Subgroups — C₁₅₁×#23162 #50328087

Order 38656.

738 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#2369×C₁₅₁	C₂	1	2	yes	maximal
3	#1987×C₁₅₁	C₂	1	2	yes	maximal
4	#1943×C₁₅₁	C₂	1	2	yes	maximal
5	#1918×C₁₅₁	C₂	1	4	yes	maximal
6	#1776×C₁₅₁	C₂	1	2	yes	maximal
7	C₄×#106×C₁₅₁	C₂	1	1	yes	maximal, characteristic
8	#1770×C₁₅₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₁₅₁	C₂²	1	2	yes	—
10	#500×C₁₅₁	C₄	1	8	yes	complemented
11	C₂×#107×C₁₅₁	C₂²	1	2	yes	—
12	C₂×#107×C₁₅₁	C₂²	1	2	yes	—
13	C₂×#106×C₁₅₁	C₂²	1	1	yes	characteristic
14	C₂×#106×C₁₅₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
16	C₂×#97×C₁₅₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
18	#391×C₁₅₁	—	2	4	no	—
19	#389×C₁₅₁	—	2	2	no	—
20	#389×C₁₅₁	C₂²	1	4	yes	—
21	C₄×C₄⋊C₄×C₁₅₁	—	2	2	no	—
22	C₄×C₄⋊C₄×C₁₅₁	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₁₅₁	C₂²	1	1	yes	characteristic
25	#340×C₁₅₁	C₂²	1	2	yes	—
26	#340×C₁₅₁	C₂²	1	4	yes	—
27	#340×C₁₅₁	C₂²	1	4	yes	—
28	#336×C₁₅₁	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₅₁	—	2	2	no	—
30	C₂²×C₈×C₁₅₁	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₁₅₁	—	2	4	no	—
34	C₄⋊Q₈×C₁₅₁	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₅₁	—	2	4	no	—
37	C₂×C₄⋊C₄×C₁₅₁	D₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₁₅₁	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₅₁	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₁₅₁	C₂³	1	2	yes	—
41	C₂×C₄²×C₁₅₁	—	2	2	no	—
42	C₂×C₄²×C₁₅₁	—	2	4	no	—
43	C₂×C₄²×C₁₅₁	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
45	#107×C₁₅₁	C₂×C₄	1	8	yes	—
46	#107×C₁₅₁	C₂×C₄	1	8	yes	—
47	#106×C₁₅₁	—	2	2	no	—
48	#106×C₁₅₁	Q₈	1	4	yes	complemented
49	C₄⋊C₈×C₁₅₁	—	2	2	no	—
50	#97×C₁₅₁	—	2	4	no	—
51	C₄×C₈×C₁₅₁	C₂×C₄	1	4	yes	—
52	#95×C₁₅₁	—	2	4	no	—
53	#95×C₁₅₁	—	2	4	no	—
54	#95×C₁₅₁	—	2	2	no	—
55	#95×C₁₅₁	D₄	1	2	yes	—
56	C₂×Q₈×C₁₅₁	—	2	8	no	—
57	C₂×Q₈×C₁₅₁	—	8	2	no	—
58	C₂×Q₈×C₁₅₁	—	8	2	no	—
59	C₂²×C₄×C₁₅₁	—	2	2	no	—
60	C₂²×C₄×C₁₅₁	—	2	4	no	—
61	C₂²×C₄×C₁₅₁	—	2	2	no	—
62	C₂²×C₄×C₁₅₁	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₁₅₁	—	2	1	no	—
67	C₂×C₈×C₁₅₁	—	2	1	no	—
68	C₂×C₈×C₁₅₁	—	2	2	no	—
69	C₂×C₈×C₁₅₁	—	2	2	no	—
70	C₂×C₈×C₁₅₁	—	2	2	no	—
71	C₂×C₈×C₁₅₁	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₁₅₁	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₁₅₁	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₁₅₁	#41	1	2	yes	—
75	C₄⋊C₄×C₁₅₁	—	2	8	no	—
76	C₄⋊C₄×C₁₅₁	—	2	4	no	—
77	C₄⋊C₄×C₁₅₁	—	2	4	no	—
78	C₄⋊C₄×C₁₅₁	—	2	2	no	—
79	C₄⋊C₄×C₁₅₁	—	2	2	no	—
80	C₄⋊C₄×C₁₅₁	—	2	2	no	—
81	C₄⋊C₄×C₁₅₁	—	8	4	no	—
82	C₄⋊C₄×C₁₅₁	C₂²×C₄	1	8	yes	—
83	C₄⋊C₄×C₁₅₁	C₂×Q₈	1	4	yes	—
84	C₄²×C₁₅₁	—	2	2	no	—
85	C₄²×C₁₅₁	—	2	2	no	—
86	C₄²×C₁₅₁	—	4	4	no	—
87	C₄²×C₁₅₁	—	8	4	no	—
88	C₄²×C₁₅₁	C₂²×C₄	1	2	yes	—
89	C₄²×C₁₅₁	#41	1	2	yes	—
90	C₂³×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
91	Q₈×C₁₅₁	—	8	8	no	—
92	C₂×C₄×C₁₅₁	—	2	2	no	—
93	C₂×C₄×C₁₅₁	—	2	8	no	—
94	C₂×C₄×C₁₅₁	—	2	4	no	—
95	C₂×C₄×C₁₅₁	—	2	4	no	—
96	C₂×C₄×C₁₅₁	—	2	4	no	—
97	C₂×C₄×C₁₅₁	—	2	4	no	—
98	C₂×C₄×C₁₅₁	—	4	2	no	—
99	C₂×C₄×C₁₅₁	—	4	4	no	—
100	C₂×C₄×C₁₅₁	—	4	4	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	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₁₅₁	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₁₅₁	#137	1	4	yes	—
108	C₂×C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₅₁	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
113	C₂×C₄×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
114	C₈×C₁₅₁	—	2	2	no	—
115	C₈×C₁₅₁	—	4	2	no	—
116	C₈×C₁₅₁	C₄×Q₈	1	4	yes	—
117	C₂²×C₁₅₁	C₂×C₄×D₄	1	1	yes	characteristic
118	C₂²×C₁₅₁	#547	1	1	yes	characteristic
119	C₂²×C₁₅₁	D₄×Q₈	1	1	yes	characteristic
120	C₂²×C₁₅₁	C₂×#137	1	2	yes	—
121	C₂²×C₁₅₁	#576	1	2	yes	—
122	C₄×C₁₅₁	—	2	2	no	—
123	C₄×C₁₅₁	—	4	4	no	—
124	C₄×C₁₅₁	—	4	4	no	—
125	C₄×C₁₅₁	—	4	8	no	—
126	C₄×C₁₅₁	—	8	2	no	—
127	C₄×C₁₅₁	#441	1	4	yes	—
128	C₄×C₁₅₁	C₂×C₄×Q₈	1	1	yes	characteristic
129	C₄×C₁₅₁	#519	1	1	yes	characteristic
130	C₄×C₁₅₁	#556	1	1	yes	characteristic
131	C₄×C₁₅₁	#557	1	1	yes	characteristic
132	C₂×C₁₅₁	#2365	1	1	yes	characteristic
133	C₂×C₁₅₁	C₂×#441	1	1	yes	characteristic
134	C₂×C₁₅₁	#2948	1	1	yes	characteristic
135	C₂×C₁₅₁	#3357	1	2	yes	—
136	C₂×C₁₅₁	#3385	1	2	yes	—
137	#23162	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₁₅₁	#23162	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×#500	C₂×C₁₅₁	1	1	yes	characteristic
140	#2369	C₂×C₁₅₁	1	2	yes	—
141	#1987	C₂×C₁₅₁	1	2	yes	—
142	#1943	C₂×C₁₅₁	1	2	yes	—
143	#1918	C₂×C₁₅₁	1	4	yes	—
144	#1776	C₂×C₁₅₁	1	2	yes	—
145	C₄×#106	C₂×C₁₅₁	1	1	yes	characteristic
146	#1770	C₂×C₁₅₁	1	1	yes	characteristic
147	C₂×C₄⋊Q₈	C₂²×C₁₅₁	1	2	yes	—
148	#500	C₄×C₁₅₁	1	8	yes	complemented
149	C₂×#107	C₂²×C₁₅₁	1	2	yes	—
150	C₂×#107	C₂²×C₁₅₁	1	2	yes	—
151	C₂×#106	C₂²×C₁₅₁	1	1	yes	characteristic
152	C₂×#106	C₂²×C₁₅₁	1	1	yes	characteristic
153	C₂×C₄⋊C₈	C₂²×C₁₅₁	1	1	yes	characteristic
154	C₂×#97	C₂²×C₁₅₁	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
156	#391	—	2	4	no	—
157	#389	—	2	2	no	—
158	#389	C₂²×C₁₅₁	1	4	yes	—
159	C₄×C₄⋊C₄	—	2	2	no	—
160	C₄×C₄⋊C₄	C₂²×C₁₅₁	1	4	yes	—
161	C₄×C₄⋊C₄	C₂²×C₁₅₁	1	2	yes	—
162	C₄×C₄⋊C₄	C₂²×C₁₅₁	1	1	yes	characteristic
163	#340	C₂²×C₁₅₁	1	2	yes	—
164	#340	C₂²×C₁₅₁	1	4	yes	—
165	#340	C₂²×C₁₅₁	1	4	yes	—
166	#336	C₂²×C₁₅₁	1	2	yes	—
167	C₂²×Q₈	—	2	2	no	—
168	C₂²×C₈	C₂³×C₁₅₁	1	2	yes	—
169	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
171	C₄⋊Q₈	—	2	4	no	—
172	C₄⋊Q₈	C₂×C₄×C₁₅₁	1	8	yes	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	4	no	—
175	C₂×C₄⋊C₄	D₄×C₁₅₁	1	2	yes	—
176	C₂×C₄⋊C₄	C₂³×C₁₅₁	1	2	yes	—
177	C₂×C₄⋊C₄	C₂³×C₁₅₁	1	2	yes	—
178	C₂×C₄⋊C₄	C₂³×C₁₅₁	1	4	yes	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	—	2	4	no	—
181	C₂×C₄²	C₂³×C₁₅₁	1	2	yes	—
182	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
183	#107	C₂×C₄×C₁₅₁	1	8	yes	—
184	#107	C₂×C₄×C₁₅₁	1	8	yes	—
185	#106	—	2	2	no	—
186	#106	Q₈×C₁₅₁	1	4	yes	complemented
187	C₄⋊C₈	—	2	2	no	—
188	#97	—	2	4	no	—
189	C₄×C₈	C₂×C₄×C₁₅₁	1	4	yes	—
190	#95	—	2	4	no	—
191	#95	—	2	4	no	—
192	#95	—	2	2	no	—
193	#95	D₄×C₁₅₁	1	2	yes	—
194	C₂×Q₈	—	2	8	no	—
195	C₂×Q₈	—	8	2	no	—
196	C₂×Q₈	—	8	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	C₂×D₄×C₁₅₁	1	4	yes	—
201	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
203	C₂²×C₄	C₂⁴×C₁₅₁	1	1	yes	characteristic
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	C₂²×C₄×C₁₅₁	1	2	yes	—
210	C₂×C₈	C₂²×C₄×C₁₅₁	1	2	yes	—
211	C₂×C₈	C₂×Q₈×C₁₅₁	1	2	yes	—
212	C₂×C₈	#41×C₁₅₁	1	2	yes	—
213	C₄⋊C₄	—	2	8	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	8	4	no	—
220	C₄⋊C₄	C₂²×C₄×C₁₅₁	1	8	yes	—
221	C₄⋊C₄	C₂×Q₈×C₁₅₁	1	4	yes	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	4	4	no	—
225	C₄²	—	8	4	no	—
226	C₄²	C₂²×C₄×C₁₅₁	1	2	yes	—
227	C₄²	#41×C₁₅₁	1	2	yes	—
228	C₂³	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
229	Q₈	—	8	8	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	C₄×D₄×C₁₅₁	1	2	yes	—
244	C₂×C₄	C₄×D₄×C₁₅₁	1	2	yes	—
245	C₂×C₄	#137×C₁₅₁	1	4	yes	—
246	C₂×C₄	C₂³×C₄×C₁₅₁	1	1	yes	characteristic
247	C₂×C₄	C₂²×Q₈×C₁₅₁	1	1	yes	characteristic
248	C₂×C₄	C₂×#41×C₁₅₁	1	1	yes	characteristic
249	C₂×C₄	C₂×#41×C₁₅₁	1	1	yes	characteristic
250	C₂×C₄	2₋¹⁺⁴×C₁₅₁	1	1	yes	characteristic
251	C₂×C₄	2₋¹⁺⁴×C₁₅₁	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	4	2	no	—
254	C₈	C₄×Q₈×C₁₅₁	1	4	yes	—
255	C₂²	C₂×C₄×D₄×C₁₅₁	1	1	yes	characteristic
256	C₂²	#547×C₁₅₁	1	1	yes	characteristic
257	C₂²	D₄×Q₈×C₁₅₁	1	1	yes	characteristic
258	C₂²	C₂×#137×C₁₅₁	1	2	yes	—
259	C₂²	#576×C₁₅₁	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	4	4	no	—
262	C₄	—	4	4	no	—
263	C₄	—	4	8	no	—
264	C₄	—	8	2	no	—
265	C₄	#441×C₁₅₁	1	4	yes	—
266	C₄	C₂×C₄×Q₈×C₁₅₁	1	1	yes	characteristic
267	C₄	#519×C₁₅₁	1	1	yes	characteristic
268	C₄	#556×C₁₅₁	1	1	yes	characteristic
269	C₄	#557×C₁₅₁	1	1	yes	characteristic
270	C₂	#2365×C₁₅₁	1	1	yes	characteristic
271	C₂	C₂×#441×C₁₅₁	1	1	yes	characteristic
272	C₂	#2948×C₁₅₁	1	1	yes	characteristic
273	C₂	#3357×C₁₅₁	1	2	yes	—
274	C₂	#3385×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
10. #500×C₁₅₁	C₄	semidirect
48. #106×C₁₅₁	Q₈	semidirect
137. #23162	C₁₅₁	direct (N × H)
138. C₁₅₁	#23162	direct (N × H)
148. #500	C₄×C₁₅₁	semidirect
186. #106	Q₈×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₅₁	2
#2369×C₁₅₁	2
#1987×C₁₅₁	2
#1943×C₁₅₁	2
#1918×C₁₅₁	2
#1776×C₁₅₁	2
C₄×#106×C₁₅₁	2
#1770×C₁₅₁	2
#23162	151

Subgroups — C151×#23162 #50328087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#23162 #50328087