Subgroups — C₂×#30632498 #44565125

Order 32128.

1460 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548762	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632498	C₂	1	16	yes	maximal, direct product factor
3	C₂×#22548745	C₂	1	3	yes	maximal, complemented
4	C₂×#22548741	C₂	1	3	yes	maximal, complemented
5	C₂×#22548735	C₂	1	3	yes	maximal, complemented
6	C₂²×#16860798	C₂	1	3	yes	maximal, complemented
7	C₂×#22548722	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×H(5,𝔽₂)×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#22548762	—	2	4	no	—
10	#22548762	C₂²	1	8	yes	complemented
11	C₂×#16860810	C₂²	1	3	yes	complemented
12	C₂×#16860809	—	2	3	no	—
13	C₂²×Dic₅₀₂	C₂²	1	3	yes	complemented
14	C₂×Q₈×D₂₅₁	—	2	1	no	—
15	#22548745	C₂²	1	24	yes	complemented
16	#22548741	C₂²	1	24	yes	complemented
17	#22548735	C₂²	1	24	yes	complemented
18	C₂×#16860800	C₂²	1	3	yes	complemented
19	C₂×#16860798	C₂²	1	3	yes	complemented
20	C₂×#16860798	C₂²	1	6	yes	complemented
21	C₂×#16860798	C₂²	1	24	yes	complemented
22	C₂×#16860797	C₂²	1	1	yes	characteristic, complemented
23	C₂×#16860795	C₂²	1	3	yes	complemented
24	#22548722	C₂²	1	8	yes	complemented
25	C₂×#16860794	C₂²	1	3	yes	complemented
26	C₂²×C₂₅₁⋊C₈	C₂²	1	3	yes	complemented
27	H(5,𝔽₂)×C₂₅₁	—	2	4	no	—
28	H(5,𝔽₂)×C₂₅₁	C₂²	1	8	yes	complemented
29	C₂×#41×C₂₅₁	—	2	1	no	—
30	C₂×#41×C₂₅₁	C₂²	1	3	yes	—
31	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
32	C₂²×D₄×C₂₅₁	—	2	3	no	—
33	C₂²×D₄×C₂₅₁	C₂²	1	3	yes	—
34	#16860810	—	2	6	no	—
35	#16860810	C₂³	1	12	yes	complemented
36	#16860809	—	2	24	no	—
37	C₂×Dic₅₀₂	—	2	3	no	—
38	C₂×Dic₅₀₂	—	2	6	no	—
39	C₂×Dic₅₀₂	C₂³	1	3	yes	—
40	C₂×Dic₅₀₂	C₂³	1	12	yes	complemented
41	Q₈×D₂₅₁	—	2	8	no	—
42	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
43	C₂×C₂²⋊D₂₅₁	—	2	3	no	—
44	C₂²×Dic₂₅₁	—	2	3	no	—
45	C₂×C₄×D₂₅₁	—	2	3	no	—
46	#16860800	C₂³	1	12	yes	complemented
47	#16860798	C₂³	1	24	yes	complemented
48	#16860798	C₂³	1	12	yes	complemented
49	#16860797	C₂³	1	4	yes	complemented
50	#16860795	C₂³	1	12	yes	complemented
51	#16860794	C₂³	1	12	yes	complemented
52	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
53	C₂×C₂₅₁⋊C₈	C₂³	1	3	yes	—
54	C₂×C₂₅₁⋊C₈	C₂³	1	12	yes	complemented
55	C₂⁴×C₂₅₁	—	2	3	no	—
56	#41×C₂₅₁	—	2	8	no	—
57	#41×C₂₅₁	—	2	6	no	—
58	#41×C₂₅₁	D₄	1	4	yes	—
59	#41×C₂₅₁	C₂³	1	12	yes	—
60	#41×C₂₅₁	C₂³	1	4	yes	—
61	C₂×Q₈×C₂₅₁	D₄	1	1	yes	characteristic
62	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₂₅₁	—	2	3	no	—
64	C₂×D₄×C₂₅₁	—	2	24	no	—
65	C₂×D₄×C₂₅₁	—	2	3	no	—
66	C₂×D₄×C₂₅₁	—	2	6	no	—
67	C₂×D₄×C₂₅₁	D₄	1	3	yes	—
68	C₂×D₄×C₂₅₁	C₂³	1	12	yes	—
69	C₂×D₄×C₂₅₁	C₂³	1	3	yes	—
70	C₂²×C₄×C₂₅₁	—	2	3	no	—
71	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
72	C₂²×D₂₅₁	—	2	1	no	—
73	Dic₅₀₂	—	2	12	no	—
74	Dic₅₀₂	—	2	3	no	—
75	Dic₅₀₂	C₂⁴	1	6	yes	—
76	D₁₀₀₄	—	2	1	no	—
77	D₁₀₀₄	C₂⁴	1	2	yes	—
78	C₂²⋊D₂₅₁	—	2	12	no	—
79	C₂×Dic₂₅₁	—	2	3	no	—
80	C₂×Dic₂₅₁	—	2	12	no	—
81	C₄×D₂₅₁	—	2	12	no	—
82	C₂₅₁⋊C₈	C₂⁴	1	2	yes	—
83	C₂₅₁⋊C₈	C₂⁴	1	6	yes	—
84	C₂³×C₂₅₁	—	2	3	no	—
85	C₂³×C₂₅₁	—	2	12	no	—
86	C₂³×C₂₅₁	—	8	6	no	—
87	C₂³×C₂₅₁	C₂×D₄	1	3	yes	—
88	Q₈×C₂₅₁	—	2	1	no	—
89	Q₈×C₂₅₁	C₂×D₄	1	4	yes	—
90	Q₈×C₂₅₁	C₂⁴	1	2	yes	—
91	D₄×C₂₅₁	—	2	12	no	—
92	D₄×C₂₅₁	—	2	12	no	—
93	D₄×C₂₅₁	—	2	3	no	—
94	D₄×C₂₅₁	C₂×D₄	1	12	yes	—
95	D₄×C₂₅₁	C₂⁴	1	6	yes	—
96	C₂×C₄×C₂₅₁	—	2	1	no	—
97	C₂×C₄×C₂₅₁	—	2	12	no	—
98	C₂×C₄×C₂₅₁	C₂×D₄	1	3	yes	—
99	C₂×C₄×C₂₅₁	C₂×D₄	1	6	yes	—
100	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
101	C₂×C₄×C₂₅₁	C₂⁴	1	6	yes	—
102	C₂²×C₂₅₁	—	2	6	no	—
103	C₂²×C₂₅₁	—	2	6	no	—
104	C₂²×C₂₅₁	—	4	6	no	—
105	C₂²×C₂₅₁	—	8	12	no	—
106	C₂²×C₂₅₁	C₂²×D₄	1	6	yes	—
107	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
108	C₂×D₂₅₁	—	2	2	no	—
109	C₂×D₂₅₁	—	4	2	no	—
110	C₄×C₂₅₁	—	2	2	no	—
111	C₄×C₂₅₁	C₂²×D₄	1	6	yes	—
112	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
113	C₄×C₂₅₁	C₂⁵	1	1	yes	characteristic
114	Dic₂₅₁	—	2	6	no	—
115	C₂×C₂₅₁	—	2	6	no	—
116	C₂×C₂₅₁	—	4	6	no	—
117	C₂×C₂₅₁	#577	1	2	yes	complemented
118	C₂×C₂₅₁	C₂³×D₄	1	1	yes	characteristic
119	D₂₅₁	—	4	2	no	—
120	C₂₅₁	C₂×#577	1	1	yes	characteristic, complemented, Sylow
121	C₂×#577	—	251	1	no	maximal, Sylow
122	C₂×2₋¹⁺⁴	—	251	1	no	—
123	C₂×H(5,𝔽₂)	D₂₅₁	1	1	yes	characteristic, complemented
124	#577	—	251	16	no	—
125	C₂×#137	—	251	3	no	—
126	C₂×AGL(1,ℤ/8)	—	251	3	no	—
127	C₂×#135	—	251	3	no	—
128	C₂²×SD₁₆	—	251	3	no	—
129	C₂×#131	—	251	1	no	—
130	2₋¹⁺⁴	—	502	4	no	—
131	2₋¹⁺⁴	—	251	8	no	—
132	H(5,𝔽₂)	—	2	4	no	—
133	H(5,𝔽₂)	C₂×D₂₅₁	1	8	yes	complemented
134	C₂×#41	—	2	1	no	—
135	C₂×#41	—	502	3	no	—
136	C₂×#41	—	251	3	no	—
137	C₂×#41	C₂×D₂₅₁	1	3	yes	—
138	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic
139	C₂²×Q₈	—	502	1	no	—
140	C₂²×Q₈	—	251	3	no	—
141	C₂²×D₄	—	2	3	no	—
142	C₂²×D₄	C₂×D₂₅₁	1	3	yes	—
143	#137	—	251	24	no	—
144	AGL(1,ℤ/8)	—	251	24	no	—
145	#135	—	251	24	no	—
146	C₂×Q₁₆	—	251	3	no	—
147	C₂×SD₁₆	—	251	1	no	—
148	C₂×SD₁₆	—	251	3	no	—
149	C₂×SD₁₆	—	251	6	no	—
150	C₂×SD₁₆	—	251	24	no	—
151	C₂×D₈	—	251	3	no	—
152	#131	—	251	8	no	—
153	C₂×M₁₆	—	251	3	no	—
154	C₂²×C₈	—	251	3	no	—
155	C₂⁴	—	2	3	no	—
156	#41	—	2	8	no	—
157	#41	—	2	6	no	—
158	#41	—	502	24	no	—
159	#41	—	502	6	no	—
160	#41	—	251	12	no	—
161	#41	C₂²⋊D₂₅₁	1	4	yes	—
162	#41	C₂²×D₂₅₁	1	12	yes	—
163	#41	C₂²×D₂₅₁	1	4	yes	—
164	C₂×Q₈	—	502	8	no	—
165	C₂×Q₈	—	502	3	no	—
166	C₂×Q₈	—	502	6	no	—
167	C₂×Q₈	—	251	3	no	—
168	C₂×Q₈	—	251	12	no	—
169	C₂×Q₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
170	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
171	C₂×D₄	—	2	3	no	—
172	C₂×D₄	—	2	24	no	—
173	C₂×D₄	—	2	3	no	—
174	C₂×D₄	—	2	6	no	—
175	C₂×D₄	—	502	3	no	—
176	C₂×D₄	—	251	1	no	—
177	C₂×D₄	C₂²⋊D₂₅₁	1	3	yes	—
178	C₂×D₄	C₂²×D₂₅₁	1	3	yes	—
179	C₂×D₄	C₂²×D₂₅₁	1	12	yes	—
180	C₂²×C₄	—	2	3	no	—
181	C₂²×C₄	—	502	3	no	—
182	C₂²×C₄	—	502	3	no	—
183	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
184	Q₁₆	—	251	12	no	—
185	SD₁₆	—	251	4	no	—
186	SD₁₆	—	251	12	no	—
187	SD₁₆	—	251	24	no	—
188	D₈	—	251	12	no	—
189	M₁₆	—	251	12	no	—
190	C₂×C₈	—	251	1	no	—
191	C₂×C₈	—	251	3	no	—
192	C₂×C₈	—	251	12	no	—
193	C₂³	—	2	3	no	—
194	C₂³	—	2	12	no	—
195	C₂³	—	8	6	no	—
196	C₂³	—	502	1	no	—
197	C₂³	C₂×C₂²⋊D₂₅₁	1	3	yes	—
198	Q₈	—	2	1	no	—
199	Q₈	—	502	12	no	—
200	Q₈	—	502	3	no	—
201	Q₈	—	251	6	no	—
202	Q₈	C₂×C₂²⋊D₂₅₁	1	4	yes	—
203	Q₈	C₂³×D₂₅₁	1	2	yes	—
204	D₄	—	2	12	no	—
205	D₄	—	2	12	no	—
206	D₄	—	2	3	no	—
207	D₄	—	502	12	no	—
208	D₄	—	502	1	no	—
209	D₄	—	251	2	no	—
210	D₄	C₂×C₂²⋊D₂₅₁	1	12	yes	—
211	D₄	C₂³×D₂₅₁	1	6	yes	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	12	no	—
214	C₂×C₄	—	502	12	no	—
215	C₂×C₄	—	502	3	no	—
216	C₂×C₄	—	502	12	no	—
217	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	6	yes	—
218	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	3	yes	—
219	C₂×C₄	C₂³×D₂₅₁	1	6	yes	—
220	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
221	C₈	—	251	2	no	—
222	C₈	—	251	6	no	—
223	C₂²	—	2	6	no	—
224	C₂²	—	2	6	no	—
225	C₂²	—	4	6	no	—
226	C₂²	—	8	12	no	—
227	C₂²	—	1004	2	no	—
228	C₂²	—	502	2	no	—
229	C₂²	C₂²×C₂²⋊D₂₅₁	1	6	yes	—
230	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	502	6	no	—
233	C₄	C₂²×C₂²⋊D₂₅₁	1	6	yes	—
234	C₄	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
235	C₄	C₂⁴×D₂₅₁	1	1	yes	characteristic
236	C₂	—	2	6	no	—
237	C₂	—	4	6	no	—
238	C₂	—	1004	2	no	—
239	C₂	#30632498	1	2	yes	direct product factor
240	C₂	C₂³×C₂²⋊D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₅₁
commutator, cosocle	C₄×C₂₅₁	C₂⁵
Frattini	C₄	C₂⁴×D₂₅₁
Fitting	C₂×H(5,𝔽₂)×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
hypercenter	C₂×H(5,𝔽₂)	D₂₅₁
hypocenter	C₂₅₁	C₂×#577
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×#22548762	C₂	semidirect
2. #30632498	C₂	direct (N × H)
3. C₂×#22548745	C₂	semidirect
4. C₂×#22548741	C₂	semidirect
5. C₂×#22548735	C₂	semidirect
6. C₂²×#16860798	C₂	semidirect
7. C₂×#22548722	C₂	semidirect
8. C₂×H(5,𝔽₂)×C₂₅₁	C₂	semidirect
10. #22548762	C₂²	semidirect
11. C₂×#16860810	C₂²	semidirect
13. C₂²×Dic₅₀₂	C₂²	semidirect
15. #22548745	C₂²	semidirect
16. #22548741	C₂²	semidirect
17. #22548735	C₂²	semidirect
18. C₂×#16860800	C₂²	semidirect
19. C₂×#16860798	C₂²	semidirect
20. C₂×#16860798	C₂²	semidirect
21. C₂×#16860798	C₂²	semidirect
22. C₂×#16860797	C₂²	semidirect
23. C₂×#16860795	C₂²	semidirect
24. #22548722	C₂²	semidirect
25. C₂×#16860794	C₂²	semidirect
26. C₂²×C₂₅₁⋊C₈	C₂²	semidirect
28. H(5,𝔽₂)×C₂₅₁	C₂²	semidirect
35. #16860810	C₂³	semidirect
40. C₂×Dic₅₀₂	C₂³	semidirect
46. #16860800	C₂³	semidirect
47. #16860798	C₂³	semidirect
48. #16860798	C₂³	semidirect
49. #16860797	C₂³	semidirect
50. #16860795	C₂³	semidirect
51. #16860794	C₂³	semidirect
54. C₂×C₂₅₁⋊C₈	C₂³	semidirect
117. C₂×C₂₅₁	#577	semidirect
120. C₂₅₁	C₂×#577	semidirect
123. C₂×H(5,𝔽₂)	D₂₅₁	semidirect
133. H(5,𝔽₂)	C₂×D₂₅₁	semidirect
239. C₂	#30632498	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×#22548762	2
#30632498	2
C₂×#22548745	2
C₂×#22548741	2
C₂×#22548735	2
C₂²×#16860798	2
C₂×#22548722	2
C₂×H(5,𝔽₂)×C₂₅₁	2
C₂×#577	251

Subgroups — C2×#30632498 #44565125

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30632498 #44565125