Subgroups — C₄×#22548741 #44559173

Order 32128.

548 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548741	C₂	1	1	yes	maximal, characteristic
2	C₄×#16860810	C₂	1	1	yes	maximal, characteristic, complemented
3	#30631797	C₂	1	2	yes	maximal, complemented
4	#30631787	C₂	1	2	yes	maximal, complemented
5	C₄×#16860798	C₂	1	2	yes	maximal, complemented
6	C₄×#16860795	C₂	1	2	yes	maximal, complemented
7	#30631727	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631692	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631686	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#16860794	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16860810	C₂²	1	1	yes	characteristic
13	#22548741	C₄	1	8	yes	direct product factor
14	C₂×#16860798	C₂²	1	2	yes	—
15	C₂×#16860795	C₂²	1	2	yes	—
16	C₂×#16860794	C₂²	1	1	yes	characteristic
17	C₂×#16860794	C₂²	1	1	yes	characteristic
18	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic, complemented
19	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic, complemented
20	C₄×D₁₀₀₄	C₂²	1	1	yes	characteristic, complemented
21	C₄×D₁₀₀₄	C₂²	1	1	yes	characteristic, complemented
22	C₄×C₂²⋊D₂₅₁	—	2	1	no	—
23	C₄×C₂²⋊D₂₅₁	—	2	1	no	—
24	#22548674	C₂²	1	1	yes	characteristic
25	#22548672	—	2	1	no	—
26	C₄²×D₂₅₁	—	2	1	no	—
27	#22548648	C₂²	1	2	yes	complemented
28	#22548646	C₂²	1	2	yes	complemented
29	#22548638	C₂²	1	2	yes	complemented
30	#22548636	C₂²	1	2	yes	—
31	#22548636	C₂²	1	2	yes	—
32	#22548635	C₂²	1	2	yes	complemented
33	#22548626	C₂²	1	2	yes	complemented
34	C₄×C₂₅₁⋊C₈	C₂²	1	2	yes	complemented
35	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
36	C₂³×C₄×C₂₅₁	—	2	1	no	—
37	C₄×D₄×C₂₅₁	—	2	1	no	—
38	C₄×D₄×C₂₅₁	—	2	1	no	—
39	C₄×D₄×C₂₅₁	C₂²	1	2	yes	—
40	C₄×D₄×C₂₅₁	C₂²	1	2	yes	—
41	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₂₅₁	—	2	1	no	—
43	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
44	#16860810	—	2	2	no	—
45	#16860810	C₂×C₄	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	#16860798	C₂×C₄	1	8	yes	complemented
53	#16860795	C₂×C₄	1	8	yes	complemented
54	#16860794	—	2	2	no	—
55	#16860794	C₂×C₄	1	4	yes	complemented
56	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
57	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
58	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
60	Dic₂₅₁⋊C₄	—	2	1	no	—
61	Dic₂₅₁⋊C₄	—	2	1	no	—
62	C₂²⋊Dic₂₅₁	—	2	1	no	—
63	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
64	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
65	C₄×Dic₂₅₁	—	2	1	no	—
66	C₄×Dic₂₅₁	—	2	1	no	—
67	C₄×Dic₂₅₁	—	2	1	no	—
68	C₂⁴×C₂₅₁	—	2	1	no	—
69	C₂×D₄×C₂₅₁	—	2	1	no	—
70	C₂×D₄×C₂₅₁	—	2	2	no	—
71	C₂×D₄×C₂₅₁	C₂×C₄	1	4	yes	complemented
72	C₂×D₄×C₂₅₁	C₂³	1	2	yes	—
73	C₂²×C₄×C₂₅₁	—	2	1	no	—
74	C₂²×C₄×C₂₅₁	—	2	2	no	—
75	C₂²×C₄×C₂₅₁	—	2	2	no	—
76	C₂²×C₄×C₂₅₁	—	4	4	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	C₂³	1	2	yes	—
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	C₂²×C₄	1	2	yes	complemented
91	D₁₀₀₄	—	2	1	no	—
92	D₁₀₀₄	C₂²×C₄	1	2	yes	complemented
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	C₂²×C₄	1	4	yes	complemented
102	C₂³×C₂₅₁	—	2	4	no	—
103	C₂³×C₂₅₁	—	2	2	no	—
104	C₂³×C₂₅₁	—	4	2	no	—
105	C₂³×C₂₅₁	—	4	2	no	—
106	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
107	D₄×C₂₅₁	—	2	4	no	—
108	D₄×C₂₅₁	—	2	2	no	—
109	D₄×C₂₅₁	C₂²×C₄	1	4	yes	—
110	C₂×C₄×C₂₅₁	—	2	2	no	—
111	C₂×C₄×C₂₅₁	—	2	2	no	—
112	C₂×C₄×C₂₅₁	—	4	1	no	—
113	C₂×C₄×C₂₅₁	—	4	4	no	—
114	C₂×C₄×C₂₅₁	—	4	2	no	—
115	C₂×C₄×C₂₅₁	—	4	4	no	—
116	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₂₅₁	#41	1	2	yes	—
123	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
124	C₂²×C₂₅₁	—	2	1	no	—
125	C₂²×C₂₅₁	—	2	1	no	—
126	C₂²×C₂₅₁	—	2	4	no	—
127	C₂²×C₂₅₁	—	4	8	no	—
128	C₂²×C₂₅₁	—	4	2	no	—
129	C₂²×C₂₅₁	—	4	2	no	—
130	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
131	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₂₅₁	—	2	2	no	—
133	C₂×D₂₅₁	—	4	1	no	—
134	C₂×D₂₅₁	—	4	1	no	—
135	C₄×C₂₅₁	—	2	1	no	—
136	C₄×C₂₅₁	—	2	1	no	—
137	C₄×C₂₅₁	—	2	1	no	—
138	C₄×C₂₅₁	—	4	2	no	—
139	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
140	C₄×C₂₅₁	AGL(1,ℤ/8)	1	2	yes	complemented
141	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
142	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
143	Dic₂₅₁	—	2	2	no	—
144	Dic₂₅₁	—	4	1	no	—
145	Dic₂₅₁	—	4	1	no	—
146	C₂×C₂₅₁	—	2	2	no	—
147	C₂×C₂₅₁	—	4	4	no	—
148	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×C₂₅₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₂₅₁	#575	1	1	yes	characteristic
151	D₂₅₁	—	4	2	no	—
152	C₂₅₁	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
153	C₄×AGL(1,ℤ/8)	—	251	1	no	maximal, Sylow
154	C₂×AGL(1,ℤ/8)	—	251	1	no	—
155	C₄×#41	—	251	1	no	—
156	C₂×C₄×D₄	D₂₅₁	1	1	yes	characteristic, complemented
157	#442	—	251	2	no	—
158	#440	—	251	2	no	—
159	C₄×SD₁₆	—	251	2	no	—
160	C₄×D₈	—	251	2	no	—
161	#428	—	251	1	no	—
162	#418	—	251	1	no	—
163	#417	—	251	1	no	—
164	C₄×M₁₆	—	251	1	no	—
165	C₂×#41	—	251	1	no	—
166	C₂²×D₄	C₂×D₂₅₁	1	1	yes	characteristic
167	C₂³×C₄	—	2	1	no	—
168	AGL(1,ℤ/8)	—	251	8	no	—
169	C₂×SD₁₆	—	251	2	no	—
170	C₂×D₈	—	251	2	no	—
171	C₂×M₁₆	—	251	1	no	—
172	C₂×M₁₆	—	251	1	no	—
173	C₄×Q₈	—	251	1	no	—
174	C₄×Q₈	—	251	1	no	—
175	C₄×D₄	—	2	1	no	—
176	C₄×D₄	—	2	1	no	—
177	C₄×D₄	—	502	1	no	—
178	C₄×D₄	—	502	1	no	—
179	C₄×D₄	—	251	1	no	—
180	C₄×D₄	—	251	1	no	—
181	C₄×D₄	C₂×D₂₅₁	1	2	yes	—
182	C₄×D₄	C₂×D₂₅₁	1	2	yes	—
183	#117	—	502	1	no	—
184	#117	—	251	1	no	—
185	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	—	2	1	no	—
187	C₂×C₄²	—	502	1	no	—
188	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
189	#107	—	251	2	no	—
190	#106	—	251	2	no	—
191	#103	—	251	2	no	—
192	#102	—	251	2	no	—
193	#102	—	251	2	no	—
194	#102	—	251	2	no	—
195	#97	—	251	2	no	—
196	C₄×C₈	—	251	2	no	—
197	C₂⁴	—	2	1	no	—
198	#41	—	502	2	no	—
199	#41	—	251	4	no	—
200	C₂×Q₈	—	251	1	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	502	1	no	—
204	C₂×D₄	—	251	1	no	—
205	C₂×D₄	C₄×D₂₅₁	1	4	yes	complemented
206	C₂×D₄	C₂²×D₂₅₁	1	2	yes	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	502	1	no	—
212	C₂²×C₄	—	502	1	no	—
213	C₂²×C₄	—	502	1	no	—
214	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
215	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
217	SD₁₆	—	251	8	no	—
218	D₈	—	251	8	no	—
219	M₁₆	—	502	2	no	—
220	M₁₆	—	251	4	no	—
221	C₂×C₈	—	251	2	no	—
222	C₂×C₈	—	251	2	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	502	1	no	—
225	C₄⋊C₄	—	502	1	no	—
226	C₄⋊C₄	—	251	1	no	—
227	C₄⋊C₄	—	251	1	no	—
228	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	502	1	no	—
232	C₂²⋊C₄	—	502	1	no	—
233	C₂²⋊C₄	—	502	1	no	—
234	C₄²	—	502	1	no	—
235	C₄²	—	502	1	no	—
236	C₄²	—	502	1	no	—
237	C₄²	C₂²⋊D₂₅₁	1	1	yes	characteristic
238	C₄²	C₂²⋊D₂₅₁	1	1	yes	characteristic
239	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
240	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
241	C₂³	—	2	2	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	502	1	no	—
246	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
247	Q₈	—	502	1	no	—
248	Q₈	—	251	2	no	—
249	D₄	—	2	2	no	—
250	D₄	—	2	4	no	—
251	D₄	—	502	4	no	—
252	D₄	—	502	1	no	—
253	D₄	—	251	2	no	—
254	D₄	C₂×C₄×D₂₅₁	1	4	yes	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	1004	2	no	—
262	C₂×C₄	—	1004	2	no	—
263	C₂×C₄	—	502	1	no	—
264	C₂×C₄	—	502	1	no	—
265	C₂×C₄	—	502	1	no	—
266	C₂×C₄	—	502	4	no	—
267	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
268	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
271	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
272	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
273	C₂×C₄	#16860810	1	2	yes	—
274	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
275	C₈	—	502	2	no	—
276	C₈	—	251	4	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	4	8	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	1004	1	no	—
284	C₂²	—	1004	1	no	—
285	C₂²	—	502	2	no	—
286	C₂²	C₄×C₂²⋊D₂₅₁	1	2	yes	—
287	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	4	2	no	—
292	C₄	—	1004	1	no	—
293	C₄	—	1004	1	no	—
294	C₄	—	502	2	no	—
295	C₄	C₄×C₂²⋊D₂₅₁	1	2	yes	—
296	C₄	#22548741	1	2	yes	direct product factor
297	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
298	C₄	C₂×#16860810	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	4	4	no	—
301	C₂	—	1004	2	no	—
302	C₂	C₂×C₄×C₂²⋊D₂₅₁	1	1	yes	characteristic
303	C₂	C₂×#22548741	1	1	yes	characteristic
304	C₂	#30632485	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	C₂×C₄×D₄×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#22548741	C₂
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₂×C₄×D₄	D₂₅₁
hypocenter	C₂₅₁	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₄×#16860810	C₂	semidirect
3. #30631797	C₂	semidirect
4. #30631787	C₂	semidirect
5. C₄×#16860798	C₂	semidirect
6. C₄×#16860795	C₂	semidirect
7. #30631727	C₂	semidirect
8. #30631692	C₂	semidirect
9. #30631686	C₂	semidirect
10. C₄×#16860794	C₂	semidirect
11. C₂×C₄×D₄×C₂₅₁	C₂	semidirect
13. #22548741	C₄	direct (N × H)
18. C₄×Dic₅₀₂	C₂²	semidirect
19. C₄×Dic₅₀₂	C₂²	semidirect
20. C₄×D₁₀₀₄	C₂²	semidirect
21. C₄×D₁₀₀₄	C₂²	semidirect
27. #22548648	C₂²	semidirect
28. #22548646	C₂²	semidirect
29. #22548638	C₂²	semidirect
32. #22548635	C₂²	semidirect
33. #22548626	C₂²	semidirect
34. C₄×C₂₅₁⋊C₈	C₂²	semidirect
45. #16860810	C₂×C₄	semidirect
52. #16860798	C₂×C₄	semidirect
53. #16860795	C₂×C₄	semidirect
55. #16860794	C₂×C₄	semidirect
71. C₂×D₄×C₂₅₁	C₂×C₄	semidirect
90. Dic₅₀₂	C₂²×C₄	semidirect
92. D₁₀₀₄	C₂²×C₄	semidirect
101. C₂₅₁⋊C₈	C₂²×C₄	semidirect
140. C₄×C₂₅₁	AGL(1,ℤ/8)	semidirect
152. C₂₅₁	C₄×AGL(1,ℤ/8)	semidirect
156. C₂×C₄×D₄	D₂₅₁	semidirect
205. C₂×D₄	C₄×D₂₅₁	semidirect
296. C₄	#22548741	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#22548741	2
C₄×#16860810	2
#30631797	2
#30631787	2
C₄×#16860798	2
C₄×#16860795	2
#30631727	2
#30631692	2
#30631686	2
C₄×#16860794	2
C₂×C₄×D₄×C₂₅₁	2
C₄×AGL(1,ℤ/8)	251

Subgroups — C4×#22548741 #44559173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#22548741 #44559173