Subgroups — C₁₅₁×#13289 #50318214

Order 38656.

342 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	C₄×M₃₂×C₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#1848×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	#1387×C₁₅₁	C₂	1	4	yes	maximal
5	C₂×#125×C₁₅₁	—	2	1	no	—
6	C₂×M₃₂×C₁₅₁	C₂²	1	2	yes	—
7	#499×C₁₅₁	—	2	1	no	—
8	#499×C₁₅₁	—	2	1	no	—
9	#499×C₁₅₁	C₄	1	2	yes	—
10	#499×C₁₅₁	C₄	1	2	yes	—
11	C₂×#108×C₁₅₁	—	2	1	no	—
12	C₂×#108×C₁₅₁	C₂²	1	2	yes	—
13	C₂×#107×C₁₅₁	C₂²	1	2	yes	—
14	C₂×#106×C₁₅₁	—	2	1	no	—
15	#423×C₁₅₁	—	2	1	no	—
16	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
17	#346×C₁₅₁	—	2	1	no	—
18	C₄×C₁₆×C₁₅₁	—	2	1	no	—
19	C₂×M₁₆×C₁₅₁	—	2	2	no	—
20	C₂²×C₈×C₁₅₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
22	#125×C₁₅₁	—	2	2	no	—
23	#125×C₁₅₁	—	2	2	no	—
24	#125×C₁₅₁	—	4	2	no	—
25	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₅₁	—	4	2	no	—
27	C₂×C₄²×C₁₅₁	D₄	1	1	yes	characteristic
28	M₃₂×C₁₅₁	—	2	2	no	—
29	M₃₂×C₁₅₁	Q₈	1	4	yes	—
30	C₂×C₁₆×C₁₅₁	—	2	2	no	—
31	#108×C₁₅₁	—	2	4	no	—
32	#108×C₁₅₁	—	4	2	no	—
33	#107×C₁₅₁	—	2	2	no	—
34	#107×C₁₅₁	C₂×C₄	1	4	yes	—
35	#106×C₁₅₁	—	2	2	no	—
36	#106×C₁₅₁	—	2	2	no	—
37	C₄⋊C₈×C₁₅₁	—	4	2	no	—
38	C₄×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₅₁	—	4	2	no	—
43	C₂²×C₄×C₁₅₁	SD₁₆	1	2	yes	—
44	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
45	M₁₆×C₁₅₁	—	4	2	no	—
46	M₁₆×C₁₅₁	—	4	2	no	—
47	C₂×C₈×C₁₅₁	—	4	2	no	—
48	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₅₁	SD₁₆	1	2	yes	—
51	C₂×C₈×C₁₅₁	Q₁₆	1	2	yes	—
52	C₂×C₈×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₅₁	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₁₅₁	#41	1	2	yes	—
56	C₄⋊C₄×C₁₅₁	—	2	4	no	—
57	C₄⋊C₄×C₁₅₁	—	4	4	no	—
58	C₄⋊C₄×C₁₅₁	—	4	4	no	—
59	C₄⋊C₄×C₁₅₁	—	4	2	no	—
60	C₄²×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₁₅₁	D₈	1	2	yes	—
63	C₁₆×C₁₅₁	—	2	4	no	—
64	C₂³×C₁₅₁	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₁₅₁	—	4	2	no	—
66	C₂×C₄×C₁₅₁	—	4	4	no	—
67	C₂×C₄×C₁₅₁	—	8	4	no	—
68	C₂×C₄×C₁₅₁	#102	1	2	yes	—
69	C₂×C₄×C₁₅₁	#102	1	2	yes	—
70	C₂×C₄×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₅₁	#124	1	1	yes	characteristic
72	C₂×C₄×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₅₁	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₁₅₁	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₅₁	C₂×Q₁₆	1	1	yes	characteristic
76	C₈×C₁₅₁	—	8	2	no	—
77	C₈×C₁₅₁	#103	1	2	yes	—
78	C₈×C₁₅₁	#103	1	2	yes	—
79	C₈×C₁₅₁	C₄×Q₈	1	2	yes	—
80	C₈×C₁₅₁	C₂²⋊Q₈	1	2	yes	—
81	C₂²×C₁₅₁	—	2	2	no	—
82	C₂²×C₁₅₁	C₂×#102	1	1	yes	characteristic
83	C₂²×C₁₅₁	#487	1	1	yes	characteristic
84	C₂²×C₁₅₁	#498	1	1	yes	characteristic
85	C₄×C₁₅₁	—	2	2	no	—
86	C₄×C₁₅₁	—	2	2	no	—
87	C₄×C₁₅₁	—	8	4	no	—
88	C₄×C₁₅₁	#391	1	1	yes	characteristic
89	C₄×C₁₅₁	C₂×#103	1	1	yes	characteristic
90	C₄×C₁₅₁	#486	1	1	yes	characteristic
91	C₄×C₁₅₁	#500	1	1	yes	characteristic
92	C₂×C₁₅₁	—	2	1	no	—
93	C₂×C₁₅₁	—	2	1	no	—
94	C₂×C₁₅₁	#1990	1	1	yes	characteristic
95	C₂×C₁₅₁	#2148	1	2	yes	—
96	#13289	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₁₅₁	#13289	1	1	yes	characteristic, direct product factor, Sylow
98	C₂×#499	C₂×C₁₅₁	1	1	yes	characteristic
99	C₄×M₃₂	C₂×C₁₅₁	1	1	yes	characteristic
100	#1848	C₂×C₁₅₁	1	1	yes	characteristic
101	#1387	C₂×C₁₅₁	1	4	yes	—
102	C₂×#125	—	2	1	no	—
103	C₂×M₃₂	C₂²×C₁₅₁	1	2	yes	—
104	#499	—	2	1	no	—
105	#499	—	2	1	no	—
106	#499	C₄×C₁₅₁	1	2	yes	—
107	#499	C₄×C₁₅₁	1	2	yes	—
108	C₂×#108	—	2	1	no	—
109	C₂×#108	C₂²×C₁₅₁	1	2	yes	—
110	C₂×#107	C₂²×C₁₅₁	1	2	yes	—
111	C₂×#106	—	2	1	no	—
112	#423	—	2	1	no	—
113	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
114	#346	—	2	1	no	—
115	C₄×C₁₆	—	2	1	no	—
116	C₂×M₁₆	—	2	2	no	—
117	C₂²×C₈	D₄×C₁₅₁	1	1	yes	characteristic
118	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
119	#125	—	2	2	no	—
120	#125	—	2	2	no	—
121	#125	—	4	2	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	4	2	no	—
124	C₂×C₄²	D₄×C₁₅₁	1	1	yes	characteristic
125	M₃₂	—	2	2	no	—
126	M₃₂	Q₈×C₁₅₁	1	4	yes	—
127	C₂×C₁₆	—	2	2	no	—
128	#108	—	2	4	no	—
129	#108	—	4	2	no	—
130	#107	—	2	2	no	—
131	#107	C₂×C₄×C₁₅₁	1	4	yes	—
132	#106	—	2	2	no	—
133	#106	—	2	2	no	—
134	C₄⋊C₈	—	4	2	no	—
135	C₄×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
136	C₄×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
137	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
138	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	SD₁₆×C₁₅₁	1	2	yes	—
141	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
146	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
147	C₂×C₈	SD₁₆×C₁₅₁	1	2	yes	—
148	C₂×C₈	Q₁₆×C₁₅₁	1	2	yes	—
149	C₂×C₈	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
150	C₂×C₈	C₂×D₄×C₁₅₁	1	1	yes	characteristic
151	C₂×C₈	C₂×Q₈×C₁₅₁	1	2	yes	—
152	C₂×C₈	#41×C₁₅₁	1	2	yes	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄²	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
158	C₄²	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
159	C₄²	D₈×C₁₅₁	1	2	yes	—
160	C₁₆	—	2	4	no	—
161	C₂³	C₂×SD₁₆×C₁₅₁	1	1	yes	characteristic
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	8	4	no	—
165	C₂×C₄	#102×C₁₅₁	1	2	yes	—
166	C₂×C₄	#102×C₁₅₁	1	2	yes	—
167	C₂×C₄	C₂×C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
168	C₂×C₄	#124×C₁₅₁	1	1	yes	characteristic
169	C₂×C₄	C₄⋊Q₈×C₁₅₁	1	1	yes	characteristic
170	C₂×C₄	C₂×D₈×C₁₅₁	1	1	yes	characteristic
171	C₂×C₄	C₂×SD₁₆×C₁₅₁	1	1	yes	characteristic
172	C₂×C₄	C₂×Q₁₆×C₁₅₁	1	1	yes	characteristic
173	C₈	—	8	2	no	—
174	C₈	#103×C₁₅₁	1	2	yes	—
175	C₈	#103×C₁₅₁	1	2	yes	—
176	C₈	C₄×Q₈×C₁₅₁	1	2	yes	—
177	C₈	C₂²⋊Q₈×C₁₅₁	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	C₂×#102×C₁₅₁	1	1	yes	characteristic
180	C₂²	#487×C₁₅₁	1	1	yes	characteristic
181	C₂²	#498×C₁₅₁	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	#391×C₁₅₁	1	1	yes	characteristic
186	C₄	C₂×#103×C₁₅₁	1	1	yes	characteristic
187	C₄	#486×C₁₅₁	1	1	yes	characteristic
188	C₄	#500×C₁₅₁	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#1990×C₁₅₁	1	1	yes	characteristic
192	C₂	#2148×C₁₅₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₅₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₅₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₅₁
prime radical	C₂³×C₁₅₁	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₂×C₄²×C₁₅₁ ◁ C₂×C₄×C₈×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
96. #13289	C₁₅₁	direct (N × H)
97. C₁₅₁	#13289	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#499×C₁₅₁	2
C₄×M₃₂×C₁₅₁	2
#1848×C₁₅₁	2
#1387×C₁₅₁	2
#13289	151

Subgroups — C151×#13289 #50318214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#13289 #50318214