Subgroups — C₁₅₁×#11628 #50316553

Order 38656.

322 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	#2965×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#2163×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2150×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#1338×C₁₅₁	C₂	1	2	yes	maximal, complemented
5	#1330×C₁₅₁	C₂	1	2	yes	maximal, complemented
6	C₄×#41×C₁₅₁	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₁₅₁	—	2	1	no	—
8	#446×C₁₅₁	C₄	1	1	yes	characteristic
9	#446×C₁₅₁	C₂²	1	1	yes	characteristic
10	#436×C₁₅₁	C₄	1	1	yes	characteristic
11	#436×C₁₅₁	C₂²	1	1	yes	characteristic
12	#435×C₁₅₁	—	2	1	no	—
13	#435×C₁₅₁	—	2	1	no	—
14	#431×C₁₅₁	—	2	1	no	—
15	#431×C₁₅₁	C₄	1	1	yes	characteristic
16	C₄×M₁₆×C₁₅₁	—	2	1	no	—
17	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₁₆×C₁₅₁	—	2	1	no	—
19	C₄⋊C₁₆×C₁₅₁	—	2	1	no	—
20	C₄⋊C₁₆×C₁₅₁	C₂²	1	2	yes	—
21	C₂²⋊C₁₆×C₁₅₁	—	2	1	no	—
22	#346×C₁₅₁	C₂²	1	2	yes	—
23	C₂×#41×C₁₅₁	C₈	1	1	yes	characteristic
24	C₂×M₁₆×C₁₅₁	—	2	1	no	—
25	C₂×M₁₆×C₁₅₁	—	2	1	no	—
26	C₂²×C₈×C₁₅₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₅₁	D₄	1	1	yes	characteristic
28	C₄×Q₈×C₁₅₁	C₈	1	1	yes	characteristic
29	C₄×Q₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
30	C₄×D₄×C₁₅₁	—	2	1	no	—
31	C₄×D₄×C₁₅₁	—	2	1	no	—
32	C₄×D₄×C₁₅₁	C₈	1	1	yes	characteristic
33	C₄×D₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
34	#117×C₁₅₁	—	2	1	no	—
35	#117×C₁₅₁	C₈	1	1	yes	characteristic
36	C₂×C₄²×C₁₅₁	—	2	1	no	—
37	C₂×C₄²×C₁₅₁	C₂×C₄	1	1	yes	characteristic
38	M₃₂×C₁₅₁	—	4	2	no	—
39	C₂×C₁₆×C₁₅₁	—	2	2	no	—
40	C₂×C₁₆×C₁₅₁	—	2	2	no	—
41	C₄⋊C₈×C₁₅₁	—	2	1	no	—
42	C₄⋊C₈×C₁₅₁	—	2	1	no	—
43	C₄⋊C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
44	C₄⋊C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
45	C₂²⋊C₈×C₁₅₁	—	2	1	no	—
46	C₂²⋊C₈×C₁₅₁	—	2	1	no	—
47	C₂²⋊C₈×C₁₅₁	—	2	1	no	—
48	#97×C₁₅₁	—	2	1	no	—
49	#97×C₁₅₁	—	2	1	no	—
50	C₄×C₈×C₁₅₁	—	2	1	no	—
51	C₄×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
55	#41×C₁₅₁	—	4	2	no	—
56	C₂×Q₈×C₁₅₁	C₂×C₈	1	1	yes	characteristic
57	C₂×D₄×C₁₅₁	—	2	1	no	—
58	C₂×D₄×C₁₅₁	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₅₁	—	2	1	no	—
60	C₂²×C₄×C₁₅₁	—	2	1	no	—
61	C₂²×C₄×C₁₅₁	—	2	1	no	—
62	C₂²×C₄×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₅₁	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₅₁	M₁₆	1	1	yes	characteristic
65	M₁₆×C₁₅₁	—	4	2	no	—
66	M₁₆×C₁₅₁	—	4	1	no	—
67	M₁₆×C₁₅₁	—	4	1	no	—
68	C₂×C₈×C₁₅₁	—	2	1	no	—
69	C₂×C₈×C₁₅₁	—	2	1	no	—
70	C₂×C₈×C₁₅₁	—	2	1	no	—
71	C₂×C₈×C₁₅₁	—	2	1	no	—
72	C₂×C₈×C₁₅₁	—	4	1	no	—
73	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₅₁	—	2	1	no	—
78	C₄⋊C₄×C₁₅₁	—	2	1	no	—
79	C₄⋊C₄×C₁₅₁	C₂×C₈	1	1	yes	characteristic
80	C₄⋊C₄×C₁₅₁	C₂×C₈	1	1	yes	characteristic
81	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
82	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
83	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
84	C₄²×C₁₅₁	—	2	1	no	—
85	C₄²×C₁₅₁	—	2	1	no	—
86	C₄²×C₁₅₁	—	2	1	no	—
87	C₄²×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₁₅₁	C₂×C₈	1	1	yes	characteristic
89	C₄²×C₁₅₁	M₁₆	1	1	yes	characteristic
90	C₄²×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
91	C₁₆×C₁₅₁	—	4	2	no	—
92	C₁₆×C₁₅₁	—	4	2	no	—
93	C₂³×C₁₅₁	—	2	1	no	—
94	C₂³×C₁₅₁	C₂²⋊C₈	1	1	yes	characteristic
95	Q₈×C₁₅₁	—	4	1	no	—
96	D₄×C₁₅₁	—	4	1	no	—
97	D₄×C₁₅₁	—	4	2	no	—
98	C₂×C₄×C₁₅₁	—	2	2	no	—
99	C₂×C₄×C₁₅₁	—	2	1	no	—
100	C₂×C₄×C₁₅₁	—	2	1	no	—
101	C₂×C₄×C₁₅₁	—	2	1	no	—
102	C₂×C₄×C₁₅₁	—	2	1	no	—
103	C₂×C₄×C₁₅₁	—	2	1	no	—
104	C₂×C₄×C₁₅₁	—	4	1	no	—
105	C₂×C₄×C₁₅₁	—	4	2	no	—
106	C₂×C₄×C₁₅₁	—	4	2	no	—
107	C₂×C₄×C₁₅₁	—	4	1	no	—
108	C₂×C₄×C₁₅₁	—	4	1	no	—
109	C₂×C₄×C₁₅₁	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₅₁	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₅₁	C₂²⋊C₈	1	1	yes	characteristic
112	C₂×C₄×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₅₁	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₅₁	C₂×M₁₆	1	1	yes	characteristic
115	C₈×C₁₅₁	—	2	1	no	—
116	C₈×C₁₅₁	—	2	1	no	—
117	C₈×C₁₅₁	—	2	1	no	—
118	C₈×C₁₅₁	—	4	1	no	—
119	C₈×C₁₅₁	—	4	1	no	—
120	C₈×C₁₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₈×C₁₅₁	#137	1	1	yes	characteristic
122	C₂²×C₁₅₁	—	2	1	no	—
123	C₂²×C₁₅₁	—	2	1	no	—
124	C₂²×C₁₅₁	—	2	1	no	—
125	C₂²×C₁₅₁	—	4	1	no	—
126	C₂²×C₁₅₁	—	4	1	no	—
127	C₂²×C₁₅₁	—	4	1	no	—
128	C₂²×C₁₅₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
129	C₄×C₁₅₁	—	2	1	no	—
130	C₄×C₁₅₁	—	2	1	no	—
131	C₄×C₁₅₁	—	2	1	no	—
132	C₄×C₁₅₁	—	2	1	no	—
133	C₄×C₁₅₁	—	4	1	no	—
134	C₄×C₁₅₁	—	4	1	no	—
135	C₄×C₁₅₁	—	4	1	no	—
136	C₄×C₁₅₁	#349	1	2	yes	—
137	C₄×C₁₅₁	#417	1	1	yes	characteristic
138	C₄×C₁₅₁	#421	1	1	yes	characteristic
139	C₂×C₁₅₁	—	4	1	no	—
140	C₂×C₁₅₁	—	8	1	no	—
141	C₂×C₁₅₁	#1477	1	1	yes	characteristic
142	C₂×C₁₅₁	C₂×#349	1	1	yes	characteristic
143	C₂×C₁₅₁	#2117	1	1	yes	characteristic
144	#11628	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₁₅₁	#11628	1	1	yes	characteristic, direct product factor, Sylow
146	#2965	C₂×C₁₅₁	1	1	yes	characteristic
147	#2163	C₂×C₁₅₁	1	1	yes	characteristic, complemented
148	#2150	C₂×C₁₅₁	1	1	yes	characteristic, complemented
149	#1338	C₂×C₁₅₁	1	2	yes	complemented
150	#1330	C₂×C₁₅₁	1	2	yes	complemented
151	C₄×#41	C₄×C₁₅₁	1	1	yes	characteristic
152	C₂×M₃₂	—	2	1	no	—
153	#446	C₄×C₁₅₁	1	1	yes	characteristic
154	#446	C₂²×C₁₅₁	1	1	yes	characteristic
155	#436	C₄×C₁₅₁	1	1	yes	characteristic
156	#436	C₂²×C₁₅₁	1	1	yes	characteristic
157	#435	—	2	1	no	—
158	#435	—	2	1	no	—
159	#431	—	2	1	no	—
160	#431	C₄×C₁₅₁	1	1	yes	characteristic
161	C₄×M₁₆	—	2	1	no	—
162	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
163	C₄⋊C₁₆	—	2	1	no	—
164	C₄⋊C₁₆	—	2	1	no	—
165	C₄⋊C₁₆	C₂²×C₁₅₁	1	2	yes	—
166	C₂²⋊C₁₆	—	2	1	no	—
167	#346	C₂²×C₁₅₁	1	2	yes	—
168	C₂×#41	C₈×C₁₅₁	1	1	yes	characteristic
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	—	2	1	no	—
171	C₂²×C₈	D₄×C₁₅₁	1	1	yes	characteristic
172	C₂²×C₈	D₄×C₁₅₁	1	1	yes	characteristic
173	C₄×Q₈	C₈×C₁₅₁	1	1	yes	characteristic
174	C₄×Q₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
175	C₄×D₄	—	2	1	no	—
176	C₄×D₄	—	2	1	no	—
177	C₄×D₄	C₈×C₁₅₁	1	1	yes	characteristic
178	C₄×D₄	C₂×C₄×C₁₅₁	1	1	yes	characteristic
179	#117	—	2	1	no	—
180	#117	C₈×C₁₅₁	1	1	yes	characteristic
181	C₂×C₄²	—	2	1	no	—
182	C₂×C₄²	C₂×C₄×C₁₅₁	1	1	yes	characteristic
183	M₃₂	—	4	2	no	—
184	C₂×C₁₆	—	2	2	no	—
185	C₂×C₁₆	—	2	2	no	—
186	C₄⋊C₈	—	2	1	no	—
187	C₄⋊C₈	—	2	1	no	—
188	C₄⋊C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
189	C₄⋊C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
190	C₂²⋊C₈	—	2	1	no	—
191	C₂²⋊C₈	—	2	1	no	—
192	C₂²⋊C₈	—	2	1	no	—
193	#97	—	2	1	no	—
194	#97	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
197	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
198	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
199	C₄×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
200	#41	—	4	2	no	—
201	C₂×Q₈	C₂×C₈×C₁₅₁	1	1	yes	characteristic
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	C₂×C₈×C₁₅₁	1	1	yes	characteristic
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₈×C₁₅₁	1	1	yes	characteristic
209	C₂²×C₄	M₁₆×C₁₅₁	1	1	yes	characteristic
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
220	C₂×C₈	C₂×D₄×C₁₅₁	1	1	yes	characteristic
221	C₂×C₈	C₂×D₄×C₁₅₁	1	1	yes	characteristic
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	C₂×C₈×C₁₅₁	1	1	yes	characteristic
225	C₄⋊C₄	C₂×C₈×C₁₅₁	1	1	yes	characteristic
226	C₂²⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₄²	—	2	1	no	—
230	C₄²	—	2	1	no	—
231	C₄²	—	2	1	no	—
232	C₄²	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
233	C₄²	C₂×C₈×C₁₅₁	1	1	yes	characteristic
234	C₄²	M₁₆×C₁₅₁	1	1	yes	characteristic
235	C₄²	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
236	C₁₆	—	4	2	no	—
237	C₁₆	—	4	2	no	—
238	C₂³	—	2	1	no	—
239	C₂³	C₂²⋊C₈×C₁₅₁	1	1	yes	characteristic
240	Q₈	—	4	1	no	—
241	D₄	—	4	1	no	—
242	D₄	—	4	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	C₂²⋊C₈×C₁₅₁	1	1	yes	characteristic
255	C₂×C₄	C₂²⋊C₈×C₁₅₁	1	1	yes	characteristic
256	C₂×C₄	C₂²⋊C₈×C₁₅₁	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₈×C₁₅₁	1	1	yes	characteristic
259	C₂×C₄	C₂×M₁₆×C₁₅₁	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	4	1	no	—
264	C₈	—	4	1	no	—
265	C₈	AGL(1,ℤ/8)×C₁₅₁	1	1	yes	characteristic
266	C₈	#137×C₁₅₁	1	1	yes	characteristic
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	C₂×C₂²⋊C₈×C₁₅₁	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	1	no	—
281	C₄	#349×C₁₅₁	1	2	yes	—
282	C₄	#417×C₁₅₁	1	1	yes	characteristic
283	C₄	#421×C₁₅₁	1	1	yes	characteristic
284	C₂	—	4	1	no	—
285	C₂	—	8	1	no	—
286	C₂	#1477×C₁₅₁	1	1	yes	characteristic
287	C₂	C₂×#349×C₁₅₁	1	1	yes	characteristic
288	C₂	#2117×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₅₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₁₅₁
Frattini, cosocle	C₄×C₈	C₂³×C₁₅₁
socle	C₂²×C₁₅₁	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₁₅₁	C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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. #2163×C₁₅₁	C₂	semidirect
3. #2150×C₁₅₁	C₂	semidirect
4. #1338×C₁₅₁	C₂	semidirect
5. #1330×C₁₅₁	C₂	semidirect
144. #11628	C₁₅₁	direct (N × H)
145. C₁₅₁	#11628	direct (N × H)
147. #2163	C₂×C₁₅₁	semidirect
148. #2150	C₂×C₁₅₁	semidirect
149. #1338	C₂×C₁₅₁	semidirect
150. #1330	C₂×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#2965×C₁₅₁	2
#2163×C₁₅₁	2
#2150×C₁₅₁	2
#1338×C₁₅₁	2
#1330×C₁₅₁	2
#11628	151

Subgroups — C151×#11628 #50316553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#11628 #50316553