Subgroups — #44553112

Order 32128.

460 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548714	C₂	1	1	yes	maximal, characteristic
2	#30631592	C₂	1	2	yes	maximal
3	#30631591	C₂	1	2	yes	maximal
4	#30631585	C₂	1	1	yes	maximal, characteristic
5	#30631569	C₂	1	2	yes	maximal
6	#30631568	C₂	1	2	yes	maximal
7	#30631552	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
9	C₄⋊C₄×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
10	#30631533	C₂	1	1	yes	maximal, characteristic
11	#389×C₂₅₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₅₀₂	C₂²	1	2	yes	—
13	#22548714	C₄	1	8	yes	complemented
14	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
17	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
19	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
20	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
22	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
23	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
24	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
25	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
26	C₂₅₁⋊#95	C₂²	1	2	yes	—
27	C₂₅₁⋊#95	C₂²	1	2	yes	—
28	C₂₅₁⋊#95	C₂²	1	2	yes	—
29	C₂₅₁⋊#95	C₂²	1	2	yes	—
30	C₂₅₁⋊#95	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
33	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
35	#95×C₂₅₁	C₂²	1	2	yes	—
36	C₂×Dic₅₀₂	—	4	2	no	—
37	C₂×Dic₅₀₂	—	4	2	no	—
38	C₂×Dic₅₀₂	C₂×C₄	1	8	yes	—
39	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
40	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
41	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
42	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
43	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
44	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
45	C₄⋊Dic₂₅₁	—	2	2	no	—
46	C₄⋊Dic₂₅₁	C₂×C₄	1	4	yes	—
47	Dic₂₅₁⋊C₄	—	4	2	no	—
48	Dic₂₅₁⋊C₄	C₂×C₄	1	8	yes	—
49	C₄×Dic₂₅₁	—	4	2	no	—
50	C₄×Dic₂₅₁	—	4	2	no	—
51	C₄×Dic₂₅₁	C₂×C₄	1	4	yes	—
52	C₄×Dic₂₅₁	Q₈	1	4	yes	complemented
53	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₂₅₁	—	2	2	no	—
59	C₄⋊C₄×C₂₅₁	—	4	2	no	—
60	C₄⋊C₄×C₂₅₁	C₂×C₄	1	4	yes	—
61	C₄²×C₂₅₁	—	2	2	no	—
62	Dic₅₀₂	—	4	8	no	—
63	C₂×Dic₂₅₁	—	2	1	no	—
64	C₂×Dic₂₅₁	—	2	1	no	—
65	C₂×Dic₂₅₁	—	2	2	no	—
66	C₂×Dic₂₅₁	—	2	2	no	—
67	C₂×Dic₂₅₁	—	2	1	no	—
68	C₂×Dic₂₅₁	—	2	1	no	—
69	C₂×Dic₂₅₁	—	2	1	no	—
70	C₂×Dic₂₅₁	—	2	1	no	—
71	C₂×Dic₂₅₁	—	2	1	no	—
72	C₂×Dic₂₅₁	—	2	1	no	—
73	C₂×Dic₂₅₁	—	2	2	no	—
74	C₂×Dic₂₅₁	—	2	2	no	—
75	C₂×Dic₂₅₁	—	2	2	no	—
76	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
77	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
78	C₂×Dic₂₅₁	C₂²×C₄	1	4	yes	—
79	C₂×Dic₂₅₁	C₂×Q₈	1	2	yes	—
80	C₂×Dic₂₅₁	C₂×Q₈	1	2	yes	—
81	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
82	C₂×C₄×C₂₅₁	—	2	2	no	—
83	C₂×C₄×C₂₅₁	—	2	2	no	—
84	C₂×C₄×C₂₅₁	—	2	1	no	—
85	C₂×C₄×C₂₅₁	—	2	1	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	1	no	—
90	C₂×C₄×C₂₅₁	—	2	2	no	—
91	C₂×C₄×C₂₅₁	—	2	1	no	—
92	C₂×C₄×C₂₅₁	—	2	1	no	—
93	C₂×C₄×C₂₅₁	C₂²×C₄	1	4	yes	—
94	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
96	C₂×C₄×C₂₅₁	#41	1	2	yes	—
97	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
98	C₂²×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
99	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
100	C₂²×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂²×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂²×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
103	C₂²×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
104	C₄×C₂₅₁	—	2	4	no	—
105	C₄×C₂₅₁	—	4	1	no	—
106	C₄×C₂₅₁	—	4	2	no	—
107	C₄×C₂₅₁	—	4	1	no	—
108	C₄×C₂₅₁	C₄×Q₈	1	4	yes	—
109	Dic₂₅₁	—	2	4	no	—
110	Dic₂₅₁	—	2	2	no	—
111	Dic₂₅₁	—	2	2	no	—
112	Dic₂₅₁	—	2	2	no	—
113	Dic₂₅₁	—	2	2	no	—
114	Dic₂₅₁	—	4	2	no	—
115	C₂×C₂₅₁	C₂×C₄×Q₈	1	1	yes	characteristic
116	C₂×C₂₅₁	#518	1	1	yes	characteristic
117	C₂×C₂₅₁	#519	1	1	yes	characteristic
118	C₂×C₂₅₁	#542	1	1	yes	characteristic
119	C₂×C₂₅₁	#542	1	1	yes	characteristic
120	C₂×C₂₅₁	#544	1	1	yes	characteristic
121	C₂×C₂₅₁	#544	1	1	yes	characteristic
122	C₂₅₁	#2331	1	1	yes	characteristic, complemented, Sylow
123	#2331	—	251	1	no	maximal, Sylow
124	C₂×C₄⋊Q₈	—	251	1	no	—
125	#391	—	251	2	no	—
126	#391	—	251	2	no	—
127	#389	—	251	1	no	—
128	#389	D₂₅₁	1	1	yes	characteristic
129	#387	—	251	2	no	—
130	#387	—	251	2	no	—
131	#382	—	251	1	no	—
132	C₄×C₄⋊C₄	—	251	1	no	—
133	C₄×C₄⋊C₄	—	251	1	no	—
134	#376	—	251	1	no	—
135	C₂²×Q₈	—	251	2	no	—
136	C₄⋊Q₈	—	251	8	no	—
137	C₂×C₄⋊C₄	—	251	2	no	—
138	C₂×C₄⋊C₄	—	251	1	no	—
139	C₂×C₄⋊C₄	—	251	1	no	—
140	C₂×C₄⋊C₄	—	251	2	no	—
141	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂×D₂₅₁	1	2	yes	—
143	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
144	C₂×C₄²	—	251	2	no	—
145	C₂×C₄²	—	251	2	no	—
146	C₂×C₄²	—	251	1	no	—
147	C₂×C₄²	—	251	1	no	—
148	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
149	#95	—	251	1	no	—
150	#95	—	251	1	no	—
151	#95	—	251	1	no	—
152	#95	—	251	1	no	—
153	#95	—	251	2	no	—
154	#95	—	251	2	no	—
155	#95	—	251	2	no	—
156	#95	—	251	2	no	—
157	#95	—	251	2	no	—
158	#95	C₂×D₂₅₁	1	2	yes	—
159	C₂×Q₈	—	1004	2	no	—
160	C₂×Q₈	—	1004	2	no	—
161	C₂×Q₈	—	251	8	no	—
162	C₂²×C₄	—	251	2	no	—
163	C₂²×C₄	—	251	1	no	—
164	C₂²×C₄	—	251	1	no	—
165	C₂²×C₄	—	251	1	no	—
166	C₂²×C₄	—	251	1	no	—
167	C₂²×C₄	—	251	2	no	—
168	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
170	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
172	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	1004	2	no	—
176	C₄⋊C₄	—	502	2	no	—
177	C₄⋊C₄	—	251	8	no	—
178	C₄⋊C₄	—	251	4	no	—
179	C₄⋊C₄	C₄×D₂₅₁	1	4	yes	—
180	C₄²	—	2	2	no	—
181	C₄²	—	1004	2	no	—
182	C₄²	—	1004	2	no	—
183	C₄²	—	251	4	no	—
184	C₄²	—	251	4	no	—
185	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
186	Q₈	—	1004	8	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	502	2	no	—
199	C₂×C₄	—	502	2	no	—
200	C₂×C₄	—	502	2	no	—
201	C₂×C₄	—	502	1	no	—
202	C₂×C₄	—	502	2	no	—
203	C₂×C₄	—	502	1	no	—
204	C₂×C₄	—	502	1	no	—
205	C₂×C₄	—	502	2	no	—
206	C₂×C₄	—	502	1	no	—
207	C₂×C₄	—	502	1	no	—
208	C₂×C₄	—	502	1	no	—
209	C₂×C₄	—	502	1	no	—
210	C₂×C₄	—	502	1	no	—
211	C₂×C₄	—	251	2	no	—
212	C₂×C₄	—	251	2	no	—
213	C₂×C₄	—	251	4	no	—
214	C₂×C₄	—	251	2	no	—
215	C₂×C₄	—	251	2	no	—
216	C₂×C₄	C₂×C₄×D₂₅₁	1	4	yes	—
217	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
218	C₂×C₄	Q₈×D₂₅₁	1	2	yes	—
219	C₂×C₄	#16860809	1	2	yes	—
220	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
221	C₂²	C₂×Q₈×D₂₅₁	1	1	yes	characteristic
222	C₂²	C₂×#16860809	1	1	yes	characteristic
223	C₂²	#22548760	1	1	yes	characteristic
224	C₂²	#22548761	1	1	yes	characteristic
225	C₂²	#22548762	1	1	yes	characteristic
226	C₂²	#22548763	1	1	yes	characteristic
227	C₄	—	2	4	no	—
228	C₄	—	4	1	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	1	no	—
231	C₄	—	1004	2	no	—
232	C₄	—	502	2	no	—
233	C₄	—	502	2	no	—
234	C₄	—	502	2	no	—
235	C₄	—	502	4	no	—
236	C₄	—	502	2	no	—
237	C₄	#22548680	1	4	yes	—
238	C₂	C₂×#22548680	1	1	yes	characteristic
239	C₂	#30632159	1	1	yes	characteristic
240	C₂	#30632163	1	1	yes	characteristic
241	C₂	#30632304	1	1	yes	characteristic
242	C₂	#30632310	1	1	yes	characteristic
243	C₂	#30632314	1	1	yes	characteristic
244	C₂	#30632317	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator	C₂²×C₂₅₁	C₂³×C₄
Fitting	#389×C₂₅₁	C₂
socle, prime radical, cosocle	C₂³×C₂₅₁	C₂⁴
hypercenter	#389	D₂₅₁
hypocenter	C₂₅₁	#2331
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
13. #22548714	C₄	semidirect
52. C₄×Dic₂₅₁	Q₈	semidirect
122. C₂₅₁	#2331	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548714	2
#30631592	2
#30631591	2
#30631585	2
#30631569	2
#30631568	2
#30631552	2
C₄⋊C₄×Dic₂₅₁	2
#30631533	2
#389×C₂₅₁	2
#2331	251