Subgroups — C₁₃₁×#33436 #45719320

Order 33536.

692 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₃₂×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2929×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#2169×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2168×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#2168×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	D₄×C₁₆×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	#2151×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#2113×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂²⋊C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#2108×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#131×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
14	#525×C₁₃₁	C₄	1	1	yes	characteristic
15	#504×C₁₃₁	—	2	4	no	—
16	C₂×M₃₂×C₁₃₁	—	2	4	no	—
17	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
19	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	complemented
21	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₁₃₁	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₁₃₁	C₂²	1	2	yes	complemented
25	#435×C₁₃₁	C₄	1	2	yes	—
26	#435×C₁₃₁	C₄	1	2	yes	—
27	C₈×D₄×C₁₃₁	C₂²	1	2	yes	—
28	C₈×D₄×C₁₃₁	C₂²	1	2	yes	—
29	#424×C₁₃₁	C₄	1	1	yes	characteristic
30	#408×C₁₃₁	C₄	1	1	yes	characteristic
31	#407×C₁₃₁	C₄	1	1	yes	characteristic
32	#405×C₁₃₁	C₂²	1	1	yes	characteristic
33	C₄⋊C₁₆×C₁₃₁	C₂²	1	2	yes	complemented
34	C₄⋊C₁₆×C₁₃₁	C₂²	1	2	yes	complemented
35	C₂²⋊C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
36	C₂²⋊C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
37	C₂²⋊C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
38	C₂²⋊C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
39	C₂²⋊C₁₆×C₁₃₁	C₂²	1	2	yes	complemented
40	C₂²⋊C₁₆×C₁₃₁	C₂²	1	2	yes	complemented
41	#346×C₁₃₁	C₂²	1	2	yes	complemented
42	C₄×C₁₆×C₁₃₁	C₂²	1	2	yes	complemented
43	C₂×#41×C₁₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
45	#131×C₁₃₁	—	2	2	no	—
46	#131×C₁₃₁	—	2	2	no	—
47	C₂×M₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×M₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×M₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₁₃₁	—	2	2	no	—
51	C₂²×C₈×C₁₃₁	—	2	2	no	—
52	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
57	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
59	#123×C₁₃₁	C₈	1	2	yes	—
60	C₂²⋊Q₈×C₁₃₁	C₈	1	1	yes	characteristic
61	C₂²⋊Q₈×C₁₃₁	C₈	1	1	yes	characteristic
62	#121×C₁₃₁	C₈	1	1	yes	characteristic
63	#121×C₁₃₁	C₈	1	1	yes	characteristic
64	#120×C₁₃₁	C₈	1	2	yes	—
65	C₄×D₄×C₁₃₁	C₂×C₄	1	2	yes	—
66	C₄×D₄×C₁₃₁	C₂×C₄	1	2	yes	—
67	#117×C₁₃₁	C₂×C₄	1	1	yes	characteristic
68	M₃₂×C₁₃₁	—	2	2	no	—
69	M₃₂×C₁₃₁	—	2	4	no	—
70	M₃₂×C₁₃₁	—	2	4	no	—
71	M₃₂×C₁₃₁	D₄	1	4	yes	complemented
72	C₂×C₁₆×C₁₃₁	—	2	2	no	—
73	C₂×C₁₆×C₁₃₁	—	2	2	no	—
74	C₂×C₁₆×C₁₃₁	—	2	4	no	—
75	C₂×C₁₆×C₁₃₁	—	2	4	no	—
76	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
78	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
82	C₄⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
83	C₄⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
84	C₂²⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
85	C₂²⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
86	C₂²⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
87	C₂²⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
88	#97×C₁₃₁	C₂×C₄	1	2	yes	—
89	C₄×C₈×C₁₃₁	C₂³	1	2	yes	—
90	C₂⁴×C₁₃₁	C₂×C₈	1	1	yes	characteristic
91	#41×C₁₃₁	—	2	2	no	—
92	#41×C₁₃₁	—	2	2	no	—
93	C₂×Q₈×C₁₃₁	C₂×C₈	1	1	yes	characteristic
94	C₂×D₄×C₁₃₁	C₂×C₈	1	2	yes	—
95	C₂×D₄×C₁₃₁	C₂×C₈	1	2	yes	—
96	C₂×D₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
97	C₂×D₄×C₁₃₁	C₂×C₈	1	2	yes	—
98	C₂²×C₄×C₁₃₁	—	2	2	no	—
99	C₂²×C₄×C₁₃₁	—	2	2	no	—
100	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
101	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
102	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
103	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
107	M₁₆×C₁₃₁	—	2	2	no	—
108	M₁₆×C₁₃₁	—	2	2	no	—
109	M₁₆×C₁₃₁	—	2	4	no	—
110	C₂×C₈×C₁₃₁	—	2	2	no	—
111	C₂×C₈×C₁₃₁	—	2	2	no	—
112	C₂×C₈×C₁₃₁	—	2	2	no	—
113	C₂×C₈×C₁₃₁	—	2	1	no	—
114	C₂×C₈×C₁₃₁	—	2	2	no	—
115	C₂×C₈×C₁₃₁	—	2	1	no	—
116	C₂×C₈×C₁₃₁	—	2	1	no	—
117	C₂×C₈×C₁₃₁	—	2	1	no	—
118	C₂×C₈×C₁₃₁	—	2	2	no	—
119	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
120	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₁₃₁	#41	1	2	yes	—
126	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
128	C₄⋊C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
129	C₄⋊C₄×C₁₃₁	C₂×C₈	1	2	yes	—
130	C₄⋊C₄×C₁₃₁	C₂×C₈	1	2	yes	—
131	C₂²⋊C₄×C₁₃₁	C₂×C₈	1	2	yes	—
132	C₂²⋊C₄×C₁₃₁	C₂×C₈	1	2	yes	—
133	C₂²⋊C₄×C₁₃₁	C₂×C₈	1	2	yes	—
134	C₂²⋊C₄×C₁₃₁	C₂×C₈	1	2	yes	—
135	C₂²⋊C₄×C₁₃₁	C₂×C₈	1	2	yes	—
136	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
137	C₁₆×C₁₃₁	—	2	1	no	—
138	C₁₆×C₁₃₁	—	2	1	no	—
139	C₁₆×C₁₃₁	—	2	2	no	—
140	C₁₆×C₁₃₁	—	2	1	no	—
141	C₁₆×C₁₃₁	—	2	1	no	—
142	C₁₆×C₁₃₁	C₂×D₄	1	4	yes	—
143	C₂³×C₁₃₁	—	2	2	no	—
144	C₂³×C₁₃₁	—	2	2	no	—
145	C₂³×C₁₃₁	—	4	1	no	—
146	C₂³×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
147	C₂³×C₁₃₁	C₂²×C₈	1	2	yes	—
148	C₂³×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
149	C₂³×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
150	Q₈×C₁₃₁	—	2	2	no	—
151	D₄×C₁₃₁	—	2	4	no	—
152	D₄×C₁₃₁	—	2	4	no	—
153	D₄×C₁₃₁	—	2	2	no	—
154	D₄×C₁₃₁	—	4	2	no	—
155	C₂×C₄×C₁₃₁	—	2	2	no	—
156	C₂×C₄×C₁₃₁	—	2	1	no	—
157	C₂×C₄×C₁₃₁	—	2	1	no	—
158	C₂×C₄×C₁₃₁	—	2	2	no	—
159	C₂×C₄×C₁₃₁	—	2	1	no	—
160	C₂×C₄×C₁₃₁	—	2	1	no	—
161	C₂×C₄×C₁₃₁	—	2	2	no	—
162	C₂×C₄×C₁₃₁	—	2	2	no	—
163	C₂×C₄×C₁₃₁	—	2	2	no	—
164	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
165	C₂×C₄×C₁₃₁	C₂²×C₈	1	2	yes	—
166	C₂×C₄×C₁₃₁	C₂²×C₈	1	2	yes	—
167	C₂×C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
168	C₂×C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
169	C₂×C₄×C₁₃₁	C₂²×C₈	1	2	yes	—
170	C₂×C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
171	C₂×C₄×C₁₃₁	#131	1	2	yes	—
172	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
173	C₈×C₁₃₁	—	2	1	no	—
174	C₈×C₁₃₁	—	2	1	no	—
175	C₈×C₁₃₁	—	2	2	no	—
176	C₈×C₁₃₁	—	2	2	no	—
177	C₈×C₁₃₁	C₄×D₄	1	2	yes	—
178	C₈×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
179	C₈×C₁₃₁	C₂×#41	1	1	yes	characteristic
180	C₂²×C₁₃₁	—	2	1	no	—
181	C₂²×C₁₃₁	—	2	1	no	—
182	C₂²×C₁₃₁	—	2	1	no	—
183	C₂²×C₁₃₁	—	2	1	no	—
184	C₂²×C₁₃₁	—	2	2	no	—
185	C₂²×C₁₃₁	—	2	1	no	—
186	C₂²×C₁₃₁	—	2	1	no	—
187	C₂²×C₁₃₁	—	2	2	no	—
188	C₂²×C₁₃₁	—	2	1	no	—
189	C₂²×C₁₃₁	—	2	2	no	—
190	C₂²×C₁₃₁	—	4	4	no	—
191	C₂²×C₁₃₁	C₈×D₄	1	2	yes	—
192	C₂²×C₁₃₁	#504	1	2	yes	complemented
193	C₂²×C₁₃₁	C₂³×C₈	1	1	yes	characteristic
194	C₄×C₁₃₁	—	2	2	no	—
195	C₄×C₁₃₁	—	2	2	no	—
196	C₄×C₁₃₁	—	2	1	no	—
197	C₄×C₁₃₁	—	2	1	no	—
198	C₄×C₁₃₁	—	2	2	no	—
199	C₄×C₁₃₁	C₈×D₄	1	2	yes	—
200	C₄×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
201	C₄×C₁₃₁	C₂×#131	1	1	yes	characteristic
202	C₂×C₁₃₁	—	2	2	no	—
203	C₂×C₁₃₁	—	2	2	no	—
204	C₂×C₁₃₁	—	4	2	no	—
205	C₂×C₁₃₁	—	4	1	no	—
206	C₂×C₁₃₁	C₂×C₈×D₄	1	1	yes	characteristic
207	C₂×C₁₃₁	C₂×#504	1	1	yes	characteristic
208	C₂×C₁₃₁	#3407	1	1	yes	characteristic
209	#33436	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₁₃₁	#33436	1	1	yes	characteristic, direct product factor, Sylow
211	C₂×#504	C₂×C₁₃₁	1	1	yes	characteristic, complemented
212	C₂²×M₃₂	C₂×C₁₃₁	1	1	yes	characteristic, complemented
213	#2929	C₂×C₁₃₁	1	1	yes	characteristic
214	#2169	C₂×C₁₃₁	1	2	yes	complemented
215	#2168	C₂×C₁₃₁	1	2	yes	complemented
216	#2168	C₂×C₁₃₁	1	2	yes	complemented
217	D₄×C₁₆	C₂×C₁₃₁	1	2	yes	complemented
218	#2151	C₂×C₁₃₁	1	1	yes	characteristic, complemented
219	#2113	C₂×C₁₃₁	1	1	yes	characteristic, complemented
220	C₂×C₂²⋊C₁₆	C₂×C₁₃₁	1	1	yes	characteristic, complemented
221	#2108	C₂×C₁₃₁	1	1	yes	characteristic, complemented
222	C₂×#131	C₂²×C₁₃₁	1	1	yes	characteristic
223	C₂³×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
224	#525	C₄×C₁₃₁	1	1	yes	characteristic
225	#504	—	2	4	no	—
226	C₂×M₃₂	—	2	4	no	—
227	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
228	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
229	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
230	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	complemented
231	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
232	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
233	C₂²×C₁₆	C₂²×C₁₃₁	1	2	yes	—
234	C₂²×C₁₆	C₂²×C₁₃₁	1	2	yes	complemented
235	#435	C₄×C₁₃₁	1	2	yes	—
236	#435	C₄×C₁₃₁	1	2	yes	—
237	C₈×D₄	C₂²×C₁₃₁	1	2	yes	—
238	C₈×D₄	C₂²×C₁₃₁	1	2	yes	—
239	#424	C₄×C₁₃₁	1	1	yes	characteristic
240	#408	C₄×C₁₃₁	1	1	yes	characteristic
241	#407	C₄×C₁₃₁	1	1	yes	characteristic
242	#405	C₂²×C₁₃₁	1	1	yes	characteristic
243	C₄⋊C₁₆	C₂²×C₁₃₁	1	2	yes	complemented
244	C₄⋊C₁₆	C₂²×C₁₃₁	1	2	yes	complemented
245	C₂²⋊C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
246	C₂²⋊C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
247	C₂²⋊C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
248	C₂²⋊C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
249	C₂²⋊C₁₆	C₂²×C₁₃₁	1	2	yes	complemented
250	C₂²⋊C₁₆	C₂²×C₁₃₁	1	2	yes	complemented
251	#346	C₂²×C₁₃₁	1	2	yes	complemented
252	C₄×C₁₆	C₂²×C₁₃₁	1	2	yes	complemented
253	C₂×#41	C₂×C₄×C₁₃₁	1	1	yes	characteristic
254	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
255	#131	—	2	2	no	—
256	#131	—	2	2	no	—
257	C₂×M₁₆	C₂×C₄×C₁₃₁	1	1	yes	characteristic
258	C₂×M₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
259	C₂×M₁₆	C₂×C₄×C₁₃₁	1	1	yes	characteristic
260	C₂²×C₈	—	2	2	no	—
261	C₂²×C₈	—	2	2	no	—
262	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
263	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
264	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
265	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
266	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
267	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
268	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
269	#123	C₈×C₁₃₁	1	2	yes	—
270	C₂²⋊Q₈	C₈×C₁₃₁	1	1	yes	characteristic
271	C₂²⋊Q₈	C₈×C₁₃₁	1	1	yes	characteristic
272	#121	C₈×C₁₃₁	1	1	yes	characteristic
273	#121	C₈×C₁₃₁	1	1	yes	characteristic
274	#120	C₈×C₁₃₁	1	2	yes	—
275	C₄×D₄	C₂×C₄×C₁₃₁	1	2	yes	—
276	C₄×D₄	C₂×C₄×C₁₃₁	1	2	yes	—
277	#117	C₂×C₄×C₁₃₁	1	1	yes	characteristic
278	M₃₂	—	2	2	no	—
279	M₃₂	—	2	4	no	—
280	M₃₂	—	2	4	no	—
281	M₃₂	D₄×C₁₃₁	1	4	yes	complemented
282	C₂×C₁₆	—	2	2	no	—
283	C₂×C₁₆	—	2	2	no	—
284	C₂×C₁₆	—	2	4	no	—
285	C₂×C₁₆	—	2	4	no	—
286	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
287	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
288	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
289	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
290	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
291	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
292	C₄⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
293	C₄⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
294	C₂²⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
295	C₂²⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
296	C₂²⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
297	C₂²⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
298	#97	C₂×C₄×C₁₃₁	1	2	yes	—
299	C₄×C₈	C₂³×C₁₃₁	1	2	yes	—
300	C₂⁴	C₂×C₈×C₁₃₁	1	1	yes	characteristic
301	#41	—	2	2	no	—
302	#41	—	2	2	no	—
303	C₂×Q₈	C₂×C₈×C₁₃₁	1	1	yes	characteristic
304	C₂×D₄	C₂×C₈×C₁₃₁	1	2	yes	—
305	C₂×D₄	C₂×C₈×C₁₃₁	1	2	yes	—
306	C₂×D₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
307	C₂×D₄	C₂×C₈×C₁₃₁	1	2	yes	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
311	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
312	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
313	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
314	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
315	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
316	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
317	M₁₆	—	2	2	no	—
318	M₁₆	—	2	2	no	—
319	M₁₆	—	2	4	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
330	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
331	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
332	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
333	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
334	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
335	C₂×C₈	#41×C₁₃₁	1	2	yes	—
336	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
337	C₄⋊C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
338	C₄⋊C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
339	C₄⋊C₄	C₂×C₈×C₁₃₁	1	2	yes	—
340	C₄⋊C₄	C₂×C₈×C₁₃₁	1	2	yes	—
341	C₂²⋊C₄	C₂×C₈×C₁₃₁	1	2	yes	—
342	C₂²⋊C₄	C₂×C₈×C₁₃₁	1	2	yes	—
343	C₂²⋊C₄	C₂×C₈×C₁₃₁	1	2	yes	—
344	C₂²⋊C₄	C₂×C₈×C₁₃₁	1	2	yes	—
345	C₂²⋊C₄	C₂×C₈×C₁₃₁	1	2	yes	—
346	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
347	C₁₆	—	2	1	no	—
348	C₁₆	—	2	1	no	—
349	C₁₆	—	2	2	no	—
350	C₁₆	—	2	1	no	—
351	C₁₆	—	2	1	no	—
352	C₁₆	C₂×D₄×C₁₃₁	1	4	yes	—
353	C₂³	—	2	2	no	—
354	C₂³	—	2	2	no	—
355	C₂³	—	4	1	no	—
356	C₂³	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
357	C₂³	C₂²×C₈×C₁₃₁	1	2	yes	—
358	C₂³	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
359	C₂³	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
360	Q₈	—	2	2	no	—
361	D₄	—	2	4	no	—
362	D₄	—	2	4	no	—
363	D₄	—	2	2	no	—
364	D₄	—	4	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
375	C₂×C₄	C₂²×C₈×C₁₃₁	1	2	yes	—
376	C₂×C₄	C₂²×C₈×C₁₃₁	1	2	yes	—
377	C₂×C₄	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
378	C₂×C₄	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
379	C₂×C₄	C₂²×C₈×C₁₃₁	1	2	yes	—
380	C₂×C₄	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
381	C₂×C₄	#131×C₁₃₁	1	2	yes	—
382	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
383	C₈	—	2	1	no	—
384	C₈	—	2	1	no	—
385	C₈	—	2	2	no	—
386	C₈	—	2	2	no	—
387	C₈	C₄×D₄×C₁₃₁	1	2	yes	—
388	C₈	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
389	C₈	C₂×#41×C₁₃₁	1	1	yes	characteristic
390	C₂²	—	2	1	no	—
391	C₂²	—	2	1	no	—
392	C₂²	—	2	1	no	—
393	C₂²	—	2	1	no	—
394	C₂²	—	2	2	no	—
395	C₂²	—	2	1	no	—
396	C₂²	—	2	1	no	—
397	C₂²	—	2	2	no	—
398	C₂²	—	2	1	no	—
399	C₂²	—	2	2	no	—
400	C₂²	—	4	4	no	—
401	C₂²	C₈×D₄×C₁₃₁	1	2	yes	—
402	C₂²	#504×C₁₃₁	1	2	yes	complemented
403	C₂²	C₂³×C₈×C₁₃₁	1	1	yes	characteristic
404	C₄	—	2	2	no	—
405	C₄	—	2	2	no	—
406	C₄	—	2	1	no	—
407	C₄	—	2	1	no	—
408	C₄	—	2	2	no	—
409	C₄	C₈×D₄×C₁₃₁	1	2	yes	—
410	C₄	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
411	C₄	C₂×#131×C₁₃₁	1	1	yes	characteristic
412	C₂	—	2	2	no	—
413	C₂	—	2	2	no	—
414	C₂	—	4	1	no	—
415	C₂	—	4	2	no	—
416	C₂	C₂×C₈×D₄×C₁₃₁	1	1	yes	characteristic
417	C₂	C₂×#504×C₁₃₁	1	1	yes	characteristic
418	C₂	#3407×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₃₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₁₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
socle	C₂²×C₁₃₁	C₂³×C₈
prime radical	#525×C₁₃₁	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#504×C₁₃₁	C₂	semidirect
2. C₂²×M₃₂×C₁₃₁	C₂	semidirect
4. #2169×C₁₃₁	C₂	semidirect
5. #2168×C₁₃₁	C₂	semidirect
6. #2168×C₁₃₁	C₂	semidirect
7. D₄×C₁₆×C₁₃₁	C₂	semidirect
8. #2151×C₁₃₁	C₂	semidirect
9. #2113×C₁₃₁	C₂	semidirect
10. C₂×C₂²⋊C₁₆×C₁₃₁	C₂	semidirect
11. #2108×C₁₃₁	C₂	semidirect
20. C₂×M₃₂×C₁₃₁	C₂²	semidirect
24. C₂²×C₁₆×C₁₃₁	C₂²	semidirect
33. C₄⋊C₁₆×C₁₃₁	C₂²	semidirect
34. C₄⋊C₁₆×C₁₃₁	C₂²	semidirect
39. C₂²⋊C₁₆×C₁₃₁	C₂²	semidirect
40. C₂²⋊C₁₆×C₁₃₁	C₂²	semidirect
41. #346×C₁₃₁	C₂²	semidirect
42. C₄×C₁₆×C₁₃₁	C₂²	semidirect
71. M₃₂×C₁₃₁	D₄	semidirect
192. C₂²×C₁₃₁	#504	semidirect
209. #33436	C₁₃₁	direct (N × H)
210. C₁₃₁	#33436	direct (N × H)
211. C₂×#504	C₂×C₁₃₁	semidirect
212. C₂²×M₃₂	C₂×C₁₃₁	semidirect
214. #2169	C₂×C₁₃₁	semidirect
215. #2168	C₂×C₁₃₁	semidirect
216. #2168	C₂×C₁₃₁	semidirect
217. D₄×C₁₆	C₂×C₁₃₁	semidirect
218. #2151	C₂×C₁₃₁	semidirect
219. #2113	C₂×C₁₃₁	semidirect
220. C₂×C₂²⋊C₁₆	C₂×C₁₃₁	semidirect
221. #2108	C₂×C₁₃₁	semidirect
230. C₂×M₃₂	C₂²×C₁₃₁	semidirect
234. C₂²×C₁₆	C₂²×C₁₃₁	semidirect
243. C₄⋊C₁₆	C₂²×C₁₃₁	semidirect
244. C₄⋊C₁₆	C₂²×C₁₃₁	semidirect
249. C₂²⋊C₁₆	C₂²×C₁₃₁	semidirect
250. C₂²⋊C₁₆	C₂²×C₁₃₁	semidirect
251. #346	C₂²×C₁₃₁	semidirect
252. C₄×C₁₆	C₂²×C₁₃₁	semidirect
281. M₃₂	D₄×C₁₃₁	semidirect
402. C₂²	#504×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#504×C₁₃₁	2
C₂²×M₃₂×C₁₃₁	2
#2929×C₁₃₁	2
#2169×C₁₃₁	2
#2168×C₁₃₁	2
D₄×C₁₆×C₁₃₁	2
#2151×C₁₃₁	2
#2113×C₁₃₁	2
C₂×C₂²⋊C₁₆×C₁₃₁	2
#2108×C₁₃₁	2
#33436	131

Subgroups — C131×#33436 #45719320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33436 #45719320