Subgroups — C₁₅₁×#13189 #50318114

Order 38656.

534 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊C₁₆×C₁₅₁	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₁₆×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₈×C₁₅₁	C₂²	1	2	yes	—
6	C₂×C₄×D₄×C₁₅₁	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₅₁	C₂²	1	2	yes	—
8	C₂²×C₁₆×C₁₅₁	C₂²	1	2	yes	—
9	C₈×D₄×C₁₅₁	—	2	2	no	—
10	C₈×D₄×C₁₅₁	—	2	2	no	—
11	C₂×C₄⋊C₈×C₁₅₁	C₄	1	1	yes	characteristic
12	C₂×C₂²⋊C₈×C₁₅₁	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₁₅₁	—	2	2	no	—
15	C₂²⋊C₁₆×C₁₅₁	—	2	8	no	—
16	C₄×C₁₆×C₁₅₁	—	2	2	no	—
17	C₂²×D₄×C₁₅₁	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₅₁	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₁₅₁	—	2	4	no	—
20	C₂²×C₈×C₁₅₁	—	2	2	no	—
21	C₂²×C₈×C₁₅₁	—	2	2	no	—
22	C₂²×C₈×C₁₅₁	—	2	4	no	—
23	C₂²×C₈×C₁₅₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
26	C₄×D₄×C₁₅₁	—	2	2	no	—
27	C₄×D₄×C₁₅₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₅₁	C₈	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₁₅₁	C₈	1	2	yes	—
30	C₂×C₄²×C₁₅₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₁₅₁	—	2	2	no	—
32	C₂×C₁₆×C₁₅₁	—	2	2	no	—
33	C₂×C₁₆×C₁₅₁	—	2	4	no	—
34	C₂×C₁₆×C₁₅₁	—	2	2	no	—
35	C₂×C₁₆×C₁₅₁	D₄	1	4	yes	complemented
36	C₄⋊C₈×C₁₅₁	—	2	2	no	—
37	C₂²⋊C₈×C₁₅₁	—	2	4	no	—
38	C₄×C₈×C₁₅₁	D₄	1	2	yes	—
39	C₄×C₈×C₁₅₁	D₄	1	2	yes	—
40	C₂⁴×C₁₅₁	C₂×C₈	1	2	yes	—
41	C₂×D₄×C₁₅₁	—	2	2	no	—
42	C₂×D₄×C₁₅₁	—	2	2	no	—
43	C₂×D₄×C₁₅₁	C₁₆	1	4	yes	complemented
44	C₂²×C₄×C₁₅₁	—	2	2	no	—
45	C₂²×C₄×C₁₅₁	—	2	4	no	—
46	C₂²×C₄×C₁₅₁	—	2	2	no	—
47	C₂²×C₄×C₁₅₁	—	2	4	no	—
48	C₂²×C₄×C₁₅₁	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₅₁	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₅₁	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₅₁	—	2	2	no	—
53	C₂×C₈×C₁₅₁	—	2	2	no	—
54	C₂×C₈×C₁₅₁	—	2	4	no	—
55	C₂×C₈×C₁₅₁	—	4	4	no	—
56	C₂×C₈×C₁₅₁	—	4	2	no	—
57	C₂×C₈×C₁₅₁	—	4	2	no	—
58	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₅₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₅₁	#41	1	2	yes	—
63	C₄⋊C₄×C₁₅₁	—	2	2	no	—
64	C₂²⋊C₄×C₁₅₁	—	2	4	no	—
65	C₄²×C₁₅₁	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₁₅₁	M₁₆	1	2	yes	—
67	C₁₆×C₁₅₁	—	2	4	no	—
68	C₁₆×C₁₅₁	—	4	2	no	—
69	C₂³×C₁₅₁	—	2	2	no	—
70	C₂³×C₁₅₁	—	2	4	no	—
71	C₂³×C₁₅₁	—	2	2	no	—
72	C₂³×C₁₅₁	—	2	4	no	—
73	C₂³×C₁₅₁	C₂×C₁₆	1	4	yes	—
74	C₂³×C₁₅₁	C₂²×C₈	1	1	yes	characteristic
75	D₄×C₁₅₁	—	2	8	no	—
76	C₂×C₄×C₁₅₁	—	2	2	no	—
77	C₂×C₄×C₁₅₁	—	2	2	no	—
78	C₂×C₄×C₁₅₁	—	2	4	no	—
79	C₂×C₄×C₁₅₁	—	4	2	no	—
80	C₂×C₄×C₁₅₁	—	4	2	no	—
81	C₂×C₄×C₁₅₁	—	4	4	no	—
82	C₂×C₄×C₁₅₁	C₂²⋊C₈	1	2	yes	—
83	C₂×C₄×C₁₅₁	C₂²⋊C₈	1	2	yes	—
84	C₂×C₄×C₁₅₁	C₂×C₁₆	1	2	yes	—
85	C₂×C₄×C₁₅₁	M₃₂	1	2	yes	—
86	C₂×C₄×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₅₁	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₁₅₁	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₅₁	#131	1	2	yes	—
90	C₈×C₁₅₁	—	2	2	no	—
91	C₈×C₁₅₁	—	4	2	no	—
92	C₈×C₁₅₁	#121	1	2	yes	—
93	C₈×C₁₅₁	#124	1	1	yes	characteristic
94	C₈×C₁₅₁	#127	1	1	yes	characteristic
95	C₂²×C₁₅₁	—	2	4	no	—
96	C₂²×C₁₅₁	—	2	8	no	—
97	C₂²×C₁₅₁	—	4	2	no	—
98	C₂²×C₁₅₁	—	4	4	no	—
99	C₂²×C₁₅₁	—	4	2	no	—
100	C₂²×C₁₅₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
101	C₂²×C₁₅₁	C₈×D₄	1	2	yes	—
102	C₂²×C₁₅₁	C₂²×C₁₆	1	1	yes	characteristic
103	C₂²×C₁₅₁	C₂×M₃₂	1	1	yes	characteristic
104	C₂²×C₁₅₁	#504	1	2	yes	—
105	C₄×C₁₅₁	—	2	2	no	—
106	C₄×C₁₅₁	—	4	2	no	—
107	C₄×C₁₅₁	C₂²⋊C₁₆	1	4	yes	complemented
108	C₄×C₁₅₁	#390	1	1	yes	characteristic
109	C₄×C₁₅₁	#408	1	1	yes	characteristic
110	C₄×C₁₅₁	#436	1	2	yes	—
111	C₂×C₁₅₁	—	4	4	no	—
112	C₂×C₁₅₁	#1954	1	1	yes	characteristic
113	C₂×C₁₅₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
114	C₂×C₁₅₁	#2113	1	1	yes	characteristic
115	C₂×C₁₅₁	D₄×C₁₆	1	2	yes	—
116	C₂×C₁₅₁	#2169	1	2	yes	—
117	#13189	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₁₅₁	#13189	1	1	yes	characteristic, direct product factor, Sylow
119	C₂×C₈×D₄	C₂×C₁₅₁	1	1	yes	characteristic
120	C₂×C₄⋊C₁₆	C₂×C₁₅₁	1	1	yes	characteristic, complemented
121	C₂×C₂²⋊C₁₆	C₂×C₁₅₁	1	4	yes	complemented
122	C₂×C₄×C₁₆	C₂×C₁₅₁	1	1	yes	characteristic, complemented
123	C₂³×C₈	C₂²×C₁₅₁	1	2	yes	—
124	C₂×C₄×D₄	C₄×C₁₅₁	1	1	yes	characteristic
125	C₂²×C₁₆	C₂²×C₁₅₁	1	2	yes	—
126	C₂²×C₁₆	C₂²×C₁₅₁	1	2	yes	—
127	C₈×D₄	—	2	2	no	—
128	C₈×D₄	—	2	2	no	—
129	C₂×C₄⋊C₈	C₄×C₁₅₁	1	1	yes	characteristic
130	C₂×C₂²⋊C₈	C₄×C₁₅₁	1	2	yes	—
131	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
132	C₄⋊C₁₆	—	2	2	no	—
133	C₂²⋊C₁₆	—	2	8	no	—
134	C₄×C₁₆	—	2	2	no	—
135	C₂²×D₄	C₈×C₁₅₁	1	1	yes	characteristic
136	C₂³×C₄	C₂×C₄×C₁₅₁	1	2	yes	—
137	C₂²×C₈	—	2	4	no	—
138	C₂²×C₈	—	2	4	no	—
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	C₂×C₄×C₁₅₁	1	2	yes	—
142	C₂²×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
144	C₄×D₄	—	2	2	no	—
145	C₄×D₄	—	2	2	no	—
146	C₂×C₄⋊C₄	C₈×C₁₅₁	1	1	yes	characteristic
147	C₂×C₂²⋊C₄	C₈×C₁₅₁	1	2	yes	—
148	C₂×C₄²	C₂×C₄×C₁₅₁	1	1	yes	characteristic
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	4	no	—
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	D₄×C₁₅₁	1	4	yes	complemented
154	C₄⋊C₈	—	2	2	no	—
155	C₂²⋊C₈	—	2	4	no	—
156	C₄×C₈	D₄×C₁₅₁	1	2	yes	—
157	C₄×C₈	D₄×C₁₅₁	1	2	yes	—
158	C₂⁴	C₂×C₈×C₁₅₁	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	C₁₆×C₁₅₁	1	4	yes	complemented
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	C₂×C₈×C₁₅₁	1	2	yes	—
167	C₂²×C₄	C₂×C₈×C₁₅₁	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₈×C₁₅₁	1	1	yes	characteristic
169	C₂²×C₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	4	4	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	2	yes	—
177	C₂×C₈	C₂×D₄×C₁₅₁	1	1	yes	characteristic
178	C₂×C₈	C₂×D₄×C₁₅₁	1	2	yes	—
179	C₂×C₈	C₂×D₄×C₁₅₁	1	1	yes	characteristic
180	C₂×C₈	#41×C₁₅₁	1	2	yes	—
181	C₄⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	4	no	—
183	C₄²	C₂²⋊C₄×C₁₅₁	1	2	yes	—
184	C₄²	M₁₆×C₁₅₁	1	2	yes	—
185	C₁₆	—	2	4	no	—
186	C₁₆	—	4	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	4	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	4	no	—
191	C₂³	C₂×C₁₆×C₁₅₁	1	4	yes	—
192	C₂³	C₂²×C₈×C₁₅₁	1	1	yes	characteristic
193	D₄	—	2	8	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	C₂²⋊C₈×C₁₅₁	1	2	yes	—
201	C₂×C₄	C₂²⋊C₈×C₁₅₁	1	2	yes	—
202	C₂×C₄	C₂×C₁₆×C₁₅₁	1	2	yes	—
203	C₂×C₄	M₃₂×C₁₅₁	1	2	yes	—
204	C₂×C₄	C₂×C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
205	C₂×C₄	C₄×D₄×C₁₅₁	1	2	yes	—
206	C₂×C₄	C₂×M₁₆×C₁₅₁	1	1	yes	characteristic
207	C₂×C₄	#131×C₁₅₁	1	2	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	4	2	no	—
210	C₈	#121×C₁₅₁	1	2	yes	—
211	C₈	#124×C₁₅₁	1	1	yes	characteristic
212	C₈	#127×C₁₅₁	1	1	yes	characteristic
213	C₂²	—	2	4	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	4	no	—
217	C₂²	—	4	2	no	—
218	C₂²	C₂×C₂²⋊C₈×C₁₅₁	1	1	yes	characteristic
219	C₂²	C₈×D₄×C₁₅₁	1	2	yes	—
220	C₂²	C₂²×C₁₆×C₁₅₁	1	1	yes	characteristic
221	C₂²	C₂×M₃₂×C₁₅₁	1	1	yes	characteristic
222	C₂²	#504×C₁₅₁	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	C₂²⋊C₁₆×C₁₅₁	1	4	yes	complemented
226	C₄	#390×C₁₅₁	1	1	yes	characteristic
227	C₄	#408×C₁₅₁	1	1	yes	characteristic
228	C₄	#436×C₁₅₁	1	2	yes	—
229	C₂	—	4	4	no	—
230	C₂	#1954×C₁₅₁	1	1	yes	characteristic
231	C₂	C₂×C₂²⋊C₁₆×C₁₅₁	1	1	yes	characteristic
232	C₂	#2113×C₁₅₁	1	1	yes	characteristic
233	C₂	D₄×C₁₆×C₁₅₁	1	2	yes	—
234	C₂	#2169×C₁₅₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₁₅₁	C₂³
commutator	C₂²	C₂²×C₁₆×C₁₅₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₅₁
socle	C₂³×C₁₅₁	C₂²×C₈
prime radical	C₂²×D₄×C₁₅₁	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊C₁₆×C₁₅₁	C₂	semidirect
3. C₂×C₂²⋊C₁₆×C₁₅₁	C₂	semidirect
4. C₂×C₄×C₁₆×C₁₅₁	C₂	semidirect
35. C₂×C₁₆×C₁₅₁	D₄	semidirect
43. C₂×D₄×C₁₅₁	C₁₆	semidirect
107. C₄×C₁₅₁	C₂²⋊C₁₆	semidirect
117. #13189	C₁₅₁	direct (N × H)
118. C₁₅₁	#13189	direct (N × H)
120. C₂×C₄⋊C₁₆	C₂×C₁₅₁	semidirect
121. C₂×C₂²⋊C₁₆	C₂×C₁₅₁	semidirect
122. C₂×C₄×C₁₆	C₂×C₁₅₁	semidirect
153. C₂×C₁₆	D₄×C₁₅₁	semidirect
161. C₂×D₄	C₁₆×C₁₅₁	semidirect
225. C₄	C₂²⋊C₁₆×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₁₅₁	2
C₂×C₄⋊C₁₆×C₁₅₁	2
C₂×C₂²⋊C₁₆×C₁₅₁	2
C₂×C₄×C₁₆×C₁₅₁	2
#13189	151

Subgroups — C151×#13189 #50318114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#13189 #50318114