Subgroups — C₂²×D₁₃×2₋¹⁺⁴ #16776807

Order 3328.

59806 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88254	C₂	1	5	yes	maximal, complemented
2	C₂²×#88257	C₂	1	10	yes	maximal, complemented
3	C₂×D₁₃×2₋¹⁺⁴	C₂	1	96	yes	maximal, direct product factor
4	C₂²×#41×D₁₃	C₂	1	10	yes	maximal, complemented
5	C₂³×Q₈×D₁₃	C₂	1	5	yes	maximal, complemented
6	C₂²×C₁₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×2₋¹⁺⁴	C₂²	1	48	yes	complemented
8	C₂²×C₁₃×#41	C₂²	1	10	yes	complemented
9	C₂³×Q₈×C₁₃	C₂²	1	5	yes	complemented
10	D₁₃×2₋¹⁺⁴	C₂²	1	1024	yes	direct product factor
11	C₂×#88257	C₂²	1	480	yes	complemented
12	C₂×#41×D₁₃	C₂²	1	480	yes	complemented
13	C₂×#88254	C₂²	1	240	yes	complemented
14	C₂²×#6131	C₂²	1	10	yes	complemented
15	C₂²×Q₈×D₁₃	C₂²	1	10	yes	—
16	C₂²×Q₈×D₁₃	C₂²	1	20	yes	complemented
17	C₂²×Q₈×D₁₃	C₂²	1	240	yes	complemented
18	C₂²×#6129	C₂²	1	30	yes	complemented
19	C₂²×D₄×D₁₃	C₂²	1	10	yes	—
20	C₂²×#6127	C₂²	1	30	yes	complemented
21	C₂³×C₄×D₁₃	C₂²	1	15	yes	—
22	C₂³×Dic₂₆	C₂²	1	15	yes	complemented
23	C₂⁴×D₁₃	C₂³	1	5	yes	—
24	C₁₃×2₋¹⁺⁴	C₂³	1	256	yes	complemented
25	C₂×C₁₃×#41	C₂³	1	240	yes	complemented
26	C₂²×Q₈×C₁₃	C₂³	1	10	yes	—
27	C₂²×Q₈×C₁₃	C₂³	1	120	yes	complemented
28	C₂²×D₄×C₁₃	C₂³	1	10	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	15	yes	—
30	C₂²×C₂²⋊D₁₃	C₂³	1	20	yes	—
31	C₂³×Dic₁₃	C₂³	1	15	yes	—
32	#88257	C₂³	1	2560	yes	complemented
33	#41×D₁₃	C₂³	1	2560	yes	complemented
34	#88254	C₂³	1	1280	yes	complemented
35	C₂×#6131	C₂³	1	240	yes	complemented
36	C₂×Q₈×D₁₃	C₂³	1	240	yes	—
37	C₂×Q₈×D₁₃	C₂³	1	480	yes	complemented
38	C₂×Q₈×D₁₃	C₂³	1	1280	yes	complemented
39	C₂×#6129	C₂³	1	720	yes	complemented
40	C₂×D₄×D₁₃	C₂³	1	240	yes	—
41	C₂×#6127	C₂³	1	720	yes	complemented
42	C₂²×D₅₂	C₂³	1	10	yes	—
43	C₂²×C₄×D₁₃	C₂³	1	10	yes	—
44	C₂²×C₄×D₁₃	C₂³	1	30	yes	—
45	C₂²×C₄×D₁₃	C₂³	1	360	yes	—
46	C₂²×Dic₂₆	C₂³	1	30	yes	—
47	C₂²×Dic₂₆	C₂³	1	360	yes	complemented
48	C₂³×2₋¹⁺⁴	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	C₂⁴	1	5	yes	—
50	C₂³×D₁₃	—	2	40	no	—
51	C₂³×D₁₃	C₂⁴	1	5	yes	—
52	C₂³×D₁₃	C₂⁴	1	60	yes	—
53	C₂³×D₁₃	C₂⁴	1	1	yes	characteristic
54	C₁₃×#41	C₂⁴	1	640	yes	complemented
55	C₂×Q₈×C₁₃	C₂⁴	1	120	yes	—
56	C₂×Q₈×C₁₃	C₂⁴	1	320	yes	complemented
57	C₂×D₄×C₁₃	C₂⁴	1	120	yes	—
58	C₂²×C₄×C₁₃	C₂⁴	1	10	yes	—
59	C₂²×C₄×C₁₃	C₂⁴	1	180	yes	—
60	C₂×C₂²⋊D₁₃	C₂⁴	1	240	yes	—
61	C₂²×Dic₁₃	C₂⁴	1	10	yes	—
62	C₂²×Dic₁₃	C₂⁴	1	180	yes	—
63	#6131	C₂⁴	1	640	yes	complemented
64	Q₈×D₁₃	C₂⁴	1	640	yes	—
65	Q₈×D₁₃	C₂⁴	1	1280	yes	complemented
66	#6129	C₂⁴	1	1920	yes	complemented
67	D₄×D₁₃	C₂⁴	1	640	yes	—
68	#6127	C₂⁴	1	1920	yes	complemented
69	C₂×D₅₂	C₂⁴	1	120	yes	—
70	C₂×C₄×D₁₃	C₂⁴	1	120	yes	—
71	C₂×C₄×D₁₃	C₂⁴	1	360	yes	—
72	C₂×C₄×D₁₃	C₂⁴	1	960	yes	—
73	C₂×Dic₂₆	C₂⁴	1	360	yes	—
74	C₂×Dic₂₆	C₂⁴	1	960	yes	complemented
75	C₂²×2₋¹⁺⁴	—	13	96	no	—
76	C₂²×2₋¹⁺⁴	—	13	10	no	—
77	C₂²×2₋¹⁺⁴	—	13	5	no	—
78	C₂²×2₋¹⁺⁴	D₁₃	1	1	yes	characteristic, direct product factor
79	C₂³×#41	—	13	10	no	—
80	C₂⁴×Q₈	—	13	5	no	—
81	C₂³×C₁₃	—	2	20	no	—
82	C₂³×C₁₃	C₂⁵	1	30	yes	—
83	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
84	C₂²×D₁₃	—	2	20	no	—
85	C₂²×D₁₃	—	2	240	no	—
86	C₂²×D₁₃	2₋¹⁺⁴	1	8	yes	direct product factor
87	C₂²×D₁₃	C₂⁵	1	30	yes	—
88	C₂²×D₁₃	C₂⁵	1	80	yes	—
89	C₂²×D₁₃	C₂⁵	1	6	yes	—
90	Q₈×C₁₃	C₂⁵	1	160	yes	—
91	D₄×C₁₃	C₂⁵	1	160	yes	—
92	C₂×C₄×C₁₃	C₂⁵	1	60	yes	—
93	C₂×C₄×C₁₃	C₂⁵	1	240	yes	—
94	C₂²⋊D₁₃	C₂⁵	1	320	yes	—
95	C₂×Dic₁₃	C₂⁵	1	60	yes	—
96	C₂×Dic₁₃	C₂⁵	1	240	yes	—
97	D₅₂	C₂⁵	1	160	yes	—
98	C₄×D₁₃	C₂⁵	1	160	yes	—
99	C₄×D₁₃	C₂⁵	1	480	yes	—
100	Dic₂₆	C₂⁵	1	480	yes	—
101	C₂×2₋¹⁺⁴	—	13	1024	no	—
102	C₂×2₋¹⁺⁴	—	13	480	no	—
103	C₂×2₋¹⁺⁴	—	13	240	no	—
104	C₂×2₋¹⁺⁴	C₂×D₁₃	1	48	yes	direct product factor
105	C₂²×#41	—	13	480	no	—
106	C₂²×#41	—	13	10	no	—
107	C₂²×#41	—	13	30	no	—
108	C₂²×#41	—	13	30	no	—
109	C₂²×#41	C₂×D₁₃	1	10	yes	complemented
110	C₂³×Q₈	—	13	10	no	—
111	C₂³×Q₈	—	13	20	no	—
112	C₂³×Q₈	—	13	240	no	—
113	C₂³×Q₈	—	13	15	no	—
114	C₂³×Q₈	C₂×D₁₃	1	5	yes	complemented
115	C₂³×D₄	—	13	10	no	—
116	C₂⁴×C₄	—	13	15	no	—
117	C₂²×C₁₃	—	2	60	no	—
118	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	4	yes	complemented
119	C₂²×C₁₃	C₂⁶	1	20	yes	—
120	C₂²×C₁₃	C₂⁶	1	3	yes	—
121	C₂×D₁₃	—	2	60	no	—
122	C₂×D₁₃	—	2	160	no	—
123	C₂×D₁₃	C₂×2₋¹⁺⁴	1	24	yes	direct product factor
124	C₂×D₁₃	C₂⁶	1	20	yes	—
125	C₂×D₁₃	C₂⁶	1	4	yes	—
126	C₄×C₁₃	C₂⁶	1	40	yes	—
127	Dic₁₃	C₂⁶	1	40	yes	—
128	C₂⁵	—	13	5	no	—
129	2₋¹⁺⁴	—	13	2560	no	—
130	2₋¹⁺⁴	—	13	1280	no	—
131	2₋¹⁺⁴	C₂²×D₁₃	1	256	yes	direct product factor
132	C₂×#41	—	13	2560	no	—
133	C₂×#41	—	13	240	no	—
134	C₂×#41	—	13	720	no	—
135	C₂×#41	—	13	720	no	—
136	C₂×#41	C₂²×D₁₃	1	240	yes	complemented
137	C₂²×Q₈	—	13	240	no	—
138	C₂²×Q₈	—	13	480	no	—
139	C₂²×Q₈	—	13	1280	no	—
140	C₂²×Q₈	—	13	30	no	—
141	C₂²×Q₈	—	13	360	no	—
142	C₂²×Q₈	C₂²×D₁₃	1	10	yes	—
143	C₂²×Q₈	C₂²×D₁₃	1	120	yes	complemented
144	C₂²×D₄	—	13	20	no	—
145	C₂²×D₄	—	13	240	no	—
146	C₂²×D₄	—	13	10	no	—
147	C₂²×D₄	C₂²×D₁₃	1	10	yes	—
148	C₂³×C₄	—	13	15	no	—
149	C₂³×C₄	—	13	10	no	—
150	C₂³×C₄	—	13	360	no	—
151	C₂³×C₄	—	13	30	no	—
152	C₂³×C₄	C₂²×D₁₃	1	15	yes	—
153	C₂×C₁₃	—	2	20	no	—
154	C₂×C₁₃	C₂²×2₋¹⁺⁴	1	6	yes	complemented
155	C₂×C₁₃	C₂⁷	1	1	yes	characteristic
156	D₁₃	—	2	20	no	—
157	D₁₃	C₂²×2₋¹⁺⁴	1	8	yes	direct product factor
158	C₂⁴	—	13	1	no	—
159	C₂⁴	—	13	60	no	—
160	C₂⁴	—	13	5	no	—
161	C₂⁴	—	26	40	no	—
162	C₂⁴	C₂³×D₁₃	1	5	yes	—
163	#41	—	13	640	no	—
164	#41	—	13	1920	no	—
165	#41	—	13	1920	no	—
166	#41	C₂³×D₁₃	1	640	yes	complemented
167	C₂×Q₈	—	13	640	no	—
168	C₂×Q₈	—	13	1280	no	—
169	C₂×Q₈	—	13	360	no	—
170	C₂×Q₈	—	13	960	no	—
171	C₂×Q₈	C₂³×D₁₃	1	120	yes	—
172	C₂×Q₈	C₂³×D₁₃	1	320	yes	complemented
173	C₂×D₄	—	13	240	no	—
174	C₂×D₄	—	13	640	no	—
175	C₂×D₄	—	13	120	no	—
176	C₂×D₄	C₂³×D₁₃	1	120	yes	—
177	C₂²×C₄	—	13	180	no	—
178	C₂²×C₄	—	13	10	no	—
179	C₂²×C₄	—	13	120	no	—
180	C₂²×C₄	—	13	360	no	—
181	C₂²×C₄	—	13	960	no	—
182	C₂²×C₄	C₂³×D₁₃	1	180	yes	—
183	C₂²×C₄	C₂³×D₁₃	1	10	yes	—
184	C₁₃	C₂³×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
185	C₂³	—	2	20	no	—
186	C₂³	—	13	8	no	—
187	C₂³	—	13	6	no	—
188	C₂³	—	13	80	no	—
189	C₂³	—	13	30	no	—
190	C₂³	—	26	240	no	—
191	C₂³	—	26	20	no	—
192	C₂³	C₂⁴×D₁₃	1	30	yes	—
193	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
194	Q₈	—	13	480	no	—
195	Q₈	C₂⁴×D₁₃	1	160	yes	—
196	D₄	—	13	320	no	—
197	D₄	—	13	160	no	—
198	D₄	C₂⁴×D₁₃	1	160	yes	—
199	C₂×C₄	—	13	60	no	—
200	C₂×C₄	—	13	240	no	—
201	C₂×C₄	—	13	160	no	—
202	C₂×C₄	—	13	480	no	—
203	C₂×C₄	C₂⁴×D₁₃	1	60	yes	—
204	C₂×C₄	C₂⁴×D₁₃	1	240	yes	—
205	C₂²	—	2	60	no	—
206	C₂²	—	13	24	no	—
207	C₂²	—	13	4	no	—
208	C₂²	—	13	20	no	—
209	C₂²	—	26	160	no	—
210	C₂²	—	26	60	no	—
211	C₂²	D₁₃×2₋¹⁺⁴	1	4	yes	direct product factor
212	C₂²	C₂⁵×D₁₃	1	20	yes	—
213	C₂²	C₂⁵×D₁₃	1	3	yes	—
214	C₄	—	13	40	no	—
215	C₄	C₂⁵×D₁₃	1	40	yes	—
216	C₂	—	2	20	no	—
217	C₂	—	13	8	no	—
218	C₂	—	26	20	no	—
219	C₂	C₂×D₁₃×2₋¹⁺⁴	1	6	yes	direct product factor
220	C₂	C₂⁶×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₁₃
commutator, cosocle	C₂×C₁₃	C₂⁷
Frattini	C₂	C₂⁶×D₁₃
Fitting	C₂²×C₁₃×2₋¹⁺⁴	C₂
socle	C₂³×C₁₃	C₂⁵
hypercenter	C₂²×2₋¹⁺⁴	D₁₃
hypocenter	C₁₃	C₂³×2₋¹⁺⁴
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂²×#88254	C₂	semidirect
2. C₂²×#88257	C₂	semidirect
3. C₂×D₁₃×2₋¹⁺⁴	C₂	direct (N × H)
4. C₂²×#41×D₁₃	C₂	semidirect
5. C₂³×Q₈×D₁₃	C₂	semidirect
6. C₂²×C₁₃×2₋¹⁺⁴	C₂	semidirect
7. C₂×C₁₃×2₋¹⁺⁴	C₂²	semidirect
8. C₂²×C₁₃×#41	C₂²	semidirect
9. C₂³×Q₈×C₁₃	C₂²	semidirect
10. D₁₃×2₋¹⁺⁴	C₂²	direct (N × H)
11. C₂×#88257	C₂²	semidirect
12. C₂×#41×D₁₃	C₂²	semidirect
13. C₂×#88254	C₂²	semidirect
14. C₂²×#6131	C₂²	semidirect
16. C₂²×Q₈×D₁₃	C₂²	semidirect
17. C₂²×Q₈×D₁₃	C₂²	semidirect
18. C₂²×#6129	C₂²	semidirect
20. C₂²×#6127	C₂²	semidirect
22. C₂³×Dic₂₆	C₂²	semidirect
24. C₁₃×2₋¹⁺⁴	C₂³	semidirect
25. C₂×C₁₃×#41	C₂³	semidirect
27. C₂²×Q₈×C₁₃	C₂³	semidirect
32. #88257	C₂³	semidirect
33. #41×D₁₃	C₂³	semidirect
34. #88254	C₂³	semidirect
35. C₂×#6131	C₂³	semidirect
37. C₂×Q₈×D₁₃	C₂³	semidirect
38. C₂×Q₈×D₁₃	C₂³	semidirect
39. C₂×#6129	C₂³	semidirect
41. C₂×#6127	C₂³	semidirect
47. C₂²×Dic₂₆	C₂³	semidirect
54. C₁₃×#41	C₂⁴	semidirect
56. C₂×Q₈×C₁₃	C₂⁴	semidirect
63. #6131	C₂⁴	semidirect
65. Q₈×D₁₃	C₂⁴	semidirect
66. #6129	C₂⁴	semidirect
68. #6127	C₂⁴	semidirect
74. C₂×Dic₂₆	C₂⁴	semidirect
78. C₂²×2₋¹⁺⁴	D₁₃	direct (N × H)
86. C₂²×D₁₃	2₋¹⁺⁴	direct (N × H)
104. C₂×2₋¹⁺⁴	C₂×D₁₃	direct (N × H)
109. C₂²×#41	C₂×D₁₃	semidirect
114. C₂³×Q₈	C₂×D₁₃	semidirect
118. C₂²×C₁₃	C₂×2₋¹⁺⁴	semidirect
123. C₂×D₁₃	C₂×2₋¹⁺⁴	direct (N × H)
131. 2₋¹⁺⁴	C₂²×D₁₃	direct (N × H)
136. C₂×#41	C₂²×D₁₃	semidirect
143. C₂²×Q₈	C₂²×D₁₃	semidirect
154. C₂×C₁₃	C₂²×2₋¹⁺⁴	semidirect
157. D₁₃	C₂²×2₋¹⁺⁴	direct (N × H)
166. #41	C₂³×D₁₃	semidirect
172. C₂×Q₈	C₂³×D₁₃	semidirect
184. C₁₃	C₂³×2₋¹⁺⁴	semidirect
211. C₂²	D₁₃×2₋¹⁺⁴	direct (N × H)
219. C₂	C₂×D₁₃×2₋¹⁺⁴	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×#88254	2
C₂²×#88257	2
C₂×D₁₃×2₋¹⁺⁴	2
C₂²×#41×D₁₃	2
C₂³×Q₈×D₁₃	2
C₂²×C₁₃×2₋¹⁺⁴	2
C₂³×2₋¹⁺⁴	13

Subgroups — C22×D13×2−1+4 #16776807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₁₃×2₋¹⁺⁴ #16776807