Subgroups — #2949×C₂₅₁ #44547536

Order 32128.

492 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SD₁₆×C₂₅₁	C₂	1	1	yes	maximal, characteristic
2	#519×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#518×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	#440×C₂₅₁	C₂	1	4	yes	maximal, complemented
5	C₄×SD₁₆×C₂₅₁	C₂	1	4	yes	maximal, complemented
6	#427×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#419×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#418×C₂₅₁	C₂	1	1	yes	maximal, characteristic
9	#405×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
12	C₂×SD₁₆×C₂₅₁	C₄	1	8	yes	complemented
13	C₂×SD₁₆×C₂₅₁	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₂₅₁	C₂²	1	2	yes	—
15	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₂₅₁	—	2	2	no	—
18	C₄×Q₈×C₂₅₁	C₂²	1	4	yes	complemented
19	C₄×D₄×C₂₅₁	—	2	2	no	—
20	C₄×D₄×C₂₅₁	C₂²	1	4	yes	—
21	#117×C₂₅₁	—	2	2	no	—
22	#117×C₂₅₁	C₂²	1	1	yes	characteristic
23	#117×C₂₅₁	C₂²	1	1	yes	characteristic
24	#117×C₂₅₁	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₂₅₁	—	2	2	no	—
26	#107×C₂₅₁	C₂²	1	2	yes	complemented
27	#106×C₂₅₁	C₂²	1	2	yes	complemented
28	#103×C₂₅₁	C₂²	1	4	yes	complemented
29	#102×C₂₅₁	C₂²	1	4	yes	—
30	#97×C₂₅₁	C₂²	1	2	yes	complemented
31	C₄×C₈×C₂₅₁	C₂²	1	2	yes	complemented
32	C₂⁴×C₂₅₁	—	2	1	no	—
33	C₂×Q₈×C₂₅₁	—	2	1	no	—
34	C₂×Q₈×C₂₅₁	—	2	2	no	—
35	C₂×Q₈×C₂₅₁	C₂×C₄	1	4	yes	complemented
36	C₂×Q₈×C₂₅₁	C₂³	1	2	yes	—
37	C₂×D₄×C₂₅₁	—	2	1	no	—
38	C₂×D₄×C₂₅₁	—	2	2	no	—
39	C₂×D₄×C₂₅₁	C₂×C₄	1	4	yes	—
40	C₂×D₄×C₂₅₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₅₁	—	2	1	no	—
42	C₂²×C₄×C₂₅₁	—	2	4	no	—
43	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
44	SD₁₆×C₂₅₁	C₂×C₄	1	8	yes	complemented
45	SD₁₆×C₂₅₁	C₂×C₄	1	8	yes	complemented
46	M₁₆×C₂₅₁	—	2	2	no	—
47	C₂×C₈×C₂₅₁	C₂×C₄	1	4	yes	complemented
48	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
49	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₂₅₁	—	2	4	no	—
52	C₄⋊C₄×C₂₅₁	D₄	1	2	yes	—
53	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
54	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
56	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
57	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
58	C₂²⋊C₄×C₂₅₁	D₄	1	2	yes	—
59	C₄²×C₂₅₁	—	2	4	no	—
60	C₄²×C₂₅₁	—	2	1	no	—
61	C₄²×C₂₅₁	—	2	1	no	—
62	C₄²×C₂₅₁	C₂³	1	2	yes	—
63	C₂³×C₂₅₁	—	2	2	no	—
64	C₂³×C₂₅₁	—	2	4	no	—
65	C₂³×C₂₅₁	—	8	2	no	—
66	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
67	Q₈×C₂₅₁	—	2	4	no	—
68	Q₈×C₂₅₁	—	2	2	no	—
69	Q₈×C₂₅₁	C₂²×C₄	1	4	yes	—
70	D₄×C₂₅₁	—	2	4	no	—
71	D₄×C₂₅₁	—	2	2	no	—
72	D₄×C₂₅₁	C₂²×C₄	1	4	yes	—
73	C₂×C₄×C₂₅₁	—	2	4	no	—
74	C₂×C₄×C₂₅₁	—	2	2	no	—
75	C₂×C₄×C₂₅₁	—	2	2	no	—
76	C₂×C₄×C₂₅₁	—	2	2	no	—
77	C₂×C₄×C₂₅₁	—	8	4	no	—
78	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₂₅₁	C₂×D₄	1	4	yes	—
80	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₅₁	#41	1	2	yes	—
82	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
83	C₈×C₂₅₁	—	2	2	no	—
84	C₈×C₂₅₁	C₂²×C₄	1	2	yes	—
85	C₈×C₂₅₁	C₂²×C₄	1	2	yes	—
86	C₂²×C₂₅₁	—	2	2	no	—
87	C₂²×C₂₅₁	—	2	4	no	—
88	C₂²×C₂₅₁	—	4	4	no	—
89	C₂²×C₂₅₁	—	8	4	no	—
90	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
91	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
92	C₄×C₂₅₁	—	2	4	no	—
93	C₄×C₂₅₁	—	2	4	no	—
94	C₄×C₂₅₁	—	4	2	no	—
95	C₄×C₂₅₁	—	4	2	no	—
96	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
97	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
98	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
99	C₂×C₂₅₁	—	2	2	no	—
100	C₂×C₂₅₁	—	4	4	no	—
101	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
102	C₂×C₂₅₁	#577	1	2	yes	—
103	C₂₅₁	#2949	1	1	yes	characteristic, direct product factor, Sylow
104	#2949	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×SD₁₆	C₂×C₂₅₁	1	1	yes	characteristic
106	#519	C₂×C₂₅₁	1	1	yes	characteristic, complemented
107	#518	C₂×C₂₅₁	1	1	yes	characteristic
108	#440	C₂×C₂₅₁	1	4	yes	complemented
109	C₄×SD₁₆	C₂×C₂₅₁	1	4	yes	complemented
110	#427	C₂×C₂₅₁	1	1	yes	characteristic, complemented
111	#419	C₂×C₂₅₁	1	1	yes	characteristic, complemented
112	#418	C₂×C₂₅₁	1	1	yes	characteristic
113	#405	C₂×C₂₅₁	1	1	yes	characteristic, complemented
114	C₂²×Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
115	C₂²×D₄	C₂²×C₂₅₁	1	1	yes	characteristic
116	C₂×SD₁₆	C₄×C₂₅₁	1	8	yes	complemented
117	C₂×SD₁₆	C₂²×C₂₅₁	1	2	yes	—
118	C₂×SD₁₆	C₂²×C₂₅₁	1	2	yes	—
119	C₂×M₁₆	C₂²×C₂₅₁	1	1	yes	characteristic
120	C₂²×C₈	C₂²×C₂₅₁	1	1	yes	characteristic
121	C₄×Q₈	—	2	2	no	—
122	C₄×Q₈	C₂²×C₂₅₁	1	4	yes	complemented
123	C₄×D₄	—	2	2	no	—
124	C₄×D₄	C₂²×C₂₅₁	1	4	yes	—
125	#117	—	2	2	no	—
126	#117	C₂²×C₂₅₁	1	1	yes	characteristic
127	#117	C₂²×C₂₅₁	1	1	yes	characteristic
128	#117	C₂²×C₂₅₁	1	1	yes	characteristic
129	C₂×C₂²⋊C₄	—	2	2	no	—
130	#107	C₂²×C₂₅₁	1	2	yes	complemented
131	#106	C₂²×C₂₅₁	1	2	yes	complemented
132	#103	C₂²×C₂₅₁	1	4	yes	complemented
133	#102	C₂²×C₂₅₁	1	4	yes	—
134	#97	C₂²×C₂₅₁	1	2	yes	complemented
135	C₄×C₈	C₂²×C₂₅₁	1	2	yes	complemented
136	C₂⁴	—	2	1	no	—
137	C₂×Q₈	—	2	1	no	—
138	C₂×Q₈	—	2	2	no	—
139	C₂×Q₈	C₂×C₄×C₂₅₁	1	4	yes	complemented
140	C₂×Q₈	C₂³×C₂₅₁	1	2	yes	—
141	C₂×D₄	—	2	1	no	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	C₂×C₄×C₂₅₁	1	4	yes	—
144	C₂×D₄	C₂³×C₂₅₁	1	2	yes	—
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
148	SD₁₆	C₂×C₄×C₂₅₁	1	8	yes	complemented
149	SD₁₆	C₂×C₄×C₂₅₁	1	8	yes	complemented
150	M₁₆	—	2	2	no	—
151	C₂×C₈	C₂×C₄×C₂₅₁	1	4	yes	complemented
152	C₂×C₈	C₂³×C₂₅₁	1	2	yes	—
153	C₂×C₈	C₂³×C₂₅₁	1	1	yes	characteristic
154	C₂×C₈	C₂³×C₂₅₁	1	1	yes	characteristic
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	D₄×C₂₅₁	1	2	yes	—
157	C₄⋊C₄	C₂³×C₂₅₁	1	2	yes	—
158	C₄⋊C₄	C₂³×C₂₅₁	1	2	yes	—
159	C₂²⋊C₄	—	2	1	no	—
160	C₂²⋊C₄	—	2	1	no	—
161	C₂²⋊C₄	—	2	4	no	—
162	C₂²⋊C₄	D₄×C₂₅₁	1	2	yes	—
163	C₄²	—	2	4	no	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	C₂³×C₂₅₁	1	2	yes	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	4	no	—
169	C₂³	—	8	2	no	—
170	C₂³	C₂×D₄×C₂₅₁	1	1	yes	characteristic
171	Q₈	—	2	4	no	—
172	Q₈	—	2	2	no	—
173	Q₈	C₂²×C₄×C₂₅₁	1	4	yes	—
174	D₄	—	2	4	no	—
175	D₄	—	2	2	no	—
176	D₄	C₂²×C₄×C₂₅₁	1	4	yes	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	8	4	no	—
182	C₂×C₄	C₂²×C₄×C₂₅₁	1	2	yes	—
183	C₂×C₄	C₂×D₄×C₂₅₁	1	4	yes	—
184	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
185	C₂×C₄	#41×C₂₅₁	1	2	yes	—
186	C₂×C₄	C₂⁴×C₂₅₁	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	C₂²×C₄×C₂₅₁	1	2	yes	—
189	C₈	C₂²×C₄×C₂₅₁	1	2	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	4	4	no	—
193	C₂²	—	8	4	no	—
194	C₂²	C₄×D₄×C₂₅₁	1	2	yes	—
195	C₂²	C₂²×D₄×C₂₅₁	1	1	yes	characteristic
196	C₄	—	2	4	no	—
197	C₄	—	2	4	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	C₄×D₄×C₂₅₁	1	2	yes	—
201	C₄	C₂³×C₄×C₂₅₁	1	1	yes	characteristic
202	C₄	C₂×#41×C₂₅₁	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	—	4	4	no	—
205	C₂	C₂×C₄×D₄×C₂₅₁	1	1	yes	characteristic
206	C₂	#577×C₂₅₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₅₁	C₂²×D₄
commutator	C₄	C₂³×C₄×C₂₅₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₅₁
prime radical	C₂²×D₄×C₂₅₁	C₂²
LCS	G ▷ 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
2. #519×C₂₅₁	C₂	semidirect
4. #440×C₂₅₁	C₂	semidirect
5. C₄×SD₁₆×C₂₅₁	C₂	semidirect
6. #427×C₂₅₁	C₂	semidirect
7. #419×C₂₅₁	C₂	semidirect
9. #405×C₂₅₁	C₂	semidirect
12. C₂×SD₁₆×C₂₅₁	C₄	semidirect
18. C₄×Q₈×C₂₅₁	C₂²	semidirect
26. #107×C₂₅₁	C₂²	semidirect
27. #106×C₂₅₁	C₂²	semidirect
28. #103×C₂₅₁	C₂²	semidirect
30. #97×C₂₅₁	C₂²	semidirect
31. C₄×C₈×C₂₅₁	C₂²	semidirect
35. C₂×Q₈×C₂₅₁	C₂×C₄	semidirect
44. SD₁₆×C₂₅₁	C₂×C₄	semidirect
45. SD₁₆×C₂₅₁	C₂×C₄	semidirect
47. C₂×C₈×C₂₅₁	C₂×C₄	semidirect
103. C₂₅₁	#2949	direct (N × H)
104. #2949	C₂₅₁	direct (N × H)
106. #519	C₂×C₂₅₁	semidirect
108. #440	C₂×C₂₅₁	semidirect
109. C₄×SD₁₆	C₂×C₂₅₁	semidirect
110. #427	C₂×C₂₅₁	semidirect
111. #419	C₂×C₂₅₁	semidirect
113. #405	C₂×C₂₅₁	semidirect
116. C₂×SD₁₆	C₄×C₂₅₁	semidirect
122. C₄×Q₈	C₂²×C₂₅₁	semidirect
130. #107	C₂²×C₂₅₁	semidirect
131. #106	C₂²×C₂₅₁	semidirect
132. #103	C₂²×C₂₅₁	semidirect
134. #97	C₂²×C₂₅₁	semidirect
135. C₄×C₈	C₂²×C₂₅₁	semidirect
139. C₂×Q₈	C₂×C₄×C₂₅₁	semidirect
148. SD₁₆	C₂×C₄×C₂₅₁	semidirect
149. SD₁₆	C₂×C₄×C₂₅₁	semidirect
151. C₂×C₈	C₂×C₄×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×SD₁₆×C₂₅₁	2
#519×C₂₅₁	2
#518×C₂₅₁	2
#440×C₂₅₁	2
C₄×SD₁₆×C₂₅₁	2
#427×C₂₅₁	2
#419×C₂₅₁	2
#418×C₂₅₁	2
#405×C₂₅₁	2
#2949	251

Subgroups — #2949×C251 #44547536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2949×C₂₅₁ #44547536