Subgroups — C₂×#11719292 #13128338

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×#120	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊Dic₁₃	C₂	1	2	yes	maximal, complemented
3	C₂×#88190	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88189	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#88189	C₂	1	2	yes	maximal, complemented
6	C₂×D₄×Dic₁₃	C₂	1	2	yes	maximal, complemented
7	#11719292	C₂	1	16	yes	maximal, direct product factor
8	C₂×#88133	C₂	1	2	yes	maximal, complemented
9	C₂×#88132	C₂	1	2	yes	maximal, complemented
10	C₂×#88131	C₂	1	2	yes	maximal, complemented
11	C₂⁵×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
13	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂³×Dic₁₃	C₂²	1	2	yes	—
15	C₂³×Dic₁₃	C₂²	1	2	yes	—
16	C₂³×Dic₁₃	C₂²	1	1	yes	characteristic, complemented
17	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic, complemented
18	C₁₃×#120	C₂²	1	8	yes	—
19	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
25	C₂×C₂²⋊Dic₁₃	C₂²	1	16	yes	complemented
26	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	complemented
27	C₂×C₂²⋊Dic₁₃	C₂²	1	1	yes	characteristic, complemented
28	#88190	C₂²	1	8	yes	complemented
29	#88189	C₂²	1	8	yes	complemented
30	#88189	C₂²	1	16	yes	complemented
31	D₄×Dic₁₃	C₂²	1	16	yes	complemented
32	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	complemented
33	C₂×#6101	C₂²	1	2	yes	complemented
34	C₂×#6101	C₂²	1	2	yes	complemented
35	C₂×#6101	C₂²	1	2	yes	complemented
36	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic, complemented
37	C₂×C₄×Dic₁₃	C₂²	1	2	yes	complemented
38	#88133	C₂²	1	16	yes	complemented
39	#88132	C₂²	1	16	yes	complemented
40	#88131	C₂²	1	16	yes	complemented
41	C₂⁴×C₁₃	—	2	8	no	—
42	C₂⁴×C₁₃	—	4	2	no	—
43	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
44	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
45	C₂⁴×C₁₃	C₂³	1	4	yes	—
46	C₂⁴×C₁₃	C₂³	1	2	yes	—
47	C₂×D₄×C₁₃	—	2	8	no	—
48	C₂×D₄×C₁₃	—	4	2	no	—
49	C₂×D₄×C₁₃	C₂³	1	8	yes	—
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×Dic₁₃	—	2	8	no	—
54	C₂²×Dic₁₃	—	4	4	no	—
55	C₂²×Dic₁₃	—	4	2	no	—
56	C₂²×Dic₁₃	C₂³	1	8	yes	—
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	2	yes	—
61	C₂²×Dic₁₃	C₂³	1	8	yes	—
62	C₂²×Dic₁₃	C₂³	1	4	yes	complemented
63	C₂×Dic₂₆	—	4	2	no	—
64	C₂×Dic₂₆	C₂³	1	4	yes	complemented
65	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
66	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
67	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
68	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
69	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
70	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
71	C₂²⋊Dic₁₃	C₂³	1	8	yes	complemented
72	C₂²⋊Dic₁₃	C₂³	1	4	yes	complemented
73	C₄⋊Dic₁₃	C₂³	1	8	yes	complemented
74	#6101	C₂³	1	8	yes	complemented
75	#6101	C₂³	1	8	yes	complemented
76	#6101	C₂³	1	8	yes	complemented
77	C₄×Dic₁₃	C₂³	1	4	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	4	no	—
83	C₂³×C₁₃	—	2	16	no	—
84	C₂³×C₁₃	—	2	4	no	—
85	C₂³×C₁₃	—	2	2	no	—
86	C₂³×C₁₃	—	2	2	no	—
87	C₂³×C₁₃	—	4	4	no	—
88	C₂³×C₁₃	—	4	16	no	—
89	C₂³×C₁₃	#41	1	8	yes	—
90	C₂³×C₁₃	C₂⁴	1	2	yes	—
91	C₂³×C₁₃	C₂⁴	1	2	yes	—
92	C₂³×C₁₃	C₂⁴	1	4	yes	—
93	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
94	D₄×C₁₃	—	2	16	no	—
95	D₄×C₁₃	—	4	4	no	—
96	C₂×C₄×C₁₃	—	2	4	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
99	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
100	C₂×Dic₁₃	—	2	16	no	—
101	C₂×Dic₁₃	—	2	4	no	—
102	C₂×Dic₁₃	—	2	4	no	—
103	C₂×Dic₁₃	—	2	4	no	—
104	C₂×Dic₁₃	—	2	4	no	—
105	C₂×Dic₁₃	—	4	8	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	C₂×Dic₁₃	C₂⁴	1	4	yes	—
111	Dic₂₆	—	4	4	no	—
112	#551	—	13	16	no	—
113	C₂×#126	—	13	2	no	—
114	C₂×#124	—	13	1	no	—
115	C₂×#123	—	13	2	no	—
116	C₂×#123	—	13	1	no	—
117	C₂×C₂²⋊Q₈	—	13	2	no	—
118	C₂×#120	D₁₃	1	1	yes	characteristic
119	C₂×C₄×D₄	—	13	2	no	—
120	C₂×#117	—	13	2	no	—
121	C₂²×C₂²⋊C₄	—	13	2	no	—
122	C₂²×C₁₃	—	2	16	no	—
123	C₂²×C₁₃	—	2	4	no	—
124	C₂²×C₁₃	—	2	4	no	—
125	C₂²×C₁₃	—	2	4	no	—
126	C₂²×C₁₃	—	2	4	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	4	16	no	—
130	C₂²×C₁₃	—	4	4	no	—
131	C₂²×C₁₃	—	4	4	no	—
132	C₂²×C₁₃	C₂×#41	1	8	yes	—
133	C₂²×C₁₃	C₂×#41	1	4	yes	—
134	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
135	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
136	C₄×C₁₃	—	2	4	no	—
137	C₄×C₁₃	—	2	2	no	—
138	Dic₁₃	—	2	4	no	—
139	Dic₁₃	—	2	4	no	—
140	Dic₁₃	—	2	4	no	—
141	Dic₁₃	—	2	4	no	—
142	C₂⁵	C₂×D₁₃	1	1	yes	characteristic
143	C₂²×Q₈	—	13	1	no	—
144	C₂²×D₄	C₂×D₁₃	1	2	yes	—
145	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
146	C₂³×C₄	—	13	2	no	—
147	C₂³×C₄	—	13	2	no	—
148	C₂³×C₄	—	13	1	no	—
149	#126	—	13	16	no	—
150	#124	—	13	8	no	—
151	#123	—	13	16	no	—
152	#123	—	13	8	no	—
153	C₂²⋊Q₈	—	13	16	no	—
154	#120	C₂×D₁₃	1	8	yes	—
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	2	no	—
163	C₂×C₂²⋊C₄	—	13	2	no	—
164	C₂×C₂²⋊C₄	—	13	2	no	—
165	C₂×C₂²⋊C₄	—	13	16	no	—
166	C₂×C₂²⋊C₄	—	13	2	no	—
167	C₂×C₂²⋊C₄	—	13	1	no	—
168	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
169	C₂×C₂²⋊C₄	C₂×D₁₃	1	1	yes	characteristic
170	C₂×C₄²	—	13	1	no	—
171	C₂×C₄²	—	13	2	no	—
172	C₂×C₁₃	—	2	8	no	—
173	C₂×C₁₃	—	4	2	no	—
174	C₂×C₁₃	—	4	2	no	—
175	C₂×C₁₃	#551	1	4	yes	complemented
176	C₂×C₁₃	C₂²×#41	1	2	yes	—
177	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
178	C₂⁴	—	2	8	no	—
179	C₂⁴	—	4	2	no	—
180	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
181	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
182	C₂⁴	C₂²×D₁₃	1	4	yes	—
183	C₂⁴	C₂²×D₁₃	1	2	yes	—
184	C₂×Q₈	—	13	4	no	—
185	C₂×Q₈	—	52	2	no	—
186	C₂×D₄	—	2	8	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	C₂²×D₁₃	1	8	yes	—
189	C₂×D₄	C₂²×D₁₃	1	4	yes	—
190	C₂²×C₄	—	13	8	no	—
191	C₂²×C₄	—	13	2	no	—
192	C₂²×C₄	—	13	2	no	—
193	C₂²×C₄	—	13	2	no	—
194	C₂²×C₄	—	13	2	no	—
195	C₂²×C₄	—	13	8	no	—
196	C₂²×C₄	—	13	4	no	—
197	C₂²×C₄	—	26	8	no	—
198	C₂²×C₄	—	52	2	no	—
199	C₂²×C₄	—	52	4	no	—
200	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
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	8	no	—
208	C₂²⋊C₄	—	13	8	no	—
209	C₂²⋊C₄	—	13	8	no	—
210	C₂²⋊C₄	—	13	8	no	—
211	C₂²⋊C₄	—	13	4	no	—
212	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
213	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
214	C₄²	—	13	4	no	—
215	C₄²	—	13	8	no	—
216	C₁₃	C₂×#551	1	1	yes	characteristic, complemented, Sylow
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	2	16	no	—
221	C₂³	—	2	4	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	4	16	no	—
226	C₂³	#6129	1	8	yes	—
227	C₂³	C₂³×D₁₃	1	2	yes	—
228	C₂³	C₂³×D₁₃	1	2	yes	—
229	C₂³	C₂³×D₁₃	1	4	yes	—
230	C₂³	C₂³×D₁₃	1	1	yes	characteristic
231	Q₈	—	52	4	no	—
232	D₄	—	2	16	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	13	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	—	26	4	no	—
245	C₂×C₄	—	52	4	no	—
246	C₂×C₄	—	52	8	no	—
247	C₂×C₄	C₂³×D₁₃	1	4	yes	—
248	C₂×C₄	C₂³×D₁₃	1	2	yes	—
249	C₂²	—	2	16	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	16	no	—
257	C₂²	—	4	4	no	—
258	C₂²	—	4	4	no	—
259	C₂²	C₂×#6129	1	8	yes	—
260	C₂²	C₂×#6129	1	4	yes	—
261	C₂²	#88251	1	2	yes	—
262	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	2	2	no	—
265	C₄	—	26	4	no	—
266	C₄	—	26	4	no	—
267	C₄	—	26	4	no	—
268	C₄	—	26	4	no	—
269	C₂	—	2	8	no	—
270	C₂	—	4	2	no	—
271	C₂	—	4	2	no	—
272	C₂	#11719292	1	4	yes	direct product factor
273	C₂	C₂²×#6129	1	2	yes	—
274	C₂	C₂×#88251	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃
Fitting, prime radical	C₂×C₁₃×#120	C₂
socle	C₂³×C₁₃	C₂⁴
hypercenter	C₂×#120	D₁₃
hypocenter	C₁₃	C₂×#551
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#120

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂²⋊Dic₁₃	C₂	semidirect
3. C₂×#88190	C₂	semidirect
4. C₂×#88189	C₂	semidirect
5. C₂×#88189	C₂	semidirect
6. C₂×D₄×Dic₁₃	C₂	semidirect
7. #11719292	C₂	direct (N × H)
8. C₂×#88133	C₂	semidirect
9. C₂×#88132	C₂	semidirect
10. C₂×#88131	C₂	semidirect
16. C₂³×Dic₁₃	C₂²	semidirect
17. C₂²×Dic₂₆	C₂²	semidirect
25. C₂×C₂²⋊Dic₁₃	C₂²	semidirect
26. C₂×C₂²⋊Dic₁₃	C₂²	semidirect
27. C₂×C₂²⋊Dic₁₃	C₂²	semidirect
28. #88190	C₂²	semidirect
29. #88189	C₂²	semidirect
30. #88189	C₂²	semidirect
31. D₄×Dic₁₃	C₂²	semidirect
32. C₂×C₄⋊Dic₁₃	C₂²	semidirect
33. C₂×#6101	C₂²	semidirect
34. C₂×#6101	C₂²	semidirect
35. C₂×#6101	C₂²	semidirect
36. C₂×C₄×Dic₁₃	C₂²	semidirect
37. C₂×C₄×Dic₁₃	C₂²	semidirect
38. #88133	C₂²	semidirect
39. #88132	C₂²	semidirect
40. #88131	C₂²	semidirect
62. C₂²×Dic₁₃	C₂³	semidirect
64. C₂×Dic₂₆	C₂³	semidirect
71. C₂²⋊Dic₁₃	C₂³	semidirect
72. C₂²⋊Dic₁₃	C₂³	semidirect
73. C₄⋊Dic₁₃	C₂³	semidirect
74. #6101	C₂³	semidirect
75. #6101	C₂³	semidirect
76. #6101	C₂³	semidirect
77. C₄×Dic₁₃	C₂³	semidirect
78. C₄×Dic₁₃	C₂³	semidirect
175. C₂×C₁₃	#551	semidirect
216. C₁₃	C₂×#551	semidirect
272. C₂	#11719292	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×#120	2
C₂²×C₂²⋊Dic₁₃	2
C₂×#88190	2
C₂×#88189	2
C₂×D₄×Dic₁₃	2
#11719292	2
C₂×#88133	2
C₂×#88132	2
C₂×#88131	2
C₂×#551	13

Subgroups — C2×#11719292 #13128338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719292 #13128338