Subgroups — C₂×#30632306 #44563741

Order 32128.

1132 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632306	C₂	1	16	yes	maximal, direct product factor
2	C₂×#22548715	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#22548708	C₂	1	2	yes	maximal, complemented
4	C₂×#22548701	C₂	1	2	yes	maximal, complemented
5	C₂×#22548699	C₂	1	2	yes	maximal, complemented
6	C₂×#22548692	C₂	1	2	yes	maximal, complemented
7	C₂×#22548691	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#22548686	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₄×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#22548672	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#124×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₅₀₂	C₂²	1	2	yes	—
14	C₂²×C₂²⋊D₂₅₁	C₂²	1	2	yes	—
15	C₂³×Dic₂₅₁	C₂²	1	2	yes	—
16	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
17	#22548715	C₂²	1	8	yes	complemented
18	#22548708	C₂²	1	16	yes	complemented
19	#22548701	C₂²	1	16	yes	complemented
20	#22548699	C₂²	1	16	yes	complemented
21	#22548692	C₂²	1	16	yes	complemented
22	#22548691	C₂²	1	8	yes	complemented
23	#22548686	C₂²	1	8	yes	—
24	Q₈×Dic₂₅₁	C₂²	1	8	yes	complemented
25	D₄×Dic₂₅₁	C₂²	1	8	yes	complemented
26	#22548672	C₂²	1	8	yes	complemented
27	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
28	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
31	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
33	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊Dic₂₅₁	C₂²	1	2	yes	—
35	C₂×C₂²⋊Dic₂₅₁	C₂²	1	2	yes	—
36	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	1	yes	characteristic
37	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	2	yes	—
38	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
40	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
42	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
43	#124×C₂₅₁	C₂²	1	8	yes	complemented
44	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
45	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
46	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
47	C₂³×D₂₅₁	C₂³	1	1	yes	characteristic
48	C₂×Dic₅₀₂	—	4	4	no	—
49	C₂×Dic₅₀₂	C₂³	1	8	yes	—
50	C₂×C₂²⋊D₂₅₁	—	4	4	no	—
51	C₂×C₂²⋊D₂₅₁	C₂³	1	8	yes	—
52	C₂²×Dic₂₅₁	—	4	4	no	—
53	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
54	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
55	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
56	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
57	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
58	C₂²×Dic₂₅₁	C₂³	1	8	yes	—
59	C₂×C₄×D₂₅₁	—	2	4	no	—
60	C₂×C₄×D₂₅₁	C₂³	1	4	yes	—
61	C₄⋊Dic₂₅₁	C₂³	1	4	yes	—
62	C₄⋊Dic₂₅₁	C₂³	1	4	yes	—
63	C₄⋊Dic₂₅₁	C₂³	1	8	yes	—
64	C₄⋊Dic₂₅₁	C₂³	1	8	yes	—
65	Dic₂₅₁⋊C₄	C₂³	1	4	yes	—
66	Dic₂₅₁⋊C₄	C₂³	1	4	yes	—
67	Dic₂₅₁⋊C₄	C₂³	1	8	yes	—
68	C₂²⋊Dic₂₅₁	C₂³	1	8	yes	—
69	C₂²⋊Dic₂₅₁	C₂³	1	8	yes	—
70	(C₂×D₂₅₁)⋊C₄	C₂³	1	4	yes	—
71	(C₂×D₂₅₁)⋊C₄	C₂³	1	4	yes	—
72	(C₂×D₂₅₁)⋊C₄	C₂³	1	8	yes	—
73	C₄×Dic₂₅₁	C₂³	1	4	yes	—
74	C₄×Dic₂₅₁	C₂³	1	8	yes	—
75	C₂⁴×C₂₅₁	C₂³	1	2	yes	—
76	C₂×Q₈×C₂₅₁	—	2	4	no	—
77	C₂×Q₈×C₂₅₁	C₂³	1	4	yes	—
78	C₂×D₄×C₂₅₁	—	2	4	no	—
79	C₂×D₄×C₂₅₁	C₂³	1	4	yes	—
80	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
83	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
84	C₂²⋊C₄×C₂₅₁	C₂³	1	8	yes	—
85	C₂²⋊C₄×C₂₅₁	C₂³	1	8	yes	—
86	C₄²×C₂₅₁	C₂³	1	4	yes	—
87	C₂²×D₂₅₁	—	2	2	no	—
88	C₂²×D₂₅₁	—	2	2	no	—
89	C₂²×D₂₅₁	—	2	2	no	—
90	C₂²×D₂₅₁	C₂⁴	1	2	yes	—
91	Dic₅₀₂	—	4	8	no	—
92	C₂²⋊D₂₅₁	—	4	8	no	—
93	C₂×Dic₂₅₁	—	2	2	no	—
94	C₂×Dic₂₅₁	—	2	2	no	—
95	C₂×Dic₂₅₁	—	2	4	no	—
96	C₂×Dic₂₅₁	—	2	4	no	—
97	C₂×Dic₂₅₁	—	2	2	no	—
98	C₂×Dic₂₅₁	—	4	8	no	—
99	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
100	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
101	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
102	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
103	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
104	C₄×D₂₅₁	—	2	8	no	—
105	C₂³×C₂₅₁	—	2	4	no	—
106	C₂³×C₂₅₁	—	2	4	no	—
107	C₂³×C₂₅₁	—	2	4	no	—
108	C₂³×C₂₅₁	C₂⁴	1	4	yes	—
109	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
110	Q₈×C₂₅₁	—	2	8	no	—
111	D₄×C₂₅₁	—	2	8	no	—
112	C₂×C₄×C₂₅₁	—	2	4	no	—
113	C₂×C₄×C₂₅₁	—	2	2	no	—
114	C₂×C₄×C₂₅₁	—	2	2	no	—
115	C₂×C₄×C₂₅₁	#41	1	4	yes	—
116	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
117	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
118	C₂×C₄×C₂₅₁	C₂⁴	1	4	yes	—
119	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
120	C₂²×C₂₅₁	—	2	4	no	—
121	C₂²×C₂₅₁	—	2	4	no	—
122	C₂²×C₂₅₁	—	2	4	no	—
123	C₂²×C₂₅₁	—	4	8	no	—
124	C₂²×C₂₅₁	C₂×#41	1	2	yes	—
125	C₂²×C₂₅₁	H(5,𝔽₂)	1	2	yes	—
126	C₂²×C₂₅₁	2₋¹⁺⁴	1	2	yes	—
127	C₂²×C₂₅₁	C₂⁵	1	1	yes	characteristic
128	C₂×D₂₅₁	—	2	2	no	—
129	C₂×D₂₅₁	—	2	2	no	—
130	C₂×D₂₅₁	—	2	2	no	—
131	C₂×D₂₅₁	—	4	4	no	—
132	C₄×C₂₅₁	—	2	4	no	—
133	C₄×C₂₅₁	—	2	2	no	—
134	C₄×C₂₅₁	—	2	2	no	—
135	C₄×C₂₅₁	C₂×#41	1	4	yes	—
136	Dic₂₅₁	—	2	2	no	—
137	Dic₂₅₁	—	2	2	no	—
138	Dic₂₅₁	—	2	4	no	—
139	Dic₂₅₁	—	2	4	no	—
140	Dic₂₅₁	—	2	2	no	—
141	C₂×C₂₅₁	—	4	4	no	—
142	C₂×C₂₅₁	#542	1	4	yes	complemented
143	C₂×C₂₅₁	C₂²×#41	1	1	yes	characteristic
144	C₂×C₂₅₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₂₅₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
146	D₂₅₁	—	4	2	no	—
147	C₂₅₁	C₂×#542	1	1	yes	characteristic, complemented, Sylow
148	C₂×#542	—	251	1	no	maximal, Sylow
149	#542	—	251	16	no	—
150	C₂×C₄⋊Q₈	—	251	1	no	—
151	C₂×#126	—	251	2	no	—
152	C₂×#124	—	251	2	no	—
153	C₂×#124	D₂₅₁	1	1	yes	characteristic, complemented
154	C₂×#123	—	251	2	no	—
155	C₂×C₂²⋊Q₈	—	251	1	no	—
156	C₂×C₂²⋊Q₈	—	251	2	no	—
157	C₂×#121	—	251	1	no	—
158	C₂×C₄×Q₈	—	251	1	no	—
159	C₂×C₄×D₄	—	251	1	no	—
160	C₂×#117	—	251	1	no	—
161	C₂²×Q₈	—	251	2	no	—
162	C₂²×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
163	C₂²×D₄	—	251	2	no	—
164	C₂²×D₄	C₂×D₂₅₁	1	1	yes	characteristic
165	C₂³×C₄	—	251	2	no	—
166	C₂³×C₄	—	251	1	no	—
167	C₄⋊Q₈	—	251	8	no	—
168	#126	—	251	16	no	—
169	#124	—	251	16	no	—
170	#124	C₂×D₂₅₁	1	8	yes	complemented
171	#123	—	251	16	no	—
172	C₂²⋊Q₈	—	251	8	no	—
173	C₂²⋊Q₈	—	251	16	no	—
174	#121	—	251	8	no	—
175	C₄×Q₈	—	251	8	no	—
176	C₄×D₄	—	251	8	no	—
177	#117	—	251	8	no	—
178	C₂×C₄⋊C₄	—	251	1	no	—
179	C₂×C₄⋊C₄	—	251	1	no	—
180	C₂×C₄⋊C₄	—	251	1	no	—
181	C₂×C₄⋊C₄	—	251	1	no	—
182	C₂×C₄⋊C₄	—	251	2	no	—
183	C₂×C₄⋊C₄	—	251	2	no	—
184	C₂×C₄⋊C₄	—	251	2	no	—
185	C₂×C₂²⋊C₄	—	251	2	no	—
186	C₂×C₂²⋊C₄	—	251	2	no	—
187	C₂×C₂²⋊C₄	—	251	1	no	—
188	C₂×C₂²⋊C₄	—	251	1	no	—
189	C₂×C₂²⋊C₄	—	251	2	no	—
190	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	2	yes	—
191	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	2	yes	—
192	C₂×C₄²	—	251	1	no	—
193	C₂×C₄²	—	251	2	no	—
194	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
195	C₂⁴	—	251	1	no	—
196	C₂⁴	C₂²×D₂₅₁	1	2	yes	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	—	1004	4	no	—
199	C₂×Q₈	—	251	8	no	—
200	C₂×Q₈	C₂²×D₂₅₁	1	4	yes	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	1004	4	no	—
203	C₂×D₄	—	251	8	no	—
204	C₂×D₄	C₂²×D₂₅₁	1	4	yes	—
205	C₂²×C₄	—	1004	4	no	—
206	C₂²×C₄	—	502	4	no	—
207	C₂²×C₄	—	251	8	no	—
208	C₂²×C₄	—	251	1	no	—
209	C₂²×C₄	—	251	1	no	—
210	C₂²×C₄	—	251	2	no	—
211	C₂²×C₄	—	251	1	no	—
212	C₂²×C₄	—	251	2	no	—
213	C₂²×C₄	—	251	4	no	—
214	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
215	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
217	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
218	C₄⋊C₄	—	251	4	no	—
219	C₄⋊C₄	—	251	4	no	—
220	C₄⋊C₄	—	251	8	no	—
221	C₄⋊C₄	—	251	4	no	—
222	C₄⋊C₄	—	251	4	no	—
223	C₄⋊C₄	—	251	8	no	—
224	C₄⋊C₄	—	251	8	no	—
225	C₂²⋊C₄	—	251	8	no	—
226	C₂²⋊C₄	—	251	8	no	—
227	C₂²⋊C₄	—	251	4	no	—
228	C₂²⋊C₄	—	251	4	no	—
229	C₂²⋊C₄	—	251	8	no	—
230	C₂²⋊C₄	C₂²×D₂₅₁	1	8	yes	—
231	C₂²⋊C₄	C₂²×D₂₅₁	1	8	yes	—
232	C₄²	—	251	4	no	—
233	C₄²	—	251	8	no	—
234	C₄²	C₂²×D₂₅₁	1	4	yes	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	2	4	no	—
238	C₂³	—	502	2	no	—
239	C₂³	—	502	2	no	—
240	C₂³	—	502	2	no	—
241	C₂³	—	251	2	no	—
242	C₂³	C₂³×D₂₅₁	1	4	yes	—
243	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
244	Q₈	—	2	8	no	—
245	Q₈	—	1004	8	no	—
246	D₄	—	2	8	no	—
247	D₄	—	1004	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	1004	8	no	—
252	C₂×C₄	—	502	2	no	—
253	C₂×C₄	—	502	2	no	—
254	C₂×C₄	—	502	4	no	—
255	C₂×C₄	—	502	4	no	—
256	C₂×C₄	—	502	2	no	—
257	C₂×C₄	—	502	8	no	—
258	C₂×C₄	—	251	2	no	—
259	C₂×C₄	—	251	2	no	—
260	C₂×C₄	—	251	4	no	—
261	C₂×C₄	—	251	2	no	—
262	C₂×C₄	—	251	4	no	—
263	C₂×C₄	#16860809	1	4	yes	—
264	C₂×C₄	C₂³×D₂₅₁	1	4	yes	—
265	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
266	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
267	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	8	no	—
272	C₂²	—	1004	4	no	—
273	C₂²	—	502	2	no	—
274	C₂²	—	502	2	no	—
275	C₂²	—	502	2	no	—
276	C₂²	C₂×#16860809	1	2	yes	—
277	C₂²	#22548761	1	2	yes	—
278	C₂²	#22548762	1	2	yes	—
279	C₂²	C₂⁴×D₂₅₁	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	502	2	no	—
284	C₄	—	502	2	no	—
285	C₄	—	502	4	no	—
286	C₄	—	502	4	no	—
287	C₄	—	502	2	no	—
288	C₄	C₂×#16860809	1	4	yes	—
289	C₂	—	4	4	no	—
290	C₂	—	1004	2	no	—
291	C₂	#30632306	1	4	yes	direct product factor
292	C₂	C₂²×#16860809	1	1	yes	characteristic
293	C₂	C₂×#22548761	1	1	yes	characteristic
294	C₂	C₂×#22548762	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₅₁
commutator, cosocle	C₂²×C₂₅₁	C₂⁵
Frattini	C₂²	C₂⁴×D₂₅₁
Fitting	C₂×#124×C₂₅₁	C₂
socle	C₂³×C₂₅₁	C₂⁴
prime radical	C₂×#22548686	C₂
hypercenter	C₂×#124	D₂₅₁
hypocenter	C₂₅₁	C₂×#542
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. #30632306	C₂	direct (N × H)
2. C₂×#22548715	C₂	semidirect
3. C₂×#22548708	C₂	semidirect
4. C₂×#22548701	C₂	semidirect
5. C₂×#22548699	C₂	semidirect
6. C₂×#22548692	C₂	semidirect
7. C₂×#22548691	C₂	semidirect
9. C₂×Q₈×Dic₂₅₁	C₂	semidirect
10. C₂×D₄×Dic₂₅₁	C₂	semidirect
11. C₂×#22548672	C₂	semidirect
12. C₂×#124×C₂₅₁	C₂	semidirect
17. #22548715	C₂²	semidirect
18. #22548708	C₂²	semidirect
19. #22548701	C₂²	semidirect
20. #22548699	C₂²	semidirect
21. #22548692	C₂²	semidirect
22. #22548691	C₂²	semidirect
24. Q₈×Dic₂₅₁	C₂²	semidirect
25. D₄×Dic₂₅₁	C₂²	semidirect
26. #22548672	C₂²	semidirect
43. #124×C₂₅₁	C₂²	semidirect
142. C₂×C₂₅₁	#542	semidirect
147. C₂₅₁	C₂×#542	semidirect
153. C₂×#124	D₂₅₁	semidirect
170. #124	C₂×D₂₅₁	semidirect
291. C₂	#30632306	direct (N × H)

Maximal subgroups

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

subgroup	index
#30632306	2
C₂×#22548715	2
C₂×#22548708	2
C₂×#22548701	2
C₂×#22548699	2
C₂×#22548692	2
C₂×#22548691	2
C₂×#22548686	2
C₂×Q₈×Dic₂₅₁	2
C₂×D₄×Dic₂₅₁	2
C₂×#22548672	2
C₂×#124×C₂₅₁	2
C₂×#542	251

Subgroups — C2×#30632306 #44563741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30632306 #44563741