Subgroups — C₁₃₁×#20565 #45706449

Order 33536.

982 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#436×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2293×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1770×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₄⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#1767×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#1753×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	#1752×C₁₃₁	C₂	1	2	yes	maximal, complemented
9	#1750×C₁₃₁	C₂	1	2	yes	maximal, complemented
10	#1744×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#1729×C₁₃₁	C₂	1	2	yes	maximal, complemented
12	C₂²×M₁₆×C₁₃₁	C₂²	1	2	yes	—
13	C₂³×C₈×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#117×C₁₃₁	C₂²	1	2	yes	—
16	#436×C₁₃₁	C₄	1	8	yes	complemented
17	C₈×D₄×C₁₃₁	C₄	1	8	yes	complemented
18	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	2	yes	—
22	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
23	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
27	#390×C₁₃₁	C₄	1	2	yes	—
28	#389×C₁₃₁	C₄	1	2	yes	—
29	#388×C₁₃₁	C₄	1	4	yes	—
30	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
32	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
33	#336×C₁₃₁	C₂²	1	1	yes	characteristic
34	#336×C₁₃₁	C₂²	1	2	yes	—
35	#336×C₁₃₁	C₂²	1	2	yes	—
36	#336×C₁₃₁	C₂²	1	2	yes	—
37	#336×C₁₃₁	C₂²	1	1	yes	characteristic
38	#336×C₁₃₁	C₂²	1	1	yes	characteristic
39	#336×C₁₃₁	C₂²	1	1	yes	characteristic
40	C₂²×D₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
42	C₂×M₁₆×C₁₃₁	—	4	2	no	—
43	C₂×M₁₆×C₁₃₁	—	4	2	no	—
44	C₂×M₁₆×C₁₃₁	C₂×C₄	1	8	yes	—
45	C₂²×C₈×C₁₃₁	—	4	2	no	—
46	C₂²×C₈×C₁₃₁	—	4	2	no	—
47	C₂²×C₈×C₁₃₁	C₂×C₄	1	8	yes	—
48	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
54	C₄×D₄×C₁₃₁	C₂×C₄	1	4	yes	—
55	C₄×D₄×C₁₃₁	C₂×C₄	1	4	yes	—
56	#117×C₁₃₁	—	4	2	no	—
57	#117×C₁₃₁	—	4	2	no	—
58	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
60	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
62	C₂×C₂²⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
63	C₂×C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
67	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₁₃₁	C₂×C₄	1	4	yes	—
71	C₄⋊C₈×C₁₃₁	C₂×C₄	1	4	yes	—
72	C₂²⋊C₈×C₁₃₁	C₂×C₄	1	8	yes	—
73	C₂²⋊C₈×C₁₃₁	C₂×C₄	1	8	yes	—
74	#97×C₁₃₁	—	4	2	no	—
75	#97×C₁₃₁	—	4	2	no	—
76	C₄×C₈×C₁₃₁	—	4	2	no	—
77	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
78	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
79	#95×C₁₃₁	C₂×C₄	1	2	yes	—
80	#95×C₁₃₁	C₂×C₄	1	4	yes	—
81	C₂⁴×C₁₃₁	C₂²×C₄	1	2	yes	—
82	C₂×D₄×C₁₃₁	—	2	4	no	—
83	C₂×D₄×C₁₃₁	C₄²	1	4	yes	—
84	C₂²×C₄×C₁₃₁	—	2	2	no	—
85	C₂²×C₄×C₁₃₁	—	2	2	no	—
86	C₂²×C₄×C₁₃₁	—	2	2	no	—
87	C₂²×C₄×C₁₃₁	—	2	2	no	—
88	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
90	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
94	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
95	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
97	M₁₆×C₁₃₁	—	4	8	no	—
98	C₂×C₈×C₁₃₁	—	2	1	no	—
99	C₂×C₈×C₁₃₁	—	2	2	no	—
100	C₂×C₈×C₁₃₁	—	2	1	no	—
101	C₂×C₈×C₁₃₁	—	2	2	no	—
102	C₂×C₈×C₁₃₁	—	2	2	no	—
103	C₂×C₈×C₁₃₁	—	2	2	no	—
104	C₂×C₈×C₁₃₁	—	2	1	no	—
105	C₂×C₈×C₁₃₁	—	2	1	no	—
106	C₂×C₈×C₁₃₁	—	2	1	no	—
107	C₂×C₈×C₁₃₁	—	2	1	no	—
108	C₂×C₈×C₁₃₁	—	2	1	no	—
109	C₂×C₈×C₁₃₁	—	2	1	no	—
110	C₂×C₈×C₁₃₁	—	4	8	no	—
111	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
116	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
117	C₄⋊C₄×C₁₃₁	—	2	4	no	—
118	C₄⋊C₄×C₁₃₁	—	4	4	no	—
119	C₄⋊C₄×C₁₃₁	C₄²	1	4	yes	—
120	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
121	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
122	C₂²⋊C₄×C₁₃₁	C₄²	1	8	yes	—
123	C₄²×C₁₃₁	—	2	4	no	—
124	C₄²×C₁₃₁	—	2	2	no	—
125	C₄²×C₁₃₁	—	2	1	no	—
126	C₄²×C₁₃₁	—	2	2	no	—
127	C₄²×C₁₃₁	—	2	1	no	—
128	C₄²×C₁₃₁	—	2	1	no	—
129	C₄²×C₁₃₁	—	2	1	no	—
130	C₄²×C₁₃₁	—	4	2	no	—
131	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
132	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
133	C₂³×C₁₃₁	—	2	2	no	—
134	C₂³×C₁₃₁	—	2	2	no	—
135	C₂³×C₁₃₁	—	2	4	no	—
136	C₂³×C₁₃₁	—	2	4	no	—
137	C₂³×C₁₃₁	C₂×C₄²	1	4	yes	—
138	C₂³×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
139	D₄×C₁₃₁	—	2	8	no	—
140	C₂×C₄×C₁₃₁	—	2	2	no	—
141	C₂×C₄×C₁₃₁	—	2	2	no	—
142	C₂×C₄×C₁₃₁	—	2	2	no	—
143	C₂×C₄×C₁₃₁	—	2	4	no	—
144	C₂×C₄×C₁₃₁	—	2	4	no	—
145	C₂×C₄×C₁₃₁	—	2	4	no	—
146	C₂×C₄×C₁₃₁	—	2	4	no	—
147	C₂×C₄×C₁₃₁	—	2	2	no	—
148	C₂×C₄×C₁₃₁	—	2	4	no	—
149	C₂×C₄×C₁₃₁	—	2	2	no	—
150	C₂×C₄×C₁₃₁	—	2	2	no	—
151	C₂×C₄×C₁₃₁	—	2	2	no	—
152	C₂×C₄×C₁₃₁	—	2	4	no	—
153	C₂×C₄×C₁₃₁	—	2	2	no	—
154	C₂×C₄×C₁₃₁	—	2	2	no	—
155	C₂×C₄×C₁₃₁	—	2	4	no	—
156	C₂×C₄×C₁₃₁	C₂×C₄²	1	2	yes	—
157	C₂×C₄×C₁₃₁	C₂×C₄²	1	4	yes	—
158	C₂×C₄×C₁₃₁	C₂×C₄²	1	2	yes	—
159	C₂×C₄×C₁₃₁	#131	1	4	yes	—
160	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
165	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
166	C₈×C₁₃₁	—	2	2	no	—
167	C₈×C₁₃₁	—	2	4	no	—
168	C₈×C₁₃₁	—	2	4	no	—
169	C₈×C₁₃₁	—	2	2	no	—
170	C₈×C₁₃₁	—	2	2	no	—
171	C₈×C₁₃₁	—	2	2	no	—
172	C₂²×C₁₃₁	—	2	4	no	—
173	C₂²×C₁₃₁	—	2	4	no	—
174	C₂²×C₁₃₁	—	2	4	no	—
175	C₂²×C₁₃₁	—	4	4	no	—
176	C₂²×C₁₃₁	—	4	4	no	—
177	C₂²×C₁₃₁	C₂²×C₄²	1	1	yes	characteristic
178	C₂²×C₁₃₁	#518	1	1	yes	characteristic
179	C₂²×C₁₃₁	#520	1	1	yes	characteristic
180	C₂²×C₁₃₁	C₂×#131	1	2	yes	—
181	C₂²×C₁₃₁	#568	1	2	yes	—
182	C₄×C₁₃₁	—	2	2	no	—
183	C₄×C₁₃₁	—	2	4	no	—
184	C₄×C₁₃₁	—	4	2	no	—
185	C₄×C₁₃₁	—	4	4	no	—
186	C₄×C₁₃₁	—	4	2	no	—
187	C₄×C₁₃₁	#405	1	4	yes	—
188	C₄×C₁₃₁	#518	1	1	yes	characteristic
189	C₄×C₁₃₁	#518	1	1	yes	characteristic
190	C₄×C₁₃₁	#520	1	1	yes	characteristic
191	C₄×C₁₃₁	#520	1	1	yes	characteristic
192	C₂×C₁₃₁	—	4	4	no	—
193	C₂×C₁₃₁	#2275	1	1	yes	characteristic
194	C₂×C₁₃₁	C₂×#405	1	1	yes	characteristic
195	C₂×C₁₃₁	#2875	1	1	yes	characteristic
196	C₂×C₁₃₁	#2981	1	2	yes	—
197	C₂×C₁₃₁	#2993	1	2	yes	—
198	#20565	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₁₃₁	#20565	1	1	yes	characteristic, direct product factor, Sylow
200	C₂×#436	C₂×C₁₃₁	1	1	yes	characteristic
201	C₂×C₈×D₄	C₂×C₁₃₁	1	1	yes	characteristic
202	#2293	C₂×C₁₃₁	1	1	yes	characteristic
203	#1770	C₂×C₁₃₁	1	1	yes	characteristic, complemented
204	C₈×C₄⋊C₄	C₂×C₁₃₁	1	1	yes	characteristic, complemented
205	#1767	C₂×C₁₃₁	1	1	yes	characteristic, complemented
206	#1753	C₂×C₁₃₁	1	2	yes	complemented
207	#1752	C₂×C₁₃₁	1	2	yes	complemented
208	#1750	C₂×C₁₃₁	1	2	yes	complemented
209	#1744	C₂×C₁₃₁	1	1	yes	characteristic, complemented
210	#1729	C₂×C₁₃₁	1	2	yes	complemented
211	C₂²×M₁₆	C₂²×C₁₃₁	1	2	yes	—
212	C₂³×C₈	C₂²×C₁₃₁	1	2	yes	—
213	C₂×C₄×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
214	C₂×#117	C₂²×C₁₃₁	1	2	yes	—
215	#436	C₄×C₁₃₁	1	8	yes	complemented
216	C₈×D₄	C₄×C₁₃₁	1	8	yes	complemented
217	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
218	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
219	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	2	yes	—
220	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	2	yes	—
221	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
222	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
223	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
224	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
225	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
226	#390	C₄×C₁₃₁	1	2	yes	—
227	#389	C₄×C₁₃₁	1	2	yes	—
228	#388	C₄×C₁₃₁	1	4	yes	—
229	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
230	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	2	yes	—
231	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
232	#336	C₂²×C₁₃₁	1	1	yes	characteristic
233	#336	C₂²×C₁₃₁	1	2	yes	—
234	#336	C₂²×C₁₃₁	1	2	yes	—
235	#336	C₂²×C₁₃₁	1	2	yes	—
236	#336	C₂²×C₁₃₁	1	1	yes	characteristic
237	#336	C₂²×C₁₃₁	1	1	yes	characteristic
238	#336	C₂²×C₁₃₁	1	1	yes	characteristic
239	C₂²×D₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
240	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
241	C₂×M₁₆	—	4	2	no	—
242	C₂×M₁₆	—	4	2	no	—
243	C₂×M₁₆	C₂×C₄×C₁₃₁	1	8	yes	—
244	C₂²×C₈	—	4	2	no	—
245	C₂²×C₈	—	4	2	no	—
246	C₂²×C₈	C₂×C₄×C₁₃₁	1	8	yes	—
247	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
248	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
249	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
250	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
251	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
252	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
253	C₄×D₄	C₂×C₄×C₁₃₁	1	4	yes	—
254	C₄×D₄	C₂×C₄×C₁₃₁	1	4	yes	—
255	#117	—	4	2	no	—
256	#117	—	4	2	no	—
257	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
258	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
259	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
260	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
261	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
262	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
263	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
264	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
265	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
266	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
267	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
268	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
269	C₄⋊C₈	C₂×C₄×C₁₃₁	1	4	yes	—
270	C₄⋊C₈	C₂×C₄×C₁₃₁	1	4	yes	—
271	C₂²⋊C₈	C₂×C₄×C₁₃₁	1	8	yes	—
272	C₂²⋊C₈	C₂×C₄×C₁₃₁	1	8	yes	—
273	#97	—	4	2	no	—
274	#97	—	4	2	no	—
275	C₄×C₈	—	4	2	no	—
276	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
277	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
278	#95	C₂×C₄×C₁₃₁	1	4	yes	—
279	#95	C₂×C₄×C₁₃₁	1	2	yes	—
280	C₂⁴	C₂²×C₄×C₁₃₁	1	2	yes	—
281	C₂×D₄	—	2	4	no	—
282	C₂×D₄	C₄²×C₁₃₁	1	4	yes	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
288	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
289	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
290	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
291	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
292	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
293	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
294	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
295	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
296	M₁₆	—	4	8	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	4	8	no	—
310	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
311	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
312	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
313	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
314	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
315	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
316	C₄⋊C₄	—	2	4	no	—
317	C₄⋊C₄	—	4	4	no	—
318	C₄⋊C₄	C₄²×C₁₃₁	1	4	yes	—
319	C₂²⋊C₄	—	4	4	no	—
320	C₂²⋊C₄	—	4	4	no	—
321	C₂²⋊C₄	C₄²×C₁₃₁	1	8	yes	—
322	C₄²	—	2	4	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	1	no	—
325	C₄²	—	2	2	no	—
326	C₄²	—	2	1	no	—
327	C₄²	—	2	1	no	—
328	C₄²	—	2	1	no	—
329	C₄²	—	4	2	no	—
330	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
331	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
332	C₂³	—	2	2	no	—
333	C₂³	—	2	2	no	—
334	C₂³	—	2	4	no	—
335	C₂³	—	2	4	no	—
336	C₂³	C₂×C₄²×C₁₃₁	1	4	yes	—
337	C₂³	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
338	D₄	—	2	8	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	4	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	4	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	4	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	4	no	—
355	C₂×C₄	C₂×C₄²×C₁₃₁	1	2	yes	—
356	C₂×C₄	C₂×C₄²×C₁₃₁	1	4	yes	—
357	C₂×C₄	C₂×C₄²×C₁₃₁	1	2	yes	—
358	C₂×C₄	#131×C₁₃₁	1	4	yes	—
359	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
360	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
361	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
362	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
363	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
364	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
365	C₈	—	2	2	no	—
366	C₈	—	2	4	no	—
367	C₈	—	2	2	no	—
368	C₈	—	2	4	no	—
369	C₈	—	2	2	no	—
370	C₈	—	2	2	no	—
371	C₂²	—	2	4	no	—
372	C₂²	—	2	4	no	—
373	C₂²	—	2	4	no	—
374	C₂²	—	4	4	no	—
375	C₂²	—	4	4	no	—
376	C₂²	C₂²×C₄²×C₁₃₁	1	1	yes	characteristic
377	C₂²	#518×C₁₃₁	1	1	yes	characteristic
378	C₂²	#520×C₁₃₁	1	1	yes	characteristic
379	C₂²	C₂×#131×C₁₃₁	1	2	yes	—
380	C₂²	#568×C₁₃₁	1	2	yes	—
381	C₄	—	2	4	no	—
382	C₄	—	2	2	no	—
383	C₄	—	4	2	no	—
384	C₄	—	4	2	no	—
385	C₄	—	4	4	no	—
386	C₄	#405×C₁₃₁	1	4	yes	—
387	C₄	#518×C₁₃₁	1	1	yes	characteristic
388	C₄	#518×C₁₃₁	1	1	yes	characteristic
389	C₄	#520×C₁₃₁	1	1	yes	characteristic
390	C₄	#520×C₁₃₁	1	1	yes	characteristic
391	C₂	—	4	4	no	—
392	C₂	#2275×C₁₃₁	1	1	yes	characteristic
393	C₂	C₂×#405×C₁₃₁	1	1	yes	characteristic
394	C₂	#2875×C₁₃₁	1	1	yes	characteristic
395	C₂	#2981×C₁₃₁	1	2	yes	—
396	C₂	#2993×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
socle	C₂³×C₁₃₁	C₂³×C₄
prime radical	C₂²×D₄×C₁₃₁	C₂×C₄
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
4. #1770×C₁₃₁	C₂	semidirect
5. C₈×C₄⋊C₄×C₁₃₁	C₂	semidirect
6. #1767×C₁₃₁	C₂	semidirect
7. #1753×C₁₃₁	C₂	semidirect
8. #1752×C₁₃₁	C₂	semidirect
9. #1750×C₁₃₁	C₂	semidirect
10. #1744×C₁₃₁	C₂	semidirect
11. #1729×C₁₃₁	C₂	semidirect
16. #436×C₁₃₁	C₄	semidirect
17. C₈×D₄×C₁₃₁	C₄	semidirect
198. #20565	C₁₃₁	direct (N × H)
199. C₁₃₁	#20565	direct (N × H)
203. #1770	C₂×C₁₃₁	semidirect
204. C₈×C₄⋊C₄	C₂×C₁₃₁	semidirect
205. #1767	C₂×C₁₃₁	semidirect
206. #1753	C₂×C₁₃₁	semidirect
207. #1752	C₂×C₁₃₁	semidirect
208. #1750	C₂×C₁₃₁	semidirect
209. #1744	C₂×C₁₃₁	semidirect
210. #1729	C₂×C₁₃₁	semidirect
215. #436	C₄×C₁₃₁	semidirect
216. C₈×D₄	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×#436×C₁₃₁	2
C₂×C₈×D₄×C₁₃₁	2
#2293×C₁₃₁	2
#1770×C₁₃₁	2
C₈×C₄⋊C₄×C₁₃₁	2
#1767×C₁₃₁	2
#1753×C₁₃₁	2
#1752×C₁₃₁	2
#1750×C₁₃₁	2
#1744×C₁₃₁	2
#1729×C₁₃₁	2
#20565	131

Subgroups — C131×#20565 #45706449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#20565 #45706449