Subgroups — C₁₃₁×#21604 #45707488

Order 33536.

834 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#405×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2396×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1985×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1940×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#1916×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#1850×C₁₃₁	C₂	1	2	yes	maximal
8	#1798×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	#1770×C₁₃₁	C₂	1	2	yes	maximal, complemented
10	C₈×C₄⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#1753×C₁₃₁	C₂	1	2	yes	maximal
12	C₂²×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
16	#446×C₁₃₁	—	2	4	no	—
17	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	2	yes	—
22	#405×C₁₃₁	—	2	4	no	—
23	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₂×#97×C₁₃₁	C₂²	1	2	yes	—
25	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
29	#400×C₁₃₁	C₄	1	1	yes	characteristic
30	#399×C₁₃₁	C₄	1	1	yes	characteristic
31	#398×C₁₃₁	C₄	1	2	yes	—
32	#396×C₁₃₁	C₄	1	2	yes	—
33	#395×C₁₃₁	C₄	1	1	yes	characteristic
34	#393×C₁₃₁	C₄	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
37	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
38	#336×C₁₃₁	C₂²	1	2	yes	—
39	#336×C₁₃₁	C₂²	1	2	yes	—
40	#336×C₁₃₁	C₂²	1	2	yes	—
41	#336×C₁₃₁	C₂²	1	2	yes	—
42	#336×C₁₃₁	C₂²	1	2	yes	—
43	C₂²×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₁₃₁	—	2	2	no	—
46	C₂×M₁₆×C₁₃₁	—	2	2	no	—
47	C₂×M₁₆×C₁₃₁	Q₈	1	4	yes	complemented
48	C₂²×C₈×C₁₃₁	—	2	2	no	—
49	C₂²×C₈×C₁₃₁	—	2	2	no	—
50	C₂²×C₈×C₁₃₁	—	2	2	no	—
51	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
57	C₄×Q₈×C₁₃₁	—	2	2	no	—
58	C₄×Q₈×C₁₃₁	—	2	2	no	—
59	#117×C₁₃₁	—	2	2	no	—
60	#117×C₁₃₁	—	2	2	no	—
61	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
74	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
75	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₁₃₁	—	2	4	no	—
77	C₄⋊C₈×C₁₃₁	—	2	4	no	—
78	C₄⋊C₈×C₁₃₁	—	2	2	no	—
79	C₄⋊C₈×C₁₃₁	—	2	2	no	—
80	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
81	#97×C₁₃₁	—	2	2	no	—
82	#97×C₁₃₁	—	2	4	no	—
83	#97×C₁₃₁	—	2	2	no	—
84	C₄×C₈×C₁₃₁	—	2	2	no	—
85	C₄×C₈×C₁₃₁	—	2	4	no	—
86	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
87	#95×C₁₃₁	C₂×C₄	1	2	yes	—
88	#95×C₁₃₁	C₂×C₄	1	1	yes	characteristic
89	#95×C₁₃₁	C₂×C₄	1	2	yes	—
90	#95×C₁₃₁	C₂×C₄	1	1	yes	characteristic
91	#95×C₁₃₁	C₂×C₄	1	2	yes	—
92	#95×C₁₃₁	C₂×C₄	1	2	yes	—
93	C₂⁴×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×Q₈×C₁₃₁	—	2	2	no	—
95	C₂×Q₈×C₁₃₁	—	4	2	no	—
96	C₂²×C₄×C₁₃₁	—	2	1	no	—
97	C₂²×C₄×C₁₃₁	—	2	1	no	—
98	C₂²×C₄×C₁₃₁	—	2	1	no	—
99	C₂²×C₄×C₁₃₁	—	2	1	no	—
100	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
110	C₂²×C₄×C₁₃₁	#41	1	2	yes	—
111	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
112	M₁₆×C₁₃₁	—	2	8	no	—
113	C₂×C₈×C₁₃₁	—	2	8	no	—
114	C₂×C₈×C₁₃₁	—	2	1	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	1	no	—
119	C₂×C₈×C₁₃₁	—	2	1	no	—
120	C₂×C₈×C₁₃₁	—	2	2	no	—
121	C₂×C₈×C₁₃₁	—	2	2	no	—
122	C₂×C₈×C₁₃₁	—	2	2	no	—
123	C₂×C₈×C₁₃₁	—	2	2	no	—
124	C₂×C₈×C₁₃₁	—	2	2	no	—
125	C₂×C₈×C₁₃₁	—	2	2	no	—
126	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
127	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
129	C₂×C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
130	C₂×C₈×C₁₃₁	#41	1	2	yes	—
131	C₂×C₈×C₁₃₁	#41	1	2	yes	—
132	C₄⋊C₄×C₁₃₁	—	2	2	no	—
133	C₄⋊C₄×C₁₃₁	—	2	2	no	—
134	C₄⋊C₄×C₁₃₁	—	2	2	no	—
135	C₄⋊C₄×C₁₃₁	—	2	4	no	—
136	C₄⋊C₄×C₁₃₁	—	2	4	no	—
137	C₄⋊C₄×C₁₃₁	—	2	4	no	—
138	C₄⋊C₄×C₁₃₁	—	4	2	no	—
139	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
140	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
141	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
142	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
143	C₄²×C₁₃₁	—	2	1	no	—
144	C₄²×C₁₃₁	—	2	2	no	—
145	C₄²×C₁₃₁	—	2	2	no	—
146	C₄²×C₁₃₁	—	2	2	no	—
147	C₄²×C₁₃₁	—	2	1	no	—
148	C₄²×C₁₃₁	—	2	2	no	—
149	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
150	C₄²×C₁₃₁	#41	1	2	yes	—
151	C₂³×C₁₃₁	—	2	1	no	—
152	C₂³×C₁₃₁	—	2	1	no	—
153	C₂³×C₁₃₁	—	2	2	no	—
154	C₂³×C₁₃₁	—	2	2	no	—
155	C₂³×C₁₃₁	C₄×Q₈	1	2	yes	—
156	C₂³×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
157	Q₈×C₁₃₁	—	4	4	no	—
158	C₂×C₄×C₁₃₁	—	2	1	no	—
159	C₂×C₄×C₁₃₁	—	2	2	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	—	2	2	no	—
165	C₂×C₄×C₁₃₁	—	2	2	no	—
166	C₂×C₄×C₁₃₁	—	2	2	no	—
167	C₂×C₄×C₁₃₁	—	2	2	no	—
168	C₂×C₄×C₁₃₁	—	2	1	no	—
169	C₂×C₄×C₁₃₁	—	2	1	no	—
170	C₂×C₄×C₁₃₁	—	2	2	no	—
171	C₂×C₄×C₁₃₁	—	2	2	no	—
172	C₂×C₄×C₁₃₁	—	2	2	no	—
173	C₂×C₄×C₁₃₁	—	2	2	no	—
174	C₂×C₄×C₁₃₁	—	2	2	no	—
175	C₂×C₄×C₁₃₁	—	2	2	no	—
176	C₂×C₄×C₁₃₁	—	2	2	no	—
177	C₂×C₄×C₁₃₁	—	2	2	no	—
178	C₂×C₄×C₁₃₁	—	2	2	no	—
179	C₂×C₄×C₁₃₁	—	2	2	no	—
180	C₂×C₄×C₁₃₁	—	2	2	no	—
181	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
182	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
183	C₂×C₄×C₁₃₁	C₄×Q₈	1	2	yes	—
184	C₂×C₄×C₁₃₁	#131	1	4	yes	—
185	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
187	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
190	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
191	C₈×C₁₃₁	—	2	2	no	—
192	C₈×C₁₃₁	—	2	2	no	—
193	C₈×C₁₃₁	—	2	2	no	—
194	C₈×C₁₃₁	—	4	2	no	—
195	C₈×C₁₃₁	—	4	2	no	—
196	C₈×C₁₃₁	C₂²⋊Q₈	1	4	yes	—
197	C₂²×C₁₃₁	—	2	4	no	—
198	C₂²×C₁₃₁	—	2	2	no	—
199	C₂²×C₁₃₁	—	4	1	no	—
200	C₂²×C₁₃₁	—	4	1	no	—
201	C₂²×C₁₃₁	—	4	2	no	—
202	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
203	C₂²×C₁₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
204	C₂²×C₁₃₁	C₄×#41	1	1	yes	characteristic
205	C₂²×C₁₃₁	C₂×#131	1	2	yes	—
206	C₂²×C₁₃₁	#568	1	2	yes	—
207	C₄×C₁₃₁	—	2	2	no	—
208	C₄×C₁₃₁	—	2	2	no	—
209	C₄×C₁₃₁	—	2	4	no	—
210	C₄×C₁₃₁	—	2	2	no	—
211	C₄×C₁₃₁	—	4	1	no	—
212	C₄×C₁₃₁	—	4	2	no	—
213	C₄×C₁₃₁	—	4	1	no	—
214	C₄×C₁₃₁	—	4	2	no	—
215	C₄×C₁₃₁	—	4	2	no	—
216	C₄×C₁₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
217	C₄×C₁₃₁	#525	1	1	yes	characteristic
218	C₄×C₁₃₁	#529	1	1	yes	characteristic
219	C₄×C₁₃₁	#533	1	1	yes	characteristic
220	C₂×C₁₃₁	—	4	2	no	—
221	C₂×C₁₃₁	C₄×C₂²⋊Q₈	1	1	yes	characteristic
222	C₂×C₁₃₁	#2930	1	1	yes	characteristic
223	C₂×C₁₃₁	#2931	1	1	yes	characteristic
224	C₂×C₁₃₁	#2962	1	2	yes	—
225	C₂×C₁₃₁	#2968	1	2	yes	—
226	#21604	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
227	C₁₃₁	#21604	1	1	yes	characteristic, direct product factor, Sylow
228	C₂×#446	C₂×C₁₃₁	1	1	yes	characteristic, complemented
229	C₂×#405	C₂×C₁₃₁	1	1	yes	characteristic
230	#2396	C₂×C₁₃₁	1	1	yes	characteristic
231	#1985	C₂×C₁₃₁	1	1	yes	characteristic, complemented
232	#1940	C₂×C₁₃₁	1	1	yes	characteristic, complemented
233	#1916	C₂×C₁₃₁	1	2	yes	complemented
234	#1850	C₂×C₁₃₁	1	2	yes	—
235	#1798	C₂×C₁₃₁	1	1	yes	characteristic
236	#1770	C₂×C₁₃₁	1	2	yes	complemented
237	C₈×C₄⋊C₄	C₂×C₁₃₁	1	1	yes	characteristic, complemented
238	#1753	C₂×C₁₃₁	1	2	yes	—
239	C₂²×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
240	C₂³×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
241	C₂×C₄×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
242	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
243	#446	—	2	4	no	—
244	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
245	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
246	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
247	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
248	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	2	yes	—
249	#405	—	2	4	no	—
250	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
251	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
252	C₂×#97	C₂²×C₁₃₁	1	2	yes	—
253	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
254	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
255	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
256	#400	C₄×C₁₃₁	1	1	yes	characteristic
257	#399	C₄×C₁₃₁	1	1	yes	characteristic
258	#398	C₄×C₁₃₁	1	2	yes	—
259	#396	C₄×C₁₃₁	1	2	yes	—
260	#395	C₄×C₁₃₁	1	1	yes	characteristic
261	#393	C₄×C₁₃₁	1	1	yes	characteristic
262	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
263	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
264	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	2	yes	—
265	#336	C₂²×C₁₃₁	1	2	yes	—
266	#336	C₂²×C₁₃₁	1	2	yes	—
267	#336	C₂²×C₁₃₁	1	2	yes	—
268	#336	C₂²×C₁₃₁	1	2	yes	—
269	#336	C₂²×C₁₃₁	1	2	yes	—
270	C₂²×Q₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
271	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
272	C₂×M₁₆	—	2	2	no	—
273	C₂×M₁₆	—	2	2	no	—
274	C₂×M₁₆	Q₈×C₁₃₁	1	4	yes	complemented
275	C₂²×C₈	—	2	2	no	—
276	C₂²×C₈	—	2	2	no	—
277	C₂²×C₈	—	2	2	no	—
278	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
279	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
280	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
281	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
282	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
283	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
284	C₄×Q₈	—	2	2	no	—
285	C₄×Q₈	—	2	2	no	—
286	#117	—	2	2	no	—
287	#117	—	2	2	no	—
288	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
289	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
291	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
292	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
293	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
294	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
295	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
296	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
297	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
298	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
299	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
300	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
301	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
302	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
303	C₄⋊C₈	—	2	4	no	—
304	C₄⋊C₈	—	2	4	no	—
305	C₄⋊C₈	—	2	2	no	—
306	C₄⋊C₈	—	2	2	no	—
307	C₂²⋊C₈	—	2	4	no	—
308	#97	—	2	2	no	—
309	#97	—	2	4	no	—
310	#97	—	2	2	no	—
311	C₄×C₈	—	2	2	no	—
312	C₄×C₈	—	2	4	no	—
313	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
314	#95	C₂×C₄×C₁₃₁	1	2	yes	—
315	#95	C₂×C₄×C₁₃₁	1	2	yes	—
316	#95	C₂×C₄×C₁₃₁	1	2	yes	—
317	#95	C₂×C₄×C₁₃₁	1	1	yes	characteristic
318	#95	C₂×C₄×C₁₃₁	1	1	yes	characteristic
319	#95	C₂×C₄×C₁₃₁	1	2	yes	—
320	C₂⁴	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
321	C₂×Q₈	—	2	2	no	—
322	C₂×Q₈	—	4	2	no	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
328	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
329	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
330	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
331	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
332	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
333	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
334	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
335	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
336	C₂²×C₄	C₂×Q₈×C₁₃₁	1	2	yes	—
337	C₂²×C₄	#41×C₁₃₁	1	2	yes	—
338	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
339	M₁₆	—	2	8	no	—
340	C₂×C₈	—	2	8	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	1	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
354	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
355	C₂×C₈	C₂×Q₈×C₁₃₁	1	2	yes	—
356	C₂×C₈	C₂×Q₈×C₁₃₁	1	2	yes	—
357	C₂×C₈	#41×C₁₃₁	1	2	yes	—
358	C₂×C₈	#41×C₁₃₁	1	2	yes	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	4	no	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	4	no	—
365	C₄⋊C₄	—	4	2	no	—
366	C₂²⋊C₄	—	2	2	no	—
367	C₂²⋊C₄	—	2	2	no	—
368	C₂²⋊C₄	—	2	4	no	—
369	C₂²⋊C₄	—	4	2	no	—
370	C₄²	—	2	1	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	2	2	no	—
373	C₄²	—	2	2	no	—
374	C₄²	—	2	1	no	—
375	C₄²	—	2	2	no	—
376	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
377	C₄²	#41×C₁₃₁	1	2	yes	—
378	C₂³	—	2	1	no	—
379	C₂³	—	2	1	no	—
380	C₂³	—	2	2	no	—
381	C₂³	—	2	2	no	—
382	C₂³	C₄×Q₈×C₁₃₁	1	2	yes	—
383	C₂³	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
384	Q₈	—	4	4	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
409	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
410	C₂×C₄	C₄×Q₈×C₁₃₁	1	2	yes	—
411	C₂×C₄	#131×C₁₃₁	1	4	yes	—
412	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
413	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
415	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
416	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
417	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
418	C₈	—	2	2	no	—
419	C₈	—	2	2	no	—
420	C₈	—	2	2	no	—
421	C₈	—	4	2	no	—
422	C₈	—	4	2	no	—
423	C₈	C₂²⋊Q₈×C₁₃₁	1	4	yes	—
424	C₂²	—	2	2	no	—
425	C₂²	—	2	4	no	—
426	C₂²	—	4	1	no	—
427	C₂²	—	4	1	no	—
428	C₂²	—	4	2	no	—
429	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
430	C₂²	C₂×C₄×Q₈×C₁₃₁	1	1	yes	characteristic
431	C₂²	C₄×#41×C₁₃₁	1	1	yes	characteristic
432	C₂²	C₂×#131×C₁₃₁	1	2	yes	—
433	C₂²	#568×C₁₃₁	1	2	yes	—
434	C₄	—	2	2	no	—
435	C₄	—	2	2	no	—
436	C₄	—	2	2	no	—
437	C₄	—	2	4	no	—
438	C₄	—	4	2	no	—
439	C₄	—	4	1	no	—
440	C₄	—	4	1	no	—
441	C₄	—	4	2	no	—
442	C₄	—	4	2	no	—
443	C₄	C₂×C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
444	C₄	#525×C₁₃₁	1	1	yes	characteristic
445	C₄	#529×C₁₃₁	1	1	yes	characteristic
446	C₄	#533×C₁₃₁	1	1	yes	characteristic
447	C₂	—	4	2	no	—
448	C₂	C₄×C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
449	C₂	#2930×C₁₃₁	1	1	yes	characteristic
450	C₂	#2931×C₁₃₁	1	1	yes	characteristic
451	C₂	#2962×C₁₃₁	1	2	yes	—
452	C₂	#2968×C₁₃₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#446×C₁₃₁	C₂	semidirect
4. #1985×C₁₃₁	C₂	semidirect
5. #1940×C₁₃₁	C₂	semidirect
6. #1916×C₁₃₁	C₂	semidirect
9. #1770×C₁₃₁	C₂	semidirect
10. C₈×C₄⋊C₄×C₁₃₁	C₂	semidirect
47. C₂×M₁₆×C₁₃₁	Q₈	semidirect
226. #21604	C₁₃₁	direct (N × H)
227. C₁₃₁	#21604	direct (N × H)
228. C₂×#446	C₂×C₁₃₁	semidirect
231. #1985	C₂×C₁₃₁	semidirect
232. #1940	C₂×C₁₃₁	semidirect
233. #1916	C₂×C₁₃₁	semidirect
236. #1770	C₂×C₁₃₁	semidirect
237. C₈×C₄⋊C₄	C₂×C₁₃₁	semidirect
274. C₂×M₁₆	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×#446×C₁₃₁	2
C₂×#405×C₁₃₁	2
#2396×C₁₃₁	2
#1985×C₁₃₁	2
#1940×C₁₃₁	2
#1916×C₁₃₁	2
#1850×C₁₃₁	2
#1798×C₁₃₁	2
#1770×C₁₃₁	2
C₈×C₄⋊C₄×C₁₃₁	2
#1753×C₁₃₁	2
#21604	131

Subgroups — C131×#21604 #45707488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#21604 #45707488