Subgroups — C₂²×#11719503 #16769500

Order 3328.

14590 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719503	C₂	1	48	yes	maximal, direct product factor
2	C₂²×D₄×Dic₁₃	C₂	1	8	yes	maximal, complemented
3	C₂²×#88180	C₂	1	2	yes	maximal, complemented
4	C₂³×C₂²⋊Dic₁₃	C₂	1	4	yes	maximal, complemented
5	C₂⁴×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂⁶×C₁₃	C₂²	1	2	yes	—
7	C₂³×D₄×C₁₃	C₄	1	32	yes	complemented
8	C₂³×D₄×C₁₃	C₂²	1	4	yes	—
9	C₂³×D₄×C₁₃	C₂²	1	24	yes	—
10	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₁₃	C₂²	1	8	yes	complemented
12	C₂²×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
13	C₂²×C₂²⋊Dic₁₃	C₂²	1	96	yes	complemented
14	C₂²×C₂²⋊Dic₁₃	C₂²	1	8	yes	complemented
15	#11719503	C₂²	1	256	yes	direct product factor
16	C₂×D₄×Dic₁₃	C₂²	1	192	yes	complemented
17	C₂×#88180	C₂²	1	48	yes	complemented
18	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	complemented
19	C₂²×C₄×Dic₁₃	C₂²	1	4	yes	complemented
20	C₂⁵×C₁₃	—	4	16	no	—
21	C₂⁵×C₁₃	C₂×C₄	1	32	yes	complemented
22	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
23	C₂⁵×C₁₃	C₂³	1	4	yes	—
24	C₂⁵×C₁₃	C₂³	1	24	yes	—
25	C₂²×D₄×C₁₃	C₂×C₄	1	64	yes	complemented
26	C₂²×D₄×C₁₃	C₂×C₄	1	384	yes	complemented
27	C₂²×D₄×C₁₃	C₂³	1	48	yes	—
28	C₂²×D₄×C₁₃	C₂³	1	64	yes	—
29	C₂³×C₄×C₁₃	C₂×C₄	1	16	yes	complemented
30	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
32	C₂³×Dic₁₃	—	4	32	no	—
33	C₂³×Dic₁₃	C₂³	1	8	yes	—
34	C₂³×Dic₁₃	C₂³	1	96	yes	complemented
35	C₂×C₂²⋊Dic₁₃	C₂³	1	48	yes	—
36	C₂×C₂²⋊Dic₁₃	C₂³	1	256	yes	complemented
37	C₂×C₂²⋊Dic₁₃	C₂³	1	96	yes	complemented
38	D₄×Dic₁₃	C₂³	1	512	yes	complemented
39	#88180	C₂³	1	128	yes	complemented
40	C₂×C₄⋊Dic₁₃	C₂³	1	48	yes	complemented
41	C₂×C₄×Dic₁₃	C₂³	1	48	yes	complemented
42	C₂²×#518	—	13	1	no	maximal, Sylow
43	C₂⁴×C₁₃	—	2	32	no	—
44	C₂⁴×C₁₃	—	2	8	no	—
45	C₂⁴×C₁₃	—	4	128	no	—
46	C₂⁴×C₁₃	—	4	96	no	—
47	C₂⁴×C₁₃	C₂²×C₄	1	8	yes	—
48	C₂⁴×C₁₃	C₂²×C₄	1	32	yes	—
49	C₂⁴×C₁₃	C₂²×C₄	1	192	yes	complemented
50	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
51	C₂⁴×C₁₃	C₂⁴	1	24	yes	—
52	C₂⁴×C₁₃	C₂⁴	1	32	yes	—
53	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
54	C₂×D₄×C₁₃	C₂²×C₄	1	384	yes	complemented
55	C₂×D₄×C₁₃	C₂²×C₄	1	512	yes	complemented
56	C₂×D₄×C₁₃	C₂⁴	1	64	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	16	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	96	yes	complemented
59	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
60	C₂²×C₄×C₁₃	C₂⁴	1	16	yes	—
61	C₂²×Dic₁₃	—	2	32	no	—
62	C₂²×Dic₁₃	—	4	192	no	—
63	C₂²×Dic₁₃	C₂⁴	1	48	yes	—
64	C₂²×Dic₁₃	C₂⁴	1	128	yes	complemented
65	C₂²⋊Dic₁₃	C₂⁴	1	64	yes	—
66	C₂²⋊Dic₁₃	C₂⁴	1	128	yes	complemented
67	C₄⋊Dic₁₃	C₂⁴	1	64	yes	complemented
68	C₄×Dic₁₃	C₂⁴	1	64	yes	complemented
69	C₂⁴×D₄	D₁₃	1	1	yes	characteristic
70	C₂×#518	—	13	48	no	—
71	C₂²×C₄×D₄	—	13	8	no	—
72	C₂²×#117	—	13	2	no	—
73	C₂³×C₂²⋊C₄	—	13	4	no	—
74	C₂³×C₁₃	—	2	32	no	—
75	C₂³×C₁₃	—	2	128	no	—
76	C₂³×C₁₃	—	2	96	no	—
77	C₂³×C₁₃	—	2	24	no	—
78	C₂³×C₁₃	—	4	384	no	—
79	C₂³×C₁₃	—	4	64	no	—
80	C₂³×C₁₃	C₂³×C₄	1	24	yes	—
81	C₂³×C₁₃	C₂³×C₄	1	96	yes	—
82	C₂³×C₁₃	C₂³×C₄	1	4	yes	—
83	C₂³×C₁₃	C₂³×C₄	1	128	yes	complemented
84	C₂³×C₁₃	H(5,𝔽₂)	1	8	yes	—
85	C₂³×C₁₃	C₂⁵	1	4	yes	—
86	C₂³×C₁₃	C₂⁵	1	16	yes	—
87	C₂³×C₁₃	C₂⁵	1	3	yes	—
88	D₄×C₁₃	C₂³×C₄	1	256	yes	complemented
89	C₂×C₄×C₁₃	C₂³×C₄	1	48	yes	—
90	C₂×C₄×C₁₃	C₂³×C₄	1	64	yes	complemented
91	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
92	C₂×Dic₁₃	—	2	96	no	—
93	C₂×Dic₁₃	—	4	128	no	—
94	C₂×Dic₁₃	C₂⁵	1	32	yes	—
95	C₂⁶	C₂×D₁₃	1	2	yes	—
96	C₂³×D₄	Dic₁₃	1	32	yes	complemented
97	C₂³×D₄	C₂×D₁₃	1	4	yes	—
98	C₂³×D₄	C₂×D₁₃	1	24	yes	—
99	C₂⁴×C₄	—	13	8	no	—
100	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
101	#518	—	13	256	no	—
102	C₂×C₄×D₄	—	13	192	no	—
103	C₂×#117	—	13	48	no	—
104	C₂²×C₄⋊C₄	—	13	4	no	—
105	C₂²×C₂²⋊C₄	—	13	4	no	—
106	C₂²×C₂²⋊C₄	—	13	8	no	—
107	C₂²×C₂²⋊C₄	—	13	96	no	—
108	C₂²×C₄²	—	13	4	no	—
109	C₂²×C₁₃	—	2	32	no	—
110	C₂²×C₁₃	—	2	8	no	—
111	C₂²×C₁₃	—	2	48	no	—
112	C₂²×C₁₃	—	2	192	no	—
113	C₂²×C₁₃	—	4	128	no	—
114	C₂²×C₁₃	#518	1	16	yes	complemented
115	C₂²×C₁₃	C₂⁴×C₄	1	8	yes	—
116	C₂²×C₁₃	C₂⁴×C₄	1	32	yes	—
117	C₂²×C₁₃	C₂⁴×C₄	1	6	yes	—
118	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	12	yes	—
119	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
120	C₄×C₁₃	C₂⁴×C₄	1	16	yes	—
121	Dic₁₃	—	2	32	no	—
122	C₂⁵	—	4	16	no	—
123	C₂⁵	C₂×Dic₁₃	1	32	yes	complemented
124	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
125	C₂⁵	C₂²×D₁₃	1	4	yes	—
126	C₂⁵	C₂²×D₁₃	1	24	yes	—
127	C₂²×D₄	C₂×Dic₁₃	1	64	yes	complemented
128	C₂²×D₄	C₂×Dic₁₃	1	384	yes	complemented
129	C₂²×D₄	C₂²×D₁₃	1	48	yes	—
130	C₂²×D₄	C₂²×D₁₃	1	64	yes	—
131	C₂³×C₄	—	13	8	no	—
132	C₂³×C₄	—	13	96	no	—
133	C₂³×C₄	—	52	32	no	—
134	C₂³×C₄	C₂×Dic₁₃	1	16	yes	complemented
135	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
136	C₂³×C₄	C₂²×D₁₃	1	12	yes	—
137	C₄×D₄	—	13	512	no	—
138	#117	—	13	128	no	—
139	C₂×C₄⋊C₄	—	13	48	no	—
140	C₂×C₂²⋊C₄	—	13	48	no	—
141	C₂×C₂²⋊C₄	—	13	96	no	—
142	C₂×C₂²⋊C₄	—	13	256	no	—
143	C₂×C₄²	—	13	48	no	—
144	C₂×C₁₃	—	2	8	no	—
145	C₂×C₁₃	—	2	32	no	—
146	C₂×C₁₃	C₂×#518	1	12	yes	complemented
147	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
148	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	2	yes	—
149	C₂⁴	—	2	32	no	—
150	C₂⁴	—	2	8	no	—
151	C₂⁴	—	4	128	no	—
152	C₂⁴	—	4	96	no	—
153	C₂⁴	C₂²×Dic₁₃	1	8	yes	—
154	C₂⁴	C₂²×Dic₁₃	1	32	yes	—
155	C₂⁴	C₂²×Dic₁₃	1	192	yes	complemented
156	C₂⁴	C₂³×D₁₃	1	6	yes	—
157	C₂⁴	C₂³×D₁₃	1	24	yes	—
158	C₂⁴	C₂³×D₁₃	1	32	yes	—
159	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
160	C₂×D₄	C₂²×Dic₁₃	1	384	yes	complemented
161	C₂×D₄	C₂²×Dic₁₃	1	512	yes	complemented
162	C₂×D₄	C₂³×D₁₃	1	64	yes	—
163	C₂²×C₄	—	13	48	no	—
164	C₂²×C₄	—	13	128	no	—
165	C₂²×C₄	—	26	32	no	—
166	C₂²×C₄	—	52	192	no	—
167	C₂²×C₄	C₂²×Dic₁₃	1	16	yes	—
168	C₂²×C₄	C₂²×Dic₁₃	1	96	yes	complemented
169	C₂²×C₄	C₂³×D₁₃	1	12	yes	—
170	C₂²×C₄	C₂³×D₁₃	1	16	yes	—
171	C₄⋊C₄	—	13	64	no	—
172	C₂²⋊C₄	—	13	64	no	—
173	C₂²⋊C₄	—	13	128	no	—
174	C₄²	—	13	64	no	—
175	C₁₃	C₂²×#518	1	1	yes	characteristic, complemented, Sylow
176	C₂³	—	2	32	no	—
177	C₂³	—	2	128	no	—
178	C₂³	—	2	96	no	—
179	C₂³	—	2	24	no	—
180	C₂³	—	4	384	no	—
181	C₂³	—	4	64	no	—
182	C₂³	#88251	1	8	yes	—
183	C₂³	C₂³×Dic₁₃	1	24	yes	—
184	C₂³	C₂³×Dic₁₃	1	96	yes	—
185	C₂³	C₂³×Dic₁₃	1	4	yes	—
186	C₂³	C₂³×Dic₁₃	1	128	yes	complemented
187	C₂³	C₂⁴×D₁₃	1	4	yes	—
188	C₂³	C₂⁴×D₁₃	1	16	yes	—
189	C₂³	C₂⁴×D₁₃	1	3	yes	—
190	D₄	C₂³×Dic₁₃	1	256	yes	complemented
191	C₂×C₄	—	13	32	no	—
192	C₂×C₄	—	26	96	no	—
193	C₂×C₄	—	52	128	no	—
194	C₂×C₄	C₂³×Dic₁₃	1	48	yes	—
195	C₂×C₄	C₂³×Dic₁₃	1	64	yes	complemented
196	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
197	C₂²	—	2	48	no	—
198	C₂²	—	2	192	no	—
199	C₂²	—	2	32	no	—
200	C₂²	—	2	8	no	—
201	C₂²	—	4	128	no	—
202	C₂²	#11719503	1	16	yes	direct product factor
203	C₂²	C₂×#88251	1	12	yes	—
204	C₂²	C₂⁴×Dic₁₃	1	8	yes	—
205	C₂²	C₂⁴×Dic₁₃	1	32	yes	—
206	C₂²	C₂⁴×Dic₁₃	1	6	yes	—
207	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
208	C₄	—	26	32	no	—
209	C₄	C₂⁴×Dic₁₃	1	16	yes	—
210	C₂	—	2	8	no	—
211	C₂	—	2	32	no	—
212	C₂	C₂×#11719503	1	12	yes	direct product factor
213	C₂	C₂⁵×Dic₁₃	1	1	yes	characteristic
214	C₂	C₂²×#88251	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator	C₂×C₁₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₃
Fitting, prime radical	C₂⁴×D₄×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂⁴
cosocle	C₂²×C₁₃	C₂⁶
hypercenter	C₂⁴×D₄	D₁₃
hypocenter	C₁₃	C₂²×#518
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. C₂×#11719503	C₂	direct (N × H)
2. C₂²×D₄×Dic₁₃	C₂	semidirect
3. C₂²×#88180	C₂	semidirect
4. C₂³×C₂²⋊Dic₁₃	C₂	semidirect
7. C₂³×D₄×C₁₃	C₄	semidirect
11. C₂⁴×Dic₁₃	C₂²	semidirect
13. C₂²×C₂²⋊Dic₁₃	C₂²	semidirect
14. C₂²×C₂²⋊Dic₁₃	C₂²	semidirect
15. #11719503	C₂²	direct (N × H)
16. C₂×D₄×Dic₁₃	C₂²	semidirect
17. C₂×#88180	C₂²	semidirect
18. C₂²×C₄⋊Dic₁₃	C₂²	semidirect
19. C₂²×C₄×Dic₁₃	C₂²	semidirect
21. C₂⁵×C₁₃	C₂×C₄	semidirect
25. C₂²×D₄×C₁₃	C₂×C₄	semidirect
26. C₂²×D₄×C₁₃	C₂×C₄	semidirect
29. C₂³×C₄×C₁₃	C₂×C₄	semidirect
34. C₂³×Dic₁₃	C₂³	semidirect
36. C₂×C₂²⋊Dic₁₃	C₂³	semidirect
37. C₂×C₂²⋊Dic₁₃	C₂³	semidirect
38. D₄×Dic₁₃	C₂³	semidirect
39. #88180	C₂³	semidirect
40. C₂×C₄⋊Dic₁₃	C₂³	semidirect
41. C₂×C₄×Dic₁₃	C₂³	semidirect
49. C₂⁴×C₁₃	C₂²×C₄	semidirect
54. C₂×D₄×C₁₃	C₂²×C₄	semidirect
55. C₂×D₄×C₁₃	C₂²×C₄	semidirect
58. C₂²×C₄×C₁₃	C₂²×C₄	semidirect
64. C₂²×Dic₁₃	C₂⁴	semidirect
66. C₂²⋊Dic₁₃	C₂⁴	semidirect
67. C₄⋊Dic₁₃	C₂⁴	semidirect
68. C₄×Dic₁₃	C₂⁴	semidirect
83. C₂³×C₁₃	C₂³×C₄	semidirect
88. D₄×C₁₃	C₂³×C₄	semidirect
90. C₂×C₄×C₁₃	C₂³×C₄	semidirect
96. C₂³×D₄	Dic₁₃	semidirect
114. C₂²×C₁₃	#518	semidirect
123. C₂⁵	C₂×Dic₁₃	semidirect
127. C₂²×D₄	C₂×Dic₁₃	semidirect
128. C₂²×D₄	C₂×Dic₁₃	semidirect
134. C₂³×C₄	C₂×Dic₁₃	semidirect
146. C₂×C₁₃	C₂×#518	semidirect
155. C₂⁴	C₂²×Dic₁₃	semidirect
160. C₂×D₄	C₂²×Dic₁₃	semidirect
161. C₂×D₄	C₂²×Dic₁₃	semidirect
168. C₂²×C₄	C₂²×Dic₁₃	semidirect
175. C₁₃	C₂²×#518	semidirect
186. C₂³	C₂³×Dic₁₃	semidirect
190. D₄	C₂³×Dic₁₃	semidirect
195. C₂×C₄	C₂³×Dic₁₃	semidirect
202. C₂²	#11719503	direct (N × H)
212. C₂	C₂×#11719503	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×#11719503	2
C₂²×D₄×Dic₁₃	2
C₂²×#88180	2
C₂³×C₂²⋊Dic₁₃	2
C₂⁴×D₄×C₁₃	2
C₂²×#518	13

Subgroups — C22×#11719503 #16769500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719503 #16769500