Subgroups — C₁₅₁×#21351 #50326276

Order 38656.

894 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#2283×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	#1939×C₁₅₁	C₂	1	4	yes	maximal, complemented
5	#1850×C₁₅₁	C₂	1	4	yes	maximal
6	#1768×C₁₅₁	C₂	1	4	yes	maximal, complemented
7	C₂³×C₈×C₁₅₁	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
9	#432×C₁₅₁	—	2	4	no	—
10	C₂×C₄⋊C₈×C₁₅₁	C₂²	1	8	yes	—
11	C₂×C₄⋊C₈×C₁₅₁	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₈×C₁₅₁	C₂²	1	2	yes	—
13	C₂×#97×C₁₅₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₅₁	—	2	4	no	—
15	C₂×C₄×C₈×C₁₅₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₅₁	C₂²	1	2	yes	—
17	#383×C₁₅₁	C₄	1	2	yes	—
18	#382×C₁₅₁	C₄	1	2	yes	—
19	#381×C₁₅₁	C₄	1	2	yes	—
20	#380×C₁₅₁	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
22	C₄×C₂²⋊C₄×C₁₅₁	C₂²	1	2	yes	—
23	#376×C₁₅₁	C₂²	1	2	yes	—
24	#336×C₁₅₁	C₂²	1	8	yes	—
25	C₂³×C₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₁₅₁	—	2	4	no	—
29	C₂²×C₈×C₁₅₁	—	2	8	no	—
30	C₂²×C₈×C₁₅₁	C₂³	1	4	yes	—
31	C₂²×C₈×C₁₅₁	C₂³	1	4	yes	—
32	C₂×C₄⋊C₄×C₁₅₁	C₂×C₄	1	4	yes	—
33	C₂×C₂²⋊C₄×C₁₅₁	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₁₅₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₅₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₅₁	D₄	1	4	yes	—
37	C₂×C₄²×C₁₅₁	Q₈	1	4	yes	—
38	C₂×C₄²×C₁₅₁	C₂³	1	4	yes	—
39	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₁₅₁	—	2	16	no	—
42	C₄⋊C₈×C₁₅₁	—	2	4	no	—
43	C₂²⋊C₈×C₁₅₁	—	2	4	no	—
44	#97×C₁₅₁	—	2	4	no	—
45	C₄×C₈×C₁₅₁	—	2	4	no	—
46	C₄×C₈×C₁₅₁	—	2	4	no	—
47	#95×C₁₅₁	C₂×C₄	1	4	yes	—
48	#95×C₁₅₁	C₂×C₄	1	4	yes	—
49	#95×C₁₅₁	C₂×C₄	1	4	yes	—
50	#95×C₁₅₁	C₂×C₄	1	4	yes	—
51	C₂⁴×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₅₁	—	2	4	no	—
53	C₂²×C₄×C₁₅₁	—	2	4	no	—
54	C₂²×C₄×C₁₅₁	—	2	4	no	—
55	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₁₅₁	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₅₁	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₁₅₁	—	2	16	no	—
67	C₂×C₈×C₁₅₁	—	2	4	no	—
68	C₂×C₈×C₁₅₁	—	2	4	no	—
69	C₂×C₈×C₁₅₁	—	2	4	no	—
70	C₂×C₈×C₁₅₁	—	2	8	no	—
71	C₂×C₈×C₁₅₁	#41	1	8	yes	—
72	C₄⋊C₄×C₁₅₁	—	2	8	no	—
73	C₂²⋊C₄×C₁₅₁	—	2	8	no	—
74	C₄²×C₁₅₁	—	2	8	no	—
75	C₄²×C₁₅₁	C₄⋊C₄	1	4	yes	—
76	C₄²×C₁₅₁	C₄⋊C₄	1	4	yes	—
77	C₄²×C₁₅₁	C₂×D₄	1	2	yes	—
78	C₄²×C₁₅₁	C₂×D₄	1	2	yes	—
79	C₄²×C₁₅₁	C₂×Q₈	1	2	yes	—
80	C₄²×C₁₅₁	C₂×Q₈	1	2	yes	—
81	C₂³×C₁₅₁	—	2	2	no	—
82	C₂³×C₁₅₁	—	2	2	no	—
83	C₂³×C₁₅₁	C₂×C₄⋊C₄	1	2	yes	—
84	C₂³×C₁₅₁	#131	1	4	yes	—
85	C₂³×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₅₁	—	2	2	no	—
87	C₂×C₄×C₁₅₁	—	2	2	no	—
88	C₂×C₄×C₁₅₁	—	2	2	no	—
89	C₂×C₄×C₁₅₁	—	2	2	no	—
90	C₂×C₄×C₁₅₁	—	2	8	no	—
91	C₂×C₄×C₁₅₁	—	2	4	no	—
92	C₂×C₄×C₁₅₁	—	2	4	no	—
93	C₂×C₄×C₁₅₁	—	2	8	no	—
94	C₂×C₄×C₁₅₁	—	2	8	no	—
95	C₂×C₄×C₁₅₁	—	2	8	no	—
96	C₂×C₄×C₁₅₁	—	2	8	no	—
97	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₁₅₁	#131	1	4	yes	—
103	C₂×C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₅₁	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₅₁	C₂×#41	1	4	yes	—
106	C₈×C₁₅₁	—	2	8	no	—
107	C₈×C₁₅₁	—	4	4	no	—
108	C₂²×C₁₅₁	—	2	2	no	—
109	C₂²×C₁₅₁	—	2	8	no	—
110	C₂²×C₁₅₁	—	2	2	no	—
111	C₂²×C₁₅₁	#424	1	4	yes	—
112	C₂²×C₁₅₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₁₅₁	C₄×#41	1	2	yes	—
114	C₂²×C₁₅₁	C₂×#131	1	2	yes	—
115	C₂²×C₁₅₁	C₂×#131	1	2	yes	—
116	C₄×C₁₅₁	—	2	2	no	—
117	C₄×C₁₅₁	—	2	2	no	—
118	C₄×C₁₅₁	—	2	2	no	—
119	C₄×C₁₅₁	—	2	2	no	—
120	C₄×C₁₅₁	—	4	8	no	—
121	C₄×C₁₅₁	#424	1	4	yes	—
122	C₄×C₁₅₁	#525	1	2	yes	—
123	C₄×C₁₅₁	#533	1	2	yes	—
124	C₂×C₁₅₁	—	2	4	no	—
125	C₂×C₁₅₁	#2296	1	1	yes	characteristic
126	C₂×C₁₅₁	C₂×#424	1	1	yes	characteristic
127	C₂×C₁₅₁	C₂×#424	1	1	yes	characteristic
128	C₂×C₁₅₁	#2966	1	4	yes	—
129	#21351	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₁₅₁	#21351	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×#432	C₂×C₁₅₁	1	1	yes	characteristic
132	C₂²×C₄×C₈	C₂×C₁₅₁	1	1	yes	characteristic
133	#2283	C₂×C₁₅₁	1	1	yes	characteristic
134	#1939	C₂×C₁₅₁	1	4	yes	complemented
135	#1850	C₂×C₁₅₁	1	4	yes	—
136	#1768	C₂×C₁₅₁	1	4	yes	complemented
137	C₂³×C₈	C₂²×C₁₅₁	1	2	yes	—
138	C₂²×C₄²	C₂²×C₁₅₁	1	1	yes	characteristic
139	#432	—	2	4	no	—
140	C₂×C₄⋊C₈	C₂²×C₁₅₁	1	8	yes	—
141	C₂×C₄⋊C₈	C₂²×C₁₅₁	1	2	yes	—
142	C₂×C₂²⋊C₈	C₂²×C₁₅₁	1	2	yes	—
143	C₂×#97	C₂²×C₁₅₁	1	2	yes	—
144	C₂×C₄×C₈	—	2	4	no	—
145	C₂×C₄×C₈	C₂²×C₁₅₁	1	2	yes	—
146	C₂×C₄×C₈	C₂²×C₁₅₁	1	2	yes	—
147	#383	C₄×C₁₅₁	1	2	yes	—
148	#382	C₄×C₁₅₁	1	2	yes	—
149	#381	C₄×C₁₅₁	1	2	yes	—
150	#380	C₄×C₁₅₁	1	2	yes	—
151	C₄×C₄⋊C₄	C₂²×C₁₅₁	1	2	yes	—
152	C₄×C₂²⋊C₄	C₂²×C₁₅₁	1	2	yes	—
153	#376	C₂²×C₁₅₁	1	2	yes	—
154	#336	C₂²×C₁₅₁	1	8	yes	—
155	C₂³×C₄	C₂×C₄×C₁₅₁	1	1	yes	characteristic
156	C₂³×C₄	C₂×C₄×C₁₅₁	1	1	yes	characteristic
157	C₂³×C₄	C₂³×C₁₅₁	1	1	yes	characteristic
158	C₂²×C₈	—	2	4	no	—
159	C₂²×C₈	—	2	8	no	—
160	C₂²×C₈	C₂³×C₁₅₁	1	4	yes	—
161	C₂²×C₈	C₂³×C₁₅₁	1	4	yes	—
162	C₂×C₄⋊C₄	C₂×C₄×C₁₅₁	1	4	yes	—
163	C₂×C₂²⋊C₄	C₂×C₄×C₁₅₁	1	4	yes	—
164	C₂×C₄²	C₂×C₄×C₁₅₁	1	1	yes	characteristic
165	C₂×C₄²	C₂×C₄×C₁₅₁	1	1	yes	characteristic
166	C₂×C₄²	D₄×C₁₅₁	1	4	yes	—
167	C₂×C₄²	Q₈×C₁₅₁	1	4	yes	—
168	C₂×C₄²	C₂³×C₁₅₁	1	4	yes	—
169	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
170	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
171	C₄⋊C₈	—	2	16	no	—
172	C₄⋊C₈	—	2	4	no	—
173	C₂²⋊C₈	—	2	4	no	—
174	#97	—	2	4	no	—
175	C₄×C₈	—	2	4	no	—
176	C₄×C₈	—	2	4	no	—
177	#95	C₂×C₄×C₁₅₁	1	4	yes	—
178	#95	C₂×C₄×C₁₅₁	1	4	yes	—
179	#95	C₂×C₄×C₁₅₁	1	4	yes	—
180	#95	C₂×C₄×C₁₅₁	1	4	yes	—
181	C₂⁴	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	4	yes	—
186	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	4	yes	—
187	C₂²×C₄	C₂²×C₄×C₁₅₁	1	8	yes	—
188	C₂²×C₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
189	C₂²×C₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
192	C₂²×C₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₁₅₁	1	2	yes	—
194	C₂²×C₄	C₂×Q₈×C₁₅₁	1	2	yes	—
195	C₂²×C₄	C₂⁴×C₁₅₁	1	1	yes	characteristic
196	C₂×C₈	—	2	16	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	#41×C₁₅₁	1	8	yes	—
202	C₄⋊C₄	—	2	8	no	—
203	C₂²⋊C₄	—	2	8	no	—
204	C₄²	—	2	8	no	—
205	C₄²	C₄⋊C₄×C₁₅₁	1	4	yes	—
206	C₄²	C₄⋊C₄×C₁₅₁	1	4	yes	—
207	C₄²	C₂×D₄×C₁₅₁	1	2	yes	—
208	C₄²	C₂×D₄×C₁₅₁	1	2	yes	—
209	C₄²	C₂×Q₈×C₁₅₁	1	2	yes	—
210	C₄²	C₂×Q₈×C₁₅₁	1	2	yes	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	C₂×C₄⋊C₄×C₁₅₁	1	2	yes	—
214	C₂³	#131×C₁₅₁	1	4	yes	—
215	C₂³	C₂³×C₄×C₁₅₁	1	1	yes	characteristic
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	2	yes	—
228	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	2	yes	—
232	C₂×C₄	#131×C₁₅₁	1	4	yes	—
233	C₂×C₄	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
234	C₂×C₄	C₂²×Q₈×C₁₅₁	1	1	yes	characteristic
235	C₂×C₄	C₂×#41×C₁₅₁	1	4	yes	—
236	C₈	—	2	8	no	—
237	C₈	—	4	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	8	no	—
240	C₂²	—	2	2	no	—
241	C₂²	#424×C₁₅₁	1	4	yes	—
242	C₂²	C₂²×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
243	C₂²	C₄×#41×C₁₅₁	1	2	yes	—
244	C₂²	C₂×#131×C₁₅₁	1	2	yes	—
245	C₂²	C₂×#131×C₁₅₁	1	2	yes	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	4	8	no	—
251	C₄	#424×C₁₅₁	1	4	yes	—
252	C₄	#525×C₁₅₁	1	2	yes	—
253	C₄	#533×C₁₅₁	1	2	yes	—
254	C₂	—	2	4	no	—
255	C₂	#2296×C₁₅₁	1	1	yes	characteristic
256	C₂	C₂×#424×C₁₅₁	1	1	yes	characteristic
257	C₂	C₂×#424×C₁₅₁	1	1	yes	characteristic
258	C₂	#2966×C₁₅₁	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. #1939×C₁₅₁	C₂	semidirect
6. #1768×C₁₅₁	C₂	semidirect
129. #21351	C₁₅₁	direct (N × H)
130. C₁₅₁	#21351	direct (N × H)
134. #1939	C₂×C₁₅₁	semidirect
136. #1768	C₂×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#432×C₁₅₁	2
C₂²×C₄×C₈×C₁₅₁	2
#2283×C₁₅₁	2
#1939×C₁₅₁	2
#1850×C₁₅₁	2
#1768×C₁₅₁	2
#21351	151

Subgroups — C151×#21351 #50326276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#21351 #50326276