Subgroups — C₂³×#88251 #16776805

Order 3328.

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×#88251	C₂	1	112	yes	maximal, direct product factor
2	C₂³×#6127	C₂	1	2	yes	maximal, complemented
3	C₂³×#6129	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₂²⋊D₁₃	C₂	1	4	yes	maximal, complemented
5	C₂³×D₄×D₁₃	C₂	1	4	yes	maximal, complemented
6	C₂⁴×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁶×C₁₃	C₂²	1	2	yes	complemented
8	C₂⁵×D₁₃	C₂²	1	4	yes	complemented
9	C₂³×D₄×C₁₃	C₂²	1	4	yes	complemented
10	C₂³×D₄×C₁₃	C₂²	1	56	yes	complemented
11	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic, complemented
12	C₂³×C₂²⋊D₁₃	C₂²	1	8	yes	complemented
13	C₂³×C₂²⋊D₁₃	C₂²	1	4	yes	complemented
14	C₂³×C₂²⋊D₁₃	C₂²	1	224	yes	complemented
15	C₂⁴×Dic₁₃	C₂²	1	4	yes	complemented
16	C₂×#88251	C₂²	1	1792	yes	direct product factor
17	C₂²×#6129	C₂²	1	224	yes	complemented
18	C₂²×D₄×D₁₃	C₂²	1	224	yes	complemented
19	C₂²×#6127	C₂²	1	112	yes	complemented
20	C₂³×D₅₂	C₂²	1	2	yes	complemented
21	C₂³×C₄×D₁₃	C₂²	1	4	yes	complemented
22	C₂³×Dic₂₆	C₂²	1	2	yes	complemented
23	C₂⁵×C₁₃	—	4	16	no	—
24	C₂⁵×C₁₃	C₂³	1	4	yes	—
25	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
26	C₂⁵×C₁₃	C₂³	1	56	yes	complemented
27	C₂⁴×D₁₃	—	4	32	no	—
28	C₂⁴×D₁₃	C₂³	1	4	yes	—
29	C₂⁴×D₁₃	C₂³	1	112	yes	complemented
30	C₂²×D₄×C₁₃	C₂³	1	112	yes	complemented
31	C₂²×D₄×C₁₃	C₂³	1	448	yes	complemented
32	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	28	yes	complemented
34	C₂²×C₂²⋊D₁₃	C₂³	1	112	yes	complemented
35	C₂²×C₂²⋊D₁₃	C₂³	1	224	yes	complemented
36	C₂²×C₂²⋊D₁₃	C₂³	1	1792	yes	complemented
37	C₂³×Dic₁₃	C₂³	1	4	yes	—
38	C₂³×Dic₁₃	C₂³	1	112	yes	complemented
39	#88251	C₂³	1	4096	yes	direct product factor
40	C₂×#6129	C₂³	1	1792	yes	complemented
41	C₂×D₄×D₁₃	C₂³	1	1792	yes	complemented
42	C₂×#6127	C₂³	1	896	yes	complemented
43	C₂²×D₅₂	C₂³	1	56	yes	complemented
44	C₂²×C₄×D₁₃	C₂³	1	112	yes	complemented
45	C₂²×Dic₂₆	C₂³	1	56	yes	complemented
46	C₂³×H(5,𝔽₂)	—	13	1	no	maximal, Sylow
47	C₂⁴×C₁₃	—	2	32	no	—
48	C₂⁴×C₁₃	—	2	8	no	—
49	C₂⁴×C₁₃	—	4	224	no	—
50	C₂⁴×C₁₃	C₂⁴	1	56	yes	—
51	C₂⁴×C₁₃	C₂⁴	1	14	yes	—
52	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
53	C₂⁴×C₁₃	C₂⁴	1	224	yes	complemented
54	C₂³×D₁₃	—	2	32	no	—
55	C₂³×D₁₃	—	4	448	no	—
56	C₂³×D₁₃	C₂⁴	1	56	yes	—
57	C₂³×D₁₃	C₂⁴	1	448	yes	complemented
58	C₂×D₄×C₁₃	C₂⁴	1	448	yes	complemented
59	C₂×D₄×C₁₃	C₂⁴	1	512	yes	complemented
60	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
61	C₂²×C₄×C₁₃	C₂⁴	1	112	yes	complemented
62	C₂×C₂²⋊D₁₃	C₂⁴	1	448	yes	complemented
63	C₂×C₂²⋊D₁₃	C₂⁴	1	896	yes	complemented
64	C₂×C₂²⋊D₁₃	C₂⁴	1	2048	yes	complemented
65	C₂²×Dic₁₃	C₂⁴	1	56	yes	—
66	C₂²×Dic₁₃	C₂⁴	1	448	yes	complemented
67	#6129	C₂⁴	1	2048	yes	complemented
68	D₄×D₁₃	C₂⁴	1	2048	yes	complemented
69	#6127	C₂⁴	1	1024	yes	complemented
70	C₂×D₅₂	C₂⁴	1	224	yes	complemented
71	C₂×C₄×D₁₃	C₂⁴	1	448	yes	complemented
72	C₂×Dic₂₆	C₂⁴	1	224	yes	complemented
73	C₂²×H(5,𝔽₂)	—	13	112	no	—
74	C₂³×#41	—	13	4	no	—
75	C₂³×#41	—	13	2	no	—
76	C₂⁴×D₄	—	13	4	no	—
77	C₂⁴×D₄	—	13	4	no	—
78	C₂⁴×D₄	D₁₃	1	1	yes	characteristic, complemented
79	C₂³×C₁₃	—	2	224	no	—
80	C₂³×C₁₃	—	2	56	no	—
81	C₂³×C₁₃	—	4	448	no	—
82	C₂³×C₁₃	H(5,𝔽₂)	1	8	yes	complemented
83	C₂³×C₁₃	C₂⁵	1	112	yes	—
84	C₂³×C₁₃	C₂⁵	1	28	yes	—
85	C₂³×C₁₃	C₂⁵	1	7	yes	—
86	C₂³×C₁₃	C₂⁵	1	128	yes	complemented
87	C₂²×D₁₃	—	2	224	no	—
88	C₂²×D₁₃	—	4	896	no	—
89	C₂²×D₁₃	C₂⁵	1	112	yes	—
90	C₂²×D₁₃	C₂⁵	1	256	yes	complemented
91	D₄×C₁₃	C₂⁵	1	256	yes	complemented
92	C₂×C₄×C₁₃	C₂⁵	1	56	yes	—
93	C₂×C₄×C₁₃	C₂⁵	1	64	yes	complemented
94	C₂²⋊D₁₃	C₂⁵	1	256	yes	complemented
95	C₂²⋊D₁₃	C₂⁵	1	512	yes	complemented
96	C₂×Dic₁₃	C₂⁵	1	112	yes	—
97	C₂×Dic₁₃	C₂⁵	1	256	yes	complemented
98	D₅₂	C₂⁵	1	128	yes	complemented
99	C₄×D₁₃	C₂⁵	1	256	yes	complemented
100	Dic₂₆	C₂⁵	1	128	yes	complemented
101	C₂⁶	—	13	4	no	—
102	C₂⁶	C₂×D₁₃	1	2	yes	complemented
103	C₂×H(5,𝔽₂)	—	13	1792	no	—
104	C₂²×#41	—	13	224	no	—
105	C₂²×#41	—	13	112	no	—
106	C₂³×Q₈	—	13	2	no	—
107	C₂³×D₄	—	13	8	no	—
108	C₂³×D₄	—	13	4	no	—
109	C₂³×D₄	—	13	224	no	—
110	C₂³×D₄	—	13	224	no	—
111	C₂³×D₄	—	13	2	no	—
112	C₂³×D₄	C₂×D₁₃	1	4	yes	complemented
113	C₂³×D₄	C₂×D₁₃	1	56	yes	complemented
114	C₂⁴×C₄	—	13	4	no	—
115	C₂⁴×C₄	—	13	4	no	—
116	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic, complemented
117	C₂²×C₁₃	—	2	224	no	—
118	C₂²×C₁₃	—	2	56	no	—
119	C₂²×C₁₃	—	4	128	no	—
120	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	28	yes	complemented
121	C₂²×C₁₃	C₂⁶	1	32	yes	—
122	C₂²×C₁₃	C₂⁶	1	8	yes	—
123	C₂²×C₁₃	C₂⁶	1	7	yes	—
124	C₂×D₁₃	—	2	224	no	—
125	C₂×D₁₃	—	4	256	no	—
126	C₂×D₁₃	C₂⁶	1	32	yes	—
127	C₄×C₁₃	C₂⁶	1	16	yes	—
128	Dic₁₃	C₂⁶	1	32	yes	—
129	C₂⁵	—	4	16	no	—
130	C₂⁵	—	13	4	no	—
131	C₂⁵	—	13	112	no	—
132	C₂⁵	—	52	32	no	—
133	C₂⁵	C₂²×D₁₃	1	4	yes	—
134	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
135	C₂⁵	C₂²×D₁₃	1	56	yes	complemented
136	H(5,𝔽₂)	—	13	4096	no	—
137	C₂×#41	—	13	1792	no	—
138	C₂×#41	—	13	896	no	—
139	C₂²×Q₈	—	13	56	no	—
140	C₂²×D₄	—	13	224	no	—
141	C₂²×D₄	—	13	112	no	—
142	C₂²×D₄	—	13	1792	no	—
143	C₂²×D₄	—	13	1792	no	—
144	C₂²×D₄	—	13	56	no	—
145	C₂²×D₄	C₂²×D₁₃	1	112	yes	complemented
146	C₂²×D₄	C₂²×D₁₃	1	448	yes	complemented
147	C₂³×C₄	—	13	112	no	—
148	C₂³×C₄	—	13	4	no	—
149	C₂³×C₄	—	13	112	no	—
150	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
151	C₂³×C₄	C₂²×D₁₃	1	28	yes	complemented
152	C₂×C₁₃	—	2	32	no	—
153	C₂×C₁₃	—	2	8	no	—
154	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	14	yes	complemented
155	C₂×C₁₃	C₂⁷	1	1	yes	characteristic
156	D₁₃	—	2	32	no	—
157	C₂⁴	—	2	8	no	—
158	C₂⁴	—	2	32	no	—
159	C₂⁴	—	4	224	no	—
160	C₂⁴	—	13	56	no	—
161	C₂⁴	—	13	448	no	—
162	C₂⁴	—	26	32	no	—
163	C₂⁴	—	52	448	no	—
164	C₂⁴	C₂³×D₁₃	1	56	yes	—
165	C₂⁴	C₂³×D₁₃	1	14	yes	—
166	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
167	C₂⁴	C₂³×D₁₃	1	224	yes	complemented
168	#41	—	13	2048	no	—
169	#41	—	13	1024	no	—
170	C₂×Q₈	—	13	224	no	—
171	C₂×D₄	—	13	896	no	—
172	C₂×D₄	—	13	448	no	—
173	C₂×D₄	—	13	2048	no	—
174	C₂×D₄	—	13	2048	no	—
175	C₂×D₄	—	13	224	no	—
176	C₂×D₄	C₂³×D₁₃	1	448	yes	complemented
177	C₂×D₄	C₂³×D₁₃	1	512	yes	complemented
178	C₂²×C₄	—	13	56	no	—
179	C₂²×C₄	—	13	448	no	—
180	C₂²×C₄	—	13	448	no	—
181	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
182	C₂²×C₄	C₂³×D₁₃	1	112	yes	complemented
183	C₁₃	C₂³×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
184	C₂³	—	2	56	no	—
185	C₂³	—	2	224	no	—
186	C₂³	—	4	448	no	—
187	C₂³	—	13	112	no	—
188	C₂³	—	13	256	no	—
189	C₂³	—	26	224	no	—
190	C₂³	—	52	896	no	—
191	C₂³	#88251	1	8	yes	direct product factor
192	C₂³	C₂⁴×D₁₃	1	112	yes	—
193	C₂³	C₂⁴×D₁₃	1	28	yes	—
194	C₂³	C₂⁴×D₁₃	1	7	yes	—
195	C₂³	C₂⁴×D₁₃	1	128	yes	complemented
196	Q₈	—	13	128	no	—
197	D₄	—	13	512	no	—
198	D₄	—	13	256	no	—
199	D₄	—	13	128	no	—
200	D₄	C₂⁴×D₁₃	1	256	yes	complemented
201	C₂×C₄	—	13	112	no	—
202	C₂×C₄	—	13	256	no	—
203	C₂×C₄	—	13	256	no	—
204	C₂×C₄	C₂⁴×D₁₃	1	56	yes	—
205	C₂×C₄	C₂⁴×D₁₃	1	64	yes	complemented
206	C₂²	—	2	56	no	—
207	C₂²	—	2	224	no	—
208	C₂²	—	4	128	no	—
209	C₂²	—	13	32	no	—
210	C₂²	—	26	224	no	—
211	C₂²	—	52	256	no	—
212	C₂²	C₂×#88251	1	28	yes	direct product factor
213	C₂²	C₂⁵×D₁₃	1	32	yes	—
214	C₂²	C₂⁵×D₁₃	1	8	yes	—
215	C₂²	C₂⁵×D₁₃	1	7	yes	—
216	C₄	—	13	32	no	—
217	C₄	C₂⁵×D₁₃	1	16	yes	—
218	C₂	—	2	8	no	—
219	C₂	—	2	32	no	—
220	C₂	—	26	32	no	—
221	C₂	C₂²×#88251	1	14	yes	direct product factor
222	C₂	C₂⁶×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂×C₁₃	C₂⁷
Frattini	C₂	C₂⁶×D₁₃
Fitting	C₂⁴×D₄×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂⁴×D₄	D₁₃
hypocenter	C₁₃	C₂³×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#88251	C₂	direct (N × H)
2. C₂³×#6127	C₂	semidirect
3. C₂³×#6129	C₂	semidirect
4. C₂⁴×C₂²⋊D₁₃	C₂	semidirect
5. C₂³×D₄×D₁₃	C₂	semidirect
6. C₂⁴×D₄×C₁₃	C₂	semidirect
7. C₂⁶×C₁₃	C₂²	semidirect
8. C₂⁵×D₁₃	C₂²	semidirect
9. C₂³×D₄×C₁₃	C₂²	semidirect
10. C₂³×D₄×C₁₃	C₂²	semidirect
11. C₂⁴×C₄×C₁₃	C₂²	semidirect
12. C₂³×C₂²⋊D₁₃	C₂²	semidirect
13. C₂³×C₂²⋊D₁₃	C₂²	semidirect
14. C₂³×C₂²⋊D₁₃	C₂²	semidirect
15. C₂⁴×Dic₁₃	C₂²	semidirect
16. C₂×#88251	C₂²	direct (N × H)
17. C₂²×#6129	C₂²	semidirect
18. C₂²×D₄×D₁₃	C₂²	semidirect
19. C₂²×#6127	C₂²	semidirect
20. C₂³×D₅₂	C₂²	semidirect
21. C₂³×C₄×D₁₃	C₂²	semidirect
22. C₂³×Dic₂₆	C₂²	semidirect
26. C₂⁵×C₁₃	C₂³	semidirect
29. C₂⁴×D₁₃	C₂³	semidirect
30. C₂²×D₄×C₁₃	C₂³	semidirect
31. C₂²×D₄×C₁₃	C₂³	semidirect
33. C₂³×C₄×C₁₃	C₂³	semidirect
34. C₂²×C₂²⋊D₁₃	C₂³	semidirect
35. C₂²×C₂²⋊D₁₃	C₂³	semidirect
36. C₂²×C₂²⋊D₁₃	C₂³	semidirect
38. C₂³×Dic₁₃	C₂³	semidirect
39. #88251	C₂³	direct (N × H)
40. C₂×#6129	C₂³	semidirect
41. C₂×D₄×D₁₃	C₂³	semidirect
42. C₂×#6127	C₂³	semidirect
43. C₂²×D₅₂	C₂³	semidirect
44. C₂²×C₄×D₁₃	C₂³	semidirect
45. C₂²×Dic₂₆	C₂³	semidirect
53. C₂⁴×C₁₃	C₂⁴	semidirect
57. C₂³×D₁₃	C₂⁴	semidirect
58. C₂×D₄×C₁₃	C₂⁴	semidirect
59. C₂×D₄×C₁₃	C₂⁴	semidirect
61. C₂²×C₄×C₁₃	C₂⁴	semidirect
62. C₂×C₂²⋊D₁₃	C₂⁴	semidirect
63. C₂×C₂²⋊D₁₃	C₂⁴	semidirect
64. C₂×C₂²⋊D₁₃	C₂⁴	semidirect
66. C₂²×Dic₁₃	C₂⁴	semidirect
67. #6129	C₂⁴	semidirect
68. D₄×D₁₃	C₂⁴	semidirect
69. #6127	C₂⁴	semidirect
70. C₂×D₅₂	C₂⁴	semidirect
71. C₂×C₄×D₁₃	C₂⁴	semidirect
72. C₂×Dic₂₆	C₂⁴	semidirect
78. C₂⁴×D₄	D₁₃	semidirect
82. C₂³×C₁₃	H(5,𝔽₂)	semidirect
86. C₂³×C₁₃	C₂⁵	semidirect
90. C₂²×D₁₃	C₂⁵	semidirect
91. D₄×C₁₃	C₂⁵	semidirect
93. C₂×C₄×C₁₃	C₂⁵	semidirect
94. C₂²⋊D₁₃	C₂⁵	semidirect
95. C₂²⋊D₁₃	C₂⁵	semidirect
97. C₂×Dic₁₃	C₂⁵	semidirect
98. D₅₂	C₂⁵	semidirect
99. C₄×D₁₃	C₂⁵	semidirect
100. Dic₂₆	C₂⁵	semidirect
102. C₂⁶	C₂×D₁₃	semidirect
112. C₂³×D₄	C₂×D₁₃	semidirect
113. C₂³×D₄	C₂×D₁₃	semidirect
116. C₂⁴×C₄	C₂×D₁₃	semidirect
120. C₂²×C₁₃	C₂×H(5,𝔽₂)	semidirect
135. C₂⁵	C₂²×D₁₃	semidirect
145. C₂²×D₄	C₂²×D₁₃	semidirect
146. C₂²×D₄	C₂²×D₁₃	semidirect
151. C₂³×C₄	C₂²×D₁₃	semidirect
154. C₂×C₁₃	C₂²×H(5,𝔽₂)	semidirect
167. C₂⁴	C₂³×D₁₃	semidirect
176. C₂×D₄	C₂³×D₁₃	semidirect
177. C₂×D₄	C₂³×D₁₃	semidirect
182. C₂²×C₄	C₂³×D₁₃	semidirect
183. C₁₃	C₂³×H(5,𝔽₂)	semidirect
191. C₂³	#88251	direct (N × H)
195. C₂³	C₂⁴×D₁₃	semidirect
200. D₄	C₂⁴×D₁₃	semidirect
205. C₂×C₄	C₂⁴×D₁₃	semidirect
212. C₂²	C₂×#88251	direct (N × H)
221. C₂	C₂²×#88251	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×#88251	2
C₂³×#6127	2
C₂³×#6129	2
C₂⁴×C₂²⋊D₁₃	2
C₂³×D₄×D₁₃	2
C₂⁴×D₄×C₁₃	2
C₂³×H(5,𝔽₂)	13

Subgroups — C23×#88251 #16776805

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88251 #16776805