Subgroups — #44558896

Order 32128.

726 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548762	C₂	1	1	yes	maximal, characteristic
2	C₂×#22548723	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632168	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631697	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631694	C₂	1	2	yes	maximal, complemented
6	#30631685	C₂	1	2	yes	maximal, complemented
7	#30631683	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631680	C₂	1	2	yes	maximal, complemented
9	C₂×#22548638	C₂	1	1	yes	maximal, characteristic, complemented
10	#103×D₂₅₁	C₂	1	2	yes	maximal, complemented
11	#417×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#22548762	—	2	4	no	—
13	#22548762	C₄	1	8	yes	complemented
14	C₂×#16860810	—	2	1	no	—
15	C₂×#16860810	C₂²	1	1	yes	characteristic
16	C₂×#16860809	—	2	2	no	—
17	C₂×#16860809	C₂²	1	2	yes	—
18	C₂²×Dic₅₀₂	—	2	1	no	—
19	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
20	C₂×Q₈×D₂₅₁	C₂²	1	2	yes	—
21	#22548723	—	2	4	no	—
22	C₂×#16860793	C₂²	1	2	yes	—
23	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₈×D₂₅₁	C₂²	1	2	yes	—
25	#22548680	—	2	1	no	—
26	#22548677	—	2	1	no	—
27	C₄×C₂²⋊D₂₅₁	—	2	2	no	—
28	#22548674	C₂²	1	1	yes	characteristic
29	#22548673	C₂²	1	2	yes	—
30	C₄⋊C₄×D₂₅₁	C₂²	1	2	yes	—
31	#22548640	C₂²	1	2	yes	—
32	#22548640	C₂²	1	2	yes	—
33	#22548639	C₂²	1	1	yes	characteristic
34	#22548639	C₂²	1	1	yes	characteristic
35	#22548638	C₂²	1	2	yes	—
36	#22548638	C₂²	1	2	yes	—
37	#22548636	C₂²	1	1	yes	characteristic
38	#22548636	C₂²	1	1	yes	characteristic
39	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
42	#103×C₂₅₁	C₂²	1	2	yes	—
43	#102×C₂₅₁	C₂²	1	2	yes	—
44	#16860810	—	2	8	no	—
45	#16860810	C₂×C₄	1	4	yes	—
46	#16860810	D₄	1	4	yes	—
47	#16860809	—	2	16	no	—
48	#16860809	—	2	4	no	—
49	#16860809	C₂×C₄	1	8	yes	complemented
50	C₂×Dic₅₀₂	—	2	8	no	—
51	C₂×Dic₅₀₂	—	2	2	no	—
52	C₂×Dic₅₀₂	—	2	1	no	—
53	C₂×Dic₅₀₂	—	2	2	no	—
54	C₂×Dic₅₀₂	C₂×C₄	1	4	yes	complemented
55	C₂×Dic₅₀₂	D₄	1	1	yes	characteristic
56	C₂×Dic₅₀₂	C₂³	1	2	yes	—
57	Q₈×D₂₅₁	—	2	4	no	—
58	Q₈×D₂₅₁	C₂×C₄	1	8	yes	complemented
59	C₂×D₁₀₀₄	D₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₂₅₁	—	2	2	no	—
61	C₂×C₂²⋊D₂₅₁	D₄	1	2	yes	—
62	C₂²×Dic₂₅₁	—	2	2	no	—
63	C₂²×Dic₂₅₁	—	2	1	no	—
64	C₂×C₄×D₂₅₁	—	2	2	no	—
65	C₂×C₄×D₂₅₁	—	2	2	no	—
66	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
67	#16860793	—	2	4	no	—
68	C₂×C₂₅₁⋊C₈	—	2	2	no	—
69	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₈×D₂₅₁	—	2	4	no	—
72	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
74	Dic₂₅₁⋊C₄	—	2	2	no	—
75	C₂²⋊Dic₂₅₁	—	2	1	no	—
76	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
77	C₄×Dic₂₅₁	—	2	2	no	—
78	C₄×Dic₂₅₁	—	2	1	no	—
79	#41×C₂₅₁	—	2	2	no	—
80	#41×C₂₅₁	C₂×C₄	1	4	yes	complemented
81	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₂₅₁	—	2	1	no	—
83	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₂₅₁	—	2	1	no	—
85	C₂²×C₄×C₂₅₁	—	2	1	no	—
86	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
87	M₁₆×C₂₅₁	—	2	2	no	—
88	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
89	C₄⋊C₄×C₂₅₁	—	2	1	no	—
90	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
91	C₂²×D₂₅₁	C₂×D₄	1	2	yes	—
92	Dic₅₀₂	—	2	4	no	—
93	Dic₅₀₂	—	2	2	no	—
94	Dic₅₀₂	—	2	8	no	—
95	Dic₅₀₂	C₂²⋊C₄	1	4	yes	—
96	Dic₅₀₂	C₂²×C₄	1	4	yes	—
97	D₁₀₀₄	C₂²⋊C₄	1	4	yes	—
98	C₂²⋊D₂₅₁	—	2	8	no	—
99	C₂²⋊D₂₅₁	C₂²⋊C₄	1	8	yes	—
100	C₂×Dic₂₅₁	—	2	4	no	—
101	C₂×Dic₂₅₁	—	2	2	no	—
102	C₂×Dic₂₅₁	—	2	8	no	—
103	C₂×Dic₂₅₁	—	2	1	no	—
104	C₂×Dic₂₅₁	—	2	1	no	—
105	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
106	C₄×D₂₅₁	—	2	8	no	—
107	C₄×D₂₅₁	—	8	2	no	—
108	C₄×D₂₅₁	C₂²×C₄	1	4	yes	—
109	C₄×D₂₅₁	C₂×D₄	1	4	yes	—
110	C₂₅₁⋊C₈	—	2	1	no	—
111	C₂₅₁⋊C₈	—	2	1	no	—
112	C₂³×C₂₅₁	—	2	1	no	—
113	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₂₅₁	—	2	1	no	—
115	Q₈×C₂₅₁	C₂²×C₄	1	2	yes	—
116	D₄×C₂₅₁	—	2	4	no	—
117	D₄×C₂₅₁	—	2	1	no	—
118	D₄×C₂₅₁	C₂²×C₄	1	2	yes	—
119	C₂×C₄×C₂₅₁	—	2	1	no	—
120	C₂×C₄×C₂₅₁	—	2	4	no	—
121	C₂×C₄×C₂₅₁	—	2	2	no	—
122	C₂×C₄×C₂₅₁	—	8	1	no	—
123	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
124	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
126	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
127	C₈×C₂₅₁	—	2	2	no	—
128	C₂²×C₂₅₁	—	2	1	no	—
129	C₂²×C₂₅₁	—	2	1	no	—
130	C₂²×C₂₅₁	—	2	2	no	—
131	C₂²×C₂₅₁	—	4	1	no	—
132	C₂²×C₂₅₁	—	4	1	no	—
133	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×D₂₅₁	—	2	2	no	—
136	C₂×D₂₅₁	—	2	2	no	—
137	C₂×D₂₅₁	C₂×C₂²⋊C₄	1	4	yes	—
138	C₄×C₂₅₁	—	2	2	no	—
139	C₄×C₂₅₁	—	4	2	no	—
140	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
141	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
142	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
143	Dic₂₅₁	—	2	4	no	—
144	Dic₂₅₁	—	4	1	no	—
145	Dic₂₅₁	—	4	1	no	—
146	Dic₂₅₁	C₂×C₂²⋊C₄	1	4	yes	—
147	C₂×C₂₅₁	—	2	2	no	—
148	C₂×C₂₅₁	—	4	2	no	—
149	C₂×C₂₅₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₂₅₁	#577	1	1	yes	characteristic
151	C₂×C₂₅₁	#578	1	1	yes	characteristic
152	D₂₅₁	—	2	4	no	—
153	C₂₅₁	#2898	1	1	yes	characteristic, complemented, Sylow
154	#2898	—	251	1	no	maximal, Sylow
155	C₂×2₋¹⁺⁴	—	251	1	no	—
156	C₂×#131	—	251	1	no	—
157	#520	—	251	1	no	—
158	#419	—	251	1	no	—
159	#419	—	251	2	no	—
160	#417	—	251	2	no	—
161	#417	D₂₅₁	1	1	yes	characteristic, complemented
162	#416	—	251	1	no	—
163	#416	—	251	2	no	—
164	C₂×#103	—	251	1	no	—
165	C₂×#103	—	251	2	no	—
166	2₋¹⁺⁴	—	502	4	no	—
167	2₋¹⁺⁴	—	251	8	no	—
168	C₂×#41	—	502	1	no	—
169	C₂×#41	—	502	2	no	—
170	C₂×#41	—	251	2	no	—
171	C₂×#41	—	251	1	no	—
172	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic
173	C₂²×Q₈	—	502	1	no	—
174	C₂²×Q₈	—	251	2	no	—
175	C₂²×Q₈	—	251	1	no	—
176	#131	—	502	4	no	—
177	C₂×M₁₆	—	251	2	no	—
178	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
179	C₂²×C₈	—	251	1	no	—
180	C₂²×C₈	—	251	2	no	—
181	C₄×Q₈	—	502	1	no	—
182	C₄×D₄	—	502	1	no	—
183	C₄×D₄	—	502	2	no	—
184	#117	—	251	1	no	—
185	#117	—	251	2	no	—
186	C₂×C₄⋊C₄	—	251	2	no	—
187	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
188	#103	—	251	1	no	—
189	#103	—	251	1	no	—
190	#103	—	251	2	no	—
191	#103	—	251	2	no	—
192	#103	—	251	2	no	—
193	#103	—	251	2	no	—
194	#103	C₂×D₂₅₁	1	2	yes	—
195	#102	—	251	1	no	—
196	#102	—	251	1	no	—
197	#102	C₂×D₂₅₁	1	2	yes	—
198	#41	—	2	2	no	—
199	#41	—	502	8	no	—
200	#41	—	502	16	no	—
201	#41	—	502	4	no	—
202	#41	—	251	8	no	—
203	#41	—	251	4	no	—
204	#41	—	251	4	no	—
205	#41	C₄×D₂₅₁	1	4	yes	complemented
206	C₂×Q₈	—	502	8	no	—
207	C₂×Q₈	—	502	2	no	—
208	C₂×Q₈	—	502	4	no	—
209	C₂×Q₈	—	502	1	no	—
210	C₂×Q₈	—	502	2	no	—
211	C₂×Q₈	—	251	8	no	—
212	C₂×Q₈	—	251	4	no	—
213	C₂×Q₈	—	251	2	no	—
214	C₂×Q₈	—	251	1	no	—
215	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	—	502	2	no	—
218	C₂×D₄	—	251	2	no	—
219	C₂×D₄	—	251	1	no	—
220	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	502	2	no	—
224	C₂²×C₄	—	502	2	no	—
225	C₂²×C₄	—	502	1	no	—
226	C₂²×C₄	—	502	2	no	—
227	C₂²×C₄	—	251	2	no	—
228	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
229	M₁₆	—	2	2	no	—
230	M₁₆	—	502	4	no	—
231	C₂×C₈	—	502	4	no	—
232	C₂×C₈	—	502	2	no	—
233	C₂×C₈	—	251	1	no	—
234	C₂×C₈	—	251	1	no	—
235	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	502	2	no	—
238	C₄⋊C₄	—	251	1	no	—
239	C₄⋊C₄	—	251	1	no	—
240	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
241	C₂²⋊C₄	—	502	2	no	—
242	C₂²⋊C₄	—	502	1	no	—
243	C₄²	—	502	2	no	—
244	C₄²	—	502	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	251	2	no	—
247	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
248	Q₈	—	2	1	no	—
249	Q₈	—	502	8	no	—
250	Q₈	—	502	2	no	—
251	Q₈	—	502	4	no	—
252	Q₈	—	251	4	no	—
253	Q₈	—	251	4	no	—
254	Q₈	C₂×C₄×D₂₅₁	1	2	yes	—
255	D₄	—	2	4	no	—
256	D₄	—	2	1	no	—
257	D₄	—	502	8	no	—
258	D₄	—	251	8	no	—
259	D₄	—	251	4	no	—
260	D₄	C₂×C₄×D₂₅₁	1	2	yes	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	8	1	no	—
265	C₂×C₄	—	2008	2	no	—
266	C₂×C₄	—	502	8	no	—
267	C₂×C₄	—	502	1	no	—
268	C₂×C₄	—	502	1	no	—
269	C₂×C₄	—	502	2	no	—
270	C₂×C₄	—	502	4	no	—
271	C₂×C₄	—	502	8	no	—
272	C₂×C₄	—	251	2	no	—
273	C₂×C₄	—	251	4	no	—
274	C₂×C₄	—	251	4	no	—
275	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
276	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
277	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
278	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	502	1	no	—
281	C₈	—	502	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	502	2	no	—
288	C₂²	—	502	2	no	—
289	C₂²	—	251	4	no	—
290	C₂²	C₂²⋊C₄×D₂₅₁	1	2	yes	—
291	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	4	2	no	—
294	C₄	—	1004	1	no	—
295	C₄	—	1004	1	no	—
296	C₄	—	502	4	no	—
297	C₄	—	251	4	no	—
298	C₄	C₂²⋊C₄×D₂₅₁	1	2	yes	—
299	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
300	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	4	2	no	—
303	C₂	—	502	4	no	—
304	C₂	C₂×C₂²⋊C₄×D₂₅₁	1	1	yes	characteristic
305	C₂	#30632497	1	1	yes	characteristic
306	C₂	#30632503	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#417×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#22548762	C₂
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#417	D₂₅₁
hypocenter	C₂₅₁	#2898
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
2. C₂×#22548723	C₂	semidirect
3. #30632168	C₂	semidirect
4. #30631697	C₂	semidirect
5. #30631694	C₂	semidirect
6. #30631685	C₂	semidirect
7. #30631683	C₂	semidirect
8. #30631680	C₂	semidirect
9. C₂×#22548638	C₂	semidirect
10. #103×D₂₅₁	C₂	semidirect
11. #417×C₂₅₁	C₂	semidirect
13. #22548762	C₄	semidirect
49. #16860809	C₂×C₄	semidirect
54. C₂×Dic₅₀₂	C₂×C₄	semidirect
58. Q₈×D₂₅₁	C₂×C₄	semidirect
80. #41×C₂₅₁	C₂×C₄	semidirect
153. C₂₅₁	#2898	semidirect
161. #417	D₂₅₁	semidirect
205. #41	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×#22548762	2
C₂×#22548723	2
#30632168	2
#30631697	2
#30631694	2
#30631685	2
#30631683	2
#30631680	2
C₂×#22548638	2
#103×D₂₅₁	2
#417×C₂₅₁	2
#2898	251