Subgroups — #44559202

Order 32128.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548746	C₂	1	1	yes	maximal, characteristic
2	#30632168	C₂	1	1	yes	maximal, characteristic
3	#30632165	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631791	C₂	1	2	yes	maximal, complemented
5	#30631790	C₂	1	2	yes	maximal, complemented
6	#30631779	C₂	1	2	yes	maximal, complemented
7	#30631772	C₂	1	2	yes	maximal, complemented
8	#30631727	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631684	C₂	1	1	yes	maximal, characteristic
10	#30631642	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#103×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16860810	C₂²	1	1	yes	characteristic
13	#22548746	C₄	1	8	yes	complemented
14	C₂×#16860800	C₂²	1	2	yes	—
15	C₂×#16860797	C₂²	1	2	yes	—
16	C₂×#16860794	C₂²	1	1	yes	characteristic
17	Q₈×Dic₂₅₁	—	2	1	no	—
18	Q₈×Dic₂₅₁	—	2	1	no	—
19	Q₈×Dic₂₅₁	C₂²	1	2	yes	—
20	Q₈×Dic₂₅₁	C₂²	1	2	yes	—
21	#22548680	C₂²	1	2	yes	—
22	#22548677	C₂²	1	2	yes	—
23	C₄×C₂²⋊D₂₅₁	—	2	2	no	—
24	#22548674	—	2	2	no	—
25	#22548674	C₂²	1	1	yes	characteristic
26	#22548674	C₂²	1	1	yes	characteristic
27	#22548673	—	2	1	no	—
28	C₄⋊C₄×D₂₅₁	—	2	1	no	—
29	#22548648	C₂²	1	2	yes	—
30	#22548646	C₂²	1	2	yes	—
31	#22548640	C₂²	1	1	yes	characteristic
32	#22548640	C₂²	1	1	yes	characteristic
33	#22548637	C₂²	1	1	yes	characteristic
34	#22548637	C₂²	1	1	yes	characteristic
35	#22548625	C₂²	1	2	yes	—
36	C₈×Dic₂₅₁	C₂²	1	2	yes	—
37	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
40	#103×C₂₅₁	C₂²	1	2	yes	—
41	#103×C₂₅₁	C₂²	1	2	yes	—
42	#16860810	—	2	2	no	—
43	#16860810	C₂×C₄	1	4	yes	—
44	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
45	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
47	C₂×C₄×D₂₅₁	—	2	2	no	—
48	C₂×C₄×D₂₅₁	—	2	1	no	—
49	#16860800	C₂×C₄	1	8	yes	complemented
50	#16860797	C₂×C₄	1	8	yes	complemented
51	#16860794	C₂×C₄	1	4	yes	complemented
52	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
53	C₄⋊Dic₂₅₁	—	2	2	no	—
54	C₄⋊Dic₂₅₁	—	2	2	no	—
55	C₄⋊Dic₂₅₁	D₄	1	1	yes	characteristic
56	C₄⋊Dic₂₅₁	D₄	1	1	yes	characteristic
57	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
59	Dic₂₅₁⋊C₄	—	2	2	no	—
60	C₂²⋊Dic₂₅₁	—	2	1	no	—
61	C₂²⋊Dic₂₅₁	D₄	1	2	yes	—
62	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
63	C₄×Dic₂₅₁	—	2	2	no	—
64	C₄×Dic₂₅₁	—	2	2	no	—
65	C₄×Dic₂₅₁	—	2	2	no	—
66	C₄×Dic₂₅₁	—	2	1	no	—
67	C₄×Dic₂₅₁	C₂³	1	2	yes	—
68	C₂×Q₈×C₂₅₁	—	2	1	no	—
69	C₂×Q₈×C₂₅₁	—	2	2	no	—
70	C₂×Q₈×C₂₅₁	C₂×C₄	1	4	yes	complemented
71	C₂×Q₈×C₂₅₁	C₂³	1	2	yes	—
72	C₂²×C₄×C₂₅₁	—	2	1	no	—
73	C₂²×C₄×C₂₅₁	—	2	1	no	—
74	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₂₅₁	—	2	2	no	—
76	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₂₅₁	—	2	1	no	—
79	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
80	C₂²×D₂₅₁	—	2	1	no	—
81	Dic₅₀₂	—	2	1	no	—
82	Dic₅₀₂	C₂²×C₄	1	2	yes	—
83	D₁₀₀₄	—	2	1	no	—
84	D₁₀₀₄	C₂²×C₄	1	2	yes	—
85	C₂²⋊D₂₅₁	—	2	4	no	—
86	C₂×Dic₂₅₁	—	2	1	no	—
87	C₂×Dic₂₅₁	—	2	1	no	—
88	C₂×Dic₂₅₁	—	2	1	no	—
89	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
90	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
91	C₄×D₂₅₁	—	2	4	no	—
92	C₄×D₂₅₁	—	8	2	no	—
93	C₂₅₁⋊C₈	C₂²×C₄	1	4	yes	—
94	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
95	Q₈×C₂₅₁	—	2	4	no	—
96	Q₈×C₂₅₁	—	2	2	no	—
97	Q₈×C₂₅₁	C₂²×C₄	1	4	yes	—
98	C₂×C₄×C₂₅₁	—	2	4	no	—
99	C₂×C₄×C₂₅₁	—	2	2	no	—
100	C₂×C₄×C₂₅₁	—	2	2	no	—
101	C₂×C₄×C₂₅₁	—	8	1	no	—
102	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₅₁	#41	1	2	yes	—
105	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
106	C₈×C₂₅₁	—	2	1	no	—
107	C₈×C₂₅₁	—	2	1	no	—
108	C₂²×C₂₅₁	—	2	1	no	—
109	C₂²×C₂₅₁	—	2	1	no	—
110	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
111	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
112	C₂×D₂₅₁	—	2	2	no	—
113	C₂×D₂₅₁	—	4	1	no	—
114	C₂×D₂₅₁	—	4	1	no	—
115	C₄×C₂₅₁	—	2	4	no	—
116	C₄×C₂₅₁	—	4	2	no	—
117	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
118	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
119	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
120	Dic₂₅₁	—	2	2	no	—
121	Dic₂₅₁	—	2	2	no	—
122	Dic₂₅₁	—	2	2	no	—
123	Dic₂₅₁	—	4	1	no	—
124	Dic₂₅₁	—	4	1	no	—
125	C₂×C₂₅₁	—	2	2	no	—
126	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂×C₂₅₁	#577	1	1	yes	characteristic
128	C₂×C₂₅₁	#578	1	1	yes	characteristic
129	D₂₅₁	—	4	2	no	—
130	C₂₅₁	#2947	1	1	yes	characteristic, complemented, Sylow
131	#2947	—	251	1	no	maximal, Sylow
132	C₂×#137	—	251	1	no	—
133	#520	—	251	1	no	—
134	#519	—	251	1	no	—
135	#441	—	251	2	no	—
136	#440	—	251	2	no	—
137	C₄×Q₁₆	—	251	2	no	—
138	C₄×SD₁₆	—	251	2	no	—
139	#428	—	251	1	no	—
140	#416	—	251	1	no	—
141	C₂×#103	D₂₅₁	1	1	yes	characteristic, complemented
142	#405	—	251	1	no	—
143	C₂×#41	—	251	1	no	—
144	C₂²×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
145	#137	—	251	8	no	—
146	C₂×Q₁₆	—	251	2	no	—
147	C₂×SD₁₆	—	251	2	no	—
148	C₂×M₁₆	—	251	1	no	—
149	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
150	C₄×Q₈	—	502	1	no	—
151	C₄×Q₈	—	502	1	no	—
152	C₄×Q₈	—	251	2	no	—
153	C₄×Q₈	—	251	2	no	—
154	C₄×Q₈	—	251	2	no	—
155	C₄×D₄	—	502	2	no	—
156	C₄×D₄	—	251	2	no	—
157	#117	—	502	1	no	—
158	#117	—	502	2	no	—
159	#117	—	251	1	no	—
160	#117	—	251	1	no	—
161	C₂×C₄⋊C₄	—	502	1	no	—
162	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
163	#107	—	251	2	no	—
164	#106	—	251	2	no	—
165	#103	—	251	1	no	—
166	#103	—	251	1	no	—
167	#103	C₂×D₂₅₁	1	2	yes	—
168	#103	C₂×D₂₅₁	1	2	yes	—
169	#102	—	251	1	no	—
170	#102	—	251	1	no	—
171	#97	—	251	2	no	—
172	C₄×C₈	—	251	2	no	—
173	#41	—	502	2	no	—
174	#41	—	251	4	no	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	251	1	no	—
178	C₂×Q₈	C₄×D₂₅₁	1	4	yes	complemented
179	C₂×Q₈	C₂²×D₂₅₁	1	2	yes	—
180	C₂×D₄	—	502	1	no	—
181	C₂×D₄	—	251	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	502	2	no	—
185	C₂²×C₄	—	502	1	no	—
186	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
187	Q₁₆	—	251	8	no	—
188	SD₁₆	—	251	8	no	—
189	M₁₆	—	251	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	251	2	no	—
192	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
193	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	502	2	no	—
196	C₄⋊C₄	—	502	2	no	—
197	C₄⋊C₄	—	502	2	no	—
198	C₄⋊C₄	—	251	1	no	—
199	C₄⋊C₄	—	251	1	no	—
200	C₄⋊C₄	—	251	1	no	—
201	C₄⋊C₄	—	251	1	no	—
202	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
203	C₂²⋊C₄	—	502	2	no	—
204	C₂²⋊C₄	—	502	1	no	—
205	C₂²⋊C₄	—	251	2	no	—
206	C₄²	—	502	2	no	—
207	C₄²	—	502	2	no	—
208	C₄²	—	502	2	no	—
209	C₄²	—	502	1	no	—
210	C₄²	—	251	2	no	—
211	C₂³	—	502	1	no	—
212	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
213	Q₈	—	2	2	no	—
214	Q₈	—	2	4	no	—
215	Q₈	—	502	1	no	—
216	Q₈	—	251	2	no	—
217	Q₈	C₂×C₄×D₂₅₁	1	4	yes	—
218	D₄	—	502	4	no	—
219	D₄	—	502	1	no	—
220	D₄	—	251	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	2008	2	no	—
226	C₂×C₄	—	502	1	no	—
227	C₂×C₄	—	502	1	no	—
228	C₂×C₄	—	502	1	no	—
229	C₂×C₄	—	502	4	no	—
230	C₂×C₄	—	251	2	no	—
231	C₂×C₄	—	251	2	no	—
232	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
233	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
234	C₂×C₄	#16860809	1	2	yes	—
235	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	251	4	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	1004	1	no	—
242	C₂²	—	1004	1	no	—
243	C₂²	—	502	2	no	—
244	C₂²	(C₂²⋊D₂₅₁)⋊C₄	1	2	yes	—
245	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	—	1004	1	no	—
249	C₄	—	1004	1	no	—
250	C₄	—	502	2	no	—
251	C₄	—	502	2	no	—
252	C₄	—	502	2	no	—
253	C₄	(C₂²⋊D₂₅₁)⋊C₄	1	2	yes	—
254	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
255	C₄	C₂×#16860809	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	1004	2	no	—
258	C₂	C₂×(C₂²⋊D₂₅₁)⋊C₄	1	1	yes	characteristic
259	C₂	#30632493	1	1	yes	characteristic
260	C₂	#30632500	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	C₂×#103×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#16860810	C₂²
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₂×#103	D₂₅₁
hypocenter	C₂₅₁	#2947
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#103

Complements

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

N	H ≅ G/N	type
3. #30632165	C₂	semidirect
4. #30631791	C₂	semidirect
5. #30631790	C₂	semidirect
6. #30631779	C₂	semidirect
7. #30631772	C₂	semidirect
8. #30631727	C₂	semidirect
10. #30631642	C₂	semidirect
11. C₂×#103×C₂₅₁	C₂	semidirect
13. #22548746	C₄	semidirect
49. #16860800	C₂×C₄	semidirect
50. #16860797	C₂×C₄	semidirect
51. #16860794	C₂×C₄	semidirect
70. C₂×Q₈×C₂₅₁	C₂×C₄	semidirect
130. C₂₅₁	#2947	semidirect
141. C₂×#103	D₂₅₁	semidirect
178. 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×#22548746	2
#30632168	2
#30632165	2
#30631791	2
#30631790	2
#30631779	2
#30631772	2
#30631727	2
#30631684	2
#30631642	2
C₂×#103×C₂₅₁	2
#2947	251