Subgroups — #44559814

Order 32128.

496 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548735	C₂	1	2	yes	maximal, complemented
2	C₄×#16860810	C₂	1	1	yes	maximal, characteristic, complemented
3	#30631906	C₂	1	2	yes	maximal, complemented
4	#30631899	C₂	1	2	yes	maximal, complemented
5	#30631894	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631889	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631884	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631879	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊(C₂₅₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	#30631683	C₂	1	2	yes	maximal, complemented
11	#532×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16860810	C₂²	1	1	yes	characteristic
13	#22548735	—	2	8	no	—
14	C₂×#16860800	C₂²	1	2	yes	—
15	C₂×#16860798	C₂²	1	2	yes	—
16	C₂×#16860797	C₂²	1	2	yes	—
17	C₂×#16860795	C₂²	1	2	yes	—
18	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
19	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
20	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
21	C₄×D₁₀₀₄	C₂²	1	1	yes	characteristic
22	C₄×D₁₀₀₄	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₂₅₁	—	2	1	no	—
24	C₄×C₂²⋊D₂₅₁	—	2	1	no	—
25	#22548674	C₂²	1	1	yes	characteristic
26	#22548672	—	2	1	no	—
27	C₄²×D₂₅₁	—	2	1	no	—
28	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
29	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
30	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
31	#22548639	C₂²	1	2	yes	—
32	#22548639	C₂²	1	2	yes	—
33	#22548636	C₂²	1	2	yes	—
34	#22548636	C₂²	1	2	yes	—
35	C₂×#41×C₂₅₁	C₂²	1	2	yes	—
36	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
37	#127×C₂₅₁	C₂²	1	1	yes	characteristic
38	#124×C₂₅₁	C₂²	1	2	yes	—
39	#121×C₂₅₁	—	2	1	no	—
40	#121×C₂₅₁	—	2	1	no	—
41	C₄×D₄×C₂₅₁	—	2	1	no	—
42	C₄×D₄×C₂₅₁	—	2	1	no	—
43	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
44	#16860810	—	2	2	no	—
45	#16860810	D₄	1	4	yes	complemented
46	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
47	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
49	C₂×C₄×D₂₅₁	—	2	1	no	—
50	C₂×C₄×D₂₅₁	—	2	1	no	—
51	C₂×C₄×D₂₅₁	—	2	1	no	—
52	#16860800	—	2	4	no	—
53	#16860798	—	2	4	no	—
54	#16860797	—	2	4	no	—
55	#16860795	—	2	4	no	—
56	C₂×C₂₅₁⋊C₈	—	2	4	no	—
57	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
58	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
59	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
60	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
61	Dic₂₅₁⋊C₄	—	2	1	no	—
62	Dic₂₅₁⋊C₄	—	2	1	no	—
63	C₂²⋊Dic₂₅₁	—	2	1	no	—
64	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
65	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
66	C₄×Dic₂₅₁	—	2	1	no	—
67	C₄×Dic₂₅₁	—	2	1	no	—
68	C₄×Dic₂₅₁	—	2	1	no	—
69	#41×C₂₅₁	—	2	4	no	—
70	#41×C₂₅₁	—	2	4	no	—
71	C₂×Q₈×C₂₅₁	C₂³	1	2	yes	—
72	C₂×D₄×C₂₅₁	—	2	2	no	—
73	C₂×D₄×C₂₅₁	—	2	1	no	—
74	C₂×D₄×C₂₅₁	—	2	1	no	—
75	C₂×D₄×C₂₅₁	C₂³	1	2	yes	—
76	C₂²×C₄×C₂₅₁	—	2	2	no	—
77	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₂₅₁	—	2	1	no	—
81	C₄⋊C₄×C₂₅₁	—	2	1	no	—
82	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
83	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
84	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
85	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
86	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
87	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
88	C₂²×D₂₅₁	—	2	1	no	—
89	Dic₅₀₂	—	2	1	no	—
90	Dic₅₀₂	C₂×D₄	1	2	yes	—
91	D₁₀₀₄	—	2	1	no	—
92	D₁₀₀₄	C₂×D₄	1	2	yes	—
93	C₂²⋊D₂₅₁	—	2	4	no	—
94	C₂×Dic₂₅₁	—	2	1	no	—
95	C₂×Dic₂₅₁	—	2	1	no	—
96	C₂×Dic₂₅₁	—	2	1	no	—
97	C₄×D₂₅₁	—	2	4	no	—
98	C₄×D₂₅₁	—	4	2	no	—
99	C₄×D₂₅₁	—	4	2	no	—
100	C₂₅₁⋊C₈	—	2	2	no	—
101	C₂₅₁⋊C₈	—	2	2	no	—
102	C₂³×C₂₅₁	—	2	2	no	—
103	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
104	Q₈×C₂₅₁	—	2	2	no	—
105	Q₈×C₂₅₁	—	2	2	no	—
106	D₄×C₂₅₁	—	2	2	no	—
107	D₄×C₂₅₁	—	2	2	no	—
108	D₄×C₂₅₁	—	4	2	no	—
109	D₄×C₂₅₁	—	4	4	no	—
110	D₄×C₂₅₁	—	4	2	no	—
111	C₂×C₄×C₂₅₁	—	2	2	no	—
112	C₂×C₄×C₂₅₁	—	2	2	no	—
113	C₂×C₄×C₂₅₁	—	4	1	no	—
114	C₂×C₄×C₂₅₁	—	4	4	no	—
115	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₂₅₁	#41	1	2	yes	—
122	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
123	C₂²×C₂₅₁	—	2	1	no	—
124	C₂²×C₂₅₁	—	2	1	no	—
125	C₂²×C₂₅₁	—	4	2	no	—
126	C₂²×C₂₅₁	—	4	2	no	—
127	C₂²×C₂₅₁	—	4	2	no	—
128	C₂²×C₂₅₁	#121	1	2	yes	—
129	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×D₂₅₁	—	2	2	no	—
131	C₂×D₂₅₁	—	4	1	no	—
132	C₂×D₂₅₁	—	4	1	no	—
133	C₄×C₂₅₁	—	2	1	no	—
134	C₄×C₂₅₁	—	2	1	no	—
135	C₄×C₂₅₁	—	2	1	no	—
136	C₄×C₂₅₁	—	4	2	no	—
137	C₄×C₂₅₁	#121	1	2	yes	—
138	C₄×C₂₅₁	#135	1	2	yes	complemented
139	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
141	Dic₂₅₁	—	2	2	no	—
142	Dic₂₅₁	—	4	1	no	—
143	Dic₂₅₁	—	4	1	no	—
144	C₂×C₂₅₁	—	2	2	no	—
145	C₂×C₂₅₁	—	8	2	no	—
146	C₂×C₂₅₁	C₂×#121	1	1	yes	characteristic
147	C₂×C₂₅₁	C₂×#135	1	1	yes	characteristic
148	C₂×C₂₅₁	#575	1	1	yes	characteristic
149	D₂₅₁	—	4	2	no	—
150	C₂₅₁	#3035	1	1	yes	characteristic, complemented, Sylow
151	#3035	—	251	1	no	maximal, Sylow
152	C₂×#135	—	251	2	no	—
153	#532	D₂₅₁	1	1	yes	characteristic, complemented
154	C₄×#41	—	251	1	no	—
155	#464	—	251	2	no	—
156	#463	—	251	2	no	—
157	#462	—	251	1	no	—
158	#461	—	251	1	no	—
159	#460	—	251	1	no	—
160	#459	—	251	1	no	—
161	C₂×C₄⋊C₈	—	251	1	no	—
162	#416	—	251	2	no	—
163	C₂×#41	—	251	1	no	—
164	C₂×#41	C₂×D₂₅₁	1	2	yes	—
165	#135	—	502	8	no	—
166	C₂×Q₁₆	—	251	2	no	—
167	C₂×SD₁₆	—	251	2	no	—
168	C₂×SD₁₆	—	251	2	no	—
169	C₂×D₈	—	251	2	no	—
170	C₂²×C₈	—	251	2	no	—
171	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
172	#127	C₂×D₂₅₁	1	1	yes	characteristic
173	#124	C₂×D₂₅₁	1	2	yes	—
174	#121	—	2	1	no	—
175	#121	—	2	1	no	—
176	C₄×Q₈	—	251	1	no	—
177	C₄×Q₈	—	251	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	2	1	no	—
180	C₄×D₄	—	502	1	no	—
181	C₄×D₄	—	502	1	no	—
182	C₄×D₄	—	251	1	no	—
183	C₄×D₄	—	251	1	no	—
184	#117	—	502	1	no	—
185	#117	—	251	1	no	—
186	C₂×C₄²	—	502	1	no	—
187	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
188	C₄⋊C₈	—	251	1	no	—
189	C₄⋊C₈	—	251	1	no	—
190	C₄⋊C₈	—	251	2	no	—
191	#103	—	251	2	no	—
192	#103	—	251	2	no	—
193	#102	—	251	2	no	—
194	#102	—	251	2	no	—
195	#41	—	2	4	no	—
196	#41	—	2	4	no	—
197	#41	—	502	2	no	—
198	#41	—	251	4	no	—
199	C₂×Q₈	—	251	1	no	—
200	C₂×Q₈	C₂²×D₂₅₁	1	2	yes	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	502	1	no	—
205	C₂×D₄	—	251	1	no	—
206	C₂×D₄	C₂²×D₂₅₁	1	2	yes	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	502	1	no	—
209	C₂²×C₄	—	502	1	no	—
210	C₂²×C₄	—	502	1	no	—
211	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
214	Q₁₆	—	502	4	no	—
215	SD₁₆	—	502	4	no	—
216	SD₁₆	—	502	4	no	—
217	D₈	—	502	4	no	—
218	C₂×C₈	—	502	4	no	—
219	C₂×C₈	—	251	2	no	—
220	C₂×C₈	—	251	2	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	502	1	no	—
224	C₄⋊C₄	—	502	1	no	—
225	C₄⋊C₄	—	251	1	no	—
226	C₄⋊C₄	—	251	1	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	502	1	no	—
230	C₂²⋊C₄	—	502	1	no	—
231	C₂²⋊C₄	—	502	1	no	—
232	C₄²	—	502	1	no	—
233	C₄²	—	502	1	no	—
234	C₄²	—	502	1	no	—
235	C₄²	C₂²⋊D₂₅₁	1	1	yes	characteristic
236	C₄²	C₂²⋊D₂₅₁	1	1	yes	characteristic
237	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
238	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
239	C₂³	—	2	2	no	—
240	C₂³	—	502	1	no	—
241	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
242	Q₈	—	2	2	no	—
243	Q₈	—	2	2	no	—
244	Q₈	—	502	1	no	—
245	Q₈	—	251	2	no	—
246	D₄	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	2	no	—
251	D₄	—	502	4	no	—
252	D₄	—	502	1	no	—
253	D₄	—	251	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	1004	2	no	—
259	C₂×C₄	—	1004	2	no	—
260	C₂×C₄	—	502	1	no	—
261	C₂×C₄	—	502	1	no	—
262	C₂×C₄	—	502	1	no	—
263	C₂×C₄	—	502	4	no	—
264	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
265	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
266	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
270	C₂×C₄	#16860809	1	2	yes	—
271	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
272	C₈	—	502	2	no	—
273	C₈	—	502	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	1004	1	no	—
280	C₂²	—	1004	1	no	—
281	C₂²	—	502	2	no	—
282	C₂²	#22548686	1	2	yes	—
283	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	2	no	—
288	C₄	—	1004	1	no	—
289	C₄	—	1004	1	no	—
290	C₄	—	502	2	no	—
291	C₄	#22548686	1	2	yes	—
292	C₄	#22548735	1	2	yes	complemented
293	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
294	C₄	C₂×#16860809	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	8	2	no	—
297	C₂	—	1004	2	no	—
298	C₂	C₂×#22548686	1	1	yes	characteristic
299	C₂	C₂×#22548735	1	1	yes	characteristic
300	C₂	#30632485	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#532×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
hypercenter	#532	D₂₅₁
hypocenter	C₂₅₁	#3035
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₂×#22548735	C₂	semidirect
2. C₄×#16860810	C₂	semidirect
3. #30631906	C₂	semidirect
4. #30631899	C₂	semidirect
5. #30631894	C₂	semidirect
6. #30631889	C₂	semidirect
7. #30631884	C₂	semidirect
8. #30631879	C₂	semidirect
9. C₂×C₄⋊(C₂₅₁⋊C₈)	C₂	semidirect
10. #30631683	C₂	semidirect
11. #532×C₂₅₁	C₂	semidirect
45. #16860810	D₄	semidirect
138. C₄×C₂₅₁	#135	semidirect
150. C₂₅₁	#3035	semidirect
153. #532	D₂₅₁	semidirect
292. C₄	#22548735	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548735	2
C₄×#16860810	2
#30631906	2
#30631899	2
#30631894	2
#30631889	2
#30631884	2
#30631879	2
C₂×C₄⋊(C₂₅₁⋊C₈)	2
#30631683	2
#532×C₂₅₁	2
#3035	251