Subgroups — #44559913

Order 32128.

472 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548745	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632258	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632170	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631907	C₂	1	2	yes	maximal, complemented
5	#30631900	C₂	1	2	yes	maximal, complemented
6	#30631895	C₂	1	2	yes	maximal, complemented
7	#30631890	C₂	1	2	yes	maximal, complemented
8	#30631713	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631688	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#22548640	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#135×C₂₅₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
13	#22548745	—	2	4	no	—
14	C₂×#16860800	C₂²	1	2	yes	—
15	C₂×#16860798	C₂²	1	2	yes	—
16	C₂×#16860794	C₂²	1	1	yes	characteristic
17	#22548713	C₂²	1	2	yes	—
18	#22548702	C₂²	1	2	yes	—
19	#22548696	—	2	2	no	—
20	#22548694	—	2	1	no	—
21	#22548694	—	2	1	no	—
22	Q₈×Dic₂₅₁	C₂²	1	2	yes	—
23	D₄×Dic₂₅₁	—	2	2	no	—
24	D₄×Dic₂₅₁	C₂²	1	2	yes	—
25	#22548674	—	2	1	no	—
26	#22548674	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₂₅₁	—	2	1	no	—
28	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
29	Dic₂₅₁⋊C₈	C₂²	1	2	yes	—
30	Dic₂₅₁⋊C₈	C₂²	1	2	yes	—
31	#22548640	C₂²	1	2	yes	—
32	#22548640	C₂²	1	2	yes	—
33	#22548639	C₂²	1	2	yes	—
34	#22548636	C₂²	1	2	yes	—
35	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
36	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
37	#135×C₂₅₁	—	2	4	no	—
38	C₂×Q₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
39	C₂×SD₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
40	C₂×SD₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
41	C₂×D₈×C₂₅₁	C₂²	1	1	yes	characteristic
42	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
43	C₂×Dic₅₀₂	—	2	1	no	—
44	C₂×Dic₅₀₂	—	2	2	no	—
45	C₂×Dic₅₀₂	—	2	2	no	—
46	C₂×Dic₅₀₂	C₂³	1	2	yes	—
47	C₂²×Dic₂₅₁	—	2	2	no	—
48	C₂²×Dic₂₅₁	—	2	1	no	—
49	C₂²×Dic₂₅₁	—	2	1	no	—
50	#16860800	—	2	4	no	—
51	#16860798	—	2	4	no	—
52	#16860794	—	2	2	no	—
53	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
54	C₄⋊Dic₂₅₁	—	2	2	no	—
55	C₄⋊Dic₂₅₁	—	2	1	no	—
56	C₄⋊Dic₂₅₁	D₄	1	1	yes	characteristic
57	C₄⋊Dic₂₅₁	D₄	1	1	yes	characteristic
58	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
59	Dic₂₅₁⋊C₄	—	2	2	no	—
60	Dic₂₅₁⋊C₄	—	2	2	no	—
61	C₂²⋊Dic₂₅₁	—	2	2	no	—
62	C₂²⋊Dic₂₅₁	—	2	2	no	—
63	C₂²⋊Dic₂₅₁	—	2	2	no	—
64	C₂²⋊Dic₂₅₁	D₄	1	2	yes	—
65	C₄×Dic₂₅₁	—	2	2	no	—
66	C₄×Dic₂₅₁	C₂³	1	2	yes	—
67	#41×C₂₅₁	—	2	2	no	—
68	#41×C₂₅₁	—	2	2	no	—
69	#41×C₂₅₁	—	2	2	no	—
70	#41×C₂₅₁	D₄	1	4	yes	—
71	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
72	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
73	C₂×D₄×C₂₅₁	—	2	1	no	—
74	C₂×D₄×C₂₅₁	—	2	1	no	—
75	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
76	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₂₅₁	—	2	1	no	—
78	C₂²×C₄×C₂₅₁	—	2	1	no	—
79	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
80	Q₁₆×C₂₅₁	—	2	2	no	—
81	SD₁₆×C₂₅₁	—	2	2	no	—
82	SD₁₆×C₂₅₁	—	2	2	no	—
83	D₈×C₂₅₁	—	2	2	no	—
84	C₂×C₈×C₂₅₁	—	2	2	no	—
85	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
87	Dic₅₀₂	—	2	2	no	—
88	Dic₅₀₂	—	2	2	no	—
89	Dic₅₀₂	—	4	2	no	—
90	C₂×Dic₂₅₁	—	2	2	no	—
91	C₂×Dic₂₅₁	—	2	2	no	—
92	C₂×Dic₂₅₁	—	4	2	no	—
93	C₂×Dic₂₅₁	—	8	2	no	—
94	C₂×Dic₂₅₁	—	8	1	no	—
95	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
96	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
97	C₂₅₁⋊C₈	—	2	2	no	—
98	C₂³×C₂₅₁	—	2	1	no	—
99	C₂³×C₂₅₁	—	2	1	no	—
100	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
101	Q₈×C₂₅₁	—	2	1	no	—
102	Q₈×C₂₅₁	—	2	1	no	—
103	Q₈×C₂₅₁	—	2	1	no	—
104	Q₈×C₂₅₁	C₂×D₄	1	2	yes	—
105	D₄×C₂₅₁	—	2	4	no	—
106	D₄×C₂₅₁	—	2	1	no	—
107	D₄×C₂₅₁	—	2	1	no	—
108	D₄×C₂₅₁	—	2	1	no	—
109	D₄×C₂₅₁	—	4	2	no	—
110	D₄×C₂₅₁	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₂₅₁	—	2	1	no	—
112	C₂×C₄×C₂₅₁	—	2	4	no	—
113	C₂×C₄×C₂₅₁	—	2	1	no	—
114	C₂×C₄×C₂₅₁	—	4	2	no	—
115	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₂₅₁	#41	1	2	yes	—
118	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
119	C₈×C₂₅₁	—	2	1	no	—
120	C₈×C₂₅₁	—	2	1	no	—
121	C₂²×C₂₅₁	—	2	1	no	—
122	C₂²×C₂₅₁	—	2	1	no	—
123	C₂²×C₂₅₁	—	2	2	no	—
124	C₂²×C₂₅₁	—	4	1	no	—
125	C₂²×C₂₅₁	—	4	1	no	—
126	C₂²×C₂₅₁	—	4	1	no	—
127	C₂²×C₂₅₁	—	4	1	no	—
128	C₂²×C₂₅₁	—	4	1	no	—
129	C₂²×C₂₅₁	#121	1	2	yes	—
130	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₂₅₁	—	2	2	no	—
132	C₄×C₂₅₁	—	4	1	no	—
133	C₄×C₂₅₁	#121	1	2	yes	—
134	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
135	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
136	Dic₂₅₁	—	2	2	no	—
137	Dic₂₅₁	—	2	2	no	—
138	Dic₂₅₁	—	4	2	no	—
139	Dic₂₅₁	—	4	2	no	—
140	C₂×C₂₅₁	—	2	2	no	—
141	C₂×C₂₅₁	—	4	2	no	—
142	C₂×C₂₅₁	—	8	1	no	—
143	C₂×C₂₅₁	C₂×#121	1	1	yes	characteristic
144	C₂×C₂₅₁	#577	1	1	yes	characteristic
145	C₂×C₂₅₁	#578	1	1	yes	characteristic
146	C₂₅₁	#3046	1	1	yes	characteristic, complemented, Sylow
147	#3046	—	251	1	no	maximal, Sylow
148	C₂×#137	—	251	1	no	—
149	C₂×#135	D₂₅₁	1	1	yes	characteristic
150	#536	—	251	1	no	—
151	#520	—	251	1	no	—
152	#464	—	251	2	no	—
153	#463	—	251	2	no	—
154	#462	—	251	2	no	—
155	#461	—	251	2	no	—
156	#424	—	251	1	no	—
157	#417	—	251	1	no	—
158	C₂×#103	—	251	1	no	—
159	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic
160	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic
161	C₂²×Q₈	—	251	1	no	—
162	#137	—	502	4	no	—
163	#135	—	2	4	no	—
164	C₂×Q₁₆	—	251	2	no	—
165	C₂×Q₁₆	C₂×D₂₅₁	1	1	yes	characteristic
166	C₂×SD₁₆	—	251	2	no	—
167	C₂×SD₁₆	C₂×D₂₅₁	1	1	yes	characteristic
168	C₂×SD₁₆	C₂×D₂₅₁	1	1	yes	characteristic
169	C₂×D₈	C₂×D₂₅₁	1	1	yes	characteristic
170	C₂×M₁₆	—	251	1	no	—
171	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
172	C₄⋊Q₈	—	251	2	no	—
173	#124	—	251	2	no	—
174	#123	—	502	2	no	—
175	C₂²⋊Q₈	—	502	1	no	—
176	C₂²⋊Q₈	—	502	1	no	—
177	C₄×Q₈	—	251	2	no	—
178	C₄×D₄	—	502	2	no	—
179	C₄×D₄	—	251	2	no	—
180	#117	—	502	1	no	—
181	#117	—	251	1	no	—
182	C₂×C₄⋊C₄	—	502	1	no	—
183	C₂×C₄⋊C₄	—	251	1	no	—
184	C₄⋊C₈	—	251	2	no	—
185	C₄⋊C₈	—	251	2	no	—
186	#103	—	251	2	no	—
187	#103	—	251	2	no	—
188	#103	—	251	2	no	—
189	#102	—	251	2	no	—
190	#41	—	2	2	no	—
191	#41	—	2	2	no	—
192	#41	—	2	2	no	—
193	#41	C₂²⋊D₂₅₁	1	4	yes	—
194	C₂×Q₈	—	502	2	no	—
195	C₂×Q₈	—	502	2	no	—
196	C₂×Q₈	—	502	1	no	—
197	C₂×Q₈	—	251	2	no	—
198	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
199	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
203	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	502	1	no	—
207	C₂²×C₄	—	502	2	no	—
208	C₂²×C₄	—	502	1	no	—
209	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
210	Q₁₆	—	2	2	no	—
211	Q₁₆	—	502	4	no	—
212	SD₁₆	—	2	2	no	—
213	SD₁₆	—	2	2	no	—
214	SD₁₆	—	502	4	no	—
215	D₈	—	2	2	no	—
216	M₁₆	—	502	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	251	2	no	—
219	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
220	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
221	C₄⋊C₄	—	502	2	no	—
222	C₄⋊C₄	—	502	2	no	—
223	C₄⋊C₄	—	502	2	no	—
224	C₄⋊C₄	—	502	1	no	—
225	C₄⋊C₄	—	251	2	no	—
226	C₄⋊C₄	—	251	1	no	—
227	C₄⋊C₄	—	251	1	no	—
228	C₂²⋊C₄	—	502	2	no	—
229	C₂²⋊C₄	—	502	2	no	—
230	C₂²⋊C₄	—	502	2	no	—
231	C₂²⋊C₄	—	251	2	no	—
232	C₄²	—	502	2	no	—
233	C₄²	—	251	2	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
237	Q₈	—	2	1	no	—
238	Q₈	—	2	1	no	—
239	Q₈	—	2	1	no	—
240	Q₈	—	1004	2	no	—
241	Q₈	—	502	2	no	—
242	Q₈	—	502	2	no	—
243	Q₈	C₂×C₂²⋊D₂₅₁	1	2	yes	—
244	D₄	—	2	4	no	—
245	D₄	—	2	1	no	—
246	D₄	—	2	1	no	—
247	D₄	—	2	1	no	—
248	D₄	—	4	2	no	—
249	D₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	2008	1	no	—
255	C₂×C₄	—	2008	2	no	—
256	C₂×C₄	—	1004	2	no	—
257	C₂×C₄	—	502	2	no	—
258	C₂×C₄	—	502	2	no	—
259	C₂×C₄	—	251	2	no	—
260	C₂×C₄	—	251	2	no	—
261	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
262	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
263	C₂×C₄	#16860809	1	2	yes	—
264	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	502	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	#22548687	1	2	yes	—
277	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	4	1	no	—
280	C₄	—	1004	2	no	—
281	C₄	—	1004	2	no	—
282	C₄	—	502	2	no	—
283	C₄	—	502	2	no	—
284	C₄	#22548687	1	2	yes	—
285	C₄	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
286	C₄	C₂×#16860809	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	4	2	no	—
289	C₂	—	8	1	no	—
290	C₂	C₂×#22548687	1	1	yes	characteristic
291	C₂	#30632494	1	1	yes	characteristic
292	C₂	#30632501	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting, prime radical	C₂×#135×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
hypercenter	C₂×#135	D₂₅₁
hypocenter	C₂₅₁	#3046
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#22548745	C₂	semidirect
2. #30632258	C₂	semidirect
3. #30632170	C₂	semidirect
4. #30631907	C₂	semidirect
5. #30631900	C₂	semidirect
6. #30631895	C₂	semidirect
7. #30631890	C₂	semidirect
8. #30631713	C₂	semidirect
9. #30631688	C₂	semidirect
10. C₂×#22548640	C₂	semidirect
146. C₂₅₁	#3046	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548745	2
#30632258	2
#30632170	2
#30631907	2
#30631900	2
#30631895	2
#30631890	2
#30631713	2
#30631688	2
C₂×#22548640	2
C₂×#135×C₂₅₁	2
#3046	251