Subgroups — #2623×C₂₅₁ #44547210

Order 32128.

526 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#123×C₂₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#400×C₂₅₁	C₂	1	2	yes	maximal, complemented
4	#399×C₂₅₁	C₂	1	2	yes	maximal, complemented
5	#398×C₂₅₁	C₂	1	2	yes	maximal, complemented
6	#397×C₂₅₁	C₂	1	2	yes	maximal, complemented
7	#386×C₂₅₁	C₂	1	2	yes	maximal
8	#385×C₂₅₁	C₂	1	2	yes	maximal, complemented
9	#381×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₅₁	C₂²	1	2	yes	—
13	#123×C₂₅₁	—	2	4	no	—
14	C₂×C₄⋊C₄×C₂₅₁	—	2	4	no	—
15	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	4	yes	—
25	#95×C₂₅₁	C₂²	1	2	yes	—
26	#95×C₂₅₁	C₂²	1	2	yes	—
27	#95×C₂₅₁	C₂²	1	4	yes	—
28	#95×C₂₅₁	C₂²	1	4	yes	—
29	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
30	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
31	C₂×D₄×C₂₅₁	—	2	2	no	—
32	C₂×D₄×C₂₅₁	—	2	4	no	—
33	C₂²×C₄×C₂₅₁	—	2	4	no	—
34	C₂²×C₄×C₂₅₁	—	2	4	no	—
35	C₂²×C₄×C₂₅₁	—	4	4	no	—
36	C₂²×C₄×C₂₅₁	D₄	1	4	yes	—
37	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₂₅₁	C₂³	1	4	yes	—
39	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₅₁	C₂³	1	4	yes	—
41	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₂₅₁	—	2	4	no	—
45	C₄⋊C₄×C₂₅₁	—	2	4	no	—
46	C₄⋊C₄×C₂₅₁	—	2	2	no	—
47	C₄⋊C₄×C₂₅₁	—	2	2	no	—
48	C₄⋊C₄×C₂₅₁	—	4	4	no	—
49	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
50	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
51	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
52	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
53	C₂²⋊C₄×C₂₅₁	—	4	4	no	—
54	C₂³×C₂₅₁	—	2	2	no	—
55	C₂³×C₂₅₁	—	2	2	no	—
56	C₂³×C₂₅₁	—	2	1	no	—
57	C₂³×C₂₅₁	—	2	2	no	—
58	C₂³×C₂₅₁	—	2	2	no	—
59	C₂³×C₂₅₁	—	2	2	no	—
60	C₂³×C₂₅₁	C₂×D₄	1	2	yes	—
61	C₂³×C₂₅₁	#41	1	4	yes	—
62	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
63	D₄×C₂₅₁	—	4	4	no	—
64	C₂×C₄×C₂₅₁	—	2	8	no	—
65	C₂×C₄×C₂₅₁	—	2	2	no	—
66	C₂×C₄×C₂₅₁	—	2	2	no	—
67	C₂×C₄×C₂₅₁	—	2	1	no	—
68	C₂×C₄×C₂₅₁	—	2	1	no	—
69	C₂×C₄×C₂₅₁	—	2	4	no	—
70	C₂×C₄×C₂₅₁	—	2	1	no	—
71	C₂×C₄×C₂₅₁	—	2	4	no	—
72	C₂×C₄×C₂₅₁	—	2	2	no	—
73	C₂×C₄×C₂₅₁	—	2	4	no	—
74	C₂×C₄×C₂₅₁	—	2	4	no	—
75	C₂×C₄×C₂₅₁	—	2	2	no	—
76	C₂×C₄×C₂₅₁	—	2	4	no	—
77	C₂×C₄×C₂₅₁	—	2	2	no	—
78	C₂×C₄×C₂₅₁	—	2	4	no	—
79	C₂×C₄×C₂₅₁	—	4	8	no	—
80	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
82	C₂²×C₂₅₁	—	2	2	no	—
83	C₂²×C₂₅₁	—	2	2	no	—
84	C₂²×C₂₅₁	—	2	8	no	—
85	C₂²×C₂₅₁	—	4	1	no	—
86	C₂²×C₂₅₁	—	4	1	no	—
87	C₂²×C₂₅₁	—	4	1	no	—
88	C₂²×C₂₅₁	—	4	4	no	—
89	C₂²×C₂₅₁	#123	1	4	yes	—
90	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
91	C₂²×C₂₅₁	C₂×#41	1	2	yes	—
92	C₂²×C₂₅₁	H(5,𝔽₂)	1	2	yes	—
93	C₂²×C₂₅₁	2₋¹⁺⁴	1	2	yes	—
94	C₄×C₂₅₁	—	2	2	no	—
95	C₄×C₂₅₁	—	2	2	no	—
96	C₄×C₂₅₁	—	4	1	no	—
97	C₄×C₂₅₁	—	4	1	no	—
98	C₄×C₂₅₁	—	4	4	no	—
99	C₄×C₂₅₁	—	4	1	no	—
100	C₄×C₂₅₁	—	4	4	no	—
101	C₂×C₂₅₁	—	2	4	no	—
102	C₂×C₂₅₁	—	8	1	no	—
103	C₂×C₂₅₁	C₂×#123	1	1	yes	characteristic
104	C₂×C₂₅₁	#535	1	1	yes	characteristic
105	C₂×C₂₅₁	#536	1	1	yes	characteristic
106	C₂×C₂₅₁	#539	1	4	yes	—
107	C₂₅₁	#2623	1	1	yes	characteristic, direct product factor, Sylow
108	#2623	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×#123	C₂×C₂₅₁	1	1	yes	characteristic
110	C₂²×C₄⋊C₄	C₂×C₂₅₁	1	1	yes	characteristic, complemented
111	#400	C₂×C₂₅₁	1	2	yes	complemented
112	#399	C₂×C₂₅₁	1	2	yes	complemented
113	#398	C₂×C₂₅₁	1	2	yes	complemented
114	#397	C₂×C₂₅₁	1	2	yes	complemented
115	#386	C₂×C₂₅₁	1	2	yes	—
116	#385	C₂×C₂₅₁	1	2	yes	complemented
117	#381	C₂×C₂₅₁	1	1	yes	characteristic, complemented
118	C₂²×D₄	C₂²×C₂₅₁	1	1	yes	characteristic
119	C₂³×C₄	C₂²×C₂₅₁	1	1	yes	characteristic
120	C₂³×C₄	C₂²×C₂₅₁	1	2	yes	—
121	#123	—	2	4	no	—
122	C₂×C₄⋊C₄	—	2	4	no	—
123	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	2	yes	—
124	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	2	yes	—
125	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
127	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	4	yes	—
128	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
129	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
130	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	2	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	4	yes	—
133	#95	C₂²×C₂₅₁	1	2	yes	—
134	#95	C₂²×C₂₅₁	1	2	yes	—
135	#95	C₂²×C₂₅₁	1	4	yes	—
136	#95	C₂²×C₂₅₁	1	4	yes	—
137	C₂⁴	C₂³×C₂₅₁	1	1	yes	characteristic
138	C₂⁴	C₂³×C₂₅₁	1	1	yes	characteristic
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	—	2	4	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	4	4	no	—
144	C₂²×C₄	D₄×C₂₅₁	1	4	yes	—
145	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
146	C₂²×C₄	C₂³×C₂₅₁	1	4	yes	—
147	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₂₅₁	1	4	yes	—
149	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
150	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
151	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	4	4	no	—
162	C₂³	—	2	2	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	1	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	2	no	—
168	C₂³	C₂×D₄×C₂₅₁	1	2	yes	—
169	C₂³	#41×C₂₅₁	1	4	yes	—
170	C₂³	C₂⁴×C₂₅₁	1	1	yes	characteristic
171	D₄	—	4	4	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	4	8	no	—
188	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
189	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	8	no	—
193	C₂²	—	4	1	no	—
194	C₂²	—	4	1	no	—
195	C₂²	—	4	1	no	—
196	C₂²	—	4	4	no	—
197	C₂²	#123×C₂₅₁	1	4	yes	—
198	C₂²	C₂²×D₄×C₂₅₁	1	1	yes	characteristic
199	C₂²	C₂×#41×C₂₅₁	1	2	yes	—
200	C₂²	H(5,𝔽₂)×C₂₅₁	1	2	yes	—
201	C₂²	2₋¹⁺⁴×C₂₅₁	1	2	yes	—
202	C₄	—	2	2	no	—
203	C₄	—	2	2	no	—
204	C₄	—	4	1	no	—
205	C₄	—	4	4	no	—
206	C₄	—	4	1	no	—
207	C₄	—	4	1	no	—
208	C₄	—	4	4	no	—
209	C₂	—	2	4	no	—
210	C₂	—	8	1	no	—
211	C₂	C₂×#123×C₂₅₁	1	1	yes	characteristic
212	C₂	#535×C₂₅₁	1	1	yes	characteristic
213	C₂	#536×C₂₅₁	1	1	yes	characteristic
214	C₂	#539×C₂₅₁	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₅₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₅₁
prime radical	C₂²×D₄×C₂₅₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊C₄×C₂₅₁	C₂	semidirect
3. #400×C₂₅₁	C₂	semidirect
4. #399×C₂₅₁	C₂	semidirect
5. #398×C₂₅₁	C₂	semidirect
6. #397×C₂₅₁	C₂	semidirect
8. #385×C₂₅₁	C₂	semidirect
9. #381×C₂₅₁	C₂	semidirect
107. C₂₅₁	#2623	direct (N × H)
108. #2623	C₂₅₁	direct (N × H)
110. C₂²×C₄⋊C₄	C₂×C₂₅₁	semidirect
111. #400	C₂×C₂₅₁	semidirect
112. #399	C₂×C₂₅₁	semidirect
113. #398	C₂×C₂₅₁	semidirect
114. #397	C₂×C₂₅₁	semidirect
116. #385	C₂×C₂₅₁	semidirect
117. #381	C₂×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#123×C₂₅₁	2
C₂²×C₄⋊C₄×C₂₅₁	2
#400×C₂₅₁	2
#399×C₂₅₁	2
#398×C₂₅₁	2
#397×C₂₅₁	2
#386×C₂₅₁	2
#385×C₂₅₁	2
#381×C₂₅₁	2
#2623	251

Subgroups — #2623×C251 #44547210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2623×C₂₅₁ #44547210