Subgroups — #2854×C₂₅₁ #44547441

Order 32128.

688 conjugacy classes of subgroups.

The subgroup lattice has 236 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	2	yes	maximal, complemented
2	C₂×#121×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄²×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#390×C₂₅₁	C₂	1	2	yes	maximal, complemented
5	#389×C₂₅₁	C₂	1	2	yes	maximal, complemented
6	#388×C₂₅₁	C₂	1	4	yes	maximal, complemented
7	#386×C₂₅₁	C₂	1	2	yes	maximal, complemented
8	#380×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₂₅₁	C₂²	1	2	yes	—
10	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
14	#123×C₂₅₁	—	2	8	no	—
15	#121×C₂₅₁	—	2	4	no	—
16	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	4	yes	—
18	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	4	yes	—
23	C₂×C₄²×C₂₅₁	—	2	4	no	—
24	C₂×C₄²×C₂₅₁	C₂²	1	2	yes	—
25	C₂×C₄²×C₂₅₁	C₂²	1	2	yes	—
26	#95×C₂₅₁	C₂²	1	2	yes	—
27	#95×C₂₅₁	C₂²	1	4	yes	—
28	C₂⁴×C₂₅₁	C₂³	1	2	yes	—
29	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
30	C₂×D₄×C₂₅₁	—	2	4	no	—
31	C₂×D₄×C₂₅₁	—	2	4	no	—
32	C₂×D₄×C₂₅₁	—	2	4	no	—
33	C₂×D₄×C₂₅₁	—	2	2	no	—
34	C₂×D₄×C₂₅₁	—	2	4	no	—
35	C₂²×C₄×C₂₅₁	—	2	2	no	—
36	C₂²×C₄×C₂₅₁	—	2	2	no	—
37	C₂²×C₄×C₂₅₁	—	2	2	no	—
38	C₂²×C₄×C₂₅₁	—	2	4	no	—
39	C₂²×C₄×C₂₅₁	—	2	4	no	—
40	C₂²×C₄×C₂₅₁	D₄	1	4	yes	—
41	C₂²×C₄×C₂₅₁	D₄	1	4	yes	complemented
42	C₂²×C₄×C₂₅₁	C₂³	1	4	yes	—
43	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₂₅₁	—	2	2	no	—
49	C₄⋊C₄×C₂₅₁	—	2	8	no	—
50	C₄⋊C₄×C₂₅₁	—	2	4	no	—
51	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
52	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
53	C₂²⋊C₄×C₂₅₁	—	2	8	no	—
54	C₂²⋊C₄×C₂₅₁	—	2	8	no	—
55	C₄²×C₂₅₁	—	2	4	no	—
56	C₄²×C₂₅₁	—	2	4	no	—
57	C₂³×C₂₅₁	—	2	2	no	—
58	C₂³×C₂₅₁	—	2	2	no	—
59	C₂³×C₂₅₁	—	2	2	no	—
60	C₂³×C₂₅₁	—	2	2	no	—
61	C₂³×C₂₅₁	—	2	4	no	—
62	C₂³×C₂₅₁	—	2	2	no	—
63	C₂³×C₂₅₁	—	2	4	no	—
64	C₂³×C₂₅₁	C₂×D₄	1	2	yes	—
65	C₂³×C₂₅₁	C₂×D₄	1	2	yes	—
66	C₂³×C₂₅₁	#41	1	2	yes	—
67	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
68	D₄×C₂₅₁	—	4	8	no	—
69	D₄×C₂₅₁	—	4	4	no	—
70	C₂×C₄×C₂₅₁	—	2	8	no	—
71	C₂×C₄×C₂₅₁	—	2	8	no	—
72	C₂×C₄×C₂₅₁	—	2	2	no	—
73	C₂×C₄×C₂₅₁	—	2	2	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	—	2	8	no	—
78	C₂×C₄×C₂₅₁	—	2	4	no	—
79	C₂×C₄×C₂₅₁	—	2	2	no	—
80	C₂×C₄×C₂₅₁	—	2	2	no	—
81	C₂×C₄×C₂₅₁	—	2	4	no	—
82	C₂×C₄×C₂₅₁	—	2	4	no	—
83	C₂×C₄×C₂₅₁	—	2	2	no	—
84	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₂₅₁	C₂×D₄	1	4	yes	—
86	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₂₅₁	#41	1	4	yes	—
88	C₂×C₄×C₂₅₁	#41	1	4	yes	—
89	C₂²×C₂₅₁	—	2	2	no	—
90	C₂²×C₂₅₁	—	2	2	no	—
91	C₂²×C₂₅₁	—	2	4	no	—
92	C₂²×C₂₅₁	—	2	4	no	—
93	C₂²×C₂₅₁	—	4	2	no	—
94	C₂²×C₂₅₁	—	4	2	no	—
95	C₂²×C₂₅₁	—	4	2	no	—
96	C₂²×C₂₅₁	—	4	4	no	—
97	C₂²×C₂₅₁	—	4	4	no	—
98	C₂²×C₂₅₁	#121	1	4	yes	—
99	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
100	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
102	C₂²×C₂₅₁	C₂×#41	1	2	yes	—
103	C₂²×C₂₅₁	C₂×#41	1	2	yes	—
104	C₄×C₂₅₁	—	2	4	no	—
105	C₄×C₂₅₁	—	2	2	no	—
106	C₄×C₂₅₁	—	2	4	no	—
107	C₄×C₂₅₁	—	4	2	no	—
108	C₄×C₂₅₁	—	4	4	no	—
109	C₄×C₂₅₁	#123	1	4	yes	complemented
110	C₂×C₂₅₁	—	2	4	no	—
111	C₂×C₂₅₁	—	8	2	no	—
112	C₂×C₂₅₁	C₂×#121	1	1	yes	characteristic
113	C₂×C₂₅₁	C₂×#123	1	1	yes	characteristic
114	C₂×C₂₅₁	#525	1	1	yes	characteristic
115	C₂×C₂₅₁	#529	1	2	yes	—
116	C₂×C₂₅₁	#532	1	2	yes	—
117	C₂₅₁	#2854	1	1	yes	characteristic, direct product factor, Sylow
118	#2854	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×#123	C₂×C₂₅₁	1	2	yes	complemented
120	C₂×#121	C₂×C₂₅₁	1	1	yes	characteristic
121	C₂²×C₄²	C₂×C₂₅₁	1	1	yes	characteristic, complemented
122	#390	C₂×C₂₅₁	1	2	yes	complemented
123	#389	C₂×C₂₅₁	1	2	yes	complemented
124	#388	C₂×C₂₅₁	1	4	yes	complemented
125	#386	C₂×C₂₅₁	1	2	yes	complemented
126	#380	C₂×C₂₅₁	1	1	yes	characteristic, complemented
127	C₂²×D₄	C₂²×C₂₅₁	1	2	yes	—
128	C₂²×D₄	C₂²×C₂₅₁	1	1	yes	characteristic
129	C₂³×C₄	C₂²×C₂₅₁	1	1	yes	characteristic
130	C₂³×C₄	C₂²×C₂₅₁	1	1	yes	characteristic
131	C₂³×C₄	C₂²×C₂₅₁	1	1	yes	characteristic
132	#123	—	2	8	no	—
133	#121	—	2	4	no	—
134	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	4	yes	—
136	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	2	yes	—
137	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	2	yes	—
138	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	2	yes	—
139	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	4	yes	—
140	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	4	yes	—
141	C₂×C₄²	—	2	4	no	—
142	C₂×C₄²	C₂²×C₂₅₁	1	2	yes	—
143	C₂×C₄²	C₂²×C₂₅₁	1	2	yes	—
144	#95	C₂²×C₂₅₁	1	2	yes	—
145	#95	C₂²×C₂₅₁	1	4	yes	—
146	C₂⁴	C₂³×C₂₅₁	1	2	yes	—
147	C₂⁴	C₂³×C₂₅₁	1	1	yes	characteristic
148	C₂×D₄	—	2	4	no	—
149	C₂×D₄	—	2	4	no	—
150	C₂×D₄	—	2	4	no	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	—	2	4	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	D₄×C₂₅₁	1	4	yes	—
159	C₂²×C₄	D₄×C₂₅₁	1	4	yes	complemented
160	C₂²×C₄	C₂³×C₂₅₁	1	4	yes	—
161	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
162	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
164	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
165	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	8	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₂²⋊C₄	—	2	4	no	—
170	C₂²⋊C₄	—	2	4	no	—
171	C₂²⋊C₄	—	2	8	no	—
172	C₂²⋊C₄	—	2	8	no	—
173	C₄²	—	2	4	no	—
174	C₄²	—	2	4	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	4	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	4	no	—
182	C₂³	C₂×D₄×C₂₅₁	1	2	yes	—
183	C₂³	C₂×D₄×C₂₅₁	1	2	yes	—
184	C₂³	#41×C₂₅₁	1	2	yes	—
185	C₂³	C₂⁴×C₂₅₁	1	1	yes	characteristic
186	D₄	—	4	8	no	—
187	D₄	—	4	4	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
203	C₂×C₄	C₂×D₄×C₂₅₁	1	4	yes	—
204	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
205	C₂×C₄	#41×C₂₅₁	1	4	yes	—
206	C₂×C₄	#41×C₂₅₁	1	4	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	4	no	—
215	C₂²	—	4	4	no	—
216	C₂²	#121×C₂₅₁	1	4	yes	—
217	C₂²	C₂²×D₄×C₂₅₁	1	1	yes	characteristic
218	C₂²	C₂²×D₄×C₂₅₁	1	1	yes	characteristic
219	C₂²	C₂×#41×C₂₅₁	1	1	yes	characteristic
220	C₂²	C₂×#41×C₂₅₁	1	2	yes	—
221	C₂²	C₂×#41×C₂₅₁	1	2	yes	—
222	C₄	—	2	4	no	—
223	C₄	—	2	2	no	—
224	C₄	—	2	4	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	4	no	—
227	C₄	#123×C₂₅₁	1	4	yes	complemented
228	C₂	—	2	4	no	—
229	C₂	—	8	2	no	—
230	C₂	C₂×#121×C₂₅₁	1	1	yes	characteristic
231	C₂	C₂×#123×C₂₅₁	1	1	yes	characteristic
232	C₂	#525×C₂₅₁	1	1	yes	characteristic
233	C₂	#529×C₂₅₁	1	2	yes	—
234	C₂	#532×C₂₅₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₅₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₅₁
prime radical	C₂×#121×C₂₅₁	C₂
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
1. C₂×#123×C₂₅₁	C₂	semidirect
3. C₂²×C₄²×C₂₅₁	C₂	semidirect
4. #390×C₂₅₁	C₂	semidirect
5. #389×C₂₅₁	C₂	semidirect
6. #388×C₂₅₁	C₂	semidirect
7. #386×C₂₅₁	C₂	semidirect
8. #380×C₂₅₁	C₂	semidirect
41. C₂²×C₄×C₂₅₁	D₄	semidirect
109. C₄×C₂₅₁	#123	semidirect
117. C₂₅₁	#2854	direct (N × H)
118. #2854	C₂₅₁	direct (N × H)
119. C₂×#123	C₂×C₂₅₁	semidirect
121. C₂²×C₄²	C₂×C₂₅₁	semidirect
122. #390	C₂×C₂₅₁	semidirect
123. #389	C₂×C₂₅₁	semidirect
124. #388	C₂×C₂₅₁	semidirect
125. #386	C₂×C₂₅₁	semidirect
126. #380	C₂×C₂₅₁	semidirect
159. C₂²×C₄	D₄×C₂₅₁	semidirect
227. C₄	#123×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
C₂×#121×C₂₅₁	2
C₂²×C₄²×C₂₅₁	2
#390×C₂₅₁	2
#389×C₂₅₁	2
#388×C₂₅₁	2
#386×C₂₅₁	2
#380×C₂₅₁	2
#2854	251

Subgroups — #2854×C251 #44547441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2854×C₂₅₁ #44547441