315	C₂×C₁₃₁	1	1	yes	characteristic, complemented
292	#2308	C₂×C₁₃₁	1	2	yes	—
293	C₂×#385	C₂×C₁₃₁	1	2	yes	complemented
294	C₂³×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
295	C₂⁴×C₄	C₂²×C₁₃₁	1	1	yes	characteristic
296	C₂⁴×C₄	C₂²×C₁₃₁	1	1	yes	characteristic
297	#543	—	2	8	no	—
298	C₂×C₂²⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
299	C₂×C₂²⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
300	C₂×C₂²⋊Q₈	C₂²×C₁₃₁	1	4	yes	complemented
301	C₂×C₂²⋊Q₈	C₂²×C₁₃₁	1	4	yes	complemented
302	C₂×C₂²⋊Q₈	C₂²×C₁₃₁	1	4	yes	complemented
303	C₂×C₄×D₄	—	2	8	no	—
304	C₂×C₄×D₄	C₂²×C₁₃₁	1	8	yes	—
305	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
306	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	complemented
307	C₂²×C₂²⋊C₄	C₂²×C₁₃₁	1	2	yes	—
308	C₂²×C₂²⋊C₄	C₂²×C₁₃₁	1	2	yes	—
309	C₂²×C₂²⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
310	C₂²×C₂²⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
311	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
312	#400	C₂²×C₁₃₁	1	4	yes	complemented
313	#399	C₂²×C₁₃₁	1	4	yes	complemented
314	#398	C₂²×C₁₃₁	1	8	yes	complemented
315	#396	C₂²×C₁₃₁	1	8	yes	complemented
316	#395	C₂²×C₁₃₁	1	4	yes	complemented
317	#393	C₂²×C₁₃₁	1	4	yes	complemented
318	#391	C₂²×C₁₃₁	1	4	yes	complemented
319	#389	C₂²×C₁₃₁	1	4	yes	complemented
320	#387	C₂²×C₁₃₁	1	8	yes	complemented
321	#386	C₂²×C₁₃₁	1	4	yes	—
322	#386	C₂²×C₁₃₁	1	8	yes	—
323	#386	C₂²×C₁₃₁	1	4	yes	—
324	#385	C₂²×C₁₃₁	1	4	yes	—
325	#385	C₂²×C₁₃₁	1	4	yes	—
326	#385	C₂²×C₁₃₁	1	8	yes	complemented
327	#385	C₂²×C₁₃₁	1	8	yes	complemented
328	#385	C₂²×C₁₃₁	1	8	yes	complemented
329	#381	C₂²×C₁₃₁	1	4	yes	—
330	#380	C₂²×C₁₃₁	1	2	yes	—
331	#380	C₂²×C₁₃₁	1	2	yes	—
332	#380	C₂²×C₁₃₁	1	4	yes	complemented
333	#379	C₂²×C₁₃₁	1	4	yes	—
334	C₂×#95	C₂²×C₁₃₁	1	4	yes	complemented
335	C₂⁵	C₂³×C₁₃₁	1	1	yes	characteristic
336	C₂⁵	C₂³×C₁₃₁	1	1	yes	characteristic
337	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
338	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
339	C₂²×D₄	—	2	4	no	—
340	C₂²×D₄	—	2	4	no	—
341	C₂²×D₄	Q₈×C₁₃₁	1	8	yes	complemented
342	C₂²×D₄	C₂³×C₁₃₁	1	4	yes	—
343	C₂³×C₄	—	2	4	no	—
344	C₂³×C₄	—	2	4	no	—
345	C₂³×C₄	—	4	4	no	—
346	C₂³×C₄	—	4	4	no	—
347	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
348	C₂³×C₄	C₂³×C₁₃₁	1	4	yes	—
349	C₂³×C₄	C₂³×C₁₃₁	1	4	yes	—
350	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
351	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
352	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
353	C₂³×C₄	C₂³×C₁₃₁	1	4	yes	—
354	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
355	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
357	C₂²⋊Q₈	—	2	8	no	—
358	C₂²⋊Q₈	—	2	8	no	—
359	C₂²⋊Q₈	—	2	16	no	—
360	C₂²⋊Q₈	—	2	16	no	—
361	C₂²⋊Q₈	—	4	8	no	—
362	C₄×D₄	—	2	32	no	—
363	C₂×C₄⋊C₄	—	2	4	no	—
364	C₂×C₄⋊C₄	—	4	8	no	—
365	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
366	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	8	yes	—
367	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
368	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
369	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
370	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
371	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	8	yes	—
372	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
373	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
374	C₂×C₂²⋊C₄	—	2	8	no	—
375	C₂×C₂²⋊C₄	—	2	8	no	—
376	C₂×C₂²⋊C₄	—	2	4	no	—
377	C₂×C₂²⋊C₄	D₄×C₁₃₁	1	8	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	4	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	4	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	4	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	4	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	4	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	8	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	8	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	4	yes	—
386	C₂×C₄²	—	2	4	no	—
387	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
388	#95	C₂³×C₁₃₁	1	8	yes	—
389	#95	C₂³×C₁₃₁	1	8	yes	—
390	#95	C₂³×C₁₃₁	1	4	yes	—
391	#95	C₂³×C₁₃₁	1	4	yes	—
392	#95	C₂³×C₁₃₁	1	8	yes	—
393	#95	C₂³×C₁₃₁	1	8	yes	—
394	C₂⁴	—	2	2	no	—
395	C₂⁴	—	2	2	no	—
396	C₂⁴	—	4	2	no	—
397	C₂⁴	—	4	2	no	—
398	C₂⁴	—	4	2	no	—
399	C₂⁴	—	4	2	no	—
400	C₂⁴	C₂×D₄×C₁₃₁	1	4	yes	—
401	C₂⁴	C₂×Q₈×C₁₃₁	1	4	yes	—
402	C₂⁴	C₂×Q₈×C₁₃₁	1	4	yes	—
403	C₂⁴	#41×C₁₃₁	1	4	yes	—
404	C₂⁴	C₂⁴×C₁₃₁	1	2	yes	—
405	C₂⁴	C₂⁴×C₁₃₁	1	1	yes	characteristic
406	C₂⁴	C₂⁴×C₁₃₁	1	2	yes	—
407	C₂×Q₈	—	2	4	no	—
408	C₂×Q₈	—	2	4	no	—
409	C₂×Q₈	—	4	4	no	—
410	C₂×Q₈	—	4	4	no	—
411	C₂×D₄	—	2	32	no	—
412	C₂×D₄	—	2	8	no	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	C₂×Q₈×C₁₃₁	1	16	yes	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	8	no	—
421	C₂²×C₄	—	2	8	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	8	no	—
425	C₂²×C₄	—	2	8	no	—
426	C₂²×C₄	—	2	8	no	—
427	C₂²×C₄	—	4	4	no	—
428	C₂²×C₄	—	4	2	no	—
429	C₂²×C₄	—	4	4	no	—
430	C₂²×C₄	—	4	4	no	—
431	C₂²×C₄	—	4	2	no	—
432	C₂²×C₄	—	4	2	no	—
433	C₂²×C₄	—	4	4	no	—
434	C₂²×C₄	—	4	4	no	—
435	C₂²×C₄	—	4	4	no	—
436	C₂²×C₄	—	4	16	no	—
437	C₂²×C₄	—	4	16	no	—
438	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
439	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
440	C₂²×C₄	C₂×Q₈×C₁₃₁	1	4	yes	—
441	C₂²×C₄	C₂⁴×C₁₃₁	1	8	yes	—
442	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
443	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
444	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
445	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
446	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
447	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
448	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
449	C₄⋊C₄	—	2	8	no	—
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	4	no	—
453	C₄⋊C₄	—	2	8	no	—
454	C₄⋊C₄	—	2	16	no	—
455	C₄⋊C₄	—	4	8	no	—
456	C₄⋊C₄	—	4	4	no	—
457	C₄⋊C₄	—	4	4	no	—
458	C₂²⋊C₄	—	2	8	no	—
459	C₂²⋊C₄	—	2	8	no	—
460	C₂²⋊C₄	—	2	8	no	—
461	C₂²⋊C₄	—	2	8	no	—
462	C₂²⋊C₄	—	2	8	no	—
463	C₂²⋊C₄	—	4	8	no	—
464	C₂²⋊C₄	—	4	8	no	—
465	C₂²⋊C₄	C₂×D₄×C₁₃₁	1	16	yes	—
466	C₄²	—	2	8	no	—
467	C₂³	—	2	4	no	—
468	C₂³	—	2	2	no	—
469	C₂³	—	2	4	no	—
470	C₂³	—	2	2	no	—
471	C₂³	—	2	8	no	—
472	C₂³	—	2	2	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	2	4	no	—
475	C₂³	—	2	4	no	—
476	C₂³	—	4	4	no	—
477	C₂³	—	4	4	no	—
478	C₂³	—	4	8	no	—
479	C₂³	—	4	8	no	—
480	C₂³	—	4	8	no	—
481	C₂³	—	4	4	no	—
482	C₂³	—	4	8	no	—
483	C₂³	—	4	4	no	—
484	C₂³	C₂²⋊Q₈×C₁₃₁	1	16	yes	complemented
485	C₂³	C₂²×D₄×C₁₃₁	1	2	yes	—
486	C₂³	C₂²×D₄×C₁₃₁	1	4	yes	—
487	C₂³	C₂²×Q₈×C₁₃₁	1	4	yes	—
488	C₂³	C₂²×Q₈×C₁₃₁	1	2	yes	—
489	C₂³	C₂²×Q₈×C₁₃₁	1	4	yes	—
490	C₂³	C₂×#41×C₁₃₁	1	2	yes	—
491	C₂³	C₂×#41×C₁₃₁	1	4	yes	—
492	C₂³	C₂⁵×C₁₃₁	1	1	yes	characteristic
493	Q₈	—	8	4	no	—
494	Q₈	—	8	4	no	—
495	D₄	—	2	32	no	—
496	C₂×C₄	—	2	8	no	—
497	C₂×C₄	—	2	8	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	8	no	—
508	C₂×C₄	—	2	8	no	—
509	C₂×C₄	—	2	8	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	8	no	—
512	C₂×C₄	—	2	8	no	—
513	C₂×C₄	—	4	4	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	—	4	16	no	—
516	C₂×C₄	—	4	16	no	—
517	C₂×C₄	—	4	4	no	—
518	C₂×C₄	—	4	16	no	—
519	C₂×C₄	—	8	4	no	—
520	C₂×C₄	—	8	4	no	—
521	C₂×C₄	C₂²×D₄×C₁₃₁	1	4	yes	—
522	C₂×C₄	C₂²×D₄×C₁₃₁	1	4	yes	—
523	C₂×C₄	C₂²×Q₈×C₁₃₁	1	4	yes	—
524	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	4	yes	—
525	C₂²	—	2	4	no	—
526	C₂²	—	2	8	no	—
527	C₂²	—	2	8	no	—
528	C₂²	—	2	4	no	—
529	C₂²	—	2	2	no	—
530	C₂²	—	2	4	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	4	no	—
533	C₂²	—	2	8	no	—
534	C₂²	—	2	4	no	—
535	C₂²	—	4	16	no	—
536	C₂²	—	4	16	no	—
537	C₂²	—	4	4	no	—
538	C₂²	—	4	2	no	—
539	C₂²	—	4	2	no	—
540	C₂²	C₂×C₂²⋊Q₈×C₁₃₁	1	4	yes	—
541	C₂²	C₂×C₂²⋊Q₈×C₁₃₁	1	8	yes	—
542	C₂²	C₂³×D₄×C₁₃₁	1	1	yes	characteristic
543	C₂²	C₂³×Q₈×C₁₃₁	1	1	yes	characteristic
544	C₂²	C₂²×#41×C₁₃₁	1	1	yes	characteristic
545	C₂²	C₂×H(5,𝔽₂)×C₁₃₁	1	2	yes	—
546	C₂²	#585×C₁₃₁	1	2	yes	—
547	C₄	—	2	4	no	—
548	C₄	—	2	4	no	—
549	C₄	—	2	4	no	—
550	C₄	—	2	4	no	—
551	C₄	—	4	8	no	—
552	C₄	—	4	2	no	—
553	C₄	—	4	4	no	—
554	C₄	—	4	4	no	—
555	C₂	—	2	4	no	—
556	C₂	—	2	8	no	—
557	C₂	—	4	4	no	—
558	C₂	C₂²×C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
559	C₂	#3459×C₁₃₁	1	2	yes	—
560	C₂	#3484×C₁₃₁	1	1	yes	characteristic
561	C₂	#3485×C₁₃₁	1	1	yes	characteristic
562	C₂	#3501×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃₁
prime radical	C₂³×D₄×C₁₃₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#543×C₁₃₁	C₂	semidirect
3. #2855×C₁₃₁	C₂	semidirect
4. #2848×C₁₃₁	C₂	semidirect
5. #2627×C₁₃₁	C₂	semidirect
6. #2626×C₁₃₁	C₂	semidirect
7. #2450×C₁₃₁	C₂	semidirect
8. #2449×C₁₃₁	C₂	semidirect
9. #2315×C₁₃₁	C₂	semidirect
11. C₂×#385×C₁₃₁	C₂	semidirect
18. C₂×C₂²⋊Q₈×C₁₃₁	C₂²	semidirect
19. C₂×C₂²⋊Q₈×C₁₃₁	C₂²	semidirect
20. C₂×C₂²⋊Q₈×C₁₃₁	C₂²	semidirect
24. C₂²×C₄⋊C₄×C₁₃₁	C₂²	semidirect
30. #400×C₁₃₁	C₂²	semidirect
31. #399×C₁₃₁	C₂²	semidirect
32. #398×C₁₃₁	C₂²	semidirect
33. #396×C₁₃₁	C₂²	semidirect
34. #395×C₁₃₁	C₂²	semidirect
35. #393×C₁₃₁	C₂²	semidirect
36. #391×C₁₃₁	C₂²	semidirect
37. #389×C₁₃₁	C₂²	semidirect
38. #387×C₁₃₁	C₂²	semidirect
44. #385×C₁₃₁	C₂²	semidirect
45. #385×C₁₃₁	C₂²	semidirect
46. #385×C₁₃₁	C₂²	semidirect
50. #380×C₁₃₁	C₂²	semidirect
52. C₂×#95×C₁₃₁	C₂²	semidirect
59. C₂²×D₄×C₁₃₁	Q₈	semidirect
202. C₂³×C₁₃₁	C₂²⋊Q₈	semidirect
281. #34985	C₁₃₁	direct (N × H)
282. C₁₃₁	#34985	direct (N × H)
283. C₂×#543	C₂×C₁₃₁	semidirect
285. #2855	C₂×C₁₃₁	semidirect
286. #2848	C₂×C₁₃₁	semidirect
287. #2627	C₂×C₁₃₁	semidirect
288. #2626	C₂×C₁₃₁	semidirect
289. #2450	C₂×C₁₃₁	semidirect
290. #2449	C₂×C₁₃₁	semidirect
291. #2315	C₂×C₁₃₁	semidirect
293. C₂×#385	C₂×C₁₃₁	semidirect
300. C₂×C₂²⋊Q₈	C₂²×C₁₃₁	semidirect
301. C₂×C₂²⋊Q₈	C₂²×C₁₃₁	semidirect
302. C₂×C₂²⋊Q₈	C₂²×C₁₃₁	semidirect
306. C₂²×C₄⋊C₄	C₂²×C₁₃₁	semidirect
312. #400	C₂²×C₁₃₁	semidirect
313. #399	C₂²×C₁₃₁	semidirect
314. #398	C₂²×C₁₃₁	semidirect
315. #396	C₂²×C₁₃₁	semidirect
316. #395	C₂²×C₁₃₁	semidirect
317. #393	C₂²×C₁₃₁	semidirect
318. #391	C₂²×C₁₃₁	semidirect
319. #389	C₂²×C₁₃₁	semidirect
320. #387	C₂²×C₁₃₁	semidirect
326. #385	C₂²×C₁₃₁	semidirect
327. #385	C₂²×C₁₃₁	semidirect
328. #385	C₂²×C₁₃₁	semidirect
332. #380	C₂²×C₁₃₁	semidirect
334. C₂×#95	C₂²×C₁₃₁	semidirect
341. C₂²×D₄	Q₈×C₁₃₁	semidirect
484. C₂³	C₂²⋊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×#543×C₁₃₁	2
C₂²×C₄×D₄×C₁₃₁	2
#2855×C₁₃₁	2
#2848×C₁₃₁	2
#2627×C₁₃₁	2
#2626×C₁₃₁	2
#2450×C₁₃₁	2
#2449×C₁₃₁	2
#2315×C₁₃₁	2
#2308×C₁₃₁	2
C₂×#385×C₁₃₁	2
#34985	131

Subgroups — C131×#34985 #45720869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#34985 #45720869