Subgroups — #44560168

Order 32128.

376 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632243	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#30631995	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631994	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631991	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631981	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631980	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631971	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631970	C₂	1	1	yes	maximal, characteristic, complemented
10	#30631728	C₂	1	1	yes	maximal, characteristic
11	#30631726	C₂	1	1	yes	maximal, characteristic
12	#30631709	C₂	1	1	yes	maximal, characteristic
13	#30631694	C₂	1	1	yes	maximal, characteristic
14	#30631694	C₂	1	1	yes	maximal, characteristic
15	#479×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₂₅₁	C₂²	1	1	yes	characteristic
17	C₂×#16860793	C₂²	1	1	yes	characteristic
18	C₂×#16860793	C₂²	1	1	yes	characteristic
19	#22548714	C₂²	1	1	yes	characteristic
20	#22548714	C₂²	1	1	yes	characteristic
21	#22548707	C₂²	1	1	yes	characteristic
22	#22548693	—	2	1	no	—
23	#22548693	—	2	1	no	—
24	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
25	C₄×Dic₅₀₂	—	2	1	no	—
26	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
27	#22548680	—	2	1	no	—
28	#22548680	—	2	1	no	—
29	#22548680	—	2	1	no	—
30	#22548680	C₂²	1	1	yes	characteristic
31	#22548673	C₂²	1	1	yes	characteristic
32	#22548673	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₂₅₁	—	2	1	no	—
34	C₄⋊C₄×D₂₅₁	C₂²	1	1	yes	characteristic
35	C₄²×D₂₅₁	—	2	1	no	—
36	C₄²×D₂₅₁	C₂²	1	1	yes	characteristic
37	#22548649	C₂²	1	1	yes	characteristic
38	#22548648	C₂²	1	1	yes	characteristic
39	#22548648	C₂²	1	1	yes	characteristic
40	#22548647	C₂²	1	1	yes	characteristic
41	#22548646	C₂²	1	1	yes	characteristic
42	#22548646	C₂²	1	1	yes	characteristic
43	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
46	#22548640	C₂²	1	1	yes	characteristic
47	#22548640	C₂²	1	1	yes	characteristic
48	#22548639	C₂²	1	1	yes	characteristic
49	#22548639	C₂²	1	1	yes	characteristic
50	#22548638	C₂²	1	1	yes	characteristic
51	#22548638	C₂²	1	1	yes	characteristic
52	#125×C₂₅₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
54	#107×C₂₅₁	C₂²	1	1	yes	characteristic
55	#106×C₂₅₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
57	#103×C₂₅₁	C₂²	1	1	yes	characteristic
58	#103×C₂₅₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₅₀₂	—	2	1	no	—
60	C₂×Dic₅₀₂	—	2	1	no	—
61	C₂×Dic₅₀₂	—	2	1	no	—
62	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
63	Q₈×D₂₅₁	—	2	2	no	—
64	Q₈×D₂₅₁	Q₈	1	4	yes	complemented
65	C₂×C₄×D₂₅₁	—	2	1	no	—
66	C₂×C₄×D₂₅₁	—	2	1	no	—
67	C₂×C₄×D₂₅₁	D₄	1	1	yes	characteristic
68	C₂×C₄×D₂₅₁	D₄	1	1	yes	characteristic
69	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
70	#16860793	—	2	2	no	—
71	#16860793	—	2	2	no	—
72	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₂₅₁	—	2	1	no	—
75	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
78	Dic₂₅₁⋊C₄	—	2	1	no	—
79	Dic₂₅₁⋊C₄	—	2	1	no	—
80	Dic₂₅₁⋊C₄	—	2	1	no	—
81	Dic₂₅₁⋊C₄	—	2	1	no	—
82	Dic₂₅₁⋊C₄	—	2	1	no	—
83	Dic₂₅₁⋊C₄	—	2	1	no	—
84	Dic₂₅₁⋊C₄	—	2	1	no	—
85	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
86	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
87	C₄×Dic₂₅₁	—	2	1	no	—
88	C₄×Dic₂₅₁	—	2	1	no	—
89	C₄×Dic₂₅₁	—	2	1	no	—
90	C₄×Dic₂₅₁	—	2	1	no	—
91	C₄×Dic₂₅₁	—	2	1	no	—
92	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
93	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
94	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₂₅₁	—	2	1	no	—
99	C₄⋊C₄×C₂₅₁	—	2	1	no	—
100	C₄⋊C₄×C₂₅₁	—	2	1	no	—
101	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
104	C₄²×C₂₅₁	—	2	1	no	—
105	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
106	C₂²×D₂₅₁	C₂×D₄	1	1	yes	characteristic
107	Dic₅₀₂	—	2	1	no	—
108	Dic₅₀₂	—	2	4	no	—
109	Dic₅₀₂	—	4	2	no	—
110	Dic₅₀₂	—	4	2	no	—
111	Dic₅₀₂	C₂×Q₈	1	2	yes	—
112	C₂×Dic₂₅₁	—	2	1	no	—
113	C₂×Dic₂₅₁	—	2	1	no	—
114	C₂×Dic₂₅₁	—	2	1	no	—
115	C₂×Dic₂₅₁	—	2	1	no	—
116	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
119	C₄×D₂₅₁	—	2	4	no	—
120	C₄×D₂₅₁	—	2	2	no	—
121	C₄×D₂₅₁	—	4	1	no	—
122	C₄×D₂₅₁	—	4	2	no	—
123	C₄×D₂₅₁	C₂×Q₈	1	2	yes	—
124	C₄×D₂₅₁	#41	1	2	yes	—
125	C₂₅₁⋊C₈	—	2	1	no	—
126	C₂₅₁⋊C₈	—	2	1	no	—
127	Q₈×C₂₅₁	—	2	1	no	—
128	Q₈×C₂₅₁	C₂×Q₈	1	2	yes	—
129	C₂×C₄×C₂₅₁	—	2	1	no	—
130	C₂×C₄×C₂₅₁	—	2	1	no	—
131	C₂×C₄×C₂₅₁	—	2	1	no	—
132	C₂×C₄×C₂₅₁	—	2	1	no	—
133	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
136	C₈×C₂₅₁	—	2	1	no	—
137	C₈×C₂₅₁	—	2	1	no	—
138	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₂₅₁	—	2	1	no	—
140	C₂×D₂₅₁	—	2	1	no	—
141	C₂×D₂₅₁	C₂²⋊Q₈	1	2	yes	—
142	C₄×C₂₅₁	—	2	2	no	—
143	C₄×C₂₅₁	—	2	1	no	—
144	C₄×C₂₅₁	—	2	1	no	—
145	C₄×C₂₅₁	—	4	1	no	—
146	C₄×C₂₅₁	—	4	1	no	—
147	C₄×C₂₅₁	—	4	1	no	—
148	C₄×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
149	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
150	Dic₂₅₁	—	2	2	no	—
151	Dic₂₅₁	—	2	1	no	—
152	Dic₂₅₁	—	4	1	no	—
153	Dic₂₅₁	—	4	1	no	—
154	Dic₂₅₁	—	4	1	no	—
155	Dic₂₅₁	C₂²⋊Q₈	1	2	yes	—
156	Dic₂₅₁	#137	1	2	yes	—
157	C₂×C₂₅₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₂×C₂₅₁	C₂×#137	1	1	yes	characteristic
159	C₂×C₂₅₁	#575	1	1	yes	characteristic
160	D₂₅₁	—	2	2	no	—
161	C₂₅₁	#3078	1	1	yes	characteristic, complemented, Sylow
162	#3078	—	251	1	no	maximal, Sylow
163	#533	—	251	1	no	—
164	C₂×C₄×Q₈	—	251	1	no	—
165	#479	—	251	1	no	—
166	#479	—	251	1	no	—
167	#479	—	251	1	no	—
168	#479	D₂₅₁	1	1	yes	characteristic, complemented
169	#477	—	251	1	no	—
170	#477	—	251	1	no	—
171	#475	—	251	1	no	—
172	#475	—	251	1	no	—
173	#428	—	251	1	no	—
174	#428	—	251	1	no	—
175	#423	—	251	1	no	—
176	#419	—	251	1	no	—
177	#419	—	251	1	no	—
178	C₂²×Q₈	—	251	1	no	—
179	C₂×M₁₆	—	251	1	no	—
180	C₂×M₁₆	—	251	1	no	—
181	C₄⋊Q₈	—	251	1	no	—
182	C₄⋊Q₈	—	251	1	no	—
183	#125	—	251	1	no	—
184	#125	C₂×D₂₅₁	1	1	yes	characteristic
185	C₂²⋊Q₈	—	502	1	no	—
186	C₂²⋊Q₈	—	502	1	no	—
187	C₄×Q₈	—	502	1	no	—
188	C₄×Q₈	—	502	1	no	—
189	C₄×Q₈	—	502	1	no	—
190	C₄×Q₈	—	502	1	no	—
191	C₄×Q₈	—	251	1	no	—
192	C₄×Q₈	—	251	1	no	—
193	C₄×Q₈	—	251	1	no	—
194	C₄×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
195	#117	—	251	1	no	—
196	#117	—	251	1	no	—
197	C₂×C₄⋊C₄	—	502	1	no	—
198	C₂×C₄⋊C₄	—	251	1	no	—
199	C₂×C₄²	—	502	1	no	—
200	C₂×C₄²	—	251	1	no	—
201	#107	—	251	1	no	—
202	#107	—	251	1	no	—
203	#107	—	251	1	no	—
204	#107	C₂×D₂₅₁	1	1	yes	characteristic
205	#106	—	251	1	no	—
206	#106	—	251	1	no	—
207	#106	—	251	1	no	—
208	#106	C₂×D₂₅₁	1	1	yes	characteristic
209	C₄⋊C₈	—	251	1	no	—
210	C₄⋊C₈	—	251	1	no	—
211	C₄⋊C₈	—	251	1	no	—
212	C₄⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
213	#103	—	251	1	no	—
214	#103	—	251	1	no	—
215	#103	—	251	1	no	—
216	#103	—	251	1	no	—
217	#103	—	251	1	no	—
218	#103	—	251	1	no	—
219	#103	C₂×D₂₅₁	1	1	yes	characteristic
220	#103	C₂×D₂₅₁	1	1	yes	characteristic
221	C₂×Q₈	—	502	1	no	—
222	C₂×Q₈	—	502	1	no	—
223	C₂×Q₈	—	502	1	no	—
224	C₂×Q₈	—	502	2	no	—
225	C₂×Q₈	—	251	4	no	—
226	C₂×Q₈	—	251	1	no	—
227	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
228	C₂²×C₄	—	502	1	no	—
229	C₂²×C₄	—	502	1	no	—
230	C₂²×C₄	—	251	1	no	—
231	C₂²×C₄	—	251	1	no	—
232	C₂²×C₄	—	251	1	no	—
233	M₁₆	—	502	2	no	—
234	M₁₆	—	502	2	no	—
235	C₂×C₈	—	251	1	no	—
236	C₂×C₈	—	251	1	no	—
237	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	502	1	no	—
243	C₄⋊C₄	—	502	1	no	—
244	C₄⋊C₄	—	502	1	no	—
245	C₄⋊C₄	—	502	1	no	—
246	C₄⋊C₄	—	502	1	no	—
247	C₄⋊C₄	—	502	1	no	—
248	C₄⋊C₄	—	502	1	no	—
249	C₄⋊C₄	—	502	1	no	—
250	C₄⋊C₄	—	251	1	no	—
251	C₄⋊C₄	—	251	1	no	—
252	C₄⋊C₄	—	251	1	no	—
253	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
256	C₂²⋊C₄	—	502	1	no	—
257	C₂²⋊C₄	—	502	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	502	1	no	—
260	C₄²	—	502	1	no	—
261	C₄²	—	502	1	no	—
262	C₄²	—	502	1	no	—
263	C₄²	—	502	1	no	—
264	C₄²	—	251	1	no	—
265	C₄²	—	251	1	no	—
266	C₄²	—	251	1	no	—
267	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
268	C₂³	—	251	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	1004	2	no	—
271	Q₈	—	1004	2	no	—
272	Q₈	—	502	4	no	—
273	Q₈	—	502	1	no	—
274	Q₈	—	251	2	no	—
275	Q₈	Q₈×D₂₅₁	1	2	yes	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	1004	1	no	—
281	C₂×C₄	—	1004	2	no	—
282	C₂×C₄	—	502	4	no	—
283	C₂×C₄	—	502	2	no	—
284	C₂×C₄	—	502	1	no	—
285	C₂×C₄	—	502	1	no	—
286	C₂×C₄	—	502	1	no	—
287	C₂×C₄	—	502	1	no	—
288	C₂×C₄	—	251	1	no	—
289	C₂×C₄	—	251	1	no	—
290	C₂×C₄	—	251	1	no	—
291	C₂×C₄	—	251	2	no	—
292	C₂×C₄	—	251	2	no	—
293	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
294	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
295	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	502	1	no	—
299	C₈	—	502	1	no	—
300	C₂²	—	502	1	no	—
301	C₂²	—	502	1	no	—
302	C₂²	—	251	2	no	—
303	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	1004	1	no	—
311	C₄	—	1004	1	no	—
312	C₄	—	1004	1	no	—
313	C₄	—	502	1	no	—
314	C₄	—	502	2	no	—
315	C₄	—	251	2	no	—
316	C₄	—	251	2	no	—
317	C₄	C₂×Q₈×D₂₅₁	1	1	yes	characteristic
318	C₄	#41×D₂₅₁	1	1	yes	characteristic
319	C₂	—	502	2	no	—
320	C₂	C₂²⋊Q₈×D₂₅₁	1	1	yes	characteristic
321	C₂	#137×D₂₅₁	1	1	yes	characteristic
322	C₂	#30632484	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#479×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂²×D₂₅₁	C₂×D₄
hypercenter	#479	D₂₅₁
hypocenter	C₂₅₁	#3078
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. #30631995	C₂	semidirect
4. #30631994	C₂	semidirect
5. #30631991	C₂	semidirect
6. #30631981	C₂	semidirect
7. #30631980	C₂	semidirect
8. #30631971	C₂	semidirect
9. #30631970	C₂	semidirect
15. #479×C₂₅₁	C₂	semidirect
64. Q₈×D₂₅₁	Q₈	semidirect
161. C₂₅₁	#3078	semidirect
168. #479	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632243	2
C₄×Q₈×D₂₅₁	2
#30631995	2
#30631994	2
#30631991	2
#30631981	2
#30631980	2
#30631971	2
#30631970	2
#30631728	2
#30631726	2
#30631709	2
#30631694	2
#479×C₂₅₁	2
#3078	251