Subgroups — #44558889

Order 32128.

778 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	C₂×#22548761	C₂	1	1	yes	maximal, characteristic
2	C₂×#22548723	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632168	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631693	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631690	C₂	1	2	yes	maximal, complemented
6	#30631687	C₂	1	2	yes	maximal, complemented
7	#30631683	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631682	C₂	1	2	yes	maximal, complemented
9	C₂×#22548635	C₂	1	1	yes	maximal, characteristic, complemented
10	#102×D₂₅₁	C₂	1	2	yes	maximal, complemented
11	#417×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#22548761	—	2	4	no	—
13	#22548761	C₄	1	8	yes	complemented
14	C₂×#16860811	C₂²	1	2	yes	—
15	C₂×#16860810	—	2	1	no	—
16	C₂×#16860810	C₂²	1	1	yes	characteristic
17	C₂²×D₁₀₀₄	—	2	1	no	—
18	C₂²×D₁₀₀₄	C₂²	1	1	yes	characteristic
19	C₂×D₄×D₂₅₁	—	2	2	no	—
20	C₂×D₄×D₂₅₁	C₂²	1	2	yes	—
21	#22548723	—	2	4	no	—
22	C₂×#16860793	C₂²	1	2	yes	complemented
23	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₈×D₂₅₁	C₂²	1	2	yes	complemented
25	#22548680	—	2	1	no	—
26	#22548677	—	2	1	no	—
27	C₄×C₂²⋊D₂₅₁	—	2	2	no	—
28	#22548674	C₂²	1	1	yes	characteristic, complemented
29	#22548673	C₂²	1	2	yes	complemented
30	C₄⋊C₄×D₂₅₁	C₂²	1	2	yes	complemented
31	#22548639	C₂²	1	1	yes	characteristic, complemented
32	#22548639	C₂²	1	1	yes	characteristic, complemented
33	#22548637	C₂²	1	2	yes	complemented
34	#22548637	C₂²	1	2	yes	complemented
35	#22548636	C₂²	1	1	yes	characteristic, complemented
36	#22548636	C₂²	1	1	yes	characteristic, complemented
37	#22548635	C₂²	1	2	yes	complemented
38	#22548635	C₂²	1	2	yes	complemented
39	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
41	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
42	#103×C₂₅₁	C₂²	1	2	yes	complemented
43	#102×C₂₅₁	C₂²	1	2	yes	complemented
44	C₂³×D₂₅₁	—	2	2	no	—
45	C₂³×D₂₅₁	—	2	1	no	—
46	#16860811	—	2	4	no	—
47	#16860811	C₂×C₄	1	8	yes	complemented
48	#16860810	—	2	8	no	—
49	#16860810	C₂×C₄	1	4	yes	—
50	#16860810	D₄	1	4	yes	—
51	C₂×Dic₅₀₂	D₄	1	1	yes	characteristic
52	C₂×D₁₀₀₄	—	2	2	no	—
53	C₂×D₁₀₀₄	—	2	8	no	—
54	C₂×D₁₀₀₄	—	2	1	no	—
55	C₂×D₁₀₀₄	—	2	2	no	—
56	C₂×D₁₀₀₄	C₂×C₄	1	4	yes	complemented
57	C₂×D₁₀₀₄	D₄	1	1	yes	characteristic
58	C₂×D₁₀₀₄	C₂³	1	2	yes	—
59	C₂×C₂²⋊D₂₅₁	—	2	2	no	—
60	C₂×C₂²⋊D₂₅₁	D₄	1	2	yes	—
61	D₄×D₂₅₁	—	2	16	no	—
62	D₄×D₂₅₁	—	2	4	no	—
63	D₄×D₂₅₁	C₂×C₄	1	8	yes	complemented
64	C₂×C₄×D₂₅₁	—	2	2	no	—
65	C₂×C₄×D₂₅₁	—	2	2	no	—
66	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
67	#16860793	—	2	4	no	—
68	C₂×C₂₅₁⋊C₈	—	2	2	no	—
69	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₈×D₂₅₁	—	2	4	no	—
72	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
74	Dic₂₅₁⋊C₄	—	2	2	no	—
75	C₂²⋊Dic₂₅₁	—	2	1	no	—
76	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
77	C₄×Dic₂₅₁	—	2	2	no	—
78	C₄×Dic₂₅₁	—	2	1	no	—
79	#41×C₂₅₁	—	2	2	no	—
80	#41×C₂₅₁	C₂×C₄	1	4	yes	complemented
81	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₂₅₁	—	2	1	no	—
83	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₂₅₁	—	2	1	no	—
85	C₂²×C₄×C₂₅₁	—	2	1	no	—
86	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
87	M₁₆×C₂₅₁	—	2	2	no	—
88	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
89	C₄⋊C₄×C₂₅₁	—	2	1	no	—
90	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
91	C₂²×D₂₅₁	—	2	2	no	—
92	C₂²×D₂₅₁	—	2	4	no	—
93	C₂²×D₂₅₁	—	2	8	no	—
94	C₂²×D₂₅₁	—	8	2	no	—
95	C₂²×D₂₅₁	—	8	2	no	—
96	C₂²×D₂₅₁	—	8	1	no	—
97	C₂²×D₂₅₁	—	8	1	no	—
98	C₂²×D₂₅₁	C₂×D₄	1	2	yes	—
99	Dic₅₀₂	C₂²⋊C₄	1	4	yes	—
100	D₁₀₀₄	—	2	8	no	—
101	D₁₀₀₄	—	2	4	no	—
102	D₁₀₀₄	—	2	2	no	—
103	D₁₀₀₄	C₂²⋊C₄	1	4	yes	—
104	D₁₀₀₄	C₂²×C₄	1	4	yes	—
105	C₂²⋊D₂₅₁	—	2	8	no	—
106	C₂²⋊D₂₅₁	C₂²⋊C₄	1	8	yes	—
107	C₂×Dic₂₅₁	—	2	1	no	—
108	C₂×Dic₂₅₁	—	2	1	no	—
109	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
110	C₄×D₂₅₁	—	2	8	no	—
111	C₄×D₂₅₁	—	8	2	no	—
112	C₄×D₂₅₁	C₂²×C₄	1	4	yes	—
113	C₄×D₂₅₁	C₂×D₄	1	4	yes	—
114	C₂₅₁⋊C₈	—	2	1	no	—
115	C₂₅₁⋊C₈	—	2	1	no	—
116	C₂³×C₂₅₁	—	2	1	no	—
117	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
118	Q₈×C₂₅₁	—	2	1	no	—
119	Q₈×C₂₅₁	C₂²×C₄	1	2	yes	—
120	D₄×C₂₅₁	—	2	4	no	—
121	D₄×C₂₅₁	—	2	1	no	—
122	D₄×C₂₅₁	C₂²×C₄	1	2	yes	—
123	C₂×C₄×C₂₅₁	—	2	1	no	—
124	C₂×C₄×C₂₅₁	—	2	4	no	—
125	C₂×C₄×C₂₅₁	—	2	2	no	—
126	C₂×C₄×C₂₅₁	—	8	1	no	—
127	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
128	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
130	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
131	C₈×C₂₅₁	—	2	2	no	—
132	C₂²×C₂₅₁	—	2	1	no	—
133	C₂²×C₂₅₁	—	2	1	no	—
134	C₂²×C₂₅₁	—	2	2	no	—
135	C₂²×C₂₅₁	—	4	1	no	—
136	C₂²×C₂₅₁	—	4	1	no	—
137	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
138	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₂₅₁	—	2	2	no	—
140	C₂×D₂₅₁	—	2	2	no	—
141	C₂×D₂₅₁	—	2	4	no	—
142	C₂×D₂₅₁	—	4	2	no	—
143	C₂×D₂₅₁	—	4	2	no	—
144	C₂×D₂₅₁	—	8	8	no	—
145	C₂×D₂₅₁	—	8	4	no	—
146	C₂×D₂₅₁	C₂×C₂²⋊C₄	1	4	yes	—
147	C₄×C₂₅₁	—	2	2	no	—
148	C₄×C₂₅₁	—	4	2	no	—
149	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
150	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
151	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
152	Dic₂₅₁	—	4	1	no	—
153	Dic₂₅₁	—	4	1	no	—
154	Dic₂₅₁	C₂×C₂²⋊C₄	1	4	yes	—
155	C₂×C₂₅₁	—	2	2	no	—
156	C₂×C₂₅₁	—	4	2	no	—
157	C₂×C₂₅₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₂₅₁	#576	1	1	yes	characteristic
159	C₂×C₂₅₁	#577	1	1	yes	characteristic
160	D₂₅₁	—	2	4	no	—
161	D₂₅₁	—	4	4	no	—
162	C₂₅₁	#2897	1	1	yes	characteristic, complemented, Sylow
163	#2897	—	251	1	no	maximal, Sylow
164	C₂×H(5,𝔽₂)	—	251	1	no	—
165	C₂×#131	—	251	1	no	—
166	#520	—	251	1	no	—
167	#418	—	251	1	no	—
168	#418	—	251	2	no	—
169	#417	—	251	2	no	—
170	#417	D₂₅₁	1	1	yes	characteristic, complemented
171	#416	—	251	1	no	—
172	#416	—	251	2	no	—
173	C₂×#102	—	251	1	no	—
174	C₂×#102	—	251	2	no	—
175	H(5,𝔽₂)	—	502	4	no	—
176	H(5,𝔽₂)	—	251	8	no	—
177	C₂×#41	—	502	1	no	—
178	C₂×#41	—	251	2	no	—
179	C₂×#41	—	251	1	no	—
180	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic
181	C₂²×D₄	—	502	1	no	—
182	C₂²×D₄	—	502	2	no	—
183	C₂²×D₄	—	251	2	no	—
184	C₂²×D₄	—	251	1	no	—
185	#131	—	502	4	no	—
186	C₂×M₁₆	—	251	2	no	—
187	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic, complemented
188	C₂²×C₈	—	251	1	no	—
189	C₂²×C₈	—	251	2	no	—
190	C₄×Q₈	—	502	1	no	—
191	C₄×D₄	—	502	1	no	—
192	C₄×D₄	—	502	2	no	—
193	#117	—	251	1	no	—
194	#117	—	251	2	no	—
195	C₂×C₄⋊C₄	—	251	2	no	—
196	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic, complemented
197	#103	—	251	1	no	—
198	#103	—	251	1	no	—
199	#103	C₂×D₂₅₁	1	2	yes	complemented
200	#102	—	251	1	no	—
201	#102	—	251	1	no	—
202	#102	—	251	2	no	—
203	#102	—	251	2	no	—
204	#102	—	251	2	no	—
205	#102	—	251	2	no	—
206	#102	C₂×D₂₅₁	1	2	yes	complemented
207	C₂⁴	—	502	2	no	—
208	C₂⁴	—	502	1	no	—
209	#41	—	2	2	no	—
210	#41	—	502	8	no	—
211	#41	—	502	4	no	—
212	#41	—	251	8	no	—
213	#41	—	251	4	no	—
214	#41	—	251	4	no	—
215	#41	C₄×D₂₅₁	1	4	yes	complemented
216	C₂×Q₈	—	251	1	no	—
217	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	502	2	no	—
220	C₂×D₄	—	502	8	no	—
221	C₂×D₄	—	502	16	no	—
222	C₂×D₄	—	502	4	no	—
223	C₂×D₄	—	502	2	no	—
224	C₂×D₄	—	502	1	no	—
225	C₂×D₄	—	502	2	no	—
226	C₂×D₄	—	251	8	no	—
227	C₂×D₄	—	251	2	no	—
228	C₂×D₄	—	251	4	no	—
229	C₂×D₄	—	251	1	no	—
230	C₂×D₄	—	251	2	no	—
231	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	502	2	no	—
235	C₂²×C₄	—	502	2	no	—
236	C₂²×C₄	—	251	2	no	—
237	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
238	M₁₆	—	2	2	no	—
239	M₁₆	—	502	4	no	—
240	C₂×C₈	—	502	4	no	—
241	C₂×C₈	—	502	2	no	—
242	C₂×C₈	—	251	1	no	—
243	C₂×C₈	—	251	1	no	—
244	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	502	2	no	—
247	C₄⋊C₄	—	251	1	no	—
248	C₄⋊C₄	—	251	1	no	—
249	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
250	C₂²⋊C₄	—	502	2	no	—
251	C₂²⋊C₄	—	502	1	no	—
252	C₄²	—	502	2	no	—
253	C₄²	—	502	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2008	1	no	—
256	C₂³	—	2008	1	no	—
257	C₂³	—	2008	2	no	—
258	C₂³	—	2008	2	no	—
259	C₂³	—	502	8	no	—
260	C₂³	—	502	4	no	—
261	C₂³	—	502	2	no	—
262	C₂³	—	251	2	no	—
263	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	—	251	4	no	—
266	Q₈	C₂×C₄×D₂₅₁	1	2	yes	—
267	D₄	—	2	4	no	—
268	D₄	—	2	1	no	—
269	D₄	—	502	8	no	—
270	D₄	—	502	8	no	—
271	D₄	—	502	2	no	—
272	D₄	—	502	4	no	—
273	D₄	—	251	8	no	—
274	D₄	—	251	4	no	—
275	D₄	—	251	4	no	—
276	D₄	C₂×C₄×D₂₅₁	1	2	yes	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	8	1	no	—
281	C₂×C₄	—	2008	2	no	—
282	C₂×C₄	—	502	8	no	—
283	C₂×C₄	—	502	1	no	—
284	C₂×C₄	—	502	1	no	—
285	C₂×C₄	—	251	2	no	—
286	C₂×C₄	—	251	4	no	—
287	C₂×C₄	—	251	4	no	—
288	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
289	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
290	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
291	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
292	C₈	—	2	2	no	—
293	C₈	—	502	1	no	—
294	C₈	—	502	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	2008	4	no	—
301	C₂²	—	2008	8	no	—
302	C₂²	—	1004	2	no	—
303	C₂²	—	1004	2	no	—
304	C₂²	—	502	4	no	—
305	C₂²	—	502	2	no	—
306	C₂²	—	502	2	no	—
307	C₂²	—	251	4	no	—
308	C₂²	C₂²⋊C₄×D₂₅₁	1	2	yes	—
309	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	1004	1	no	—
313	C₄	—	1004	1	no	—
314	C₄	—	251	4	no	—
315	C₄	C₂²⋊C₄×D₂₅₁	1	2	yes	—
316	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
317	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	2	no	—
320	C₂	—	1004	4	no	—
321	C₂	—	502	4	no	—
322	C₂	C₂×C₂²⋊C₄×D₂₅₁	1	1	yes	characteristic
323	C₂	#30632491	1	1	yes	characteristic
324	C₂	#30632496	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#417×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#22548761	C₂
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#417	D₂₅₁
hypocenter	C₂₅₁	#2897
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
2. C₂×#22548723	C₂	semidirect
3. #30632168	C₂	semidirect
4. #30631693	C₂	semidirect
5. #30631690	C₂	semidirect
6. #30631687	C₂	semidirect
7. #30631683	C₂	semidirect
8. #30631682	C₂	semidirect
9. C₂×#22548635	C₂	semidirect
10. #102×D₂₅₁	C₂	semidirect
11. #417×C₂₅₁	C₂	semidirect
13. #22548761	C₄	semidirect
22. C₂×#16860793	C₂²	semidirect
23. C₂²×C₂₅₁⋊C₈	C₂²	semidirect
24. C₂×C₈×D₂₅₁	C₂²	semidirect
28. #22548674	C₂²	semidirect
29. #22548673	C₂²	semidirect
30. C₄⋊C₄×D₂₅₁	C₂²	semidirect
31. #22548639	C₂²	semidirect
32. #22548639	C₂²	semidirect
33. #22548637	C₂²	semidirect
34. #22548637	C₂²	semidirect
35. #22548636	C₂²	semidirect
36. #22548636	C₂²	semidirect
37. #22548635	C₂²	semidirect
38. #22548635	C₂²	semidirect
40. C₂×M₁₆×C₂₅₁	C₂²	semidirect
41. C₂×C₄⋊C₄×C₂₅₁	C₂²	semidirect
42. #103×C₂₅₁	C₂²	semidirect
43. #102×C₂₅₁	C₂²	semidirect
47. #16860811	C₂×C₄	semidirect
56. C₂×D₁₀₀₄	C₂×C₄	semidirect
63. D₄×D₂₅₁	C₂×C₄	semidirect
80. #41×C₂₅₁	C₂×C₄	semidirect
162. C₂₅₁	#2897	semidirect
170. #417	D₂₅₁	semidirect
187. C₂×M₁₆	C₂×D₂₅₁	semidirect
196. C₂×C₄⋊C₄	C₂×D₂₅₁	semidirect
199. #103	C₂×D₂₅₁	semidirect
206. #102	C₂×D₂₅₁	semidirect
215. #41	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×#22548761	2
C₂×#22548723	2
#30632168	2
#30631693	2
#30631690	2
#30631687	2
#30631683	2
#30631682	2
C₂×#22548635	2
#102×D₂₅₁	2
#417×C₂₅₁	2
#2897	251