Subgroups — #44553294

Order 32128.

660 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548674	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄⋊Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#30631580	C₂	1	2	yes	maximal, complemented
4	#30631573	C₂	1	2	yes	maximal, complemented
5	#30631567	C₂	1	2	yes	maximal
6	#30631553	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631547	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631547	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₄⋊Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊C₄×Dic₂₅₁	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×D₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₂₅₁	C₂²	1	2	yes	—
13	#22548674	—	2	4	no	—
14	C₂×C₄⋊Dic₂₅₁	—	2	4	no	—
15	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Dic₂₅₁	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₂₅₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₂₅₁	C₂²	1	2	yes	—
23	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
24	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
25	C₂₅₁⋊#95	C₂²	1	2	yes	—
26	C₂₅₁⋊#95	C₂²	1	2	yes	—
27	C₂₅₁⋊#95	C₂²	1	2	yes	—
28	C₂₅₁⋊#95	C₂²	1	2	yes	—
29	C₂₅₁⋊#95	C₂²	1	2	yes	—
30	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
31	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
32	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
33	C₄×D₄×C₂₅₁	C₄	1	8	yes	complemented
34	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
38	C₂²×Dic₂₅₁	—	2	4	no	—
39	C₂²×Dic₂₅₁	—	2	4	no	—
40	C₂²×Dic₂₅₁	—	4	2	no	—
41	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
42	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
43	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
44	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
45	C₄⋊Dic₂₅₁	—	2	4	no	—
46	C₄⋊Dic₂₅₁	—	2	2	no	—
47	C₄⋊Dic₂₅₁	—	2	2	no	—
48	C₄⋊Dic₂₅₁	—	2	2	no	—
49	C₄⋊Dic₂₅₁	D₄	1	4	yes	complemented
50	C₂²⋊Dic₂₅₁	—	2	4	no	—
51	C₂²⋊Dic₂₅₁	—	2	4	no	—
52	C₂²⋊Dic₂₅₁	—	4	2	no	—
53	C₄×Dic₂₅₁	—	2	4	no	—
54	C₄×Dic₂₅₁	—	4	2	no	—
55	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
56	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
57	C₂×D₄×C₂₅₁	—	2	2	no	—
58	C₂×D₄×C₂₅₁	—	2	2	no	—
59	C₂×D₄×C₂₅₁	C₂×C₄	1	4	yes	—
60	C₂²×C₄×C₂₅₁	—	2	2	no	—
61	C₂²×C₄×C₂₅₁	—	2	2	no	—
62	C₂²×C₄×C₂₅₁	—	2	2	no	—
63	C₂²×C₄×C₂₅₁	—	2	2	no	—
64	C₂²×C₄×C₂₅₁	C₂×C₄	1	4	yes	—
65	C₂²×C₄×C₂₅₁	C₂×C₄	1	4	yes	complemented
66	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₂₅₁	C₂×C₄	1	4	yes	complemented
72	C₂²⋊C₄×C₂₅₁	C₂×C₄	1	4	yes	—
73	C₂²⋊C₄×C₂₅₁	C₂×C₄	1	4	yes	complemented
74	C₄²×C₂₅₁	C₂×C₄	1	4	yes	complemented
75	C₂×Dic₂₅₁	—	2	2	no	—
76	C₂×Dic₂₅₁	—	2	2	no	—
77	C₂×Dic₂₅₁	—	2	2	no	—
78	C₂×Dic₂₅₁	—	2	2	no	—
79	C₂×Dic₂₅₁	—	2	2	no	—
80	C₂×Dic₂₅₁	—	2	2	no	—
81	C₂×Dic₂₅₁	—	2	2	no	—
82	C₂×Dic₂₅₁	—	2	2	no	—
83	C₂×Dic₂₅₁	—	2	2	no	—
84	C₂×Dic₂₅₁	—	2	2	no	—
85	C₂×Dic₂₅₁	—	2	2	no	—
86	C₂×Dic₂₅₁	—	4	8	no	—
87	C₂×Dic₂₅₁	C₂×D₄	1	4	yes	—
88	C₂³×C₂₅₁	—	2	1	no	—
89	C₂³×C₂₅₁	—	2	1	no	—
90	C₂³×C₂₅₁	—	2	1	no	—
91	C₂³×C₂₅₁	—	2	1	no	—
92	C₂³×C₂₅₁	—	2	1	no	—
93	C₂³×C₂₅₁	—	2	1	no	—
94	C₂³×C₂₅₁	—	2	1	no	—
95	C₂³×C₂₅₁	—	2	1	no	—
96	C₂³×C₂₅₁	—	2	1	no	—
97	C₂³×C₂₅₁	—	2	1	no	—
98	C₂³×C₂₅₁	C₂²×C₄	1	2	yes	—
99	C₂³×C₂₅₁	C₂²×C₄	1	2	yes	—
100	C₂³×C₂₅₁	#41	1	2	yes	—
101	C₂³×C₂₅₁	#41	1	2	yes	—
102	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
103	D₄×C₂₅₁	—	2	8	no	—
104	C₂×C₄×C₂₅₁	—	2	1	no	—
105	C₂×C₄×C₂₅₁	—	2	1	no	—
106	C₂×C₄×C₂₅₁	—	2	1	no	—
107	C₂×C₄×C₂₅₁	—	2	1	no	—
108	C₂×C₄×C₂₅₁	—	2	1	no	—
109	C₂×C₄×C₂₅₁	—	2	1	no	—
110	C₂×C₄×C₂₅₁	—	2	1	no	—
111	C₂×C₄×C₂₅₁	—	2	1	no	—
112	C₂×C₄×C₂₅₁	—	2	8	no	—
113	C₂×C₄×C₂₅₁	—	2	8	no	—
114	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
115	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
116	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
117	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
118	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
119	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
120	C₂×C₄×C₂₅₁	#41	1	2	yes	—
121	C₂²×C₂₅₁	—	2	2	no	—
122	C₂²×C₂₅₁	—	2	2	no	—
123	C₂²×C₂₅₁	—	2	2	no	—
124	C₂²×C₂₅₁	—	2	2	no	—
125	C₂²×C₂₅₁	—	2	2	no	—
126	C₂²×C₂₅₁	—	2	2	no	—
127	C₂²×C₂₅₁	—	2	2	no	—
128	C₂²×C₂₅₁	—	2	2	no	—
129	C₂²×C₂₅₁	—	2	2	no	—
130	C₂²×C₂₅₁	—	4	1	no	—
131	C₂²×C₂₅₁	—	4	1	no	—
132	C₂²×C₂₅₁	—	4	1	no	—
133	C₂²×C₂₅₁	—	4	1	no	—
134	C₂²×C₂₅₁	#117	1	4	yes	complemented
135	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
136	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
137	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
138	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
139	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
140	C₂²×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂²×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
142	C₄×C₂₅₁	—	2	2	no	—
143	C₄×C₂₅₁	—	2	2	no	—
144	C₄×C₂₅₁	—	2	2	no	—
145	C₄×C₂₅₁	—	2	2	no	—
146	C₄×C₂₅₁	C₄×D₄	1	4	yes	—
147	Dic₂₅₁	—	2	4	no	—
148	Dic₂₅₁	—	4	2	no	—
149	Dic₂₅₁	—	4	2	no	—
150	Dic₂₅₁	—	4	2	no	—
151	C₂×C₂₅₁	—	2	4	no	—
152	C₂×C₂₅₁	—	4	2	no	—
153	C₂×C₂₅₁	C₂×#117	1	1	yes	characteristic
154	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂×C₂₅₁	#520	1	1	yes	characteristic
156	C₂×C₂₅₁	#546	1	1	yes	characteristic
157	C₂×C₂₅₁	#548	1	1	yes	characteristic
158	C₂×C₂₅₁	#552	1	1	yes	characteristic
159	C₂×C₂₅₁	#553	1	1	yes	characteristic
160	C₂₅₁	#2352	1	1	yes	characteristic, complemented, Sylow
161	#2352	—	251	1	no	maximal, Sylow
162	C₂×C₄×D₄	D₂₅₁	1	1	yes	characteristic
163	C₂×#117	—	251	1	no	—
164	C₂²×C₄⋊C₄	—	251	1	no	—
165	#388	—	251	2	no	—
166	#387	—	251	2	no	—
167	#386	—	251	2	no	—
168	#382	—	251	1	no	—
169	#380	—	251	1	no	—
170	#380	—	251	1	no	—
171	C₄×C₄⋊C₄	—	251	1	no	—
172	C₄×C₂²⋊C₄	—	251	2	no	—
173	C₂²×D₄	C₂×D₂₅₁	1	1	yes	characteristic
174	C₂³×C₄	—	251	2	no	—
175	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
176	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
177	C₄×D₄	Dic₂₅₁	1	8	yes	complemented
178	#117	—	502	4	no	—
179	C₂×C₄⋊C₄	—	502	4	no	—
180	C₂×C₄⋊C₄	—	251	1	no	—
181	C₂×C₄⋊C₄	—	251	2	no	—
182	C₂×C₄⋊C₄	—	251	1	no	—
183	C₂×C₄⋊C₄	—	251	1	no	—
184	C₂×C₄⋊C₄	—	251	1	no	—
185	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	—	251	2	no	—
187	C₂×C₂²⋊C₄	—	251	2	no	—
188	C₂×C₂²⋊C₄	—	251	2	no	—
189	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
191	C₂×C₄²	—	251	2	no	—
192	C₂×C₄²	—	251	2	no	—
193	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
194	#95	—	251	2	no	—
195	#95	—	251	2	no	—
196	#95	—	251	2	no	—
197	#95	—	251	2	no	—
198	#95	—	251	2	no	—
199	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
200	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	C₂×Dic₂₅₁	1	4	yes	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	1004	2	no	—
209	C₂²×C₄	—	502	4	no	—
210	C₂²×C₄	—	502	4	no	—
211	C₂²×C₄	—	251	2	no	—
212	C₂²×C₄	—	251	2	no	—
213	C₂²×C₄	—	251	2	no	—
214	C₂²×C₄	—	251	2	no	—
215	C₂²×C₄	C₂×Dic₂₅₁	1	4	yes	—
216	C₂²×C₄	C₂×Dic₂₅₁	1	4	yes	complemented
217	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
219	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
221	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
222	C₄⋊C₄	—	502	4	no	—
223	C₄⋊C₄	—	502	2	no	—
224	C₄⋊C₄	—	502	2	no	—
225	C₄⋊C₄	—	502	2	no	—
226	C₄⋊C₄	—	251	4	no	—
227	C₄⋊C₄	C₂×Dic₂₅₁	1	4	yes	complemented
228	C₂²⋊C₄	—	1004	2	no	—
229	C₂²⋊C₄	—	502	4	no	—
230	C₂²⋊C₄	—	502	4	no	—
231	C₂²⋊C₄	C₂×Dic₂₅₁	1	4	yes	—
232	C₂²⋊C₄	C₂×Dic₂₅₁	1	4	yes	complemented
233	C₄²	—	1004	2	no	—
234	C₄²	—	502	4	no	—
235	C₄²	C₂×Dic₂₅₁	1	4	yes	complemented
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	C₂²×Dic₂₅₁	1	2	yes	—
247	C₂³	C₂²×Dic₂₅₁	1	2	yes	—
248	C₂³	#16860810	1	2	yes	—
249	C₂³	#16860810	1	2	yes	—
250	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
251	D₄	—	2	8	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	1004	8	no	—
263	C₂×C₄	—	502	2	no	—
264	C₂×C₄	—	502	2	no	—
265	C₂×C₄	—	502	2	no	—
266	C₂×C₄	—	502	2	no	—
267	C₂×C₄	—	502	2	no	—
268	C₂×C₄	—	502	2	no	—
269	C₂×C₄	—	502	2	no	—
270	C₂×C₄	—	502	2	no	—
271	C₂×C₄	—	502	2	no	—
272	C₂×C₄	—	502	2	no	—
273	C₂×C₄	—	502	2	no	—
274	C₂×C₄	—	251	4	no	—
275	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
276	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
277	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
278	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
279	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
280	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
281	C₂×C₄	#16860809	1	2	yes	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	#22548674	1	4	yes	complemented
296	C₂²	C₂³×Dic₂₅₁	1	1	yes	characteristic
297	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
298	C₂²	C₂×#16860809	1	1	yes	characteristic
299	C₂²	C₂×#16860810	1	1	yes	characteristic
300	C₂²	C₂×#16860810	1	1	yes	characteristic
301	C₂²	#22548761	1	1	yes	characteristic
302	C₂²	#22548762	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	1004	2	no	—
308	C₄	—	1004	2	no	—
309	C₄	—	1004	2	no	—
310	C₄	—	502	4	no	—
311	C₄	D₄×Dic₂₅₁	1	4	yes	—
312	C₂	—	2	4	no	—
313	C₂	—	4	2	no	—
314	C₂	C₂×#22548674	1	1	yes	characteristic
315	C₂	C₂×D₄×Dic₂₅₁	1	1	yes	characteristic
316	C₂	#30632170	1	1	yes	characteristic
317	C₂	#30632333	1	1	yes	characteristic
318	C₂	#30632346	1	1	yes	characteristic
319	C₂	#30632377	1	1	yes	characteristic
320	C₂	#30632383	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator	C₂²×C₂₅₁	C₂³×C₄
Fitting	C₂×C₄×D₄×C₂₅₁	C₂
socle, cosocle	C₂³×C₂₅₁	C₂⁴
prime radical	C₂²×D₄×C₂₅₁	C₂²
hypercenter	C₂×C₄×D₄	D₂₅₁
hypocenter	C₂₅₁	#2352
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#22548674	C₂	semidirect
2. C₂²×C₄⋊Dic₂₅₁	C₂	semidirect
3. #30631580	C₂	semidirect
4. #30631573	C₂	semidirect
6. #30631553	C₂	semidirect
7. #30631547	C₂	semidirect
8. #30631547	C₂	semidirect
9. C₄×C₄⋊Dic₂₅₁	C₂	semidirect
10. C₂²⋊C₄×Dic₂₅₁	C₂	semidirect
33. C₄×D₄×C₂₅₁	C₄	semidirect
49. C₄⋊Dic₂₅₁	D₄	semidirect
65. C₂²×C₄×C₂₅₁	C₂×C₄	semidirect
71. C₄⋊C₄×C₂₅₁	C₂×C₄	semidirect
73. C₂²⋊C₄×C₂₅₁	C₂×C₄	semidirect
74. C₄²×C₂₅₁	C₂×C₄	semidirect
134. C₂²×C₂₅₁	#117	semidirect
160. C₂₅₁	#2352	semidirect
177. C₄×D₄	Dic₂₅₁	semidirect
216. C₂²×C₄	C₂×Dic₂₅₁	semidirect
227. C₄⋊C₄	C₂×Dic₂₅₁	semidirect
232. C₂²⋊C₄	C₂×Dic₂₅₁	semidirect
235. C₄²	C₂×Dic₂₅₁	semidirect
295. C₂²	#22548674	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548674	2
C₂²×C₄⋊Dic₂₅₁	2
#30631580	2
#30631573	2
#30631567	2
#30631553	2
#30631547	2
C₄×C₄⋊Dic₂₅₁	2
C₂²⋊C₄×Dic₂₅₁	2
C₂×C₄×D₄×C₂₅₁	2
#2352	251