Subgroups — C₂³×#14519822 #30593899

Order 15616.

65726 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519822	C₂	1	112	yes	maximal, direct product factor
2	C₂³×#11758222	C₂	1	2	yes	maximal, complemented
3	C₂³×#11758224	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₂²⋊D₆₁	C₂	1	4	yes	maximal, complemented
5	C₂³×D₄×D₆₁	C₂	1	4	yes	maximal, complemented
6	C₂⁴×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₆₁	C₂²	1	4	yes	complemented
8	C₂×#14519822	C₂²	1	1792	yes	direct product factor
9	C₂²×#11758222	C₂²	1	112	yes	complemented
10	C₂²×#11758224	C₂²	1	224	yes	complemented
11	C₂³×Dic₁₂₂	C₂²	1	2	yes	complemented
12	C₂³×D₂₄₄	C₂²	1	2	yes	complemented
13	C₂³×C₂²⋊D₆₁	C₂²	1	8	yes	complemented
14	C₂³×C₂²⋊D₆₁	C₂²	1	4	yes	complemented
15	C₂³×C₂²⋊D₆₁	C₂²	1	224	yes	complemented
16	C₂²×D₄×D₆₁	C₂²	1	224	yes	complemented
17	C₂⁴×Dic₆₁	C₂²	1	4	yes	complemented
18	C₂³×C₄×D₆₁	C₂²	1	4	yes	complemented
19	C₂⁶×C₆₁	C₂²	1	2	yes	complemented
20	C₂³×D₄×C₆₁	C₂²	1	4	yes	complemented
21	C₂³×D₄×C₆₁	C₂²	1	56	yes	complemented
22	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic, complemented
23	C₂⁴×D₆₁	—	4	32	no	—
24	C₂⁴×D₆₁	C₂³	1	4	yes	—
25	C₂⁴×D₆₁	C₂³	1	112	yes	complemented
26	#14519822	C₂³	1	4096	yes	direct product factor
27	C₂×#11758222	C₂³	1	896	yes	complemented
28	C₂×#11758224	C₂³	1	1792	yes	complemented
29	C₂²×Dic₁₂₂	C₂³	1	56	yes	complemented
30	C₂²×D₂₄₄	C₂³	1	56	yes	complemented
31	C₂²×C₂²⋊D₆₁	C₂³	1	224	yes	complemented
32	C₂²×C₂²⋊D₆₁	C₂³	1	112	yes	complemented
33	C₂²×C₂²⋊D₆₁	C₂³	1	1792	yes	complemented
34	C₂×D₄×D₆₁	C₂³	1	1792	yes	complemented
35	C₂³×Dic₆₁	C₂³	1	4	yes	—
36	C₂³×Dic₆₁	C₂³	1	112	yes	complemented
37	C₂²×C₄×D₆₁	C₂³	1	112	yes	complemented
38	C₂⁵×C₆₁	—	4	16	no	—
39	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
40	C₂⁵×C₆₁	C₂³	1	4	yes	—
41	C₂⁵×C₆₁	C₂³	1	56	yes	complemented
42	C₂²×D₄×C₆₁	C₂³	1	112	yes	complemented
43	C₂²×D₄×C₆₁	C₂³	1	448	yes	complemented
44	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
45	C₂³×C₄×C₆₁	C₂³	1	28	yes	complemented
46	C₂⁴×C₆₁	—	2	32	no	—
47	C₂⁴×C₆₁	—	2	8	no	—
48	C₂⁴×C₆₁	—	4	224	no	—
49	C₂⁴×C₆₁	C₂⁴	1	56	yes	—
50	C₂⁴×C₆₁	C₂⁴	1	14	yes	—
51	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
52	C₂⁴×C₆₁	C₂⁴	1	224	yes	complemented
53	C₂³×D₆₁	—	2	32	no	—
54	C₂³×D₆₁	—	4	448	no	—
55	C₂³×D₆₁	C₂⁴	1	56	yes	—
56	C₂³×D₆₁	C₂⁴	1	448	yes	complemented
57	C₂×D₄×C₆₁	C₂⁴	1	448	yes	complemented
58	C₂×D₄×C₆₁	C₂⁴	1	512	yes	complemented
59	C₂²×C₄×C₆₁	C₂⁴	1	28	yes	—
60	C₂²×C₄×C₆₁	C₂⁴	1	112	yes	complemented
61	C₂×C₂²⋊D₆₁	C₂⁴	1	896	yes	complemented
62	C₂×C₂²⋊D₆₁	C₂⁴	1	448	yes	complemented
63	C₂×C₂²⋊D₆₁	C₂⁴	1	2048	yes	complemented
64	C₂²×Dic₆₁	C₂⁴	1	56	yes	—
65	C₂²×Dic₆₁	C₂⁴	1	448	yes	complemented
66	#11758224	C₂⁴	1	2048	yes	complemented
67	D₄×D₆₁	C₂⁴	1	2048	yes	complemented
68	#11758222	C₂⁴	1	1024	yes	complemented
69	C₂×D₂₄₄	C₂⁴	1	224	yes	complemented
70	C₂×C₄×D₆₁	C₂⁴	1	448	yes	complemented
71	C₂×Dic₁₂₂	C₂⁴	1	224	yes	complemented
72	C₂³×C₆₁	—	2	224	no	—
73	C₂³×C₆₁	—	2	56	no	—
74	C₂³×C₆₁	—	4	448	no	—
75	C₂³×C₆₁	H(5,𝔽₂)	1	8	yes	complemented
76	C₂³×C₆₁	C₂⁵	1	112	yes	—
77	C₂³×C₆₁	C₂⁵	1	28	yes	—
78	C₂³×C₆₁	C₂⁵	1	7	yes	—
79	C₂³×C₆₁	C₂⁵	1	128	yes	complemented
80	C₂²×D₆₁	—	2	224	no	—
81	C₂²×D₆₁	—	4	896	no	—
82	C₂²×D₆₁	C₂⁵	1	112	yes	—
83	C₂²×D₆₁	C₂⁵	1	256	yes	complemented
84	D₄×C₆₁	C₂⁵	1	256	yes	complemented
85	C₂×C₄×C₆₁	C₂⁵	1	56	yes	—
86	C₂×C₄×C₆₁	C₂⁵	1	64	yes	complemented
87	C₂²⋊D₆₁	C₂⁵	1	256	yes	complemented
88	C₂²⋊D₆₁	C₂⁵	1	512	yes	complemented
89	C₂×Dic₆₁	C₂⁵	1	112	yes	—
90	C₂×Dic₆₁	C₂⁵	1	256	yes	complemented
91	D₂₄₄	C₂⁵	1	128	yes	complemented
92	C₄×D₆₁	C₂⁵	1	256	yes	complemented
93	Dic₁₂₂	C₂⁵	1	128	yes	complemented
94	C₂³×H(5,𝔽₂)	—	61	1	no	maximal, Sylow
95	C₂²×C₆₁	—	2	224	no	—
96	C₂²×C₆₁	—	2	56	no	—
97	C₂²×C₆₁	—	4	128	no	—
98	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	28	yes	complemented
99	C₂²×C₆₁	C₂⁶	1	32	yes	—
100	C₂²×C₆₁	C₂⁶	1	8	yes	—
101	C₂²×C₆₁	C₂⁶	1	7	yes	—
102	C₂×D₆₁	—	2	224	no	—
103	C₂×D₆₁	—	4	256	no	—
104	C₂×D₆₁	C₂⁶	1	32	yes	—
105	C₄×C₆₁	C₂⁶	1	16	yes	—
106	Dic₆₁	C₂⁶	1	32	yes	—
107	C₂²×H(5,𝔽₂)	—	61	112	no	—
108	C₂³×#41	—	61	4	no	—
109	C₂³×#41	—	61	2	no	—
110	C₂⁴×D₄	—	61	4	no	—
111	C₂⁴×D₄	—	61	4	no	—
112	C₂⁴×D₄	D₆₁	1	1	yes	characteristic, complemented
113	C₂×C₆₁	—	2	32	no	—
114	C₂×C₆₁	—	2	8	no	—
115	C₂×C₆₁	C₂²×H(5,𝔽₂)	1	14	yes	complemented
116	C₂×C₆₁	C₂⁷	1	1	yes	characteristic
117	D₆₁	—	2	32	no	—
118	C₂⁶	—	61	4	no	—
119	C₂⁶	C₂×D₆₁	1	2	yes	complemented
120	C₂×H(5,𝔽₂)	—	61	1792	no	—
121	C₂²×#41	—	61	224	no	—
122	C₂²×#41	—	61	112	no	—
123	C₂³×Q₈	—	61	2	no	—
124	C₂³×D₄	—	61	8	no	—
125	C₂³×D₄	—	61	4	no	—
126	C₂³×D₄	—	61	224	no	—
127	C₂³×D₄	—	61	224	no	—
128	C₂³×D₄	—	61	2	no	—
129	C₂³×D₄	C₂×D₆₁	1	4	yes	complemented
130	C₂³×D₄	C₂×D₆₁	1	56	yes	complemented
131	C₂⁴×C₄	—	61	4	no	—
132	C₂⁴×C₄	—	61	4	no	—
133	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic, complemented
134	C₆₁	C₂³×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
135	C₂⁵	—	4	16	no	—
136	C₂⁵	—	61	4	no	—
137	C₂⁵	—	61	112	no	—
138	C₂⁵	—	244	32	no	—
139	C₂⁵	C₂²×D₆₁	1	4	yes	—
140	C₂⁵	C₂²×D₆₁	1	1	yes	characteristic
141	C₂⁵	C₂²×D₆₁	1	56	yes	complemented
142	H(5,𝔽₂)	—	61	4096	no	—
143	C₂×#41	—	61	1792	no	—
144	C₂×#41	—	61	896	no	—
145	C₂²×Q₈	—	61	56	no	—
146	C₂²×D₄	—	61	224	no	—
147	C₂²×D₄	—	61	112	no	—
148	C₂²×D₄	—	61	1792	no	—
149	C₂²×D₄	—	61	1792	no	—
150	C₂²×D₄	—	61	56	no	—
151	C₂²×D₄	C₂²×D₆₁	1	112	yes	complemented
152	C₂²×D₄	C₂²×D₆₁	1	448	yes	complemented
153	C₂³×C₄	—	61	112	no	—
154	C₂³×C₄	—	61	4	no	—
155	C₂³×C₄	—	61	112	no	—
156	C₂³×C₄	C₂²×D₆₁	1	2	yes	—
157	C₂³×C₄	C₂²×D₆₁	1	28	yes	complemented
158	C₂⁴	—	2	8	no	—
159	C₂⁴	—	2	32	no	—
160	C₂⁴	—	4	224	no	—
161	C₂⁴	—	61	56	no	—
162	C₂⁴	—	61	448	no	—
163	C₂⁴	—	122	32	no	—
164	C₂⁴	—	244	448	no	—
165	C₂⁴	C₂³×D₆₁	1	56	yes	—
166	C₂⁴	C₂³×D₆₁	1	14	yes	—
167	C₂⁴	C₂³×D₆₁	1	1	yes	characteristic
168	C₂⁴	C₂³×D₆₁	1	224	yes	complemented
169	#41	—	61	2048	no	—
170	#41	—	61	1024	no	—
171	C₂×Q₈	—	61	224	no	—
172	C₂×D₄	—	61	896	no	—
173	C₂×D₄	—	61	448	no	—
174	C₂×D₄	—	61	2048	no	—
175	C₂×D₄	—	61	2048	no	—
176	C₂×D₄	—	61	224	no	—
177	C₂×D₄	C₂³×D₆₁	1	448	yes	complemented
178	C₂×D₄	C₂³×D₆₁	1	512	yes	complemented
179	C₂²×C₄	—	61	56	no	—
180	C₂²×C₄	—	61	448	no	—
181	C₂²×C₄	—	61	448	no	—
182	C₂²×C₄	C₂³×D₆₁	1	28	yes	—
183	C₂²×C₄	C₂³×D₆₁	1	112	yes	complemented
184	C₂³	—	2	56	no	—
185	C₂³	—	2	224	no	—
186	C₂³	—	4	448	no	—
187	C₂³	—	61	112	no	—
188	C₂³	—	61	256	no	—
189	C₂³	—	122	224	no	—
190	C₂³	—	244	896	no	—
191	C₂³	#14519822	1	8	yes	direct product factor
192	C₂³	C₂⁴×D₆₁	1	112	yes	—
193	C₂³	C₂⁴×D₆₁	1	28	yes	—
194	C₂³	C₂⁴×D₆₁	1	7	yes	—
195	C₂³	C₂⁴×D₆₁	1	128	yes	complemented
196	Q₈	—	61	128	no	—
197	D₄	—	61	512	no	—
198	D₄	—	61	256	no	—
199	D₄	—	61	128	no	—
200	D₄	C₂⁴×D₆₁	1	256	yes	complemented
201	C₂×C₄	—	61	112	no	—
202	C₂×C₄	—	61	256	no	—
203	C₂×C₄	—	61	256	no	—
204	C₂×C₄	C₂⁴×D₆₁	1	56	yes	—
205	C₂×C₄	C₂⁴×D₆₁	1	64	yes	complemented
206	C₂²	—	2	56	no	—
207	C₂²	—	2	224	no	—
208	C₂²	—	4	128	no	—
209	C₂²	—	61	32	no	—
210	C₂²	—	122	224	no	—
211	C₂²	—	244	256	no	—
212	C₂²	C₂×#14519822	1	28	yes	direct product factor
213	C₂²	C₂⁵×D₆₁	1	32	yes	—
214	C₂²	C₂⁵×D₆₁	1	8	yes	—
215	C₂²	C₂⁵×D₆₁	1	7	yes	—
216	C₄	—	61	32	no	—
217	C₄	C₂⁵×D₆₁	1	16	yes	—
218	C₂	—	2	8	no	—
219	C₂	—	2	32	no	—
220	C₂	—	122	32	no	—
221	C₂	C₂²×#14519822	1	14	yes	direct product factor
222	C₂	C₂⁶×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₆₁
commutator, cosocle	C₂×C₆₁	C₂⁷
Frattini	C₂	C₂⁶×D₆₁
Fitting	C₂⁴×D₄×C₆₁	C₂
socle	C₂⁴×C₆₁	C₂⁴
hypercenter	C₂⁴×D₄	D₆₁
hypocenter	C₆₁	C₂³×H(5,𝔽₂)
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14519822	C₂	direct (N × H)
2. C₂³×#11758222	C₂	semidirect
3. C₂³×#11758224	C₂	semidirect
4. C₂⁴×C₂²⋊D₆₁	C₂	semidirect
5. C₂³×D₄×D₆₁	C₂	semidirect
6. C₂⁴×D₄×C₆₁	C₂	semidirect
7. C₂⁵×D₆₁	C₂²	semidirect
8. C₂×#14519822	C₂²	direct (N × H)
9. C₂²×#11758222	C₂²	semidirect
10. C₂²×#11758224	C₂²	semidirect
11. C₂³×Dic₁₂₂	C₂²	semidirect
12. C₂³×D₂₄₄	C₂²	semidirect
13. C₂³×C₂²⋊D₆₁	C₂²	semidirect
14. C₂³×C₂²⋊D₆₁	C₂²	semidirect
15. C₂³×C₂²⋊D₆₁	C₂²	semidirect
16. C₂²×D₄×D₆₁	C₂²	semidirect
17. C₂⁴×Dic₆₁	C₂²	semidirect
18. C₂³×C₄×D₆₁	C₂²	semidirect
19. C₂⁶×C₆₁	C₂²	semidirect
20. C₂³×D₄×C₆₁	C₂²	semidirect
21. C₂³×D₄×C₆₁	C₂²	semidirect
22. C₂⁴×C₄×C₆₁	C₂²	semidirect
25. C₂⁴×D₆₁	C₂³	semidirect
26. #14519822	C₂³	direct (N × H)
27. C₂×#11758222	C₂³	semidirect
28. C₂×#11758224	C₂³	semidirect
29. C₂²×Dic₁₂₂	C₂³	semidirect
30. C₂²×D₂₄₄	C₂³	semidirect
31. C₂²×C₂²⋊D₆₁	C₂³	semidirect
32. C₂²×C₂²⋊D₆₁	C₂³	semidirect
33. C₂²×C₂²⋊D₆₁	C₂³	semidirect
34. C₂×D₄×D₆₁	C₂³	semidirect
36. C₂³×Dic₆₁	C₂³	semidirect
37. C₂²×C₄×D₆₁	C₂³	semidirect
41. C₂⁵×C₆₁	C₂³	semidirect
42. C₂²×D₄×C₆₁	C₂³	semidirect
43. C₂²×D₄×C₆₁	C₂³	semidirect
45. C₂³×C₄×C₆₁	C₂³	semidirect
52. C₂⁴×C₆₁	C₂⁴	semidirect
56. C₂³×D₆₁	C₂⁴	semidirect
57. C₂×D₄×C₆₁	C₂⁴	semidirect
58. C₂×D₄×C₆₁	C₂⁴	semidirect
60. C₂²×C₄×C₆₁	C₂⁴	semidirect
61. C₂×C₂²⋊D₆₁	C₂⁴	semidirect
62. C₂×C₂²⋊D₆₁	C₂⁴	semidirect
63. C₂×C₂²⋊D₆₁	C₂⁴	semidirect
65. C₂²×Dic₆₁	C₂⁴	semidirect
66. #11758224	C₂⁴	semidirect
67. D₄×D₆₁	C₂⁴	semidirect
68. #11758222	C₂⁴	semidirect
69. C₂×D₂₄₄	C₂⁴	semidirect
70. C₂×C₄×D₆₁	C₂⁴	semidirect
71. C₂×Dic₁₂₂	C₂⁴	semidirect
75. C₂³×C₆₁	H(5,𝔽₂)	semidirect
79. C₂³×C₆₁	C₂⁵	semidirect
83. C₂²×D₆₁	C₂⁵	semidirect
84. D₄×C₆₁	C₂⁵	semidirect
86. C₂×C₄×C₆₁	C₂⁵	semidirect
87. C₂²⋊D₆₁	C₂⁵	semidirect
88. C₂²⋊D₆₁	C₂⁵	semidirect
90. C₂×Dic₆₁	C₂⁵	semidirect
91. D₂₄₄	C₂⁵	semidirect
92. C₄×D₆₁	C₂⁵	semidirect
93. Dic₁₂₂	C₂⁵	semidirect
98. C₂²×C₆₁	C₂×H(5,𝔽₂)	semidirect
112. C₂⁴×D₄	D₆₁	semidirect
115. C₂×C₆₁	C₂²×H(5,𝔽₂)	semidirect
119. C₂⁶	C₂×D₆₁	semidirect
129. C₂³×D₄	C₂×D₆₁	semidirect
130. C₂³×D₄	C₂×D₆₁	semidirect
133. C₂⁴×C₄	C₂×D₆₁	semidirect
134. C₆₁	C₂³×H(5,𝔽₂)	semidirect
141. C₂⁵	C₂²×D₆₁	semidirect
151. C₂²×D₄	C₂²×D₆₁	semidirect
152. C₂²×D₄	C₂²×D₆₁	semidirect
157. C₂³×C₄	C₂²×D₆₁	semidirect
168. C₂⁴	C₂³×D₆₁	semidirect
177. C₂×D₄	C₂³×D₆₁	semidirect
178. C₂×D₄	C₂³×D₆₁	semidirect
183. C₂²×C₄	C₂³×D₆₁	semidirect
191. C₂³	#14519822	direct (N × H)
195. C₂³	C₂⁴×D₆₁	semidirect
200. D₄	C₂⁴×D₆₁	semidirect
205. C₂×C₄	C₂⁴×D₆₁	semidirect
212. C₂²	C₂×#14519822	direct (N × H)
221. C₂	C₂²×#14519822	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×#14519822	2
C₂³×#11758222	2
C₂³×#11758224	2
C₂⁴×C₂²⋊D₆₁	2
C₂³×D₄×D₆₁	2
C₂⁴×D₄×C₆₁	2
C₂³×H(5,𝔽₂)	61

Subgroups — C23×#14519822 #30593899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14519822 #30593899