Subgroups — C₂×#30632255 #44563675

Order 32128.

1896 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16860811	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄×D₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632255	C₂	1	16	yes	maximal, direct product factor
4	C₂×#22548712	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#22548711	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#22548704	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#22548688	C₂	1	4	yes	maximal, complemented
8	C₂×#22548685	C₂	1	4	yes	maximal, complemented
9	C₂×#22548672	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#124×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₂₅₁	C₂²	1	2	yes	—
12	C₂×#16860811	—	2	8	no	—
13	C₂×#16860811	C₂²	1	8	yes	complemented
14	C₂²×D₁₀₀₄	C₂²	1	1	yes	characteristic
15	C₂²×D₁₀₀₄	C₂²	1	2	yes	—
16	C₂²×D₁₀₀₄	C₂²	1	1	yes	characteristic, complemented
17	C₂²×D₁₀₀₄	C₂²	1	4	yes	complemented
18	C₂²×C₂²⋊D₂₅₁	C₂²	1	2	yes	—
19	C₂²×C₂²⋊D₂₅₁	C₂²	1	4	yes	complemented
20	C₂×D₄×D₂₅₁	—	2	8	no	—
21	C₂×D₄×D₂₅₁	C₂²	1	8	yes	complemented
22	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄×D₂₅₁	C₂²	1	2	yes	complemented
24	#22548712	C₂²	1	8	yes	complemented
25	#22548711	C₂²	1	8	yes	complemented
26	#22548704	C₂²	1	8	yes	complemented
27	#22548688	C₂²	1	32	yes	complemented
28	#22548685	C₂²	1	32	yes	complemented
29	#22548672	C₂²	1	8	yes	complemented
30	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	complemented
31	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	2	yes	—
32	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	4	yes	complemented
33	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic, complemented
34	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
35	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
36	#124×C₂₅₁	C₂²	1	8	yes	complemented
37	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	4	yes	complemented
38	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
39	C₂³×D₂₅₁	—	2	8	no	—
40	C₂³×D₂₅₁	—	4	8	no	—
41	C₂³×D₂₅₁	C₂³	1	2	yes	—
42	C₂³×D₂₅₁	C₂³	1	2	yes	—
43	C₂³×D₂₅₁	C₂³	1	1	yes	characteristic
44	C₂³×D₂₅₁	C₂³	1	8	yes	—
45	#16860811	—	2	32	no	—
46	C₂×D₁₀₀₄	—	2	8	no	—
47	C₂×D₁₀₀₄	—	2	4	no	—
48	C₂×D₁₀₀₄	—	2	4	no	—
49	C₂×D₁₀₀₄	—	4	8	no	—
50	C₂×D₁₀₀₄	C₂³	1	4	yes	—
51	C₂×D₁₀₀₄	C₂³	1	8	yes	—
52	C₂×D₁₀₀₄	C₂³	1	4	yes	complemented
53	C₂×D₁₀₀₄	C₂³	1	16	yes	complemented
54	C₂×C₂²⋊D₂₅₁	—	2	8	no	—
55	C₂×C₂²⋊D₂₅₁	—	4	8	no	—
56	C₂×C₂²⋊D₂₅₁	C₂³	1	8	yes	—
57	C₂×C₂²⋊D₂₅₁	C₂³	1	16	yes	complemented
58	D₄×D₂₅₁	—	2	32	no	—
59	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
60	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₂₅₁	—	2	8	no	—
62	C₂×C₄×D₂₅₁	D₄	1	8	yes	complemented
63	C₂×C₄×D₂₅₁	C₂³	1	4	yes	—
64	C₂×C₄×D₂₅₁	C₂³	1	8	yes	complemented
65	Dic₂₅₁⋊C₄	C₂³	1	8	yes	complemented
66	(C₂×D₂₅₁)⋊C₄	C₂³	1	8	yes	—
67	(C₂×D₂₅₁)⋊C₄	C₂³	1	16	yes	complemented
68	C₄×Dic₂₅₁	C₂³	1	4	yes	complemented
69	C₂⁴×C₂₅₁	C₂³	1	2	yes	—
70	C₂×Q₈×C₂₅₁	—	2	4	no	—
71	C₂×Q₈×C₂₅₁	C₂³	1	4	yes	complemented
72	C₂×D₄×C₂₅₁	—	2	4	no	—
73	C₂×D₄×C₂₅₁	C₂³	1	4	yes	—
74	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
75	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
76	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₂₅₁	C₂³	1	16	yes	complemented
78	C₄²×C₂₅₁	C₂³	1	4	yes	complemented
79	C₂²×D₂₅₁	—	2	4	no	—
80	C₂²×D₂₅₁	—	2	4	no	—
81	C₂²×D₂₅₁	—	2	4	no	—
82	C₂²×D₂₅₁	—	2	8	no	—
83	C₂²×D₂₅₁	—	2	8	no	—
84	C₂²×D₂₅₁	—	2	16	no	—
85	C₂²×D₂₅₁	—	4	16	no	—
86	C₂²×D₂₅₁	—	4	32	no	—
87	C₂²×D₂₅₁	C₂×D₄	1	4	yes	—
88	C₂²×D₂₅₁	C₂⁴	1	4	yes	—
89	C₂²×D₂₅₁	C₂⁴	1	4	yes	—
90	C₂²×D₂₅₁	C₂⁴	1	2	yes	—
91	D₁₀₀₄	—	2	16	no	—
92	D₁₀₀₄	—	2	8	no	—
93	D₁₀₀₄	—	2	8	no	—
94	D₁₀₀₄	—	4	16	no	—
95	C₂²⋊D₂₅₁	—	2	16	no	—
96	C₂²⋊D₂₅₁	—	4	16	no	—
97	C₂×Dic₂₅₁	—	2	4	no	—
98	C₂×Dic₂₅₁	C₂×D₄	1	4	yes	—
99	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
100	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
101	C₄×D₂₅₁	—	2	16	no	—
102	C₄×D₂₅₁	C₂×D₄	1	16	yes	complemented
103	C₂³×C₂₅₁	—	2	4	no	—
104	C₂³×C₂₅₁	—	2	8	no	—
105	C₂³×C₂₅₁	C₂⁴	1	4	yes	—
106	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
107	Q₈×C₂₅₁	—	2	8	no	—
108	D₄×C₂₅₁	—	2	8	no	—
109	C₂×C₄×C₂₅₁	—	2	4	no	—
110	C₂×C₄×C₂₅₁	—	2	4	no	—
111	C₂×C₄×C₂₅₁	C₂×D₄	1	4	yes	—
112	C₂×C₄×C₂₅₁	C₂⁴	1	4	yes	—
113	C₂×C₄×C₂₅₁	C₂⁴	1	4	yes	—
114	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
115	C₂²×C₂₅₁	—	2	4	no	—
116	C₂²×C₂₅₁	—	2	8	no	—
117	C₂²×C₂₅₁	—	4	8	no	—
118	C₂²×C₂₅₁	C₂²×D₄	1	2	yes	—
119	C₂²×C₂₅₁	H(5,𝔽₂)	1	4	yes	—
120	C₂²×C₂₅₁	C₂⁵	1	1	yes	characteristic
121	C₂×D₂₅₁	—	2	8	no	—
122	C₂×D₂₅₁	—	2	4	no	—
123	C₂×D₂₅₁	—	2	4	no	—
124	C₂×D₂₅₁	—	2	4	no	—
125	C₂×D₂₅₁	—	2	8	no	—
126	C₂×D₂₅₁	—	2	8	no	—
127	C₂×D₂₅₁	—	4	32	no	—
128	C₂×D₂₅₁	—	4	8	no	—
129	C₂×D₂₅₁	—	4	8	no	—
130	C₂×D₂₅₁	C₂²×D₄	1	4	yes	—
131	C₄×C₂₅₁	—	2	4	no	—
132	C₄×C₂₅₁	—	2	4	no	—
133	C₄×C₂₅₁	C₂²×D₄	1	4	yes	—
134	Dic₂₅₁	—	2	4	no	—
135	Dic₂₅₁	C₂²×D₄	1	4	yes	—
136	C₂×C₂₅₁	—	4	4	no	—
137	C₂×C₂₅₁	#535	1	4	yes	complemented
138	C₂×C₂₅₁	C₂³×D₄	1	1	yes	characteristic
139	C₂×C₂₅₁	C₂×H(5,𝔽₂)	1	2	yes	—
140	D₂₅₁	—	2	4	no	—
141	D₂₅₁	—	4	4	no	—
142	D₂₅₁	—	4	4	no	—
143	C₂₅₁	C₂×#535	1	1	yes	characteristic, complemented, Sylow
144	C₂×#535	—	251	1	no	maximal, Sylow
145	C₂²×#41	—	251	1	no	—
146	C₂³×D₄	—	251	1	no	—
147	#535	—	251	16	no	—
148	C₂×#127	—	251	1	no	—
149	C₂×#127	—	251	1	no	—
150	C₂×#124	—	251	1	no	—
151	C₂×#124	D₂₅₁	1	1	yes	characteristic, complemented
152	C₂×#121	—	251	4	no	—
153	C₂×#120	—	251	4	no	—
154	C₂×#117	—	251	1	no	—
155	C₂⁵	—	251	2	no	—
156	C₂×#41	—	502	8	no	—
157	C₂×#41	—	251	8	no	—
158	C₂²×Q₈	C₂×D₂₅₁	1	1	yes	characteristic, complemented
159	C₂²×D₄	—	502	8	no	—
160	C₂²×D₄	—	251	4	no	—
161	C₂²×D₄	—	251	2	no	—
162	C₂²×D₄	—	251	8	no	—
163	C₂²×D₄	—	251	4	no	—
164	C₂²×D₄	—	251	2	no	—
165	C₂²×D₄	—	251	1	no	—
166	C₂²×D₄	—	251	1	no	—
167	C₂²×D₄	C₂×D₂₅₁	1	1	yes	characteristic
168	C₂³×C₄	—	251	2	no	—
169	C₂³×C₄	—	251	1	no	—
170	#127	—	251	8	no	—
171	#127	—	251	8	no	—
172	#124	—	251	8	no	—
173	#124	C₂×D₂₅₁	1	8	yes	complemented
174	#121	—	251	32	no	—
175	#120	—	251	32	no	—
176	#117	—	251	8	no	—
177	C₂×C₄⋊C₄	—	251	2	no	—
178	C₂×C₂²⋊C₄	—	251	2	no	—
179	C₂×C₂²⋊C₄	—	251	4	no	—
180	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	4	yes	complemented
181	C₂×C₄²	—	251	1	no	—
182	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic, complemented
183	C₂⁴	—	1004	8	no	—
184	C₂⁴	—	502	8	no	—
185	C₂⁴	—	251	8	no	—
186	C₂⁴	—	251	2	no	—
187	C₂⁴	—	251	2	no	—
188	C₂⁴	—	251	1	no	—
189	C₂⁴	C₂²×D₂₅₁	1	2	yes	—
190	#41	—	502	32	no	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	C₂²×D₂₅₁	1	4	yes	complemented
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	1004	8	no	—
195	C₂×D₄	—	1004	8	no	—
196	C₂×D₄	—	502	8	no	—
197	C₂×D₄	—	502	32	no	—
198	C₂×D₄	—	502	8	no	—
199	C₂×D₄	—	502	4	no	—
200	C₂×D₄	—	502	4	no	—
201	C₂×D₄	—	251	16	no	—
202	C₂×D₄	—	251	8	no	—
203	C₂×D₄	—	251	16	no	—
204	C₂×D₄	—	251	8	no	—
205	C₂×D₄	—	251	4	no	—
206	C₂×D₄	—	251	4	no	—
207	C₂×D₄	C₂²×D₂₅₁	1	4	yes	—
208	C₂²×C₄	—	502	8	no	—
209	C₂²×C₄	—	251	2	no	—
210	C₂²×C₄	—	251	8	no	—
211	C₂²×C₄	—	251	1	no	—
212	C₂²×C₄	—	251	8	no	—
213	C₂²×C₄	—	251	4	no	—
214	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
215	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
216	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
217	C₄⋊C₄	—	251	8	no	—
218	C₂²⋊C₄	—	251	8	no	—
219	C₂²⋊C₄	—	251	16	no	—
220	C₂²⋊C₄	C₂²×D₂₅₁	1	16	yes	complemented
221	C₄²	—	251	4	no	—
222	C₄²	C₂²×D₂₅₁	1	4	yes	complemented
223	C₂³	—	2	4	no	—
224	C₂³	—	2	8	no	—
225	C₂³	—	1004	16	no	—
226	C₂³	—	1004	32	no	—
227	C₂³	—	502	16	no	—
228	C₂³	—	502	4	no	—
229	C₂³	—	502	4	no	—
230	C₂³	—	502	8	no	—
231	C₂³	—	502	8	no	—
232	C₂³	—	502	4	no	—
233	C₂³	—	251	4	no	—
234	C₂³	—	251	4	no	—
235	C₂³	—	251	4	no	—
236	C₂³	—	251	2	no	—
237	C₂³	C₂³×D₂₅₁	1	4	yes	—
238	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
239	Q₈	—	2	8	no	—
240	D₄	—	2	8	no	—
241	D₄	—	1004	16	no	—
242	D₄	—	1004	16	no	—
243	D₄	—	502	16	no	—
244	D₄	—	502	16	no	—
245	D₄	—	502	8	no	—
246	D₄	—	502	8	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	502	16	no	—
250	C₂×C₄	—	502	4	no	—
251	C₂×C₄	—	251	4	no	—
252	C₂×C₄	—	251	4	no	—
253	C₂×C₄	—	251	2	no	—
254	C₂×C₄	—	251	16	no	—
255	C₂×C₄	D₄×D₂₅₁	1	4	yes	—
256	C₂×C₄	C₂³×D₂₅₁	1	4	yes	—
257	C₂×C₄	C₂³×D₂₅₁	1	4	yes	—
258	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
259	C₂²	—	2	4	no	—
260	C₂²	—	2	8	no	—
261	C₂²	—	4	8	no	—
262	C₂²	—	1004	32	no	—
263	C₂²	—	1004	8	no	—
264	C₂²	—	1004	8	no	—
265	C₂²	—	502	4	no	—
266	C₂²	—	502	4	no	—
267	C₂²	—	502	8	no	—
268	C₂²	—	502	8	no	—
269	C₂²	—	502	8	no	—
270	C₂²	—	502	4	no	—
271	C₂²	—	251	4	no	—
272	C₂²	C₂×D₄×D₂₅₁	1	2	yes	—
273	C₂²	#22548761	1	4	yes	—
274	C₂²	C₂⁴×D₂₅₁	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	2	4	no	—
277	C₄	—	502	4	no	—
278	C₄	—	251	4	no	—
279	C₄	C₂×D₄×D₂₅₁	1	4	yes	—
280	C₂	—	4	4	no	—
281	C₂	—	1004	4	no	—
282	C₂	—	1004	4	no	—
283	C₂	—	502	4	no	—
284	C₂	#30632255	1	4	yes	direct product factor
285	C₂	C₂²×D₄×D₂₅₁	1	1	yes	characteristic
286	C₂	C₂×#22548761	1	2	yes	—

	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
hypercenter	C₂×#124	D₂₅₁
hypocenter	C₂₅₁	C₂×#535
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. C₂²×#16860811	C₂	semidirect
2. C₂²×D₄×D₂₅₁	C₂	semidirect
3. #30632255	C₂	direct (N × H)
4. C₂×#22548712	C₂	semidirect
5. C₂×#22548711	C₂	semidirect
6. C₂×#22548704	C₂	semidirect
7. C₂×#22548688	C₂	semidirect
8. C₂×#22548685	C₂	semidirect
9. C₂×#22548672	C₂	semidirect
10. C₂×#124×C₂₅₁	C₂	semidirect
13. C₂×#16860811	C₂²	semidirect
16. C₂²×D₁₀₀₄	C₂²	semidirect
17. C₂²×D₁₀₀₄	C₂²	semidirect
19. C₂²×C₂²⋊D₂₅₁	C₂²	semidirect
21. C₂×D₄×D₂₅₁	C₂²	semidirect
23. C₂²×C₄×D₂₅₁	C₂²	semidirect
24. #22548712	C₂²	semidirect
25. #22548711	C₂²	semidirect
26. #22548704	C₂²	semidirect
27. #22548688	C₂²	semidirect
28. #22548685	C₂²	semidirect
29. #22548672	C₂²	semidirect
30. C₂×Dic₂₅₁⋊C₄	C₂²	semidirect
32. C₂×(C₂×D₂₅₁)⋊C₄	C₂²	semidirect
33. C₂×C₄×Dic₂₅₁	C₂²	semidirect
34. C₂²×Q₈×C₂₅₁	C₂²	semidirect
36. #124×C₂₅₁	C₂²	semidirect
37. C₂×C₂²⋊C₄×C₂₅₁	C₂²	semidirect
38. C₂×C₄²×C₂₅₁	C₂²	semidirect
52. C₂×D₁₀₀₄	C₂³	semidirect
53. C₂×D₁₀₀₄	C₂³	semidirect
57. C₂×C₂²⋊D₂₅₁	C₂³	semidirect
62. C₂×C₄×D₂₅₁	D₄	semidirect
64. C₂×C₄×D₂₅₁	C₂³	semidirect
65. Dic₂₅₁⋊C₄	C₂³	semidirect
67. (C₂×D₂₅₁)⋊C₄	C₂³	semidirect
68. C₄×Dic₂₅₁	C₂³	semidirect
71. C₂×Q₈×C₂₅₁	C₂³	semidirect
77. C₂²⋊C₄×C₂₅₁	C₂³	semidirect
78. C₄²×C₂₅₁	C₂³	semidirect
102. C₄×D₂₅₁	C₂×D₄	semidirect
137. C₂×C₂₅₁	#535	semidirect
143. C₂₅₁	C₂×#535	semidirect
151. C₂×#124	D₂₅₁	semidirect
158. C₂²×Q₈	C₂×D₂₅₁	semidirect
173. #124	C₂×D₂₅₁	semidirect
180. C₂×C₂²⋊C₄	C₂×D₂₅₁	semidirect
182. C₂×C₄²	C₂×D₂₅₁	semidirect
192. C₂×Q₈	C₂²×D₂₅₁	semidirect
220. C₂²⋊C₄	C₂²×D₂₅₁	semidirect
222. C₄²	C₂²×D₂₅₁	semidirect
284. C₂	#30632255	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#16860811	2
C₂²×D₄×D₂₅₁	2
#30632255	2
C₂×#22548712	2
C₂×#22548711	2
C₂×#22548704	2
C₂×#22548688	2
C₂×#22548685	2
C₂×#22548672	2
C₂×#124×C₂₅₁	2
C₂×#535	251

Subgroups — C2×#30632255 #44563675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30632255 #44563675