Subgroups — C₁₅₁×#13180 #50318105

Order 38656.

294 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2159×C₁₅₁	C₂	1	2	yes	maximal
2	C₂×#364×C₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#1846×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	#1772×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	#1387×C₁₅₁	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₁₅₁	C₂²	1	2	yes	—
7	C₂×#108×C₁₅₁	C₂²	1	1	yes	characteristic
8	#427×C₁₅₁	C₂²	1	1	yes	characteristic
9	C₂×#107×C₁₅₁	C₂²	1	2	yes	—
10	C₂×#106×C₁₅₁	—	2	1	no	—
11	#405×C₁₅₁	—	2	1	no	—
12	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
13	#382×C₁₅₁	—	2	1	no	—
14	#367×C₁₅₁	—	2	2	no	—
15	#366×C₁₅₁	—	2	2	no	—
16	#364×C₁₅₁	C₄	1	4	yes	complemented
17	#339×C₁₅₁	—	2	1	no	—
18	C₂×M₁₆×C₁₅₁	—	2	1	no	—
19	C₂²×C₈×C₁₅₁	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
21	#117×C₁₅₁	—	2	1	no	—
22	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
23	C₂×C₄²×C₁₅₁	D₄	1	1	yes	characteristic
24	M₃₂×C₁₅₁	—	2	2	no	—
25	M₃₂×C₁₅₁	C₂×C₄	1	4	yes	—
26	C₂×C₁₆×C₁₅₁	—	2	2	no	—
27	#108×C₁₅₁	—	2	1	no	—
28	#108×C₁₅₁	Q₈	1	2	yes	—
29	#107×C₁₅₁	—	2	2	no	—
30	#107×C₁₅₁	—	2	2	no	—
31	#107×C₁₅₁	D₄	1	2	yes	—
32	#106×C₁₅₁	—	2	1	no	—
33	#106×C₁₅₁	—	4	1	no	—
34	#106×C₁₅₁	—	4	1	no	—
35	#97×C₁₅₁	—	4	1	no	—
36	C₄×C₈×C₁₅₁	—	4	1	no	—
37	C₄×C₈×C₁₅₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₅₁	Q₈	1	2	yes	—
39	#95×C₁₅₁	—	4	2	no	—
40	C₂²×C₄×C₁₅₁	—	2	1	no	—
41	C₂²×C₄×C₁₅₁	—	4	2	no	—
42	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
43	M₁₆×C₁₅₁	—	2	2	no	—
44	M₁₆×C₁₅₁	—	4	1	no	—
45	C₂×C₈×C₁₅₁	—	4	1	no	—
46	C₂×C₈×C₁₅₁	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₅₁	D₈	1	1	yes	characteristic
48	C₂×C₈×C₁₅₁	D₈	1	1	yes	characteristic
49	C₂×C₈×C₁₅₁	Q₁₆	1	2	yes	—
50	C₂×C₈×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₁₅₁	#41	1	1	yes	characteristic
55	C₂×C₈×C₁₅₁	#41	1	1	yes	characteristic
56	C₄⋊C₄×C₁₅₁	—	2	2	no	—
57	C₄⋊C₄×C₁₅₁	—	4	2	no	—
58	C₄⋊C₄×C₁₅₁	—	4	2	no	—
59	C₂²⋊C₄×C₁₅₁	—	4	1	no	—
60	C₄²×C₁₅₁	—	2	1	no	—
61	C₄²×C₁₅₁	—	4	1	no	—
62	C₄²×C₁₅₁	C₄⋊C₄	1	2	yes	—
63	C₁₆×C₁₅₁	—	2	4	no	—
64	C₂³×C₁₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂×C₄×C₁₅₁	—	2	2	no	—
66	C₂×C₄×C₁₅₁	—	4	2	no	—
67	C₂×C₄×C₁₅₁	—	4	2	no	—
68	C₂×C₄×C₁₅₁	—	8	4	no	—
69	C₂×C₄×C₁₅₁	—	8	2	no	—
70	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₅₁	#121	1	1	yes	characteristic
72	C₂×C₄×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₅₁	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₁₅₁	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₅₁	#137	1	1	yes	characteristic
76	C₈×C₁₅₁	—	4	2	no	—
77	C₈×C₁₅₁	#107	1	2	yes	—
78	C₈×C₁₅₁	#107	1	2	yes	—
79	C₈×C₁₅₁	C₄×D₄	1	1	yes	characteristic
80	C₈×C₁₅₁	C₄×Q₈	1	1	yes	characteristic
81	C₈×C₁₅₁	#125	1	1	yes	characteristic
82	C₈×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
83	C₂²×C₁₅₁	—	2	2	no	—
84	C₂²×C₁₅₁	#428	1	1	yes	characteristic
85	C₂²×C₁₅₁	#459	1	1	yes	characteristic
86	C₂²×C₁₅₁	#477	1	1	yes	characteristic
87	C₄×C₁₅₁	—	4	2	no	—
88	C₄×C₁₅₁	—	8	2	no	—
89	C₄×C₁₅₁	—	16	2	no	—
90	C₄×C₁₅₁	#389	1	1	yes	characteristic
91	C₄×C₁₅₁	C₂×#107	1	1	yes	characteristic
92	C₄×C₁₅₁	#462	1	1	yes	characteristic
93	C₄×C₁₅₁	#474	1	1	yes	characteristic
94	C₂×C₁₅₁	—	2	1	no	—
95	C₂×C₁₅₁	—	2	1	no	—
96	C₂×C₁₅₁	#1947	1	1	yes	characteristic
97	C₂×C₁₅₁	#2212	1	2	yes	—
98	#13180	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₁₅₁	#13180	1	1	yes	characteristic, direct product factor, Sylow
100	#2159	C₂×C₁₅₁	1	2	yes	—
101	C₂×#364	C₂×C₁₅₁	1	1	yes	characteristic
102	#1846	C₂×C₁₅₁	1	1	yes	characteristic
103	#1772	C₂×C₁₅₁	1	1	yes	characteristic
104	#1387	C₂×C₁₅₁	1	2	yes	—
105	C₂×M₃₂	C₂²×C₁₅₁	1	2	yes	—
106	C₂×#108	C₂²×C₁₅₁	1	1	yes	characteristic
107	#427	C₂²×C₁₅₁	1	1	yes	characteristic
108	C₂×#107	C₂²×C₁₅₁	1	2	yes	—
109	C₂×#106	—	2	1	no	—
110	#405	—	2	1	no	—
111	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
112	#382	—	2	1	no	—
113	#367	—	2	2	no	—
114	#366	—	2	2	no	—
115	#364	C₄×C₁₅₁	1	4	yes	complemented
116	#339	—	2	1	no	—
117	C₂×M₁₆	—	2	1	no	—
118	C₂²×C₈	D₄×C₁₅₁	1	1	yes	characteristic
119	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
120	#117	—	2	1	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄²	D₄×C₁₅₁	1	1	yes	characteristic
123	M₃₂	—	2	2	no	—
124	M₃₂	C₂×C₄×C₁₅₁	1	4	yes	—
125	C₂×C₁₆	—	2	2	no	—
126	#108	—	2	1	no	—
127	#108	Q₈×C₁₅₁	1	2	yes	—
128	#107	—	2	2	no	—
129	#107	—	2	2	no	—
130	#107	D₄×C₁₅₁	1	2	yes	—
131	#106	—	2	1	no	—
132	#106	—	4	1	no	—
133	#106	—	4	1	no	—
134	#97	—	4	1	no	—
135	C₄×C₈	—	4	1	no	—
136	C₄×C₈	C₂×C₄×C₁₅₁	1	2	yes	—
137	C₄×C₈	Q₈×C₁₅₁	1	2	yes	—
138	#95	—	4	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	M₁₆	—	4	1	no	—
144	C₂×C₈	—	4	1	no	—
145	C₂×C₈	C₄⋊C₄×C₁₅₁	1	2	yes	—
146	C₂×C₈	D₈×C₁₅₁	1	1	yes	characteristic
147	C₂×C₈	D₈×C₁₅₁	1	1	yes	characteristic
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	1	yes	characteristic
152	C₂×C₈	C₂×Q₈×C₁₅₁	1	1	yes	characteristic
153	C₂×C₈	#41×C₁₅₁	1	1	yes	characteristic
154	C₂×C₈	#41×C₁₅₁	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₂²⋊C₄	—	4	1	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	4	1	no	—
161	C₄²	C₄⋊C₄×C₁₅₁	1	2	yes	—
162	C₁₆	—	2	4	no	—
163	C₂³	AGL(1,ℤ/8)×C₁₅₁	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	8	4	no	—
168	C₂×C₄	—	8	2	no	—
169	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
170	C₂×C₄	#121×C₁₅₁	1	1	yes	characteristic
171	C₂×C₄	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
172	C₂×C₄	C₂×D₈×C₁₅₁	1	1	yes	characteristic
173	C₂×C₄	C₂×Q₁₆×C₁₅₁	1	1	yes	characteristic
174	C₂×C₄	#137×C₁₅₁	1	1	yes	characteristic
175	C₈	—	4	2	no	—
176	C₈	#107×C₁₅₁	1	2	yes	—
177	C₈	#107×C₁₅₁	1	2	yes	—
178	C₈	C₄×D₄×C₁₅₁	1	1	yes	characteristic
179	C₈	C₄×Q₈×C₁₅₁	1	1	yes	characteristic
180	C₈	#125×C₁₅₁	1	1	yes	characteristic
181	C₈	C₄⋊Q₈×C₁₅₁	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	#428×C₁₅₁	1	1	yes	characteristic
184	C₂²	#459×C₁₅₁	1	1	yes	characteristic
185	C₂²	#477×C₁₅₁	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	8	2	no	—
188	C₄	—	16	2	no	—
189	C₄	#389×C₁₅₁	1	1	yes	characteristic
190	C₄	C₂×#107×C₁₅₁	1	1	yes	characteristic
191	C₄	#462×C₁₅₁	1	1	yes	characteristic
192	C₄	#474×C₁₅₁	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#1947×C₁₅₁	1	1	yes	characteristic
196	C₂	#2212×C₁₅₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₅₁	#459
commutator	C₂×C₈	C₂²×C₄×C₁₅₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₅₁
prime radical	C₂³×C₁₅₁	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₂²×C₄×C₁₅₁ ◁ C₂×C₄×C₈×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
16. #364×C₁₅₁	C₄	semidirect
98. #13180	C₁₅₁	direct (N × H)
99. C₁₅₁	#13180	direct (N × H)
115. #364	C₄×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#2159×C₁₅₁	2
C₂×#364×C₁₅₁	2
#1846×C₁₅₁	2
#1772×C₁₅₁	2
#1387×C₁₅₁	2
#13180	151

Subgroups — C151×#13180 #50318105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#13180 #50318105