Subgroups — C₁₅₁×#12953 #50317878

Order 38656.

416 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2948×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#2155×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#361×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#361×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	#1953×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#1389×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#1378×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₅₁	C₂²	1	1	yes	characteristic
9	#518×C₁₅₁	—	2	1	no	—
10	C₂×M₃₂×C₁₅₁	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₁₅₁	C₂²	1	1	yes	characteristic
12	#442×C₁₅₁	—	2	2	no	—
13	C₄×D₈×C₁₅₁	—	2	2	no	—
14	#430×C₁₅₁	—	2	1	no	—
15	C₂×#108×C₁₅₁	C₂²	1	1	yes	characteristic
16	C₂×#108×C₁₅₁	C₂²	1	1	yes	characteristic
17	#427×C₁₅₁	C₂²	1	1	yes	characteristic
18	#418×C₁₅₁	—	2	1	no	—
19	#405×C₁₅₁	C₂²	1	1	yes	characteristic
20	#365×C₁₅₁	—	2	2	no	—
21	#361×C₁₅₁	—	2	2	no	—
22	#361×C₁₅₁	C₄	1	4	yes	complemented
23	#344×C₁₅₁	—	2	1	no	—
24	C₂²×D₄×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	C₂×C₄	1	2	yes	—
29	C₂×D₈×C₁₅₁	D₄	1	2	yes	—
30	C₂×D₈×C₁₅₁	Q₈	1	2	yes	—
31	C₂×M₁₆×C₁₅₁	—	2	1	no	—
32	C₂×M₁₆×C₁₅₁	—	2	1	no	—
33	C₂×M₁₆×C₁₅₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
35	C₄×D₄×C₁₅₁	—	2	4	no	—
36	C₄×D₄×C₁₅₁	—	4	2	no	—
37	#117×C₁₅₁	—	2	1	no	—
38	#117×C₁₅₁	D₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₁₅₁	—	4	2	no	—
40	M₃₂×C₁₅₁	—	2	1	no	—
41	M₃₂×C₁₅₁	—	2	1	no	—
42	M₃₂×C₁₅₁	—	2	1	no	—
43	M₃₂×C₁₅₁	C₂×C₄	1	2	yes	complemented
44	C₂×C₁₆×C₁₅₁	—	2	1	no	—
45	C₂×C₁₆×C₁₅₁	—	2	1	no	—
46	#108×C₁₅₁	—	2	1	no	—
47	#108×C₁₅₁	—	2	1	no	—
48	#108×C₁₅₁	—	2	1	no	—
49	#108×C₁₅₁	—	4	2	no	—
50	#108×C₁₅₁	C₂×C₄	1	2	yes	complemented
51	#107×C₁₅₁	—	2	1	no	—
52	#106×C₁₅₁	D₄	1	2	yes	—
53	#102×C₁₅₁	—	2	4	no	—
54	#97×C₁₅₁	—	2	1	no	—
55	C₄×C₈×C₁₅₁	—	2	1	no	—
56	C₂⁴×C₁₅₁	—	4	1	no	—
57	C₂×D₄×C₁₅₁	—	2	2	no	—
58	C₂×D₄×C₁₅₁	—	2	2	no	—
59	C₂×D₄×C₁₅₁	—	2	2	no	—
60	C₂×D₄×C₁₅₁	—	4	1	no	—
61	C₂×D₄×C₁₅₁	—	4	1	no	—
62	C₂×D₄×C₁₅₁	—	4	1	no	—
63	C₂²×C₄×C₁₅₁	—	4	4	no	—
64	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
65	D₈×C₁₅₁	—	2	2	no	—
66	D₈×C₁₅₁	—	2	2	no	—
67	D₈×C₁₅₁	—	2	2	no	—
68	D₈×C₁₅₁	C₄⋊C₄	1	4	yes	—
69	M₁₆×C₁₅₁	—	2	2	no	—
70	M₁₆×C₁₅₁	—	2	1	no	—
71	M₁₆×C₁₅₁	—	2	1	no	—
72	M₁₆×C₁₅₁	—	4	1	no	—
73	M₁₆×C₁₅₁	—	4	1	no	—
74	M₁₆×C₁₅₁	—	4	1	no	—
75	C₂×C₈×C₁₅₁	—	2	1	no	—
76	C₂×C₈×C₁₅₁	—	4	1	no	—
77	C₂×C₈×C₁₅₁	—	4	1	no	—
78	C₂×C₈×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈×C₁₅₁	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₅₁	#41	1	1	yes	characteristic
84	C₄⋊C₄×C₁₅₁	—	2	1	no	—
85	C₄⋊C₄×C₁₅₁	—	2	2	no	—
86	C₂²⋊C₄×C₁₅₁	—	4	4	no	—
87	C₂²⋊C₄×C₁₅₁	—	4	1	no	—
88	C₂²⋊C₄×C₁₅₁	SD₁₆	1	2	yes	—
89	C₄²×C₁₅₁	—	2	1	no	—
90	C₄²×C₁₅₁	—	4	1	no	—
91	C₁₆×C₁₅₁	—	2	2	no	—
92	C₁₆×C₁₅₁	—	4	1	no	—
93	C₂³×C₁₅₁	—	4	2	no	—
94	C₂³×C₁₅₁	—	4	2	no	—
95	C₂³×C₁₅₁	—	4	2	no	—
96	C₂³×C₁₅₁	—	16	2	no	—
97	C₂³×C₁₅₁	C₂×SD₁₆	1	1	yes	characteristic
98	D₄×C₁₅₁	—	2	4	no	—
99	D₄×C₁₅₁	—	4	2	no	—
100	D₄×C₁₅₁	—	4	2	no	—
101	D₄×C₁₅₁	—	4	2	no	—
102	C₂×C₄×C₁₅₁	—	2	2	no	—
103	C₂×C₄×C₁₅₁	—	4	2	no	—
104	C₂×C₄×C₁₅₁	—	16	4	no	—
105	C₂×C₄×C₁₅₁	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₅₁	#120	1	1	yes	characteristic
107	C₂×C₄×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₁₅₁	#135	1	1	yes	characteristic
109	C₂×C₄×C₁₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₁₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₈×C₁₅₁	—	4	2	no	—
112	C₈×C₁₅₁	—	4	1	no	—
113	C₈×C₁₅₁	—	8	1	no	—
114	C₈×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₈×C₁₅₁	C₄×D₄	1	1	yes	characteristic
116	C₈×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
117	C₈×C₁₅₁	#123	1	1	yes	characteristic
118	C₂²×C₁₅₁	—	4	1	no	—
119	C₂²×C₁₅₁	—	4	4	no	—
120	C₂²×C₁₅₁	—	8	4	no	—
121	C₂²×C₁₅₁	—	16	2	no	—
122	C₂²×C₁₅₁	—	16	2	no	—
123	C₂²×C₁₅₁	C₄×SD₁₆	1	1	yes	characteristic
124	C₂²×C₁₅₁	#450	1	1	yes	characteristic
125	C₂²×C₁₅₁	#476	1	1	yes	characteristic
126	C₄×C₁₅₁	—	4	2	no	—
127	C₄×C₁₅₁	—	8	2	no	—
128	C₄×C₁₅₁	#385	1	1	yes	characteristic
129	C₄×C₁₅₁	#442	1	1	yes	characteristic
130	C₄×C₁₅₁	#449	1	1	yes	characteristic
131	C₄×C₁₅₁	#478	1	1	yes	characteristic
132	C₂×C₁₅₁	—	2	1	no	—
133	C₂×C₁₅₁	—	2	1	no	—
134	C₂×C₁₅₁	—	2	1	no	—
135	C₂×C₁₅₁	—	8	4	no	—
136	C₂×C₁₅₁	#1864	1	1	yes	characteristic
137	#12953	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₁₅₁	#12953	1	1	yes	characteristic, direct product factor, Sylow
139	#2948	C₂×C₁₅₁	1	1	yes	characteristic
140	#2155	C₂×C₁₅₁	1	1	yes	characteristic, complemented
141	C₂×#361	C₂×C₁₅₁	1	1	yes	characteristic
142	C₂×#361	C₂×C₁₅₁	1	1	yes	characteristic
143	#1953	C₂×C₁₅₁	1	1	yes	characteristic, complemented
144	#1389	C₂×C₁₅₁	1	1	yes	characteristic, complemented
145	#1378	C₂×C₁₅₁	1	1	yes	characteristic, complemented
146	C₂²×D₈	C₂²×C₁₅₁	1	1	yes	characteristic
147	#518	—	2	1	no	—
148	C₂×M₃₂	C₂²×C₁₅₁	1	1	yes	characteristic
149	C₂×M₃₂	C₂²×C₁₅₁	1	1	yes	characteristic
150	#442	—	2	2	no	—
151	C₄×D₈	—	2	2	no	—
152	#430	—	2	1	no	—
153	C₂×#108	C₂²×C₁₅₁	1	1	yes	characteristic
154	C₂×#108	C₂²×C₁₅₁	1	1	yes	characteristic
155	#427	C₂²×C₁₅₁	1	1	yes	characteristic
156	#418	—	2	1	no	—
157	#405	C₂²×C₁₅₁	1	1	yes	characteristic
158	#365	—	2	2	no	—
159	#361	—	2	2	no	—
160	#361	C₄×C₁₅₁	1	4	yes	complemented
161	#344	—	2	1	no	—
162	C₂²×D₄	—	2	1	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	—	2	1	no	—
166	C₂×D₈	C₂×C₄×C₁₅₁	1	2	yes	—
167	C₂×D₈	D₄×C₁₅₁	1	2	yes	—
168	C₂×D₈	Q₈×C₁₅₁	1	2	yes	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	—	2	1	no	—
171	C₂×M₁₆	D₄×C₁₅₁	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
173	C₄×D₄	—	2	4	no	—
174	C₄×D₄	—	4	2	no	—
175	#117	—	2	1	no	—
176	#117	D₄×C₁₅₁	1	1	yes	characteristic
177	C₂×C₂²⋊C₄	—	4	2	no	—
178	M₃₂	—	2	1	no	—
179	M₃₂	—	2	1	no	—
180	M₃₂	—	2	1	no	—
181	M₃₂	C₂×C₄×C₁₅₁	1	2	yes	complemented
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	#108	—	2	1	no	—
185	#108	—	2	1	no	—
186	#108	—	2	1	no	—
187	#108	—	4	2	no	—
188	#108	C₂×C₄×C₁₅₁	1	2	yes	complemented
189	#107	—	2	1	no	—
190	#106	D₄×C₁₅₁	1	2	yes	—
191	#102	—	2	4	no	—
192	#97	—	2	1	no	—
193	C₄×C₈	—	2	1	no	—
194	C₂⁴	—	4	1	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
203	D₈	—	2	2	no	—
204	D₈	—	2	2	no	—
205	D₈	—	2	2	no	—
206	D₈	C₄⋊C₄×C₁₅₁	1	4	yes	—
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	1	no	—
209	M₁₆	—	2	1	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
217	C₂×C₈	C₂×D₄×C₁₅₁	1	1	yes	characteristic
218	C₂×C₈	C₂×D₄×C₁₅₁	1	1	yes	characteristic
219	C₂×C₈	C₂×Q₈×C₁₅₁	1	1	yes	characteristic
220	C₂×C₈	#41×C₁₅₁	1	1	yes	characteristic
221	C₂×C₈	#41×C₁₅₁	1	1	yes	characteristic
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	4	4	no	—
225	C₂²⋊C₄	—	4	1	no	—
226	C₂²⋊C₄	SD₁₆×C₁₅₁	1	2	yes	—
227	C₄²	—	2	1	no	—
228	C₄²	—	4	1	no	—
229	C₁₆	—	2	2	no	—
230	C₁₆	—	4	1	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	—	4	2	no	—
234	C₂³	—	16	2	no	—
235	C₂³	C₂×SD₁₆×C₁₅₁	1	1	yes	characteristic
236	D₄	—	2	4	no	—
237	D₄	—	4	2	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	16	4	no	—
243	C₂×C₄	C₄×D₄×C₁₅₁	1	1	yes	characteristic
244	C₂×C₄	#120×C₁₅₁	1	1	yes	characteristic
245	C₂×C₄	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
246	C₂×C₄	#135×C₁₅₁	1	1	yes	characteristic
247	C₂×C₄	AGL(1,ℤ/8)×C₁₅₁	1	1	yes	characteristic
248	C₂×C₄	AGL(1,ℤ/8)×C₁₅₁	1	1	yes	characteristic
249	C₈	—	4	2	no	—
250	C₈	—	4	1	no	—
251	C₈	—	8	1	no	—
252	C₈	C₂×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
253	C₈	C₄×D₄×C₁₅₁	1	1	yes	characteristic
254	C₈	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
255	C₈	#123×C₁₅₁	1	1	yes	characteristic
256	C₂²	—	4	1	no	—
257	C₂²	—	4	4	no	—
258	C₂²	—	8	4	no	—
259	C₂²	—	16	2	no	—
260	C₂²	—	16	2	no	—
261	C₂²	C₄×SD₁₆×C₁₅₁	1	1	yes	characteristic
262	C₂²	#450×C₁₅₁	1	1	yes	characteristic
263	C₂²	#476×C₁₅₁	1	1	yes	characteristic
264	C₄	—	4	2	no	—
265	C₄	—	8	2	no	—
266	C₄	#385×C₁₅₁	1	1	yes	characteristic
267	C₄	#442×C₁₅₁	1	1	yes	characteristic
268	C₄	#449×C₁₅₁	1	1	yes	characteristic
269	C₄	#478×C₁₅₁	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	8	4	no	—
274	C₂	#1864×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₅₁	#1864
commutator	C₂×C₈	C₂²×C₄×C₁₅₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₅₁
prime radical	C₂²×D₈×C₁₅₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₅₁ ◁ C₂²×C₄×C₁₅₁ ◁ #405×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2155×C₁₅₁	C₂	semidirect
5. #1953×C₁₅₁	C₂	semidirect
6. #1389×C₁₅₁	C₂	semidirect
7. #1378×C₁₅₁	C₂	semidirect
22. #361×C₁₅₁	C₄	semidirect
43. M₃₂×C₁₅₁	C₂×C₄	semidirect
50. #108×C₁₅₁	C₂×C₄	semidirect
137. #12953	C₁₅₁	direct (N × H)
138. C₁₅₁	#12953	direct (N × H)
140. #2155	C₂×C₁₅₁	semidirect
143. #1953	C₂×C₁₅₁	semidirect
144. #1389	C₂×C₁₅₁	semidirect
145. #1378	C₂×C₁₅₁	semidirect
160. #361	C₄×C₁₅₁	semidirect
181. M₃₂	C₂×C₄×C₁₅₁	semidirect
188. #108	C₂×C₄×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#2948×C₁₅₁	2
#2155×C₁₅₁	2
C₂×#361×C₁₅₁	2
#1953×C₁₅₁	2
#1389×C₁₅₁	2
#1378×C₁₅₁	2
#12953	151

Subgroups — C151×#12953 #50317878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#12953 #50317878