Subgroups — C₁₅₁×#23187 #50328112

Order 38656.

946 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#441×C₁₅₁	C₂	1	2	yes	maximal
3	#2365×C₁₅₁	C₂	1	2	yes	maximal
4	#1981×C₁₅₁	C₂	1	2	yes	maximal
5	#1971×C₁₅₁	C₂	1	2	yes	maximal
6	#1878×C₁₅₁	C₂	1	4	yes	maximal
7	C₄×#106×C₁₅₁	C₂	1	1	yes	maximal, characteristic
8	#1770×C₁₅₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₁₅₁	C₂²	1	2	yes	—
10	C₂²×Q₁₆×C₁₅₁	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₁₅₁	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₁₅₁	C₂²	1	4	yes	—
13	#494×C₁₅₁	C₄	1	8	yes	complemented
14	#441×C₁₅₁	—	2	8	no	—
15	C₂×#106×C₁₅₁	C₂²	1	1	yes	characteristic
16	C₂×#106×C₁₅₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
18	C₂×#103×C₁₅₁	C₂²	1	4	yes	—
19	C₂×#103×C₁₅₁	C₂²	1	4	yes	—
20	C₂×#97×C₁₅₁	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
22	#391×C₁₅₁	C₂²	1	4	yes	—
23	#389×C₁₅₁	—	2	2	no	—
24	#387×C₁₅₁	—	2	4	no	—
25	C₄×C₄⋊C₄×C₁₅₁	—	2	2	no	—
26	C₄×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₁₅₁	C₂²	1	1	yes	characteristic
28	#340×C₁₅₁	C₂²	1	2	yes	—
29	#336×C₁₅₁	C₂²	1	2	yes	—
30	C₂²×Q₈×C₁₅₁	C₂³	1	4	yes	—
31	C₂×Q₁₆×C₁₅₁	—	2	4	no	—
32	C₂×Q₁₆×C₁₅₁	—	2	4	no	—
33	C₂×Q₁₆×C₁₅₁	—	4	2	no	—
34	C₂×Q₁₆×C₁₅₁	—	4	2	no	—
35	C₂×Q₁₆×C₁₅₁	C₂×C₄	1	8	yes	—
36	C₂×Q₁₆×C₁₅₁	C₂×C₄	1	8	yes	—
37	C₂²×C₈×C₁₅₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₁₅₁	—	2	4	no	—
41	C₄⋊Q₈×C₁₅₁	C₂×C₄	1	8	yes	—
42	C₄×Q₈×C₁₅₁	—	2	8	no	—
43	C₄×Q₈×C₁₅₁	—	4	4	no	—
44	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₅₁	—	2	4	no	—
47	C₂×C₄⋊C₄×C₁₅₁	D₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₁₅₁	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₁₅₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₅₁	—	2	2	no	—
51	C₂×C₄²×C₁₅₁	—	2	4	no	—
52	C₂×C₄²×C₁₅₁	C₂³	1	2	yes	—
53	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
54	#106×C₁₅₁	—	2	2	no	—
55	#106×C₁₅₁	D₄	1	4	yes	—
56	C₄⋊C₈×C₁₅₁	—	2	2	no	—
57	#103×C₁₅₁	—	2	8	no	—
58	#103×C₁₅₁	—	4	4	no	—
59	#97×C₁₅₁	—	2	4	no	—
60	C₄×C₈×C₁₅₁	C₂×C₄	1	4	yes	—
61	#95×C₁₅₁	—	2	4	no	—
62	#95×C₁₅₁	—	2	2	no	—
63	#95×C₁₅₁	—	2	4	no	—
64	#95×C₁₅₁	D₄	1	2	yes	—
65	C₂×Q₈×C₁₅₁	—	2	4	no	—
66	C₂×Q₈×C₁₅₁	—	2	4	no	—
67	C₂×Q₈×C₁₅₁	—	2	4	no	—
68	C₂×Q₈×C₁₅₁	—	2	4	no	—
69	C₂×Q₈×C₁₅₁	—	2	4	no	—
70	C₂×Q₈×C₁₅₁	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₁₅₁	—	2	4	no	—
72	C₂²×C₄×C₁₅₁	—	2	2	no	—
73	C₂²×C₄×C₁₅₁	—	2	2	no	—
74	C₂²×C₄×C₁₅₁	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
78	Q₁₆×C₁₅₁	—	2	16	no	—
79	Q₁₆×C₁₅₁	—	4	8	no	—
80	C₂×C₈×C₁₅₁	—	2	1	no	—
81	C₂×C₈×C₁₅₁	—	2	1	no	—
82	C₂×C₈×C₁₅₁	—	2	2	no	—
83	C₂×C₈×C₁₅₁	—	2	2	no	—
84	C₂×C₈×C₁₅₁	—	2	2	no	—
85	C₂×C₈×C₁₅₁	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₅₁	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₅₁	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₅₁	#41	1	2	yes	—
89	C₄⋊C₄×C₁₅₁	—	2	8	no	—
90	C₄⋊C₄×C₁₅₁	—	2	8	no	—
91	C₄⋊C₄×C₁₅₁	—	2	4	no	—
92	C₄⋊C₄×C₁₅₁	—	2	2	no	—
93	C₄⋊C₄×C₁₅₁	—	2	2	no	—
94	C₄⋊C₄×C₁₅₁	—	2	2	no	—
95	C₄⋊C₄×C₁₅₁	—	4	8	no	—
96	C₄⋊C₄×C₁₅₁	C₂×D₄	1	4	yes	—
97	C₄²×C₁₅₁	—	2	2	no	—
98	C₄²×C₁₅₁	—	2	2	no	—
99	C₄²×C₁₅₁	—	4	8	no	—
100	C₄²×C₁₅₁	—	4	4	no	—
101	C₄²×C₁₅₁	C₂²×C₄	1	2	yes	—
102	C₄²×C₁₅₁	#41	1	2	yes	—
103	C₂³×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₁₅₁	—	2	8	no	—
105	Q₈×C₁₅₁	—	2	8	no	—
106	Q₈×C₁₅₁	—	4	4	no	—
107	Q₈×C₁₅₁	—	4	4	no	—
108	C₂×C₄×C₁₅₁	—	2	2	no	—
109	C₂×C₄×C₁₅₁	—	2	8	no	—
110	C₂×C₄×C₁₅₁	—	2	4	no	—
111	C₂×C₄×C₁₅₁	—	2	4	no	—
112	C₂×C₄×C₁₅₁	—	2	4	no	—
113	C₂×C₄×C₁₅₁	—	2	4	no	—
114	C₂×C₄×C₁₅₁	—	4	4	no	—
115	C₂×C₄×C₁₅₁	—	4	4	no	—
116	C₂×C₄×C₁₅₁	—	4	2	no	—
117	C₂×C₄×C₁₅₁	—	4	2	no	—
118	C₂×C₄×C₁₅₁	—	4	2	no	—
119	C₂×C₄×C₁₅₁	—	4	2	no	—
120	C₂×C₄×C₁₅₁	—	4	2	no	—
121	C₂×C₄×C₁₅₁	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₅₁	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₅₁	#137	1	4	yes	—
124	C₂×C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₁₅₁	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₄×C₁₅₁	H(5,𝔽₂)	1	1	yes	characteristic
130	C₈×C₁₅₁	—	2	2	no	—
131	C₈×C₁₅₁	—	4	2	no	—
132	C₈×C₁₅₁	C₄×D₄	1	4	yes	—
133	C₂²×C₁₅₁	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²×C₁₅₁	D₄²	1	1	yes	characteristic
135	C₂²×C₁₅₁	#550	1	1	yes	characteristic
136	C₂²×C₁₅₁	C₂×#137	1	2	yes	—
137	C₂²×C₁₅₁	#578	1	2	yes	—
138	C₄×C₁₅₁	—	2	2	no	—
139	C₄×C₁₅₁	—	4	8	no	—
140	C₄×C₁₅₁	—	4	4	no	—
141	C₄×C₁₅₁	—	4	4	no	—
142	C₄×C₁₅₁	—	8	2	no	—
143	C₄×C₁₅₁	#441	1	4	yes	complemented
144	C₄×C₁₅₁	C₂×C₄×D₄	1	1	yes	characteristic
145	C₄×C₁₅₁	#518	1	1	yes	characteristic
146	C₄×C₁₅₁	#550	1	1	yes	characteristic
147	C₄×C₁₅₁	#559	1	1	yes	characteristic
148	C₂×C₁₅₁	#2366	1	1	yes	characteristic
149	C₂×C₁₅₁	C₂×#441	1	1	yes	characteristic
150	C₂×C₁₅₁	#2950	1	1	yes	characteristic
151	C₂×C₁₅₁	#3276	1	2	yes	—
152	C₂×C₁₅₁	#3398	1	2	yes	—
153	#23187	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₁₅₁	#23187	1	1	yes	characteristic, direct product factor, Sylow
155	C₂×#494	C₂×C₁₅₁	1	1	yes	characteristic
156	C₂×#441	C₂×C₁₅₁	1	2	yes	—
157	#2365	C₂×C₁₅₁	1	2	yes	—
158	#1981	C₂×C₁₅₁	1	2	yes	—
159	#1971	C₂×C₁₅₁	1	2	yes	—
160	#1878	C₂×C₁₅₁	1	4	yes	—
161	C₄×#106	C₂×C₁₅₁	1	1	yes	characteristic
162	#1770	C₂×C₁₅₁	1	1	yes	characteristic
163	C₂²×Q₁₆	C₂²×C₁₅₁	1	2	yes	—
164	C₂²×Q₁₆	C₂²×C₁₅₁	1	2	yes	—
165	C₂×C₄⋊Q₈	C₂²×C₁₅₁	1	2	yes	—
166	C₂×C₄×Q₈	C₂²×C₁₅₁	1	4	yes	—
167	#494	C₄×C₁₅₁	1	8	yes	complemented
168	#441	—	2	8	no	—
169	C₂×#106	C₂²×C₁₅₁	1	1	yes	characteristic
170	C₂×#106	C₂²×C₁₅₁	1	1	yes	characteristic
171	C₂×C₄⋊C₈	C₂²×C₁₅₁	1	1	yes	characteristic
172	C₂×#103	C₂²×C₁₅₁	1	4	yes	—
173	C₂×#103	C₂²×C₁₅₁	1	4	yes	—
174	C₂×#97	C₂²×C₁₅₁	1	2	yes	—
175	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
176	#391	C₂²×C₁₅₁	1	4	yes	—
177	#389	—	2	2	no	—
178	#387	—	2	4	no	—
179	C₄×C₄⋊C₄	—	2	2	no	—
180	C₄×C₄⋊C₄	C₂²×C₁₅₁	1	2	yes	—
181	C₄×C₄⋊C₄	C₂²×C₁₅₁	1	1	yes	characteristic
182	#340	C₂²×C₁₅₁	1	2	yes	—
183	#336	C₂²×C₁₅₁	1	2	yes	—
184	C₂²×Q₈	C₂³×C₁₅₁	1	4	yes	—
185	C₂×Q₁₆	—	2	4	no	—
186	C₂×Q₁₆	—	2	4	no	—
187	C₂×Q₁₆	—	4	2	no	—
188	C₂×Q₁₆	—	4	2	no	—
189	C₂×Q₁₆	C₂×C₄×C₁₅₁	1	8	yes	—
190	C₂×Q₁₆	C₂×C₄×C₁₅₁	1	8	yes	—
191	C₂²×C₈	C₂³×C₁₅₁	1	2	yes	—
192	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
194	C₄⋊Q₈	—	2	4	no	—
195	C₄⋊Q₈	C₂×C₄×C₁₅₁	1	8	yes	—
196	C₄×Q₈	—	2	8	no	—
197	C₄×Q₈	—	4	4	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	D₄×C₁₅₁	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₁₅₁	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₅₁	1	2	yes	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	C₂³×C₁₅₁	1	2	yes	—
207	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
208	#106	—	2	2	no	—
209	#106	D₄×C₁₅₁	1	4	yes	—
210	C₄⋊C₈	—	2	2	no	—
211	#103	—	2	8	no	—
212	#103	—	4	4	no	—
213	#97	—	2	4	no	—
214	C₄×C₈	C₂×C₄×C₁₅₁	1	4	yes	—
215	#95	—	2	4	no	—
216	#95	—	2	2	no	—
217	#95	—	2	4	no	—
218	#95	D₄×C₁₅₁	1	2	yes	—
219	C₂×Q₈	—	2	4	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	C₂²×C₄×C₁₅₁	1	8	yes	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂×D₄×C₁₅₁	1	4	yes	—
229	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
231	C₂²×C₄	C₂⁴×C₁₅₁	1	1	yes	characteristic
232	Q₁₆	—	2	16	no	—
233	Q₁₆	—	4	8	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	C₂²×C₄×C₁₅₁	1	2	yes	—
240	C₂×C₈	C₂²×C₄×C₁₅₁	1	2	yes	—
241	C₂×C₈	C₂×D₄×C₁₅₁	1	2	yes	—
242	C₂×C₈	#41×C₁₅₁	1	2	yes	—
243	C₄⋊C₄	—	2	8	no	—
244	C₄⋊C₄	—	2	8	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	8	no	—
250	C₄⋊C₄	C₂×D₄×C₁₅₁	1	4	yes	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	4	8	no	—
254	C₄²	—	4	4	no	—
255	C₄²	C₂²×C₄×C₁₅₁	1	2	yes	—
256	C₄²	#41×C₁₅₁	1	2	yes	—
257	C₂³	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
258	Q₈	—	2	8	no	—
259	Q₈	—	2	8	no	—
260	Q₈	—	4	4	no	—
261	Q₈	—	4	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	C₄×D₄×C₁₅₁	1	2	yes	—
276	C₂×C₄	C₄×D₄×C₁₅₁	1	2	yes	—
277	C₂×C₄	#137×C₁₅₁	1	4	yes	—
278	C₂×C₄	C₂³×C₄×C₁₅₁	1	1	yes	characteristic
279	C₂×C₄	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
280	C₂×C₄	C₂×#41×C₁₅₁	1	1	yes	characteristic
281	C₂×C₄	C₂×#41×C₁₅₁	1	1	yes	characteristic
282	C₂×C₄	H(5,𝔽₂)×C₁₅₁	1	1	yes	characteristic
283	C₂×C₄	H(5,𝔽₂)×C₁₅₁	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	4	2	no	—
286	C₈	C₄×D₄×C₁₅₁	1	4	yes	—
287	C₂²	C₂×C₄×D₄×C₁₅₁	1	1	yes	characteristic
288	C₂²	D₄²×C₁₅₁	1	1	yes	characteristic
289	C₂²	#550×C₁₅₁	1	1	yes	characteristic
290	C₂²	C₂×#137×C₁₅₁	1	2	yes	—
291	C₂²	#578×C₁₅₁	1	2	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	4	8	no	—
294	C₄	—	4	4	no	—
295	C₄	—	4	4	no	—
296	C₄	—	8	2	no	—
297	C₄	#441×C₁₅₁	1	4	yes	complemented
298	C₄	C₂×C₄×D₄×C₁₅₁	1	1	yes	characteristic
299	C₄	#518×C₁₅₁	1	1	yes	characteristic
300	C₄	#550×C₁₅₁	1	1	yes	characteristic
301	C₄	#559×C₁₅₁	1	1	yes	characteristic
302	C₂	#2366×C₁₅₁	1	1	yes	characteristic
303	C₂	C₂×#441×C₁₅₁	1	1	yes	characteristic
304	C₂	#2950×C₁₅₁	1	1	yes	characteristic
305	C₂	#3276×C₁₅₁	1	2	yes	—
306	C₂	#3398×C₁₅₁	1	2	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 ◁ C₄×C₄⋊C₄×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. #494×C₁₅₁	C₄	semidirect
143. C₄×C₁₅₁	#441	semidirect
153. #23187	C₁₅₁	direct (N × H)
154. C₁₅₁	#23187	direct (N × H)
167. #494	C₄×C₁₅₁	semidirect
297. C₄	#441×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494×C₁₅₁	2
C₂×#441×C₁₅₁	2
#2365×C₁₅₁	2
#1981×C₁₅₁	2
#1971×C₁₅₁	2
#1878×C₁₅₁	2
C₄×#106×C₁₅₁	2
#1770×C₁₅₁	2
#23187	151

Subgroups — C151×#23187 #50328112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#23187 #50328112