Subgroups — #2712×C₂₅₁ #44547299

Order 32128.

556 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#124×C₂₅₁	C₂	1	2	yes	maximal, complemented
2	#397×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#394×C₂₅₁	C₂	1	2	yes	maximal, complemented
4	#393×C₂₅₁	C₂	1	2	yes	maximal, complemented
5	#392×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	#391×C₂₅₁	C₂	1	2	yes	maximal, complemented
7	#388×C₂₅₁	C₂	1	2	yes	maximal, complemented
8	#386×C₂₅₁	C₂	1	2	yes	maximal, complemented
9	#385×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₂₅₁	C₂²	1	2	yes	—
11	C₂²×D₄×C₂₅₁	C₂²	1	2	yes	—
12	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
14	#124×C₂₅₁	—	2	8	no	—
15	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	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	2	yes	—
23	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₅₁	C₂²	1	2	yes	—
26	#95×C₂₅₁	C₂²	1	2	yes	—
27	#95×C₂₅₁	C₂²	1	2	yes	—
28	#95×C₂₅₁	C₂²	1	2	yes	—
29	#95×C₂₅₁	C₂²	1	1	yes	characteristic
30	#95×C₂₅₁	C₂²	1	2	yes	—
31	#95×C₂₅₁	C₂²	1	1	yes	characteristic
32	C₂⁴×C₂₅₁	C₂³	1	2	yes	—
33	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
34	C₂×Q₈×C₂₅₁	—	2	4	no	—
35	C₂×Q₈×C₂₅₁	—	2	4	no	—
36	C₂×Q₈×C₂₅₁	—	4	2	no	—
37	C₂×D₄×C₂₅₁	—	2	4	no	—
38	C₂×D₄×C₂₅₁	—	2	4	no	—
39	C₂×D₄×C₂₅₁	—	4	2	no	—
40	C₂×D₄×C₂₅₁	—	4	1	no	—
41	C₂×D₄×C₂₅₁	—	4	2	no	—
42	C₂²×C₄×C₂₅₁	—	4	1	no	—
43	C₂²×C₄×C₂₅₁	—	4	2	no	—
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	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
51	C₄⋊C₄×C₂₅₁	—	4	2	no	—
52	C₄⋊C₄×C₂₅₁	—	4	1	no	—
53	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
54	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
55	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
56	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
57	C₂²⋊C₄×C₂₅₁	—	4	2	no	—
58	C₂²⋊C₄×C₂₅₁	—	4	1	no	—
59	C₂²⋊C₄×C₂₅₁	—	4	2	no	—
60	C₂²⋊C₄×C₂₅₁	D₄	1	4	yes	complemented
61	C₄²×C₂₅₁	—	2	4	no	—
62	C₂³×C₂₅₁	—	2	2	no	—
63	C₂³×C₂₅₁	—	2	1	no	—
64	C₂³×C₂₅₁	—	2	2	no	—
65	C₂³×C₂₅₁	—	2	1	no	—
66	C₂³×C₂₅₁	—	2	2	no	—
67	C₂³×C₂₅₁	—	2	2	no	—
68	C₂³×C₂₅₁	—	2	2	no	—
69	C₂³×C₂₅₁	—	2	2	no	—
70	C₂³×C₂₅₁	—	2	2	no	—
71	C₂³×C₂₅₁	—	2	2	no	—
72	C₂³×C₂₅₁	—	2	2	no	—
73	C₂³×C₂₅₁	C₂×D₄	1	2	yes	—
74	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
75	Q₈×C₂₅₁	—	4	8	no	—
76	D₄×C₂₅₁	—	4	8	no	—
77	D₄×C₂₅₁	—	4	4	no	—
78	C₂×C₄×C₂₅₁	—	2	2	no	—
79	C₂×C₄×C₂₅₁	—	2	2	no	—
80	C₂×C₄×C₂₅₁	—	2	2	no	—
81	C₂×C₄×C₂₅₁	—	2	2	no	—
82	C₂×C₄×C₂₅₁	—	2	2	no	—
83	C₂×C₄×C₂₅₁	—	2	2	no	—
84	C₂×C₄×C₂₅₁	—	2	2	no	—
85	C₂×C₄×C₂₅₁	—	2	2	no	—
86	C₂×C₄×C₂₅₁	—	2	2	no	—
87	C₂×C₄×C₂₅₁	—	2	1	no	—
88	C₂×C₄×C₂₅₁	—	2	2	no	—
89	C₂×C₄×C₂₅₁	—	2	2	no	—
90	C₂×C₄×C₂₅₁	—	2	1	no	—
91	C₂×C₄×C₂₅₁	—	2	2	no	—
92	C₂×C₄×C₂₅₁	—	2	2	no	—
93	C₂×C₄×C₂₅₁	—	2	2	no	—
94	C₂×C₄×C₂₅₁	—	2	2	no	—
95	C₂×C₄×C₂₅₁	—	4	4	no	—
96	C₂×C₄×C₂₅₁	C₂×D₄	1	4	yes	—
97	C₂×C₄×C₂₅₁	#41	1	4	yes	—
98	C₂²×C₂₅₁	—	2	2	no	—
99	C₂²×C₂₅₁	—	2	4	no	—
100	C₂²×C₂₅₁	—	4	2	no	—
101	C₂²×C₂₅₁	—	4	1	no	—
102	C₂²×C₂₅₁	—	4	1	no	—
103	C₂²×C₂₅₁	—	4	2	no	—
104	C₂²×C₂₅₁	—	4	2	no	—
105	C₂²×C₂₅₁	—	4	2	no	—
106	C₂²×C₂₅₁	—	4	2	no	—
107	C₂²×C₂₅₁	—	4	2	no	—
108	C₂²×C₂₅₁	—	4	2	no	—
109	C₂²×C₂₅₁	—	4	2	no	—
110	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
111	C₂²×C₂₅₁	C₂×#41	1	2	yes	—
112	C₂²×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂²×C₂₅₁	H(5,𝔽₂)	1	2	yes	—
114	C₂²×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
115	C₄×C₂₅₁	—	2	4	no	—
116	C₄×C₂₅₁	—	2	4	no	—
117	C₄×C₂₅₁	—	4	2	no	—
118	C₄×C₂₅₁	—	4	2	no	—
119	C₄×C₂₅₁	—	4	1	no	—
120	C₄×C₂₅₁	—	4	2	no	—
121	C₄×C₂₅₁	—	4	1	no	—
122	C₂×C₂₅₁	—	4	2	no	—
123	C₂×C₂₅₁	—	8	2	no	—
124	C₂×C₂₅₁	#537	1	1	yes	characteristic
125	C₂×C₂₅₁	#542	1	2	yes	—
126	C₂×C₂₅₁	#546	1	2	yes	—
127	C₂×C₂₅₁	#559	1	1	yes	characteristic
128	C₂×C₂₅₁	H(3,𝔽₄)	1	1	yes	characteristic
129	C₂₅₁	#2712	1	1	yes	characteristic, direct product factor, Sylow
130	#2712	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×#124	C₂×C₂₅₁	1	2	yes	complemented
132	#397	C₂×C₂₅₁	1	1	yes	characteristic, complemented
133	#394	C₂×C₂₅₁	1	2	yes	complemented
134	#393	C₂×C₂₅₁	1	2	yes	complemented
135	#392	C₂×C₂₅₁	1	1	yes	characteristic
136	#391	C₂×C₂₅₁	1	2	yes	complemented
137	#388	C₂×C₂₅₁	1	2	yes	complemented
138	#386	C₂×C₂₅₁	1	2	yes	complemented
139	#385	C₂×C₂₅₁	1	1	yes	characteristic, complemented
140	C₂²×Q₈	C₂²×C₂₅₁	1	2	yes	—
141	C₂²×D₄	C₂²×C₂₅₁	1	2	yes	—
142	C₂²×D₄	C₂²×C₂₅₁	1	1	yes	characteristic
143	C₂³×C₄	C₂²×C₂₅₁	1	1	yes	characteristic
144	#124	—	2	8	no	—
145	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
147	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	2	yes	—
148	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	2	yes	—
149	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
152	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	2	yes	—
154	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
155	C₂×C₄²	C₂²×C₂₅₁	1	2	yes	—
156	#95	C₂²×C₂₅₁	1	2	yes	—
157	#95	C₂²×C₂₅₁	1	2	yes	—
158	#95	C₂²×C₂₅₁	1	2	yes	—
159	#95	C₂²×C₂₅₁	1	1	yes	characteristic
160	#95	C₂²×C₂₅₁	1	2	yes	—
161	#95	C₂²×C₂₅₁	1	1	yes	characteristic
162	C₂⁴	C₂³×C₂₅₁	1	2	yes	—
163	C₂⁴	C₂³×C₂₅₁	1	1	yes	characteristic
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	2	4	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	—	2	4	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	4	1	no	—
171	C₂×D₄	—	4	2	no	—
172	C₂²×C₄	—	4	1	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
176	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
177	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
179	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
180	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₂²⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	4	2	no	—
188	C₂²⋊C₄	—	4	1	no	—
189	C₂²⋊C₄	—	4	2	no	—
190	C₂²⋊C₄	D₄×C₂₅₁	1	4	yes	complemented
191	C₄²	—	2	4	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	C₂×D₄×C₂₅₁	1	2	yes	—
204	C₂³	C₂⁴×C₂₅₁	1	1	yes	characteristic
205	Q₈	—	4	8	no	—
206	D₄	—	4	8	no	—
207	D₄	—	4	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	C₂×D₄×C₂₅₁	1	4	yes	—
227	C₂×C₄	#41×C₂₅₁	1	4	yes	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	1	no	—
232	C₂²	—	4	1	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	C₂²×D₄×C₂₅₁	1	1	yes	characteristic
241	C₂²	C₂×#41×C₂₅₁	1	2	yes	—
242	C₂²	H(5,𝔽₂)×C₂₅₁	1	1	yes	characteristic
243	C₂²	H(5,𝔽₂)×C₂₅₁	1	2	yes	—
244	C₂²	2₋¹⁺⁴×C₂₅₁	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	1	no	—
252	C₂	—	4	2	no	—
253	C₂	—	8	2	no	—
254	C₂	#537×C₂₅₁	1	1	yes	characteristic
255	C₂	#542×C₂₅₁	1	2	yes	—
256	C₂	#546×C₂₅₁	1	2	yes	—
257	C₂	#559×C₂₅₁	1	1	yes	characteristic
258	C₂	H(3,𝔽₄)×C₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₅₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₅₁
prime radical	#392×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×#124×C₂₅₁	C₂	semidirect
2. #397×C₂₅₁	C₂	semidirect
3. #394×C₂₅₁	C₂	semidirect
4. #393×C₂₅₁	C₂	semidirect
6. #391×C₂₅₁	C₂	semidirect
7. #388×C₂₅₁	C₂	semidirect
8. #386×C₂₅₁	C₂	semidirect
9. #385×C₂₅₁	C₂	semidirect
60. C₂²⋊C₄×C₂₅₁	D₄	semidirect
129. C₂₅₁	#2712	direct (N × H)
130. #2712	C₂₅₁	direct (N × H)
131. C₂×#124	C₂×C₂₅₁	semidirect
132. #397	C₂×C₂₅₁	semidirect
133. #394	C₂×C₂₅₁	semidirect
134. #393	C₂×C₂₅₁	semidirect
136. #391	C₂×C₂₅₁	semidirect
137. #388	C₂×C₂₅₁	semidirect
138. #386	C₂×C₂₅₁	semidirect
139. #385	C₂×C₂₅₁	semidirect
190. C₂²⋊C₄	D₄×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#124×C₂₅₁	2
#397×C₂₅₁	2
#394×C₂₅₁	2
#393×C₂₅₁	2
#392×C₂₅₁	2
#391×C₂₅₁	2
#388×C₂₅₁	2
#386×C₂₅₁	2
#385×C₂₅₁	2
#2712	251

Subgroups — #2712×C251 #44547299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2712×C₂₅₁ #44547299