Subgroups — C₁₅₁×#17814 #50322739

Order 38656.

730 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×#431×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₁₅₁	C₂	1	2	yes	maximal
3	#1571×C₁₅₁	C₂	1	8	yes	maximal, complemented
4	#1567×C₁₅₁	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₁₅₁	—	2	2	no	—
6	C₂²×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
7	#431×C₁₅₁	C₄	1	4	yes	—
8	#431×C₁₅₁	C₂²	1	4	yes	—
9	#423×C₁₅₁	—	2	2	no	—
10	#423×C₁₅₁	—	2	2	no	—
11	#423×C₁₅₁	C₂²	1	4	yes	—
12	#423×C₁₅₁	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₁₅₁	—	2	2	no	—
14	C₂×C₄⋊C₈×C₁₅₁	C₄	1	2	yes	—
15	C₂×C₄⋊C₈×C₁₅₁	C₂²	1	4	yes	—
16	C₂×C₂²⋊C₈×C₁₅₁	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₁₅₁	C₂²	1	2	yes	—
18	#335×C₁₅₁	C₂²	1	16	yes	complemented
19	C₂³×C₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₅₁	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₅₁	Q₈	1	1	yes	characteristic
22	C₂×M₁₆×C₁₅₁	—	2	4	no	—
23	C₂×M₁₆×C₁₅₁	—	2	4	no	—
24	C₂×M₁₆×C₁₅₁	—	4	8	no	—
25	C₂²×C₈×C₁₅₁	—	2	4	no	—
26	C₂²×C₈×C₁₅₁	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₅₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₅₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₅₁	D₄	1	2	yes	—
30	C₂×C₄²×C₁₅₁	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₅₁	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₅₁	Q₈	1	2	yes	—
33	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₅₁	C₂³	1	2	yes	—
35	C₄⋊C₈×C₁₅₁	—	2	4	no	—
36	C₄⋊C₈×C₁₅₁	—	2	4	no	—
37	C₄⋊C₈×C₁₅₁	—	2	4	no	—
38	C₄⋊C₈×C₁₅₁	C₂×C₄	1	8	yes	—
39	C₄⋊C₈×C₁₅₁	C₂³	1	8	yes	—
40	C₂²⋊C₈×C₁₅₁	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₁₅₁	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₁₅₁	C₂³	1	4	yes	—
43	C₂⁴×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₅₁	—	4	2	no	—
45	C₂²×C₄×C₁₅₁	—	4	2	no	—
46	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₅₁	D₈	1	4	yes	—
53	C₂²×C₄×C₁₅₁	Q₁₆	1	4	yes	—
54	C₂²×C₄×C₁₅₁	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₅₁	C₂×Q₈	1	2	yes	—
60	M₁₆×C₁₅₁	—	4	8	no	—
61	M₁₆×C₁₅₁	—	4	8	no	—
62	C₂×C₈×C₁₅₁	—	2	8	no	—
63	C₂×C₈×C₁₅₁	—	4	4	no	—
64	C₂×C₈×C₁₅₁	—	4	8	no	—
65	C₂×C₈×C₁₅₁	C₂²×C₄	1	8	yes	—
66	C₄²×C₁₅₁	C₄⋊C₄	1	2	yes	—
67	C₄²×C₁₅₁	C₄⋊C₄	1	2	yes	—
68	C₄²×C₁₅₁	C₄⋊C₄	1	2	yes	—
69	C₄²×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₁₅₁	C₂²×C₄	1	2	yes	—
71	C₄²×C₁₅₁	C₂×D₄	1	2	yes	—
72	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₁₅₁	C₂×Q₈	1	2	yes	—
75	C₄²×C₁₅₁	C₂⁴	1	1	yes	characteristic
76	C₂³×C₁₅₁	—	4	1	no	—
77	C₂³×C₁₅₁	—	4	1	no	—
78	C₂³×C₁₅₁	#107	1	4	yes	—
79	C₂³×C₁₅₁	C₂×C₄⋊C₄	1	2	yes	—
80	C₂³×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₅₁	—	2	2	no	—
82	C₂×C₄×C₁₅₁	—	2	4	no	—
83	C₂×C₄×C₁₅₁	—	2	4	no	—
84	C₂×C₄×C₁₅₁	—	2	2	no	—
85	C₂×C₄×C₁₅₁	—	4	2	no	—
86	C₂×C₄×C₁₅₁	—	4	1	no	—
87	C₂×C₄×C₁₅₁	—	4	1	no	—
88	C₂×C₄×C₁₅₁	—	4	2	no	—
89	C₂×C₄×C₁₅₁	#107	1	4	yes	—
90	C₂×C₄×C₁₅₁	#107	1	8	yes	—
91	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₅₁	C₂×D₈	1	4	yes	—
98	C₂×C₄×C₁₅₁	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₁₅₁	C₂×Q₁₆	1	4	yes	—
100	C₂×C₄×C₁₅₁	C₂×Q₁₆	1	2	yes	—
101	C₂×C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₅₁	C₂²×Q₈	1	1	yes	characteristic
104	C₈×C₁₅₁	—	2	8	no	—
105	C₈×C₁₅₁	—	4	8	no	—
106	C₂²×C₁₅₁	—	2	1	no	—
107	C₂²×C₁₅₁	—	2	2	no	—
108	C₂²×C₁₅₁	—	2	2	no	—
109	C₂²×C₁₅₁	—	2	1	no	—
110	C₂²×C₁₅₁	—	4	4	no	—
111	C₂²×C₁₅₁	C₂×#107	1	4	yes	—
112	C₂²×C₁₅₁	C₂×#107	1	2	yes	—
113	C₂²×C₁₅₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₄×C₁₅₁	—	2	1	no	—
115	C₄×C₁₅₁	—	2	2	no	—
116	C₄×C₁₅₁	—	2	2	no	—
117	C₄×C₁₅₁	—	2	1	no	—
118	C₄×C₁₅₁	C₂×#107	1	4	yes	—
119	C₄×C₁₅₁	C₂×#107	1	2	yes	—
120	C₄×C₁₅₁	#430	1	2	yes	—
121	C₄×C₁₅₁	#568	1	2	yes	—
122	C₄×C₁₅₁	C₂²×D₈	1	1	yes	characteristic
123	C₄×C₁₅₁	C₂²×Q₁₆	1	1	yes	characteristic
124	C₂×C₁₅₁	—	2	2	no	—
125	C₂×C₁₅₁	—	2	4	no	—
126	C₂×C₁₅₁	#2905	1	1	yes	characteristic
127	C₂×C₁₅₁	C₂²×#107	1	1	yes	characteristic
128	C₂×C₁₅₁	C₂×#430	1	1	yes	characteristic
129	#17814	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₁₅₁	#17814	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×#431	C₂×C₁₅₁	1	1	yes	characteristic
132	C₂×#423	C₂×C₁₅₁	1	2	yes	—
133	#1571	C₂×C₁₅₁	1	8	yes	complemented
134	#1567	C₂×C₁₅₁	1	4	yes	complemented
135	C₂²×M₁₆	—	2	2	no	—
136	C₂²×C₄²	C₂²×C₁₅₁	1	1	yes	characteristic
137	#431	C₄×C₁₅₁	1	4	yes	—
138	#431	C₂²×C₁₅₁	1	4	yes	—
139	#423	—	2	2	no	—
140	#423	—	2	2	no	—
141	#423	C₂²×C₁₅₁	1	4	yes	—
142	#423	C₂²×C₁₅₁	1	4	yes	—
143	C₂×C₄⋊C₈	—	2	2	no	—
144	C₂×C₄⋊C₈	C₄×C₁₅₁	1	2	yes	—
145	C₂×C₄⋊C₈	C₂²×C₁₅₁	1	4	yes	—
146	C₂×C₂²⋊C₈	C₄×C₁₅₁	1	2	yes	—
147	C₂×C₄×C₈	C₂²×C₁₅₁	1	2	yes	—
148	#335	C₂²×C₁₅₁	1	16	yes	complemented
149	C₂³×C₄	C₂×C₄×C₁₅₁	1	1	yes	characteristic
150	C₂³×C₄	D₄×C₁₅₁	1	1	yes	characteristic
151	C₂³×C₄	Q₈×C₁₅₁	1	1	yes	characteristic
152	C₂×M₁₆	—	2	4	no	—
153	C₂×M₁₆	—	2	4	no	—
154	C₂×M₁₆	—	4	8	no	—
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	C₂×C₄×C₁₅₁	1	4	yes	—
157	C₂×C₄²	C₂×C₄×C₁₅₁	1	1	yes	characteristic
158	C₂×C₄²	C₂×C₄×C₁₅₁	1	2	yes	—
159	C₂×C₄²	D₄×C₁₅₁	1	2	yes	—
160	C₂×C₄²	D₄×C₁₅₁	1	1	yes	characteristic
161	C₂×C₄²	Q₈×C₁₅₁	1	1	yes	characteristic
162	C₂×C₄²	Q₈×C₁₅₁	1	2	yes	—
163	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
164	C₂×C₄²	C₂³×C₁₅₁	1	2	yes	—
165	C₄⋊C₈	—	2	4	no	—
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	—	2	4	no	—
168	C₄⋊C₈	C₂×C₄×C₁₅₁	1	8	yes	—
169	C₄⋊C₈	C₂³×C₁₅₁	1	8	yes	—
170	C₂²⋊C₈	C₂×C₄×C₁₅₁	1	8	yes	—
171	C₄×C₈	C₂×C₄×C₁₅₁	1	4	yes	—
172	C₄×C₈	C₂³×C₁₅₁	1	4	yes	—
173	C₂⁴	C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	2	yes	—
177	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	2	yes	—
179	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	2	yes	—
182	C₂²×C₄	D₈×C₁₅₁	1	4	yes	—
183	C₂²×C₄	Q₁₆×C₁₅₁	1	4	yes	—
184	C₂²×C₄	C₂²×C₄×C₁₅₁	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₁₅₁	1	2	yes	—
187	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
188	C₂²×C₄	C₂×Q₈×C₁₅₁	1	1	yes	characteristic
189	C₂²×C₄	C₂×Q₈×C₁₅₁	1	2	yes	—
190	M₁₆	—	4	8	no	—
191	M₁₆	—	4	8	no	—
192	C₂×C₈	—	2	8	no	—
193	C₂×C₈	—	4	4	no	—
194	C₂×C₈	—	4	8	no	—
195	C₂×C₈	C₂²×C₄×C₁₅₁	1	8	yes	—
196	C₄²	C₄⋊C₄×C₁₅₁	1	2	yes	—
197	C₄²	C₄⋊C₄×C₁₅₁	1	2	yes	—
198	C₄²	C₄⋊C₄×C₁₅₁	1	2	yes	—
199	C₄²	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
200	C₄²	C₂²×C₄×C₁₅₁	1	2	yes	—
201	C₄²	C₂×D₄×C₁₅₁	1	2	yes	—
202	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
203	C₄²	C₂×Q₈×C₁₅₁	1	1	yes	characteristic
204	C₄²	C₂×Q₈×C₁₅₁	1	2	yes	—
205	C₄²	C₂⁴×C₁₅₁	1	1	yes	characteristic
206	C₂³	—	4	1	no	—
207	C₂³	—	4	1	no	—
208	C₂³	#107×C₁₅₁	1	4	yes	—
209	C₂³	C₂×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
210	C₂³	C₂×C₄⋊C₄×C₁₅₁	1	2	yes	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	#107×C₁₅₁	1	4	yes	—
220	C₂×C₄	#107×C₁₅₁	1	8	yes	—
221	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	2	yes	—
222	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	2	yes	—
224	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	2	yes	—
226	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
227	C₂×C₄	C₂×D₈×C₁₅₁	1	4	yes	—
228	C₂×C₄	C₂×D₈×C₁₅₁	1	2	yes	—
229	C₂×C₄	C₂×Q₁₆×C₁₅₁	1	4	yes	—
230	C₂×C₄	C₂×Q₁₆×C₁₅₁	1	2	yes	—
231	C₂×C₄	C₂³×C₄×C₁₅₁	1	1	yes	characteristic
232	C₂×C₄	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
233	C₂×C₄	C₂²×Q₈×C₁₅₁	1	1	yes	characteristic
234	C₈	—	2	8	no	—
235	C₈	—	4	8	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	4	4	no	—
241	C₂²	C₂×#107×C₁₅₁	1	4	yes	—
242	C₂²	C₂×#107×C₁₅₁	1	2	yes	—
243	C₂²	C₂²×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	1	no	—
248	C₄	C₂×#107×C₁₅₁	1	4	yes	—
249	C₄	C₂×#107×C₁₅₁	1	2	yes	—
250	C₄	#430×C₁₅₁	1	2	yes	—
251	C₄	#568×C₁₅₁	1	2	yes	—
252	C₄	C₂²×D₈×C₁₅₁	1	1	yes	characteristic
253	C₄	C₂²×Q₁₆×C₁₅₁	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	2	4	no	—
256	C₂	#2905×C₁₅₁	1	1	yes	characteristic
257	C₂	C₂²×#107×C₁₅₁	1	1	yes	characteristic
258	C₂	C₂×#430×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₅₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₅₁
Frattini, cosocle	C₄²	C₂⁴×C₁₅₁
socle	C₂²×C₁₅₁	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₅₁	C₄⋊C₄
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
3. #1571×C₁₅₁	C₂	semidirect
4. #1567×C₁₅₁	C₂	semidirect
18. #335×C₁₅₁	C₂²	semidirect
129. #17814	C₁₅₁	direct (N × H)
130. C₁₅₁	#17814	direct (N × H)
133. #1571	C₂×C₁₅₁	semidirect
134. #1567	C₂×C₁₅₁	semidirect
148. #335	C₂²×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431×C₁₅₁	2
C₂×#423×C₁₅₁	2
#1571×C₁₅₁	2
#1567×C₁₅₁	2
#17814	151

Subgroups — C151×#17814 #50322739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#17814 #50322739