Subgroups — #44559472

Order 32128.

432 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#16860809	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₅₀₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#30631811	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₂₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₂²⋊D₂₅₁	C₂	1	2	yes	maximal, complemented
6	(C₂²⋊D₂₅₁)⋊C₈	C₂	1	2	yes	maximal, complemented
7	#30631738	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631737	C₂	1	2	yes	maximal, complemented
9	C₂×Dic₂₅₁⋊C₈	C₂	1	2	yes	maximal, complemented
10	C₄⋊C₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16860809	C₄	1	1	yes	characteristic
13	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
14	C₂×C₈×D₂₅₁	C₂²	1	2	yes	—
15	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
16	#22548680	C₄	1	1	yes	characteristic
17	D₄×Dic₂₅₁	C₄	1	1	yes	characteristic
18	C₄×C₂²⋊D₂₅₁	C₂²	1	2	yes	—
19	(C₂²⋊D₂₅₁)⋊C₄	C₄	1	2	yes	—
20	#22548673	C₄	1	1	yes	characteristic
21	#22548671	C₄	1	2	yes	—
22	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
23	C₄²×D₂₅₁	C₂²	1	1	yes	characteristic
24	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
25	Dic₂₅₁⋊C₈	C₂²	1	2	yes	—
26	Dic₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
27	Dic₂₅₁⋊C₈	C₂²	1	2	yes	—
28	Dic₂₅₁⋊C₈	C₂²	1	2	yes	—
29	Dic₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
30	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
31	(C₂×D₂₅₁)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₂₅₁)⋊C₈	C₂²	1	1	yes	characteristic
33	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
34	C₈×Dic₂₅₁	C₂²	1	2	yes	—
35	C₈×Dic₂₅₁	C₂²	1	2	yes	—
36	C₂²×C₈×C₂₅₁	C₂²	1	2	yes	—
37	C₄×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₂₅₁	C₂²	1	2	yes	—
40	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
41	#16860809	C₈	1	8	yes	complemented
42	C₂×Dic₅₀₂	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₂₅₁	C₂×C₄	1	2	yes	—
44	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
45	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₂₅₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₂₅₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
49	C₂×C₂₅₁⋊C₈	—	4	2	no	—
50	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
51	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
52	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
53	C₈×D₂₅₁	—	4	2	no	—
54	C₄⋊Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
55	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
56	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
58	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
59	C₄×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₂₅₁	C₂×C₄	1	2	yes	—
62	C₄×Dic₂₅₁	C₂×C₄	1	2	yes	—
63	C₄×Dic₂₅₁	C₂³	1	2	yes	—
64	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
67	C₂×C₈×C₂₅₁	—	4	2	no	—
68	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
71	C₄⋊C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₂₅₁	C₂×C₄	1	2	yes	—
73	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
74	C₂²×D₂₅₁	C₂²×C₄	1	1	yes	characteristic
75	Dic₅₀₂	C₂×C₈	1	4	yes	complemented
76	C₂²⋊D₂₅₁	C₂×C₈	1	8	yes	complemented
77	C₂×Dic₂₅₁	—	4	2	no	—
78	C₂×Dic₂₅₁	C₂×C₈	1	8	yes	complemented
79	C₂×Dic₂₅₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₂₅₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
82	C₂×Dic₂₅₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
84	C₄×D₂₅₁	—	2	1	no	—
85	C₄×D₂₅₁	—	2	1	no	—
86	C₄×D₂₅₁	—	4	1	no	—
87	C₄×D₂₅₁	C₂×C₈	1	4	yes	complemented
88	C₂₅₁⋊C₈	—	2	2	no	—
89	C₂₅₁⋊C₈	—	2	1	no	—
90	C₂₅₁⋊C₈	—	2	1	no	—
91	C₂³×C₂₅₁	C₂²×C₄	1	2	yes	—
92	D₄×C₂₅₁	C₂×C₈	1	4	yes	complemented
93	C₂×C₄×C₂₅₁	—	2	2	no	—
94	C₂×C₄×C₂₅₁	—	2	2	no	—
95	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
99	C₈×C₂₅₁	—	2	1	no	—
100	C₈×C₂₅₁	—	2	2	no	—
101	C₈×C₂₅₁	—	2	1	no	—
102	C₂²×C₂₅₁	—	2	2	no	—
103	C₂²×C₂₅₁	—	2	2	no	—
104	C₂²×C₂₅₁	C₂²×C₈	1	4	yes	—
105	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
106	C₂×D₂₅₁	—	2	1	no	—
107	C₂×D₂₅₁	—	2	1	no	—
108	C₂×D₂₅₁	C₂²×C₈	1	2	yes	—
109	C₄×C₂₅₁	—	2	2	no	—
110	C₄×C₂₅₁	—	2	1	no	—
111	C₄×C₂₅₁	C₂²×C₈	1	2	yes	—
112	C₄×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
113	C₄×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
114	Dic₂₅₁	—	2	1	no	—
115	Dic₂₅₁	—	2	1	no	—
116	Dic₂₅₁	—	2	2	no	—
117	Dic₂₅₁	C₂²×C₈	1	2	yes	—
118	Dic₂₅₁	C₂²×C₈	1	4	yes	—
119	C₂×C₂₅₁	—	2	4	no	—
120	C₂×C₂₅₁	#520	1	1	yes	characteristic
121	C₂×C₂₅₁	C₂³×C₈	1	1	yes	characteristic
122	C₂×C₂₅₁	#568	1	1	yes	characteristic
123	D₂₅₁	—	2	2	no	—
124	C₂₅₁	#2988	1	1	yes	characteristic, complemented, Sylow
125	#2988	—	251	1	no	maximal, Sylow
126	C₄×#41	—	251	1	no	—
127	C₈×Q₈	—	251	1	no	—
128	C₈×Q₈	—	251	1	no	—
129	C₈×D₄	—	251	2	no	—
130	C₈×D₄	—	251	1	no	—
131	C₈×D₄	—	251	2	no	—
132	C₈×D₄	D₂₅₁	1	1	yes	characteristic, complemented
133	#431	—	251	2	no	—
134	#431	—	251	1	no	—
135	C₂×C₄⋊C₈	—	251	2	no	—
136	C₂×C₄⋊C₈	—	251	1	no	—
137	C₂×#41	—	251	1	no	—
138	C₂²×C₈	—	251	2	no	—
139	C₂²×C₈	—	251	2	no	—
140	C₂²×C₈	C₂×D₂₅₁	1	2	yes	—
141	C₄×Q₈	—	251	1	no	—
142	C₄×Q₈	—	251	1	no	—
143	C₄×D₄	—	251	1	no	—
144	C₄×D₄	—	251	2	no	—
145	C₄×D₄	—	251	2	no	—
146	C₄×D₄	C₂×D₂₅₁	1	1	yes	characteristic
147	#117	—	251	1	no	—
148	#117	—	251	2	no	—
149	C₂×C₄²	—	251	2	no	—
150	C₂×C₄²	—	251	1	no	—
151	C₄⋊C₈	—	251	2	no	—
152	C₄⋊C₈	—	251	1	no	—
153	C₄⋊C₈	—	251	2	no	—
154	C₄⋊C₈	—	251	2	no	—
155	C₄⋊C₈	—	251	1	no	—
156	C₄⋊C₈	—	251	1	no	—
157	C₄⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
158	C₂²⋊C₈	—	251	2	no	—
159	C₂²⋊C₈	—	251	1	no	—
160	C₂²⋊C₈	—	251	1	no	—
161	C₂²⋊C₈	C₂×D₂₅₁	1	2	yes	—
162	C₄×C₈	—	251	2	no	—
163	C₄×C₈	—	251	2	no	—
164	C₄×C₈	—	251	1	no	—
165	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
166	#41	—	251	8	no	—
167	C₂×Q₈	—	251	1	no	—
168	C₂×D₄	—	251	2	no	—
169	C₂×D₄	C₄×D₂₅₁	1	1	yes	characteristic
170	C₂²×C₄	—	251	2	no	—
171	C₂²×C₄	—	251	1	no	—
172	C₂²×C₄	—	251	1	no	—
173	C₂²×C₄	—	251	2	no	—
174	C₂²×C₄	—	251	1	no	—
175	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	1004	2	no	—
178	C₂×C₈	—	1004	2	no	—
179	C₂×C₈	—	251	1	no	—
180	C₂×C₈	—	251	2	no	—
181	C₂×C₈	—	251	1	no	—
182	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
184	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
185	C₄⋊C₄	—	251	2	no	—
186	C₄⋊C₄	—	251	2	no	—
187	C₄⋊C₄	—	251	1	no	—
188	C₄⋊C₄	C₄×D₂₅₁	1	1	yes	characteristic
189	C₂²⋊C₄	—	251	2	no	—
190	C₂²⋊C₄	—	251	2	no	—
191	C₂²⋊C₄	C₄×D₂₅₁	1	2	yes	—
192	C₄²	—	251	2	no	—
193	C₄²	—	251	1	no	—
194	C₄²	—	251	1	no	—
195	C₄²	—	251	2	no	—
196	C₄²	—	251	2	no	—
197	C₄²	—	251	1	no	—
198	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
199	C₂³	—	251	1	no	—
200	C₂³	C₂×C₄×D₂₅₁	1	2	yes	—
201	Q₈	—	251	4	no	—
202	D₄	—	251	8	no	—
203	D₄	C₈×D₂₅₁	1	4	yes	complemented
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	1004	1	no	—
207	C₂×C₄	—	1004	2	no	—
208	C₂×C₄	—	502	1	no	—
209	C₂×C₄	—	502	1	no	—
210	C₂×C₄	—	251	1	no	—
211	C₂×C₄	—	251	2	no	—
212	C₂×C₄	—	251	1	no	—
213	C₂×C₄	—	251	1	no	—
214	C₂×C₄	—	251	2	no	—
215	C₂×C₄	—	251	8	no	—
216	C₂×C₄	—	251	4	no	—
217	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	—	502	2	no	—
225	C₈	—	502	1	no	—
226	C₈	—	502	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	502	1	no	—
230	C₂²	—	502	1	no	—
231	C₂²	—	251	2	no	—
232	C₂²	C₂×C₈×D₂₅₁	1	4	yes	—
233	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	2	no	—
236	C₄	—	502	1	no	—
237	C₄	—	502	2	no	—
238	C₄	—	502	1	no	—
239	C₄	—	251	2	no	—
240	C₄	—	251	4	no	—
241	C₄	C₂×C₈×D₂₅₁	1	2	yes	—
242	C₄	#22548760	1	1	yes	characteristic
243	C₄	#22548762	1	1	yes	characteristic
244	C₂	—	2	4	no	—
245	C₂	—	502	2	no	—
246	C₂	#30632166	1	1	yes	characteristic
247	C₂	C₂²×C₈×D₂₅₁	1	1	yes	characteristic
248	C₂	#30632452	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₅₁
commutator	C₂×C₂₅₁	C₂³×C₈
Fitting	C₈×D₄×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂³×C₄
prime radical	C₂×#16860809	C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₈×D₄	D₂₅₁
hypocenter	C₂₅₁	#2988
LCS	G ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₅₀₂	C₂	semidirect
3. #30631811	C₂	semidirect
4. D₄×C₂₅₁⋊C₈	C₂	semidirect
5. C₈×C₂²⋊D₂₅₁	C₂	semidirect
6. (C₂²⋊D₂₅₁)⋊C₈	C₂	semidirect
7. #30631738	C₂	semidirect
8. #30631737	C₂	semidirect
9. C₂×Dic₂₅₁⋊C₈	C₂	semidirect
10. C₄⋊C₈×D₂₅₁	C₂	semidirect
11. C₈×D₄×C₂₅₁	C₂	semidirect
41. #16860809	C₈	semidirect
75. Dic₅₀₂	C₂×C₈	semidirect
76. C₂²⋊D₂₅₁	C₂×C₈	semidirect
78. C₂×Dic₂₅₁	C₂×C₈	semidirect
87. C₄×D₂₅₁	C₂×C₈	semidirect
92. D₄×C₂₅₁	C₂×C₈	semidirect
124. C₂₅₁	#2988	semidirect
132. C₈×D₄	D₂₅₁	semidirect
203. D₄	C₈×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#16860809	2
C₈×Dic₅₀₂	2
#30631811	2
D₄×C₂₅₁⋊C₈	2
C₈×C₂²⋊D₂₅₁	2
(C₂²⋊D₂₅₁)⋊C₈	2
#30631738	2
#30631737	2
C₂×Dic₂₅₁⋊C₈	2
C₄⋊C₈×D₂₅₁	2
C₈×D₄×C₂₅₁	2
#2988	251