Subgroups — C₁₅₁×#8687 #50313612

Order 38656.

346 conjugacy classes of subgroups.

The subgroup lattice has 242 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	#1837×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	#424×C₁₅₁	—	2	1	no	—
12	#423×C₁₅₁	C₂²	1	2	yes	—
13	#423×C₁₅₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₁₅₁	C₂²	1	2	yes	—
15	#405×C₁₅₁	—	2	1	no	—
16	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
17	#341×C₁₅₁	—	2	1	no	—
18	#341×C₁₅₁	—	2	1	no	—
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	C₄⋊C₈×C₁₅₁	—	2	2	no	—
38	C₄⋊C₈×C₁₅₁	—	2	2	no	—
39	C₄⋊C₈×C₁₅₁	—	2	1	no	—
40	C₄⋊C₈×C₁₅₁	—	2	2	no	—
41	C₄⋊C₈×C₁₅₁	—	2	2	no	—
42	C₄⋊C₈×C₁₅₁	—	2	2	no	—
43	C₄⋊C₈×C₁₅₁	—	2	2	no	—
44	C₄⋊C₈×C₁₅₁	C₂×C₄	1	2	yes	—
45	#97×C₁₅₁	—	2	2	no	—
46	C₄×C₈×C₁₅₁	—	2	2	no	—
47	C₄×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
51	C₂×Q₈×C₁₅₁	—	2	1	no	—
52	C₂×Q₈×C₁₅₁	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₅₁	#41	1	1	yes	characteristic
56	M₁₆×C₁₅₁	—	4	2	no	—
57	M₁₆×C₁₅₁	—	4	2	no	—
58	M₁₆×C₁₅₁	—	4	2	no	—
59	C₂×C₈×C₁₅₁	—	2	2	no	—
60	C₂×C₈×C₁₅₁	—	2	2	no	—
61	C₂×C₈×C₁₅₁	—	2	2	no	—
62	C₂×C₈×C₁₅₁	—	4	2	no	—
63	C₂×C₈×C₁₅₁	—	4	2	no	—
64	C₂×C₈×C₁₅₁	—	8	2	no	—
65	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₁₅₁	—	2	2	no	—
70	C₄⋊C₄×C₁₅₁	—	2	1	no	—
71	C₄⋊C₄×C₁₅₁	—	2	1	no	—
72	C₄⋊C₄×C₁₅₁	—	2	1	no	—
73	C₄⋊C₄×C₁₅₁	—	2	2	no	—
74	C₄⋊C₄×C₁₅₁	C₄⋊C₄	1	2	yes	—
75	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
76	C₄²×C₁₅₁	—	2	2	no	—
77	C₄²×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₅₁	#41	1	1	yes	characteristic
81	C₂³×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
82	Q₈×C₁₅₁	—	4	2	no	—
83	Q₈×C₁₅₁	—	4	2	no	—
84	C₂×C₄×C₁₅₁	—	2	2	no	—
85	C₂×C₄×C₁₅₁	—	2	2	no	—
86	C₂×C₄×C₁₅₁	—	2	2	no	—
87	C₂×C₄×C₁₅₁	—	4	1	no	—
88	C₂×C₄×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₅₁	#117	1	1	yes	characteristic
91	C₂×C₄×C₁₅₁	#121	1	1	yes	characteristic
92	C₂×C₄×C₁₅₁	#121	1	1	yes	characteristic
93	C₂×C₄×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₁₅₁	#135	1	2	yes	—
95	C₂×C₄×C₁₅₁	#137	1	2	yes	—
96	C₈×C₁₅₁	—	2	1	no	—
97	C₈×C₁₅₁	—	2	1	no	—
98	C₈×C₁₅₁	—	2	1	no	—
99	C₈×C₁₅₁	—	2	1	no	—
100	C₈×C₁₅₁	—	4	2	no	—
101	C₈×C₁₅₁	—	8	2	no	—
102	C₈×C₁₅₁	—	8	2	no	—
103	C₂²×C₁₅₁	—	2	1	no	—
104	C₂²×C₁₅₁	—	2	1	no	—
105	C₂²×C₁₅₁	#380	1	1	yes	characteristic
106	C₂²×C₁₅₁	#479	1	2	yes	—
107	C₄×C₁₅₁	—	2	1	no	—
108	C₄×C₁₅₁	—	2	1	no	—
109	C₄×C₁₅₁	—	2	1	no	—
110	C₄×C₁₅₁	—	4	2	no	—
111	C₄×C₁₅₁	—	4	2	no	—
112	C₄×C₁₅₁	#416	1	1	yes	characteristic
113	C₄×C₁₅₁	#419	1	1	yes	characteristic
114	C₄×C₁₅₁	#464	1	2	yes	—
115	C₄×C₁₅₁	#471	1	2	yes	—
116	C₂×C₁₅₁	—	2	2	no	—
117	C₂×C₁₅₁	#1805	1	1	yes	characteristic
118	C₂×C₁₅₁	#1813	1	1	yes	characteristic
119	C₂×C₁₅₁	#1815	1	1	yes	characteristic
120	#8687	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₁₅₁	#8687	1	1	yes	characteristic, direct product factor, Sylow
122	#2960	C₂×C₁₅₁	1	1	yes	characteristic
123	#1837	C₂×C₁₅₁	1	1	yes	characteristic
124	C₂×#334	C₂×C₁₅₁	1	1	yes	characteristic
125	#1550	C₂×C₁₅₁	1	2	yes	—
126	#1292	C₂×C₁₅₁	1	2	yes	—
127	#533	C₂²×C₁₅₁	1	1	yes	characteristic
128	#446	—	2	1	no	—
129	#446	C₄×C₁₅₁	1	2	yes	—
130	C₈×Q₈	—	2	1	no	—
131	C₈×Q₈	C₄×C₁₅₁	1	2	yes	—
132	#424	—	2	1	no	—
133	#423	C₂²×C₁₅₁	1	2	yes	—
134	#423	C₂²×C₁₅₁	1	1	yes	characteristic
135	C₂×C₄⋊C₈	C₂²×C₁₅₁	1	2	yes	—
136	#405	—	2	1	no	—
137	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
138	#341	—	2	1	no	—
139	#341	—	2	1	no	—
140	#334	—	2	1	no	—
141	#334	—	2	1	no	—
142	#333	—	2	2	no	—
143	#332	—	2	2	no	—
144	C₂×M₁₆	—	2	1	no	—
145	C₂×M₁₆	—	2	2	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	D₄×C₁₅₁	1	1	yes	characteristic
148	C₂²×C₈	D₄×C₁₅₁	1	1	yes	characteristic
149	C₄⋊Q₈	D₄×C₁₅₁	1	1	yes	characteristic
150	C₄⋊Q₈	D₄×C₁₅₁	1	1	yes	characteristic
151	#125	—	2	1	no	—
152	C₂²⋊Q₈	—	2	1	no	—
153	C₂²⋊Q₈	Q₈×C₁₅₁	1	2	yes	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	C₂×C₄×C₁₅₁	1	2	yes	—
156	#117	—	2	1	no	—
157	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	—	2	1	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	C₂×C₄×C₁₅₁	1	2	yes	—
166	#97	—	2	2	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
169	C₄×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
170	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
171	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	C₄⋊C₄×C₁₅₁	1	2	yes	—
174	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
175	C₂²×C₄	C₂×Q₈×C₁₅₁	1	1	yes	characteristic
176	C₂²×C₄	#41×C₁₅₁	1	1	yes	characteristic
177	M₁₆	—	4	2	no	—
178	M₁₆	—	4	2	no	—
179	M₁₆	—	4	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	8	2	no	—
186	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
187	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
188	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	C₄⋊C₄×C₁₅₁	1	2	yes	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
199	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
200	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
201	C₄²	#41×C₁₅₁	1	1	yes	characteristic
202	C₂³	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	C₂×C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
211	C₂×C₄	#117×C₁₅₁	1	1	yes	characteristic
212	C₂×C₄	#121×C₁₅₁	1	1	yes	characteristic
213	C₂×C₄	#121×C₁₅₁	1	1	yes	characteristic
214	C₂×C₄	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
215	C₂×C₄	#135×C₁₅₁	1	2	yes	—
216	C₂×C₄	#137×C₁₅₁	1	2	yes	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	4	2	no	—
222	C₈	—	8	2	no	—
223	C₈	—	8	2	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	#380×C₁₅₁	1	1	yes	characteristic
227	C₂²	#479×C₁₅₁	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	#416×C₁₅₁	1	1	yes	characteristic
234	C₄	#419×C₁₅₁	1	1	yes	characteristic
235	C₄	#464×C₁₅₁	1	2	yes	—
236	C₄	#471×C₁₅₁	1	2	yes	—
237	C₂	—	2	2	no	—
238	C₂	#1805×C₁₅₁	1	1	yes	characteristic
239	C₂	#1813×C₁₅₁	1	1	yes	characteristic
240	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
120. #8687	C₁₅₁	direct (N × H)
121. C₁₅₁	#8687	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
#1837×C₁₅₁	2
C₂×#334×C₁₅₁	2
#1550×C₁₅₁	2
#1292×C₁₅₁	2
#8687	151

Subgroups — C151×#8687 #50313612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#8687 #50313612