Subgroups — C₁₅₁×#8006 #50312931

Order 38656.

542 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2920×C₁₅₁	C₂	1	2	yes	maximal
2	C₂×#431×C₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#1293×C₁₅₁	C₂	1	4	yes	maximal
4	C₂²×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
5	#432×C₁₅₁	—	2	2	no	—
6	#432×C₁₅₁	C₄	1	4	yes	—
7	#431×C₁₅₁	—	2	2	no	—
8	#431×C₁₅₁	—	2	2	no	—
9	#431×C₁₅₁	—	2	2	no	—
10	#431×C₁₅₁	C₄	1	4	yes	—
11	#423×C₁₅₁	—	2	2	no	—
12	C₂×C₄⋊C₈×C₁₅₁	C₄	1	2	yes	—
13	#407×C₁₅₁	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₈×C₁₅₁	C₂²	1	2	yes	—
15	C₄×M₁₆×C₁₅₁	—	2	2	no	—
16	C₂×C₄×C₈×C₁₅₁	C₄	1	2	yes	—
17	C₂³×C₄×C₁₅₁	Q₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₅₁	Q₈	1	1	yes	characteristic
19	C₂³×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₅₁	—	2	4	no	—
21	C₂²×C₈×C₁₅₁	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₁₅₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₅₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₅₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₅₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₅₁	D₄	1	2	yes	—
27	C₂×C₄²×C₁₅₁	D₄	1	2	yes	—
28	C₂×C₄²×C₁₅₁	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₅₁	D₄	1	1	yes	characteristic
30	C₄⋊C₈×C₁₅₁	—	2	4	no	—
31	C₄⋊C₈×C₁₅₁	—	2	4	no	—
32	C₄⋊C₈×C₁₅₁	—	2	4	no	—
33	C₂²⋊C₈×C₁₅₁	—	2	4	no	—
34	C₂²⋊C₈×C₁₅₁	—	2	4	no	—
35	C₂²⋊C₈×C₁₅₁	C₂×C₄	1	8	yes	—
36	#97×C₁₅₁	—	2	4	no	—
37	C₄×C₈×C₁₅₁	—	2	2	no	—
38	C₄×C₈×C₁₅₁	—	2	2	no	—
39	C₄×C₈×C₁₅₁	—	2	4	no	—
40	C₂⁴×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₅₁	—	2	4	no	—
42	C₂²×C₄×C₁₅₁	—	4	2	no	—
43	C₂²×C₄×C₁₅₁	—	4	2	no	—
44	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₅₁	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
56	M₁₆×C₁₅₁	—	4	8	no	—
57	C₂×C₈×C₁₅₁	—	2	4	no	—
58	C₂×C₈×C₁₅₁	—	2	8	no	—
59	C₂×C₈×C₁₅₁	—	4	4	no	—
60	C₄²×C₁₅₁	C₄²	1	1	yes	characteristic
61	C₄²×C₁₅₁	C₄²	1	2	yes	—
62	C₄²×C₁₅₁	C₄²	1	1	yes	characteristic
63	C₄²×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₁₅₁	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₁₅₁	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₁₅₁	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₁₅₁	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₁₅₁	C₂²⋊C₄	1	2	yes	—
70	C₂³×C₁₅₁	—	2	2	no	—
71	C₂³×C₁₅₁	—	4	1	no	—
72	C₂³×C₁₅₁	—	4	1	no	—
73	C₂³×C₁₅₁	C₂×C₄⋊C₄	1	2	yes	—
74	C₂³×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₅₁	—	2	2	no	—
76	C₂×C₄×C₁₅₁	—	2	2	no	—
77	C₂×C₄×C₁₅₁	—	2	4	no	—
78	C₂×C₄×C₁₅₁	—	2	4	no	—
79	C₂×C₄×C₁₅₁	—	2	4	no	—
80	C₂×C₄×C₁₅₁	—	2	2	no	—
81	C₂×C₄×C₁₅₁	—	2	4	no	—
82	C₂×C₄×C₁₅₁	—	4	1	no	—
83	C₂×C₄×C₁₅₁	—	4	2	no	—
84	C₂×C₄×C₁₅₁	—	4	2	no	—
85	C₂×C₄×C₁₅₁	—	4	1	no	—
86	C₂×C₄×C₁₅₁	#95	1	1	yes	characteristic
87	C₂×C₄×C₁₅₁	#95	1	2	yes	—
88	C₂×C₄×C₁₅₁	#95	1	1	yes	characteristic
89	C₂×C₄×C₁₅₁	#95	1	1	yes	characteristic
90	C₂×C₄×C₁₅₁	#95	1	2	yes	—
91	C₂×C₄×C₁₅₁	#95	1	1	yes	characteristic
92	C₂×C₄×C₁₅₁	#108	1	4	yes	—
93	C₂×C₄×C₁₅₁	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₁₅₁	C₂×C₂²⋊C₄	1	2	yes	—
95	C₂×C₄×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₈×C₁₅₁	—	4	8	no	—
97	C₈×C₁₅₁	—	4	4	no	—
98	C₂²×C₁₅₁	—	2	1	no	—
99	C₂²×C₁₅₁	—	2	1	no	—
100	C₂²×C₁₅₁	—	2	2	no	—
101	C₂²×C₁₅₁	—	2	2	no	—
102	C₂²×C₁₅₁	—	2	2	no	—
103	C₂²×C₁₅₁	—	4	4	no	—
104	C₂²×C₁₅₁	C₂×#95	1	1	yes	characteristic
105	C₂²×C₁₅₁	C₂×#108	1	2	yes	—
106	C₄×C₁₅₁	—	2	1	no	—
107	C₄×C₁₅₁	—	2	2	no	—
108	C₄×C₁₅₁	—	2	1	no	—
109	C₄×C₁₅₁	—	2	1	no	—
110	C₄×C₁₅₁	—	2	2	no	—
111	C₄×C₁₅₁	—	2	1	no	—
112	C₄×C₁₅₁	#338	1	2	yes	—
113	C₄×C₁₅₁	#341	1	4	yes	—
114	C₄×C₁₅₁	#410	1	2	yes	—
115	C₂×C₁₅₁	—	2	2	no	—
116	C₂×C₁₅₁	—	4	2	no	—
117	C₂×C₁₅₁	C₂×#338	1	1	yes	characteristic
118	C₂×C₁₅₁	C₂×#341	1	1	yes	characteristic
119	C₂×C₁₅₁	#1740	1	1	yes	characteristic
120	#8006	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₁₅₁	#8006	1	1	yes	characteristic, direct product factor, Sylow
122	#2920	C₂×C₁₅₁	1	2	yes	—
123	C₂×#431	C₂×C₁₅₁	1	1	yes	characteristic
124	#1293	C₂×C₁₅₁	1	4	yes	—
125	C₂²×C₄²	C₂²×C₁₅₁	1	1	yes	characteristic
126	#432	—	2	2	no	—
127	#432	C₄×C₁₅₁	1	4	yes	—
128	#431	—	2	2	no	—
129	#431	—	2	2	no	—
130	#431	—	2	2	no	—
131	#431	C₄×C₁₅₁	1	4	yes	—
132	#423	—	2	2	no	—
133	C₂×C₄⋊C₈	C₄×C₁₅₁	1	2	yes	—
134	#407	C₂²×C₁₅₁	1	4	yes	—
135	C₂×C₂²⋊C₈	C₂²×C₁₅₁	1	2	yes	—
136	C₄×M₁₆	—	2	2	no	—
137	C₂×C₄×C₈	C₄×C₁₅₁	1	2	yes	—
138	C₂³×C₄	Q₈×C₁₅₁	1	1	yes	characteristic
139	C₂³×C₄	Q₈×C₁₅₁	1	1	yes	characteristic
140	C₂³×C₄	C₂³×C₁₅₁	1	1	yes	characteristic
141	C₂×M₁₆	—	2	4	no	—
142	C₂²×C₈	C₂×C₄×C₁₅₁	1	4	yes	—
143	C₂×C₄²	C₂×C₄×C₁₅₁	1	1	yes	characteristic
144	C₂×C₄²	C₂×C₄×C₁₅₁	1	1	yes	characteristic
145	C₂×C₄²	C₂×C₄×C₁₅₁	1	2	yes	—
146	C₂×C₄²	C₂×C₄×C₁₅₁	1	2	yes	—
147	C₂×C₄²	D₄×C₁₅₁	1	2	yes	—
148	C₂×C₄²	D₄×C₁₅₁	1	2	yes	—
149	C₂×C₄²	D₄×C₁₅₁	1	1	yes	characteristic
150	C₂×C₄²	D₄×C₁₅₁	1	1	yes	characteristic
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	—	2	4	no	—
153	C₄⋊C₈	—	2	4	no	—
154	C₂²⋊C₈	—	2	4	no	—
155	C₂²⋊C₈	—	2	4	no	—
156	C₂²⋊C₈	C₂×C₄×C₁₅₁	1	8	yes	—
157	#97	—	2	4	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	2	4	no	—
161	C₂⁴	C₂×Q₈×C₁₅₁	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
166	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	2	yes	—
167	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
168	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	2	yes	—
169	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	2	yes	—
170	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	2	yes	—
171	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
172	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
173	C₂²×C₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
174	C₂²×C₄	C₂²×C₄×C₁₅₁	1	2	yes	—
175	C₂²×C₄	C₂×D₄×C₁₅₁	1	2	yes	—
176	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
177	M₁₆	—	4	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	—	4	4	no	—
181	C₄²	C₄²×C₁₅₁	1	1	yes	characteristic
182	C₄²	C₄²×C₁₅₁	1	1	yes	characteristic
183	C₄²	C₄²×C₁₅₁	1	2	yes	—
184	C₄²	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
185	C₄²	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
186	C₄²	C₂²⋊C₄×C₁₅₁	1	2	yes	—
187	C₄²	C₂²⋊C₄×C₁₅₁	1	2	yes	—
188	C₄²	C₂²⋊C₄×C₁₅₁	1	2	yes	—
189	C₄²	C₂²⋊C₄×C₁₅₁	1	2	yes	—
190	C₄²	C₂²⋊C₄×C₁₅₁	1	2	yes	—
191	C₂³	—	2	2	no	—
192	C₂³	—	4	1	no	—
193	C₂³	—	4	1	no	—
194	C₂³	C₂×C₄⋊C₄×C₁₅₁	1	2	yes	—
195	C₂³	C₂×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	#95×C₁₅₁	1	1	yes	characteristic
208	C₂×C₄	#95×C₁₅₁	1	2	yes	—
209	C₂×C₄	#95×C₁₅₁	1	1	yes	characteristic
210	C₂×C₄	#95×C₁₅₁	1	1	yes	characteristic
211	C₂×C₄	#95×C₁₅₁	1	1	yes	characteristic
212	C₂×C₄	#95×C₁₅₁	1	2	yes	—
213	C₂×C₄	#108×C₁₅₁	1	4	yes	—
214	C₂×C₄	C₂×C₄²×C₁₅₁	1	1	yes	characteristic
215	C₂×C₄	C₂×C₂²⋊C₄×C₁₅₁	1	2	yes	—
216	C₂×C₄	C₂×C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
217	C₈	—	4	8	no	—
218	C₈	—	4	4	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	4	no	—
225	C₂²	C₂×#95×C₁₅₁	1	1	yes	characteristic
226	C₂²	C₂×#108×C₁₅₁	1	2	yes	—
227	C₄	—	2	1	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	#338×C₁₅₁	1	2	yes	—
234	C₄	#341×C₁₅₁	1	4	yes	—
235	C₄	#410×C₁₅₁	1	2	yes	—
236	C₂	—	2	2	no	—
237	C₂	—	4	2	no	—
238	C₂	C₂×#338×C₁₅₁	1	1	yes	characteristic
239	C₂	C₂×#341×C₁₅₁	1	1	yes	characteristic
240	C₂	#1740×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₅₁	#95
commutator	C₂×C₄	C₂×C₄²×C₁₅₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₅₁
socle	C₂²×C₁₅₁	C₂×#95
prime radical	C₂⁴×C₁₅₁	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₅₁ ◁ C₂²×C₄²×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
120. #8006	C₁₅₁	direct (N × H)
121. C₁₅₁	#8006	direct (N × H)

Maximal subgroups

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

subgroup	index
#2920×C₁₅₁	2
C₂×#431×C₁₅₁	2
#1293×C₁₅₁	2
#8006	151

Subgroups — C151×#8006 #50312931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#8006 #50312931