Subgroups — #44553432

Order 32128.

632 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548704	C₂	1	1	yes	maximal, characteristic
2	C₂×#22548677	C₂	1	1	yes	maximal, characteristic
3	#30631594	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631589	C₂	1	1	yes	maximal, characteristic
5	#30631578	C₂	1	2	yes	maximal, complemented
6	#30631577	C₂	1	2	yes	maximal, complemented
7	#30631558	C₂	1	2	yes	maximal, complemented
8	C₄⋊C₄×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×Dic₂₅₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×(C₂×D₂₅₁)⋊C₄	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₁₀₀₄	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₂₅₁	C₂²	1	2	yes	—
14	#22548704	C₄	1	8	yes	complemented
15	#22548677	—	2	4	no	—
16	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	2	yes	—
20	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	2	yes	—
21	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	2	yes	—
22	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	2	yes	—
23	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	2	yes	—
24	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
26	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
27	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
28	C₂₅₁⋊#95	C₂²	1	2	yes	—
29	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
30	C₂₅₁⋊#95	C₂²	1	2	yes	—
31	C₂₅₁⋊#95	C₂²	1	2	yes	—
32	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₂₅₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
35	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₂₅₁	C₂²	1	2	yes	—
37	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
38	C₂³×D₂₅₁	C₂³	1	2	yes	—
39	C₂×D₁₀₀₄	—	2	2	no	—
40	C₂×D₁₀₀₄	—	2	2	no	—
41	C₂×D₁₀₀₄	C₂×C₄	1	4	yes	—
42	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
43	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
44	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
45	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
46	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₂₅₁	—	2	4	no	—
48	C₂×C₄×D₂₅₁	—	2	4	no	—
49	C₂×C₄×D₂₅₁	—	4	2	no	—
50	C₄⋊Dic₂₅₁	—	2	2	no	—
51	Dic₂₅₁⋊C₄	—	2	2	no	—
52	Dic₂₅₁⋊C₄	—	2	2	no	—
53	(C₂×D₂₅₁)⋊C₄	—	2	4	no	—
54	(C₂×D₂₅₁)⋊C₄	—	2	4	no	—
55	(C₂×D₂₅₁)⋊C₄	—	4	2	no	—
56	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	8	yes	complemented
57	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	8	yes	complemented
58	C₄×Dic₂₅₁	—	2	2	no	—
59	C₄×Dic₂₅₁	—	4	2	no	—
60	C₄×Dic₂₅₁	C₂×C₄	1	4	yes	complemented
61	C₂×Q₈×C₂₅₁	—	2	2	no	—
62	C₂×Q₈×C₂₅₁	—	2	2	no	—
63	C₂×Q₈×C₂₅₁	C₂×C₄	1	4	yes	complemented
64	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
66	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
67	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₂₅₁	—	2	4	no	—
70	C₄⋊C₄×C₂₅₁	D₄	1	4	yes	—
71	C₄²×C₂₅₁	—	2	4	no	—
72	C₄²×C₂₅₁	—	2	2	no	—
73	C₂²×D₂₅₁	—	2	2	no	—
74	C₂²×D₂₅₁	—	2	2	no	—
75	C₂²×D₂₅₁	—	2	2	no	—
76	C₂²×D₂₅₁	—	2	2	no	—
77	C₂²×D₂₅₁	—	2	2	no	—
78	C₂²×D₂₅₁	—	2	2	no	—
79	C₂²×D₂₅₁	C₂²×C₄	1	4	yes	—
80	D₁₀₀₄	—	2	8	no	—
81	C₂×Dic₂₅₁	—	2	1	no	—
82	C₂×Dic₂₅₁	—	2	1	no	—
83	C₂×Dic₂₅₁	—	2	1	no	—
84	C₂×Dic₂₅₁	—	2	1	no	—
85	C₂×Dic₂₅₁	—	2	1	no	—
86	C₂×Dic₂₅₁	—	2	1	no	—
87	C₂×Dic₂₅₁	—	2	1	no	—
88	C₂×Dic₂₅₁	—	2	2	no	—
89	C₂×Dic₂₅₁	—	2	1	no	—
90	C₂×Dic₂₅₁	—	2	2	no	—
91	C₂×Dic₂₅₁	—	2	2	no	—
92	C₂×Dic₂₅₁	—	2	1	no	—
93	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
94	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
95	C₂×Dic₂₅₁	#41	1	2	yes	—
96	C₄×D₂₅₁	—	4	8	no	—
97	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
98	Q₈×C₂₅₁	—	2	8	no	—
99	C₂×C₄×C₂₅₁	—	2	1	no	—
100	C₂×C₄×C₂₅₁	—	2	2	no	—
101	C₂×C₄×C₂₅₁	—	2	2	no	—
102	C₂×C₄×C₂₅₁	—	2	2	no	—
103	C₂×C₄×C₂₅₁	—	2	2	no	—
104	C₂×C₄×C₂₅₁	—	2	1	no	—
105	C₂×C₄×C₂₅₁	—	2	1	no	—
106	C₂×C₄×C₂₅₁	—	2	1	no	—
107	C₂×C₄×C₂₅₁	—	2	1	no	—
108	C₂×C₄×C₂₅₁	C₂²×C₄	1	4	yes	—
109	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₂₅₁	C₂×D₄	1	4	yes	—
111	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₂₅₁	#41	1	2	yes	—
113	C₂×C₄×C₂₅₁	#41	1	2	yes	—
114	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
115	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
116	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
117	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
118	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
119	C₂²×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂²×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
121	C₂×D₂₅₁	—	2	4	no	—
122	C₂×D₂₅₁	—	2	4	no	—
123	C₂×D₂₅₁	—	2	4	no	—
124	C₂×D₂₅₁	—	4	2	no	—
125	C₂×D₂₅₁	—	4	2	no	—
126	C₂×D₂₅₁	—	4	2	no	—
127	C₂×D₂₅₁	—	4	2	no	—
128	C₄×C₂₅₁	—	2	4	no	—
129	C₄×C₂₅₁	—	2	4	no	—
130	C₄×C₂₅₁	—	2	2	no	—
131	C₄×C₂₅₁	—	4	1	no	—
132	C₄×C₂₅₁	C₄×D₄	1	4	yes	—
133	Dic₂₅₁	—	2	2	no	—
134	Dic₂₅₁	—	2	2	no	—
135	Dic₂₅₁	—	2	2	no	—
136	Dic₂₅₁	—	4	1	no	—
137	Dic₂₅₁	—	4	2	no	—
138	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂×C₂₅₁	C₄×#41	1	1	yes	characteristic
140	C₂×C₂₅₁	#520	1	1	yes	characteristic
141	C₂×C₂₅₁	#546	1	1	yes	characteristic
142	C₂×C₂₅₁	#548	1	1	yes	characteristic
143	C₂×C₂₅₁	#556	1	1	yes	characteristic
144	C₂×C₂₅₁	#559	1	1	yes	characteristic
145	D₂₅₁	—	4	4	no	—
146	C₂₅₁	#2367	1	1	yes	characteristic, complemented, Sylow
147	#2367	—	251	1	no	maximal, Sylow
148	C₂×#124	—	251	1	no	—
149	C₂×C₄×Q₈	D₂₅₁	1	1	yes	characteristic, complemented
150	C₂×C₄×D₄	—	251	1	no	—
151	#391	—	251	1	no	—
152	#390	—	251	1	no	—
153	#388	—	251	2	no	—
154	#388	—	251	2	no	—
155	#385	—	251	2	no	—
156	C₄×C₄⋊C₄	—	251	1	no	—
157	C₄×C₄⋊C₄	—	251	1	no	—
158	C₄×C₂²⋊C₄	—	251	2	no	—
159	C₂²×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
160	C₂²×D₄	—	251	1	no	—
161	C₂³×C₄	—	251	2	no	—
162	#124	—	251	8	no	—
163	C₄×Q₈	—	2	4	no	—
164	C₄×D₄	—	502	4	no	—
165	C₂×C₄⋊C₄	—	251	1	no	—
166	C₂×C₄⋊C₄	—	251	1	no	—
167	C₂×C₄⋊C₄	—	251	1	no	—
168	C₂×C₄⋊C₄	C₂×D₂₅₁	1	2	yes	—
169	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	251	2	no	—
171	C₂×C₂²⋊C₄	—	251	2	no	—
172	C₂×C₂²⋊C₄	—	251	2	no	—
173	C₂×C₂²⋊C₄	—	251	2	no	—
174	C₂×C₂²⋊C₄	—	251	2	no	—
175	C₂×C₄²	—	251	2	no	—
176	C₂×C₄²	—	251	1	no	—
177	C₂×C₄²	—	251	1	no	—
178	C₂×C₄²	C₂×D₂₅₁	1	2	yes	—
179	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
180	#95	—	251	1	no	—
181	#95	—	251	2	no	—
182	#95	—	251	2	no	—
183	#95	—	251	1	no	—
184	#95	—	251	2	no	—
185	C₂⁴	—	251	2	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	C₄×D₂₅₁	1	4	yes	complemented
189	C₂×D₄	—	502	2	no	—
190	C₂×D₄	—	502	2	no	—
191	C₂×D₄	—	251	4	no	—
192	C₂²×C₄	—	1004	2	no	—
193	C₂²×C₄	—	502	4	no	—
194	C₂²×C₄	—	502	4	no	—
195	C₂²×C₄	—	251	1	no	—
196	C₂²×C₄	—	251	2	no	—
197	C₂²×C₄	—	251	1	no	—
198	C₂²×C₄	—	251	1	no	—
199	C₂²×C₄	—	251	1	no	—
200	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
202	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
203	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	502	2	no	—
207	C₄⋊C₄	—	502	2	no	—
208	C₄⋊C₄	—	502	2	no	—
209	C₄⋊C₄	C₂²⋊D₂₅₁	1	4	yes	—
210	C₂²⋊C₄	—	1004	2	no	—
211	C₂²⋊C₄	—	502	4	no	—
212	C₂²⋊C₄	—	502	4	no	—
213	C₂²⋊C₄	—	251	8	no	—
214	C₂²⋊C₄	—	251	8	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	1004	2	no	—
218	C₄²	—	502	2	no	—
219	C₄²	—	251	4	no	—
220	C₂³	—	502	2	no	—
221	C₂³	—	502	2	no	—
222	C₂³	—	502	2	no	—
223	C₂³	—	502	2	no	—
224	C₂³	—	502	2	no	—
225	C₂³	—	502	2	no	—
226	C₂³	—	251	4	no	—
227	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
228	Q₈	—	2	8	no	—
229	D₄	—	502	8	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	1004	8	no	—
240	C₂×C₄	—	502	1	no	—
241	C₂×C₄	—	502	2	no	—
242	C₂×C₄	—	502	2	no	—
243	C₂×C₄	—	502	1	no	—
244	C₂×C₄	—	502	2	no	—
245	C₂×C₄	—	502	1	no	—
246	C₂×C₄	—	502	1	no	—
247	C₂×C₄	—	502	1	no	—
248	C₂×C₄	—	502	1	no	—
249	C₂×C₄	—	502	1	no	—
250	C₂×C₄	—	502	1	no	—
251	C₂×C₄	—	502	1	no	—
252	C₂×C₄	—	251	2	no	—
253	C₂×C₄	—	251	2	no	—
254	C₂×C₄	—	251	2	no	—
255	C₂×C₄	C₂×C₄×D₂₅₁	1	4	yes	—
256	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
257	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	4	yes	—
258	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
259	C₂×C₄	#16860810	1	2	yes	—
260	C₂×C₄	#16860811	1	2	yes	—
261	C₂²	—	1004	2	no	—
262	C₂²	—	1004	2	no	—
263	C₂²	—	1004	2	no	—
264	C₂²	—	1004	2	no	—
265	C₂²	—	502	4	no	—
266	C₂²	—	502	4	no	—
267	C₂²	—	502	4	no	—
268	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
269	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
270	C₂²	#41×D₂₅₁	1	1	yes	characteristic
271	C₂²	C₂×#16860810	1	1	yes	characteristic
272	C₂²	C₂×#16860811	1	1	yes	characteristic
273	C₂²	#22548761	1	1	yes	characteristic
274	C₂²	#22548763	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	4	no	—
278	C₄	—	4	1	no	—
279	C₄	—	1004	1	no	—
280	C₄	—	1004	2	no	—
281	C₄	—	502	2	no	—
282	C₄	—	502	2	no	—
283	C₄	—	502	2	no	—
284	C₄	C₄×C₂²⋊D₂₅₁	1	4	yes	—
285	C₂	—	1004	4	no	—
286	C₂	C₂×C₄×C₂²⋊D₂₅₁	1	1	yes	characteristic
287	C₂	C₄×#16860811	1	1	yes	characteristic
288	C₂	#30632169	1	1	yes	characteristic
289	C₂	#30632325	1	1	yes	characteristic
290	C₂	#30632348	1	1	yes	characteristic
291	C₂	#30632409	1	1	yes	characteristic
292	C₂	#30632422	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator	C₂²×C₂₅₁	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₂₅₁	C₂
socle, cosocle	C₂³×C₂₅₁	C₂⁴
prime radical	C₂²×D₁₀₀₄	C₂²
hypercenter	C₂×C₄×Q₈	D₂₅₁
hypocenter	C₂₅₁	#2367
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
3. #30631594	C₂	semidirect
5. #30631578	C₂	semidirect
6. #30631577	C₂	semidirect
7. #30631558	C₂	semidirect
8. C₄⋊C₄×Dic₂₅₁	C₂	semidirect
9. C₄×Dic₂₅₁⋊C₄	C₂	semidirect
10. C₄×(C₂×D₂₅₁)⋊C₄	C₂	semidirect
11. C₂×C₄×Q₈×C₂₅₁	C₂	semidirect
14. #22548704	C₄	semidirect
56. (C₂×D₂₅₁)⋊C₄	C₂×C₄	semidirect
57. (C₂×D₂₅₁)⋊C₄	C₂×C₄	semidirect
60. C₄×Dic₂₅₁	C₂×C₄	semidirect
63. C₂×Q₈×C₂₅₁	C₂×C₄	semidirect
146. C₂₅₁	#2367	semidirect
149. C₂×C₄×Q₈	D₂₅₁	semidirect
188. C₂×Q₈	C₄×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548704	2
C₂×#22548677	2
#30631594	2
#30631589	2
#30631578	2
#30631577	2
#30631558	2
C₄⋊C₄×Dic₂₅₁	2
C₄×Dic₂₅₁⋊C₄	2
C₄×(C₂×D₂₅₁)⋊C₄	2
C₂×C₄×Q₈×C₂₅₁	2
#2367	251