Subgroups — C₁₃₁×#34207 #45720091

Order 33536.

3254 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#520×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#2361×C₁₃₁	C₂	1	8	yes	maximal, complemented
4	#2354×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	D₄×C₄⋊C₄×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#2326×C₁₃₁	C₂	1	4	yes	maximal, complemented
7	#2311×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	C₄×C₄⋊Q₈×C₁₃₁	C₂	1	2	yes	maximal, complemented
9	C₄×#127×C₁₃₁	C₂	1	2	yes	maximal, complemented
10	C₂×#384×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#117×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₁₃₁	C₂²	1	2	yes	—
13	#532×C₁₃₁	C₄	1	16	yes	complemented
14	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#127×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×#124×C₁₃₁	C₂²	1	2	yes	—
17	C₂×#121×C₁₃₁	C₂²	1	4	yes	—
18	#520×C₁₃₁	—	2	16	no	—
19	C₂×C₄×Q₈×C₁₃₁	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₁₃₁	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₁₃₁	C₂²	1	8	yes	—
22	C₂×C₄×D₄×C₁₃₁	C₂²	1	8	yes	—
23	C₂×C₄×D₄×C₁₃₁	C₂²	1	4	yes	—
24	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
25	C₂×#117×C₁₃₁	C₂²	1	4	yes	—
26	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	—
29	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	—
30	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
31	#391×C₁₃₁	C₂²	1	4	yes	—
32	#390×C₁₃₁	C₂²	1	4	yes	—
33	#390×C₁₃₁	C₂²	1	8	yes	—
34	#389×C₁₃₁	C₂²	1	8	yes	—
35	#388×C₁₃₁	C₂²	1	16	yes	—
36	#386×C₁₃₁	C₂²	1	8	yes	—
37	#385×C₁₃₁	C₂²	1	16	yes	—
38	#384×C₁₃₁	C₂²	1	2	yes	—
39	#384×C₁₃₁	C₂²	1	2	yes	—
40	#384×C₁₃₁	C₂²	1	4	yes	—
41	#380×C₁₃₁	C₂²	1	8	yes	—
42	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
43	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
44	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	8	yes	—
45	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	4	yes	—
46	#376×C₁₃₁	C₂²	1	2	yes	—
47	C₄³×C₁₃₁	C₂²	1	2	yes	—
48	C₂×#41×C₁₃₁	—	2	8	no	—
49	C₂×#41×C₁₃₁	—	2	8	no	—
50	C₂×#41×C₁₃₁	C₂×C₄	1	16	yes	complemented
51	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
52	C₂²×D₄×C₁₃₁	C₂³	1	4	yes	—
53	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
54	C₂²×D₄×C₁₃₁	C₂³	1	4	yes	—
55	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
56	C₂³×C₄×C₁₃₁	C₂³	1	4	yes	—
57	C₂³×C₄×C₁₃₁	C₂³	1	4	yes	—
58	C₂³×C₄×C₁₃₁	C₂³	1	8	yes	—
59	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₁₃₁	C₂×C₄	1	8	yes	complemented
61	#127×C₁₃₁	C₂×C₄	1	8	yes	complemented
62	#124×C₁₃₁	C₂×C₄	1	16	yes	complemented
63	#121×C₁₃₁	C₂×C₄	1	32	yes	complemented
64	C₄×Q₈×C₁₃₁	—	2	16	no	—
65	C₄×D₄×C₁₃₁	—	2	32	no	—
66	C₄×D₄×C₁₃₁	—	2	16	no	—
67	C₄×D₄×C₁₃₁	—	4	16	no	—
68	C₄×D₄×C₁₃₁	C₂×C₄	1	32	yes	complemented
69	#117×C₁₃₁	—	2	4	no	—
70	#117×C₁₃₁	—	2	16	no	—
71	#117×C₁₃₁	D₄	1	8	yes	complemented
72	C₂×C₄⋊C₄×C₁₃₁	—	2	8	no	—
73	C₂×C₄⋊C₄×C₁₃₁	—	2	16	no	—
74	C₂×C₄⋊C₄×C₁₃₁	—	4	8	no	—
75	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	8	yes	—
76	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	8	yes	—
77	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	8	yes	—
81	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
86	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	16	yes	—
87	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	8	yes	—
88	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
89	C₂×C₄²×C₁₃₁	C₂×C₄	1	8	yes	complemented
90	C₂×C₄²×C₁₃₁	C₂³	1	8	yes	—
91	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
92	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
93	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
94	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
95	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
98	#95×C₁₃₁	C₂³	1	4	yes	—
99	#95×C₁₃₁	C₂³	1	16	yes	—
100	#95×C₁₃₁	C₂³	1	4	yes	—
101	C₂⁴×C₁₃₁	C₂⁴	1	4	yes	—
102	C₂⁴×C₁₃₁	C₂⁴	1	1	yes	characteristic
103	#41×C₁₃₁	—	2	64	no	—
104	C₂×Q₈×C₁₃₁	—	2	4	no	—
105	C₂×Q₈×C₁₃₁	—	2	4	no	—
106	C₂×Q₈×C₁₃₁	C₂²×C₄	1	8	yes	—
107	C₂×D₄×C₁₃₁	—	2	8	no	—
108	C₂×D₄×C₁₃₁	—	2	8	no	—
109	C₂×D₄×C₁₃₁	—	2	4	no	—
110	C₂×D₄×C₁₃₁	—	2	4	no	—
111	C₂×D₄×C₁₃₁	—	2	8	no	—
112	C₂×D₄×C₁₃₁	—	2	8	no	—
113	C₂×D₄×C₁₃₁	C₂²×C₄	1	16	yes	—
114	C₂×D₄×C₁₃₁	C₂²×C₄	1	16	yes	—
115	C₂×D₄×C₁₃₁	C₂²×C₄	1	8	yes	—
116	C₂²×C₄×C₁₃₁	—	2	8	no	—
117	C₂²×C₄×C₁₃₁	—	2	8	no	—
118	C₂²×C₄×C₁₃₁	—	2	4	no	—
119	C₂²×C₄×C₁₃₁	—	2	4	no	—
120	C₂²×C₄×C₁₃₁	—	2	8	no	—
121	C₂²×C₄×C₁₃₁	—	2	16	no	—
122	C₂²×C₄×C₁₃₁	—	4	8	no	—
123	C₂²×C₄×C₁₃₁	—	4	16	no	—
124	C₂²×C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
125	C₂²×C₄×C₁₃₁	C₂²×C₄	1	16	yes	—
126	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
127	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
128	C₂²×C₄×C₁₃₁	#41	1	4	yes	—
129	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₁₃₁	C₂⁴	1	8	yes	—
131	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
132	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
133	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
134	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
135	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
136	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
137	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
138	C₄⋊C₄×C₁₃₁	—	2	8	no	—
139	C₄⋊C₄×C₁₃₁	—	2	16	no	—
140	C₄⋊C₄×C₁₃₁	—	2	8	no	—
141	C₄⋊C₄×C₁₃₁	—	2	4	no	—
142	C₄⋊C₄×C₁₃₁	—	2	16	no	—
143	C₄⋊C₄×C₁₃₁	—	4	8	no	—
144	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	16	yes	—
145	C₄⋊C₄×C₁₃₁	C₂×D₄	1	4	yes	—
146	C₄⋊C₄×C₁₃₁	C₂×D₄	1	4	yes	—
147	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
148	C₂²⋊C₄×C₁₃₁	—	2	16	no	—
149	C₂²⋊C₄×C₁₃₁	—	4	16	no	—
150	C₂²⋊C₄×C₁₃₁	C₂²×C₄	1	32	yes	—
151	C₂²⋊C₄×C₁₃₁	C₂×D₄	1	8	yes	—
152	C₄²×C₁₃₁	—	2	16	no	—
153	C₄²×C₁₃₁	—	2	4	no	—
154	C₄²×C₁₃₁	—	2	8	no	—
155	C₄²×C₁₃₁	—	2	8	no	—
156	C₄²×C₁₃₁	C₂²×C₄	1	8	yes	—
157	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
158	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
159	C₄²×C₁₃₁	C₂×D₄	1	8	yes	—
160	C₂³×C₁₃₁	—	2	2	no	—
161	C₂³×C₁₃₁	—	2	2	no	—
162	C₂³×C₁₃₁	—	2	4	no	—
163	C₂³×C₁₃₁	—	2	4	no	—
164	C₂³×C₁₃₁	—	2	8	no	—
165	C₂³×C₁₃₁	—	2	8	no	—
166	C₂³×C₁₃₁	C₂³×C₄	1	8	yes	—
167	C₂³×C₁₃₁	C₂³×C₄	1	2	yes	—
168	C₂³×C₁₃₁	C₂²×D₄	1	2	yes	—
169	C₂³×C₁₃₁	C₂×#41	1	2	yes	—
170	C₂³×C₁₃₁	C₂⁵	1	1	yes	characteristic
171	Q₈×C₁₃₁	—	2	16	no	—
172	D₄×C₁₃₁	—	2	32	no	—
173	D₄×C₁₃₁	—	2	16	no	—
174	D₄×C₁₃₁	—	2	32	no	—
175	C₂×C₄×C₁₃₁	—	2	4	no	—
176	C₂×C₄×C₁₃₁	—	2	4	no	—
177	C₂×C₄×C₁₃₁	—	2	4	no	—
178	C₂×C₄×C₁₃₁	—	2	2	no	—
179	C₂×C₄×C₁₃₁	—	2	4	no	—
180	C₂×C₄×C₁₃₁	—	2	2	no	—
181	C₂×C₄×C₁₃₁	—	2	32	no	—
182	C₂×C₄×C₁₃₁	—	2	16	no	—
183	C₂×C₄×C₁₃₁	—	2	8	no	—
184	C₂×C₄×C₁₃₁	—	2	4	no	—
185	C₂×C₄×C₁₃₁	—	2	4	no	—
186	C₂×C₄×C₁₃₁	—	2	16	no	—
187	C₂×C₄×C₁₃₁	—	2	4	no	—
188	C₂×C₄×C₁₃₁	—	8	16	no	—
189	C₂×C₄×C₁₃₁	—	8	8	no	—
190	C₂×C₄×C₁₃₁	C₄×D₄	1	16	yes	complemented
191	C₂×C₄×C₁₃₁	C₂³×C₄	1	4	yes	—
192	C₂×C₄×C₁₃₁	C₂³×C₄	1	8	yes	—
193	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
194	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
195	C₂×C₄×C₁₃₁	C₂³×C₄	1	4	yes	—
196	C₂×C₄×C₁₃₁	C₂²×D₄	1	4	yes	—
197	C₂×C₄×C₁₃₁	C₂²×D₄	1	4	yes	—
198	C₂×C₄×C₁₃₁	C₂²×D₄	1	2	yes	—
199	C₂×C₄×C₁₃₁	C₂²×D₄	1	2	yes	—
200	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
201	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
202	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	4	yes	—
203	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	4	yes	—
204	C₂²×C₁₃₁	—	2	4	no	—
205	C₂²×C₁₃₁	—	2	2	no	—
206	C₂²×C₁₃₁	—	2	2	no	—
207	C₂²×C₁₃₁	—	2	4	no	—
208	C₂²×C₁₃₁	—	2	8	no	—
209	C₂²×C₁₃₁	—	2	16	no	—
210	C₂²×C₁₃₁	—	4	4	no	—
211	C₂²×C₁₃₁	—	4	4	no	—
212	C₂²×C₁₃₁	—	4	8	no	—
213	C₂²×C₁₃₁	C₂×C₄×D₄	1	4	yes	—
214	C₂²×C₁₃₁	C₂⁴×C₄	1	1	yes	characteristic
215	C₂²×C₁₃₁	C₂³×D₄	1	1	yes	characteristic
216	C₂²×C₁₃₁	C₂²×#41	1	1	yes	characteristic
217	C₂²×C₁₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
218	C₂²×C₁₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
219	C₄×C₁₃₁	—	2	8	no	—
220	C₄×C₁₃₁	—	2	4	no	—
221	C₄×C₁₃₁	—	2	4	no	—
222	C₄×C₁₃₁	—	2	4	no	—
223	C₄×C₁₃₁	—	4	8	no	—
224	C₄×C₁₃₁	—	4	4	no	—
225	C₄×C₁₃₁	C₂×C₄×D₄	1	4	yes	—
226	C₄×C₁₃₁	C₂×C₄×D₄	1	4	yes	—
227	C₄×C₁₃₁	#520	1	8	yes	complemented
228	C₂×C₁₃₁	—	2	4	no	—
229	C₂×C₁₃₁	—	4	8	no	—
230	C₂×C₁₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
231	C₂×C₁₃₁	C₂×#520	1	2	yes	—
232	C₂×C₁₃₁	#3498	1	1	yes	characteristic
233	C₂×C₁₃₁	#3499	1	1	yes	characteristic
234	C₂×C₁₃₁	#3511	1	2	yes	—
235	#34207	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₁₃₁	#34207	1	1	yes	characteristic, direct product factor, Sylow
237	C₂×#532	C₂×C₁₃₁	1	1	yes	characteristic
238	C₂×#520	C₂×C₁₃₁	1	2	yes	complemented
239	#2361	C₂×C₁₃₁	1	8	yes	complemented
240	#2354	C₂×C₁₃₁	1	4	yes	complemented
241	D₄×C₄⋊C₄	C₂×C₁₃₁	1	4	yes	complemented
242	#2326	C₂×C₁₃₁	1	4	yes	complemented
243	#2311	C₂×C₁₃₁	1	2	yes	complemented
244	C₄×C₄⋊Q₈	C₂×C₁₃₁	1	2	yes	complemented
245	C₄×#127	C₂×C₁₃₁	1	2	yes	complemented
246	C₂×#384	C₂×C₁₃₁	1	1	yes	characteristic, complemented
247	C₄×#117	C₂×C₁₃₁	1	1	yes	characteristic, complemented
248	C₂²×#41	C₂²×C₁₃₁	1	2	yes	—
249	#532	C₄×C₁₃₁	1	16	yes	complemented
250	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
251	C₂×#127	C₂²×C₁₃₁	1	1	yes	characteristic
252	C₂×#124	C₂²×C₁₃₁	1	2	yes	—
253	C₂×#121	C₂²×C₁₃₁	1	4	yes	—
254	#520	—	2	16	no	—
255	C₂×C₄×Q₈	C₂²×C₁₃₁	1	4	yes	—
256	C₂×C₄×D₄	C₂²×C₁₃₁	1	4	yes	—
257	C₂×C₄×D₄	C₂²×C₁₃₁	1	8	yes	—
258	C₂×C₄×D₄	C₂²×C₁₃₁	1	8	yes	—
259	C₂×C₄×D₄	C₂²×C₁₃₁	1	4	yes	—
260	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
261	C₂×#117	C₂²×C₁₃₁	1	4	yes	—
262	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
263	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
264	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	—
265	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	—
266	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
267	#391	C₂²×C₁₃₁	1	4	yes	—
268	#390	C₂²×C₁₃₁	1	4	yes	—
269	#390	C₂²×C₁₃₁	1	8	yes	—
270	#389	C₂²×C₁₃₁	1	8	yes	—
271	#388	C₂²×C₁₃₁	1	16	yes	—
272	#386	C₂²×C₁₃₁	1	8	yes	—
273	#385	C₂²×C₁₃₁	1	16	yes	—
274	#384	C₂²×C₁₃₁	1	2	yes	—
275	#384	C₂²×C₁₃₁	1	2	yes	—
276	#384	C₂²×C₁₃₁	1	4	yes	—
277	#380	C₂²×C₁₃₁	1	8	yes	—
278	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	8	yes	—
279	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
280	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
281	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	4	yes	—
282	#376	C₂²×C₁₃₁	1	2	yes	—
283	C₄³	C₂²×C₁₃₁	1	2	yes	—
284	C₂×#41	—	2	8	no	—
285	C₂×#41	—	2	8	no	—
286	C₂×#41	C₂×C₄×C₁₃₁	1	16	yes	complemented
287	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
288	C₂²×D₄	C₂³×C₁₃₁	1	4	yes	—
289	C₂²×D₄	C₂³×C₁₃₁	1	4	yes	—
290	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
291	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
292	C₂³×C₄	C₂³×C₁₃₁	1	4	yes	—
293	C₂³×C₄	C₂³×C₁₃₁	1	4	yes	—
294	C₂³×C₄	C₂³×C₁₃₁	1	8	yes	—
295	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
296	C₄⋊Q₈	C₂×C₄×C₁₃₁	1	8	yes	complemented
297	#127	C₂×C₄×C₁₃₁	1	8	yes	complemented
298	#124	C₂×C₄×C₁₃₁	1	16	yes	complemented
299	#121	C₂×C₄×C₁₃₁	1	32	yes	complemented
300	C₄×Q₈	—	2	16	no	—
301	C₄×D₄	—	2	32	no	—
302	C₄×D₄	—	2	16	no	—
303	C₄×D₄	—	4	16	no	—
304	C₄×D₄	C₂×C₄×C₁₃₁	1	32	yes	complemented
305	#117	—	2	4	no	—
306	#117	—	2	16	no	—
307	#117	D₄×C₁₃₁	1	8	yes	complemented
308	C₂×C₄⋊C₄	—	2	8	no	—
309	C₂×C₄⋊C₄	—	2	16	no	—
310	C₂×C₄⋊C₄	—	4	8	no	—
311	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
312	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
313	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	8	yes	—
314	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	8	yes	—
315	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
316	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	8	yes	—
317	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
318	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
319	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
320	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	8	yes	—
321	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	2	yes	—
322	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	16	yes	—
323	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	8	yes	—
324	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	2	yes	—
325	C₂×C₄²	C₂×C₄×C₁₃₁	1	8	yes	complemented
326	C₂×C₄²	C₂³×C₁₃₁	1	8	yes	—
327	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
328	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
329	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
330	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
331	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
332	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
333	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
334	#95	C₂³×C₁₃₁	1	4	yes	—
335	#95	C₂³×C₁₃₁	1	4	yes	—
336	#95	C₂³×C₁₃₁	1	16	yes	—
337	C₂⁴	C₂⁴×C₁₃₁	1	4	yes	—
338	C₂⁴	C₂⁴×C₁₃₁	1	1	yes	characteristic
339	#41	—	2	64	no	—
340	C₂×Q₈	—	2	4	no	—
341	C₂×Q₈	—	2	4	no	—
342	C₂×Q₈	C₂²×C₄×C₁₃₁	1	8	yes	—
343	C₂×D₄	—	2	8	no	—
344	C₂×D₄	—	2	8	no	—
345	C₂×D₄	—	2	4	no	—
346	C₂×D₄	—	2	4	no	—
347	C₂×D₄	—	2	8	no	—
348	C₂×D₄	—	2	8	no	—
349	C₂×D₄	C₂²×C₄×C₁₃₁	1	16	yes	—
350	C₂×D₄	C₂²×C₄×C₁₃₁	1	16	yes	—
351	C₂×D₄	C₂²×C₄×C₁₃₁	1	8	yes	—
352	C₂²×C₄	—	2	8	no	—
353	C₂²×C₄	—	2	8	no	—
354	C₂²×C₄	—	2	4	no	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	2	8	no	—
357	C₂²×C₄	—	2	16	no	—
358	C₂²×C₄	—	4	8	no	—
359	C₂²×C₄	—	4	16	no	—
360	C₂²×C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
361	C₂²×C₄	C₂²×C₄×C₁₃₁	1	16	yes	—
362	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
363	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
364	C₂²×C₄	#41×C₁₃₁	1	4	yes	—
365	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
366	C₂²×C₄	C₂⁴×C₁₃₁	1	8	yes	—
367	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
368	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
369	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
370	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
371	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
372	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
373	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
374	C₄⋊C₄	—	2	8	no	—
375	C₄⋊C₄	—	2	16	no	—
376	C₄⋊C₄	—	2	8	no	—
377	C₄⋊C₄	—	2	4	no	—
378	C₄⋊C₄	—	2	16	no	—
379	C₄⋊C₄	—	4	8	no	—
380	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	16	yes	—
381	C₄⋊C₄	C₂×D₄×C₁₃₁	1	4	yes	—
382	C₄⋊C₄	C₂×D₄×C₁₃₁	1	4	yes	—
383	C₂²⋊C₄	—	2	4	no	—
384	C₂²⋊C₄	—	2	16	no	—
385	C₂²⋊C₄	—	4	16	no	—
386	C₂²⋊C₄	C₂²×C₄×C₁₃₁	1	32	yes	—
387	C₂²⋊C₄	C₂×D₄×C₁₃₁	1	8	yes	—
388	C₄²	—	2	16	no	—
389	C₄²	—	2	4	no	—
390	C₄²	—	2	8	no	—
391	C₄²	—	2	8	no	—
392	C₄²	C₂²×C₄×C₁₃₁	1	8	yes	—
393	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
394	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
395	C₄²	C₂×D₄×C₁₃₁	1	8	yes	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	2	no	—
398	C₂³	—	2	4	no	—
399	C₂³	—	2	4	no	—
400	C₂³	—	2	8	no	—
401	C₂³	—	2	8	no	—
402	C₂³	C₂³×C₄×C₁₃₁	1	8	yes	—
403	C₂³	C₂³×C₄×C₁₃₁	1	2	yes	—
404	C₂³	C₂²×D₄×C₁₃₁	1	2	yes	—
405	C₂³	C₂×#41×C₁₃₁	1	2	yes	—
406	C₂³	C₂⁵×C₁₃₁	1	1	yes	characteristic
407	Q₈	—	2	16	no	—
408	D₄	—	2	32	no	—
409	D₄	—	2	16	no	—
410	D₄	—	2	32	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	32	no	—
418	C₂×C₄	—	2	16	no	—
419	C₂×C₄	—	2	8	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	4	no	—
422	C₂×C₄	—	2	16	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	8	16	no	—
425	C₂×C₄	—	8	8	no	—
426	C₂×C₄	C₄×D₄×C₁₃₁	1	16	yes	complemented
427	C₂×C₄	C₂³×C₄×C₁₃₁	1	4	yes	—
428	C₂×C₄	C₂³×C₄×C₁₃₁	1	8	yes	—
429	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
430	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
431	C₂×C₄	C₂³×C₄×C₁₃₁	1	4	yes	—
432	C₂×C₄	C₂²×D₄×C₁₃₁	1	4	yes	—
433	C₂×C₄	C₂²×D₄×C₁₃₁	1	4	yes	—
434	C₂×C₄	C₂²×D₄×C₁₃₁	1	2	yes	—
435	C₂×C₄	C₂²×D₄×C₁₃₁	1	2	yes	—
436	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
437	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
438	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	4	yes	—
439	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	4	yes	—
440	C₂²	—	2	2	no	—
441	C₂²	—	2	2	no	—
442	C₂²	—	2	4	no	—
443	C₂²	—	2	4	no	—
444	C₂²	—	2	8	no	—
445	C₂²	—	2	16	no	—
446	C₂²	—	4	4	no	—
447	C₂²	—	4	4	no	—
448	C₂²	—	4	8	no	—
449	C₂²	C₂×C₄×D₄×C₁₃₁	1	4	yes	—
450	C₂²	C₂⁴×C₄×C₁₃₁	1	1	yes	characteristic
451	C₂²	C₂³×D₄×C₁₃₁	1	1	yes	characteristic
452	C₂²	C₂²×#41×C₁₃₁	1	1	yes	characteristic
453	C₂²	C₂×H(5,𝔽₂)×C₁₃₁	1	2	yes	—
454	C₂²	C₂×2₋¹⁺⁴×C₁₃₁	1	2	yes	—
455	C₄	—	2	4	no	—
456	C₄	—	2	4	no	—
457	C₄	—	2	8	no	—
458	C₄	—	2	4	no	—
459	C₄	—	4	8	no	—
460	C₄	—	4	4	no	—
461	C₄	C₂×C₄×D₄×C₁₃₁	1	4	yes	—
462	C₄	C₂×C₄×D₄×C₁₃₁	1	4	yes	—
463	C₄	#520×C₁₃₁	1	8	yes	complemented
464	C₂	—	2	4	no	—
465	C₂	—	4	8	no	—
466	C₂	C₂²×C₄×D₄×C₁₃₁	1	1	yes	characteristic
467	C₂	C₂×#520×C₁₃₁	1	2	yes	—
468	C₂	#3498×C₁₃₁	1	1	yes	characteristic
469	C₂	#3499×C₁₃₁	1	1	yes	characteristic
470	C₂	#3511×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₁₃₁
Frattini, cosocle	C₂³	C₂⁵×C₁₃₁
prime radical	C₂×#532×C₁₃₁	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#520×C₁₃₁	C₂	semidirect
3. #2361×C₁₃₁	C₂	semidirect
4. #2354×C₁₃₁	C₂	semidirect
5. D₄×C₄⋊C₄×C₁₃₁	C₂	semidirect
6. #2326×C₁₃₁	C₂	semidirect
7. #2311×C₁₃₁	C₂	semidirect
8. C₄×C₄⋊Q₈×C₁₃₁	C₂	semidirect
9. C₄×#127×C₁₃₁	C₂	semidirect
10. C₂×#384×C₁₃₁	C₂	semidirect
11. C₄×#117×C₁₃₁	C₂	semidirect
13. #532×C₁₃₁	C₄	semidirect
50. C₂×#41×C₁₃₁	C₂×C₄	semidirect
60. C₄⋊Q₈×C₁₃₁	C₂×C₄	semidirect
61. #127×C₁₃₁	C₂×C₄	semidirect
62. #124×C₁₃₁	C₂×C₄	semidirect
63. #121×C₁₃₁	C₂×C₄	semidirect
68. C₄×D₄×C₁₃₁	C₂×C₄	semidirect
71. #117×C₁₃₁	D₄	semidirect
89. C₂×C₄²×C₁₃₁	C₂×C₄	semidirect
190. C₂×C₄×C₁₃₁	C₄×D₄	semidirect
227. C₄×C₁₃₁	#520	semidirect
235. #34207	C₁₃₁	direct (N × H)
236. C₁₃₁	#34207	direct (N × H)
238. C₂×#520	C₂×C₁₃₁	semidirect
239. #2361	C₂×C₁₃₁	semidirect
240. #2354	C₂×C₁₃₁	semidirect
241. D₄×C₄⋊C₄	C₂×C₁₃₁	semidirect
242. #2326	C₂×C₁₃₁	semidirect
243. #2311	C₂×C₁₃₁	semidirect
244. C₄×C₄⋊Q₈	C₂×C₁₃₁	semidirect
245. C₄×#127	C₂×C₁₃₁	semidirect
246. C₂×#384	C₂×C₁₃₁	semidirect
247. C₄×#117	C₂×C₁₃₁	semidirect
249. #532	C₄×C₁₃₁	semidirect
286. C₂×#41	C₂×C₄×C₁₃₁	semidirect
296. C₄⋊Q₈	C₂×C₄×C₁₃₁	semidirect
297. #127	C₂×C₄×C₁₃₁	semidirect
298. #124	C₂×C₄×C₁₃₁	semidirect
299. #121	C₂×C₄×C₁₃₁	semidirect
304. C₄×D₄	C₂×C₄×C₁₃₁	semidirect
307. #117	D₄×C₁₃₁	semidirect
325. C₂×C₄²	C₂×C₄×C₁₃₁	semidirect
426. C₂×C₄	C₄×D₄×C₁₃₁	semidirect
463. C₄	#520×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#532×C₁₃₁	2
C₂×#520×C₁₃₁	2
#2361×C₁₃₁	2
#2354×C₁₃₁	2
D₄×C₄⋊C₄×C₁₃₁	2
#2326×C₁₃₁	2
#2311×C₁₃₁	2
C₄×C₄⋊Q₈×C₁₃₁	2
C₄×#127×C₁₃₁	2
C₂×#384×C₁₃₁	2
C₄×#117×C₁₃₁	2
#34207	131

Subgroups — C131×#34207 #45720091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#34207 #45720091