Subgroups — C₂×#11719384 #13128337

Order 1664.

1328 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88200	C₂	1	2	yes	maximal, complemented
3	C₂×#88191	C₂	1	1	yes	maximal, characteristic
4	C₂×#88189	C₂	1	1	yes	maximal, characteristic, complemented
5	#11719384	C₂	1	16	yes	maximal, direct product factor
6	C₂×#88137	C₂	1	2	yes	maximal, complemented
7	C₂×#88135	C₂	1	2	yes	maximal, complemented
8	C₂×C₂²⋊C₄×D₁₃	C₂	1	2	yes	maximal, complemented
9	C₂×#88133	C₂	1	2	yes	maximal, complemented
10	C₂×#88126	C₂	1	2	yes	maximal, complemented
11	C₂⁴×D₁₃	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₁₃	C₂²	1	2	yes	—
15	C₂³×Dic₁₃	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
17	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
18	C₁₃×#124	C₂²	1	8	yes	complemented
19	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
21	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₂²⋊Dic₁₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊Dic₁₃	C₂²	1	1	yes	characteristic, complemented
25	#88200	C₂²	1	16	yes	complemented
26	#88191	C₂²	1	8	yes	—
27	#88189	C₂²	1	8	yes	complemented
28	C₂×#6103	C₂²	1	2	yes	—
29	C₂×#6103	C₂²	1	2	yes	—
30	C₂×#6103	C₂²	1	2	yes	—
31	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	complemented
32	C₂×#6101	C₂²	1	2	yes	complemented
33	C₂×#6101	C₂²	1	2	yes	complemented
34	C₂×#6101	C₂²	1	2	yes	complemented
35	C₂×C₄×Dic₁₃	C₂²	1	2	yes	complemented
36	#88137	C₂²	1	16	yes	complemented
37	#88135	C₂²	1	16	yes	complemented
38	C₂²⋊C₄×D₁₃	C₂²	1	16	yes	complemented
39	#88133	C₂²	1	16	yes	complemented
40	#88126	C₂²	1	16	yes	complemented
41	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
42	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
43	C₂³×D₁₃	—	2	8	no	—
44	C₂³×D₁₃	—	4	2	no	—
45	C₂³×D₁₃	C₂³	1	4	yes	—
46	C₂³×D₁₃	C₂³	1	2	yes	—
47	C₂×Q₈×C₁₃	—	4	2	no	—
48	C₂×Q₈×C₁₃	C₂³	1	4	yes	complemented
49	C₂×D₄×C₁₃	—	4	2	no	—
50	C₂×D₄×C₁₃	C₂³	1	4	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
54	C₂×C₂²⋊D₁₃	—	2	8	no	—
55	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
56	C₂²×Dic₁₃	—	4	2	no	—
57	C₂²×Dic₁₃	C₂³	1	2	yes	—
58	C₂²×Dic₁₃	C₂³	1	2	yes	—
59	C₂²×Dic₁₃	C₂³	1	2	yes	—
60	C₂²×Dic₁₃	C₂³	1	4	yes	complemented
61	C₂×C₄×D₁₃	—	2	8	no	—
62	C₂×C₄×D₁₃	—	4	4	no	—
63	C₂×C₄×D₁₃	C₂³	1	8	yes	—
64	C₂×C₄×D₁₃	C₂³	1	8	yes	—
65	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
66	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	complemented
67	C₄²×C₁₃	C₂³	1	4	yes	complemented
68	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
69	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
70	C₂²⋊Dic₁₃	C₂³	1	4	yes	complemented
71	#6103	C₂³	1	8	yes	—
72	#6103	C₂³	1	8	yes	—
73	#6103	C₂³	1	8	yes	—
74	C₄⋊Dic₁₃	C₂³	1	8	yes	complemented
75	#6101	C₂³	1	8	yes	complemented
76	#6101	C₂³	1	8	yes	complemented
77	#6101	C₂³	1	8	yes	complemented
78	C₄×Dic₁₃	C₂³	1	8	yes	complemented
79	C₂×#551	—	13	1	no	maximal, Sylow
80	C₂³×C₁₃	—	2	4	no	—
81	C₂³×C₁₃	—	2	4	no	—
82	C₂³×C₁₃	—	2	2	no	—
83	C₂³×C₁₃	—	2	2	no	—
84	C₂³×C₁₃	C₂⁴	1	2	yes	—
85	C₂³×C₁₃	C₂⁴	1	2	yes	—
86	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₂²×D₁₃	—	2	4	no	—
88	C₂²×D₁₃	—	2	4	no	—
89	C₂²×D₁₃	—	2	16	no	—
90	C₂²×D₁₃	—	4	4	no	—
91	C₂²×D₁₃	—	4	16	no	—
92	C₂²×D₁₃	#41	1	8	yes	—
93	C₂²×D₁₃	C₂⁴	1	4	yes	—
94	Q₈×C₁₃	—	4	4	no	—
95	D₄×C₁₃	—	4	4	no	—
96	C₂×C₄×C₁₃	—	2	4	no	—
97	C₂×C₄×C₁₃	—	2	4	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
100	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
101	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
102	C₂²⋊D₁₃	—	2	16	no	—
103	C₂×Dic₁₃	—	2	4	no	—
104	C₂×Dic₁₃	—	2	4	no	—
105	C₂×Dic₁₃	—	2	4	no	—
106	C₂×Dic₁₃	—	4	4	no	—
107	C₂×Dic₁₃	C₂⁴	1	4	yes	—
108	C₂×Dic₁₃	C₂⁴	1	4	yes	—
109	C₂×Dic₁₃	C₂⁴	1	4	yes	—
110	C₄×D₁₃	—	2	16	no	—
111	C₄×D₁₃	—	4	8	no	—
112	#551	—	13	16	no	—
113	C₂×#126	—	13	2	no	—
114	C₂×#124	D₁₃	1	1	yes	characteristic, complemented
115	C₂×#123	—	13	1	no	—
116	C₂×#123	—	13	2	no	—
117	C₂×C₂²⋊Q₈	—	13	2	no	—
118	C₂×#120	—	13	1	no	—
119	C₂×C₄×D₄	—	13	2	no	—
120	C₂×#117	—	13	2	no	—
121	C₂²×C₂²⋊C₄	—	13	2	no	—
122	C₂²×C₁₃	—	2	4	no	—
123	C₂²×C₁₃	—	2	4	no	—
124	C₂²×C₁₃	—	2	2	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	4	4	no	—
127	C₂²×C₁₃	—	4	4	no	—
128	C₂²×C₁₃	C₂×#41	1	4	yes	—
129	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
130	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
131	C₂×D₁₃	—	2	16	no	—
132	C₂×D₁₃	—	2	4	no	—
133	C₂×D₁₃	—	2	4	no	—
134	C₂×D₁₃	—	4	16	no	—
135	C₂×D₁₃	C₂×#41	1	8	yes	—
136	C₄×C₁₃	—	2	4	no	—
137	C₄×C₁₃	—	2	4	no	—
138	C₄×C₁₃	—	2	2	no	—
139	Dic₁₃	—	2	4	no	—
140	Dic₁₃	—	2	4	no	—
141	Dic₁₃	—	2	4	no	—
142	C₂⁵	—	13	1	no	—
143	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
144	C₂²×D₄	—	13	2	no	—
145	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
146	C₂³×C₄	—	13	1	no	—
147	C₂³×C₄	—	13	2	no	—
148	C₂³×C₄	—	13	2	no	—
149	#126	—	13	16	no	—
150	#124	C₂×D₁₃	1	8	yes	complemented
151	#123	—	13	8	no	—
152	#123	—	13	16	no	—
153	C₂²⋊Q₈	—	13	16	no	—
154	#120	—	13	8	no	—
155	C₄×D₄	—	13	16	no	—
156	#117	—	13	16	no	—
157	C₂×C₄⋊C₄	—	13	2	no	—
158	C₂×C₄⋊C₄	—	13	2	no	—
159	C₂×C₄⋊C₄	—	13	2	no	—
160	C₂×C₄⋊C₄	—	13	2	no	—
161	C₂×C₂²⋊C₄	—	13	2	no	—
162	C₂×C₂²⋊C₄	—	13	1	no	—
163	C₂×C₂²⋊C₄	—	13	1	no	—
164	C₂×C₂²⋊C₄	—	13	2	no	—
165	C₂×C₂²⋊C₄	—	13	2	no	—
166	C₂×C₂²⋊C₄	—	13	2	no	—
167	C₂×C₂²⋊C₄	—	13	16	no	—
168	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
169	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	complemented
170	C₂×C₄²	—	13	2	no	—
171	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic, complemented
172	C₂×C₁₃	—	4	2	no	—
173	C₂×C₁₃	—	4	2	no	—
174	C₂×C₁₃	#551	1	4	yes	complemented
175	C₂×C₁₃	C₂²×#41	1	2	yes	—
176	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
177	D₁₃	—	2	8	no	—
178	C₂⁴	—	13	2	no	—
179	C₂⁴	—	13	4	no	—
180	C₂⁴	—	26	8	no	—
181	C₂⁴	—	52	2	no	—
182	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
183	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	C₂²×D₁₃	1	4	yes	complemented
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	13	8	no	—
188	C₂×D₄	—	26	8	no	—
189	C₂×D₄	C₂²×D₁₃	1	4	yes	—
190	C₂²×C₄	—	13	2	no	—
191	C₂²×C₄	—	13	2	no	—
192	C₂²×C₄	—	13	2	no	—
193	C₂²×C₄	—	13	4	no	—
194	C₂²×C₄	—	13	8	no	—
195	C₂²×C₄	—	13	8	no	—
196	C₂²×C₄	—	26	8	no	—
197	C₂²×C₄	—	52	4	no	—
198	C₂²×C₄	—	52	2	no	—
199	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
200	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
202	C₄⋊C₄	—	13	8	no	—
203	C₄⋊C₄	—	13	8	no	—
204	C₄⋊C₄	—	13	8	no	—
205	C₄⋊C₄	—	13	8	no	—
206	C₂²⋊C₄	—	13	8	no	—
207	C₂²⋊C₄	—	13	4	no	—
208	C₂²⋊C₄	—	13	4	no	—
209	C₂²⋊C₄	—	13	8	no	—
210	C₂²⋊C₄	—	13	8	no	—
211	C₂²⋊C₄	—	13	8	no	—
212	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
213	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	complemented
214	C₄²	—	13	8	no	—
215	C₄²	C₂²×D₁₃	1	4	yes	complemented
216	C₁₃	C₂×#551	1	1	yes	characteristic, complemented, Sylow
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	13	8	no	—
222	C₂³	—	13	4	no	—
223	C₂³	—	26	16	no	—
224	C₂³	—	26	4	no	—
225	C₂³	—	26	4	no	—
226	C₂³	—	52	4	no	—
227	C₂³	—	52	16	no	—
228	C₂³	C₂³×D₁₃	1	2	yes	—
229	C₂³	C₂³×D₁₃	1	2	yes	—
230	C₂³	C₂³×D₁₃	1	1	yes	characteristic
231	Q₈	—	4	4	no	—
232	D₄	—	4	4	no	—
233	D₄	—	26	16	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	13	4	no	—
238	C₂×C₄	—	13	4	no	—
239	C₂×C₄	—	13	4	no	—
240	C₂×C₄	—	26	16	no	—
241	C₂×C₄	—	26	4	no	—
242	C₂×C₄	—	26	4	no	—
243	C₂×C₄	—	26	4	no	—
244	C₂×C₄	—	52	8	no	—
245	C₂×C₄	—	52	4	no	—
246	C₂×C₄	C₂³×D₁₃	1	4	yes	—
247	C₂×C₄	C₂³×D₁₃	1	2	yes	—
248	C₂×C₄	C₂³×D₁₃	1	4	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	13	8	no	—
256	C₂²	—	26	16	no	—
257	C₂²	—	26	4	no	—
258	C₂²	—	26	4	no	—
259	C₂²	—	52	16	no	—
260	C₂²	#88251	1	2	yes	—
261	C₂²	#41×D₁₃	1	4	yes	—
262	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	4	no	—
266	C₄	—	26	4	no	—
267	C₄	—	26	4	no	—
268	C₄	—	26	4	no	—
269	C₂	—	4	2	no	—
270	C₂	—	4	2	no	—
271	C₂	—	26	8	no	—
272	C₂	#11719384	1	4	yes	direct product factor
273	C₂	C₂×#88251	1	1	yes	characteristic
274	C₂	C₂×#41×D₁₃	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#124	C₂
socle	C₂³×C₁₃	C₂⁴
prime radical	C₂×#88191	C₂
hypercenter	C₂×#124	D₁₃
hypocenter	C₁₃	C₂×#551
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#124	C₂	semidirect
2. C₂×#88200	C₂	semidirect
4. C₂×#88189	C₂	semidirect
5. #11719384	C₂	direct (N × H)
6. C₂×#88137	C₂	semidirect
7. C₂×#88135	C₂	semidirect
8. C₂×C₂²⋊C₄×D₁₃	C₂	semidirect
9. C₂×#88133	C₂	semidirect
10. C₂×#88126	C₂	semidirect
12. C₂²×Q₈×C₁₃	C₂²	semidirect
15. C₂³×Dic₁₃	C₂²	semidirect
18. C₁₃×#124	C₂²	semidirect
20. C₂×C₁₃×C₂²⋊C₄	C₂²	semidirect
21. C₂×C₄²×C₁₃	C₂²	semidirect
24. C₂×C₂²⋊Dic₁₃	C₂²	semidirect
25. #88200	C₂²	semidirect
27. #88189	C₂²	semidirect
31. C₂×C₄⋊Dic₁₃	C₂²	semidirect
32. C₂×#6101	C₂²	semidirect
33. C₂×#6101	C₂²	semidirect
34. C₂×#6101	C₂²	semidirect
35. C₂×C₄×Dic₁₃	C₂²	semidirect
36. #88137	C₂²	semidirect
37. #88135	C₂²	semidirect
38. C₂²⋊C₄×D₁₃	C₂²	semidirect
39. #88133	C₂²	semidirect
40. #88126	C₂²	semidirect
48. C₂×Q₈×C₁₃	C₂³	semidirect
60. C₂²×Dic₁₃	C₂³	semidirect
66. C₁₃×C₂²⋊C₄	C₂³	semidirect
67. C₄²×C₁₃	C₂³	semidirect
70. C₂²⋊Dic₁₃	C₂³	semidirect
74. C₄⋊Dic₁₃	C₂³	semidirect
75. #6101	C₂³	semidirect
76. #6101	C₂³	semidirect
77. #6101	C₂³	semidirect
78. C₄×Dic₁₃	C₂³	semidirect
114. C₂×#124	D₁₃	semidirect
143. C₂²×Q₈	C₂×D₁₃	semidirect
150. #124	C₂×D₁₃	semidirect
169. C₂×C₂²⋊C₄	C₂×D₁₃	semidirect
171. C₂×C₄²	C₂×D₁₃	semidirect
174. C₂×C₁₃	#551	semidirect
185. C₂×Q₈	C₂²×D₁₃	semidirect
213. C₂²⋊C₄	C₂²×D₁₃	semidirect
215. C₄²	C₂²×D₁₃	semidirect
216. C₁₃	C₂×#551	semidirect
272. C₂	#11719384	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×C₁₃×#124	2
C₂×#88200	2
C₂×#88191	2
C₂×#88189	2
#11719384	2
C₂×#88137	2
C₂×#88135	2
C₂×C₂²⋊C₄×D₁₃	2
C₂×#88133	2
C₂×#88126	2
C₂×#551	13

Subgroups — C2×#11719384 #13128337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719384 #13128337