Subgroups — C₁₅₁×#22236 #50327161

Order 38656.

678 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3131×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#2961×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2932×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	#1995×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1963×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	#1929×C₁₅₁	C₂	1	4	yes	maximal, complemented
7	#1866×C₁₅₁	C₂	1	4	yes	maximal, complemented
8	#1848×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#1746×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131×C₁₅₁	C₂²	1	2	yes	—
11	#533×C₁₅₁	C₂²	1	1	yes	characteristic
12	#532×C₁₅₁	C₂²	1	1	yes	characteristic
13	#488×C₁₅₁	C₄	1	4	yes	complemented
14	#487×C₁₅₁	C₄	1	2	yes	complemented
15	#486×C₁₅₁	C₄	1	2	yes	complemented
16	#446×C₁₅₁	—	2	4	no	—
17	#435×C₁₅₁	—	2	4	no	—
18	C₂×#108×C₁₅₁	C₂²	1	4	yes	—
19	#424×C₁₅₁	C₂²	1	2	yes	—
20	#423×C₁₅₁	C₂²	1	1	yes	characteristic
21	#423×C₁₅₁	C₂²	1	1	yes	characteristic
22	C₂×#104×C₁₅₁	C₂²	1	4	yes	—
23	#416×C₁₅₁	C₂²	1	4	yes	—
24	#408×C₁₅₁	C₂²	1	2	yes	—
25	#405×C₁₅₁	C₂²	1	2	yes	—
26	C₂×#97×C₁₅₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
28	#376×C₁₅₁	C₂²	1	1	yes	characteristic
29	#339×C₁₅₁	C₂²	1	4	yes	—
30	#336×C₁₅₁	C₂²	1	2	yes	—
31	#336×C₁₅₁	C₂²	1	2	yes	—
32	C₂×#41×C₁₅₁	C₂³	1	2	yes	—
33	#131×C₁₅₁	—	4	4	no	—
34	C₂×M₁₆×C₁₅₁	—	2	2	no	—
35	C₂×M₁₆×C₁₅₁	C₂³	1	2	yes	—
36	C₂×M₁₆×C₁₅₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₅₁	—	2	2	no	—
38	C₂²×C₈×C₁₅₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₁₅₁	C₂×C₄	1	2	yes	—
42	C₄⋊Q₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
43	#127×C₁₅₁	C₂×C₄	1	1	yes	characteristic
44	#125×C₁₅₁	C₂×C₄	1	2	yes	—
45	#124×C₁₅₁	C₂×C₄	1	2	yes	—
46	C₂²⋊Q₈×C₁₅₁	—	2	2	no	—
47	#121×C₁₅₁	—	2	2	no	—
48	C₄×Q₈×C₁₅₁	—	2	2	no	—
49	C₄×D₄×C₁₅₁	—	2	2	no	—
50	#117×C₁₅₁	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₅₁	C₂³	1	2	yes	—
52	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
53	#108×C₁₅₁	—	4	4	no	—
54	C₄⋊C₈×C₁₅₁	—	2	4	no	—
55	C₄⋊C₈×C₁₅₁	—	2	2	no	—
56	C₄⋊C₈×C₁₅₁	—	2	2	no	—
57	#104×C₁₅₁	—	4	4	no	—
58	#103×C₁₅₁	C₂×C₄	1	8	yes	—
59	#102×C₁₅₁	C₂×C₄	1	8	yes	—
60	C₂²⋊C₈×C₁₅₁	—	2	4	no	—
61	#97×C₁₅₁	—	2	2	no	—
62	#97×C₁₅₁	D₄	1	4	yes	—
63	C₄×C₈×C₁₅₁	—	2	2	no	—
64	C₄×C₈×C₁₅₁	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₁₅₁	C₂×C₄	1	2	yes	—
66	#95×C₁₅₁	—	2	2	no	—
67	#41×C₁₅₁	—	2	4	no	—
68	#41×C₁₅₁	—	2	4	no	—
69	C₂×Q₈×C₁₅₁	—	2	2	no	—
70	C₂×Q₈×C₁₅₁	C₂²×C₄	1	2	yes	—
71	C₂×D₄×C₁₅₁	—	2	2	no	—
72	C₂×D₄×C₁₅₁	—	2	2	no	—
73	C₂×D₄×C₁₅₁	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₁₅₁	—	2	2	no	—
75	C₂²×C₄×C₁₅₁	—	2	2	no	—
76	C₂²×C₄×C₁₅₁	#41	1	2	yes	—
77	C₂²×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
78	M₁₆×C₁₅₁	—	2	4	no	—
79	M₁₆×C₁₅₁	—	2	4	no	—
80	M₁₆×C₁₅₁	—	4	4	no	—
81	C₂×C₈×C₁₅₁	—	2	2	no	—
82	C₂×C₈×C₁₅₁	—	2	2	no	—
83	C₂×C₈×C₁₅₁	—	2	2	no	—
84	C₂×C₈×C₁₅₁	—	2	2	no	—
85	C₂×C₈×C₁₅₁	—	2	2	no	—
86	C₂×C₈×C₁₅₁	—	2	2	no	—
87	C₂×C₈×C₁₅₁	—	4	4	no	—
88	C₂×C₈×C₁₅₁	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₁₅₁	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₅₁	C₂×D₄	1	2	yes	—
91	C₄⋊C₄×C₁₅₁	—	2	2	no	—
92	C₄⋊C₄×C₁₅₁	—	2	4	no	—
93	C₄⋊C₄×C₁₅₁	—	2	2	no	—
94	C₄⋊C₄×C₁₅₁	C₂²×C₄	1	4	yes	—
95	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
96	C₂²⋊C₄×C₁₅₁	—	2	4	no	—
97	C₄²×C₁₅₁	—	2	2	no	—
98	C₄²×C₁₅₁	—	2	2	no	—
99	C₄²×C₁₅₁	—	2	2	no	—
100	C₄²×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₅₁	C₂×D₄	1	2	yes	—
103	C₂³×C₁₅₁	—	2	2	no	—
104	C₂³×C₁₅₁	C₂×#41	1	1	yes	characteristic
105	Q₈×C₁₅₁	—	2	4	no	—
106	Q₈×C₁₅₁	—	8	2	no	—
107	D₄×C₁₅₁	—	2	4	no	—
108	D₄×C₁₅₁	—	4	4	no	—
109	D₄×C₁₅₁	—	8	2	no	—
110	C₂×C₄×C₁₅₁	—	2	4	no	—
111	C₂×C₄×C₁₅₁	—	2	2	no	—
112	C₂×C₄×C₁₅₁	—	4	2	no	—
113	C₂×C₄×C₁₅₁	—	4	2	no	—
114	C₂×C₄×C₁₅₁	—	4	2	no	—
115	C₂×C₄×C₁₅₁	—	4	4	no	—
116	C₂×C₄×C₁₅₁	—	4	2	no	—
117	C₂×C₄×C₁₅₁	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₅₁	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₅₁	H(5,𝔽₂)	1	2	yes	—
122	C₂×C₄×C₁₅₁	2₋¹⁺⁴	1	2	yes	—
123	C₈×C₁₅₁	—	2	4	no	—
124	C₈×C₁₅₁	—	2	2	no	—
125	C₈×C₁₅₁	—	2	2	no	—
126	C₈×C₁₅₁	—	4	2	no	—
127	C₈×C₁₅₁	—	4	2	no	—
128	C₂²×C₁₅₁	—	2	2	no	—
129	C₂²×C₁₅₁	—	4	2	no	—
130	C₂²×C₁₅₁	—	4	4	no	—
131	C₂²×C₁₅₁	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂²×C₁₅₁	#542	1	2	yes	—
133	C₄×C₁₅₁	—	2	2	no	—
134	C₄×C₁₅₁	—	2	2	no	—
135	C₄×C₁₅₁	—	4	2	no	—
136	C₄×C₁₅₁	—	4	4	no	—
137	C₄×C₁₅₁	—	8	2	no	—
138	C₄×C₁₅₁	#520	1	2	yes	—
139	C₄×C₁₅₁	#535	1	1	yes	characteristic
140	C₄×C₁₅₁	#536	1	1	yes	characteristic
141	C₂×C₁₅₁	—	2	2	no	—
142	C₂×C₁₅₁	—	8	2	no	—
143	C₂×C₁₅₁	#2326	1	1	yes	characteristic
144	C₂×C₁₅₁	#2957	1	2	yes	—
145	#22236	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₁₅₁	#22236	1	1	yes	characteristic, direct product factor, Sylow
147	#3131	C₂×C₁₅₁	1	1	yes	characteristic
148	#2961	C₂×C₁₅₁	1	1	yes	characteristic, complemented
149	#2932	C₂×C₁₅₁	1	1	yes	characteristic
150	#1995	C₂×C₁₅₁	1	1	yes	characteristic, complemented
151	#1963	C₂×C₁₅₁	1	1	yes	characteristic
152	#1929	C₂×C₁₅₁	1	4	yes	complemented
153	#1866	C₂×C₁₅₁	1	4	yes	complemented
154	#1848	C₂×C₁₅₁	1	1	yes	characteristic, complemented
155	#1746	C₂×C₁₅₁	1	1	yes	characteristic, complemented
156	C₂×#131	C₂²×C₁₅₁	1	2	yes	—
157	#533	C₂²×C₁₅₁	1	1	yes	characteristic
158	#532	C₂²×C₁₅₁	1	1	yes	characteristic
159	#488	C₄×C₁₅₁	1	4	yes	complemented
160	#487	C₄×C₁₅₁	1	2	yes	complemented
161	#486	C₄×C₁₅₁	1	2	yes	complemented
162	#446	—	2	4	no	—
163	#435	—	2	4	no	—
164	C₂×#108	C₂²×C₁₅₁	1	4	yes	—
165	#424	C₂²×C₁₅₁	1	2	yes	—
166	#423	C₂²×C₁₅₁	1	1	yes	characteristic
167	#423	C₂²×C₁₅₁	1	1	yes	characteristic
168	C₂×#104	C₂²×C₁₅₁	1	4	yes	—
169	#416	C₂²×C₁₅₁	1	4	yes	—
170	#408	C₂²×C₁₅₁	1	2	yes	—
171	#405	C₂²×C₁₅₁	1	2	yes	—
172	C₂×#97	C₂²×C₁₅₁	1	1	yes	characteristic
173	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
174	#376	C₂²×C₁₅₁	1	1	yes	characteristic
175	#339	C₂²×C₁₅₁	1	4	yes	—
176	#336	C₂²×C₁₅₁	1	2	yes	—
177	#336	C₂²×C₁₅₁	1	2	yes	—
178	C₂×#41	C₂³×C₁₅₁	1	2	yes	—
179	#131	—	4	4	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	C₂³×C₁₅₁	1	2	yes	—
182	C₂×M₁₆	C₂³×C₁₅₁	1	2	yes	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	C₂³×C₁₅₁	1	2	yes	—
185	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
187	C₄⋊Q₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
188	C₄⋊Q₈	C₂×C₄×C₁₅₁	1	2	yes	—
189	#127	C₂×C₄×C₁₅₁	1	1	yes	characteristic
190	#125	C₂×C₄×C₁₅₁	1	2	yes	—
191	#124	C₂×C₄×C₁₅₁	1	2	yes	—
192	C₂²⋊Q₈	—	2	2	no	—
193	#121	—	2	2	no	—
194	C₄×Q₈	—	2	2	no	—
195	C₄×D₄	—	2	2	no	—
196	#117	C₂³×C₁₅₁	1	2	yes	—
197	C₂×C₄²	C₂³×C₁₅₁	1	2	yes	—
198	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
199	#108	—	4	4	no	—
200	C₄⋊C₈	—	2	4	no	—
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	2	2	no	—
203	#104	—	4	4	no	—
204	#103	C₂×C₄×C₁₅₁	1	8	yes	—
205	#102	C₂×C₄×C₁₅₁	1	8	yes	—
206	C₂²⋊C₈	—	2	4	no	—
207	#97	—	2	2	no	—
208	#97	D₄×C₁₅₁	1	4	yes	—
209	C₄×C₈	—	2	2	no	—
210	C₄×C₈	C₂×C₄×C₁₅₁	1	2	yes	—
211	C₄×C₈	C₂×C₄×C₁₅₁	1	2	yes	—
212	#95	—	2	2	no	—
213	#41	—	2	4	no	—
214	#41	—	2	4	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	C₂²×C₄×C₁₅₁	1	2	yes	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	C₂²×C₄×C₁₅₁	1	2	yes	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	#41×C₁₅₁	1	2	yes	—
223	C₂²×C₄	C₂⁴×C₁₅₁	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	4	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	4	4	no	—
234	C₂×C₈	C₂²×C₄×C₁₅₁	1	4	yes	—
235	C₂×C₈	C₂×D₄×C₁₅₁	1	2	yes	—
236	C₂×C₈	C₂×D₄×C₁₅₁	1	2	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	C₂²×C₄×C₁₅₁	1	4	yes	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	4	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
247	C₄²	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
248	C₄²	C₂×D₄×C₁₅₁	1	2	yes	—
249	C₂³	—	2	2	no	—
250	C₂³	C₂×#41×C₁₅₁	1	1	yes	characteristic
251	Q₈	—	2	4	no	—
252	Q₈	—	8	2	no	—
253	D₄	—	2	4	no	—
254	D₄	—	4	4	no	—
255	D₄	—	8	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	C₄×D₄×C₁₅₁	1	2	yes	—
264	C₂×C₄	C₄×D₄×C₁₅₁	1	2	yes	—
265	C₂×C₄	C₂³×C₄×C₁₅₁	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
267	C₂×C₄	H(5,𝔽₂)×C₁₅₁	1	2	yes	—
268	C₂×C₄	2₋¹⁺⁴×C₁₅₁	1	2	yes	—
269	C₈	—	2	4	no	—
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	—	4	2	no	—
273	C₈	—	4	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	—	4	2	no	—
277	C₂²	C₂×C₄×D₄×C₁₅₁	1	1	yes	characteristic
278	C₂²	#542×C₁₅₁	1	2	yes	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	4	no	—
282	C₄	—	4	2	no	—
283	C₄	—	8	2	no	—
284	C₄	#520×C₁₅₁	1	2	yes	—
285	C₄	#535×C₁₅₁	1	1	yes	characteristic
286	C₄	#536×C₁₅₁	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	8	2	no	—
289	C₂	#2326×C₁₅₁	1	1	yes	characteristic
290	C₂	#2957×C₁₅₁	1	2	yes	—

	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	#532×C₁₅₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₅₁ ◁ C₂×#97×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2961×C₁₅₁	C₂	semidirect
4. #1995×C₁₅₁	C₂	semidirect
6. #1929×C₁₅₁	C₂	semidirect
7. #1866×C₁₅₁	C₂	semidirect
8. #1848×C₁₅₁	C₂	semidirect
9. #1746×C₁₅₁	C₂	semidirect
13. #488×C₁₅₁	C₄	semidirect
14. #487×C₁₅₁	C₄	semidirect
15. #486×C₁₅₁	C₄	semidirect
145. #22236	C₁₅₁	direct (N × H)
146. C₁₅₁	#22236	direct (N × H)
148. #2961	C₂×C₁₅₁	semidirect
150. #1995	C₂×C₁₅₁	semidirect
152. #1929	C₂×C₁₅₁	semidirect
153. #1866	C₂×C₁₅₁	semidirect
154. #1848	C₂×C₁₅₁	semidirect
155. #1746	C₂×C₁₅₁	semidirect
159. #488	C₄×C₁₅₁	semidirect
160. #487	C₄×C₁₅₁	semidirect
161. #486	C₄×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#3131×C₁₅₁	2
#2961×C₁₅₁	2
#2932×C₁₅₁	2
#1995×C₁₅₁	2
#1963×C₁₅₁	2
#1929×C₁₅₁	2
#1866×C₁₅₁	2
#1848×C₁₅₁	2
#1746×C₁₅₁	2
#22236	151

Subgroups — C151×#22236 #50327161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#22236 #50327161