Subgroups — #3252×C₂₅₁ #44547839

Order 32128.

348 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#529×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#479×C₂₅₁	C₂	1	2	yes	maximal, complemented
4	#477×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#476×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#470×C₂₅₁	C₂	1	2	yes	maximal, complemented
7	#468×C₂₅₁	C₂	1	2	yes	maximal
8	#464×C₂₅₁	C₂	1	2	yes	maximal, complemented
9	#462×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#461×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#423×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×C₂₅₁	C₂²	1	2	yes	—
13	C₂×SD₁₆×C₂₅₁	C₂²	1	2	yes	—
14	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
17	#126×C₂₅₁	—	2	1	no	—
18	#125×C₂₅₁	—	2	1	no	—
19	#125×C₂₅₁	C₂²	1	1	yes	characteristic
20	#124×C₂₅₁	C₂²	1	1	yes	characteristic
21	#123×C₂₅₁	—	2	1	no	—
22	C₂²⋊Q₈×C₂₅₁	—	2	1	no	—
23	C₂²⋊Q₈×C₂₅₁	C₂²	1	2	yes	—
24	C₂²⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
25	#121×C₂₅₁	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₂₅₁	—	2	1	no	—
27	C₄×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
28	C₄×Q₈×C₂₅₁	C₂²	1	2	yes	—
29	C₄×D₄×C₂₅₁	—	2	1	no	—
30	C₄×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
31	#117×C₂₅₁	—	2	1	no	—
32	#117×C₂₅₁	—	2	1	no	—
33	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
34	#107×C₂₅₁	C₂²	1	2	yes	—
35	#106×C₂₅₁	C₂²	1	2	yes	—
36	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	—
39	#103×C₂₅₁	C₂²	1	2	yes	—
40	#103×C₂₅₁	C₂²	1	2	yes	—
41	#103×C₂₅₁	C₂²	1	2	yes	—
42	#102×C₂₅₁	C₂²	1	2	yes	—
43	C₂×Q₈×C₂₅₁	—	2	1	no	—
44	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₂₅₁	C₂³	1	2	yes	—
46	C₂×D₄×C₂₅₁	—	2	1	no	—
47	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₅₁	—	2	1	no	—
49	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
52	Q₁₆×C₂₅₁	—	4	2	no	—
53	SD₁₆×C₂₅₁	—	4	2	no	—
54	M₁₆×C₂₅₁	—	4	2	no	—
55	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
56	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
57	C₄⋊C₄×C₂₅₁	—	2	1	no	—
58	C₄⋊C₄×C₂₅₁	—	2	1	no	—
59	C₄⋊C₄×C₂₅₁	—	2	1	no	—
60	C₄⋊C₄×C₂₅₁	—	2	1	no	—
61	C₄⋊C₄×C₂₅₁	—	2	1	no	—
62	C₄⋊C₄×C₂₅₁	—	2	1	no	—
63	C₄⋊C₄×C₂₅₁	—	2	2	no	—
64	C₄⋊C₄×C₂₅₁	—	2	2	no	—
65	C₄⋊C₄×C₂₅₁	—	2	1	no	—
66	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
67	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
70	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
71	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
72	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
73	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
74	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
75	C₄²×C₂₅₁	—	2	1	no	—
76	C₄²×C₂₅₁	—	2	2	no	—
77	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
78	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
79	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
80	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
81	C₂³×C₂₅₁	—	2	1	no	—
82	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₂₅₁	—	2	2	no	—
84	Q₈×C₂₅₁	—	2	1	no	—
85	Q₈×C₂₅₁	—	2	1	no	—
86	Q₈×C₂₅₁	—	2	2	no	—
87	Q₈×C₂₅₁	—	4	2	no	—
88	D₄×C₂₅₁	—	2	1	no	—
89	D₄×C₂₅₁	—	2	1	no	—
90	D₄×C₂₅₁	—	4	2	no	—
91	C₂×C₄×C₂₅₁	—	2	2	no	—
92	C₂×C₄×C₂₅₁	—	2	1	no	—
93	C₂×C₄×C₂₅₁	—	2	1	no	—
94	C₂×C₄×C₂₅₁	—	2	1	no	—
95	C₂×C₄×C₂₅₁	—	2	2	no	—
96	C₂×C₄×C₂₅₁	—	2	1	no	—
97	C₂×C₄×C₂₅₁	—	2	2	no	—
98	C₂×C₄×C₂₅₁	—	2	1	no	—
99	C₂×C₄×C₂₅₁	—	4	1	no	—
100	C₂×C₄×C₂₅₁	—	4	2	no	—
101	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
107	C₈×C₂₅₁	—	2	2	no	—
108	C₈×C₂₅₁	—	2	2	no	—
109	C₂²×C₂₅₁	—	2	1	no	—
110	C₂²×C₂₅₁	—	2	1	no	—
111	C₂²×C₂₅₁	—	2	1	no	—
112	C₂²×C₂₅₁	—	4	1	no	—
113	C₂²×C₂₅₁	—	4	1	no	—
114	C₂²×C₂₅₁	—	4	1	no	—
115	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
116	C₄×C₂₅₁	—	2	1	no	—
117	C₄×C₂₅₁	—	2	1	no	—
118	C₄×C₂₅₁	—	2	1	no	—
119	C₄×C₂₅₁	—	2	1	no	—
120	C₄×C₂₅₁	—	4	1	no	—
121	C₄×C₂₅₁	—	4	2	no	—
122	C₄×C₂₅₁	—	4	1	no	—
123	C₄×C₂₅₁	—	4	1	no	—
124	C₄×C₂₅₁	—	4	2	no	—
125	C₄×C₂₅₁	#137	1	2	yes	—
126	C₄×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
127	C₄×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×C₂₅₁	—	4	1	no	—
129	C₂×C₂₅₁	—	8	1	no	—
130	C₂×C₂₅₁	#537	1	1	yes	characteristic
131	C₂×C₂₅₁	C₂×#137	1	1	yes	characteristic
132	C₂×C₂₅₁	#575	1	1	yes	characteristic
133	C₂₅₁	#3252	1	1	yes	characteristic, direct product factor, Sylow
134	#3252	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	#533	C₂×C₂₅₁	1	1	yes	characteristic, complemented
136	#529	C₂×C₂₅₁	1	1	yes	characteristic
137	#479	C₂×C₂₅₁	1	2	yes	complemented
138	#477	C₂×C₂₅₁	1	1	yes	characteristic, complemented
139	#476	C₂×C₂₅₁	1	1	yes	characteristic, complemented
140	#470	C₂×C₂₅₁	1	2	yes	complemented
141	#468	C₂×C₂₅₁	1	2	yes	—
142	#464	C₂×C₂₅₁	1	2	yes	complemented
143	#462	C₂×C₂₅₁	1	1	yes	characteristic, complemented
144	#461	C₂×C₂₅₁	1	1	yes	characteristic, complemented
145	#423	C₂×C₂₅₁	1	1	yes	characteristic, complemented
146	C₂×Q₁₆	C₂²×C₂₅₁	1	2	yes	—
147	C₂×SD₁₆	C₂²×C₂₅₁	1	2	yes	—
148	C₂×M₁₆	C₂²×C₂₅₁	1	2	yes	—
149	C₄⋊Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
150	C₄⋊Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
151	#126	—	2	1	no	—
152	#125	—	2	1	no	—
153	#125	C₂²×C₂₅₁	1	1	yes	characteristic
154	#124	C₂²×C₂₅₁	1	1	yes	characteristic
155	#123	—	2	1	no	—
156	C₂²⋊Q₈	—	2	1	no	—
157	C₂²⋊Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
158	C₂²⋊Q₈	C₂²×C₂₅₁	1	2	yes	—
159	#121	C₂²×C₂₅₁	1	1	yes	characteristic
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
162	C₄×Q₈	C₂²×C₂₅₁	1	2	yes	—
163	C₄×D₄	—	2	1	no	—
164	C₄×D₄	C₂²×C₂₅₁	1	1	yes	characteristic
165	#117	—	2	1	no	—
166	#117	—	2	1	no	—
167	C₂×C₄²	C₂²×C₂₅₁	1	1	yes	characteristic
168	#107	C₂²×C₂₅₁	1	2	yes	—
169	#106	C₂²×C₂₅₁	1	2	yes	—
170	C₄⋊C₈	C₂²×C₂₅₁	1	1	yes	characteristic
171	C₄⋊C₈	C₂²×C₂₅₁	1	1	yes	characteristic
172	C₄⋊C₈	C₂²×C₂₅₁	1	2	yes	—
173	#103	C₂²×C₂₅₁	1	2	yes	—
174	#103	C₂²×C₂₅₁	1	2	yes	—
175	#103	C₂²×C₂₅₁	1	2	yes	—
176	#102	C₂²×C₂₅₁	1	2	yes	—
177	C₂×Q₈	—	2	1	no	—
178	C₂×Q₈	C₂³×C₂₅₁	1	1	yes	characteristic
179	C₂×Q₈	C₂³×C₂₅₁	1	2	yes	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	C₂³×C₂₅₁	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	D₄×C₂₅₁	1	1	yes	characteristic
184	C₂²×C₄	D₄×C₂₅₁	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
186	Q₁₆	—	4	2	no	—
187	SD₁₆	—	4	2	no	—
188	M₁₆	—	4	2	no	—
189	C₂×C₈	C₂³×C₂₅₁	1	2	yes	—
190	C₂×C₈	C₂³×C₂₅₁	1	2	yes	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	C₂³×C₂₅₁	1	2	yes	—
201	C₄⋊C₄	C₂³×C₂₅₁	1	1	yes	characteristic
202	C₄⋊C₄	C₂³×C₂₅₁	1	1	yes	characteristic
203	C₂²⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	2	no	—
211	C₄²	D₄×C₂₅₁	1	1	yes	characteristic
212	C₄²	D₄×C₂₅₁	1	1	yes	characteristic
213	C₄²	C₂³×C₂₅₁	1	1	yes	characteristic
214	C₄²	C₂³×C₂₅₁	1	1	yes	characteristic
215	C₂³	—	2	1	no	—
216	C₂³	C₂×D₄×C₂₅₁	1	1	yes	characteristic
217	Q₈	—	2	2	no	—
218	Q₈	—	2	1	no	—
219	Q₈	—	2	1	no	—
220	Q₈	—	2	2	no	—
221	Q₈	—	4	2	no	—
222	D₄	—	2	1	no	—
223	D₄	—	2	1	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
238	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
239	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
240	C₂×C₄	C₂⁴×C₂₅₁	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	C₂²×D₄×C₂₅₁	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	2	no	—
259	C₄	#137×C₂₅₁	1	2	yes	—
260	C₄	H(5,𝔽₂)×C₂₅₁	1	1	yes	characteristic
261	C₄	2₋¹⁺⁴×C₂₅₁	1	1	yes	characteristic
262	C₂	—	4	1	no	—
263	C₂	—	8	1	no	—
264	C₂	#537×C₂₅₁	1	1	yes	characteristic
265	C₂	C₂×#137×C₂₅₁	1	1	yes	characteristic
266	C₂	#575×C₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₅₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₅₁
prime radical	#121×C₂₅₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅₁ ◁ C₂×C₄²×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #533×C₂₅₁	C₂	semidirect
3. #479×C₂₅₁	C₂	semidirect
4. #477×C₂₅₁	C₂	semidirect
5. #476×C₂₅₁	C₂	semidirect
6. #470×C₂₅₁	C₂	semidirect
8. #464×C₂₅₁	C₂	semidirect
9. #462×C₂₅₁	C₂	semidirect
10. #461×C₂₅₁	C₂	semidirect
11. #423×C₂₅₁	C₂	semidirect
133. C₂₅₁	#3252	direct (N × H)
134. #3252	C₂₅₁	direct (N × H)
135. #533	C₂×C₂₅₁	semidirect
137. #479	C₂×C₂₅₁	semidirect
138. #477	C₂×C₂₅₁	semidirect
139. #476	C₂×C₂₅₁	semidirect
140. #470	C₂×C₂₅₁	semidirect
142. #464	C₂×C₂₅₁	semidirect
143. #462	C₂×C₂₅₁	semidirect
144. #461	C₂×C₂₅₁	semidirect
145. #423	C₂×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#533×C₂₅₁	2
#529×C₂₅₁	2
#479×C₂₅₁	2
#477×C₂₅₁	2
#476×C₂₅₁	2
#470×C₂₅₁	2
#468×C₂₅₁	2
#464×C₂₅₁	2
#462×C₂₅₁	2
#461×C₂₅₁	2
#423×C₂₅₁	2
#3252	251

Subgroups — #3252×C251 #44547839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3252×C₂₅₁ #44547839