Subgroups — #44553047

Order 32128.

556 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548694	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₅₀₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#30631593	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631585	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631582	C₂	1	2	yes	maximal, complemented
6	#30631581	C₂	1	2	yes	maximal
7	#30631575	C₂	1	2	yes	maximal
8	#30631568	C₂	1	2	yes	maximal, complemented
9	#30631547	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#117×C₂₅₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
13	#22548694	C₄	1	8	yes	complemented
14	C₄×Dic₅₀₂	—	2	4	no	—
15	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
19	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₂₅₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₂₅₁	C₂²	1	2	yes	—
23	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
24	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
25	C₂₅₁⋊#95	C₂²	1	2	yes	—
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	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
31	#117×C₂₅₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
38	C₂×Dic₅₀₂	—	2	2	no	—
39	C₂×Dic₅₀₂	—	2	2	no	—
40	C₂×Dic₅₀₂	C₂×C₄	1	4	yes	—
41	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
42	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
43	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
44	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
45	C₄⋊Dic₂₅₁	—	2	2	no	—
46	C₄⋊Dic₂₅₁	—	2	2	no	—
47	C₄⋊Dic₂₅₁	C₂×C₄	1	4	yes	—
48	Dic₂₅₁⋊C₄	—	2	4	no	—
49	Dic₂₅₁⋊C₄	—	4	2	no	—
50	Dic₂₅₁⋊C₄	C₂×C₄	1	8	yes	—
51	C₂²⋊Dic₂₅₁	—	4	2	no	—
52	C₂²⋊Dic₂₅₁	C₂×C₄	1	8	yes	—
53	C₄×Dic₂₅₁	—	2	4	no	—
54	C₄×Dic₂₅₁	—	4	2	no	—
55	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₅₁	—	2	2	no	—
57	C₂²×C₄×C₂₅₁	—	2	2	no	—
58	C₂²×C₄×C₂₅₁	C₂×C₄	1	4	yes	—
59	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₂₅₁	—	2	2	no	—
66	C₄⋊C₄×C₂₅₁	—	2	2	no	—
67	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
68	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
69	C₄²×C₂₅₁	—	2	2	no	—
70	C₄²×C₂₅₁	D₄	1	4	yes	—
71	Dic₅₀₂	—	2	8	no	—
72	C₂×Dic₂₅₁	—	2	2	no	—
73	C₂×Dic₂₅₁	—	2	2	no	—
74	C₂×Dic₂₅₁	—	2	2	no	—
75	C₂×Dic₂₅₁	—	2	2	no	—
76	C₂×Dic₂₅₁	—	2	2	no	—
77	C₂×Dic₂₅₁	—	2	2	no	—
78	C₂×Dic₂₅₁	—	2	2	no	—
79	C₂×Dic₂₅₁	—	2	2	no	—
80	C₂×Dic₂₅₁	—	2	2	no	—
81	C₂×Dic₂₅₁	—	2	2	no	—
82	C₂×Dic₂₅₁	C₂²×C₄	1	4	yes	—
83	C₂×Dic₂₅₁	C₂²×C₄	1	4	yes	—
84	C₂³×C₂₅₁	—	2	1	no	—
85	C₂³×C₂₅₁	—	2	1	no	—
86	C₂³×C₂₅₁	—	2	1	no	—
87	C₂³×C₂₅₁	—	2	1	no	—
88	C₂³×C₂₅₁	—	2	1	no	—
89	C₂³×C₂₅₁	—	2	1	no	—
90	C₂³×C₂₅₁	C₂²×C₄	1	2	yes	—
91	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
92	C₂×C₄×C₂₅₁	—	2	1	no	—
93	C₂×C₄×C₂₅₁	—	2	1	no	—
94	C₂×C₄×C₂₅₁	—	2	1	no	—
95	C₂×C₄×C₂₅₁	—	2	1	no	—
96	C₂×C₄×C₂₅₁	—	2	1	no	—
97	C₂×C₄×C₂₅₁	—	2	1	no	—
98	C₂×C₄×C₂₅₁	—	2	8	no	—
99	C₂×C₄×C₂₅₁	—	2	1	no	—
100	C₂×C₄×C₂₅₁	—	2	1	no	—
101	C₂×C₄×C₂₅₁	—	2	1	no	—
102	C₂×C₄×C₂₅₁	—	2	1	no	—
103	C₂×C₄×C₂₅₁	—	2	1	no	—
104	C₂×C₄×C₂₅₁	—	2	1	no	—
105	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₂₅₁	#41	1	2	yes	—
111	C₂²×C₂₅₁	—	2	2	no	—
112	C₂²×C₂₅₁	—	2	2	no	—
113	C₂²×C₂₅₁	—	2	2	no	—
114	C₂²×C₂₅₁	—	4	1	no	—
115	C₂²×C₂₅₁	—	4	1	no	—
116	C₂²×C₂₅₁	—	4	1	no	—
117	C₂²×C₂₅₁	—	4	1	no	—
118	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
119	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
120	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
121	C₂²×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂²×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂²×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂²×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
125	C₄×C₂₅₁	—	2	2	no	—
126	C₄×C₂₅₁	—	2	2	no	—
127	C₄×C₂₅₁	—	2	2	no	—
128	C₄×C₂₅₁	—	4	1	no	—
129	C₄×C₂₅₁	—	4	1	no	—
130	C₄×C₂₅₁	C₄×D₄	1	4	yes	—
131	Dic₂₅₁	—	2	4	no	—
132	Dic₂₅₁	—	2	4	no	—
133	Dic₂₅₁	—	4	2	no	—
134	Dic₂₅₁	—	4	2	no	—
135	C₂×C₂₅₁	—	4	2	no	—
136	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂×C₂₅₁	#519	1	1	yes	characteristic
138	C₂×C₂₅₁	#520	1	1	yes	characteristic
139	C₂×C₂₅₁	#536	1	1	yes	characteristic
140	C₂×C₂₅₁	#537	1	1	yes	characteristic
141	C₂×C₂₅₁	#541	1	1	yes	characteristic
142	C₂×C₂₅₁	#542	1	1	yes	characteristic
143	C₂₅₁	#2323	1	1	yes	characteristic, complemented, Sylow
144	#2323	—	251	1	no	maximal, Sylow
145	C₂×C₂²⋊Q₈	—	251	1	no	—
146	C₂×C₄×Q₈	—	251	1	no	—
147	C₂×#117	D₂₅₁	1	1	yes	characteristic
148	#391	—	251	1	no	—
149	#389	—	251	2	no	—
150	#389	—	251	1	no	—
151	#388	—	251	2	no	—
152	#388	—	251	2	no	—
153	#387	—	251	2	no	—
154	#380	—	251	1	no	—
155	C₄×C₄⋊C₄	—	251	1	no	—
156	C₂²×Q₈	—	251	1	no	—
157	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
158	C₂²⋊Q₈	—	251	8	no	—
159	C₄×Q₈	—	502	4	no	—
160	#117	—	2	4	no	—
161	C₂×C₄⋊C₄	—	251	2	no	—
162	C₂×C₄⋊C₄	—	251	1	no	—
163	C₂×C₄⋊C₄	—	251	2	no	—
164	C₂×C₄⋊C₄	—	251	1	no	—
165	C₂×C₄⋊C₄	—	251	1	no	—
166	C₂×C₄⋊C₄	—	251	2	no	—
167	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
168	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	—	251	2	no	—
170	C₂×C₂²⋊C₄	—	251	2	no	—
171	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
173	C₂×C₄²	—	251	2	no	—
174	C₂×C₄²	—	251	2	no	—
175	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
176	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
177	#95	—	251	2	no	—
178	#95	—	251	2	no	—
179	#95	—	251	2	no	—
180	#95	—	251	2	no	—
181	#95	—	251	2	no	—
182	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
183	C₂×Q₈	—	502	2	no	—
184	C₂×Q₈	—	502	2	no	—
185	C₂×Q₈	—	251	4	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	251	2	no	—
189	C₂²×C₄	—	251	2	no	—
190	C₂²×C₄	—	251	2	no	—
191	C₂²×C₄	—	251	2	no	—
192	C₂²×C₄	C₄×D₂₅₁	1	4	yes	—
193	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
194	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	1004	2	no	—
202	C₄⋊C₄	—	502	4	no	—
203	C₄⋊C₄	—	502	2	no	—
204	C₄⋊C₄	—	502	2	no	—
205	C₄⋊C₄	—	251	8	no	—
206	C₄⋊C₄	—	251	4	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	1004	2	no	—
210	C₂²⋊C₄	—	251	8	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	1004	2	no	—
213	C₄²	—	502	4	no	—
214	C₄²	C₂²⋊D₂₅₁	1	4	yes	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	C₂×C₄×D₂₅₁	1	2	yes	—
222	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
223	Q₈	—	502	8	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	502	2	no	—
238	C₂×C₄	—	502	2	no	—
239	C₂×C₄	—	502	2	no	—
240	C₂×C₄	—	502	2	no	—
241	C₂×C₄	—	502	2	no	—
242	C₂×C₄	—	502	2	no	—
243	C₂×C₄	—	502	2	no	—
244	C₂×C₄	—	502	2	no	—
245	C₂×C₄	—	502	2	no	—
246	C₂×C₄	—	502	2	no	—
247	C₂×C₄	—	251	4	no	—
248	C₂×C₄	—	251	4	no	—
249	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
250	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
251	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
252	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
253	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
254	C₂×C₄	#16860810	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
263	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
264	C₂²	C₂×#16860810	1	1	yes	characteristic
265	C₂²	#22548761	1	1	yes	characteristic
266	C₂²	#22548762	1	1	yes	characteristic
267	C₂²	#22548762	1	1	yes	characteristic
268	C₂²	#22548762	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	1004	2	no	—
275	C₄	—	1004	2	no	—
276	C₄	—	502	4	no	—
277	C₄	—	502	4	no	—
278	C₄	C₄×C₂²⋊D₂₅₁	1	4	yes	—
279	C₂	—	4	2	no	—
280	C₂	C₂×C₄×C₂²⋊D₂₅₁	1	1	yes	characteristic
281	C₂	#30632164	1	1	yes	characteristic
282	C₂	#30632167	1	1	yes	characteristic
283	C₂	#30632258	1	1	yes	characteristic
284	C₂	#30632265	1	1	yes	characteristic
285	C₂	#30632299	1	1	yes	characteristic
286	C₂	#30632308	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator	C₂²×C₂₅₁	C₂³×C₄
Fitting	C₂×#117×C₂₅₁	C₂
socle, cosocle	C₂³×C₂₅₁	C₂⁴
prime radical	C₂⁴×C₂₅₁	C₂³
hypercenter	C₂×#117	D₂₅₁
hypocenter	C₂₅₁	#2323
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₅₀₂	C₂	semidirect
3. #30631593	C₂	semidirect
4. #30631585	C₂	semidirect
5. #30631582	C₂	semidirect
8. #30631568	C₂	semidirect
10. C₄×C₄⋊Dic₂₅₁	C₂	semidirect
13. #22548694	C₄	semidirect
143. C₂₅₁	#2323	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548694	2
C₂×C₄×Dic₅₀₂	2
#30631593	2
#30631585	2
#30631582	2
#30631581	2
#30631575	2
#30631568	2
#30631547	2
C₄×C₄⋊Dic₂₅₁	2
C₂×#117×C₂₅₁	2
#2323	251