Subgroups — C₁₃₁×#34210 #45720094

Order 33536.

4006 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#518×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#2366×C₁₃₁	C₂	1	8	yes	maximal, complemented
4	D₄×C₂²⋊C₄×C₁₃₁	C₂	1	8	yes	maximal, complemented
5	#2328×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#2309×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	C₄×#127×C₁₃₁	C₂	1	4	yes	maximal, complemented
8	#2285×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#117×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₁₃₁	C₂²	1	2	yes	—
11	C₂³×D₄×C₁₃₁	C₂²	1	4	yes	—
12	C₂×#127×C₁₃₁	C₄	1	16	yes	complemented
13	C₂×#127×C₁₃₁	C₂²	1	2	yes	—
14	C₂×#127×C₁₃₁	C₂²	1	2	yes	—
15	C₂×#127×C₁₃₁	C₂²	1	4	yes	—
16	#518×C₁₃₁	—	2	16	no	—
17	C₂×C₄×D₄×C₁₃₁	C₂²	1	8	yes	complemented
18	C₂×C₄×D₄×C₁₃₁	C₂²	1	8	yes	complemented
19	C₂×C₄×D₄×C₁₃₁	C₂²	1	16	yes	complemented
20	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂×#117×C₁₃₁	C₂²	1	2	yes	—
22	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₁₃₁	C₂²	1	8	yes	—
24	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
25	#390×C₁₃₁	C₂²	1	8	yes	complemented
26	#390×C₁₃₁	C₂²	1	16	yes	complemented
27	#390×C₁₃₁	C₂²	1	8	yes	complemented
28	#386×C₁₃₁	C₂²	1	32	yes	complemented
29	#384×C₁₃₁	C₂²	1	2	yes	complemented
30	#382×C₁₃₁	C₂²	1	2	yes	complemented
31	#380×C₁₃₁	C₂²	1	4	yes	—
32	#379×C₁₃₁	C₂²	1	8	yes	—
33	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	complemented
34	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	complemented
35	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	4	yes	—
36	#376×C₁₃₁	C₂²	1	2	yes	complemented
37	C₄³×C₁₃₁	C₂²	1	2	yes	complemented
38	C₂⁵×C₁₃₁	C₂³	1	4	yes	—
39	C₂²×D₄×C₁₃₁	—	2	8	no	—
40	C₂²×D₄×C₁₃₁	—	2	8	no	—
41	C₂²×D₄×C₁₃₁	—	4	16	no	—
42	C₂²×D₄×C₁₃₁	C₂×C₄	1	16	yes	complemented
43	C₂²×D₄×C₁₃₁	C₂×C₄	1	32	yes	complemented
44	C₂²×D₄×C₁₃₁	C₂³	1	4	yes	—
45	C₂²×D₄×C₁₃₁	C₂³	1	4	yes	—
46	C₂²×D₄×C₁₃₁	C₂³	1	8	yes	—
47	C₂²×D₄×C₁₃₁	C₂³	1	8	yes	—
48	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
49	C₂³×C₄×C₁₃₁	C₂³	1	16	yes	—
50	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
51	#127×C₁₃₁	C₂×C₄	1	16	yes	complemented
52	#127×C₁₃₁	C₂×C₄	1	32	yes	complemented
53	#127×C₁₃₁	C₂×C₄	1	16	yes	complemented
54	C₄×D₄×C₁₃₁	—	2	32	no	—
55	C₄×D₄×C₁₃₁	—	2	32	no	—
56	C₄×D₄×C₁₃₁	—	4	32	no	—
57	#117×C₁₃₁	—	2	8	no	—
58	#117×C₁₃₁	—	2	4	no	—
59	#117×C₁₃₁	D₄	1	8	yes	complemented
60	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
62	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	8	yes	—
63	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
64	C₂×C₂²⋊C₄×C₁₃₁	—	2	32	no	—
65	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	16	yes	—
66	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	4	yes	—
67	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	32	yes	—
69	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
70	C₂×C₄²×C₁₃₁	C₂×C₄	1	8	yes	complemented
71	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
72	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
73	C₂×C₄²×C₁₃₁	C₂³	1	8	yes	—
74	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
77	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
78	#95×C₁₃₁	C₂³	1	8	yes	—
79	#95×C₁₃₁	C₂³	1	8	yes	—
80	C₂⁴×C₁₃₁	—	2	8	no	—
81	C₂⁴×C₁₃₁	—	2	8	no	—
82	C₂⁴×C₁₃₁	—	4	16	no	—
83	C₂⁴×C₁₃₁	C₂²×C₄	1	16	yes	—
84	C₂⁴×C₁₃₁	C₂⁴	1	8	yes	—
85	C₂⁴×C₁₃₁	C₂⁴	1	1	yes	characteristic
86	C₂×D₄×C₁₃₁	—	2	64	no	—
87	C₂×D₄×C₁₃₁	—	2	8	no	—
88	C₂×D₄×C₁₃₁	—	2	8	no	—
89	C₂×D₄×C₁₃₁	—	2	8	no	—
90	C₂×D₄×C₁₃₁	—	2	8	no	—
91	C₂×D₄×C₁₃₁	—	2	32	no	—
92	C₂×D₄×C₁₃₁	—	4	64	no	—
93	C₂×D₄×C₁₃₁	—	4	16	no	—
94	C₂×D₄×C₁₃₁	C₂²×C₄	1	16	yes	—
95	C₂×D₄×C₁₃₁	C₂²×C₄	1	16	yes	—
96	C₂×D₄×C₁₃₁	C₂²×C₄	1	32	yes	—
97	C₂×D₄×C₁₃₁	C₂²×C₄	1	32	yes	—
98	C₂²×C₄×C₁₃₁	—	2	8	no	—
99	C₂²×C₄×C₁₃₁	—	2	32	no	—
100	C₂²×C₄×C₁₃₁	—	4	32	no	—
101	C₂²×C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
102	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
103	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
104	C₂²×C₄×C₁₃₁	#41	1	4	yes	—
105	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
106	C₂²×C₄×C₁₃₁	C₂⁴	1	8	yes	—
107	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
108	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
109	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
110	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
111	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
112	C₄⋊C₄×C₁₃₁	—	2	8	no	—
113	C₄⋊C₄×C₁₃₁	—	2	4	no	—
114	C₄⋊C₄×C₁₃₁	—	2	16	no	—
115	C₄⋊C₄×C₁₃₁	C₂×D₄	1	8	yes	complemented
116	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
117	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
118	C₂²⋊C₄×C₁₃₁	—	2	32	no	—
119	C₂²⋊C₄×C₁₃₁	—	4	32	no	—
120	C₂²⋊C₄×C₁₃₁	C₂×D₄	1	8	yes	—
121	C₄²×C₁₃₁	—	2	8	no	—
122	C₄²×C₁₃₁	—	2	4	no	—
123	C₄²×C₁₃₁	—	2	16	no	—
124	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
125	C₄²×C₁₃₁	C₂²×C₄	1	8	yes	—
126	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
127	C₄²×C₁₃₁	C₂×D₄	1	8	yes	complemented
128	C₂³×C₁₃₁	—	2	4	no	—
129	C₂³×C₁₃₁	—	2	8	no	—
130	C₂³×C₁₃₁	—	2	8	no	—
131	C₂³×C₁₃₁	—	2	32	no	—
132	C₂³×C₁₃₁	—	2	32	no	—
133	C₂³×C₁₃₁	—	4	32	no	—
134	C₂³×C₁₃₁	—	8	16	no	—
135	C₂³×C₁₃₁	—	8	16	no	—
136	C₂³×C₁₃₁	C₂³×C₄	1	16	yes	—
137	C₂³×C₁₃₁	C₂³×C₄	1	2	yes	—
138	C₂³×C₁₃₁	C₂²×D₄	1	2	yes	—
139	C₂³×C₁₃₁	C₂×#41	1	2	yes	—
140	C₂³×C₁₃₁	C₂⁵	1	1	yes	characteristic
141	D₄×C₁₃₁	—	2	32	no	—
142	D₄×C₁₃₁	—	2	32	no	—
143	D₄×C₁₃₁	—	2	64	no	—
144	D₄×C₁₃₁	—	4	32	no	—
145	C₂×C₄×C₁₃₁	—	2	8	no	—
146	C₂×C₄×C₁₃₁	—	2	4	no	—
147	C₂×C₄×C₁₃₁	—	2	16	no	—
148	C₂×C₄×C₁₃₁	—	2	8	no	—
149	C₂×C₄×C₁₃₁	—	2	4	no	—
150	C₂×C₄×C₁₃₁	—	2	8	no	—
151	C₂×C₄×C₁₃₁	—	2	16	no	—
152	C₂×C₄×C₁₃₁	—	8	32	no	—
153	C₂×C₄×C₁₃₁	C₄×D₄	1	16	yes	complemented
154	C₂×C₄×C₁₃₁	C₂³×C₄	1	4	yes	—
155	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
156	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
157	C₂×C₄×C₁₃₁	C₂³×C₄	1	4	yes	—
158	C₂×C₄×C₁₃₁	C₂²×D₄	1	8	yes	—
159	C₂×C₄×C₁₃₁	C₂²×D₄	1	2	yes	—
160	C₂×C₄×C₁₃₁	C₂²×D₄	1	2	yes	—
161	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
162	C₂×C₄×C₁₃₁	C₂×#41	1	2	yes	—
163	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	8	yes	—
164	C₂²×C₁₃₁	—	2	4	no	—
165	C₂²×C₁₃₁	—	2	4	no	—
166	C₂²×C₁₃₁	—	2	4	no	—
167	C₂²×C₁₃₁	—	2	16	no	—
168	C₂²×C₁₃₁	—	2	32	no	—
169	C₂²×C₁₃₁	—	4	16	no	—
170	C₂²×C₁₃₁	—	4	16	no	—
171	C₂²×C₁₃₁	—	8	32	no	—
172	C₂²×C₁₃₁	C₂×C₄×D₄	1	4	yes	—
173	C₂²×C₁₃₁	C₂⁴×C₄	1	1	yes	characteristic
174	C₂²×C₁₃₁	C₂³×D₄	1	1	yes	characteristic
175	C₂²×C₁₃₁	C₂²×#41	1	1	yes	characteristic
176	C₂²×C₁₃₁	C₂×H(5,𝔽₂)	1	4	yes	—
177	C₄×C₁₃₁	—	2	4	no	—
178	C₄×C₁₃₁	—	2	8	no	—
179	C₄×C₁₃₁	—	4	8	no	—
180	C₄×C₁₃₁	—	4	4	no	—
181	C₄×C₁₃₁	C₂×C₄×D₄	1	4	yes	—
182	C₄×C₁₃₁	C₂×C₄×D₄	1	4	yes	—
183	C₄×C₁₃₁	#518	1	8	yes	complemented
184	C₂×C₁₃₁	—	2	4	no	—
185	C₂×C₁₃₁	—	4	16	no	—
186	C₂×C₁₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
187	C₂×C₁₃₁	C₂×#518	1	2	yes	—
188	C₂×C₁₃₁	#3498	1	2	yes	—
189	C₂×C₁₃₁	#3508	1	2	yes	—
190	#34210	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₁₃₁	#34210	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²×#127	C₂×C₁₃₁	1	1	yes	characteristic
193	C₂×#518	C₂×C₁₃₁	1	2	yes	complemented
194	#2366	C₂×C₁₃₁	1	8	yes	complemented
195	D₄×C₂²⋊C₄	C₂×C₁₃₁	1	8	yes	complemented
196	#2328	C₂×C₁₃₁	1	4	yes	complemented
197	#2309	C₂×C₁₃₁	1	2	yes	complemented
198	C₄×#127	C₂×C₁₃₁	1	4	yes	complemented
199	#2285	C₂×C₁₃₁	1	1	yes	characteristic, complemented
200	C₄×#117	C₂×C₁₃₁	1	1	yes	characteristic, complemented
201	C₂³×D₄	C₂²×C₁₃₁	1	2	yes	—
202	C₂³×D₄	C₂²×C₁₃₁	1	4	yes	—
203	C₂×#127	C₄×C₁₃₁	1	16	yes	complemented
204	C₂×#127	C₂²×C₁₃₁	1	2	yes	—
205	C₂×#127	C₂²×C₁₃₁	1	2	yes	—
206	C₂×#127	C₂²×C₁₃₁	1	4	yes	—
207	#518	—	2	16	no	—
208	C₂×C₄×D₄	C₂²×C₁₃₁	1	8	yes	complemented
209	C₂×C₄×D₄	C₂²×C₁₃₁	1	8	yes	complemented
210	C₂×C₄×D₄	C₂²×C₁₃₁	1	16	yes	complemented
211	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
212	C₂×#117	C₂²×C₁₃₁	1	2	yes	—
213	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
214	C₂²×C₂²⋊C₄	C₂²×C₁₃₁	1	8	yes	—
215	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
216	#390	C₂²×C₁₃₁	1	8	yes	complemented
217	#390	C₂²×C₁₃₁	1	16	yes	complemented
218	#390	C₂²×C₁₃₁	1	8	yes	complemented
219	#386	C₂²×C₁₃₁	1	32	yes	complemented
220	#384	C₂²×C₁₃₁	1	2	yes	complemented
221	#382	C₂²×C₁₃₁	1	2	yes	complemented
222	#380	C₂²×C₁₃₁	1	4	yes	—
223	#379	C₂²×C₁₃₁	1	8	yes	—
224	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	complemented
225	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	complemented
226	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	4	yes	—
227	#376	C₂²×C₁₃₁	1	2	yes	complemented
228	C₄³	C₂²×C₁₃₁	1	2	yes	complemented
229	C₂⁵	C₂³×C₁₃₁	1	4	yes	—
230	C₂²×D₄	—	2	8	no	—
231	C₂²×D₄	—	2	8	no	—
232	C₂²×D₄	—	4	16	no	—
233	C₂²×D₄	C₂×C₄×C₁₃₁	1	16	yes	complemented
234	C₂²×D₄	C₂×C₄×C₁₃₁	1	32	yes	complemented
235	C₂²×D₄	C₂³×C₁₃₁	1	4	yes	—
236	C₂²×D₄	C₂³×C₁₃₁	1	4	yes	—
237	C₂²×D₄	C₂³×C₁₃₁	1	8	yes	—
238	C₂²×D₄	C₂³×C₁₃₁	1	8	yes	—
239	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
240	C₂³×C₄	C₂³×C₁₃₁	1	16	yes	—
241	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
242	#127	C₂×C₄×C₁₃₁	1	16	yes	complemented
243	#127	C₂×C₄×C₁₃₁	1	32	yes	complemented
244	#127	C₂×C₄×C₁₃₁	1	16	yes	complemented
245	C₄×D₄	—	2	32	no	—
246	C₄×D₄	—	2	32	no	—
247	C₄×D₄	—	4	32	no	—
248	#117	—	2	8	no	—
249	#117	—	2	4	no	—
250	#117	D₄×C₁₃₁	1	8	yes	complemented
251	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
252	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
253	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	8	yes	—
254	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
255	C₂×C₂²⋊C₄	—	2	32	no	—
256	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	4	yes	—
257	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	16	yes	—
258	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	2	yes	—
259	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	32	yes	—
260	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	2	yes	—
261	C₂×C₄²	C₂×C₄×C₁₃₁	1	8	yes	complemented
262	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
263	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
264	C₂×C₄²	C₂³×C₁₃₁	1	8	yes	—
265	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
266	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
267	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
268	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
269	#95	C₂³×C₁₃₁	1	8	yes	—
270	#95	C₂³×C₁₃₁	1	8	yes	—
271	C₂⁴	—	2	8	no	—
272	C₂⁴	—	2	8	no	—
273	C₂⁴	—	4	16	no	—
274	C₂⁴	C₂²×C₄×C₁₃₁	1	16	yes	—
275	C₂⁴	C₂⁴×C₁₃₁	1	8	yes	—
276	C₂⁴	C₂⁴×C₁₃₁	1	1	yes	characteristic
277	C₂×D₄	—	2	64	no	—
278	C₂×D₄	—	2	8	no	—
279	C₂×D₄	—	2	8	no	—
280	C₂×D₄	—	2	8	no	—
281	C₂×D₄	—	2	8	no	—
282	C₂×D₄	—	2	32	no	—
283	C₂×D₄	—	4	64	no	—
284	C₂×D₄	—	4	16	no	—
285	C₂×D₄	C₂²×C₄×C₁₃₁	1	16	yes	—
286	C₂×D₄	C₂²×C₄×C₁₃₁	1	16	yes	—
287	C₂×D₄	C₂²×C₄×C₁₃₁	1	32	yes	—
288	C₂×D₄	C₂²×C₄×C₁₃₁	1	32	yes	—
289	C₂²×C₄	—	2	8	no	—
290	C₂²×C₄	—	2	32	no	—
291	C₂²×C₄	—	4	32	no	—
292	C₂²×C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
293	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
294	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
295	C₂²×C₄	#41×C₁₃₁	1	4	yes	—
296	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
297	C₂²×C₄	C₂⁴×C₁₃₁	1	8	yes	—
298	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
299	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
300	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
301	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
302	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
303	C₄⋊C₄	—	2	8	no	—
304	C₄⋊C₄	—	2	4	no	—
305	C₄⋊C₄	—	2	16	no	—
306	C₄⋊C₄	C₂×D₄×C₁₃₁	1	8	yes	complemented
307	C₂²⋊C₄	—	2	8	no	—
308	C₂²⋊C₄	—	2	4	no	—
309	C₂²⋊C₄	—	2	32	no	—
310	C₂²⋊C₄	—	4	32	no	—
311	C₂²⋊C₄	C₂×D₄×C₁₃₁	1	8	yes	—
312	C₄²	—	2	8	no	—
313	C₄²	—	2	4	no	—
314	C₄²	—	2	16	no	—
315	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
316	C₄²	C₂²×C₄×C₁₃₁	1	8	yes	—
317	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
318	C₄²	C₂×D₄×C₁₃₁	1	8	yes	complemented
319	C₂³	—	2	4	no	—
320	C₂³	—	2	8	no	—
321	C₂³	—	2	8	no	—
322	C₂³	—	2	32	no	—
323	C₂³	—	2	32	no	—
324	C₂³	—	4	32	no	—
325	C₂³	—	8	16	no	—
326	C₂³	—	8	16	no	—
327	C₂³	C₂³×C₄×C₁₃₁	1	16	yes	—
328	C₂³	C₂³×C₄×C₁₃₁	1	2	yes	—
329	C₂³	C₂²×D₄×C₁₃₁	1	2	yes	—
330	C₂³	C₂×#41×C₁₃₁	1	2	yes	—
331	C₂³	C₂⁵×C₁₃₁	1	1	yes	characteristic
332	D₄	—	2	32	no	—
333	D₄	—	2	32	no	—
334	D₄	—	2	64	no	—
335	D₄	—	4	32	no	—
336	C₂×C₄	—	2	8	no	—
337	C₂×C₄	—	2	4	no	—
338	C₂×C₄	—	2	16	no	—
339	C₂×C₄	—	2	8	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	8	no	—
342	C₂×C₄	—	2	16	no	—
343	C₂×C₄	—	8	32	no	—
344	C₂×C₄	C₄×D₄×C₁₃₁	1	16	yes	complemented
345	C₂×C₄	C₂³×C₄×C₁₃₁	1	4	yes	—
346	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
347	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
348	C₂×C₄	C₂³×C₄×C₁₃₁	1	4	yes	—
349	C₂×C₄	C₂²×D₄×C₁₃₁	1	8	yes	—
350	C₂×C₄	C₂²×D₄×C₁₃₁	1	2	yes	—
351	C₂×C₄	C₂²×D₄×C₁₃₁	1	2	yes	—
352	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
353	C₂×C₄	C₂×#41×C₁₃₁	1	2	yes	—
354	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	8	yes	—
355	C₂²	—	2	4	no	—
356	C₂²	—	2	4	no	—
357	C₂²	—	2	4	no	—
358	C₂²	—	2	16	no	—
359	C₂²	—	2	32	no	—
360	C₂²	—	4	16	no	—
361	C₂²	—	4	16	no	—
362	C₂²	—	8	32	no	—
363	C₂²	C₂×C₄×D₄×C₁₃₁	1	4	yes	—
364	C₂²	C₂⁴×C₄×C₁₃₁	1	1	yes	characteristic
365	C₂²	C₂³×D₄×C₁₃₁	1	1	yes	characteristic
366	C₂²	C₂²×#41×C₁₃₁	1	1	yes	characteristic
367	C₂²	C₂×H(5,𝔽₂)×C₁₃₁	1	4	yes	—
368	C₄	—	2	8	no	—
369	C₄	—	2	4	no	—
370	C₄	—	4	8	no	—
371	C₄	—	4	4	no	—
372	C₄	C₂×C₄×D₄×C₁₃₁	1	4	yes	—
373	C₄	C₂×C₄×D₄×C₁₃₁	1	4	yes	—
374	C₄	#518×C₁₃₁	1	8	yes	complemented
375	C₂	—	2	4	no	—
376	C₂	—	4	16	no	—
377	C₂	C₂²×C₄×D₄×C₁₃₁	1	1	yes	characteristic
378	C₂	C₂×#518×C₁₃₁	1	2	yes	—
379	C₂	#3498×C₁₃₁	1	2	yes	—
380	C₂	#3508×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₁₃₁
Frattini, cosocle	C₂³	C₂⁵×C₁₃₁
prime radical	C₂²×#127×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×#518×C₁₃₁	C₂	semidirect
3. #2366×C₁₃₁	C₂	semidirect
4. D₄×C₂²⋊C₄×C₁₃₁	C₂	semidirect
5. #2328×C₁₃₁	C₂	semidirect
6. #2309×C₁₃₁	C₂	semidirect
7. C₄×#127×C₁₃₁	C₂	semidirect
8. #2285×C₁₃₁	C₂	semidirect
9. C₄×#117×C₁₃₁	C₂	semidirect
12. C₂×#127×C₁₃₁	C₄	semidirect
17. C₂×C₄×D₄×C₁₃₁	C₂²	semidirect
18. C₂×C₄×D₄×C₁₃₁	C₂²	semidirect
19. C₂×C₄×D₄×C₁₃₁	C₂²	semidirect
25. #390×C₁₃₁	C₂²	semidirect
26. #390×C₁₃₁	C₂²	semidirect
27. #390×C₁₃₁	C₂²	semidirect
28. #386×C₁₃₁	C₂²	semidirect
29. #384×C₁₃₁	C₂²	semidirect
30. #382×C₁₃₁	C₂²	semidirect
33. C₄×C₄⋊C₄×C₁₃₁	C₂²	semidirect
34. C₄×C₄⋊C₄×C₁₃₁	C₂²	semidirect
36. #376×C₁₃₁	C₂²	semidirect
37. C₄³×C₁₃₁	C₂²	semidirect
42. C₂²×D₄×C₁₃₁	C₂×C₄	semidirect
43. C₂²×D₄×C₁₃₁	C₂×C₄	semidirect
51. #127×C₁₃₁	C₂×C₄	semidirect
52. #127×C₁₃₁	C₂×C₄	semidirect
53. #127×C₁₃₁	C₂×C₄	semidirect
59. #117×C₁₃₁	D₄	semidirect
70. C₂×C₄²×C₁₃₁	C₂×C₄	semidirect
115. C₄⋊C₄×C₁₃₁	C₂×D₄	semidirect
127. C₄²×C₁₃₁	C₂×D₄	semidirect
153. C₂×C₄×C₁₃₁	C₄×D₄	semidirect
183. C₄×C₁₃₁	#518	semidirect
190. #34210	C₁₃₁	direct (N × H)
191. C₁₃₁	#34210	direct (N × H)
193. C₂×#518	C₂×C₁₃₁	semidirect
194. #2366	C₂×C₁₃₁	semidirect
195. D₄×C₂²⋊C₄	C₂×C₁₃₁	semidirect
196. #2328	C₂×C₁₃₁	semidirect
197. #2309	C₂×C₁₃₁	semidirect
198. C₄×#127	C₂×C₁₃₁	semidirect
199. #2285	C₂×C₁₃₁	semidirect
200. C₄×#117	C₂×C₁₃₁	semidirect
203. C₂×#127	C₄×C₁₃₁	semidirect
208. C₂×C₄×D₄	C₂²×C₁₃₁	semidirect
209. C₂×C₄×D₄	C₂²×C₁₃₁	semidirect
210. C₂×C₄×D₄	C₂²×C₁₃₁	semidirect
216. #390	C₂²×C₁₃₁	semidirect
217. #390	C₂²×C₁₃₁	semidirect
218. #390	C₂²×C₁₃₁	semidirect
219. #386	C₂²×C₁₃₁	semidirect
220. #384	C₂²×C₁₃₁	semidirect
221. #382	C₂²×C₁₃₁	semidirect
224. C₄×C₄⋊C₄	C₂²×C₁₃₁	semidirect
225. C₄×C₄⋊C₄	C₂²×C₁₃₁	semidirect
227. #376	C₂²×C₁₃₁	semidirect
228. C₄³	C₂²×C₁₃₁	semidirect
233. C₂²×D₄	C₂×C₄×C₁₃₁	semidirect
234. C₂²×D₄	C₂×C₄×C₁₃₁	semidirect
242. #127	C₂×C₄×C₁₃₁	semidirect
243. #127	C₂×C₄×C₁₃₁	semidirect
244. #127	C₂×C₄×C₁₃₁	semidirect
250. #117	D₄×C₁₃₁	semidirect
261. C₂×C₄²	C₂×C₄×C₁₃₁	semidirect
306. C₄⋊C₄	C₂×D₄×C₁₃₁	semidirect
318. C₄²	C₂×D₄×C₁₃₁	semidirect
344. C₂×C₄	C₄×D₄×C₁₃₁	semidirect
374. C₄	#518×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#127×C₁₃₁	2
C₂×#518×C₁₃₁	2
#2366×C₁₃₁	2
D₄×C₂²⋊C₄×C₁₃₁	2
#2328×C₁₃₁	2
#2309×C₁₃₁	2
C₄×#127×C₁₃₁	2
#2285×C₁₃₁	2
C₄×#117×C₁₃₁	2
#34210	131

Subgroups — C131×#34210 #45720094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#34210 #45720094