Subgroups — C₂²×D₁₃×#126 #16769588

Order 3328.

9486 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88152	C₂	1	3	yes	maximal, complemented
2	C₂²×#88130	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#88133	C₂	1	3	yes	maximal, complemented
4	C₂×D₁₃×#126	C₂	1	48	yes	maximal, direct product factor
5	C₂²×C₄⋊C₄×D₁₃	C₂	1	3	yes	maximal, complemented
6	C₂²×C₂²⋊C₄×D₁₃	C₂	1	3	yes	maximal
7	C₂²×C₄²×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×#126	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₁₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×D₁₃	C₂²	1	3	yes	—
11	C₂³×C₄×D₁₃	C₂²	1	3	yes	—
12	C₂×C₁₃×#126	C₂²	1	24	yes	complemented
13	C₂²×C₁₃×C₄⋊C₄	C₂²	1	3	yes	complemented
14	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
15	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₂²⋊Dic₁₃	C₂²	1	3	yes	—
17	C₂²×#6103	C₂²	1	3	yes	—
18	C₂²×#6103	C₂²	1	3	yes	—
19	C₂²×C₄⋊Dic₁₃	C₂²	1	3	yes	complemented
20	C₂²×#6101	C₂²	1	3	yes	complemented
21	C₂²×#6101	C₂²	1	3	yes	complemented
22	C₂²×C₄×Dic₁₃	C₂²	1	3	yes	complemented
23	D₁₃×#126	C₂²	1	256	yes	direct product factor
24	C₂×#88152	C₂²	1	72	yes	complemented
25	C₂×C₄⋊C₄×D₁₃	C₂²	1	72	yes	complemented
26	C₂×C₂²⋊C₄×D₁₃	C₂²	1	72	yes	—
27	C₂×#88133	C₂²	1	72	yes	complemented
28	C₂×#88130	C₂²	1	24	yes	complemented
29	C₂×C₄²×D₁₃	C₂²	1	24	yes	complemented
30	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
31	C₂⁴×D₁₃	—	2	24	no	—
32	C₂⁴×D₁₃	C₂³	1	12	yes	—
33	C₂⁴×D₁₃	C₂³	1	1	yes	characteristic
34	C₂⁴×D₁₃	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
36	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
37	C₂³×Dic₁₃	C₂³	1	3	yes	—
38	C₂³×Dic₁₃	C₂³	1	3	yes	—
39	C₂²×C₄×D₁₃	—	2	24	no	—
40	C₂²×C₄×D₁₃	—	2	24	no	—
41	C₂²×C₄×D₁₃	C₂³	1	36	yes	—
42	C₂²×C₄×D₁₃	C₂³	1	36	yes	—
43	C₁₃×#126	C₂³	1	64	yes	complemented
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	36	yes	complemented
45	C₂×C₁₃×C₂²⋊C₄	C₂³	1	36	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	12	yes	complemented
47	C₂×C₂²⋊Dic₁₃	C₂³	1	36	yes	—
48	C₂×#6103	C₂³	1	36	yes	—
49	C₂×#6103	C₂³	1	36	yes	—
50	C₂×C₄⋊Dic₁₃	C₂³	1	36	yes	complemented
51	C₂×#6101	C₂³	1	36	yes	complemented
52	C₂×#6101	C₂³	1	36	yes	complemented
53	C₂×C₄×Dic₁₃	C₂³	1	36	yes	complemented
54	#88152	C₂³	1	192	yes	complemented
55	C₄⋊C₄×D₁₃	C₂³	1	192	yes	complemented
56	C₂²⋊C₄×D₁₃	C₂³	1	192	yes	—
57	#88133	C₂³	1	192	yes	complemented
58	#88130	C₂³	1	64	yes	complemented
59	C₄²×D₁₃	C₂³	1	64	yes	complemented
60	C₂³×#126	—	13	1	no	maximal, Sylow
61	C₂⁴×C₁₃	—	2	12	no	—
62	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
63	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₂³×D₁₃	—	2	12	no	—
65	C₂³×D₁₃	—	2	144	no	—
66	C₂³×D₁₃	—	4	64	no	—
67	C₂³×D₁₃	#41	1	24	yes	—
68	C₂³×D₁₃	C₂⁴	1	16	yes	—
69	C₂³×D₁₃	C₂⁴	1	6	yes	—
70	C₂³×D₁₃	C₂⁴	1	6	yes	—
71	C₂²×C₄×C₁₃	—	2	12	no	—
72	C₂²×C₄×C₁₃	—	2	12	no	—
73	C₂²×C₄×C₁₃	C₂⁴	1	18	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	18	yes	—
75	C₂²×Dic₁₃	—	2	12	no	—
76	C₂²×Dic₁₃	—	2	12	no	—
77	C₂²×Dic₁₃	C₂⁴	1	18	yes	—
78	C₂²×Dic₁₃	C₂⁴	1	18	yes	—
79	C₂×C₄×D₁₃	—	2	144	no	—
80	C₂×C₄×D₁₃	—	2	144	no	—
81	C₂×C₄×D₁₃	C₂⁴	1	48	yes	—
82	C₂×C₄×D₁₃	C₂⁴	1	48	yes	—
83	C₁₃×C₄⋊C₄	C₂⁴	1	48	yes	complemented
84	C₁₃×C₂²⋊C₄	C₂⁴	1	48	yes	—
85	C₄²×C₁₃	C₂⁴	1	16	yes	complemented
86	C₂²⋊Dic₁₃	C₂⁴	1	48	yes	—
87	#6103	C₂⁴	1	48	yes	—
88	#6103	C₂⁴	1	48	yes	—
89	C₄⋊Dic₁₃	C₂⁴	1	48	yes	complemented
90	#6101	C₂⁴	1	48	yes	complemented
91	#6101	C₂⁴	1	48	yes	complemented
92	C₄×Dic₁₃	C₂⁴	1	48	yes	complemented
93	C₂²×#126	—	13	1	no	—
94	C₂²×#126	—	13	48	no	—
95	C₂²×#126	—	13	3	no	—
96	C₂²×#126	—	13	3	no	—
97	C₂²×#126	D₁₃	1	1	yes	characteristic, direct product factor
98	C₂³×C₄⋊C₄	—	13	3	no	—
99	C₂³×C₂²⋊C₄	—	13	3	no	—
100	C₂³×C₄²	—	13	1	no	—
101	C₂³×C₁₃	—	2	36	no	—
102	C₂³×C₁₃	—	4	16	no	—
103	C₂³×C₁₃	C₂×#41	1	12	yes	—
104	C₂³×C₁₃	C₂⁵	1	4	yes	—
105	C₂³×C₁₃	C₂⁵	1	3	yes	—
106	C₂²×D₁₃	—	2	36	no	—
107	C₂²×D₁₃	—	2	96	no	—
108	C₂²×D₁₃	—	4	16	no	—
109	C₂²×D₁₃	—	4	192	no	—
110	C₂²×D₁₃	#126	1	64	yes	direct product factor
111	C₂²×D₁₃	C₂×#41	1	72	yes	—
112	C₂²×D₁₃	C₂⁵	1	4	yes	—
113	C₂²×D₁₃	C₂⁵	1	4	yes	—
114	C₂×C₄×C₁₃	—	2	36	no	—
115	C₂×C₄×C₁₃	—	2	36	no	—
116	C₂×C₄×C₁₃	C₂⁵	1	12	yes	—
117	C₂×C₄×C₁₃	C₂⁵	1	12	yes	—
118	C₂×Dic₁₃	—	2	36	no	—
119	C₂×Dic₁₃	—	2	36	no	—
120	C₂×Dic₁₃	C₂⁵	1	12	yes	—
121	C₂×Dic₁₃	C₂⁵	1	12	yes	—
122	C₄×D₁₃	—	2	96	no	—
123	C₄×D₁₃	—	2	96	no	—
124	C₂⁶	—	13	1	no	—
125	C₂⁴×C₄	—	13	3	no	—
126	C₂⁴×C₄	—	13	3	no	—
127	C₂×#126	—	13	256	no	—
128	C₂×#126	—	13	72	no	—
129	C₂×#126	—	13	72	no	—
130	C₂×#126	—	13	24	no	—
131	C₂×#126	C₂×D₁₃	1	24	yes	direct product factor
132	C₂²×C₄⋊C₄	—	13	3	no	—
133	C₂²×C₄⋊C₄	—	13	3	no	—
134	C₂²×C₄⋊C₄	—	13	3	no	—
135	C₂²×C₄⋊C₄	—	13	72	no	—
136	C₂²×C₄⋊C₄	C₂×D₁₃	1	3	yes	complemented
137	C₂²×C₂²⋊C₄	—	13	3	no	—
138	C₂²×C₂²⋊C₄	—	13	3	no	—
139	C₂²×C₂²⋊C₄	—	13	3	no	—
140	C₂²×C₂²⋊C₄	—	13	72	no	—
141	C₂²×C₂²⋊C₄	C₂×D₁₃	1	3	yes	—
142	C₂²×C₄²	—	13	3	no	—
143	C₂²×C₄²	—	13	24	no	—
144	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic, complemented
145	C₂²×C₁₃	—	2	12	no	—
146	C₂²×C₁₃	—	4	24	no	—
147	C₂²×C₁₃	C₂×#126	1	16	yes	complemented
148	C₂²×C₁₃	C₂²×#41	1	18	yes	—
149	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
150	C₂×D₁₃	—	2	12	no	—
151	C₂×D₁₃	—	4	24	no	—
152	C₂×D₁₃	—	4	64	no	—
153	C₂×D₁₃	C₂×#126	1	96	yes	direct product factor
154	C₂×D₁₃	C₂²×#41	1	24	yes	—
155	C₄×C₁₃	—	2	12	no	—
156	C₄×C₁₃	—	2	12	no	—
157	Dic₁₃	—	2	12	no	—
158	Dic₁₃	—	2	12	no	—
159	C₂⁵	—	13	1	no	—
160	C₂⁵	—	13	12	no	—
161	C₂⁵	—	13	1	no	—
162	C₂⁵	—	26	24	no	—
163	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
164	C₂³×C₄	—	13	3	no	—
165	C₂³×C₄	—	13	3	no	—
166	C₂³×C₄	—	13	36	no	—
167	C₂³×C₄	—	13	36	no	—
168	C₂³×C₄	—	26	24	no	—
169	C₂³×C₄	—	26	24	no	—
170	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
171	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
172	#126	—	13	192	no	—
173	#126	—	13	192	no	—
174	#126	—	13	64	no	—
175	#126	C₂²×D₁₃	1	64	yes	direct product factor
176	C₂×C₄⋊C₄	—	13	36	no	—
177	C₂×C₄⋊C₄	—	13	36	no	—
178	C₂×C₄⋊C₄	—	13	36	no	—
179	C₂×C₄⋊C₄	—	13	192	no	—
180	C₂×C₄⋊C₄	C₂²×D₁₃	1	36	yes	complemented
181	C₂×C₂²⋊C₄	—	13	36	no	—
182	C₂×C₂²⋊C₄	—	13	36	no	—
183	C₂×C₂²⋊C₄	—	13	36	no	—
184	C₂×C₂²⋊C₄	—	13	192	no	—
185	C₂×C₂²⋊C₄	C₂²×D₁₃	1	36	yes	—
186	C₂×C₄²	—	13	36	no	—
187	C₂×C₄²	—	13	64	no	—
188	C₂×C₄²	C₂²×D₁₃	1	12	yes	complemented
189	C₂×C₁₃	—	4	4	no	—
190	C₂×C₁₃	C₂²×#126	1	12	yes	complemented
191	C₂×C₁₃	C₂³×#41	1	3	yes	—
192	D₁₃	—	4	4	no	—
193	D₁₃	C₂²×#126	1	16	yes	direct product factor
194	C₂⁴	—	2	12	no	—
195	C₂⁴	—	13	24	no	—
196	C₂⁴	—	13	6	no	—
197	C₂⁴	—	13	16	no	—
198	C₂⁴	—	13	6	no	—
199	C₂⁴	—	26	144	no	—
200	C₂⁴	—	26	12	no	—
201	C₂⁴	—	52	64	no	—
202	C₂⁴	C₂³×D₁₃	1	6	yes	—
203	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
204	C₂²×C₄	—	2	12	no	—
205	C₂²×C₄	—	2	12	no	—
206	C₂²×C₄	—	13	18	no	—
207	C₂²×C₄	—	13	18	no	—
208	C₂²×C₄	—	13	48	no	—
209	C₂²×C₄	—	13	48	no	—
210	C₂²×C₄	—	26	144	no	—
211	C₂²×C₄	—	26	144	no	—
212	C₂²×C₄	—	26	12	no	—
213	C₂²×C₄	—	26	12	no	—
214	C₂²×C₄	C₂³×D₁₃	1	18	yes	—
215	C₂²×C₄	C₂³×D₁₃	1	18	yes	—
216	C₄⋊C₄	—	13	48	no	—
217	C₄⋊C₄	—	13	48	no	—
218	C₄⋊C₄	—	13	48	no	—
219	C₄⋊C₄	C₂³×D₁₃	1	48	yes	complemented
220	C₂²⋊C₄	—	13	48	no	—
221	C₂²⋊C₄	—	13	48	no	—
222	C₂²⋊C₄	—	13	48	no	—
223	C₂²⋊C₄	C₂³×D₁₃	1	48	yes	—
224	C₄²	—	13	48	no	—
225	C₄²	C₂³×D₁₃	1	16	yes	complemented
226	C₁₃	C₂³×#126	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	2	36	no	—
228	C₂³	—	4	16	no	—
229	C₂³	—	13	72	no	—
230	C₂³	—	13	4	no	—
231	C₂³	—	13	64	no	—
232	C₂³	—	13	4	no	—
233	C₂³	—	26	96	no	—
234	C₂³	—	26	36	no	—
235	C₂³	—	52	192	no	—
236	C₂³	—	52	16	no	—
237	C₂³	#41×D₁₃	1	12	yes	—
238	C₂³	C₂⁴×D₁₃	1	4	yes	—
239	C₂³	C₂⁴×D₁₃	1	3	yes	—
240	C₂×C₄	—	2	36	no	—
241	C₂×C₄	—	2	36	no	—
242	C₂×C₄	—	13	12	no	—
243	C₂×C₄	—	13	12	no	—
244	C₂×C₄	—	26	96	no	—
245	C₂×C₄	—	26	96	no	—
246	C₂×C₄	—	26	36	no	—
247	C₂×C₄	—	26	36	no	—
248	C₂×C₄	C₂⁴×D₁₃	1	12	yes	—
249	C₂×C₄	C₂⁴×D₁₃	1	12	yes	—
250	C₂²	—	2	12	no	—
251	C₂²	—	4	24	no	—
252	C₂²	—	13	96	no	—
253	C₂²	—	13	24	no	—
254	C₂²	—	26	12	no	—
255	C₂²	—	52	64	no	—
256	C₂²	—	52	24	no	—
257	C₂²	D₁₃×#126	1	16	yes	direct product factor
258	C₂²	C₂×#41×D₁₃	1	18	yes	—
259	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
260	C₄	—	2	12	no	—
261	C₄	—	2	12	no	—
262	C₄	—	26	12	no	—
263	C₄	—	26	12	no	—
264	C₂	—	4	4	no	—
265	C₂	—	13	16	no	—
266	C₂	—	52	4	no	—
267	C₂	C₂×D₁₃×#126	1	12	yes	direct product factor
268	C₂	C₂²×#41×D₁₃	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂²×C₁₃×#126	C₂
socle	C₂⁴×C₁₃	C₂⁴
prime radical	C₂⁵×D₁₃	C₂²
hypercenter	C₂²×#126	D₁₃
hypocenter	C₁₃	C₂³×#126
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#126

Complements

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

N	H ≅ G/N	type
1. C₂²×#88152	C₂	semidirect
2. C₂²×#88130	C₂	semidirect
3. C₂²×#88133	C₂	semidirect
4. C₂×D₁₃×#126	C₂	direct (N × H)
5. C₂²×C₄⋊C₄×D₁₃	C₂	semidirect
7. C₂²×C₄²×D₁₃	C₂	semidirect
8. C₂²×C₁₃×#126	C₂	semidirect
12. C₂×C₁₃×#126	C₂²	semidirect
13. C₂²×C₁₃×C₄⋊C₄	C₂²	semidirect
15. C₂²×C₄²×C₁₃	C₂²	semidirect
19. C₂²×C₄⋊Dic₁₃	C₂²	semidirect
20. C₂²×#6101	C₂²	semidirect
21. C₂²×#6101	C₂²	semidirect
22. C₂²×C₄×Dic₁₃	C₂²	semidirect
23. D₁₃×#126	C₂²	direct (N × H)
24. C₂×#88152	C₂²	semidirect
25. C₂×C₄⋊C₄×D₁₃	C₂²	semidirect
27. C₂×#88133	C₂²	semidirect
28. C₂×#88130	C₂²	semidirect
29. C₂×C₄²×D₁₃	C₂²	semidirect
43. C₁₃×#126	C₂³	semidirect
44. C₂×C₁₃×C₄⋊C₄	C₂³	semidirect
46. C₂×C₄²×C₁₃	C₂³	semidirect
50. C₂×C₄⋊Dic₁₃	C₂³	semidirect
51. C₂×#6101	C₂³	semidirect
52. C₂×#6101	C₂³	semidirect
53. C₂×C₄×Dic₁₃	C₂³	semidirect
54. #88152	C₂³	semidirect
55. C₄⋊C₄×D₁₃	C₂³	semidirect
57. #88133	C₂³	semidirect
58. #88130	C₂³	semidirect
59. C₄²×D₁₃	C₂³	semidirect
83. C₁₃×C₄⋊C₄	C₂⁴	semidirect
85. C₄²×C₁₃	C₂⁴	semidirect
89. C₄⋊Dic₁₃	C₂⁴	semidirect
90. #6101	C₂⁴	semidirect
91. #6101	C₂⁴	semidirect
92. C₄×Dic₁₃	C₂⁴	semidirect
97. C₂²×#126	D₁₃	direct (N × H)
110. C₂²×D₁₃	#126	direct (N × H)
131. C₂×#126	C₂×D₁₃	direct (N × H)
136. C₂²×C₄⋊C₄	C₂×D₁₃	semidirect
144. C₂²×C₄²	C₂×D₁₃	semidirect
147. C₂²×C₁₃	C₂×#126	semidirect
153. C₂×D₁₃	C₂×#126	direct (N × H)
175. #126	C₂²×D₁₃	direct (N × H)
180. C₂×C₄⋊C₄	C₂²×D₁₃	semidirect
188. C₂×C₄²	C₂²×D₁₃	semidirect
190. C₂×C₁₃	C₂²×#126	semidirect
193. D₁₃	C₂²×#126	direct (N × H)
219. C₄⋊C₄	C₂³×D₁₃	semidirect
225. C₄²	C₂³×D₁₃	semidirect
226. C₁₃	C₂³×#126	semidirect
257. C₂²	D₁₃×#126	direct (N × H)
267. C₂	C₂×D₁₃×#126	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×#88152	2
C₂²×#88130	2
C₂²×#88133	2
C₂×D₁₃×#126	2
C₂²×C₄⋊C₄×D₁₃	2
C₂²×C₂²⋊C₄×D₁₃	2
C₂²×C₄²×D₁₃	2
C₂²×C₁₃×#126	2
C₂³×#126	13

Subgroups — C22×D13×#126 #16769588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₁₃×#126 #16769588