Subgroups — C₁₅₁×#22254 #50327179

Order 38656.

674 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#2331×C₁₅₁	C₂	1	2	yes	maximal
3	#1993×C₁₅₁	C₂	1	2	yes	maximal
4	#1921×C₁₅₁	C₂	1	4	yes	maximal
5	#1918×C₁₅₁	C₂	1	4	yes	maximal
6	#1835×C₁₅₁	C₂	1	1	yes	maximal, characteristic
7	#1746×C₁₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₁₅₁	C₂²	1	2	yes	—
9	#500×C₁₅₁	C₄	1	8	yes	complemented
10	C₂×#107×C₁₅₁	C₂²	1	2	yes	—
11	C₂×#107×C₁₅₁	C₂²	1	2	yes	—
12	C₂×#97×C₁₅₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
14	#391×C₁₅₁	—	2	4	no	—
15	#389×C₁₅₁	C₂²	1	4	yes	—
16	#387×C₁₅₁	—	2	4	no	—
17	#382×C₁₅₁	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₁₅₁	C₂²	1	4	yes	—
19	#376×C₁₅₁	C₂²	1	1	yes	characteristic
20	#340×C₁₅₁	C₂²	1	2	yes	—
21	#340×C₁₅₁	C₂²	1	2	yes	—
22	#340×C₁₅₁	C₂²	1	8	yes	—
23	#336×C₁₅₁	C₂²	1	2	yes	—
24	#336×C₁₅₁	C₂²	1	2	yes	—
25	C₂²×Q₈×C₁₅₁	—	2	2	no	—
26	C₂²×C₈×C₁₅₁	C₂³	1	2	yes	—
27	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₁₅₁	—	2	4	no	—
30	C₄⋊Q₈×C₁₅₁	C₂×C₄	1	8	yes	—
31	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₅₁	—	2	4	no	—
33	C₂×C₄⋊C₄×C₁₅₁	C₂³	1	4	yes	—
34	C₂×C₄⋊C₄×C₁₅₁	C₂³	1	4	yes	—
35	C₂×C₄²×C₁₅₁	—	2	4	no	—
36	C₂×C₄²×C₁₅₁	C₂³	1	2	yes	—
37	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
38	#107×C₁₅₁	C₂×C₄	1	8	yes	—
39	#107×C₁₅₁	C₂×C₄	1	8	yes	—
40	#97×C₁₅₁	—	2	2	no	—
41	C₄×C₈×C₁₅₁	C₂×C₄	1	4	yes	—
42	#95×C₁₅₁	—	2	4	no	—
43	#95×C₁₅₁	—	2	2	no	—
44	#95×C₁₅₁	—	2	2	no	—
45	#95×C₁₅₁	—	2	4	no	—
46	#95×C₁₅₁	D₄	1	2	yes	—
47	#95×C₁₅₁	D₄	1	2	yes	—
48	C₂×Q₈×C₁₅₁	—	2	8	no	—
49	C₂×Q₈×C₁₅₁	—	8	4	no	—
50	C₂²×C₄×C₁₅₁	—	2	4	no	—
51	C₂²×C₄×C₁₅₁	—	2	4	no	—
52	C₂²×C₄×C₁₅₁	C₂×D₄	1	4	yes	—
53	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
56	C₂×C₈×C₁₅₁	—	2	2	no	—
57	C₂×C₈×C₁₅₁	—	2	2	no	—
58	C₂×C₈×C₁₅₁	—	2	2	no	—
59	C₂×C₈×C₁₅₁	—	2	2	no	—
60	C₂×C₈×C₁₅₁	—	2	2	no	—
61	C₂×C₈×C₁₅₁	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₁₅₁	C₂²×C₄	1	2	yes	—
63	C₄⋊C₄×C₁₅₁	—	2	8	no	—
64	C₄⋊C₄×C₁₅₁	—	2	4	no	—
65	C₄⋊C₄×C₁₅₁	—	2	4	no	—
66	C₄⋊C₄×C₁₅₁	—	2	4	no	—
67	C₄⋊C₄×C₁₅₁	—	8	4	no	—
68	C₄⋊C₄×C₁₅₁	C₂²×C₄	1	8	yes	—
69	C₄²×C₁₅₁	—	2	2	no	—
70	C₄²×C₁₅₁	—	2	2	no	—
71	C₄²×C₁₅₁	—	8	4	no	—
72	C₄²×C₁₅₁	C₂²×C₄	1	2	yes	—
73	C₄²×C₁₅₁	#41	1	2	yes	—
74	C₂³×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
75	Q₈×C₁₅₁	—	8	8	no	—
76	C₂×C₄×C₁₅₁	—	2	2	no	—
77	C₂×C₄×C₁₅₁	—	2	2	no	—
78	C₂×C₄×C₁₅₁	—	2	8	no	—
79	C₂×C₄×C₁₅₁	—	2	4	no	—
80	C₂×C₄×C₁₅₁	—	2	4	no	—
81	C₂×C₄×C₁₅₁	—	2	4	no	—
82	C₂×C₄×C₁₅₁	—	4	4	no	—
83	C₂×C₄×C₁₅₁	—	4	4	no	—
84	C₂×C₄×C₁₅₁	—	4	4	no	—
85	C₂×C₄×C₁₅₁	—	4	4	no	—
86	C₂×C₄×C₁₅₁	—	4	4	no	—
87	C₂×C₄×C₁₅₁	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₁₅₁	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
91	C₂×C₄×C₁₅₁	2₋¹⁺⁴	1	2	yes	—
92	C₂×C₄×C₁₅₁	2₋¹⁺⁴	1	2	yes	—
93	C₈×C₁₅₁	—	2	2	no	—
94	C₈×C₁₅₁	—	2	2	no	—
95	C₈×C₁₅₁	—	4	2	no	—
96	C₂²×C₁₅₁	C₂×C₄×D₄	1	1	yes	characteristic
97	C₂²×C₁₅₁	#536	1	2	yes	—
98	C₂²×C₁₅₁	#576	1	2	yes	—
99	C₂²×C₁₅₁	#578	1	2	yes	—
100	C₄×C₁₅₁	—	2	2	no	—
101	C₄×C₁₅₁	—	2	2	no	—
102	C₄×C₁₅₁	—	4	8	no	—
103	C₄×C₁₅₁	—	4	4	no	—
104	C₄×C₁₅₁	—	8	4	no	—
105	C₄×C₁₅₁	#519	1	1	yes	characteristic
106	C₄×C₁₅₁	#519	1	1	yes	characteristic
107	C₄×C₁₅₁	#541	1	2	yes	—
108	C₂×C₁₅₁	#2327	1	1	yes	characteristic
109	C₂×C₁₅₁	#2948	1	1	yes	characteristic
110	C₂×C₁₅₁	#2950	1	1	yes	characteristic
111	C₂×C₁₅₁	#3243	1	4	yes	—
112	#22254	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₁₅₁	#22254	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×#500	C₂×C₁₅₁	1	1	yes	characteristic
115	#2331	C₂×C₁₅₁	1	2	yes	—
116	#1993	C₂×C₁₅₁	1	2	yes	—
117	#1921	C₂×C₁₅₁	1	4	yes	—
118	#1918	C₂×C₁₅₁	1	4	yes	—
119	#1835	C₂×C₁₅₁	1	1	yes	characteristic
120	#1746	C₂×C₁₅₁	1	1	yes	characteristic
121	C₂×C₄⋊Q₈	C₂²×C₁₅₁	1	2	yes	—
122	#500	C₄×C₁₅₁	1	8	yes	complemented
123	C₂×#107	C₂²×C₁₅₁	1	2	yes	—
124	C₂×#107	C₂²×C₁₅₁	1	2	yes	—
125	C₂×#97	C₂²×C₁₅₁	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
127	#391	—	2	4	no	—
128	#389	C₂²×C₁₅₁	1	4	yes	—
129	#387	—	2	4	no	—
130	#382	C₂²×C₁₅₁	1	2	yes	—
131	C₄×C₄⋊C₄	C₂²×C₁₅₁	1	4	yes	—
132	#376	C₂²×C₁₅₁	1	1	yes	characteristic
133	#340	C₂²×C₁₅₁	1	2	yes	—
134	#340	C₂²×C₁₅₁	1	2	yes	—
135	#340	C₂²×C₁₅₁	1	8	yes	—
136	#336	C₂²×C₁₅₁	1	2	yes	—
137	#336	C₂²×C₁₅₁	1	2	yes	—
138	C₂²×Q₈	—	2	2	no	—
139	C₂²×C₈	C₂³×C₁₅₁	1	2	yes	—
140	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
141	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
142	C₄⋊Q₈	—	2	4	no	—
143	C₄⋊Q₈	C₂×C₄×C₁₅₁	1	8	yes	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	2	4	no	—
146	C₂×C₄⋊C₄	C₂³×C₁₅₁	1	4	yes	—
147	C₂×C₄⋊C₄	C₂³×C₁₅₁	1	4	yes	—
148	C₂×C₄²	—	2	4	no	—
149	C₂×C₄²	C₂³×C₁₅₁	1	2	yes	—
150	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
151	#107	C₂×C₄×C₁₅₁	1	8	yes	—
152	#107	C₂×C₄×C₁₅₁	1	8	yes	—
153	#97	—	2	2	no	—
154	C₄×C₈	C₂×C₄×C₁₅₁	1	4	yes	—
155	#95	—	2	4	no	—
156	#95	—	2	2	no	—
157	#95	—	2	2	no	—
158	#95	—	2	4	no	—
159	#95	D₄×C₁₅₁	1	2	yes	—
160	#95	D₄×C₁₅₁	1	2	yes	—
161	C₂×Q₈	—	2	8	no	—
162	C₂×Q₈	—	8	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	C₂×D₄×C₁₅₁	1	4	yes	—
166	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
168	C₂²×C₄	C₂⁴×C₁₅₁	1	1	yes	characteristic
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₂²×C₄×C₁₅₁	1	2	yes	—
175	C₂×C₈	C₂²×C₄×C₁₅₁	1	2	yes	—
176	C₄⋊C₄	—	2	8	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	8	4	no	—
181	C₄⋊C₄	C₂²×C₄×C₁₅₁	1	8	yes	—
182	C₄²	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	8	4	no	—
185	C₄²	C₂²×C₄×C₁₅₁	1	2	yes	—
186	C₄²	#41×C₁₅₁	1	2	yes	—
187	C₂³	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
188	Q₈	—	8	8	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	C₄×D₄×C₁₅₁	1	2	yes	—
201	C₂×C₄	C₄×D₄×C₁₅₁	1	2	yes	—
202	C₂×C₄	C₂³×C₄×C₁₅₁	1	1	yes	characteristic
203	C₂×C₄	C₂×#41×C₁₅₁	1	1	yes	characteristic
204	C₂×C₄	2₋¹⁺⁴×C₁₅₁	1	2	yes	—
205	C₂×C₄	2₋¹⁺⁴×C₁₅₁	1	2	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	4	2	no	—
209	C₂²	C₂×C₄×D₄×C₁₅₁	1	1	yes	characteristic
210	C₂²	#536×C₁₅₁	1	2	yes	—
211	C₂²	#576×C₁₅₁	1	2	yes	—
212	C₂²	#578×C₁₅₁	1	2	yes	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	8	no	—
216	C₄	—	4	4	no	—
217	C₄	—	8	4	no	—
218	C₄	#519×C₁₅₁	1	1	yes	characteristic
219	C₄	#519×C₁₅₁	1	1	yes	characteristic
220	C₄	#541×C₁₅₁	1	2	yes	—
221	C₂	#2327×C₁₅₁	1	1	yes	characteristic
222	C₂	#2948×C₁₅₁	1	1	yes	characteristic
223	C₂	#2950×C₁₅₁	1	1	yes	characteristic
224	C₂	#3243×C₁₅₁	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₅₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₅₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₅₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₅₁ ◁ #376×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
9. #500×C₁₅₁	C₄	semidirect
112. #22254	C₁₅₁	direct (N × H)
113. C₁₅₁	#22254	direct (N × H)
122. #500	C₄×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₅₁	2
#2331×C₁₅₁	2
#1993×C₁₅₁	2
#1921×C₁₅₁	2
#1918×C₁₅₁	2
#1835×C₁₅₁	2
#1746×C₁₅₁	2
#22254	151

Subgroups — C151×#22254 #50327179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#22254 #50327179