Subgroups — C₁₅₁×#23820 #50328745

Order 38656.

726 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	C₄×#125×C₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#2099×C₁₅₁	C₂	1	2	yes	maximal
4	#2074×C₁₅₁	C₂	1	2	yes	maximal
5	#2043×C₁₅₁	C₂	1	2	yes	maximal
6	#2026×C₁₅₁	C₂	1	2	yes	maximal
7	C₄×#103×C₁₅₁	C₂	1	4	yes	maximal
8	#1744×C₁₅₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
10	C₂×#125×C₁₅₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₅₁	C₂²	1	2	yes	—
12	C₂×#103×C₁₅₁	C₂²	1	4	yes	—
13	C₂×#103×C₁₅₁	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₁₅₁	C₂²	1	2	yes	—
15	#391×C₁₅₁	C₂²	1	2	yes	—
16	#389×C₁₅₁	—	2	2	no	—
17	#389×C₁₅₁	C₂²	1	2	yes	—
18	#387×C₁₅₁	—	2	2	no	—
19	#384×C₁₅₁	C₂²	1	1	yes	characteristic
20	#382×C₁₅₁	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₁₅₁	—	2	2	no	—
22	C₄×C₄⋊C₄×C₁₅₁	—	2	2	no	—
23	C₄×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
24	C₄³×C₁₅₁	C₂²	1	1	yes	characteristic
25	#340×C₁₅₁	C₂²	1	4	yes	—
26	#340×C₁₅₁	C₂²	1	4	yes	—
27	#336×C₁₅₁	C₂²	1	2	yes	—
28	#336×C₁₅₁	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₅₁	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₅₁	C₂³	1	4	yes	—
31	C₄⋊Q₈×C₁₅₁	—	2	4	no	—
32	#125×C₁₅₁	—	2	4	no	—
33	C₄×Q₈×C₁₅₁	—	2	8	no	—
34	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₅₁	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₅₁	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₅₁	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₅₁	—	2	2	no	—
43	C₂×C₄²×C₁₅₁	—	2	2	no	—
44	C₂×C₄²×C₁₅₁	D₄	1	2	yes	—
45	C₂×C₄²×C₁₅₁	D₄	1	2	yes	—
46	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
49	#103×C₁₅₁	—	2	8	no	—
50	#103×C₁₅₁	—	2	8	no	—
51	C₄×C₈×C₁₅₁	—	2	4	no	—
52	#95×C₁₅₁	—	2	2	no	—
53	#95×C₁₅₁	—	2	2	no	—
54	#95×C₁₅₁	—	2	4	no	—
55	#95×C₁₅₁	—	2	2	no	—
56	C₂×Q₈×C₁₅₁	—	2	2	no	—
57	C₂×Q₈×C₁₅₁	—	2	4	no	—
58	C₂×Q₈×C₁₅₁	—	2	4	no	—
59	C₂×Q₈×C₁₅₁	#41	1	4	yes	—
60	C₂²×C₄×C₁₅₁	—	2	2	no	—
61	C₂²×C₄×C₁₅₁	—	2	2	no	—
62	C₂²×C₄×C₁₅₁	—	2	2	no	—
63	C₂²×C₄×C₁₅₁	—	2	2	no	—
64	C₂²×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₁₅₁	—	2	4	no	—
70	C₂×C₈×C₁₅₁	—	2	4	no	—
71	C₂×C₈×C₁₅₁	—	2	4	no	—
72	C₄⋊C₄×C₁₅₁	—	2	8	no	—
73	C₄⋊C₄×C₁₅₁	—	2	8	no	—
74	C₄⋊C₄×C₁₅₁	—	2	2	no	—
75	C₄⋊C₄×C₁₅₁	—	2	4	no	—
76	C₄⋊C₄×C₁₅₁	—	2	4	no	—
77	C₄⋊C₄×C₁₅₁	—	4	4	no	—
78	C₄⋊C₄×C₁₅₁	—	4	4	no	—
79	C₄⋊C₄×C₁₅₁	—	4	4	no	—
80	C₄⋊C₄×C₁₅₁	#41	1	4	yes	—
81	C₄²×C₁₅₁	—	2	4	no	—
82	C₄²×C₁₅₁	—	2	4	no	—
83	C₄²×C₁₅₁	—	2	2	no	—
84	C₄²×C₁₅₁	—	2	2	no	—
85	C₄²×C₁₅₁	—	4	4	no	—
86	C₄²×C₁₅₁	—	4	4	no	—
87	C₄²×C₁₅₁	#41	1	4	yes	—
88	C₂³×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
89	Q₈×C₁₅₁	—	2	8	no	—
90	Q₈×C₁₅₁	—	4	4	no	—
91	C₂×C₄×C₁₅₁	—	2	2	no	—
92	C₂×C₄×C₁₅₁	—	2	2	no	—
93	C₂×C₄×C₁₅₁	—	2	2	no	—
94	C₂×C₄×C₁₅₁	—	2	4	no	—
95	C₂×C₄×C₁₅₁	—	2	2	no	—
96	C₂×C₄×C₁₅₁	—	2	4	no	—
97	C₂×C₄×C₁₅₁	—	2	2	no	—
98	C₂×C₄×C₁₅₁	—	2	2	no	—
99	C₂×C₄×C₁₅₁	—	4	2	no	—
100	C₂×C₄×C₁₅₁	—	4	2	no	—
101	C₂×C₄×C₁₅₁	—	4	2	no	—
102	C₂×C₄×C₁₅₁	—	4	2	no	—
103	C₂×C₄×C₁₅₁	—	4	2	no	—
104	C₂×C₄×C₁₅₁	—	4	2	no	—
105	C₂×C₄×C₁₅₁	—	4	2	no	—
106	C₂×C₄×C₁₅₁	—	4	4	no	—
107	C₂×C₄×C₁₅₁	—	4	2	no	—
108	C₂×C₄×C₁₅₁	#124	1	2	yes	—
109	C₂×C₄×C₁₅₁	#124	1	2	yes	—
110	C₂×C₄×C₁₅₁	#135	1	4	yes	—
111	C₂×C₄×C₁₅₁	#137	1	4	yes	—
112	C₂×C₄×C₁₅₁	C₂×#41	1	2	yes	—
113	C₂×C₄×C₁₅₁	C₂×#41	1	2	yes	—
114	C₂×C₄×C₁₅₁	C₂×#41	1	2	yes	—
115	C₈×C₁₅₁	—	4	4	no	—
116	C₂²×C₁₅₁	#525	1	1	yes	characteristic
117	C₂²×C₁₅₁	#525	1	1	yes	characteristic
118	C₂²×C₁₅₁	C₂×#124	1	1	yes	characteristic
119	C₂²×C₁₅₁	C₂×#135	1	2	yes	—
120	C₂²×C₁₅₁	C₂×#137	1	2	yes	—
121	C₄×C₁₅₁	—	2	4	no	—
122	C₄×C₁₅₁	—	2	2	no	—
123	C₄×C₁₅₁	—	4	2	no	—
124	C₄×C₁₅₁	—	4	4	no	—
125	C₄×C₁₅₁	—	4	4	no	—
126	C₄×C₁₅₁	—	8	2	no	—
127	C₄×C₁₅₁	—	8	2	no	—
128	C₄×C₁₅₁	#491	1	4	yes	—
129	C₄×C₁₅₁	#529	1	4	yes	—
130	C₂×C₁₅₁	#2398	1	1	yes	characteristic
131	C₂×C₁₅₁	#3107	1	2	yes	—
132	C₂×C₁₅₁	#3108	1	2	yes	—
133	C₂×C₁₅₁	#3131	1	1	yes	characteristic
134	C₂×C₁₅₁	C₂×#491	1	1	yes	characteristic
135	#23820	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₁₅₁	#23820	1	1	yes	characteristic, direct product factor, Sylow
137	C₄×C₄⋊Q₈	C₂×C₁₅₁	1	1	yes	characteristic
138	C₄×#125	C₂×C₁₅₁	1	1	yes	characteristic
139	#2099	C₂×C₁₅₁	1	2	yes	—
140	#2074	C₂×C₁₅₁	1	2	yes	—
141	#2043	C₂×C₁₅₁	1	2	yes	—
142	#2026	C₂×C₁₅₁	1	2	yes	—
143	C₄×#103	C₂×C₁₅₁	1	4	yes	—
144	#1744	C₂×C₁₅₁	1	1	yes	characteristic
145	C₂×C₄⋊Q₈	C₂²×C₁₅₁	1	1	yes	characteristic
146	C₂×#125	C₂²×C₁₅₁	1	1	yes	characteristic
147	C₂×C₄×Q₈	C₂²×C₁₅₁	1	2	yes	—
148	C₂×#103	C₂²×C₁₅₁	1	4	yes	—
149	C₂×#103	C₂²×C₁₅₁	1	4	yes	—
150	C₂×C₄×C₈	C₂²×C₁₅₁	1	2	yes	—
151	#391	C₂²×C₁₅₁	1	2	yes	—
152	#389	—	2	2	no	—
153	#389	C₂²×C₁₅₁	1	2	yes	—
154	#387	—	2	2	no	—
155	#384	C₂²×C₁₅₁	1	1	yes	characteristic
156	#382	C₂²×C₁₅₁	1	1	yes	characteristic
157	C₄×C₄⋊C₄	—	2	2	no	—
158	C₄×C₄⋊C₄	—	2	2	no	—
159	C₄×C₄⋊C₄	C₂²×C₁₅₁	1	2	yes	—
160	C₄³	C₂²×C₁₅₁	1	1	yes	characteristic
161	#340	C₂²×C₁₅₁	1	4	yes	—
162	#340	C₂²×C₁₅₁	1	4	yes	—
163	#336	C₂²×C₁₅₁	1	2	yes	—
164	#336	C₂²×C₁₅₁	1	2	yes	—
165	C₂²×Q₈	C₂³×C₁₅₁	1	2	yes	—
166	C₂²×C₈	C₂³×C₁₅₁	1	4	yes	—
167	C₄⋊Q₈	—	2	4	no	—
168	#125	—	2	4	no	—
169	C₄×Q₈	—	2	8	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	C₂³×C₁₅₁	1	2	yes	—
176	C₂×C₄⋊C₄	C₂³×C₁₅₁	1	2	yes	—
177	C₂×C₄⋊C₄	C₂³×C₁₅₁	1	2	yes	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	D₄×C₁₅₁	1	2	yes	—
181	C₂×C₄²	D₄×C₁₅₁	1	2	yes	—
182	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
185	#103	—	2	8	no	—
186	#103	—	2	8	no	—
187	C₄×C₈	—	2	4	no	—
188	#95	—	2	2	no	—
189	#95	—	2	2	no	—
190	#95	—	2	4	no	—
191	#95	—	2	2	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	#41×C₁₅₁	1	4	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	C₂×D₄×C₁₅₁	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₁₅₁	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
204	C₂²×C₄	C₂⁴×C₁₅₁	1	1	yes	characteristic
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	2	8	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₄⋊C₄	#41×C₁₅₁	1	4	yes	—
217	C₄²	—	2	4	no	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	4	4	no	—
222	C₄²	—	4	4	no	—
223	C₄²	#41×C₁₅₁	1	4	yes	—
224	C₂³	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
225	Q₈	—	2	8	no	—
226	Q₈	—	4	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	#124×C₁₅₁	1	2	yes	—
245	C₂×C₄	#124×C₁₅₁	1	2	yes	—
246	C₂×C₄	#135×C₁₅₁	1	4	yes	—
247	C₂×C₄	#137×C₁₅₁	1	4	yes	—
248	C₂×C₄	C₂×#41×C₁₅₁	1	2	yes	—
249	C₂×C₄	C₂×#41×C₁₅₁	1	2	yes	—
250	C₂×C₄	C₂×#41×C₁₅₁	1	2	yes	—
251	C₈	—	4	4	no	—
252	C₂²	#525×C₁₅₁	1	1	yes	characteristic
253	C₂²	#525×C₁₅₁	1	1	yes	characteristic
254	C₂²	C₂×#124×C₁₅₁	1	1	yes	characteristic
255	C₂²	C₂×#135×C₁₅₁	1	2	yes	—
256	C₂²	C₂×#137×C₁₅₁	1	2	yes	—
257	C₄	—	2	4	no	—
258	C₄	—	2	2	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	4	no	—
261	C₄	—	4	4	no	—
262	C₄	—	8	2	no	—
263	C₄	—	8	2	no	—
264	C₄	#491×C₁₅₁	1	4	yes	—
265	C₄	#529×C₁₅₁	1	4	yes	—
266	C₂	#2398×C₁₅₁	1	1	yes	characteristic
267	C₂	#3107×C₁₅₁	1	2	yes	—
268	C₂	#3108×C₁₅₁	1	2	yes	—
269	C₂	#3131×C₁₅₁	1	1	yes	characteristic
270	C₂	C₂×#491×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₅₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₅₁
socle, prime radical	C₂³×C₁₅₁	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₅₁ ◁ C₄³×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
135. #23820	C₁₅₁	direct (N × H)
136. C₁₅₁	#23820	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₁₅₁	2
C₄×#125×C₁₅₁	2
#2099×C₁₅₁	2
#2074×C₁₅₁	2
#2043×C₁₅₁	2
#2026×C₁₅₁	2
C₄×#103×C₁₅₁	2
#1744×C₁₅₁	2
#23820	151

Subgroups — C151×#23820 #50328745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#23820 #50328745