Subgroups — #44553247

Order 32128.

504 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548671	C₂	1	1	yes	maximal, characteristic
2	#30631584	C₂	1	1	yes	maximal, characteristic, complemented
3	#30631582	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631581	C₂	1	1	yes	maximal, characteristic
5	#30631580	C₂	1	1	yes	maximal, characteristic
6	#30631575	C₂	1	1	yes	maximal, characteristic
7	#30631552	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631551	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631545	C₂	1	1	yes	maximal, characteristic
10	C₄×Dic₂₅₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	#30631538	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²⋊C₄×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
13	#30631533	C₂	1	1	yes	maximal, characteristic, complemented
14	#30631532	C₂	1	1	yes	maximal, characteristic, complemented
15	#388×C₂₅₁	C₂	1	1	yes	maximal, characteristic
16	C₂³×Dic₂₅₁	C₂²	1	1	yes	characteristic
17	#22548671	C₄	1	8	yes	complemented
18	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
35	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
36	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
37	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
38	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
39	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
40	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
41	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
42	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
43	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
44	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
49	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
50	#95×C₂₅₁	C₂²	1	1	yes	characteristic
51	#95×C₂₅₁	C₂²	1	1	yes	characteristic
52	C₂²×Dic₂₅₁	—	2	2	no	—
53	C₂²×Dic₂₅₁	—	2	2	no	—
54	C₂²×Dic₂₅₁	C₂×C₄	1	4	yes	—
55	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
56	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
57	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
58	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
59	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
60	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
61	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
62	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
63	C₄⋊Dic₂₅₁	—	4	1	no	—
64	Dic₂₅₁⋊C₄	—	2	2	no	—
65	Dic₂₅₁⋊C₄	—	2	2	no	—
66	Dic₂₅₁⋊C₄	—	4	1	no	—
67	Dic₂₅₁⋊C₄	—	4	1	no	—
68	Dic₂₅₁⋊C₄	C₂×C₄	1	4	yes	—
69	Dic₂₅₁⋊C₄	C₂×C₄	1	4	yes	—
70	C₂²⋊Dic₂₅₁	—	2	2	no	—
71	C₂²⋊Dic₂₅₁	—	4	1	no	—
72	C₂²⋊Dic₂₅₁	C₂×C₄	1	4	yes	—
73	C₄×Dic₂₅₁	—	2	2	no	—
74	C₄×Dic₂₅₁	—	2	2	no	—
75	C₄×Dic₂₅₁	—	2	2	no	—
76	C₄×Dic₂₅₁	—	2	2	no	—
77	C₄×Dic₂₅₁	—	2	2	no	—
78	C₄×Dic₂₅₁	C₂×C₄	1	4	yes	—
79	C₄×Dic₂₅₁	C₂×C₄	1	4	yes	—
80	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₂₅₁	—	4	1	no	—
88	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
89	C₂²⋊C₄×C₂₅₁	—	4	1	no	—
90	C₂²⋊C₄×C₂₅₁	C₂×C₄	1	4	yes	—
91	C₄²×C₂₅₁	—	2	2	no	—
92	C₂×Dic₂₅₁	—	2	1	no	—
93	C₂×Dic₂₅₁	—	2	1	no	—
94	C₂×Dic₂₅₁	—	2	1	no	—
95	C₂×Dic₂₅₁	—	2	1	no	—
96	C₂×Dic₂₅₁	—	2	1	no	—
97	C₂×Dic₂₅₁	—	2	1	no	—
98	C₂×Dic₂₅₁	—	2	1	no	—
99	C₂×Dic₂₅₁	—	2	1	no	—
100	C₂×Dic₂₅₁	—	2	8	no	—
101	C₂×Dic₂₅₁	—	2	1	no	—
102	C₂×Dic₂₅₁	—	2	1	no	—
103	C₂×Dic₂₅₁	—	2	1	no	—
104	C₂×Dic₂₅₁	—	2	1	no	—
105	C₂×Dic₂₅₁	—	2	1	no	—
106	C₂×Dic₂₅₁	—	2	1	no	—
107	C₂×Dic₂₅₁	—	2	1	no	—
108	C₂×Dic₂₅₁	—	2	1	no	—
109	C₂×Dic₂₅₁	—	2	1	no	—
110	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
111	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
112	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
113	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
114	C₂×Dic₂₅₁	#41	1	2	yes	—
115	C₂×Dic₂₅₁	#41	1	2	yes	—
116	C₂×Dic₂₅₁	#41	1	2	yes	—
117	C₂³×C₂₅₁	—	2	1	no	—
118	C₂³×C₂₅₁	—	2	1	no	—
119	C₂³×C₂₅₁	—	2	1	no	—
120	C₂³×C₂₅₁	—	2	1	no	—
121	C₂³×C₂₅₁	—	2	1	no	—
122	C₂³×C₂₅₁	—	2	1	no	—
123	C₂³×C₂₅₁	C₂²×C₄	1	2	yes	—
124	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
125	C₂×C₄×C₂₅₁	—	2	1	no	—
126	C₂×C₄×C₂₅₁	—	2	1	no	—
127	C₂×C₄×C₂₅₁	—	2	1	no	—
128	C₂×C₄×C₂₅₁	—	2	1	no	—
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	—	2	1	no	—
134	C₂×C₄×C₂₅₁	—	2	1	no	—
135	C₂×C₄×C₂₅₁	—	2	1	no	—
136	C₂×C₄×C₂₅₁	—	2	1	no	—
137	C₂×C₄×C₂₅₁	—	2	1	no	—
138	C₂×C₄×C₂₅₁	—	2	1	no	—
139	C₂×C₄×C₂₅₁	—	2	1	no	—
140	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
141	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
142	C₂×C₄×C₂₅₁	#41	1	2	yes	—
143	C₂²×C₂₅₁	—	2	2	no	—
144	C₂²×C₂₅₁	—	2	2	no	—
145	C₂²×C₂₅₁	—	2	2	no	—
146	C₂²×C₂₅₁	—	4	1	no	—
147	C₂²×C₂₅₁	—	4	1	no	—
148	C₂²×C₂₅₁	—	4	1	no	—
149	C₂²×C₂₅₁	—	4	1	no	—
150	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
151	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
152	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
153	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
154	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
155	C₂²×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
157	C₄×C₂₅₁	—	2	2	no	—
158	C₄×C₂₅₁	—	2	2	no	—
159	C₄×C₂₅₁	—	2	2	no	—
160	C₄×C₂₅₁	—	4	1	no	—
161	C₄×C₂₅₁	—	4	1	no	—
162	C₄×C₂₅₁	—	4	1	no	—
163	Dic₂₅₁	—	2	2	no	—
164	Dic₂₅₁	—	2	2	no	—
165	Dic₂₅₁	—	2	2	no	—
166	Dic₂₅₁	—	2	2	no	—
167	Dic₂₅₁	—	4	1	no	—
168	Dic₂₅₁	—	4	1	no	—
169	Dic₂₅₁	—	4	1	no	—
170	Dic₂₅₁	#117	1	4	yes	—
171	C₂×C₂₅₁	—	4	2	no	—
172	C₂×C₂₅₁	C₂×#117	1	1	yes	characteristic
173	C₂×C₂₅₁	C₄×#41	1	1	yes	characteristic
174	C₂×C₂₅₁	#520	1	1	yes	characteristic
175	C₂×C₂₅₁	#552	1	1	yes	characteristic
176	C₂×C₂₅₁	#552	1	1	yes	characteristic
177	C₂×C₂₅₁	#555	1	1	yes	characteristic
178	C₂×C₂₅₁	#555	1	1	yes	characteristic
179	C₂₅₁	#2346	1	1	yes	characteristic, complemented, Sylow
180	#2346	—	251	1	no	maximal, Sylow
181	C₂×#117	—	251	1	no	—
182	#389	—	251	1	no	—
183	#389	—	251	1	no	—
184	#388	—	251	1	no	—
185	#388	—	251	1	no	—
186	#388	—	251	1	no	—
187	#388	D₂₅₁	1	1	yes	characteristic
188	#382	—	251	1	no	—
189	#382	—	251	1	no	—
190	#380	—	251	1	no	—
191	C₄×C₄⋊C₄	—	251	1	no	—
192	C₄×C₄⋊C₄	—	251	1	no	—
193	C₄×C₂²⋊C₄	—	251	1	no	—
194	#376	—	251	1	no	—
195	#376	—	251	1	no	—
196	C₂³×C₄	—	251	1	no	—
197	#117	—	251	8	no	—
198	C₂×C₄⋊C₄	—	251	1	no	—
199	C₂×C₄⋊C₄	—	251	1	no	—
200	C₂×C₄⋊C₄	—	251	1	no	—
201	C₂×C₄⋊C₄	—	251	1	no	—
202	C₂×C₄⋊C₄	—	251	1	no	—
203	C₂×C₄⋊C₄	—	251	1	no	—
204	C₂×C₄⋊C₄	—	251	1	no	—
205	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
206	C₂×C₂²⋊C₄	—	251	1	no	—
207	C₂×C₂²⋊C₄	—	251	1	no	—
208	C₂×C₂²⋊C₄	—	251	1	no	—
209	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
212	C₂×C₄²	—	251	1	no	—
213	C₂×C₄²	—	251	1	no	—
214	C₂×C₄²	—	251	1	no	—
215	C₂×C₄²	—	251	1	no	—
216	C₂×C₄²	—	251	1	no	—
217	C₂×C₄²	—	251	1	no	—
218	C₂×C₄²	—	251	1	no	—
219	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
220	#95	—	251	1	no	—
221	#95	—	251	1	no	—
222	#95	—	251	1	no	—
223	#95	—	251	1	no	—
224	#95	—	251	1	no	—
225	#95	—	251	1	no	—
226	#95	—	251	1	no	—
227	#95	—	251	1	no	—
228	#95	—	251	1	no	—
229	#95	—	251	1	no	—
230	#95	C₂×D₂₅₁	1	1	yes	characteristic
231	#95	C₂×D₂₅₁	1	1	yes	characteristic
232	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
233	C₂²×C₄	—	502	2	no	—
234	C₂²×C₄	—	502	2	no	—
235	C₂²×C₄	—	251	1	no	—
236	C₂²×C₄	—	251	1	no	—
237	C₂²×C₄	—	251	1	no	—
238	C₂²×C₄	—	251	4	no	—
239	C₂²×C₄	—	251	1	no	—
240	C₂²×C₄	—	251	1	no	—
241	C₂²×C₄	—	251	1	no	—
242	C₂²×C₄	—	251	1	no	—
243	C₂²×C₄	—	251	1	no	—
244	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
245	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
246	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
247	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	1004	1	no	—
252	C₄⋊C₄	—	1004	1	no	—
253	C₄⋊C₄	—	1004	1	no	—
254	C₄⋊C₄	—	502	2	no	—
255	C₄⋊C₄	—	502	2	no	—
256	C₄⋊C₄	—	251	4	no	—
257	C₄⋊C₄	—	251	4	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	4	1	no	—
260	C₂²⋊C₄	—	1004	1	no	—
261	C₂²⋊C₄	—	502	2	no	—
262	C₂²⋊C₄	—	251	4	no	—
263	C₂²⋊C₄	C₄×D₂₅₁	1	4	yes	—
264	C₄²	—	2	2	no	—
265	C₄²	—	502	2	no	—
266	C₄²	—	502	2	no	—
267	C₄²	—	502	2	no	—
268	C₄²	—	502	2	no	—
269	C₄²	—	502	2	no	—
270	C₄²	—	251	4	no	—
271	C₄²	—	251	4	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	C₂×C₄×D₂₅₁	1	2	yes	—
279	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
280	C₂×C₄	—	2	1	no	—
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	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	502	1	no	—
296	C₂×C₄	—	502	1	no	—
297	C₂×C₄	—	502	1	no	—
298	C₂×C₄	—	502	1	no	—
299	C₂×C₄	—	502	1	no	—
300	C₂×C₄	—	502	1	no	—
301	C₂×C₄	—	502	1	no	—
302	C₂×C₄	—	502	1	no	—
303	C₂×C₄	—	502	1	no	—
304	C₂×C₄	—	502	1	no	—
305	C₂×C₄	—	502	1	no	—
306	C₂×C₄	—	502	1	no	—
307	C₂×C₄	—	502	1	no	—
308	C₂×C₄	—	502	1	no	—
309	C₂×C₄	—	502	1	no	—
310	C₂×C₄	—	502	1	no	—
311	C₂×C₄	—	502	1	no	—
312	C₂×C₄	—	502	8	no	—
313	C₂×C₄	—	251	2	no	—
314	C₂×C₄	—	251	2	no	—
315	C₂×C₄	—	251	2	no	—
316	C₂×C₄	—	251	2	no	—
317	C₂×C₄	—	251	2	no	—
318	C₂×C₄	—	251	2	no	—
319	C₂×C₄	—	251	2	no	—
320	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
321	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
322	C₂×C₄	#16860809	1	2	yes	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
331	C₂²	#41×D₂₅₁	1	1	yes	characteristic
332	C₂²	#41×D₂₅₁	1	1	yes	characteristic
333	C₂²	#41×D₂₅₁	1	1	yes	characteristic
334	C₂²	C₂×#16860809	1	1	yes	characteristic
335	C₂²	#22548760	1	1	yes	characteristic
336	C₂²	#22548762	1	1	yes	characteristic
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	1004	1	no	—
344	C₄	—	1004	1	no	—
345	C₄	—	1004	1	no	—
346	C₄	—	502	2	no	—
347	C₄	—	502	2	no	—
348	C₄	—	502	2	no	—
349	C₄	—	502	2	no	—
350	C₄	—	251	4	no	—
351	C₂	—	4	2	no	—
352	C₂	#117×D₂₅₁	1	1	yes	characteristic
353	C₂	C₄×#16860809	1	1	yes	characteristic
354	C₂	#30632166	1	1	yes	characteristic
355	C₂	#30632373	1	1	yes	characteristic
356	C₂	#30632374	1	1	yes	characteristic
357	C₂	#30632399	1	1	yes	characteristic
358	C₂	#30632401	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator	C₂²×C₂₅₁	C₂³×C₄
Fitting	#388×C₂₅₁	C₂
socle, cosocle	C₂³×C₂₅₁	C₂⁴
prime radical	C₂⁴×C₂₅₁	C₂³
hypercenter	#388	D₂₅₁
hypocenter	C₂₅₁	#2346
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ #388

Complements

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

N	H ≅ G/N	type
2. #30631584	C₂	semidirect
3. #30631582	C₂	semidirect
7. #30631552	C₂	semidirect
8. #30631551	C₂	semidirect
10. C₄×Dic₂₅₁⋊C₄	C₂	semidirect
11. #30631538	C₂	semidirect
13. #30631533	C₂	semidirect
14. #30631532	C₂	semidirect
17. #22548671	C₄	semidirect
179. C₂₅₁	#2346	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548671	2
#30631584	2
#30631582	2
#30631581	2
#30631580	2
#30631575	2
#30631552	2
#30631551	2
#30631545	2
C₄×Dic₂₅₁⋊C₄	2
#30631538	2
C₂²⋊C₄×Dic₂₅₁	2
#30631533	2
#30631532	2
#388×C₂₅₁	2
#2346	251