Subgroups — C₂×#30632490 #44565116

Order 32128.

1512 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548761	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632490	C₂	1	16	yes	maximal, direct product factor
3	C₂×#22548741	C₂	1	3	yes	maximal, complemented
4	C₂×#22548740	C₂	1	3	yes	maximal, complemented
5	C₂×#22548735	C₂	1	3	yes	maximal, complemented
6	C₂²×#16860795	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	#22548761	—	2	4	no	—
10	#22548761	C₂²	1	8	yes	complemented
11	C₂×#16860811	—	2	1	no	—
12	C₂×#16860810	C₂²	1	3	yes	complemented
13	C₂²×D₁₀₀₄	C₂²	1	3	yes	complemented
14	C₂×D₄×D₂₅₁	—	2	3	no	—
15	#22548741	C₂²	1	24	yes	complemented
16	#22548740	C₂²	1	24	yes	complemented
17	#22548735	C₂²	1	24	yes	complemented
18	C₂×#16860800	C₂²	1	1	yes	characteristic, complemented
19	C₂×#16860798	C₂²	1	3	yes	complemented
20	C₂×#16860797	C₂²	1	3	yes	complemented
21	C₂×#16860795	C₂²	1	3	yes	complemented
22	C₂×#16860795	C₂²	1	6	yes	complemented
23	C₂×#16860795	C₂²	1	24	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	1	yes	characteristic
31	C₂×#41×C₂₅₁	C₂²	1	3	yes	complemented
32	C₂²×D₄×C₂₅₁	—	2	3	no	—
33	C₂²×D₄×C₂₅₁	C₂²	1	3	yes	complemented
34	C₂³×D₂₅₁	—	2	3	no	—
35	#16860811	—	2	8	no	—
36	#16860810	—	2	6	no	—
37	#16860810	C₂³	1	12	yes	complemented
38	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
39	C₂×D₁₀₀₄	—	2	3	no	—
40	C₂×D₁₀₀₄	—	2	6	no	—
41	C₂×D₁₀₀₄	C₂³	1	3	yes	—
42	C₂×D₁₀₀₄	C₂³	1	12	yes	complemented
43	C₂×C₂²⋊D₂₅₁	—	2	3	no	—
44	D₄×D₂₅₁	—	2	24	no	—
45	C₂×C₄×D₂₅₁	—	2	3	no	—
46	#16860800	C₂³	1	4	yes	complemented
47	#16860798	C₂³	1	12	yes	complemented
48	#16860797	C₂³	1	12	yes	complemented
49	#16860795	C₂³	1	24	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	4	yes	—
60	#41×C₂₅₁	C₂³	1	12	yes	complemented
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	3	yes	—
69	C₂×D₄×C₂₅₁	C₂³	1	12	yes	complemented
70	C₂²×C₄×C₂₅₁	—	2	3	no	—
71	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
72	C₂²×D₂₅₁	—	2	3	no	—
73	C₂²×D₂₅₁	—	2	12	no	—
74	C₂²×D₂₅₁	—	8	6	no	—
75	Dic₅₀₂	—	2	1	no	—
76	Dic₅₀₂	C₂⁴	1	2	yes	—
77	D₁₀₀₄	—	2	12	no	—
78	D₁₀₀₄	—	2	3	no	—
79	D₁₀₀₄	C₂⁴	1	6	yes	—
80	C₂²⋊D₂₅₁	—	2	12	no	—
81	C₂×Dic₂₅₁	—	2	1	no	—
82	C₄×D₂₅₁	—	2	12	no	—
83	C₂₅₁⋊C₈	C₂⁴	1	2	yes	—
84	C₂₅₁⋊C₈	C₂⁴	1	6	yes	—
85	C₂³×C₂₅₁	—	2	3	no	—
86	C₂³×C₂₅₁	—	2	12	no	—
87	C₂³×C₂₅₁	—	8	6	no	—
88	C₂³×C₂₅₁	C₂×D₄	1	3	yes	—
89	Q₈×C₂₅₁	—	2	1	no	—
90	Q₈×C₂₅₁	C₂×D₄	1	4	yes	—
91	Q₈×C₂₅₁	C₂⁴	1	2	yes	—
92	D₄×C₂₅₁	—	2	12	no	—
93	D₄×C₂₅₁	—	2	3	no	—
94	D₄×C₂₅₁	—	2	12	no	—
95	D₄×C₂₅₁	C₂×D₄	1	12	yes	—
96	D₄×C₂₅₁	C₂⁴	1	6	yes	—
97	C₂×C₄×C₂₅₁	—	2	1	no	—
98	C₂×C₄×C₂₅₁	—	2	12	no	—
99	C₂×C₄×C₂₅₁	C₂×D₄	1	3	yes	—
100	C₂×C₄×C₂₅₁	C₂×D₄	1	6	yes	—
101	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
102	C₂×C₄×C₂₅₁	C₂⁴	1	6	yes	—
103	C₂²×C₂₅₁	—	2	6	no	—
104	C₂²×C₂₅₁	—	2	6	no	—
105	C₂²×C₂₅₁	—	4	6	no	—
106	C₂²×C₂₅₁	—	8	12	no	—
107	C₂²×C₂₅₁	C₂²×D₄	1	6	yes	—
108	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×D₂₅₁	—	2	6	no	—
110	C₂×D₂₅₁	—	4	6	no	—
111	C₂×D₂₅₁	—	8	12	no	—
112	C₄×C₂₅₁	—	2	2	no	—
113	C₄×C₂₅₁	C₂²×D₄	1	6	yes	—
114	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₂₅₁	C₂⁵	1	1	yes	characteristic
116	Dic₂₅₁	—	2	2	no	—
117	C₂×C₂₅₁	—	2	6	no	—
118	C₂×C₂₅₁	—	4	6	no	—
119	C₂×C₂₅₁	#576	1	2	yes	complemented
120	C₂×C₂₅₁	C₂³×D₄	1	1	yes	characteristic
121	D₂₅₁	—	4	6	no	—
122	C₂₅₁	C₂×#576	1	1	yes	characteristic, complemented, Sylow
123	C₂×#576	—	251	1	no	maximal, Sylow
124	C₂×H(5,𝔽₂)	—	251	1	no	—
125	C₂×H(5,𝔽₂)	D₂₅₁	1	1	yes	characteristic, complemented
126	#576	—	251	16	no	—
127	C₂×AGL(1,ℤ/8)	—	251	3	no	—
128	C₂×AGL(1,ℤ/8)	—	251	3	no	—
129	C₂×#135	—	251	3	no	—
130	C₂²×D₈	—	251	3	no	—
131	C₂×#131	—	251	1	no	—
132	H(5,𝔽₂)	—	2	4	no	—
133	H(5,𝔽₂)	—	502	4	no	—
134	H(5,𝔽₂)	—	251	8	no	—
135	H(5,𝔽₂)	C₂×D₂₅₁	1	8	yes	complemented
136	C₂×#41	—	2	1	no	—
137	C₂×#41	—	502	1	no	—
138	C₂×#41	—	251	3	no	—
139	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic
140	C₂×#41	C₂×D₂₅₁	1	3	yes	complemented
141	C₂²×D₄	—	2	3	no	—
142	C₂²×D₄	—	502	3	no	—
143	C₂²×D₄	—	251	3	no	—
144	C₂²×D₄	C₂×D₂₅₁	1	3	yes	complemented
145	AGL(1,ℤ/8)	—	251	24	no	—
146	AGL(1,ℤ/8)	—	251	24	no	—
147	#135	—	251	24	no	—
148	C₂×Q₁₆	—	251	1	no	—
149	C₂×SD₁₆	—	251	3	no	—
150	C₂×SD₁₆	—	251	3	no	—
151	C₂×D₈	—	251	3	no	—
152	C₂×D₈	—	251	6	no	—
153	C₂×D₈	—	251	24	no	—
154	#131	—	251	8	no	—
155	C₂×M₁₆	—	251	3	no	—
156	C₂²×C₈	—	251	3	no	—
157	C₂⁴	—	2	3	no	—
158	C₂⁴	—	502	3	no	—
159	#41	—	2	8	no	—
160	#41	—	2	6	no	—
161	#41	—	502	8	no	—
162	#41	—	502	6	no	—
163	#41	—	251	12	no	—
164	#41	C₂²⋊D₂₅₁	1	4	yes	—
165	#41	C₂²×D₂₅₁	1	4	yes	—
166	#41	C₂²×D₂₅₁	1	12	yes	complemented
167	C₂×Q₈	—	251	1	no	—
168	C₂×Q₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
169	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
170	C₂×D₄	—	2	3	no	—
171	C₂×D₄	—	2	24	no	—
172	C₂×D₄	—	2	3	no	—
173	C₂×D₄	—	2	6	no	—
174	C₂×D₄	—	502	3	no	—
175	C₂×D₄	—	502	24	no	—
176	C₂×D₄	—	502	3	no	—
177	C₂×D₄	—	502	6	no	—
178	C₂×D₄	—	251	3	no	—
179	C₂×D₄	—	251	12	no	—
180	C₂×D₄	C₂²⋊D₂₅₁	1	3	yes	—
181	C₂×D₄	C₂²×D₂₅₁	1	3	yes	—
182	C₂×D₄	C₂²×D₂₅₁	1	12	yes	complemented
183	C₂²×C₄	—	2	3	no	—
184	C₂²×C₄	—	502	3	no	—
185	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
186	Q₁₆	—	251	4	no	—
187	SD₁₆	—	251	12	no	—
188	SD₁₆	—	251	12	no	—
189	D₈	—	251	12	no	—
190	D₈	—	251	24	no	—
191	M₁₆	—	251	12	no	—
192	C₂×C₈	—	251	1	no	—
193	C₂×C₈	—	251	3	no	—
194	C₂×C₈	—	251	12	no	—
195	C₂³	—	2	3	no	—
196	C₂³	—	2	12	no	—
197	C₂³	—	8	6	no	—
198	C₂³	—	2008	6	no	—
199	C₂³	—	502	12	no	—
200	C₂³	—	502	3	no	—
201	C₂³	C₂×C₂²⋊D₂₅₁	1	3	yes	—
202	Q₈	—	2	1	no	—
203	Q₈	—	502	1	no	—
204	Q₈	—	251	2	no	—
205	Q₈	C₂×C₂²⋊D₂₅₁	1	4	yes	—
206	Q₈	C₂³×D₂₅₁	1	2	yes	—
207	D₄	—	2	12	no	—
208	D₄	—	2	3	no	—
209	D₄	—	2	12	no	—
210	D₄	—	502	12	no	—
211	D₄	—	502	12	no	—
212	D₄	—	502	3	no	—
213	D₄	—	251	6	no	—
214	D₄	C₂×C₂²⋊D₂₅₁	1	12	yes	—
215	D₄	C₂³×D₂₅₁	1	6	yes	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	12	no	—
218	C₂×C₄	—	502	1	no	—
219	C₂×C₄	—	502	12	no	—
220	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	6	yes	—
221	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	3	yes	—
222	C₂×C₄	C₂³×D₂₅₁	1	6	yes	—
223	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
224	C₈	—	251	2	no	—
225	C₈	—	251	6	no	—
226	C₂²	—	2	6	no	—
227	C₂²	—	2	6	no	—
228	C₂²	—	4	6	no	—
229	C₂²	—	8	12	no	—
230	C₂²	—	2008	12	no	—
231	C₂²	—	1004	6	no	—
232	C₂²	—	502	6	no	—
233	C₂²	C₂²×C₂²⋊D₂₅₁	1	6	yes	—
234	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	502	2	no	—
237	C₄	C₂²×C₂²⋊D₂₅₁	1	6	yes	—
238	C₄	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
239	C₄	C₂⁴×D₂₅₁	1	1	yes	characteristic
240	C₂	—	2	6	no	—
241	C₂	—	4	6	no	—
242	C₂	—	1004	6	no	—
243	C₂	#30632490	1	2	yes	direct product factor
244	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×#576
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×#22548761	C₂	semidirect
2. #30632490	C₂	direct (N × H)
3. C₂×#22548741	C₂	semidirect
4. C₂×#22548740	C₂	semidirect
5. C₂×#22548735	C₂	semidirect
6. C₂²×#16860795	C₂	semidirect
7. C₂×#22548722	C₂	semidirect
8. C₂×H(5,𝔽₂)×C₂₅₁	C₂	semidirect
10. #22548761	C₂²	semidirect
12. C₂×#16860810	C₂²	semidirect
13. C₂²×D₁₀₀₄	C₂²	semidirect
15. #22548741	C₂²	semidirect
16. #22548740	C₂²	semidirect
17. #22548735	C₂²	semidirect
18. C₂×#16860800	C₂²	semidirect
19. C₂×#16860798	C₂²	semidirect
20. C₂×#16860797	C₂²	semidirect
21. C₂×#16860795	C₂²	semidirect
22. C₂×#16860795	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
31. C₂×#41×C₂₅₁	C₂²	semidirect
33. C₂²×D₄×C₂₅₁	C₂²	semidirect
37. #16860810	C₂³	semidirect
42. C₂×D₁₀₀₄	C₂³	semidirect
46. #16860800	C₂³	semidirect
47. #16860798	C₂³	semidirect
48. #16860797	C₂³	semidirect
49. #16860795	C₂³	semidirect
50. #16860795	C₂³	semidirect
51. #16860794	C₂³	semidirect
54. C₂×C₂₅₁⋊C₈	C₂³	semidirect
60. #41×C₂₅₁	C₂³	semidirect
69. C₂×D₄×C₂₅₁	C₂³	semidirect
119. C₂×C₂₅₁	#576	semidirect
122. C₂₅₁	C₂×#576	semidirect
125. C₂×H(5,𝔽₂)	D₂₅₁	semidirect
135. H(5,𝔽₂)	C₂×D₂₅₁	semidirect
140. C₂×#41	C₂×D₂₅₁	semidirect
144. C₂²×D₄	C₂×D₂₅₁	semidirect
166. #41	C₂²×D₂₅₁	semidirect
182. C₂×D₄	C₂²×D₂₅₁	semidirect
243. C₂	#30632490	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×#22548761	2
#30632490	2
C₂×#22548741	2
C₂×#22548740	2
C₂×#22548735	2
C₂²×#16860795	2
C₂×#22548722	2
C₂×H(5,𝔽₂)×C₂₅₁	2
C₂×#576	251

Subgroups — C2×#30632490 #44565116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30632490 #44565116