Subgroups — C₁₅₁×#22783 #50327708

Order 38656.

752 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#135×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#2925×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2906×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#1937×C₁₅₁	C₂	1	2	yes	maximal, complemented
5	#1922×C₁₅₁	C₂	1	2	yes	maximal, complemented
6	#1876×C₁₅₁	C₂	1	2	yes	maximal
7	#1837×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#1806×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#1806×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×C₄⋊C₄×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#1764×C₁₅₁	C₂	1	2	yes	maximal, complemented
12	C₂×#135×C₁₅₁	C₂²	1	1	yes	characteristic
13	C₂×#131×C₁₅₁	C₂²	1	2	yes	—
14	C₄×#41×C₁₅₁	C₂²	1	1	yes	characteristic
15	C₄×#41×C₁₅₁	C₂²	1	1	yes	characteristic
16	C₄×Q₁₆×C₁₅₁	C₄	1	1	yes	characteristic, complemented
17	C₄×Q₁₆×C₁₅₁	C₄	1	1	yes	characteristic, complemented
18	C₄×SD₁₆×C₁₅₁	C₄	1	1	yes	characteristic, complemented
19	C₄×SD₁₆×C₁₅₁	C₄	1	1	yes	characteristic, complemented
20	C₄×SD₁₆×C₁₅₁	C₄	1	1	yes	characteristic, complemented
21	C₄×SD₁₆×C₁₅₁	C₄	1	1	yes	characteristic, complemented
22	C₄×D₈×C₁₅₁	C₄	1	1	yes	characteristic, complemented
23	C₄×D₈×C₁₅₁	C₄	1	1	yes	characteristic, complemented
24	#433×C₁₅₁	—	2	2	no	—
25	#432×C₁₅₁	—	2	1	no	—
26	#431×C₁₅₁	—	2	1	no	—
27	#427×C₁₅₁	C₂²	1	1	yes	characteristic
28	#424×C₁₅₁	—	2	2	no	—
29	#424×C₁₅₁	C₂²	1	2	yes	—
30	#423×C₁₅₁	—	2	1	no	—
31	#423×C₁₅₁	C₂²	1	1	yes	characteristic
32	#423×C₁₅₁	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈×C₁₅₁	—	2	1	no	—
34	C₂×C₄⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
35	C₂×#104×C₁₅₁	C₂²	1	2	yes	—
36	C₂×#104×C₁₅₁	C₂²	1	2	yes	—
37	#416×C₁₅₁	C₂²	1	1	yes	characteristic
38	#416×C₁₅₁	C₂²	1	1	yes	characteristic
39	#408×C₁₅₁	C₂²	1	2	yes	—
40	#405×C₁₅₁	C₂²	1	2	yes	—
41	C₂×C₄×C₈×C₁₅₁	C₂²	1	2	yes	—
42	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄×C₁₅₁	C₂²	1	1	yes	characteristic
44	#341×C₁₅₁	C₂²	1	2	yes	—
45	#341×C₁₅₁	C₂²	1	2	yes	—
46	#339×C₁₅₁	C₂²	1	2	yes	—
47	#339×C₁₅₁	C₂²	1	2	yes	—
48	#336×C₁₅₁	C₂²	1	2	yes	—
49	C₂×#41×C₁₅₁	C₂³	1	1	yes	characteristic
50	C₂×#41×C₁₅₁	C₂³	1	1	yes	characteristic
51	#135×C₁₅₁	—	2	2	no	—
52	#135×C₁₅₁	—	2	2	no	—
53	C₂×Q₁₆×C₁₅₁	C₂×C₄	1	1	yes	characteristic
54	C₂×SD₁₆×C₁₅₁	C₂×C₄	1	1	yes	characteristic
55	C₂×SD₁₆×C₁₅₁	C₂×C₄	1	1	yes	characteristic
56	C₂×D₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
57	#131×C₁₅₁	—	2	4	no	—
58	#131×C₁₅₁	—	4	2	no	—
59	C₂×M₁₆×C₁₅₁	—	2	2	no	—
60	C₂×M₁₆×C₁₅₁	C₂³	1	2	yes	—
61	C₂×M₁₆×C₁₅₁	C₂³	1	2	yes	—
62	C₂²×C₈×C₁₅₁	—	2	2	no	—
63	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₁₅₁	C₂³	1	2	yes	—
66	C₄×Q₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
67	C₄×Q₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
68	C₄×Q₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
69	C₄×Q₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
70	C₄×D₄×C₁₅₁	—	2	1	no	—
71	C₄×D₄×C₁₅₁	—	2	1	no	—
72	C₄×D₄×C₁₅₁	—	2	1	no	—
73	C₄×D₄×C₁₅₁	—	2	1	no	—
74	C₄×D₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
75	C₄×D₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
76	C₄×D₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
77	C₄×D₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
78	#117×C₁₅₁	—	2	1	no	—
79	#117×C₁₅₁	—	2	1	no	—
80	#117×C₁₅₁	C₂³	1	1	yes	characteristic
81	#117×C₁₅₁	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₁₅₁	—	2	1	no	—
83	C₂×C₄²×C₁₅₁	—	2	1	no	—
84	C₂×C₄²×C₁₅₁	—	2	1	no	—
85	C₂×C₄²×C₁₅₁	C₂³	1	2	yes	—
86	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
87	#107×C₁₅₁	C₂×C₄	1	1	yes	characteristic
88	#107×C₁₅₁	C₂×C₄	1	1	yes	characteristic
89	#106×C₁₅₁	C₂×C₄	1	1	yes	characteristic
90	#106×C₁₅₁	C₂×C₄	1	1	yes	characteristic
91	C₄⋊C₈×C₁₅₁	—	2	2	no	—
92	C₄⋊C₈×C₁₅₁	—	2	2	no	—
93	C₄⋊C₈×C₁₅₁	—	2	1	no	—
94	C₄⋊C₈×C₁₅₁	—	2	1	no	—
95	C₄⋊C₈×C₁₅₁	—	2	1	no	—
96	C₄⋊C₈×C₁₅₁	—	2	1	no	—
97	C₄⋊C₈×C₁₅₁	D₄	1	1	yes	characteristic
98	C₄⋊C₈×C₁₅₁	D₄	1	1	yes	characteristic
99	C₄⋊C₈×C₁₅₁	D₄	1	2	yes	—
100	#104×C₁₅₁	—	2	4	no	—
101	#104×C₁₅₁	—	4	2	no	—
102	#103×C₁₅₁	C₂×C₄	1	1	yes	characteristic
103	#103×C₁₅₁	C₂×C₄	1	1	yes	characteristic
104	#103×C₁₅₁	C₂×C₄	1	1	yes	characteristic
105	#103×C₁₅₁	C₂×C₄	1	1	yes	characteristic
106	#102×C₁₅₁	C₂×C₄	1	1	yes	characteristic
107	#102×C₁₅₁	C₂×C₄	1	1	yes	characteristic
108	#102×C₁₅₁	C₂×C₄	1	1	yes	characteristic
109	#102×C₁₅₁	C₂×C₄	1	1	yes	characteristic
110	C₂²⋊C₈×C₁₅₁	—	2	2	no	—
111	C₂²⋊C₈×C₁₅₁	—	2	2	no	—
112	#97×C₁₅₁	—	2	2	no	—
113	C₄×C₈×C₁₅₁	—	2	2	no	—
114	C₄×C₈×C₁₅₁	—	4	2	no	—
115	C₄×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
116	C₄×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
117	C₄×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
118	C₄×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
119	#95×C₁₅₁	—	2	1	no	—
120	#41×C₁₅₁	—	2	1	no	—
121	#41×C₁₅₁	—	2	1	no	—
122	#41×C₁₅₁	—	2	1	no	—
123	#41×C₁₅₁	—	2	1	no	—
124	#41×C₁₅₁	—	2	1	no	—
125	#41×C₁₅₁	—	2	1	no	—
126	#41×C₁₅₁	#41	1	2	yes	—
127	#41×C₁₅₁	#41	1	2	yes	—
128	C₂×Q₈×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
129	C₂×Q₈×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
130	C₂×D₄×C₁₅₁	—	2	1	no	—
131	C₂×D₄×C₁₅₁	—	2	1	no	—
132	C₂×D₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
133	C₂×D₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₁₅₁	—	2	1	no	—
135	C₂²×C₄×C₁₅₁	—	2	2	no	—
136	C₂²×C₄×C₁₅₁	—	2	1	no	—
137	C₂²×C₄×C₁₅₁	—	2	1	no	—
138	C₂²×C₄×C₁₅₁	—	2	1	no	—
139	C₂²×C₄×C₁₅₁	—	2	1	no	—
140	C₂²×C₄×C₁₅₁	—	2	1	no	—
141	C₂²×C₄×C₁₅₁	#41	1	1	yes	characteristic
142	C₂²×C₄×C₁₅₁	#41	1	1	yes	characteristic
143	C₂²×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
144	Q₁₆×C₁₅₁	—	2	2	no	—
145	SD₁₆×C₁₅₁	—	2	2	no	—
146	SD₁₆×C₁₅₁	—	2	2	no	—
147	D₈×C₁₅₁	—	2	2	no	—
148	M₁₆×C₁₅₁	—	2	2	no	—
149	M₁₆×C₁₅₁	—	2	2	no	—
150	M₁₆×C₁₅₁	—	2	2	no	—
151	M₁₆×C₁₅₁	—	2	2	no	—
152	M₁₆×C₁₅₁	—	4	4	no	—
153	C₂×C₈×C₁₅₁	—	2	2	no	—
154	C₂×C₈×C₁₅₁	—	2	1	no	—
155	C₂×C₈×C₁₅₁	—	2	1	no	—
156	C₂×C₈×C₁₅₁	—	2	1	no	—
157	C₂×C₈×C₁₅₁	—	2	2	no	—
158	C₂×C₈×C₁₅₁	—	2	1	no	—
159	C₂×C₈×C₁₅₁	—	2	2	no	—
160	C₂×C₈×C₁₅₁	—	2	2	no	—
161	C₂×C₈×C₁₅₁	—	4	4	no	—
162	C₂×C₈×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
163	C₂×C₈×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
164	C₂×C₈×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
165	C₂×C₈×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
166	C₂×C₈×C₁₅₁	C₂×D₄	1	2	yes	—
167	C₂×C₈×C₁₅₁	C₂×D₄	1	2	yes	—
168	C₄⋊C₄×C₁₅₁	—	2	4	no	—
169	C₄⋊C₄×C₁₅₁	—	2	1	no	—
170	C₄⋊C₄×C₁₅₁	—	2	1	no	—
171	C₄⋊C₄×C₁₅₁	—	2	1	no	—
172	C₄⋊C₄×C₁₅₁	—	2	1	no	—
173	C₄⋊C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
174	C₄⋊C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
175	C₄⋊C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
176	C₄⋊C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
177	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
178	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
179	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
180	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
181	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
182	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
183	C₄²×C₁₅₁	—	2	1	no	—
184	C₄²×C₁₅₁	—	2	1	no	—
185	C₄²×C₁₅₁	—	2	1	no	—
186	C₄²×C₁₅₁	—	2	1	no	—
187	C₄²×C₁₅₁	—	2	1	no	—
188	C₄²×C₁₅₁	—	2	1	no	—
189	C₄²×C₁₅₁	—	2	2	no	—
190	C₄²×C₁₅₁	—	2	2	no	—
191	C₄²×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
192	C₄²×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
193	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
195	C₂³×C₁₅₁	—	2	1	no	—
196	C₂³×C₁₅₁	—	2	1	no	—
197	C₂³×C₁₅₁	C₂×#41	1	1	yes	characteristic
198	Q₈×C₁₅₁	—	2	1	no	—
199	Q₈×C₁₅₁	—	2	1	no	—
200	Q₈×C₁₅₁	—	2	1	no	—
201	Q₈×C₁₅₁	#117	1	2	yes	—
202	D₄×C₁₅₁	—	2	2	no	—
203	D₄×C₁₅₁	—	2	2	no	—
204	D₄×C₁₅₁	—	2	1	no	—
205	D₄×C₁₅₁	—	2	1	no	—
206	D₄×C₁₅₁	—	2	1	no	—
207	D₄×C₁₅₁	—	4	1	no	—
208	D₄×C₁₅₁	—	4	1	no	—
209	D₄×C₁₅₁	#117	1	2	yes	—
210	C₂×C₄×C₁₅₁	—	2	1	no	—
211	C₂×C₄×C₁₅₁	—	2	2	no	—
212	C₂×C₄×C₁₅₁	—	2	2	no	—
213	C₂×C₄×C₁₅₁	—	2	2	no	—
214	C₂×C₄×C₁₅₁	—	2	4	no	—
215	C₂×C₄×C₁₅₁	—	2	1	no	—
216	C₂×C₄×C₁₅₁	—	2	1	no	—
217	C₂×C₄×C₁₅₁	—	2	1	no	—
218	C₂×C₄×C₁₅₁	—	2	1	no	—
219	C₂×C₄×C₁₅₁	—	2	1	no	—
220	C₂×C₄×C₁₅₁	—	4	1	no	—
221	C₂×C₄×C₁₅₁	—	4	2	no	—
222	C₂×C₄×C₁₅₁	—	4	2	no	—
223	C₂×C₄×C₁₅₁	—	4	1	no	—
224	C₂×C₄×C₁₅₁	—	4	1	no	—
225	C₂×C₄×C₁₅₁	—	4	2	no	—
226	C₂×C₄×C₁₅₁	—	4	2	no	—
227	C₂×C₄×C₁₅₁	—	8	1	no	—
228	C₂×C₄×C₁₅₁	—	8	1	no	—
229	C₂×C₄×C₁₅₁	C₄×D₄	1	1	yes	characteristic
230	C₂×C₄×C₁₅₁	C₄×D₄	1	1	yes	characteristic
231	C₂×C₄×C₁₅₁	C₄×D₄	1	1	yes	characteristic
232	C₂×C₄×C₁₅₁	C₄×D₄	1	1	yes	characteristic
233	C₂×C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
234	C₂×C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
236	C₂×C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
237	C₂×C₄×C₁₅₁	H(5,𝔽₂)	1	1	yes	characteristic
238	C₂×C₄×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
239	C₈×C₁₅₁	—	2	1	no	—
240	C₈×C₁₅₁	—	2	1	no	—
241	C₈×C₁₅₁	—	2	1	no	—
242	C₈×C₁₅₁	—	2	1	no	—
243	C₈×C₁₅₁	—	2	2	no	—
244	C₈×C₁₅₁	—	2	2	no	—
245	C₈×C₁₅₁	—	4	2	no	—
246	C₈×C₁₅₁	—	4	2	no	—
247	C₂²×C₁₅₁	—	2	1	no	—
248	C₂²×C₁₅₁	—	2	1	no	—
249	C₂²×C₁₅₁	—	2	2	no	—
250	C₂²×C₁₅₁	—	4	1	no	—
251	C₂²×C₁₅₁	—	4	1	no	—
252	C₂²×C₁₅₁	—	4	1	no	—
253	C₂²×C₁₅₁	—	4	1	no	—
254	C₂²×C₁₅₁	—	4	1	no	—
255	C₂²×C₁₅₁	C₂×C₄×D₄	1	1	yes	characteristic
256	C₂²×C₁₅₁	#552	1	1	yes	characteristic
257	C₂²×C₁₅₁	#553	1	1	yes	characteristic
258	C₄×C₁₅₁	—	2	2	no	—
259	C₄×C₁₅₁	—	2	1	no	—
260	C₄×C₁₅₁	—	2	1	no	—
261	C₄×C₁₅₁	—	2	1	no	—
262	C₄×C₁₅₁	—	4	4	no	—
263	C₄×C₁₅₁	—	4	1	no	—
264	C₄×C₁₅₁	—	4	1	no	—
265	C₄×C₁₅₁	—	4	1	no	—
266	C₄×C₁₅₁	—	4	1	no	—
267	C₄×C₁₅₁	—	4	1	no	—
268	C₄×C₁₅₁	#443	1	2	yes	—
269	C₄×C₁₅₁	C₂×#117	1	1	yes	characteristic
270	C₄×C₁₅₁	#520	1	1	yes	characteristic
271	C₄×C₁₅₁	#546	1	1	yes	characteristic
272	C₄×C₁₅₁	#548	1	1	yes	characteristic
273	C₂×C₁₅₁	—	2	1	no	—
274	C₂×C₁₅₁	—	2	1	no	—
275	C₂×C₁₅₁	—	4	2	no	—
276	C₂×C₁₅₁	—	8	1	no	—
277	C₂×C₁₅₁	#2352	1	1	yes	characteristic
278	C₂×C₁₅₁	C₂×#443	1	1	yes	characteristic
279	C₂×C₁₅₁	#2957	1	1	yes	characteristic
280	#22783	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
281	C₁₅₁	#22783	1	1	yes	characteristic, direct product factor, Sylow
282	C₄×#135	C₂×C₁₅₁	1	1	yes	characteristic
283	#2925	C₂×C₁₅₁	1	1	yes	characteristic, complemented
284	#2906	C₂×C₁₅₁	1	1	yes	characteristic, complemented
285	#1937	C₂×C₁₅₁	1	2	yes	complemented
286	#1922	C₂×C₁₅₁	1	2	yes	complemented
287	#1876	C₂×C₁₅₁	1	2	yes	—
288	#1837	C₂×C₁₅₁	1	1	yes	characteristic, complemented
289	#1806	C₂×C₁₅₁	1	1	yes	characteristic, complemented
290	#1806	C₂×C₁₅₁	1	1	yes	characteristic, complemented
291	C₈×C₄⋊C₄	C₂×C₁₅₁	1	1	yes	characteristic, complemented
292	#1764	C₂×C₁₅₁	1	2	yes	complemented
293	C₂×#135	C₂²×C₁₅₁	1	1	yes	characteristic
294	C₂×#131	C₂²×C₁₅₁	1	2	yes	—
295	C₄×#41	C₂²×C₁₅₁	1	1	yes	characteristic
296	C₄×#41	C₂²×C₁₅₁	1	1	yes	characteristic
297	C₄×Q₁₆	C₄×C₁₅₁	1	1	yes	characteristic, complemented
298	C₄×Q₁₆	C₄×C₁₅₁	1	1	yes	characteristic, complemented
299	C₄×SD₁₆	C₄×C₁₅₁	1	1	yes	characteristic, complemented
300	C₄×SD₁₆	C₄×C₁₅₁	1	1	yes	characteristic, complemented
301	C₄×SD₁₆	C₄×C₁₅₁	1	1	yes	characteristic, complemented
302	C₄×SD₁₆	C₄×C₁₅₁	1	1	yes	characteristic, complemented
303	C₄×D₈	C₄×C₁₅₁	1	1	yes	characteristic, complemented
304	C₄×D₈	C₄×C₁₅₁	1	1	yes	characteristic, complemented
305	#433	—	2	2	no	—
306	#432	—	2	1	no	—
307	#431	—	2	1	no	—
308	#427	C₂²×C₁₅₁	1	1	yes	characteristic
309	#424	—	2	2	no	—
310	#424	C₂²×C₁₅₁	1	2	yes	—
311	#423	—	2	1	no	—
312	#423	C₂²×C₁₅₁	1	1	yes	characteristic
313	#423	C₂²×C₁₅₁	1	1	yes	characteristic
314	C₂×C₄⋊C₈	—	2	1	no	—
315	C₂×C₄⋊C₈	C₂²×C₁₅₁	1	1	yes	characteristic
316	C₂×#104	C₂²×C₁₅₁	1	2	yes	—
317	C₂×#104	C₂²×C₁₅₁	1	2	yes	—
318	#416	C₂²×C₁₅₁	1	1	yes	characteristic
319	#416	C₂²×C₁₅₁	1	1	yes	characteristic
320	#408	C₂²×C₁₅₁	1	2	yes	—
321	#405	C₂²×C₁₅₁	1	2	yes	—
322	C₂×C₄×C₈	C₂²×C₁₅₁	1	2	yes	—
323	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
324	C₄×C₄⋊C₄	C₂²×C₁₅₁	1	1	yes	characteristic
325	#341	C₂²×C₁₅₁	1	2	yes	—
326	#341	C₂²×C₁₅₁	1	2	yes	—
327	#339	C₂²×C₁₅₁	1	2	yes	—
328	#339	C₂²×C₁₅₁	1	2	yes	—
329	#336	C₂²×C₁₅₁	1	2	yes	—
330	C₂×#41	C₂³×C₁₅₁	1	1	yes	characteristic
331	C₂×#41	C₂³×C₁₅₁	1	1	yes	characteristic
332	#135	—	2	2	no	—
333	#135	—	2	2	no	—
334	C₂×Q₁₆	C₂×C₄×C₁₅₁	1	1	yes	characteristic
335	C₂×SD₁₆	C₂×C₄×C₁₅₁	1	1	yes	characteristic
336	C₂×SD₁₆	C₂×C₄×C₁₅₁	1	1	yes	characteristic
337	C₂×D₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
338	#131	—	2	4	no	—
339	#131	—	4	2	no	—
340	C₂×M₁₆	—	2	2	no	—
341	C₂×M₁₆	C₂³×C₁₅₁	1	2	yes	—
342	C₂×M₁₆	C₂³×C₁₅₁	1	2	yes	—
343	C₂²×C₈	—	2	2	no	—
344	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
346	C₂²×C₈	C₂³×C₁₅₁	1	2	yes	—
347	C₄×Q₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
348	C₄×Q₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
349	C₄×Q₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
350	C₄×Q₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
351	C₄×D₄	—	2	1	no	—
352	C₄×D₄	—	2	1	no	—
353	C₄×D₄	—	2	1	no	—
354	C₄×D₄	—	2	1	no	—
355	C₄×D₄	C₂×C₄×C₁₅₁	1	1	yes	characteristic
356	C₄×D₄	C₂×C₄×C₁₅₁	1	1	yes	characteristic
357	C₄×D₄	C₂×C₄×C₁₅₁	1	1	yes	characteristic
358	C₄×D₄	C₂×C₄×C₁₅₁	1	1	yes	characteristic
359	#117	—	2	1	no	—
360	#117	—	2	1	no	—
361	#117	C₂³×C₁₅₁	1	1	yes	characteristic
362	#117	C₂³×C₁₅₁	1	1	yes	characteristic
363	C₂×C₄⋊C₄	—	2	1	no	—
364	C₂×C₄²	—	2	1	no	—
365	C₂×C₄²	—	2	1	no	—
366	C₂×C₄²	C₂³×C₁₅₁	1	2	yes	—
367	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
368	#107	C₂×C₄×C₁₅₁	1	1	yes	characteristic
369	#107	C₂×C₄×C₁₅₁	1	1	yes	characteristic
370	#106	C₂×C₄×C₁₅₁	1	1	yes	characteristic
371	#106	C₂×C₄×C₁₅₁	1	1	yes	characteristic
372	C₄⋊C₈	—	2	2	no	—
373	C₄⋊C₈	—	2	2	no	—
374	C₄⋊C₈	—	2	1	no	—
375	C₄⋊C₈	—	2	1	no	—
376	C₄⋊C₈	—	2	1	no	—
377	C₄⋊C₈	—	2	1	no	—
378	C₄⋊C₈	D₄×C₁₅₁	1	1	yes	characteristic
379	C₄⋊C₈	D₄×C₁₅₁	1	1	yes	characteristic
380	C₄⋊C₈	D₄×C₁₅₁	1	2	yes	—
381	#104	—	2	4	no	—
382	#104	—	4	2	no	—
383	#103	C₂×C₄×C₁₅₁	1	1	yes	characteristic
384	#103	C₂×C₄×C₁₅₁	1	1	yes	characteristic
385	#103	C₂×C₄×C₁₅₁	1	1	yes	characteristic
386	#103	C₂×C₄×C₁₅₁	1	1	yes	characteristic
387	#102	C₂×C₄×C₁₅₁	1	1	yes	characteristic
388	#102	C₂×C₄×C₁₅₁	1	1	yes	characteristic
389	#102	C₂×C₄×C₁₅₁	1	1	yes	characteristic
390	#102	C₂×C₄×C₁₅₁	1	1	yes	characteristic
391	C₂²⋊C₈	—	2	2	no	—
392	C₂²⋊C₈	—	2	2	no	—
393	#97	—	2	2	no	—
394	C₄×C₈	—	2	2	no	—
395	C₄×C₈	—	4	2	no	—
396	C₄×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
397	C₄×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
398	C₄×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
399	C₄×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
400	#95	—	2	1	no	—
401	#41	—	2	1	no	—
402	#41	—	2	1	no	—
403	#41	—	2	1	no	—
404	#41	—	2	1	no	—
405	#41	—	2	1	no	—
406	#41	—	2	1	no	—
407	#41	#41×C₁₅₁	1	2	yes	—
408	#41	#41×C₁₅₁	1	2	yes	—
409	C₂×Q₈	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
410	C₂×Q₈	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
411	C₂×D₄	—	2	1	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
414	C₂×D₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	#41×C₁₅₁	1	1	yes	characteristic
423	C₂²×C₄	#41×C₁₅₁	1	1	yes	characteristic
424	C₂²×C₄	C₂⁴×C₁₅₁	1	1	yes	characteristic
425	Q₁₆	—	2	2	no	—
426	SD₁₆	—	2	2	no	—
427	SD₁₆	—	2	2	no	—
428	D₈	—	2	2	no	—
429	M₁₆	—	2	2	no	—
430	M₁₆	—	2	2	no	—
431	M₁₆	—	2	2	no	—
432	M₁₆	—	2	2	no	—
433	M₁₆	—	4	4	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	1	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	2	no	—
442	C₂×C₈	—	4	4	no	—
443	C₂×C₈	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
444	C₂×C₈	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
445	C₂×C₈	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
446	C₂×C₈	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
447	C₂×C₈	C₂×D₄×C₁₅₁	1	2	yes	—
448	C₂×C₈	C₂×D₄×C₁₅₁	1	2	yes	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
455	C₄⋊C₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
456	C₄⋊C₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
457	C₄⋊C₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
458	C₂²⋊C₄	—	2	1	no	—
459	C₂²⋊C₄	—	2	1	no	—
460	C₂²⋊C₄	—	2	1	no	—
461	C₂²⋊C₄	—	2	1	no	—
462	C₂²⋊C₄	—	2	1	no	—
463	C₂²⋊C₄	—	2	1	no	—
464	C₄²	—	2	1	no	—
465	C₄²	—	2	1	no	—
466	C₄²	—	2	1	no	—
467	C₄²	—	2	1	no	—
468	C₄²	—	2	1	no	—
469	C₄²	—	2	1	no	—
470	C₄²	—	2	2	no	—
471	C₄²	—	2	2	no	—
472	C₄²	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
473	C₄²	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
474	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
475	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
476	C₂³	—	2	1	no	—
477	C₂³	—	2	1	no	—
478	C₂³	C₂×#41×C₁₅₁	1	1	yes	characteristic
479	Q₈	—	2	1	no	—
480	Q₈	—	2	1	no	—
481	Q₈	—	2	1	no	—
482	Q₈	#117×C₁₅₁	1	2	yes	—
483	D₄	—	2	2	no	—
484	D₄	—	2	2	no	—
485	D₄	—	2	1	no	—
486	D₄	—	2	1	no	—
487	D₄	—	2	1	no	—
488	D₄	—	4	1	no	—
489	D₄	—	4	1	no	—
490	D₄	#117×C₁₅₁	1	2	yes	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	1	no	—
507	C₂×C₄	—	4	1	no	—
508	C₂×C₄	—	8	1	no	—
509	C₂×C₄	—	8	1	no	—
510	C₂×C₄	C₄×D₄×C₁₅₁	1	1	yes	characteristic
511	C₂×C₄	C₄×D₄×C₁₅₁	1	1	yes	characteristic
512	C₂×C₄	C₄×D₄×C₁₅₁	1	1	yes	characteristic
513	C₂×C₄	C₄×D₄×C₁₅₁	1	1	yes	characteristic
514	C₂×C₄	C₂³×C₄×C₁₅₁	1	1	yes	characteristic
515	C₂×C₄	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
516	C₂×C₄	C₂×#41×C₁₅₁	1	1	yes	characteristic
517	C₂×C₄	C₂×#41×C₁₅₁	1	1	yes	characteristic
518	C₂×C₄	H(5,𝔽₂)×C₁₅₁	1	1	yes	characteristic
519	C₂×C₄	2₋¹⁺⁴×C₁₅₁	1	1	yes	characteristic
520	C₈	—	2	1	no	—
521	C₈	—	2	1	no	—
522	C₈	—	2	1	no	—
523	C₈	—	2	1	no	—
524	C₈	—	2	2	no	—
525	C₈	—	2	2	no	—
526	C₈	—	4	2	no	—
527	C₈	—	4	2	no	—
528	C₂²	—	2	1	no	—
529	C₂²	—	2	1	no	—
530	C₂²	—	2	2	no	—
531	C₂²	—	4	1	no	—
532	C₂²	—	4	1	no	—
533	C₂²	—	4	1	no	—
534	C₂²	—	4	1	no	—
535	C₂²	—	4	1	no	—
536	C₂²	C₂×C₄×D₄×C₁₅₁	1	1	yes	characteristic
537	C₂²	#552×C₁₅₁	1	1	yes	characteristic
538	C₂²	#553×C₁₅₁	1	1	yes	characteristic
539	C₄	—	2	2	no	—
540	C₄	—	2	1	no	—
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	4	4	no	—
544	C₄	—	4	1	no	—
545	C₄	—	4	1	no	—
546	C₄	—	4	1	no	—
547	C₄	—	4	1	no	—
548	C₄	—	4	1	no	—
549	C₄	#443×C₁₅₁	1	2	yes	—
550	C₄	C₂×#117×C₁₅₁	1	1	yes	characteristic
551	C₄	#520×C₁₅₁	1	1	yes	characteristic
552	C₄	#546×C₁₅₁	1	1	yes	characteristic
553	C₄	#548×C₁₅₁	1	1	yes	characteristic
554	C₂	—	2	1	no	—
555	C₂	—	2	1	no	—
556	C₂	—	4	2	no	—
557	C₂	—	8	1	no	—
558	C₂	#2352×C₁₅₁	1	1	yes	characteristic
559	C₂	C₂×#443×C₁₅₁	1	1	yes	characteristic
560	C₂	#2957×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₅₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₅₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₅₁
socle	C₂²×C₁₅₁	C₂×C₄×D₄
prime radical	C₂×#135×C₁₅₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₅₁ ◁ C₂×C₄⋊C₈×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2925×C₁₅₁	C₂	semidirect
3. #2906×C₁₅₁	C₂	semidirect
4. #1937×C₁₅₁	C₂	semidirect
5. #1922×C₁₅₁	C₂	semidirect
7. #1837×C₁₅₁	C₂	semidirect
8. #1806×C₁₅₁	C₂	semidirect
9. #1806×C₁₅₁	C₂	semidirect
10. C₈×C₄⋊C₄×C₁₅₁	C₂	semidirect
11. #1764×C₁₅₁	C₂	semidirect
16. C₄×Q₁₆×C₁₅₁	C₄	semidirect
17. C₄×Q₁₆×C₁₅₁	C₄	semidirect
18. C₄×SD₁₆×C₁₅₁	C₄	semidirect
19. C₄×SD₁₆×C₁₅₁	C₄	semidirect
20. C₄×SD₁₆×C₁₅₁	C₄	semidirect
21. C₄×SD₁₆×C₁₅₁	C₄	semidirect
22. C₄×D₈×C₁₅₁	C₄	semidirect
23. C₄×D₈×C₁₅₁	C₄	semidirect
280. #22783	C₁₅₁	direct (N × H)
281. C₁₅₁	#22783	direct (N × H)
283. #2925	C₂×C₁₅₁	semidirect
284. #2906	C₂×C₁₅₁	semidirect
285. #1937	C₂×C₁₅₁	semidirect
286. #1922	C₂×C₁₅₁	semidirect
288. #1837	C₂×C₁₅₁	semidirect
289. #1806	C₂×C₁₅₁	semidirect
290. #1806	C₂×C₁₅₁	semidirect
291. C₈×C₄⋊C₄	C₂×C₁₅₁	semidirect
292. #1764	C₂×C₁₅₁	semidirect
297. C₄×Q₁₆	C₄×C₁₅₁	semidirect
298. C₄×Q₁₆	C₄×C₁₅₁	semidirect
299. C₄×SD₁₆	C₄×C₁₅₁	semidirect
300. C₄×SD₁₆	C₄×C₁₅₁	semidirect
301. C₄×SD₁₆	C₄×C₁₅₁	semidirect
302. C₄×SD₁₆	C₄×C₁₅₁	semidirect
303. C₄×D₈	C₄×C₁₅₁	semidirect
304. C₄×D₈	C₄×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#135×C₁₅₁	2
#2925×C₁₅₁	2
#2906×C₁₅₁	2
#1937×C₁₅₁	2
#1922×C₁₅₁	2
#1876×C₁₅₁	2
#1837×C₁₅₁	2
#1806×C₁₅₁	2
C₈×C₄⋊C₄×C₁₅₁	2
#1764×C₁₅₁	2
#22783	151

Subgroups — C151×#22783 #50327708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#22783 #50327708