Subgroups — #44559183

Order 32128.

484 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×#22548748	C₂	1	1	yes	maximal, characteristic
2	#30632155	C₂	1	1	yes	maximal, characteristic
3	C₂×#22548680	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631793	C₂	1	2	yes	maximal, complemented
5	#30631786	C₂	1	2	yes	maximal, complemented
6	#30631781	C₂	1	2	yes	maximal, complemented
7	#30631774	C₂	1	2	yes	maximal, complemented
8	#30631695	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631686	C₂	1	1	yes	maximal, characteristic
10	M₁₆×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#428×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16860810	C₂²	1	1	yes	characteristic
13	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
14	#22548748	C₄	1	8	yes	complemented
15	C₂×Dic₁₀₀₄	C₂²	1	2	yes	—
16	C₂×#16860799	C₂²	1	2	yes	—
17	C₂×#16860794	C₂²	1	1	yes	characteristic
18	#22548680	—	2	1	no	—
19	#22548680	—	2	1	no	—
20	#22548680	C₂²	1	2	yes	—
21	#22548680	C₂²	1	1	yes	characteristic
22	#22548680	C₂²	1	2	yes	—
23	#22548680	C₂²	1	1	yes	characteristic
24	#22548677	C₂²	1	1	yes	characteristic
25	#22548677	C₂²	1	1	yes	characteristic
26	(C₂²⋊D₂₅₁)⋊C₄	—	2	1	no	—
27	(C₂²⋊D₂₅₁)⋊C₄	—	2	1	no	—
28	#22548672	—	2	1	no	—
29	C₂×Dic₂₅₁⋊C₄	—	2	1	no	—
30	C₂×C₄×Dic₂₅₁	—	2	1	no	—
31	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
32	C₄²×D₂₅₁	—	2	1	no	—
33	#22548638	C₂²	1	2	yes	—
34	#22548638	C₂²	1	2	yes	—
35	#22548638	C₂²	1	2	yes	—
36	#22548635	C₂²	1	2	yes	—
37	#22548625	C₂²	1	2	yes	—
38	C₈×Dic₂₅₁	C₂²	1	2	yes	—
39	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
40	#117×C₂₅₁	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
42	#107×C₂₅₁	C₂²	1	2	yes	—
43	#106×C₂₅₁	C₂²	1	2	yes	—
44	#16860810	—	2	2	no	—
45	#16860810	C₂×C₄	1	4	yes	—
46	C₂×Dic₅₀₂	—	2	1	no	—
47	C₂×Dic₅₀₂	—	2	2	no	—
48	C₂×Dic₅₀₂	C₂×C₄	1	4	yes	complemented
49	C₂×Dic₅₀₂	C₂³	1	2	yes	—
50	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
51	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
53	C₂²×Dic₂₅₁	—	2	1	no	—
54	C₂²×Dic₂₅₁	D₄	1	1	yes	characteristic
55	C₂²×Dic₂₅₁	D₄	1	1	yes	characteristic
56	C₂×C₄×D₂₅₁	—	2	1	no	—
57	C₂×C₄×D₂₅₁	—	2	1	no	—
58	C₂×C₄×D₂₅₁	—	2	1	no	—
59	Dic₁₀₀₄	C₂×C₄	1	8	yes	complemented
60	#16860799	C₂×C₄	1	8	yes	complemented
61	#16860794	—	2	2	no	—
62	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
63	Dic₂₅₁⋊C₄	—	2	2	no	—
64	Dic₂₅₁⋊C₄	—	2	2	no	—
65	Dic₂₅₁⋊C₄	—	2	1	no	—
66	Dic₂₅₁⋊C₄	—	2	1	no	—
67	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
68	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
69	C₄×Dic₂₅₁	—	2	1	no	—
70	C₄×Dic₂₅₁	—	2	1	no	—
71	C₄×Dic₂₅₁	—	2	2	no	—
72	C₄×Dic₂₅₁	—	2	2	no	—
73	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
74	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
75	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
76	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₂₅₁	—	2	1	no	—
78	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
79	M₁₆×C₂₅₁	C₂×C₄	1	4	yes	complemented
80	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
81	C₄⋊C₄×C₂₅₁	—	2	1	no	—
82	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
84	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
86	C₄²×C₂₅₁	—	2	1	no	—
87	C₂²×D₂₅₁	—	2	1	no	—
88	Dic₅₀₂	—	2	4	no	—
89	Dic₅₀₂	—	2	2	no	—
90	Dic₅₀₂	—	2	1	no	—
91	Dic₅₀₂	C₂²×C₄	1	4	yes	—
92	Dic₅₀₂	C₂²×C₄	1	2	yes	—
93	D₁₀₀₄	—	2	1	no	—
94	D₁₀₀₄	C₂²×C₄	1	2	yes	—
95	C₂²⋊D₂₅₁	—	2	4	no	—
96	C₂×Dic₂₅₁	—	2	4	no	—
97	C₂×Dic₂₅₁	—	2	1	no	—
98	C₂×Dic₂₅₁	—	2	2	no	—
99	C₂×Dic₂₅₁	—	2	2	no	—
100	C₂×Dic₂₅₁	—	4	1	no	—
101	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
102	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
103	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
104	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
105	C₄×D₂₅₁	—	2	4	no	—
106	C₄×D₂₅₁	—	4	2	no	—
107	C₄×D₂₅₁	—	4	2	no	—
108	C₂₅₁⋊C₈	—	2	2	no	—
109	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₂₅₁	—	2	1	no	—
111	C₂×C₄×C₂₅₁	—	2	1	no	—
112	C₂×C₄×C₂₅₁	—	2	2	no	—
113	C₂×C₄×C₂₅₁	—	4	2	no	—
114	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
115	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₂₅₁	#41	1	2	yes	—
117	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
118	C₈×C₂₅₁	C₂²×C₄	1	4	yes	—
119	C₂²×C₂₅₁	—	2	1	no	—
120	C₂²×C₂₅₁	—	2	1	no	—
121	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
122	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₂₅₁	—	2	2	no	—
124	C₂×D₂₅₁	—	4	1	no	—
125	C₂×D₂₅₁	—	4	1	no	—
126	C₄×C₂₅₁	—	4	1	no	—
127	C₄×C₂₅₁	—	4	2	no	—
128	C₄×C₂₅₁	—	4	1	no	—
129	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
130	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
131	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
132	Dic₂₅₁	—	2	2	no	—
133	Dic₂₅₁	—	2	4	no	—
134	Dic₂₅₁	—	2	1	no	—
135	Dic₂₅₁	—	2	1	no	—
136	Dic₂₅₁	—	2	1	no	—
137	Dic₂₅₁	#137	1	2	yes	complemented
138	C₂×C₂₅₁	—	2	2	no	—
139	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₂₅₁	C₂×#137	1	1	yes	characteristic
141	C₂×C₂₅₁	#575	1	1	yes	characteristic
142	D₂₅₁	—	4	2	no	—
143	C₂₅₁	C₄×#137	1	1	yes	characteristic, complemented, Sylow
144	C₄×#137	—	251	1	no	maximal, Sylow
145	C₂×#137	—	251	1	no	—
146	C₄×#41	—	251	1	no	—
147	C₂×C₄×Q₈	—	251	1	no	—
148	#441	—	251	2	no	—
149	#440	—	251	2	no	—
150	C₄×Q₁₆	—	251	2	no	—
151	C₄×SD₁₆	—	251	2	no	—
152	#428	D₂₅₁	1	1	yes	characteristic, complemented
153	#419	—	251	1	no	—
154	#417	—	251	1	no	—
155	C₄×M₁₆	—	251	1	no	—
156	C₂×#41	—	251	1	no	—
157	C₂²×Q₈	—	251	1	no	—
158	#137	—	251	8	no	—
159	C₂×Q₁₆	—	251	2	no	—
160	C₂×SD₁₆	—	251	2	no	—
161	C₂×M₁₆	—	251	1	no	—
162	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
163	C₄×Q₈	—	502	1	no	—
164	C₄×Q₈	—	502	1	no	—
165	C₄×Q₈	—	251	2	no	—
166	C₄×Q₈	—	251	2	no	—
167	C₄×Q₈	—	251	1	no	—
168	C₄×Q₈	—	251	1	no	—
169	C₄×D₄	—	502	1	no	—
170	C₄×D₄	—	502	1	no	—
171	C₄×D₄	—	251	1	no	—
172	C₄×D₄	—	251	1	no	—
173	#117	—	502	1	no	—
174	#117	C₂×D₂₅₁	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	502	1	no	—
176	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
177	C₂×C₄²	—	502	1	no	—
178	C₂×C₄²	—	502	1	no	—
179	C₂×C₄²	—	251	1	no	—
180	#107	C₂×D₂₅₁	1	2	yes	—
181	#106	C₂×D₂₅₁	1	2	yes	—
182	#103	—	251	2	no	—
183	#103	—	251	2	no	—
184	#103	—	251	2	no	—
185	#102	—	251	2	no	—
186	#97	—	251	2	no	—
187	C₄×C₈	—	251	2	no	—
188	#41	—	502	2	no	—
189	#41	—	251	4	no	—
190	C₂×Q₈	—	502	1	no	—
191	C₂×Q₈	—	502	2	no	—
192	C₂×Q₈	—	251	4	no	—
193	C₂×Q₈	—	251	2	no	—
194	C₂×Q₈	—	251	1	no	—
195	C₂×D₄	—	502	1	no	—
196	C₂×D₄	—	251	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	502	1	no	—
199	C₂²×C₄	—	502	1	no	—
200	C₂²×C₄	—	502	1	no	—
201	C₂²×C₄	—	502	1	no	—
202	C₂²×C₄	—	251	1	no	—
203	C₂²×C₄	—	251	1	no	—
204	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
205	Q₁₆	—	251	8	no	—
206	SD₁₆	—	251	8	no	—
207	M₁₆	—	502	2	no	—
208	M₁₆	C₄×D₂₅₁	1	4	yes	complemented
209	C₂×C₈	—	251	2	no	—
210	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	502	2	no	—
213	C₄⋊C₄	—	502	2	no	—
214	C₄⋊C₄	—	502	1	no	—
215	C₄⋊C₄	—	502	1	no	—
216	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
217	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
218	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	502	1	no	—
221	C₂²⋊C₄	—	502	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	502	1	no	—
224	C₄²	—	502	1	no	—
225	C₄²	—	502	2	no	—
226	C₄²	—	502	2	no	—
227	C₄²	—	251	1	no	—
228	C₄²	—	251	1	no	—
229	C₄²	—	251	1	no	—
230	C₄²	—	251	1	no	—
231	C₂³	—	502	1	no	—
232	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
233	Q₈	—	502	2	no	—
234	Q₈	—	502	1	no	—
235	Q₈	—	502	4	no	—
236	Q₈	—	251	4	no	—
237	Q₈	—	251	2	no	—
238	D₄	—	502	4	no	—
239	D₄	—	502	1	no	—
240	D₄	—	251	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	1004	2	no	—
246	C₂×C₄	—	1004	2	no	—
247	C₂×C₄	—	1004	1	no	—
248	C₂×C₄	—	502	1	no	—
249	C₂×C₄	—	502	2	no	—
250	C₂×C₄	—	502	4	no	—
251	C₂×C₄	—	502	2	no	—
252	C₂×C₄	—	502	4	no	—
253	C₂×C₄	—	251	1	no	—
254	C₂×C₄	—	251	1	no	—
255	C₂×C₄	—	251	1	no	—
256	C₂×C₄	—	251	1	no	—
257	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
258	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
259	C₂×C₄	#16860811	1	2	yes	—
260	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
261	C₈	—	502	2	no	—
262	C₈	C₂×C₄×D₂₅₁	1	4	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	1004	1	no	—
266	C₂²	—	1004	1	no	—
267	C₂²	—	502	2	no	—
268	C₂²	#22548677	1	2	yes	—
269	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
270	C₄	—	4	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	502	1	no	—
274	C₄	—	502	1	no	—
275	C₄	—	502	1	no	—
276	C₄	—	502	2	no	—
277	C₄	—	502	4	no	—
278	C₄	—	251	2	no	—
279	C₄	#22548677	1	2	yes	—
280	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
281	C₄	C₂×#16860811	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	1004	2	no	—
284	C₂	C₂×#22548677	1	1	yes	characteristic
285	C₂	#137×D₂₅₁	1	1	yes	characteristic
286	C₂	#30632483	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#428×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#16860810	C₂²
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#428	D₂₅₁
hypocenter	C₂₅₁	C₄×#137
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
3. C₂×#22548680	C₂	semidirect
4. #30631793	C₂	semidirect
5. #30631786	C₂	semidirect
6. #30631781	C₂	semidirect
7. #30631774	C₂	semidirect
8. #30631695	C₂	semidirect
10. M₁₆×Dic₂₅₁	C₂	semidirect
11. #428×C₂₅₁	C₂	semidirect
14. #22548748	C₄	semidirect
48. C₂×Dic₅₀₂	C₂×C₄	semidirect
59. Dic₁₀₀₄	C₂×C₄	semidirect
60. #16860799	C₂×C₄	semidirect
79. M₁₆×C₂₅₁	C₂×C₄	semidirect
137. Dic₂₅₁	#137	semidirect
143. C₂₅₁	C₄×#137	semidirect
152. #428	D₂₅₁	semidirect
208. M₁₆	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×#22548748	2
#30632155	2
C₂×#22548680	2
#30631793	2
#30631786	2
#30631781	2
#30631774	2
#30631695	2
#30631686	2
M₁₆×Dic₂₅₁	2
#428×C₂₅₁	2
C₄×#137	251