Subgroups — #3081×C₂₅₁ #44547668

Order 32128.

398 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#544×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#518×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#478×C₂₅₁	C₂	1	2	yes	maximal, complemented
4	#478×C₂₅₁	C₂	1	2	yes	maximal, complemented
5	#476×C₂₅₁	C₂	1	2	yes	maximal, complemented
6	#474×C₂₅₁	C₂	1	2	yes	maximal, complemented
7	#428×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#427×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#424×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#418×C₂₅₁	C₂	1	1	yes	maximal, characteristic
11	C₂×#102×C₂₅₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₂₅₁	—	2	1	no	—
16	C₄⋊Q₈×C₂₅₁	C₂²	1	2	yes	complemented
17	#125×C₂₅₁	—	2	1	no	—
18	#125×C₂₅₁	C₂²	1	2	yes	complemented
19	C₂²⋊Q₈×C₂₅₁	—	2	2	no	—
20	C₄×D₄×C₂₅₁	—	2	1	no	—
21	C₄×D₄×C₂₅₁	—	2	1	no	—
22	C₄×D₄×C₂₅₁	C₂²	1	2	yes	—
23	C₄×D₄×C₂₅₁	C₂²	1	2	yes	—
24	#117×C₂₅₁	C₂²	1	1	yes	characteristic
25	#117×C₂₅₁	C₂²	1	1	yes	characteristic
26	#117×C₂₅₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₂₅₁	—	2	2	no	—
29	#107×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
30	#107×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
31	#107×C₂₅₁	C₂²	1	2	yes	complemented
32	#106×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
33	#106×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
34	#106×C₂₅₁	C₂²	1	2	yes	complemented
35	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	complemented
36	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	complemented
37	#102×C₂₅₁	C₂²	1	2	yes	—
38	#102×C₂₅₁	C₂²	1	2	yes	—
39	#102×C₂₅₁	C₂²	1	2	yes	—
40	#102×C₂₅₁	C₂²	1	2	yes	—
41	C₂⁴×C₂₅₁	—	2	1	no	—
42	C₂×Q₈×C₂₅₁	—	2	2	no	—
43	C₂×D₄×C₂₅₁	—	2	1	no	—
44	C₂×D₄×C₂₅₁	—	2	2	no	—
45	C₂×D₄×C₂₅₁	Q₈	1	4	yes	complemented
46	C₂×D₄×C₂₅₁	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₅₁	—	2	1	no	—
48	C₂²×C₄×C₂₅₁	—	2	2	no	—
49	C₂²×C₄×C₂₅₁	—	2	2	no	—
50	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
51	M₁₆×C₂₅₁	—	2	2	no	—
52	C₂×C₈×C₂₅₁	—	2	2	no	—
53	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
55	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₂₅₁	—	2	1	no	—
57	C₄⋊C₄×C₂₅₁	—	2	2	no	—
58	C₄⋊C₄×C₂₅₁	—	2	2	no	—
59	C₄⋊C₄×C₂₅₁	—	2	2	no	—
60	C₄⋊C₄×C₂₅₁	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₂₅₁	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
63	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
67	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
68	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
69	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
70	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
71	C₂²⋊C₄×C₂₅₁	D₄	1	2	yes	—
72	C₄²×C₂₅₁	—	2	1	no	—
73	C₄²×C₂₅₁	—	2	1	no	—
74	C₄²×C₂₅₁	C₂³	1	2	yes	—
75	C₂³×C₂₅₁	—	2	2	no	—
76	C₂³×C₂₅₁	—	2	4	no	—
77	C₂³×C₂₅₁	—	8	1	no	—
78	C₂³×C₂₅₁	—	8	1	no	—
79	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
80	Q₈×C₂₅₁	—	8	2	no	—
81	D₄×C₂₅₁	—	2	4	no	—
82	D₄×C₂₅₁	—	2	2	no	—
83	D₄×C₂₅₁	C₂×Q₈	1	4	yes	—
84	C₂×C₄×C₂₅₁	—	2	1	no	—
85	C₂×C₄×C₂₅₁	—	2	1	no	—
86	C₂×C₄×C₂₅₁	—	2	1	no	—
87	C₂×C₄×C₂₅₁	—	2	1	no	—
88	C₂×C₄×C₂₅₁	—	2	2	no	—
89	C₂×C₄×C₂₅₁	—	8	2	no	—
90	C₂×C₄×C₂₅₁	—	8	2	no	—
91	C₂×C₄×C₂₅₁	—	8	1	no	—
92	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
96	C₂×C₄×C₂₅₁	#41	1	2	yes	—
97	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
98	C₈×C₂₅₁	—	2	1	no	—
99	C₈×C₂₅₁	—	2	1	no	—
100	C₈×C₂₅₁	—	2	2	no	—
101	C₂²×C₂₅₁	—	2	1	no	—
102	C₂²×C₂₅₁	—	2	1	no	—
103	C₂²×C₂₅₁	—	2	4	no	—
104	C₂²×C₂₅₁	—	4	2	no	—
105	C₂²×C₂₅₁	—	4	2	no	—
106	C₂²×C₂₅₁	—	8	4	no	—
107	C₂²×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
108	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
109	C₄×C₂₅₁	—	2	2	no	—
110	C₄×C₂₅₁	—	2	2	no	—
111	C₄×C₂₅₁	—	4	1	no	—
112	C₄×C₂₅₁	—	4	2	no	—
113	C₄×C₂₅₁	—	4	1	no	—
114	C₄×C₂₅₁	—	4	1	no	—
115	C₄×C₂₅₁	—	4	1	no	—
116	C₄×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
117	C₄×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
118	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
119	C₂×C₂₅₁	—	2	2	no	—
120	C₂×C₂₅₁	—	4	4	no	—
121	C₂×C₂₅₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₂₅₁	#576	1	1	yes	characteristic
123	C₂×C₂₅₁	#577	1	1	yes	characteristic
124	C₂₅₁	#3081	1	1	yes	characteristic, direct product factor, Sylow
125	#3081	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	#544	C₂×C₂₅₁	1	1	yes	characteristic, complemented
127	#518	C₂×C₂₅₁	1	1	yes	characteristic
128	#478	C₂×C₂₅₁	1	2	yes	complemented
129	#478	C₂×C₂₅₁	1	2	yes	complemented
130	#476	C₂×C₂₅₁	1	2	yes	complemented
131	#474	C₂×C₂₅₁	1	2	yes	complemented
132	#428	C₂×C₂₅₁	1	1	yes	characteristic, complemented
133	#427	C₂×C₂₅₁	1	1	yes	characteristic, complemented
134	#424	C₂×C₂₅₁	1	1	yes	characteristic, complemented
135	#418	C₂×C₂₅₁	1	1	yes	characteristic
136	C₂×#102	C₂×C₂₅₁	1	1	yes	characteristic
137	C₂²×D₄	C₂²×C₂₅₁	1	1	yes	characteristic
138	C₂×M₁₆	C₂²×C₂₅₁	1	1	yes	characteristic
139	C₂²×C₈	C₂²×C₂₅₁	1	1	yes	characteristic
140	C₄⋊Q₈	—	2	1	no	—
141	C₄⋊Q₈	C₂²×C₂₅₁	1	2	yes	complemented
142	#125	—	2	1	no	—
143	#125	C₂²×C₂₅₁	1	2	yes	complemented
144	C₂²⋊Q₈	—	2	2	no	—
145	C₄×D₄	—	2	1	no	—
146	C₄×D₄	—	2	1	no	—
147	C₄×D₄	C₂²×C₂₅₁	1	2	yes	—
148	C₄×D₄	C₂²×C₂₅₁	1	2	yes	—
149	#117	C₂²×C₂₅₁	1	1	yes	characteristic
150	#117	C₂²×C₂₅₁	1	1	yes	characteristic
151	#117	C₂²×C₂₅₁	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
153	C₂×C₂²⋊C₄	—	2	2	no	—
154	#107	C₂²×C₂₅₁	1	1	yes	characteristic, complemented
155	#107	C₂²×C₂₅₁	1	1	yes	characteristic, complemented
156	#107	C₂²×C₂₅₁	1	2	yes	complemented
157	#106	C₂²×C₂₅₁	1	1	yes	characteristic, complemented
158	#106	C₂²×C₂₅₁	1	1	yes	characteristic, complemented
159	#106	C₂²×C₂₅₁	1	2	yes	complemented
160	C₄⋊C₈	C₂²×C₂₅₁	1	2	yes	complemented
161	C₄⋊C₈	C₂²×C₂₅₁	1	2	yes	complemented
162	#102	C₂²×C₂₅₁	1	2	yes	—
163	#102	C₂²×C₂₅₁	1	2	yes	—
164	#102	C₂²×C₂₅₁	1	2	yes	—
165	#102	C₂²×C₂₅₁	1	2	yes	—
166	C₂⁴	—	2	1	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×D₄	—	2	1	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	Q₈×C₂₅₁	1	4	yes	complemented
171	C₂×D₄	C₂³×C₂₅₁	1	2	yes	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
176	M₁₆	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	C₂³×C₂₅₁	1	1	yes	characteristic
179	C₂×C₈	C₂³×C₂₅₁	1	2	yes	—
180	C₂×C₈	C₂³×C₂₅₁	1	1	yes	characteristic
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	D₄×C₂₅₁	1	1	yes	characteristic
186	C₄⋊C₄	D₄×C₂₅₁	1	1	yes	characteristic
187	C₄⋊C₄	C₂³×C₂₅₁	1	2	yes	—
188	C₄⋊C₄	C₂³×C₂₅₁	1	1	yes	characteristic
189	C₄⋊C₄	C₂³×C₂₅₁	1	1	yes	characteristic
190	C₄⋊C₄	C₂³×C₂₅₁	1	1	yes	characteristic
191	C₄⋊C₄	C₂³×C₂₅₁	1	1	yes	characteristic
192	C₂²⋊C₄	—	2	1	no	—
193	C₂²⋊C₄	—	2	1	no	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	D₄×C₂₅₁	1	2	yes	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	C₂³×C₂₅₁	1	2	yes	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	4	no	—
202	C₂³	—	8	1	no	—
203	C₂³	—	8	1	no	—
204	C₂³	C₂×D₄×C₂₅₁	1	1	yes	characteristic
205	Q₈	—	8	2	no	—
206	D₄	—	2	4	no	—
207	D₄	—	2	2	no	—
208	D₄	C₂×Q₈×C₂₅₁	1	4	yes	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
218	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
219	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
220	C₂×C₄	C₂×Q₈×C₂₅₁	1	2	yes	—
221	C₂×C₄	#41×C₂₅₁	1	2	yes	—
222	C₂×C₄	C₂⁴×C₂₅₁	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	2	2	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	8	4	no	—
232	C₂²	C₂²⋊Q₈×C₂₅₁	1	2	yes	—
233	C₂²	C₂²×D₄×C₂₅₁	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	4	1	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	1	no	—
240	C₄	—	4	1	no	—
241	C₄	C₂²⋊Q₈×C₂₅₁	1	2	yes	—
242	C₄	C₂²×Q₈×C₂₅₁	1	1	yes	characteristic
243	C₄	C₂×#41×C₂₅₁	1	1	yes	characteristic
244	C₂	—	2	2	no	—
245	C₂	—	4	4	no	—
246	C₂	C₂×C₂²⋊Q₈×C₂₅₁	1	1	yes	characteristic
247	C₂	#576×C₂₅₁	1	1	yes	characteristic
248	C₂	#577×C₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₅₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₅₁
prime radical	C₂²×D₄×C₂₅₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅₁ ◁ #117×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #544×C₂₅₁	C₂	semidirect
3. #478×C₂₅₁	C₂	semidirect
4. #478×C₂₅₁	C₂	semidirect
5. #476×C₂₅₁	C₂	semidirect
6. #474×C₂₅₁	C₂	semidirect
7. #428×C₂₅₁	C₂	semidirect
8. #427×C₂₅₁	C₂	semidirect
9. #424×C₂₅₁	C₂	semidirect
16. C₄⋊Q₈×C₂₅₁	C₂²	semidirect
18. #125×C₂₅₁	C₂²	semidirect
29. #107×C₂₅₁	C₂²	semidirect
30. #107×C₂₅₁	C₂²	semidirect
31. #107×C₂₅₁	C₂²	semidirect
32. #106×C₂₅₁	C₂²	semidirect
33. #106×C₂₅₁	C₂²	semidirect
34. #106×C₂₅₁	C₂²	semidirect
35. C₄⋊C₈×C₂₅₁	C₂²	semidirect
36. C₄⋊C₈×C₂₅₁	C₂²	semidirect
45. C₂×D₄×C₂₅₁	Q₈	semidirect
124. C₂₅₁	#3081	direct (N × H)
125. #3081	C₂₅₁	direct (N × H)
126. #544	C₂×C₂₅₁	semidirect
128. #478	C₂×C₂₅₁	semidirect
129. #478	C₂×C₂₅₁	semidirect
130. #476	C₂×C₂₅₁	semidirect
131. #474	C₂×C₂₅₁	semidirect
132. #428	C₂×C₂₅₁	semidirect
133. #427	C₂×C₂₅₁	semidirect
134. #424	C₂×C₂₅₁	semidirect
141. C₄⋊Q₈	C₂²×C₂₅₁	semidirect
143. #125	C₂²×C₂₅₁	semidirect
154. #107	C₂²×C₂₅₁	semidirect
155. #107	C₂²×C₂₅₁	semidirect
156. #107	C₂²×C₂₅₁	semidirect
157. #106	C₂²×C₂₅₁	semidirect
158. #106	C₂²×C₂₅₁	semidirect
159. #106	C₂²×C₂₅₁	semidirect
160. C₄⋊C₈	C₂²×C₂₅₁	semidirect
161. C₄⋊C₈	C₂²×C₂₅₁	semidirect
170. C₂×D₄	Q₈×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#544×C₂₅₁	2
#518×C₂₅₁	2
#478×C₂₅₁	2
#476×C₂₅₁	2
#474×C₂₅₁	2
#428×C₂₅₁	2
#427×C₂₅₁	2
#424×C₂₅₁	2
#418×C₂₅₁	2
C₂×#102×C₂₅₁	2
#3081	251

Subgroups — #3081×C251 #44547668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3081×C₂₅₁ #44547668