Subgroups — C₂³×#88256 #16776806

Order 3328.

65726 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88256	C₂	1	112	yes	maximal, direct product factor
2	C₂³×#6131	C₂	1	2	yes	maximal, complemented
3	C₂³×#6127	C₂	1	3	yes	maximal, complemented
4	C₂⁴×D₅₂	C₂	1	3	yes	maximal, complemented
5	C₂³×D₄×D₁₃	C₂	1	6	yes	maximal, complemented
6	C₂³×C₁₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₃	C₂²	1	6	yes	complemented
8	C₂²×C₁₃×#41	C₂²	1	56	yes	complemented
9	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
10	C₂³×D₄×C₁₃	C₂²	1	3	yes	complemented
11	C₂⁴×C₄×C₁₃	C₂²	1	3	yes	complemented
12	C₂³×C₂²⋊D₁₃	C₂²	1	6	yes	complemented
13	C₂×#88256	C₂²	1	1792	yes	direct product factor
14	C₂²×#6131	C₂²	1	112	yes	complemented
15	C₂²×D₄×D₁₃	C₂²	1	336	yes	complemented
16	C₂²×#6127	C₂²	1	168	yes	complemented
17	C₂³×D₅₂	C₂²	1	6	yes	complemented
18	C₂³×D₅₂	C₂²	1	3	yes	complemented
19	C₂³×D₅₂	C₂²	1	168	yes	complemented
20	C₂³×C₄×D₁₃	C₂²	1	6	yes	complemented
21	C₂³×Dic₂₆	C₂²	1	1	yes	characteristic, complemented
22	C₂⁵×C₁₃	C₂³	1	3	yes	—
23	C₂⁴×D₁₃	—	4	48	no	—
24	C₂⁴×D₁₃	C₂³	1	6	yes	—
25	C₂⁴×D₁₃	C₂³	1	168	yes	complemented
26	C₂×C₁₃×#41	C₂³	1	448	yes	complemented
27	C₂²×Q₈×C₁₃	C₂³	1	28	yes	complemented
28	C₂²×D₄×C₁₃	C₂³	1	84	yes	complemented
29	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	84	yes	complemented
32	C₂²×C₂²⋊D₁₃	C₂³	1	168	yes	complemented
33	C₂³×Dic₁₃	C₂³	1	2	yes	—
34	#88256	C₂³	1	4096	yes	direct product factor
35	C₂×#6131	C₂³	1	896	yes	complemented
36	C₂×D₄×D₁₃	C₂³	1	2688	yes	complemented
37	C₂×#6127	C₂³	1	1344	yes	complemented
38	C₂²×D₅₂	C₂³	1	84	yes	complemented
39	C₂²×D₅₂	C₂³	1	168	yes	complemented
40	C₂²×D₅₂	C₂³	1	1344	yes	complemented
41	C₂²×C₄×D₁₃	C₂³	1	168	yes	complemented
42	C₂²×Dic₂₆	C₂³	1	28	yes	complemented
43	C₂³×H(5,𝔽₂)	—	13	1	no	maximal, Sylow
44	C₂⁴×C₁₃	—	2	24	no	—
45	C₂⁴×C₁₃	C₂⁴	1	42	yes	—
46	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
47	C₂³×D₁₃	—	2	48	no	—
48	C₂³×D₁₃	—	4	672	no	—
49	C₂³×D₁₃	C₂⁴	1	84	yes	—
50	C₂³×D₁₃	C₂⁴	1	672	yes	complemented
51	C₁₃×#41	C₂⁴	1	512	yes	complemented
52	C₂×Q₈×C₁₃	C₂⁴	1	112	yes	complemented
53	C₂×D₄×C₁₃	C₂⁴	1	336	yes	complemented
54	C₂²×C₄×C₁₃	C₂⁴	1	14	yes	—
55	C₂²×C₄×C₁₃	C₂⁴	1	42	yes	—
56	C₂²×C₄×C₁₃	C₂⁴	1	336	yes	complemented
57	C₂×C₂²⋊D₁₃	C₂⁴	1	672	yes	complemented
58	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
59	#6131	C₂⁴	1	1024	yes	complemented
60	D₄×D₁₃	C₂⁴	1	3072	yes	complemented
61	#6127	C₂⁴	1	1536	yes	complemented
62	C₂×D₅₂	C₂⁴	1	336	yes	complemented
63	C₂×D₅₂	C₂⁴	1	672	yes	complemented
64	C₂×D₅₂	C₂⁴	1	1536	yes	complemented
65	C₂×C₄×D₁₃	C₂⁴	1	672	yes	complemented
66	C₂×Dic₂₆	C₂⁴	1	112	yes	complemented
67	C₂²×H(5,𝔽₂)	—	13	112	no	—
68	C₂³×#41	—	13	2	no	—
69	C₂³×#41	—	13	3	no	—
70	C₂³×#41	D₁₃	1	1	yes	characteristic, complemented
71	C₂⁴×D₄	—	13	6	no	—
72	C₂⁴×D₄	—	13	3	no	—
73	C₂³×C₁₃	—	2	168	no	—
74	C₂³×C₁₃	H(5,𝔽₂)	1	8	yes	complemented
75	C₂³×C₁₃	C₂⁵	1	84	yes	—
76	C₂³×C₁₃	C₂⁵	1	7	yes	—
77	C₂²×D₁₃	—	2	336	no	—
78	C₂²×D₁₃	—	4	1344	no	—
79	C₂²×D₁₃	C₂⁵	1	168	yes	—
80	C₂²×D₁₃	C₂⁵	1	384	yes	complemented
81	Q₈×C₁₃	C₂⁵	1	64	yes	complemented
82	D₄×C₁₃	C₂⁵	1	192	yes	complemented
83	C₂×C₄×C₁₃	C₂⁵	1	28	yes	—
84	C₂×C₄×C₁₃	C₂⁵	1	84	yes	—
85	C₂×C₄×C₁₃	C₂⁵	1	192	yes	complemented
86	C₂²⋊D₁₃	C₂⁵	1	384	yes	complemented
87	C₂×Dic₁₃	C₂⁵	1	56	yes	—
88	D₅₂	C₂⁵	1	192	yes	complemented
89	D₅₂	C₂⁵	1	384	yes	complemented
90	C₄×D₁₃	C₂⁵	1	384	yes	complemented
91	Dic₂₆	C₂⁵	1	64	yes	complemented
92	C₂⁶	—	13	6	no	—
93	C₂×H(5,𝔽₂)	—	13	1792	no	—
94	C₂²×#41	—	13	112	no	—
95	C₂²×#41	—	13	168	no	—
96	C₂²×#41	C₂×D₁₃	1	56	yes	complemented
97	C₂³×Q₈	—	13	1	no	—
98	C₂³×Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
99	C₂³×D₄	—	13	6	no	—
100	C₂³×D₄	—	13	336	no	—
101	C₂³×D₄	—	13	6	no	—
102	C₂³×D₄	—	13	3	no	—
103	C₂³×D₄	—	13	168	no	—
104	C₂³×D₄	C₂×D₁₃	1	3	yes	complemented
105	C₂⁴×C₄	—	13	6	no	—
106	C₂⁴×C₄	C₂×D₁₃	1	3	yes	complemented
107	C₂²×C₁₃	—	2	168	no	—
108	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	28	yes	complemented
109	C₂²×C₁₃	C₂⁶	1	24	yes	—
110	C₂²×C₁₃	C₂⁶	1	7	yes	—
111	C₂×D₁₃	—	2	336	no	—
112	C₂×D₁₃	—	4	384	no	—
113	C₂×D₁₃	C₂⁶	1	48	yes	—
114	C₄×C₁₃	C₂⁶	1	8	yes	—
115	C₄×C₁₃	C₂⁶	1	24	yes	—
116	Dic₁₃	C₂⁶	1	16	yes	—
117	C₂⁵	—	13	6	no	—
118	C₂⁵	—	13	168	no	—
119	C₂⁵	—	52	48	no	—
120	C₂⁵	C₂²×D₁₃	1	3	yes	—
121	H(5,𝔽₂)	—	13	4096	no	—
122	C₂×#41	—	13	896	no	—
123	C₂×#41	—	13	1344	no	—
124	C₂×#41	C₂²×D₁₃	1	448	yes	complemented
125	C₂²×Q₈	—	13	28	no	—
126	C₂²×Q₈	C₂²×D₁₃	1	28	yes	complemented
127	C₂²×D₄	—	13	168	no	—
128	C₂²×D₄	—	13	2688	no	—
129	C₂²×D₄	—	13	168	no	—
130	C₂²×D₄	—	13	84	no	—
131	C₂²×D₄	—	13	1344	no	—
132	C₂²×D₄	C₂²×D₁₃	1	84	yes	complemented
133	C₂³×C₄	—	13	2	no	—
134	C₂³×C₄	—	13	168	no	—
135	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
136	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
137	C₂³×C₄	C₂²×D₁₃	1	84	yes	complemented
138	C₂×C₁₃	—	2	24	no	—
139	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	14	yes	complemented
140	C₂×C₁₃	C₂⁷	1	1	yes	characteristic
141	D₁₃	—	2	48	no	—
142	C₂⁴	—	2	24	no	—
143	C₂⁴	—	13	84	no	—
144	C₂⁴	—	13	672	no	—
145	C₂⁴	—	26	48	no	—
146	C₂⁴	—	52	672	no	—
147	C₂⁴	C₂³×D₁₃	1	42	yes	—
148	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
149	#41	—	13	1024	no	—
150	#41	—	13	1536	no	—
151	#41	C₂³×D₁₃	1	512	yes	complemented
152	C₂×Q₈	—	13	112	no	—
153	C₂×Q₈	C₂³×D₁₃	1	112	yes	complemented
154	C₂×D₄	—	13	672	no	—
155	C₂×D₄	—	13	3072	no	—
156	C₂×D₄	—	13	672	no	—
157	C₂×D₄	—	13	336	no	—
158	C₂×D₄	—	13	1536	no	—
159	C₂×D₄	C₂³×D₁₃	1	336	yes	complemented
160	C₂²×C₄	—	13	28	no	—
161	C₂²×C₄	—	13	672	no	—
162	C₂²×C₄	C₂³×D₁₃	1	42	yes	—
163	C₂²×C₄	C₂³×D₁₃	1	14	yes	—
164	C₂²×C₄	C₂³×D₁₃	1	336	yes	complemented
165	C₁₃	C₂³×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
166	C₂³	—	2	168	no	—
167	C₂³	—	13	168	no	—
168	C₂³	—	13	384	no	—
169	C₂³	—	26	336	no	—
170	C₂³	—	52	1344	no	—
171	C₂³	#88256	1	8	yes	direct product factor
172	C₂³	C₂⁴×D₁₃	1	84	yes	—
173	C₂³	C₂⁴×D₁₃	1	7	yes	—
174	Q₈	—	13	64	no	—
175	Q₈	C₂⁴×D₁₃	1	64	yes	complemented
176	D₄	—	13	384	no	—
177	D₄	—	13	384	no	—
178	D₄	—	13	192	no	—
179	D₄	C₂⁴×D₁₃	1	192	yes	complemented
180	C₂×C₄	—	13	56	no	—
181	C₂×C₄	—	13	384	no	—
182	C₂×C₄	C₂⁴×D₁₃	1	84	yes	—
183	C₂×C₄	C₂⁴×D₁₃	1	28	yes	—
184	C₂×C₄	C₂⁴×D₁₃	1	192	yes	complemented
185	C₂²	—	2	168	no	—
186	C₂²	—	13	48	no	—
187	C₂²	—	26	336	no	—
188	C₂²	—	52	384	no	—
189	C₂²	C₂×#88256	1	28	yes	direct product factor
190	C₂²	C₂⁵×D₁₃	1	24	yes	—
191	C₂²	C₂⁵×D₁₃	1	7	yes	—
192	C₄	—	13	16	no	—
193	C₄	C₂⁵×D₁₃	1	24	yes	—
194	C₄	C₂⁵×D₁₃	1	8	yes	—
195	C₂	—	2	24	no	—
196	C₂	—	26	48	no	—
197	C₂	C₂²×#88256	1	14	yes	direct product factor
198	C₂	C₂⁶×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂×C₁₃	C₂⁷
Frattini	C₂	C₂⁶×D₁₃
Fitting	C₂³×C₁₃×#41	C₂
socle	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂³×#41	D₁₃
hypocenter	C₁₃	C₂³×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#88256	C₂	direct (N × H)
2. C₂³×#6131	C₂	semidirect
3. C₂³×#6127	C₂	semidirect
4. C₂⁴×D₅₂	C₂	semidirect
5. C₂³×D₄×D₁₃	C₂	semidirect
6. C₂³×C₁₃×#41	C₂	semidirect
7. C₂⁵×D₁₃	C₂²	semidirect
8. C₂²×C₁₃×#41	C₂²	semidirect
9. C₂³×Q₈×C₁₃	C₂²	semidirect
10. C₂³×D₄×C₁₃	C₂²	semidirect
11. C₂⁴×C₄×C₁₃	C₂²	semidirect
12. C₂³×C₂²⋊D₁₃	C₂²	semidirect
13. C₂×#88256	C₂²	direct (N × H)
14. C₂²×#6131	C₂²	semidirect
15. C₂²×D₄×D₁₃	C₂²	semidirect
16. C₂²×#6127	C₂²	semidirect
17. C₂³×D₅₂	C₂²	semidirect
18. C₂³×D₅₂	C₂²	semidirect
19. C₂³×D₅₂	C₂²	semidirect
20. C₂³×C₄×D₁₃	C₂²	semidirect
21. C₂³×Dic₂₆	C₂²	semidirect
25. C₂⁴×D₁₃	C₂³	semidirect
26. C₂×C₁₃×#41	C₂³	semidirect
27. C₂²×Q₈×C₁₃	C₂³	semidirect
28. C₂²×D₄×C₁₃	C₂³	semidirect
31. C₂³×C₄×C₁₃	C₂³	semidirect
32. C₂²×C₂²⋊D₁₃	C₂³	semidirect
34. #88256	C₂³	direct (N × H)
35. C₂×#6131	C₂³	semidirect
36. C₂×D₄×D₁₃	C₂³	semidirect
37. C₂×#6127	C₂³	semidirect
38. C₂²×D₅₂	C₂³	semidirect
39. C₂²×D₅₂	C₂³	semidirect
40. C₂²×D₅₂	C₂³	semidirect
41. C₂²×C₄×D₁₃	C₂³	semidirect
42. C₂²×Dic₂₆	C₂³	semidirect
50. C₂³×D₁₃	C₂⁴	semidirect
51. C₁₃×#41	C₂⁴	semidirect
52. C₂×Q₈×C₁₃	C₂⁴	semidirect
53. C₂×D₄×C₁₃	C₂⁴	semidirect
56. C₂²×C₄×C₁₃	C₂⁴	semidirect
57. C₂×C₂²⋊D₁₃	C₂⁴	semidirect
59. #6131	C₂⁴	semidirect
60. D₄×D₁₃	C₂⁴	semidirect
61. #6127	C₂⁴	semidirect
62. C₂×D₅₂	C₂⁴	semidirect
63. C₂×D₅₂	C₂⁴	semidirect
64. C₂×D₅₂	C₂⁴	semidirect
65. C₂×C₄×D₁₃	C₂⁴	semidirect
66. C₂×Dic₂₆	C₂⁴	semidirect
70. C₂³×#41	D₁₃	semidirect
74. C₂³×C₁₃	H(5,𝔽₂)	semidirect
80. C₂²×D₁₃	C₂⁵	semidirect
81. Q₈×C₁₃	C₂⁵	semidirect
82. D₄×C₁₃	C₂⁵	semidirect
85. C₂×C₄×C₁₃	C₂⁵	semidirect
86. C₂²⋊D₁₃	C₂⁵	semidirect
88. D₅₂	C₂⁵	semidirect
89. D₅₂	C₂⁵	semidirect
90. C₄×D₁₃	C₂⁵	semidirect
91. Dic₂₆	C₂⁵	semidirect
96. C₂²×#41	C₂×D₁₃	semidirect
98. C₂³×Q₈	C₂×D₁₃	semidirect
104. C₂³×D₄	C₂×D₁₃	semidirect
106. C₂⁴×C₄	C₂×D₁₃	semidirect
108. C₂²×C₁₃	C₂×H(5,𝔽₂)	semidirect
124. C₂×#41	C₂²×D₁₃	semidirect
126. C₂²×Q₈	C₂²×D₁₃	semidirect
132. C₂²×D₄	C₂²×D₁₃	semidirect
137. C₂³×C₄	C₂²×D₁₃	semidirect
139. C₂×C₁₃	C₂²×H(5,𝔽₂)	semidirect
151. #41	C₂³×D₁₃	semidirect
153. C₂×Q₈	C₂³×D₁₃	semidirect
159. C₂×D₄	C₂³×D₁₃	semidirect
164. C₂²×C₄	C₂³×D₁₃	semidirect
165. C₁₃	C₂³×H(5,𝔽₂)	semidirect
171. C₂³	#88256	direct (N × H)
175. Q₈	C₂⁴×D₁₃	semidirect
179. D₄	C₂⁴×D₁₃	semidirect
184. C₂×C₄	C₂⁴×D₁₃	semidirect
189. C₂²	C₂×#88256	direct (N × H)
197. C₂	C₂²×#88256	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×#88256	2
C₂³×#6131	2
C₂³×#6127	2
C₂⁴×D₅₂	2
C₂³×D₄×D₁₃	2
C₂³×C₁₃×#41	2
C₂³×H(5,𝔽₂)	13

Subgroups — C23×#88256 #16776806

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88256 #16776806