Subgroups — C₁₅₁×#19063 #50323988

Order 38656.

806 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3556×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#3235×C₁₅₁	C₂	1	2	yes	maximal
3	#3135×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	#1663×C₁₅₁	C₂	1	4	yes	maximal, complemented
5	#1658×C₁₅₁	C₂	1	4	yes	maximal, complemented
6	#1537×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#1531×C₁₅₁	C₂	1	2	yes	maximal, complemented
8	#564×C₁₅₁	C₂²	1	4	yes	—
9	#557×C₁₅₁	—	2	2	no	—
10	#557×C₁₅₁	—	2	2	no	—
11	#557×C₁₅₁	—	2	2	no	—
12	#541×C₁₅₁	—	2	2	no	—
13	#541×C₁₅₁	—	2	4	no	—
14	#533×C₁₅₁	C₂²	1	1	yes	characteristic
15	#533×C₁₅₁	C₂²	1	2	yes	—
16	#532×C₁₅₁	C₂²	1	1	yes	characteristic
17	#501×C₁₅₁	C₂²	1	4	yes	—
18	#491×C₁₅₁	C₂²	1	2	yes	—
19	#490×C₁₅₁	C₂²	1	2	yes	—
20	#490×C₁₅₁	C₂²	1	4	yes	—
21	#489×C₁₅₁	—	2	2	no	—
22	#482×C₁₅₁	—	2	4	no	—
23	#476×C₁₅₁	—	2	2	no	—
24	#474×C₁₅₁	—	2	2	no	—
25	#432×C₁₅₁	C₂²	1	2	yes	—
26	#416×C₁₅₁	—	2	2	no	—
27	C₂×#97×C₁₅₁	C₂²	1	1	yes	characteristic
28	#330×C₁₅₁	C₂²	1	8	yes	—
29	#330×C₁₅₁	C₂²	1	4	yes	—
30	C₂×#41×C₁₅₁	—	2	1	no	—
31	C₂²×C₈×C₁₅₁	—	2	1	no	—
32	C₄⋊Q₈×C₁₅₁	—	2	4	no	—
33	C₄⋊Q₈×C₁₅₁	—	2	2	no	—
34	C₄⋊Q₈×C₁₅₁	D₄	1	2	yes	—
35	C₄⋊Q₈×C₁₅₁	D₄	1	2	yes	—
36	C₄⋊Q₈×C₁₅₁	D₄	1	2	yes	—
37	C₄⋊Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
38	#127×C₁₅₁	C₂³	1	1	yes	characteristic
39	#126×C₁₅₁	—	2	8	no	—
40	#125×C₁₅₁	—	2	4	no	—
41	#125×C₁₅₁	—	2	2	no	—
42	#125×C₁₅₁	—	2	8	no	—
43	#125×C₁₅₁	—	2	8	no	—
44	#125×C₁₅₁	—	2	8	no	—
45	#125×C₁₅₁	C₂³	1	2	yes	—
46	#125×C₁₅₁	C₂³	1	4	yes	—
47	#124×C₁₅₁	—	2	1	no	—
48	C₂²⋊Q₈×C₁₅₁	—	2	2	no	—
49	C₂²⋊Q₈×C₁₅₁	—	2	2	no	—
50	C₂²⋊Q₈×C₁₅₁	—	2	2	no	—
51	#121×C₁₅₁	—	2	2	no	—
52	C₄×Q₈×C₁₅₁	—	2	4	no	—
53	C₄×Q₈×C₁₅₁	—	2	4	no	—
54	C₄×Q₈×C₁₅₁	—	2	4	no	—
55	C₄×Q₈×C₁₅₁	—	2	2	no	—
56	C₄×Q₈×C₁₅₁	—	2	2	no	—
57	C₄×Q₈×C₁₅₁	—	2	2	no	—
58	C₄×D₄×C₁₅₁	—	2	2	no	—
59	#117×C₁₅₁	D₄	1	2	yes	—
60	#117×C₁₅₁	D₄	1	4	yes	—
61	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
62	#107×C₁₅₁	—	2	4	no	—
63	#106×C₁₅₁	—	2	4	no	—
64	C₄⋊C₈×C₁₅₁	—	2	2	no	—
65	#103×C₁₅₁	—	2	4	no	—
66	#102×C₁₅₁	—	2	4	no	—
67	#102×C₁₅₁	—	2	4	no	—
68	#102×C₁₅₁	—	2	4	no	—
69	C₂²⋊C₈×C₁₅₁	—	2	2	no	—
70	#97×C₁₅₁	—	2	1	no	—
71	#97×C₁₅₁	C₂³	1	2	yes	—
72	#97×C₁₅₁	C₂³	1	4	yes	—
73	#41×C₁₅₁	—	4	4	no	—
74	C₂×Q₈×C₁₅₁	—	2	2	no	—
75	C₂×Q₈×C₁₅₁	—	2	2	no	—
76	C₂×Q₈×C₁₅₁	—	2	2	no	—
77	C₂×Q₈×C₁₅₁	—	2	1	no	—
78	C₂×D₄×C₁₅₁	—	2	2	no	—
79	C₂×D₄×C₁₅₁	—	2	1	no	—
80	C₂×D₄×C₁₅₁	—	4	1	no	—
81	C₂²×C₄×C₁₅₁	—	4	1	no	—
82	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₅₁	—	2	2	no	—
85	C₂×C₈×C₁₅₁	—	2	4	no	—
86	C₂×C₈×C₁₅₁	—	4	2	no	—
87	C₄⋊C₄×C₁₅₁	—	2	8	no	—
88	C₄⋊C₄×C₁₅₁	—	2	8	no	—
89	C₄⋊C₄×C₁₅₁	—	2	8	no	—
90	C₄⋊C₄×C₁₅₁	—	2	2	no	—
91	C₄⋊C₄×C₁₅₁	—	2	2	no	—
92	C₄⋊C₄×C₁₅₁	—	2	2	no	—
93	C₄⋊C₄×C₁₅₁	—	2	4	no	—
94	C₄⋊C₄×C₁₅₁	—	2	4	no	—
95	C₄⋊C₄×C₁₅₁	—	2	4	no	—
96	C₄⋊C₄×C₁₅₁	—	2	2	no	—
97	C₄⋊C₄×C₁₅₁	C₂×D₄	1	4	yes	—
98	C₄⋊C₄×C₁₅₁	C₂×D₄	1	4	yes	—
99	C₄⋊C₄×C₁₅₁	C₂×D₄	1	4	yes	—
100	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
101	C₂²⋊C₄×C₁₅₁	—	2	4	no	—
102	C₂²⋊C₄×C₁₅₁	—	4	2	no	—
103	C₄²×C₁₅₁	—	2	8	no	—
104	C₄²×C₁₅₁	—	2	4	no	—
105	C₄²×C₁₅₁	—	2	2	no	—
106	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₅₁	C₂×D₄	1	2	yes	—
108	C₄²×C₁₅₁	C₂⁴	1	1	yes	characteristic
109	C₂³×C₁₅₁	—	4	1	no	—
110	C₂³×C₁₅₁	#120	1	1	yes	characteristic
111	Q₈×C₁₅₁	—	4	4	no	—
112	Q₈×C₁₅₁	—	4	4	no	—
113	Q₈×C₁₅₁	—	4	2	no	—
114	Q₈×C₁₅₁	—	4	4	no	—
115	D₄×C₁₅₁	—	4	2	no	—
116	D₄×C₁₅₁	—	4	4	no	—
117	D₄×C₁₅₁	—	8	2	no	—
118	C₂×C₄×C₁₅₁	—	2	2	no	—
119	C₂×C₄×C₁₅₁	—	2	4	no	—
120	C₂×C₄×C₁₅₁	—	2	4	no	—
121	C₂×C₄×C₁₅₁	—	2	4	no	—
122	C₂×C₄×C₁₅₁	—	4	2	no	—
123	C₂×C₄×C₁₅₁	—	4	1	no	—
124	C₂×C₄×C₁₅₁	—	8	2	no	—
125	C₂×C₄×C₁₅₁	#120	1	2	yes	—
126	C₂×C₄×C₁₅₁	#120	1	1	yes	characteristic
127	C₂×C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₅₁	C₂²×D₄	1	2	yes	—
129	C₈×C₁₅₁	—	4	4	no	—
130	C₈×C₁₅₁	—	4	2	no	—
131	C₂²×C₁₅₁	—	2	1	no	—
132	C₂²×C₁₅₁	—	2	2	no	—
133	C₂²×C₁₅₁	—	8	1	no	—
134	C₂²×C₁₅₁	—	8	2	no	—
135	C₂²×C₁₅₁	C₂×#120	1	1	yes	characteristic
136	C₄×C₁₅₁	—	2	1	no	—
137	C₄×C₁₅₁	—	2	2	no	—
138	C₄×C₁₅₁	—	2	2	no	—
139	C₄×C₁₅₁	—	4	4	no	—
140	C₄×C₁₅₁	—	4	4	no	—
141	C₄×C₁₅₁	—	4	4	no	—
142	C₄×C₁₅₁	—	8	1	no	—
143	C₄×C₁₅₁	#454	1	2	yes	—
144	C₂×C₁₅₁	—	4	1	no	—
145	C₂×C₁₅₁	—	16	1	no	—
146	C₂×C₁₅₁	C₂×#454	1	1	yes	characteristic
147	C₂×C₁₅₁	#3020	1	2	yes	—
148	#19063	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₁₅₁	#19063	1	1	yes	characteristic, direct product factor, Sylow
150	#3556	C₂×C₁₅₁	1	1	yes	characteristic, complemented
151	#3235	C₂×C₁₅₁	1	2	yes	—
152	#3135	C₂×C₁₅₁	1	1	yes	characteristic
153	#1663	C₂×C₁₅₁	1	4	yes	complemented
154	#1658	C₂×C₁₅₁	1	4	yes	complemented
155	#1537	C₂×C₁₅₁	1	1	yes	characteristic, complemented
156	#1531	C₂×C₁₅₁	1	2	yes	complemented
157	#564	C₂²×C₁₅₁	1	4	yes	—
158	#557	—	2	2	no	—
159	#557	—	2	2	no	—
160	#557	—	2	2	no	—
161	#541	—	2	2	no	—
162	#541	—	2	4	no	—
163	#533	C₂²×C₁₅₁	1	1	yes	characteristic
164	#533	C₂²×C₁₅₁	1	2	yes	—
165	#532	C₂²×C₁₅₁	1	1	yes	characteristic
166	#501	C₂²×C₁₅₁	1	4	yes	—
167	#491	C₂²×C₁₅₁	1	2	yes	—
168	#490	C₂²×C₁₅₁	1	2	yes	—
169	#490	C₂²×C₁₅₁	1	4	yes	—
170	#489	—	2	2	no	—
171	#482	—	2	4	no	—
172	#476	—	2	2	no	—
173	#474	—	2	2	no	—
174	#432	C₂²×C₁₅₁	1	2	yes	—
175	#416	—	2	2	no	—
176	C₂×#97	C₂²×C₁₅₁	1	1	yes	characteristic
177	#330	C₂²×C₁₅₁	1	4	yes	—
178	#330	C₂²×C₁₅₁	1	8	yes	—
179	C₂×#41	—	2	1	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₄⋊Q₈	—	2	4	no	—
182	C₄⋊Q₈	—	2	2	no	—
183	C₄⋊Q₈	D₄×C₁₅₁	1	2	yes	—
184	C₄⋊Q₈	D₄×C₁₅₁	1	2	yes	—
185	C₄⋊Q₈	D₄×C₁₅₁	1	2	yes	—
186	C₄⋊Q₈	C₂³×C₁₅₁	1	1	yes	characteristic
187	#127	C₂³×C₁₅₁	1	1	yes	characteristic
188	#126	—	2	8	no	—
189	#125	—	2	4	no	—
190	#125	—	2	2	no	—
191	#125	—	2	8	no	—
192	#125	—	2	8	no	—
193	#125	—	2	8	no	—
194	#125	C₂³×C₁₅₁	1	2	yes	—
195	#125	C₂³×C₁₅₁	1	4	yes	—
196	#124	—	2	1	no	—
197	C₂²⋊Q₈	—	2	2	no	—
198	C₂²⋊Q₈	—	2	2	no	—
199	C₂²⋊Q₈	—	2	2	no	—
200	#121	—	2	2	no	—
201	C₄×Q₈	—	2	4	no	—
202	C₄×Q₈	—	2	4	no	—
203	C₄×Q₈	—	2	4	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×D₄	—	2	2	no	—
208	#117	D₄×C₁₅₁	1	2	yes	—
209	#117	D₄×C₁₅₁	1	4	yes	—
210	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
211	#107	—	2	4	no	—
212	#106	—	2	4	no	—
213	C₄⋊C₈	—	2	2	no	—
214	#103	—	2	4	no	—
215	#102	—	2	4	no	—
216	#102	—	2	4	no	—
217	#102	—	2	4	no	—
218	C₂²⋊C₈	—	2	2	no	—
219	#97	—	2	1	no	—
220	#97	C₂³×C₁₅₁	1	2	yes	—
221	#97	C₂³×C₁₅₁	1	4	yes	—
222	#41	—	4	4	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	1	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	—	4	1	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₁₅₁	1	2	yes	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	8	no	—
238	C₄⋊C₄	—	2	8	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	C₂×D₄×C₁₅₁	1	4	yes	—
247	C₄⋊C₄	C₂×D₄×C₁₅₁	1	4	yes	—
248	C₄⋊C₄	C₂×D₄×C₁₅₁	1	4	yes	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	4	2	no	—
252	C₄²	—	2	8	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
256	C₄²	C₂×D₄×C₁₅₁	1	2	yes	—
257	C₄²	C₂⁴×C₁₅₁	1	1	yes	characteristic
258	C₂³	—	4	1	no	—
259	C₂³	#120×C₁₅₁	1	1	yes	characteristic
260	Q₈	—	4	4	no	—
261	Q₈	—	4	4	no	—
262	Q₈	—	4	2	no	—
263	Q₈	—	4	4	no	—
264	D₄	—	4	2	no	—
265	D₄	—	4	4	no	—
266	D₄	—	8	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	#120×C₁₅₁	1	2	yes	—
275	C₂×C₄	#120×C₁₅₁	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₄×C₁₅₁	1	2	yes	—
277	C₂×C₄	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
278	C₈	—	4	2	no	—
279	C₈	—	4	4	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	8	1	no	—
283	C₂²	—	8	2	no	—
284	C₂²	C₂×#120×C₁₅₁	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	4	4	no	—
290	C₄	—	4	4	no	—
291	C₄	—	8	1	no	—
292	C₄	#454×C₁₅₁	1	2	yes	—
293	C₂	—	4	1	no	—
294	C₂	—	16	1	no	—
295	C₂	C₂×#454×C₁₅₁	1	1	yes	characteristic
296	C₂	#3020×C₁₅₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₅₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₅₁
prime radical	#532×C₁₅₁	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₂×C₄²×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3556×C₁₅₁	C₂	semidirect
4. #1663×C₁₅₁	C₂	semidirect
5. #1658×C₁₅₁	C₂	semidirect
6. #1537×C₁₅₁	C₂	semidirect
7. #1531×C₁₅₁	C₂	semidirect
148. #19063	C₁₅₁	direct (N × H)
149. C₁₅₁	#19063	direct (N × H)
150. #3556	C₂×C₁₅₁	semidirect
153. #1663	C₂×C₁₅₁	semidirect
154. #1658	C₂×C₁₅₁	semidirect
155. #1537	C₂×C₁₅₁	semidirect
156. #1531	C₂×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#3556×C₁₅₁	2
#3235×C₁₅₁	2
#3135×C₁₅₁	2
#1663×C₁₅₁	2
#1658×C₁₅₁	2
#1537×C₁₅₁	2
#1531×C₁₅₁	2
#19063	151

Subgroups — C151×#19063 #50323988

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#19063 #50323988