Subgroups — #44560663

Order 32128.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632298	C₂	1	1	yes	maximal, characteristic
2	#30632163	C₂	1	1	yes	maximal, characteristic
3	#30632027	C₂	1	1	yes	maximal, characteristic
4	#30632026	C₂	1	1	yes	maximal, characteristic
5	#30631995	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631994	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631894	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631841	C₂	1	1	yes	maximal, characteristic
9	#30631841	C₂	1	1	yes	maximal, characteristic
10	#30631794	C₂	1	1	yes	maximal, characteristic, complemented
11	#30631791	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₁₆×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
13	#30631779	C₂	1	1	yes	maximal, characteristic, complemented
14	#30631747	C₂	1	1	yes	maximal, characteristic
15	#462×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₂₅₁	C₂²	1	1	yes	characteristic
17	C₂×#16860800	C₂²	1	1	yes	characteristic
18	C₂×#16860800	C₂²	1	1	yes	characteristic
19	#22548710	—	2	1	no	—
20	#22548710	—	2	1	no	—
21	#22548707	C₂²	1	1	yes	characteristic
22	#22548705	—	2	1	no	—
23	#22548705	—	2	1	no	—
24	#22548691	C₂²	1	1	yes	characteristic
25	#22548691	C₂²	1	1	yes	characteristic
26	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
27	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
28	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
29	C₄×Dic₅₀₂	—	2	1	no	—
30	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
31	#22548680	—	2	1	no	—
32	#22548680	C₂²	1	1	yes	characteristic
33	#22548673	—	2	1	no	—
34	#22548673	C₂²	1	1	yes	characteristic
35	#22548672	—	2	1	no	—
36	#22548672	C₂²	1	1	yes	characteristic
37	#22548649	C₂²	1	1	yes	characteristic
38	#22548648	C₂²	1	1	yes	characteristic
39	#22548647	C₂²	1	1	yes	characteristic
40	#22548646	C₂²	1	1	yes	characteristic
41	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
42	#22548640	C₂²	1	1	yes	characteristic
43	#22548640	C₂²	1	1	yes	characteristic
44	#22548639	C₂²	1	1	yes	characteristic
45	#22548639	C₂²	1	1	yes	characteristic
46	#22548639	C₂²	1	1	yes	characteristic
47	#22548639	C₂²	1	1	yes	characteristic
48	(C₂×D₂₅₁)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₂₅₁)⋊C₈	C₂²	1	1	yes	characteristic
50	#22548625	C₂²	1	1	yes	characteristic
51	C₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
54	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
55	C₄×Q₈×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	C₂³	1	1	yes	characteristic
61	Q₈×D₂₅₁	—	2	2	no	—
62	Q₈×D₂₅₁	—	2	2	no	—
63	C₂×C₄×D₂₅₁	—	2	1	no	—
64	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
65	#16860800	—	2	2	no	—
66	#16860800	—	2	2	no	—
67	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₂₅₁	—	2	1	no	—
70	C₄⋊Dic₂₅₁	—	2	1	no	—
71	C₄⋊Dic₂₅₁	—	2	1	no	—
72	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
75	Dic₂₅₁⋊C₄	—	2	1	no	—
76	Dic₂₅₁⋊C₄	—	2	1	no	—
77	Dic₂₅₁⋊C₄	—	2	1	no	—
78	Dic₂₅₁⋊C₄	—	2	1	no	—
79	Dic₂₅₁⋊C₄	—	2	1	no	—
80	Dic₂₅₁⋊C₄	—	2	1	no	—
81	Dic₂₅₁⋊C₄	D₄	1	1	yes	characteristic
82	Dic₂₅₁⋊C₄	D₄	1	1	yes	characteristic
83	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
84	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
85	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
86	(C₂×D₂₅₁)⋊C₄	D₄	1	1	yes	characteristic
87	(C₂×D₂₅₁)⋊C₄	D₄	1	1	yes	characteristic
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	—	2	1	no	—
93	C₄×Dic₂₅₁	—	2	1	no	—
94	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
98	Q₁₆×C₂₅₁	—	2	2	no	—
99	Q₁₆×C₂₅₁	—	2	2	no	—
100	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₂₅₁	—	2	1	no	—
103	C₄⋊C₄×C₂₅₁	—	2	1	no	—
104	C₄⋊C₄×C₂₅₁	—	2	1	no	—
105	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
106	C₄²×C₂₅₁	—	2	1	no	—
107	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
108	C₂²×D₂₅₁	C₂×D₄	1	1	yes	characteristic
109	Dic₅₀₂	—	2	4	no	—
110	Dic₅₀₂	—	2	1	no	—
111	Dic₅₀₂	#41	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	1	no	—
121	C₄×D₂₅₁	—	2	1	no	—
122	C₂₅₁⋊C₈	—	2	1	no	—
123	C₂₅₁⋊C₈	—	2	1	no	—
124	Q₈×C₂₅₁	—	2	1	no	—
125	Q₈×C₂₅₁	—	2	1	no	—
126	Q₈×C₂₅₁	—	2	1	no	—
127	Q₈×C₂₅₁	—	2	1	no	—
128	Q₈×C₂₅₁	—	2	1	no	—
129	Q₈×C₂₅₁	#41	1	2	yes	—
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	—	2	1	no	—
134	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
137	C₈×C₂₅₁	—	2	1	no	—
138	C₈×C₂₅₁	—	2	1	no	—
139	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₂₅₁	—	2	1	no	—
141	C₂×D₂₅₁	—	2	1	no	—
142	C₂×D₂₅₁	—	2	1	no	—
143	C₄×C₂₅₁	—	2	2	no	—
144	C₄×C₂₅₁	—	2	1	no	—
145	C₄×C₂₅₁	—	2	1	no	—
146	C₄×C₂₅₁	—	4	1	no	—
147	C₄×C₂₅₁	—	4	1	no	—
148	C₄×C₂₅₁	—	4	1	no	—
149	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
150	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
151	Dic₂₅₁	—	2	2	no	—
152	Dic₂₅₁	—	2	1	no	—
153	Dic₂₅₁	—	2	1	no	—
154	Dic₂₅₁	—	2	1	no	—
155	Dic₂₅₁	—	4	1	no	—
156	Dic₂₅₁	—	4	1	no	—
157	Dic₂₅₁	—	4	1	no	—
158	C₂×C₂₅₁	#525	1	1	yes	characteristic
159	C₂×C₂₅₁	#578	1	1	yes	characteristic
160	C₂×C₂₅₁	#578	1	1	yes	characteristic
161	D₂₅₁	—	4	1	no	—
162	C₂₅₁	#3127	1	1	yes	characteristic, complemented, Sylow
163	#3127	—	251	1	no	maximal, Sylow
164	#541	—	251	1	no	—
165	#519	—	251	1	no	—
166	#485	—	251	1	no	—
167	#485	—	251	1	no	—
168	#479	—	251	1	no	—
169	#479	—	251	1	no	—
170	#462	—	251	1	no	—
171	#462	D₂₅₁	1	1	yes	characteristic, complemented
172	#451	—	251	1	no	—
173	#451	—	251	1	no	—
174	#441	—	251	1	no	—
175	#441	—	251	1	no	—
176	C₄×Q₁₆	—	251	1	no	—
177	C₄×Q₁₆	—	251	1	no	—
178	#433	—	251	1	no	—
179	C₂²×Q₈	—	251	1	no	—
180	C₂×Q₁₆	—	251	1	no	—
181	C₂×Q₁₆	—	251	1	no	—
182	C₂×Q₁₆	C₂×D₂₅₁	1	1	yes	characteristic
183	C₂×Q₁₆	C₂×D₂₅₁	1	1	yes	characteristic
184	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
185	#126	—	502	1	no	—
186	#126	—	502	1	no	—
187	#125	—	502	1	no	—
188	#125	—	502	1	no	—
189	#125	—	251	1	no	—
190	C₂²⋊Q₈	—	251	1	no	—
191	C₂²⋊Q₈	—	251	1	no	—
192	C₄×Q₈	—	502	1	no	—
193	C₄×Q₈	—	502	1	no	—
194	C₄×Q₈	—	251	1	no	—
195	C₄×Q₈	—	251	1	no	—
196	C₄×Q₈	—	251	1	no	—
197	C₄×Q₈	—	251	1	no	—
198	C₄×Q₈	—	251	1	no	—
199	C₄×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
200	#117	—	502	1	no	—
201	#117	—	502	1	no	—
202	#117	—	251	1	no	—
203	#117	—	251	1	no	—
204	#107	—	251	1	no	—
205	#107	—	251	1	no	—
206	#106	—	251	1	no	—
207	#106	—	251	1	no	—
208	C₄⋊C₈	—	251	1	no	—
209	C₄⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
210	#103	—	251	1	no	—
211	#103	—	251	1	no	—
212	#103	—	251	1	no	—
213	#103	—	251	1	no	—
214	#103	—	251	1	no	—
215	#103	—	251	1	no	—
216	#103	C₂×D₂₅₁	1	1	yes	characteristic
217	#103	C₂×D₂₅₁	1	1	yes	characteristic
218	C₂²⋊C₈	—	251	1	no	—
219	C₂²⋊C₈	—	251	1	no	—
220	#97	—	251	1	no	—
221	C₄×C₈	—	251	1	no	—
222	C₂×Q₈	—	502	1	no	—
223	C₂×Q₈	—	502	2	no	—
224	C₂×Q₈	—	502	2	no	—
225	C₂×Q₈	—	251	1	no	—
226	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
227	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
228	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
229	C₂²×C₄	—	502	1	no	—
230	C₂²×C₄	—	251	1	no	—
231	Q₁₆	—	2	2	no	—
232	Q₁₆	—	2	2	no	—
233	Q₁₆	—	502	2	no	—
234	Q₁₆	—	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	—	502	1	no	—
251	C₄⋊C₄	—	251	1	no	—
252	C₄⋊C₄	—	251	1	no	—
253	C₄⋊C₄	—	251	1	no	—
254	C₄⋊C₄	—	251	1	no	—
255	C₄⋊C₄	—	251	1	no	—
256	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
257	C₂²⋊C₄	—	502	1	no	—
258	C₂²⋊C₄	—	502	1	no	—
259	C₂²⋊C₄	—	502	1	no	—
260	C₂²⋊C₄	—	251	1	no	—
261	C₂²⋊C₄	—	251	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	502	1	no	—
264	C₄²	—	502	1	no	—
265	C₄²	—	502	1	no	—
266	C₄²	—	502	1	no	—
267	C₄²	—	502	1	no	—
268	C₄²	—	502	1	no	—
269	C₄²	—	251	1	no	—
270	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
271	C₂³	—	251	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	502	4	no	—
278	Q₈	—	502	1	no	—
279	Q₈	—	251	2	no	—
280	Q₈	#16860809	1	2	yes	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	502	4	no	—
286	C₂×C₄	—	502	1	no	—
287	C₂×C₄	—	502	1	no	—
288	C₂×C₄	—	502	1	no	—
289	C₂×C₄	—	502	1	no	—
290	C₂×C₄	—	502	1	no	—
291	C₂×C₄	—	502	1	no	—
292	C₂×C₄	—	251	1	no	—
293	C₂×C₄	—	251	1	no	—
294	C₂×C₄	—	251	1	no	—
295	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
296	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	502	1	no	—
301	C₈	—	502	1	no	—
302	C₂²	—	502	1	no	—
303	C₂²	—	502	1	no	—
304	C₂²	—	502	1	no	—
305	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	1004	1	no	—
313	C₄	—	1004	1	no	—
314	C₄	—	1004	1	no	—
315	C₄	—	502	2	no	—
316	C₄	—	502	1	no	—
317	C₄	—	502	1	no	—
318	C₄	—	502	1	no	—
319	C₄	#41×D₂₅₁	1	1	yes	characteristic
320	C₄	C₂×#16860809	1	1	yes	characteristic
321	C₂	—	1004	1	no	—
322	C₂	#30632197	1	1	yes	characteristic
323	C₂	#30632500	1	1	yes	characteristic
324	C₂	#30632503	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#462×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂²×D₂₅₁	C₂×D₄
hypercenter	#462	D₂₅₁
hypocenter	C₂₅₁	#3127
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

Complements

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

N	H ≅ G/N	type
5. #30631995	C₂	semidirect
6. #30631994	C₂	semidirect
7. #30631894	C₂	semidirect
10. #30631794	C₂	semidirect
11. #30631791	C₂	semidirect
12. Q₁₆×Dic₂₅₁	C₂	semidirect
13. #30631779	C₂	semidirect
15. #462×C₂₅₁	C₂	semidirect
162. C₂₅₁	#3127	semidirect
171. #462	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632298	2
#30632163	2
#30632027	2
#30632026	2
#30631995	2
#30631994	2
#30631894	2
#30631841	2
#30631794	2
#30631791	2
Q₁₆×Dic₂₅₁	2
#30631779	2
#30631747	2
#462×C₂₅₁	2
#3127	251