Subgroups — C₁₅₁×#8683 #50313608

Order 38656.

362 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2960×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#1848×C₁₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#334×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	#1550×C₁₅₁	C₂	1	2	yes	maximal
5	#1292×C₁₅₁	C₂	1	2	yes	maximal
6	#533×C₁₅₁	C₂²	1	1	yes	characteristic
7	#446×C₁₅₁	—	2	1	no	—
8	#446×C₁₅₁	C₄	1	2	yes	—
9	C₈×Q₈×C₁₅₁	—	2	1	no	—
10	C₈×Q₈×C₁₅₁	C₄	1	2	yes	—
11	C₂×#108×C₁₅₁	—	2	1	no	—
12	C₂×#108×C₁₅₁	—	2	1	no	—
13	#424×C₁₅₁	—	2	1	no	—
14	#423×C₁₅₁	C₂²	1	2	yes	—
15	#423×C₁₅₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₁₅₁	C₂²	1	2	yes	—
17	#405×C₁₅₁	—	2	1	no	—
18	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
19	#334×C₁₅₁	—	2	1	no	—
20	#334×C₁₅₁	—	2	1	no	—
21	#333×C₁₅₁	—	2	2	no	—
22	#332×C₁₅₁	—	2	2	no	—
23	C₂×M₁₆×C₁₅₁	—	2	1	no	—
24	C₂×M₁₆×C₁₅₁	—	2	2	no	—
25	C₂²×C₈×C₁₅₁	—	2	2	no	—
26	C₂²×C₈×C₁₅₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₅₁	D₄	1	1	yes	characteristic
28	C₄⋊Q₈×C₁₅₁	D₄	1	1	yes	characteristic
29	C₄⋊Q₈×C₁₅₁	D₄	1	1	yes	characteristic
30	#125×C₁₅₁	—	2	1	no	—
31	C₂²⋊Q₈×C₁₅₁	—	2	1	no	—
32	C₂²⋊Q₈×C₁₅₁	Q₈	1	2	yes	—
33	C₄×Q₈×C₁₅₁	—	2	1	no	—
34	C₄×Q₈×C₁₅₁	C₂×C₄	1	2	yes	—
35	#117×C₁₅₁	—	2	1	no	—
36	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
37	#108×C₁₅₁	—	4	2	no	—
38	#108×C₁₅₁	—	4	2	no	—
39	C₄⋊C₈×C₁₅₁	—	2	2	no	—
40	C₄⋊C₈×C₁₅₁	—	2	2	no	—
41	C₄⋊C₈×C₁₅₁	—	2	1	no	—
42	C₄⋊C₈×C₁₅₁	—	2	2	no	—
43	C₄⋊C₈×C₁₅₁	—	2	2	no	—
44	C₄⋊C₈×C₁₅₁	—	2	2	no	—
45	C₄⋊C₈×C₁₅₁	—	2	2	no	—
46	C₄⋊C₈×C₁₅₁	C₂×C₄	1	2	yes	—
47	#97×C₁₅₁	—	2	2	no	—
48	C₄×C₈×C₁₅₁	—	2	2	no	—
49	C₄×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
53	C₂×Q₈×C₁₅₁	—	2	1	no	—
54	C₂×Q₈×C₁₅₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₅₁	#41	1	1	yes	characteristic
58	M₁₆×C₁₅₁	—	4	1	no	—
59	M₁₆×C₁₅₁	—	4	1	no	—
60	M₁₆×C₁₅₁	—	4	2	no	—
61	M₁₆×C₁₅₁	—	4	2	no	—
62	C₂×C₈×C₁₅₁	—	2	1	no	—
63	C₂×C₈×C₁₅₁	—	2	1	no	—
64	C₂×C₈×C₁₅₁	—	2	2	no	—
65	C₂×C₈×C₁₅₁	—	4	2	no	—
66	C₂×C₈×C₁₅₁	—	4	2	no	—
67	C₂×C₈×C₁₅₁	—	8	2	no	—
68	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₅₁	SD₁₆	1	2	yes	—
73	C₂×C₈×C₁₅₁	Q₁₆	1	2	yes	—
74	C₄⋊C₄×C₁₅₁	—	2	2	no	—
75	C₄⋊C₄×C₁₅₁	—	2	1	no	—
76	C₄⋊C₄×C₁₅₁	—	2	1	no	—
77	C₄⋊C₄×C₁₅₁	—	2	1	no	—
78	C₄⋊C₄×C₁₅₁	—	2	2	no	—
79	C₄⋊C₄×C₁₅₁	C₄⋊C₄	1	2	yes	—
80	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
81	C₄²×C₁₅₁	—	2	2	no	—
82	C₄²×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₅₁	#41	1	1	yes	characteristic
86	C₂³×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
87	Q₈×C₁₅₁	—	4	2	no	—
88	Q₈×C₁₅₁	—	4	2	no	—
89	C₂×C₄×C₁₅₁	—	2	2	no	—
90	C₂×C₄×C₁₅₁	—	2	2	no	—
91	C₂×C₄×C₁₅₁	—	2	2	no	—
92	C₂×C₄×C₁₅₁	—	4	1	no	—
93	C₂×C₄×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₅₁	#117	1	1	yes	characteristic
96	C₂×C₄×C₁₅₁	#121	1	1	yes	characteristic
97	C₂×C₄×C₁₅₁	#121	1	1	yes	characteristic
98	C₂×C₄×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₁₅₁	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₅₁	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₁₅₁	#137	1	1	yes	characteristic
103	C₈×C₁₅₁	—	2	1	no	—
104	C₈×C₁₅₁	—	2	1	no	—
105	C₈×C₁₅₁	—	4	2	no	—
106	C₈×C₁₅₁	—	8	2	no	—
107	C₈×C₁₅₁	—	8	2	no	—
108	C₈×C₁₅₁	#103	1	2	yes	—
109	C₈×C₁₅₁	#103	1	2	yes	—
110	C₂²×C₁₅₁	—	2	1	no	—
111	C₂²×C₁₅₁	—	2	1	no	—
112	C₂²×C₁₅₁	#380	1	1	yes	characteristic
113	C₂²×C₁₅₁	#475	1	1	yes	characteristic
114	C₂²×C₁₅₁	#477	1	1	yes	characteristic
115	C₄×C₁₅₁	—	2	1	no	—
116	C₄×C₁₅₁	—	2	1	no	—
117	C₄×C₁₅₁	—	2	1	no	—
118	C₄×C₁₅₁	—	4	2	no	—
119	C₄×C₁₅₁	—	4	2	no	—
120	C₄×C₁₅₁	C₂×#103	1	1	yes	characteristic
121	C₄×C₁₅₁	#417	1	1	yes	characteristic
122	C₄×C₁₅₁	#460	1	1	yes	characteristic
123	C₄×C₁₅₁	#462	1	1	yes	characteristic
124	C₄×C₁₅₁	#471	1	2	yes	—
125	C₂×C₁₅₁	—	2	1	no	—
126	C₂×C₁₅₁	—	2	1	no	—
127	C₂×C₁₅₁	#1803	1	1	yes	characteristic
128	C₂×C₁₅₁	#1813	1	1	yes	characteristic
129	C₂×C₁₅₁	#1815	1	1	yes	characteristic
130	#8683	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₁₅₁	#8683	1	1	yes	characteristic, direct product factor, Sylow
132	#2960	C₂×C₁₅₁	1	1	yes	characteristic
133	#1848	C₂×C₁₅₁	1	1	yes	characteristic
134	C₂×#334	C₂×C₁₅₁	1	1	yes	characteristic
135	#1550	C₂×C₁₅₁	1	2	yes	—
136	#1292	C₂×C₁₅₁	1	2	yes	—
137	#533	C₂²×C₁₅₁	1	1	yes	characteristic
138	#446	—	2	1	no	—
139	#446	C₄×C₁₅₁	1	2	yes	—
140	C₈×Q₈	—	2	1	no	—
141	C₈×Q₈	C₄×C₁₅₁	1	2	yes	—
142	C₂×#108	—	2	1	no	—
143	C₂×#108	—	2	1	no	—
144	#424	—	2	1	no	—
145	#423	C₂²×C₁₅₁	1	2	yes	—
146	#423	C₂²×C₁₅₁	1	1	yes	characteristic
147	C₂×C₄⋊C₈	C₂²×C₁₅₁	1	2	yes	—
148	#405	—	2	1	no	—
149	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
150	#334	—	2	1	no	—
151	#334	—	2	1	no	—
152	#333	—	2	2	no	—
153	#332	—	2	2	no	—
154	C₂×M₁₆	—	2	1	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	D₄×C₁₅₁	1	1	yes	characteristic
158	C₂²×C₈	D₄×C₁₅₁	1	1	yes	characteristic
159	C₄⋊Q₈	D₄×C₁₅₁	1	1	yes	characteristic
160	C₄⋊Q₈	D₄×C₁₅₁	1	1	yes	characteristic
161	#125	—	2	1	no	—
162	C₂²⋊Q₈	—	2	1	no	—
163	C₂²⋊Q₈	Q₈×C₁₅₁	1	2	yes	—
164	C₄×Q₈	—	2	1	no	—
165	C₄×Q₈	C₂×C₄×C₁₅₁	1	2	yes	—
166	#117	—	2	1	no	—
167	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
168	#108	—	4	2	no	—
169	#108	—	4	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	2	1	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	C₂×C₄×C₁₅₁	1	2	yes	—
178	#97	—	2	2	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
181	C₄×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
182	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
183	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
184	C₂×Q₈	—	2	1	no	—
185	C₂×Q₈	C₄⋊C₄×C₁₅₁	1	2	yes	—
186	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
187	C₂²×C₄	C₂×Q₈×C₁₅₁	1	1	yes	characteristic
188	C₂²×C₄	#41×C₁₅₁	1	1	yes	characteristic
189	M₁₆	—	4	1	no	—
190	M₁₆	—	4	1	no	—
191	M₁₆	—	4	2	no	—
192	M₁₆	—	4	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	8	2	no	—
199	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
201	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
203	C₂×C₈	SD₁₆×C₁₅₁	1	2	yes	—
204	C₂×C₈	Q₁₆×C₁₅₁	1	2	yes	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	C₄⋊C₄×C₁₅₁	1	2	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
214	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
215	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
216	C₄²	#41×C₁₅₁	1	1	yes	characteristic
217	C₂³	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	Q₈	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	C₂×C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
226	C₂×C₄	#117×C₁₅₁	1	1	yes	characteristic
227	C₂×C₄	#121×C₁₅₁	1	1	yes	characteristic
228	C₂×C₄	#121×C₁₅₁	1	1	yes	characteristic
229	C₂×C₄	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
230	C₂×C₄	C₂×SD₁₆×C₁₅₁	1	1	yes	characteristic
231	C₂×C₄	C₂×Q₁₆×C₁₅₁	1	1	yes	characteristic
232	C₂×C₄	AGL(1,ℤ/8)×C₁₅₁	1	1	yes	characteristic
233	C₂×C₄	#137×C₁₅₁	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	4	2	no	—
237	C₈	—	8	2	no	—
238	C₈	—	8	2	no	—
239	C₈	#103×C₁₅₁	1	2	yes	—
240	C₈	#103×C₁₅₁	1	2	yes	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	#380×C₁₅₁	1	1	yes	characteristic
244	C₂²	#475×C₁₅₁	1	1	yes	characteristic
245	C₂²	#477×C₁₅₁	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	C₂×#103×C₁₅₁	1	1	yes	characteristic
252	C₄	#417×C₁₅₁	1	1	yes	characteristic
253	C₄	#460×C₁₅₁	1	1	yes	characteristic
254	C₄	#462×C₁₅₁	1	1	yes	characteristic
255	C₄	#471×C₁₅₁	1	2	yes	—
256	C₂	—	2	1	no	—
257	C₂	—	2	1	no	—
258	C₂	#1803×C₁₅₁	1	1	yes	characteristic
259	C₂	#1813×C₁₅₁	1	1	yes	characteristic
260	C₂	#1815×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₅₁	#380
commutator	C₄²	C₂²×C₄×C₁₅₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₅₁
prime radical	C₂³×C₁₅₁	C₂²⋊Q₈
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
130. #8683	C₁₅₁	direct (N × H)
131. C₁₅₁	#8683	direct (N × H)

Maximal subgroups

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

subgroup	index
#2960×C₁₅₁	2
#1848×C₁₅₁	2
C₂×#334×C₁₅₁	2
#1550×C₁₅₁	2
#1292×C₁₅₁	2
#8683	151

Subgroups — C151×#8683 #50313608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#8683 #50313608