Subgroups — C₁₅₁×#13528 #50318453

Order 38656.

378 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#367×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#366×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#357×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#357×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	#1970×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	#1846×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#1374×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₅₁	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
11	C₂×#107×C₁₅₁	C₂²	1	1	yes	characteristic
12	C₂×#107×C₁₅₁	C₂²	1	1	yes	characteristic
13	C₂×#107×C₁₅₁	C₂²	1	1	yes	characteristic
14	C₂×#106×C₁₅₁	—	2	1	no	—
15	C₂×#102×C₁₅₁	—	2	1	no	—
16	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
17	#390×C₁₅₁	—	2	1	no	—
18	#382×C₁₅₁	—	2	1	no	—
19	#367×C₁₅₁	—	2	2	no	—
20	#366×C₁₅₁	—	2	2	no	—
21	#357×C₁₅₁	—	2	2	no	—
22	#357×C₁₅₁	—	2	2	no	—
23	C₂²×D₄×C₁₅₁	—	2	1	no	—
24	C₂×D₈×C₁₅₁	—	2	1	no	—
25	C₂×D₈×C₁₅₁	—	2	1	no	—
26	C₂×D₈×C₁₅₁	—	2	1	no	—
27	C₂×D₈×C₁₅₁	—	2	1	no	—
28	C₂×D₈×C₁₅₁	—	2	1	no	—
29	C₂×D₈×C₁₅₁	Q₈	1	2	yes	—
30	C₂²×C₈×C₁₅₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₁₅₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₅₁	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₅₁	—	2	1	no	—
35	C₂×C₂²⋊C₄×C₁₅₁	—	4	2	no	—
36	C₂×C₄²×C₁₅₁	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₁₅₁	—	2	1	no	—
38	C₂×C₁₆×C₁₅₁	—	2	1	no	—
39	C₂×C₁₆×C₁₅₁	—	2	1	no	—
40	C₂×C₁₆×C₁₅₁	—	2	1	no	—
41	C₂×C₁₆×C₁₅₁	D₄	1	2	yes	—
42	C₂×C₁₆×C₁₅₁	D₄	1	2	yes	—
43	#107×C₁₅₁	—	2	1	no	—
44	#107×C₁₅₁	—	2	1	no	—
45	#107×C₁₅₁	—	2	1	no	—
46	#107×C₁₅₁	—	2	1	no	—
47	#107×C₁₅₁	—	2	1	no	—
48	#107×C₁₅₁	—	2	1	no	—
49	#106×C₁₅₁	—	4	2	no	—
50	#102×C₁₅₁	—	4	2	no	—
51	C₄×C₈×C₁₅₁	—	2	1	no	—
52	C₄×C₈×C₁₅₁	—	2	1	no	—
53	#95×C₁₅₁	—	4	2	no	—
54	C₂⁴×C₁₅₁	—	4	1	no	—
55	C₂×D₄×C₁₅₁	—	2	2	no	—
56	C₂×D₄×C₁₅₁	—	4	1	no	—
57	C₂×D₄×C₁₅₁	—	4	1	no	—
58	C₂×D₄×C₁₅₁	—	4	1	no	—
59	C₂×D₄×C₁₅₁	—	4	1	no	—
60	C₂×D₄×C₁₅₁	—	4	1	no	—
61	C₂²×C₄×C₁₅₁	—	4	1	no	—
62	C₂²×C₄×C₁₅₁	—	4	1	no	—
63	C₂²×C₄×C₁₅₁	—	4	1	no	—
64	C₂²×C₄×C₁₅₁	D₈	1	2	yes	—
65	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
66	D₈×C₁₅₁	—	2	2	no	—
67	D₈×C₁₅₁	—	2	2	no	—
68	D₈×C₁₅₁	—	4	1	no	—
69	D₈×C₁₅₁	—	4	1	no	—
70	C₂×C₈×C₁₅₁	—	2	1	no	—
71	C₂×C₈×C₁₅₁	—	2	1	no	—
72	C₂×C₈×C₁₅₁	D₈	1	2	yes	—
73	C₂×C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₈×C₁₅₁	#41	1	1	yes	characteristic
77	C₂×C₈×C₁₅₁	#41	1	1	yes	characteristic
78	C₂×C₈×C₁₅₁	#41	1	1	yes	characteristic
79	C₄⋊C₄×C₁₅₁	—	4	1	no	—
80	C₄⋊C₄×C₁₅₁	—	4	1	no	—
81	C₄⋊C₄×C₁₅₁	—	4	1	no	—
82	C₄⋊C₄×C₁₅₁	—	4	1	no	—
83	C₄⋊C₄×C₁₅₁	—	4	1	no	—
84	C₄⋊C₄×C₁₅₁	—	4	1	no	—
85	C₂²⋊C₄×C₁₅₁	—	8	4	no	—
86	C₄²×C₁₅₁	—	2	1	no	—
87	C₄²×C₁₅₁	—	2	1	no	—
88	C₁₆×C₁₅₁	—	2	2	no	—
89	C₁₆×C₁₅₁	—	2	2	no	—
90	C₂³×C₁₅₁	—	4	2	no	—
91	C₂³×C₁₅₁	—	8	1	no	—
92	C₂³×C₁₅₁	—	8	1	no	—
93	C₂³×C₁₅₁	—	8	2	no	—
94	C₂³×C₁₅₁	—	8	2	no	—
95	C₂³×C₁₅₁	C₂×D₈	1	1	yes	characteristic
96	D₄×C₁₅₁	—	4	2	no	—
97	D₄×C₁₅₁	—	4	2	no	—
98	D₄×C₁₅₁	—	8	1	no	—
99	D₄×C₁₅₁	—	8	1	no	—
100	C₂×C₄×C₁₅₁	—	2	2	no	—
101	C₂×C₄×C₁₅₁	—	2	2	no	—
102	C₂×C₄×C₁₅₁	—	2	2	no	—
103	C₂×C₄×C₁₅₁	—	8	1	no	—
104	C₂×C₄×C₁₅₁	—	8	1	no	—
105	C₂×C₄×C₁₅₁	—	8	1	no	—
106	C₂×C₄×C₁₅₁	—	8	2	no	—
107	C₂×C₄×C₁₅₁	—	8	2	no	—
108	C₂×C₄×C₁₅₁	—	8	2	no	—
109	C₂×C₄×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₅₁	#123	1	1	yes	characteristic
111	C₂×C₄×C₁₅₁	#127	1	1	yes	characteristic
112	C₂×C₄×C₁₅₁	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₁₅₁	#137	1	1	yes	characteristic
114	C₂×C₄×C₁₅₁	#137	1	1	yes	characteristic
115	C₈×C₁₅₁	—	2	2	no	—
116	C₈×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
117	C₈×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
118	C₈×C₁₅₁	#123	1	1	yes	characteristic
119	C₈×C₁₅₁	#123	1	1	yes	characteristic
120	C₂²×C₁₅₁	—	8	2	no	—
121	C₂²×C₁₅₁	—	8	4	no	—
122	C₂²×C₁₅₁	—	16	2	no	—
123	C₂²×C₁₅₁	—	16	1	no	—
124	C₂²×C₁₅₁	—	16	1	no	—
125	C₂²×C₁₅₁	#474	1	1	yes	characteristic
126	C₂²×C₁₅₁	#480	1	1	yes	characteristic
127	C₂²×C₁₅₁	#493	1	1	yes	characteristic
128	C₂²×C₁₅₁	#508	1	2	yes	—
129	C₂²×C₁₅₁	#509	1	2	yes	—
130	C₄×C₁₅₁	—	4	2	no	—
131	C₄×C₁₅₁	—	16	1	no	—
132	C₄×C₁₅₁	—	16	1	no	—
133	C₄×C₁₅₁	—	16	1	no	—
134	C₄×C₁₅₁	#399	1	1	yes	characteristic
135	C₄×C₁₅₁	#474	1	1	yes	characteristic
136	C₄×C₁₅₁	#480	1	1	yes	characteristic
137	C₄×C₁₅₁	#497	1	1	yes	characteristic
138	C₂×C₁₅₁	—	16	2	no	—
139	C₂×C₁₅₁	#2093	1	1	yes	characteristic
140	C₂×C₁₅₁	#2228	1	2	yes	—
141	C₂×C₁₅₁	#2234	1	2	yes	—
142	C₂×C₁₅₁	#2249	1	1	yes	characteristic
143	C₂×C₁₅₁	#2250	1	1	yes	characteristic
144	#13528	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₁₅₁	#13528	1	1	yes	characteristic, direct product factor, Sylow
146	C₂×#367	C₂×C₁₅₁	1	1	yes	characteristic, complemented
147	C₂×#366	C₂×C₁₅₁	1	1	yes	characteristic, complemented
148	C₂×#357	C₂×C₁₅₁	1	1	yes	characteristic
149	C₂×#357	C₂×C₁₅₁	1	1	yes	characteristic
150	#1970	C₂×C₁₅₁	1	1	yes	characteristic
151	#1846	C₂×C₁₅₁	1	1	yes	characteristic, complemented
152	#1374	C₂×C₁₅₁	1	1	yes	characteristic, complemented
153	C₂²×D₈	C₂²×C₁₅₁	1	1	yes	characteristic
154	C₂²×C₁₆	C₂²×C₁₅₁	1	1	yes	characteristic
155	C₂²×C₁₆	C₂²×C₁₅₁	1	1	yes	characteristic
156	C₂×#107	C₂²×C₁₅₁	1	1	yes	characteristic
157	C₂×#107	C₂²×C₁₅₁	1	1	yes	characteristic
158	C₂×#107	C₂²×C₁₅₁	1	1	yes	characteristic
159	C₂×#106	—	2	1	no	—
160	C₂×#102	—	2	1	no	—
161	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
162	#390	—	2	1	no	—
163	#382	—	2	1	no	—
164	#367	—	2	2	no	—
165	#366	—	2	2	no	—
166	#357	—	2	2	no	—
167	#357	—	2	2	no	—
168	C₂²×D₄	—	2	1	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	—	2	1	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	—	2	1	no	—
173	C₂×D₈	—	2	1	no	—
174	C₂×D₈	Q₈×C₁₅₁	1	2	yes	—
175	C₂²×C₈	D₄×C₁₅₁	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₂²⋊C₄	—	4	2	no	—
181	C₂×C₄²	D₄×C₁₅₁	1	1	yes	characteristic
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	—	2	1	no	—
186	C₂×C₁₆	D₄×C₁₅₁	1	2	yes	—
187	C₂×C₁₆	D₄×C₁₅₁	1	2	yes	—
188	#107	—	2	1	no	—
189	#107	—	2	1	no	—
190	#107	—	2	1	no	—
191	#107	—	2	1	no	—
192	#107	—	2	1	no	—
193	#107	—	2	1	no	—
194	#106	—	4	2	no	—
195	#102	—	4	2	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	#95	—	4	2	no	—
199	C₂⁴	—	4	1	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	4	1	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂×D₄	—	4	1	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	D₈×C₁₅₁	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
211	D₈	—	2	2	no	—
212	D₈	—	2	2	no	—
213	D₈	—	4	1	no	—
214	D₈	—	4	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	D₈×C₁₅₁	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₁₅₁	1	1	yes	characteristic
219	C₂×C₈	C₂×D₄×C₁₅₁	1	1	yes	characteristic
220	C₂×C₈	C₂×Q₈×C₁₅₁	1	1	yes	characteristic
221	C₂×C₈	#41×C₁₅₁	1	1	yes	characteristic
222	C₂×C₈	#41×C₁₅₁	1	1	yes	characteristic
223	C₂×C₈	#41×C₁₅₁	1	1	yes	characteristic
224	C₄⋊C₄	—	4	1	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₂²⋊C₄	—	8	4	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₁₆	—	2	2	no	—
234	C₁₆	—	2	2	no	—
235	C₂³	—	4	2	no	—
236	C₂³	—	8	1	no	—
237	C₂³	—	8	1	no	—
238	C₂³	—	8	2	no	—
239	C₂³	—	8	2	no	—
240	C₂³	C₂×D₈×C₁₅₁	1	1	yes	characteristic
241	D₄	—	4	2	no	—
242	D₄	—	4	2	no	—
243	D₄	—	8	1	no	—
244	D₄	—	8	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	8	1	no	—
249	C₂×C₄	—	8	1	no	—
250	C₂×C₄	—	8	1	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
255	C₂×C₄	#123×C₁₅₁	1	1	yes	characteristic
256	C₂×C₄	#127×C₁₅₁	1	1	yes	characteristic
257	C₂×C₄	C₂×D₈×C₁₅₁	1	1	yes	characteristic
258	C₂×C₄	#137×C₁₅₁	1	1	yes	characteristic
259	C₂×C₄	#137×C₁₅₁	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
262	C₈	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
263	C₈	#123×C₁₅₁	1	1	yes	characteristic
264	C₈	#123×C₁₅₁	1	1	yes	characteristic
265	C₂²	—	8	2	no	—
266	C₂²	—	8	4	no	—
267	C₂²	—	16	2	no	—
268	C₂²	—	16	1	no	—
269	C₂²	—	16	1	no	—
270	C₂²	#474×C₁₅₁	1	1	yes	characteristic
271	C₂²	#480×C₁₅₁	1	1	yes	characteristic
272	C₂²	#493×C₁₅₁	1	1	yes	characteristic
273	C₂²	#508×C₁₅₁	1	2	yes	—
274	C₂²	#509×C₁₅₁	1	2	yes	—
275	C₄	—	4	2	no	—
276	C₄	—	16	1	no	—
277	C₄	—	16	1	no	—
278	C₄	—	16	1	no	—
279	C₄	#399×C₁₅₁	1	1	yes	characteristic
280	C₄	#474×C₁₅₁	1	1	yes	characteristic
281	C₄	#480×C₁₅₁	1	1	yes	characteristic
282	C₄	#497×C₁₅₁	1	1	yes	characteristic
283	C₂	—	16	2	no	—
284	C₂	#2093×C₁₅₁	1	1	yes	characteristic
285	C₂	#2228×C₁₅₁	1	2	yes	—
286	C₂	#2234×C₁₅₁	1	2	yes	—
287	C₂	#2249×C₁₅₁	1	1	yes	characteristic
288	C₂	#2250×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₅₁	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₅₁
prime radical	C₂²×D₈×C₁₅₁	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₅₁ ◁ C₂×C₄²×C₁₅₁ ◁ C₂×C₄×C₈×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#367×C₁₅₁	C₂	semidirect
2. C₂×#366×C₁₅₁	C₂	semidirect
6. #1846×C₁₅₁	C₂	semidirect
7. #1374×C₁₅₁	C₂	semidirect
144. #13528	C₁₅₁	direct (N × H)
145. C₁₅₁	#13528	direct (N × H)
146. C₂×#367	C₂×C₁₅₁	semidirect
147. C₂×#366	C₂×C₁₅₁	semidirect
151. #1846	C₂×C₁₅₁	semidirect
152. #1374	C₂×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#367×C₁₅₁	2
C₂×#366×C₁₅₁	2
C₂×#357×C₁₅₁	2
#1970×C₁₅₁	2
#1846×C₁₅₁	2
#1374×C₁₅₁	2
#13528	151

Subgroups — C151×#13528 #50318453

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#13528 #50318453