Subgroups — C₁₅₁×#12942 #50317867

Order 38656.

380 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2957×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#2161×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2148×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#359×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	#1953×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#1381×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#1379×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₁₅₁	C₂²	1	1	yes	characteristic
9	#568×C₁₅₁	—	2	1	no	—
10	C₂×M₃₂×C₁₅₁	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
12	#444×C₁₅₁	—	2	2	no	—
13	#443×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	C₂×#108×C₁₅₁	C₂²	1	1	yes	characteristic
18	#421×C₁₅₁	—	2	1	no	—
19	#405×C₁₅₁	C₂²	1	1	yes	characteristic
20	#368×C₁₅₁	—	2	2	no	—
21	#362×C₁₅₁	C₄	1	2	yes	complemented
22	#361×C₁₅₁	C₄	1	2	yes	complemented
23	#359×C₁₅₁	—	2	2	no	—
24	#344×C₁₅₁	—	2	1	no	—
25	C₂×#41×C₁₅₁	—	2	1	no	—
26	#135×C₁₅₁	—	2	1	no	—
27	#135×C₁₅₁	—	2	1	no	—
28	#135×C₁₅₁	D₄	1	2	yes	—
29	#135×C₁₅₁	Q₈	1	2	yes	—
30	C₂×Q₁₆×C₁₅₁	C₂×C₄	1	1	yes	characteristic
31	C₂×SD₁₆×C₁₅₁	—	2	1	no	—
32	C₂×D₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
33	#131×C₁₅₁	—	2	4	no	—
34	#131×C₁₅₁	—	4	2	no	—
35	C₂×M₁₆×C₁₅₁	—	2	1	no	—
36	C₂×M₁₆×C₁₅₁	—	2	1	no	—
37	C₂×M₁₆×C₁₅₁	—	2	1	no	—
38	C₂×M₁₆×C₁₅₁	—	4	2	no	—
39	C₂×M₁₆×C₁₅₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
41	#117×C₁₅₁	D₄	1	1	yes	characteristic
42	M₃₂×C₁₅₁	—	2	1	no	—
43	M₃₂×C₁₅₁	C₂×C₄	1	2	yes	—
44	C₂×C₁₆×C₁₅₁	—	2	1	no	—
45	C₂×C₁₆×C₁₅₁	—	2	1	no	—
46	C₂×C₁₆×C₁₅₁	—	2	1	no	—
47	C₂×C₁₆×C₁₅₁	—	2	1	no	—
48	#108×C₁₅₁	—	2	1	no	—
49	#108×C₁₅₁	—	2	1	no	—
50	#108×C₁₅₁	—	2	1	no	—
51	#108×C₁₅₁	—	2	1	no	—
52	#108×C₁₅₁	—	4	2	no	—
53	#108×C₁₅₁	C₂×C₄	1	2	yes	—
54	#108×C₁₅₁	D₄	1	2	yes	—
55	#104×C₁₅₁	—	2	4	no	—
56	#97×C₁₅₁	—	2	1	no	—
57	C₄×C₈×C₁₅₁	—	2	1	no	—
58	#41×C₁₅₁	—	2	2	no	—
59	#41×C₁₅₁	—	2	2	no	—
60	#41×C₁₅₁	—	4	1	no	—
61	#41×C₁₅₁	—	4	1	no	—
62	C₂×Q₈×C₁₅₁	—	2	1	no	—
63	C₂×D₄×C₁₅₁	—	2	1	no	—
64	C₂×D₄×C₁₅₁	—	4	1	no	—
65	C₂²×C₄×C₁₅₁	—	4	1	no	—
66	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₁₅₁	—	2	1	no	—
68	Q₁₆×C₁₅₁	C₄⋊C₄	1	2	yes	—
69	SD₁₆×C₁₅₁	—	2	2	no	—
70	SD₁₆×C₁₅₁	—	2	2	no	—
71	D₈×C₁₅₁	—	2	1	no	—
72	D₈×C₁₅₁	C₄⋊C₄	1	2	yes	—
73	M₁₆×C₁₅₁	—	2	1	no	—
74	M₁₆×C₁₅₁	—	2	2	no	—
75	M₁₆×C₁₅₁	—	2	2	no	—
76	M₁₆×C₁₅₁	—	4	4	no	—
77	M₁₆×C₁₅₁	—	4	1	no	—
78	M₁₆×C₁₅₁	—	4	1	no	—
79	M₁₆×C₁₅₁	—	4	1	no	—
80	M₁₆×C₁₅₁	—	4	1	no	—
81	M₁₆×C₁₅₁	D₈	1	2	yes	—
82	C₂×C₈×C₁₅₁	—	2	1	no	—
83	C₂×C₈×C₁₅₁	—	4	4	no	—
84	C₂×C₈×C₁₅₁	—	4	1	no	—
85	C₂×C₈×C₁₅₁	—	4	1	no	—
86	C₂×C₈×C₁₅₁	—	4	1	no	—
87	C₂×C₈×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₈×C₁₅₁	#41	1	1	yes	characteristic
92	C₂×C₈×C₁₅₁	#41	1	1	yes	characteristic
93	C₄⋊C₄×C₁₅₁	—	2	1	no	—
94	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
95	C₄²×C₁₅₁	—	2	1	no	—
96	C₁₆×C₁₅₁	—	2	2	no	—
97	C₁₆×C₁₅₁	—	4	1	no	—
98	C₂³×C₁₅₁	—	4	1	no	—
99	C₂³×C₁₅₁	#137	1	1	yes	characteristic
100	Q₈×C₁₅₁	—	2	2	no	—
101	Q₈×C₁₅₁	—	4	1	no	—
102	D₄×C₁₅₁	—	2	2	no	—
103	D₄×C₁₅₁	—	4	2	no	—
104	D₄×C₁₅₁	—	4	2	no	—
105	D₄×C₁₅₁	—	4	1	no	—
106	C₂×C₄×C₁₅₁	—	4	1	no	—
107	C₂×C₄×C₁₅₁	—	4	2	no	—
108	C₂×C₄×C₁₅₁	—	4	2	no	—
109	C₂×C₄×C₁₅₁	—	4	1	no	—
110	C₂×C₄×C₁₅₁	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₅₁	#120	1	1	yes	characteristic
112	C₂×C₄×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₅₁	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₁₅₁	#135	1	1	yes	characteristic
115	C₂×C₄×C₁₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₈×C₁₅₁	—	2	2	no	—
117	C₈×C₁₅₁	—	4	2	no	—
118	C₈×C₁₅₁	—	4	2	no	—
119	C₈×C₁₅₁	—	8	1	no	—
120	C₈×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₈×C₁₅₁	C₄×D₄	1	1	yes	characteristic
122	C₈×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
123	C₈×C₁₅₁	#123	1	1	yes	characteristic
124	C₂²×C₁₅₁	—	4	1	no	—
125	C₂²×C₁₅₁	—	4	2	no	—
126	C₂²×C₁₅₁	—	16	1	no	—
127	C₂²×C₁₅₁	#440	1	1	yes	characteristic
128	C₂²×C₁₅₁	#452	1	1	yes	characteristic
129	C₂²×C₁₅₁	#474	1	1	yes	characteristic
130	C₄×C₁₅₁	—	4	2	no	—
131	C₄×C₁₅₁	—	8	1	no	—
132	C₄×C₁₅₁	#385	1	1	yes	characteristic
133	C₄×C₁₅₁	C₄×D₈	1	1	yes	characteristic
134	C₄×C₁₅₁	#447	1	1	yes	characteristic
135	C₄×C₁₅₁	#478	1	1	yes	characteristic
136	C₂×C₁₅₁	—	2	1	no	—
137	C₂×C₁₅₁	—	2	1	no	—
138	C₂×C₁₅₁	—	2	1	no	—
139	C₂×C₁₅₁	—	8	2	no	—
140	C₂×C₁₅₁	#1862	1	1	yes	characteristic
141	#12942	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₁₅₁	#12942	1	1	yes	characteristic, direct product factor, Sylow
143	#2957	C₂×C₁₅₁	1	1	yes	characteristic
144	#2161	C₂×C₁₅₁	1	1	yes	characteristic, complemented
145	#2148	C₂×C₁₅₁	1	1	yes	characteristic
146	C₂×#359	C₂×C₁₅₁	1	1	yes	characteristic
147	#1953	C₂×C₁₅₁	1	1	yes	characteristic, complemented
148	#1381	C₂×C₁₅₁	1	1	yes	characteristic, complemented
149	#1379	C₂×C₁₅₁	1	1	yes	characteristic, complemented
150	C₂×#135	C₂²×C₁₅₁	1	1	yes	characteristic
151	#568	—	2	1	no	—
152	C₂×M₃₂	C₂²×C₁₅₁	1	1	yes	characteristic
153	C₂²×C₁₆	C₂²×C₁₅₁	1	1	yes	characteristic
154	#444	—	2	2	no	—
155	#443	—	2	2	no	—
156	#430	—	2	1	no	—
157	C₂×#108	C₂²×C₁₅₁	1	1	yes	characteristic
158	C₂×#108	C₂²×C₁₅₁	1	1	yes	characteristic
159	C₂×#108	C₂²×C₁₅₁	1	1	yes	characteristic
160	#421	—	2	1	no	—
161	#405	C₂²×C₁₅₁	1	1	yes	characteristic
162	#368	—	2	2	no	—
163	#362	C₄×C₁₅₁	1	2	yes	complemented
164	#361	C₄×C₁₅₁	1	2	yes	complemented
165	#359	—	2	2	no	—
166	#344	—	2	1	no	—
167	C₂×#41	—	2	1	no	—
168	#135	—	2	1	no	—
169	#135	—	2	1	no	—
170	#135	D₄×C₁₅₁	1	2	yes	—
171	#135	Q₈×C₁₅₁	1	2	yes	—
172	C₂×Q₁₆	C₂×C₄×C₁₅₁	1	1	yes	characteristic
173	C₂×SD₁₆	—	2	1	no	—
174	C₂×D₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
175	#131	—	2	4	no	—
176	#131	—	4	2	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	4	2	no	—
181	C₂×M₁₆	D₄×C₁₅₁	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
183	#117	D₄×C₁₅₁	1	1	yes	characteristic
184	M₃₂	—	2	1	no	—
185	M₃₂	C₂×C₄×C₁₅₁	1	2	yes	—
186	C₂×C₁₆	—	2	1	no	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	2	1	no	—
190	#108	—	2	1	no	—
191	#108	—	2	1	no	—
192	#108	—	2	1	no	—
193	#108	—	2	1	no	—
194	#108	—	4	2	no	—
195	#108	C₂×C₄×C₁₅₁	1	2	yes	—
196	#108	D₄×C₁₅₁	1	2	yes	—
197	#104	—	2	4	no	—
198	#97	—	2	1	no	—
199	C₄×C₈	—	2	1	no	—
200	#41	—	2	2	no	—
201	#41	—	2	2	no	—
202	#41	—	4	1	no	—
203	#41	—	4	1	no	—
204	C₂×Q₈	—	2	1	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
209	Q₁₆	—	2	1	no	—
210	Q₁₆	C₄⋊C₄×C₁₅₁	1	2	yes	—
211	SD₁₆	—	2	2	no	—
212	SD₁₆	—	2	2	no	—
213	D₈	—	2	1	no	—
214	D₈	C₄⋊C₄×C₁₅₁	1	2	yes	—
215	M₁₆	—	2	1	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	—	2	2	no	—
218	M₁₆	—	4	4	no	—
219	M₁₆	—	4	1	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	1	no	—
223	M₁₆	D₈×C₁₅₁	1	2	yes	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	4	4	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
230	C₂×C₈	C₂×D₄×C₁₅₁	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄×C₁₅₁	1	1	yes	characteristic
232	C₂×C₈	C₂×Q₈×C₁₅₁	1	1	yes	characteristic
233	C₂×C₈	#41×C₁₅₁	1	1	yes	characteristic
234	C₂×C₈	#41×C₁₅₁	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₂²⋊C₄	—	2	1	no	—
237	C₄²	—	2	1	no	—
238	C₁₆	—	2	2	no	—
239	C₁₆	—	4	1	no	—
240	C₂³	—	4	1	no	—
241	C₂³	#137×C₁₅₁	1	1	yes	characteristic
242	Q₈	—	2	2	no	—
243	Q₈	—	4	1	no	—
244	D₄	—	2	2	no	—
245	D₄	—	4	2	no	—
246	D₄	—	4	2	no	—
247	D₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	C₄×D₄×C₁₅₁	1	1	yes	characteristic
253	C₂×C₄	#120×C₁₅₁	1	1	yes	characteristic
254	C₂×C₄	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
255	C₂×C₄	C₂×D₈×C₁₅₁	1	1	yes	characteristic
256	C₂×C₄	#135×C₁₅₁	1	1	yes	characteristic
257	C₂×C₄	AGL(1,ℤ/8)×C₁₅₁	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	4	2	no	—
260	C₈	—	4	2	no	—
261	C₈	—	8	1	no	—
262	C₈	C₂×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
263	C₈	C₄×D₄×C₁₅₁	1	1	yes	characteristic
264	C₈	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
265	C₈	#123×C₁₅₁	1	1	yes	characteristic
266	C₂²	—	4	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	16	1	no	—
269	C₂²	#440×C₁₅₁	1	1	yes	characteristic
270	C₂²	#452×C₁₅₁	1	1	yes	characteristic
271	C₂²	#474×C₁₅₁	1	1	yes	characteristic
272	C₄	—	4	2	no	—
273	C₄	—	8	1	no	—
274	C₄	#385×C₁₅₁	1	1	yes	characteristic
275	C₄	C₄×D₈×C₁₅₁	1	1	yes	characteristic
276	C₄	#447×C₁₅₁	1	1	yes	characteristic
277	C₄	#478×C₁₅₁	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	8	2	no	—
282	C₂	#1862×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₅₁	#447
commutator	C₂×C₈	C₂²×C₄×C₁₅₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₅₁
socle	C₂×C₁₅₁	#1862
prime radical	C₂×#135×C₁₅₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₅₁ ◁ C₂²×C₄×C₁₅₁ ◁ #405×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2161×C₁₅₁	C₂	semidirect
5. #1953×C₁₅₁	C₂	semidirect
6. #1381×C₁₅₁	C₂	semidirect
7. #1379×C₁₅₁	C₂	semidirect
21. #362×C₁₅₁	C₄	semidirect
22. #361×C₁₅₁	C₄	semidirect
141. #12942	C₁₅₁	direct (N × H)
142. C₁₅₁	#12942	direct (N × H)
144. #2161	C₂×C₁₅₁	semidirect
147. #1953	C₂×C₁₅₁	semidirect
148. #1381	C₂×C₁₅₁	semidirect
149. #1379	C₂×C₁₅₁	semidirect
163. #362	C₄×C₁₅₁	semidirect
164. #361	C₄×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#2957×C₁₅₁	2
#2161×C₁₅₁	2
#2148×C₁₅₁	2
C₂×#359×C₁₅₁	2
#1953×C₁₅₁	2
#1381×C₁₅₁	2
#1379×C₁₅₁	2
#12942	151

Subgroups — C151×#12942 #50317867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#12942 #50317867