Subgroups — C₁₅₁×#22256 #50327181

Order 38656.

614 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3160×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#2960×C₁₅₁	C₂	1	2	yes	maximal
3	#1994×C₁₅₁	C₂	1	2	yes	maximal
4	#1922×C₁₅₁	C₂	1	8	yes	maximal
5	#1837×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	#1744×C₁₅₁	C₂	1	1	yes	maximal, characteristic
7	#533×C₁₅₁	C₂²	1	2	yes	—
8	#500×C₁₅₁	C₄	1	1	yes	characteristic, complemented
9	#500×C₁₅₁	C₄	1	1	yes	characteristic, complemented
10	#499×C₁₅₁	C₄	1	4	yes	complemented
11	#498×C₁₅₁	C₄	1	2	yes	complemented
12	#446×C₁₅₁	—	2	2	no	—
13	#446×C₁₅₁	—	2	2	no	—
14	C₈×Q₈×C₁₅₁	—	2	2	no	—
15	C₈×Q₈×C₁₅₁	—	2	2	no	—
16	#427×C₁₅₁	C₂²	1	4	yes	—
17	#424×C₁₅₁	C₂²	1	4	yes	—
18	#423×C₁₅₁	C₂²	1	2	yes	—
19	#405×C₁₅₁	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
22	C₄³×C₁₅₁	C₂²	1	1	yes	characteristic
23	#341×C₁₅₁	C₂²	1	4	yes	—
24	#339×C₁₅₁	C₂²	1	8	yes	—
25	#336×C₁₅₁	C₂²	1	4	yes	—
26	C₂×M₁₆×C₁₅₁	C₂³	1	4	yes	—
27	C₂²×C₈×C₁₅₁	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₅₁	C₂³	1	2	yes	—
29	C₄⋊Q₈×C₁₅₁	C₂×C₄	1	2	yes	—
30	C₄⋊Q₈×C₁₅₁	C₂×C₄	1	2	yes	—
31	#125×C₁₅₁	C₂×C₄	1	4	yes	—
32	C₂²⋊Q₈×C₁₅₁	—	2	2	no	—
33	C₂²⋊Q₈×C₁₅₁	—	2	2	no	—
34	C₄×Q₈×C₁₅₁	—	2	2	no	—
35	C₄×Q₈×C₁₅₁	—	2	2	no	—
36	#117×C₁₅₁	C₂³	1	4	yes	—
37	C₂×C₄²×C₁₅₁	—	2	1	no	—
38	C₂×C₄²×C₁₅₁	—	2	1	no	—
39	C₂×C₄²×C₁₅₁	C₂³	1	2	yes	—
40	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
41	#107×C₁₅₁	C₂×C₄	1	4	yes	—
42	#107×C₁₅₁	C₂×C₄	1	4	yes	—
43	#106×C₁₅₁	C₂×C₄	1	8	yes	—
44	C₄⋊C₈×C₁₅₁	—	2	4	no	—
45	C₄⋊C₈×C₁₅₁	—	2	4	no	—
46	C₄⋊C₈×C₁₅₁	—	2	2	no	—
47	C₄⋊C₈×C₁₅₁	—	2	2	no	—
48	#97×C₁₅₁	—	2	4	no	—
49	C₄×C₈×C₁₅₁	—	2	4	no	—
50	C₄×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₁₅₁	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₅₁	D₄	1	2	yes	—
55	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
56	C₂×Q₈×C₁₅₁	—	2	2	no	—
57	C₂×Q₈×C₁₅₁	—	2	2	no	—
58	C₂²×C₄×C₁₅₁	—	2	2	no	—
59	C₂²×C₄×C₁₅₁	#41	1	1	yes	characteristic
60	C₂²×C₄×C₁₅₁	#41	1	1	yes	characteristic
61	C₂²×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
62	M₁₆×C₁₅₁	—	2	8	no	—
63	C₂×C₈×C₁₅₁	—	2	4	no	—
64	C₂×C₈×C₁₅₁	—	2	4	no	—
65	C₂×C₈×C₁₅₁	—	2	4	no	—
66	C₂×C₈×C₁₅₁	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₁₅₁	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₅₁	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₅₁	C₂×D₄	1	2	yes	—
70	C₄⋊C₄×C₁₅₁	—	2	4	no	—
71	C₄⋊C₄×C₁₅₁	—	2	4	no	—
72	C₄⋊C₄×C₁₅₁	—	2	2	no	—
73	C₄⋊C₄×C₁₅₁	—	2	2	no	—
74	C₄⋊C₄×C₁₅₁	C₂²×C₄	1	4	yes	—
75	C₄⋊C₄×C₁₅₁	C₂²×C₄	1	4	yes	—
76	C₂²⋊C₄×C₁₅₁	—	2	4	no	—
77	C₄²×C₁₅₁	—	2	4	no	—
78	C₄²×C₁₅₁	—	2	4	no	—
79	C₄²×C₁₅₁	—	2	4	no	—
80	C₄²×C₁₅₁	—	4	2	no	—
81	C₄²×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
85	C₂³×C₁₅₁	C₂×#41	1	1	yes	characteristic
86	Q₈×C₁₅₁	—	4	4	no	—
87	Q₈×C₁₅₁	—	4	4	no	—
88	C₂×C₄×C₁₅₁	—	2	2	no	—
89	C₂×C₄×C₁₅₁	—	2	2	no	—
90	C₂×C₄×C₁₅₁	—	2	4	no	—
91	C₂×C₄×C₁₅₁	—	2	4	no	—
92	C₂×C₄×C₁₅₁	—	2	4	no	—
93	C₂×C₄×C₁₅₁	—	4	2	no	—
94	C₂×C₄×C₁₅₁	—	4	2	no	—
95	C₂×C₄×C₁₅₁	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₅₁	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₅₁	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₅₁	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₅₁	2₋¹⁺⁴	1	4	yes	—
102	C₈×C₁₅₁	—	2	2	no	—
103	C₈×C₁₅₁	—	2	2	no	—
104	C₈×C₁₅₁	—	2	2	no	—
105	C₈×C₁₅₁	—	2	2	no	—
106	C₈×C₁₅₁	—	4	4	no	—
107	C₂²×C₁₅₁	—	2	1	no	—
108	C₂²×C₁₅₁	—	2	1	no	—
109	C₂²×C₁₅₁	C₂×C₄×D₄	1	1	yes	characteristic
110	C₂²×C₁₅₁	#541	1	2	yes	—
111	C₄×C₁₅₁	—	2	1	no	—
112	C₄×C₁₅₁	—	2	1	no	—
113	C₄×C₁₅₁	—	4	4	no	—
114	C₄×C₁₅₁	—	4	4	no	—
115	C₄×C₁₅₁	—	4	4	no	—
116	C₄×C₁₅₁	#443	1	2	yes	—
117	C₄×C₁₅₁	#444	1	2	yes	—
118	C₄×C₁₅₁	#519	1	2	yes	—
119	C₄×C₁₅₁	#536	1	2	yes	—
120	C₂×C₁₅₁	—	2	2	no	—
121	C₂×C₁₅₁	#2327	1	1	yes	characteristic
122	C₂×C₁₅₁	C₂×#443	1	1	yes	characteristic
123	C₂×C₁₅₁	C₂×#444	1	1	yes	characteristic
124	#22256	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₁₅₁	#22256	1	1	yes	characteristic, direct product factor, Sylow
126	#3160	C₂×C₁₅₁	1	1	yes	characteristic
127	#2960	C₂×C₁₅₁	1	2	yes	—
128	#1994	C₂×C₁₅₁	1	2	yes	—
129	#1922	C₂×C₁₅₁	1	8	yes	—
130	#1837	C₂×C₁₅₁	1	1	yes	characteristic
131	#1744	C₂×C₁₅₁	1	1	yes	characteristic
132	#533	C₂²×C₁₅₁	1	2	yes	—
133	#500	C₄×C₁₅₁	1	1	yes	characteristic, complemented
134	#500	C₄×C₁₅₁	1	1	yes	characteristic, complemented
135	#499	C₄×C₁₅₁	1	4	yes	complemented
136	#498	C₄×C₁₅₁	1	2	yes	complemented
137	#446	—	2	2	no	—
138	#446	—	2	2	no	—
139	C₈×Q₈	—	2	2	no	—
140	C₈×Q₈	—	2	2	no	—
141	#427	C₂²×C₁₅₁	1	4	yes	—
142	#424	C₂²×C₁₅₁	1	4	yes	—
143	#423	C₂²×C₁₅₁	1	2	yes	—
144	#405	C₂²×C₁₅₁	1	4	yes	—
145	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
147	C₄³	C₂²×C₁₅₁	1	1	yes	characteristic
148	#341	C₂²×C₁₅₁	1	4	yes	—
149	#339	C₂²×C₁₅₁	1	8	yes	—
150	#336	C₂²×C₁₅₁	1	4	yes	—
151	C₂×M₁₆	C₂³×C₁₅₁	1	4	yes	—
152	C₂²×C₈	C₂³×C₁₅₁	1	2	yes	—
153	C₂²×C₈	C₂³×C₁₅₁	1	2	yes	—
154	C₄⋊Q₈	C₂×C₄×C₁₅₁	1	2	yes	—
155	C₄⋊Q₈	C₂×C₄×C₁₅₁	1	2	yes	—
156	#125	C₂×C₄×C₁₅₁	1	4	yes	—
157	C₂²⋊Q₈	—	2	2	no	—
158	C₂²⋊Q₈	—	2	2	no	—
159	C₄×Q₈	—	2	2	no	—
160	C₄×Q₈	—	2	2	no	—
161	#117	C₂³×C₁₅₁	1	4	yes	—
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	C₂³×C₁₅₁	1	2	yes	—
165	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
166	#107	C₂×C₄×C₁₅₁	1	4	yes	—
167	#107	C₂×C₄×C₁₅₁	1	4	yes	—
168	#106	C₂×C₄×C₁₅₁	1	8	yes	—
169	C₄⋊C₈	—	2	4	no	—
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	2	2	no	—
173	#97	—	2	4	no	—
174	C₄×C₈	—	2	4	no	—
175	C₄×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
176	C₄×C₈	C₂×C₄×C₁₅₁	1	2	yes	—
177	C₄×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
178	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
179	C₄×C₈	D₄×C₁₅₁	1	2	yes	—
180	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	#41×C₁₅₁	1	1	yes	characteristic
185	C₂²×C₄	#41×C₁₅₁	1	1	yes	characteristic
186	C₂²×C₄	C₂⁴×C₁₅₁	1	1	yes	characteristic
187	M₁₆	—	2	8	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	C₂²×C₄×C₁₅₁	1	2	yes	—
192	C₂×C₈	C₂²×C₄×C₁₅₁	1	2	yes	—
193	C₂×C₈	C₂×D₄×C₁₅₁	1	2	yes	—
194	C₂×C₈	C₂×D₄×C₁₅₁	1	2	yes	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	C₂²×C₄×C₁₅₁	1	4	yes	—
200	C₄⋊C₄	C₂²×C₄×C₁₅₁	1	4	yes	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	—	4	2	no	—
206	C₄²	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
207	C₄²	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
208	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
209	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
210	C₂³	C₂×#41×C₁₅₁	1	1	yes	characteristic
211	Q₈	—	4	4	no	—
212	Q₈	—	4	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	C₄×D₄×C₁₅₁	1	1	yes	characteristic
221	C₂×C₄	C₄×D₄×C₁₅₁	1	1	yes	characteristic
222	C₂×C₄	C₄×D₄×C₁₅₁	1	1	yes	characteristic
223	C₂×C₄	C₄×D₄×C₁₅₁	1	1	yes	characteristic
224	C₂×C₄	C₂³×C₄×C₁₅₁	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
226	C₂×C₄	2₋¹⁺⁴×C₁₅₁	1	4	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₈	—	4	4	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	C₂×C₄×D₄×C₁₅₁	1	1	yes	characteristic
235	C₂²	#541×C₁₅₁	1	2	yes	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	4	no	—
239	C₄	—	4	4	no	—
240	C₄	—	4	4	no	—
241	C₄	#443×C₁₅₁	1	2	yes	—
242	C₄	#444×C₁₅₁	1	2	yes	—
243	C₄	#519×C₁₅₁	1	2	yes	—
244	C₄	#536×C₁₅₁	1	2	yes	—
245	C₂	—	2	2	no	—
246	C₂	#2327×C₁₅₁	1	1	yes	characteristic
247	C₂	C₂×#443×C₁₅₁	1	1	yes	characteristic
248	C₂	C₂×#444×C₁₅₁	1	1	yes	characteristic

	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	C₂³×C₁₅₁	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
8. #500×C₁₅₁	C₄	semidirect
9. #500×C₁₅₁	C₄	semidirect
10. #499×C₁₅₁	C₄	semidirect
11. #498×C₁₅₁	C₄	semidirect
124. #22256	C₁₅₁	direct (N × H)
125. C₁₅₁	#22256	direct (N × H)
133. #500	C₄×C₁₅₁	semidirect
134. #500	C₄×C₁₅₁	semidirect
135. #499	C₄×C₁₅₁	semidirect
136. #498	C₄×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#3160×C₁₅₁	2
#2960×C₁₅₁	2
#1994×C₁₅₁	2
#1922×C₁₅₁	2
#1837×C₁₅₁	2
#1744×C₁₅₁	2
#22256	151

Subgroups — C151×#22256 #50327181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#22256 #50327181