Subgroups — #1864×C₂₅₁ #44546451

Order 32128.

362 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#106×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#102×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#102×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	#389×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#340×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#336×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₂₅₁	—	2	1	no	—
10	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
12	C₄×D₄×C₂₅₁	—	2	2	no	—
13	C₄×D₄×C₂₅₁	—	2	2	no	—
14	C₂×C₄⋊C₄×C₂₅₁	—	2	1	no	—
15	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₂₅₁	—	2	1	no	—
19	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
20	#106×C₂₅₁	—	2	2	no	—
21	#102×C₂₅₁	—	2	2	no	—
22	#102×C₂₅₁	C₄	1	4	yes	complemented
23	#95×C₂₅₁	—	2	1	no	—
24	C₂⁴×C₂₅₁	—	2	1	no	—
25	C₂×D₄×C₂₅₁	—	2	1	no	—
26	C₂×D₄×C₂₅₁	—	2	1	no	—
27	C₂×D₄×C₂₅₁	—	2	1	no	—
28	C₂×D₄×C₂₅₁	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₂₅₁	D₄	1	2	yes	—
30	C₂×D₄×C₂₅₁	Q₈	1	2	yes	—
31	C₂²×C₄×C₂₅₁	—	2	4	no	—
32	C₂²×C₄×C₂₅₁	—	2	1	no	—
33	C₂²×C₄×C₂₅₁	—	2	1	no	—
34	C₂²×C₄×C₂₅₁	—	2	1	no	—
35	C₂²×C₄×C₂₅₁	—	4	2	no	—
36	C₂²×C₄×C₂₅₁	—	4	2	no	—
37	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₂₅₁	—	2	1	no	—
41	C₂×C₈×C₂₅₁	—	2	1	no	—
42	C₂×C₈×C₂₅₁	—	2	1	no	—
43	C₂×C₈×C₂₅₁	—	2	1	no	—
44	C₂×C₈×C₂₅₁	—	2	1	no	—
45	C₂×C₈×C₂₅₁	C₂×C₄	1	2	yes	complemented
46	C₄⋊C₄×C₂₅₁	—	2	1	no	—
47	C₄⋊C₄×C₂₅₁	—	2	1	no	—
48	C₄⋊C₄×C₂₅₁	—	2	1	no	—
49	C₄⋊C₄×C₂₅₁	—	2	1	no	—
50	C₄⋊C₄×C₂₅₁	—	4	2	no	—
51	C₄⋊C₄×C₂₅₁	C₂×C₄	1	2	yes	complemented
52	C₄⋊C₄×C₂₅₁	D₄	1	2	yes	—
53	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
54	C₄²×C₂₅₁	—	2	1	no	—
55	C₄²×C₂₅₁	—	2	1	no	—
56	C₂³×C₂₅₁	—	2	2	no	—
57	C₂³×C₂₅₁	—	2	2	no	—
58	C₂³×C₂₅₁	—	2	2	no	—
59	C₂³×C₂₅₁	—	4	1	no	—
60	C₂³×C₂₅₁	—	4	1	no	—
61	C₂³×C₂₅₁	—	4	1	no	—
62	C₂³×C₂₅₁	—	4	1	no	—
63	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
64	D₄×C₂₅₁	—	2	2	no	—
65	D₄×C₂₅₁	—	2	2	no	—
66	D₄×C₂₅₁	—	2	2	no	—
67	D₄×C₂₅₁	C₄⋊C₄	1	4	yes	complemented
68	C₂×C₄×C₂₅₁	—	2	1	no	—
69	C₂×C₄×C₂₅₁	—	2	1	no	—
70	C₂×C₄×C₂₅₁	—	2	2	no	—
71	C₂×C₄×C₂₅₁	—	2	1	no	—
72	C₂×C₄×C₂₅₁	—	2	2	no	—
73	C₂×C₄×C₂₅₁	—	2	1	no	—
74	C₂×C₄×C₂₅₁	—	2	1	no	—
75	C₂×C₄×C₂₅₁	—	4	8	no	—
76	C₂×C₄×C₂₅₁	—	4	1	no	—
77	C₂×C₄×C₂₅₁	—	4	1	no	—
78	C₂×C₄×C₂₅₁	—	4	1	no	—
79	C₂×C₄×C₂₅₁	—	4	1	no	—
80	C₂×C₄×C₂₅₁	—	4	1	no	—
81	C₂×C₄×C₂₅₁	—	4	1	no	—
82	C₂×C₄×C₂₅₁	—	4	1	no	—
83	C₂×C₄×C₂₅₁	SD₁₆	1	2	yes	—
84	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₂₅₁	#41	1	1	yes	characteristic
89	C₂×C₄×C₂₅₁	#41	1	1	yes	characteristic
90	C₈×C₂₅₁	—	2	2	no	—
91	C₈×C₂₅₁	—	4	1	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	2	no	—
97	C₂²×C₂₅₁	—	4	2	no	—
98	C₂²×C₂₅₁	—	4	2	no	—
99	C₂²×C₂₅₁	C₄×D₄	1	1	yes	characteristic
100	C₂²×C₂₅₁	#120	1	1	yes	characteristic
101	C₂²×C₂₅₁	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂²×C₂₅₁	C₂×SD₁₆	1	1	yes	characteristic
103	C₂²×C₂₅₁	#135	1	1	yes	characteristic
104	C₂²×C₂₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂²×C₂₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₄×C₂₅₁	—	2	2	no	—
107	C₄×C₂₅₁	—	4	2	no	—
108	C₄×C₂₅₁	—	4	1	no	—
109	C₄×C₂₅₁	—	4	2	no	—
110	C₄×C₂₅₁	—	8	1	no	—
111	C₄×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₄×C₂₅₁	C₄×D₄	1	1	yes	characteristic
113	C₄×C₂₅₁	C₂²⋊Q₈	1	1	yes	characteristic
114	C₄×C₂₅₁	#123	1	1	yes	characteristic
115	C₂×C₂₅₁	—	4	4	no	—
116	C₂×C₂₅₁	#385	1	1	yes	characteristic
117	C₂×C₂₅₁	C₄×SD₁₆	1	1	yes	characteristic
118	C₂×C₂₅₁	#442	1	1	yes	characteristic
119	C₂×C₂₅₁	#449	1	1	yes	characteristic
120	C₂×C₂₅₁	#450	1	1	yes	characteristic
121	C₂×C₂₅₁	#476	1	1	yes	characteristic
122	C₂×C₂₅₁	#478	1	1	yes	characteristic
123	C₂₅₁	#1864	1	1	yes	characteristic, direct product factor, Sylow
124	#1864	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₄×D₄	C₂×C₂₅₁	1	1	yes	characteristic
126	C₂×#106	C₂×C₂₅₁	1	1	yes	characteristic, complemented
127	C₂×#102	C₂×C₂₅₁	1	1	yes	characteristic
128	C₂×#102	C₂×C₂₅₁	1	1	yes	characteristic
129	#389	C₂×C₂₅₁	1	1	yes	characteristic, complemented
130	#340	C₂×C₂₅₁	1	1	yes	characteristic, complemented
131	#336	C₂×C₂₅₁	1	1	yes	characteristic, complemented
132	C₂²×D₄	C₂²×C₂₅₁	1	1	yes	characteristic
133	C₂³×C₄	—	2	1	no	—
134	C₂²×C₈	C₂²×C₂₅₁	1	1	yes	characteristic
135	C₂²×C₈	C₂²×C₂₅₁	1	1	yes	characteristic
136	C₄×D₄	—	2	2	no	—
137	C₄×D₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
142	C₂×C₂²⋊C₄	—	2	1	no	—
143	C₂×C₄²	C₂²×C₂₅₁	1	1	yes	characteristic
144	#106	—	2	2	no	—
145	#102	—	2	2	no	—
146	#102	C₄×C₂₅₁	1	4	yes	complemented
147	#95	—	2	1	no	—
148	C₂⁴	—	2	1	no	—
149	C₂×D₄	—	2	1	no	—
150	C₂×D₄	—	2	1	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	C₂×C₄×C₂₅₁	1	2	yes	—
153	C₂×D₄	D₄×C₂₅₁	1	2	yes	—
154	C₂×D₄	Q₈×C₂₅₁	1	2	yes	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	D₄×C₂₅₁	1	1	yes	characteristic
162	C₂²×C₄	D₄×C₂₅₁	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	C₂×C₄×C₂₅₁	1	2	yes	complemented
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	C₂×C₄×C₂₅₁	1	2	yes	complemented
176	C₄⋊C₄	D₄×C₂₅₁	1	2	yes	—
177	C₂²⋊C₄	—	2	4	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	2	1	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	4	1	no	—
184	C₂³	—	4	1	no	—
185	C₂³	—	4	1	no	—
186	C₂³	—	4	1	no	—
187	C₂³	C₂×D₄×C₂₅₁	1	1	yes	characteristic
188	D₄	—	2	2	no	—
189	D₄	—	2	2	no	—
190	D₄	—	2	2	no	—
191	D₄	C₄⋊C₄×C₂₅₁	1	4	yes	complemented
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	8	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	SD₁₆×C₂₅₁	1	2	yes	—
208	C₂×C₄	C₂²×C₄×C₂₅₁	1	1	yes	characteristic
209	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
210	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
211	C₂×C₄	C₂×Q₈×C₂₅₁	1	1	yes	characteristic
212	C₂×C₄	#41×C₂₅₁	1	1	yes	characteristic
213	C₂×C₄	#41×C₂₅₁	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	4	1	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	C₄×D₄×C₂₅₁	1	1	yes	characteristic
224	C₂²	#120×C₂₅₁	1	1	yes	characteristic
225	C₂²	C₂²⋊Q₈×C₂₅₁	1	1	yes	characteristic
226	C₂²	C₂×SD₁₆×C₂₅₁	1	1	yes	characteristic
227	C₂²	#135×C₂₅₁	1	1	yes	characteristic
228	C₂²	AGL(1,ℤ/8)×C₂₅₁	1	1	yes	characteristic
229	C₂²	AGL(1,ℤ/8)×C₂₅₁	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	8	1	no	—
235	C₄	C₂×C₄⋊C₄×C₂₅₁	1	1	yes	characteristic
236	C₄	C₄×D₄×C₂₅₁	1	1	yes	characteristic
237	C₄	C₂²⋊Q₈×C₂₅₁	1	1	yes	characteristic
238	C₄	#123×C₂₅₁	1	1	yes	characteristic
239	C₂	—	4	4	no	—
240	C₂	#385×C₂₅₁	1	1	yes	characteristic
241	C₂	C₄×SD₁₆×C₂₅₁	1	1	yes	characteristic
242	C₂	#442×C₂₅₁	1	1	yes	characteristic
243	C₂	#449×C₂₅₁	1	1	yes	characteristic
244	C₂	#450×C₂₅₁	1	1	yes	characteristic
245	C₂	#476×C₂₅₁	1	1	yes	characteristic
246	C₂	#478×C₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₅₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₅₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₅₁
prime radical	C₂²×D₄×C₂₅₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₅₁ ◁ C₂×C₄²×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#106×C₂₅₁	C₂	semidirect
5. #389×C₂₅₁	C₂	semidirect
6. #340×C₂₅₁	C₂	semidirect
7. #336×C₂₅₁	C₂	semidirect
22. #102×C₂₅₁	C₄	semidirect
45. C₂×C₈×C₂₅₁	C₂×C₄	semidirect
51. C₄⋊C₄×C₂₅₁	C₂×C₄	semidirect
67. D₄×C₂₅₁	C₄⋊C₄	semidirect
123. C₂₅₁	#1864	direct (N × H)
124. #1864	C₂₅₁	direct (N × H)
126. C₂×#106	C₂×C₂₅₁	semidirect
129. #389	C₂×C₂₅₁	semidirect
130. #340	C₂×C₂₅₁	semidirect
131. #336	C₂×C₂₅₁	semidirect
146. #102	C₄×C₂₅₁	semidirect
169. C₂×C₈	C₂×C₄×C₂₅₁	semidirect
175. C₄⋊C₄	C₂×C₄×C₂₅₁	semidirect
191. D₄	C₄⋊C₄×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₂₅₁	2
C₂×#106×C₂₅₁	2
C₂×#102×C₂₅₁	2
#389×C₂₅₁	2
#340×C₂₅₁	2
#336×C₂₅₁	2
#1864	251

Subgroups — #1864×C251 #44546451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1864×C₂₅₁ #44546451