Subgroups — C₂×#11719623 #13131813

Order 1664.

1920 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊#261	C₂	1	4	yes	maximal, complemented
3	#11719623	C₂	1	16	yes	maximal, direct product factor
4	C₂×D₄×#261	C₂	1	8	yes	maximal, complemented
5	C₂×#88237	C₂	1	2	yes	maximal, complemented
6	C₂⁴×D₁₃	C₂²	1	2	yes	—
7	C₂³×#261	C₂²	1	8	yes	complemented
8	C₂²×D₄×C₁₃	C₄	1	1	yes	characteristic, complemented
9	C₂²×C₂²⋊D₁₃	C₄	1	2	yes	complemented
10	C₂×D₄×D₁₃	C₄	1	8	yes	complemented
11	C₂×D₄×D₁₃	C₄	1	4	yes	complemented
12	C₂×D₄×D₁₃	C₂²	1	4	yes	—
13	C₂×D₄×D₁₃	C₂²	1	8	yes	—
14	C₂²×D₅₂	C₄	1	1	yes	characteristic, complemented
15	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊#261	C₂²	1	4	yes	—
17	C₂×C₂²⋊#261	C₂²	1	32	yes	complemented
18	C₂×C₂²⋊#261	C₂²	1	8	yes	complemented
19	D₄×#261	C₂²	1	64	yes	complemented
20	#88237	C₂²	1	16	yes	complemented
21	C₂×C₄⋊#261	C₂²	1	4	yes	complemented
22	C₂×C₄×#261	C₂²	1	4	yes	complemented
23	C₂⁴×C₁₃	C₂×C₄	1	2	yes	complemented
24	C₂³×D₁₃	—	4	8	no	—
25	C₂³×D₁₃	C₂×C₄	1	8	yes	complemented
26	C₂³×D₁₃	C₂×C₄	1	2	yes	complemented
27	C₂³×D₁₃	C₂×C₄	1	4	yes	complemented
28	C₂³×D₁₃	C₂³	1	1	yes	characteristic
29	C₂³×D₁₃	C₂³	1	4	yes	—
30	C₂³×D₁₃	C₂³	1	8	yes	—
31	C₂²×#261	—	4	16	no	—
32	C₂²×#261	C₂³	1	8	yes	—
33	C₂²×#261	C₂³	1	32	yes	complemented
34	C₂×D₄×C₁₃	C₂×C₄	1	4	yes	complemented
35	C₂×D₄×C₁₃	C₂×C₄	1	8	yes	complemented
36	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic, complemented
37	C₂×C₂²⋊D₁₃	C₂×C₄	1	8	yes	complemented
38	C₂×C₂²⋊D₁₃	C₂×C₄	1	16	yes	complemented
39	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic, complemented
40	D₄×D₁₃	C₂×C₄	1	16	yes	complemented
41	D₄×D₁₃	C₂×C₄	1	32	yes	complemented
42	D₄×D₁₃	C₂³	1	16	yes	—
43	C₂×D₅₂	C₂×C₄	1	4	yes	complemented
44	C₂×D₅₂	C₂×C₄	1	8	yes	complemented
45	C₂×C₄×D₁₃	C₂×C₄	1	4	yes	complemented
46	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	complemented
47	C₂×C₄×D₁₃	C₂³	1	2	yes	—
48	C₂×C₄×D₁₃	C₂³	1	4	yes	—
49	C₂²⋊#261	C₂³	1	16	yes	—
50	C₂²⋊#261	C₂³	1	32	yes	complemented
51	C₄⋊#261	C₂³	1	16	yes	complemented
52	C₄×#261	C₂³	1	16	yes	complemented
53	C₂×#518	—	13	1	no	maximal, Sylow
54	C₂³×C₁₃	—	4	4	no	—
55	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂³×C₁₃	C₂²×C₄	1	4	yes	—
57	C₂³×C₁₃	C₂²×C₄	1	8	yes	complemented
58	C₂²×D₁₃	—	2	16	no	—
59	C₂²×D₁₃	—	2	4	no	—
60	C₂²×D₁₃	—	4	16	no	—
61	C₂²×D₁₃	—	4	16	no	—
62	C₂²×D₁₃	—	4	8	no	—
63	C₂²×D₁₃	—	4	4	no	—
64	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×D₁₃	C₂²×C₄	1	4	yes	—
67	C₂²×D₁₃	C₂²×C₄	1	8	yes	—
68	C₂²×D₁₃	C₂²×C₄	1	8	yes	complemented
69	C₂²×D₁₃	C₂²×C₄	1	16	yes	complemented
70	C₂²×D₁₃	C₂⁴	1	2	yes	—
71	C₂²×D₁₃	C₂⁴	1	8	yes	—
72	C₂²×D₁₃	C₂⁴	1	1	yes	characteristic
73	C₂×#261	—	2	16	no	—
74	C₂×#261	—	4	32	no	—
75	C₂×#261	C₂⁴	1	16	yes	—
76	D₄×C₁₃	C₂²×C₄	1	16	yes	complemented
77	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	complemented
79	C₂²⋊D₁₃	C₂²×C₄	1	32	yes	complemented
80	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
81	C₂×Dic₁₃	C₂²×C₄	1	4	yes	complemented
82	D₅₂	C₂²×C₄	1	16	yes	complemented
83	C₄×D₁₃	C₂²×C₄	1	4	yes	—
84	C₄×D₁₃	C₂²×C₄	1	8	yes	complemented
85	C₄×D₁₃	C₂⁴	1	4	yes	—
86	C₂³×D₄	—	13	1	no	—
87	#518	—	13	16	no	—
88	C₂×C₄×D₄	—	13	8	no	—
89	C₂×#117	—	13	2	no	—
90	C₂²×C₂²⋊C₄	—	13	4	no	—
91	C₂²×C₁₃	—	2	2	no	—
92	C₂²×C₁₃	—	2	8	no	—
93	C₂²×C₁₃	—	4	8	no	—
94	C₂²×C₁₃	C₂³×C₄	1	2	yes	—
95	C₂²×C₁₃	C₂³×C₄	1	8	yes	—
96	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×D₁₃	—	2	2	no	—
98	C₂×D₁₃	—	2	8	no	—
99	C₂×D₁₃	—	2	4	no	—
100	C₂×D₁₃	—	2	16	no	—
101	C₂×D₁₃	—	2	16	no	—
102	C₂×D₁₃	—	2	4	no	—
103	C₂×D₁₃	—	4	16	no	—
104	C₂×D₁₃	—	4	8	no	—
105	C₂×D₁₃	C₂³×C₄	1	2	yes	—
106	C₂×D₁₃	C₂³×C₄	1	8	yes	—
107	C₂×D₁₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×D₁₃	H(5,𝔽₂)	1	4	yes	—
109	C₂×D₁₃	C₂⁵	1	1	yes	characteristic
110	#261	—	2	16	no	—
111	C₄×C₁₃	C₂³×C₄	1	4	yes	—
112	Dic₁₃	C₂³×C₄	1	4	yes	—
113	C₂⁵	—	13	2	no	—
114	C₂²×D₄	—	13	2	no	—
115	C₂²×D₄	—	13	4	no	—
116	C₂²×D₄	—	13	8	no	—
117	C₂²×D₄	—	13	4	no	—
118	C₂²×D₄	—	13	8	no	—
119	C₂²×D₄	—	13	1	no	—
120	C₂²×D₄	#261	1	1	yes	characteristic, complemented
121	C₂³×C₄	—	13	1	no	—
122	C₂³×C₄	—	13	8	no	—
123	C₄×D₄	—	13	64	no	—
124	#117	—	13	16	no	—
125	C₂×C₄⋊C₄	—	13	4	no	—
126	C₂×C₂²⋊C₄	—	13	4	no	—
127	C₂×C₂²⋊C₄	—	13	8	no	—
128	C₂×C₂²⋊C₄	—	13	32	no	—
129	C₂×C₄²	—	13	4	no	—
130	C₂×C₁₃	—	2	2	no	—
131	C₂×C₁₃	—	2	8	no	—
132	C₂×C₁₃	#518	1	2	yes	complemented
133	C₂×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
134	D₁₃	—	2	2	no	—
135	D₁₃	—	2	8	no	—
136	D₁₃	#518	1	2	yes	complemented
137	D₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
138	C₂⁴	—	13	1	no	—
139	C₂⁴	—	13	4	no	—
140	C₂⁴	—	13	4	no	—
141	C₂⁴	—	13	2	no	—
142	C₂⁴	—	13	8	no	—
143	C₂⁴	—	13	8	no	—
144	C₂⁴	—	52	8	no	—
145	C₂⁴	C₂×#261	1	2	yes	complemented
146	C₂×D₄	—	13	8	no	—
147	C₂×D₄	—	13	16	no	—
148	C₂×D₄	—	13	16	no	—
149	C₂×D₄	—	13	16	no	—
150	C₂×D₄	—	13	32	no	—
151	C₂×D₄	—	13	4	no	—
152	C₂×D₄	—	13	8	no	—
153	C₂×D₄	C₂×#261	1	4	yes	complemented
154	C₂×D₄	C₂×#261	1	8	yes	complemented
155	C₂²×C₄	—	13	1	no	—
156	C₂²×C₄	—	13	2	no	—
157	C₂²×C₄	—	13	2	no	—
158	C₂²×C₄	—	13	4	no	—
159	C₂²×C₄	—	13	4	no	—
160	C₂²×C₄	—	13	8	no	—
161	C₂²×C₄	—	13	32	no	—
162	C₂²×C₄	—	52	16	no	—
163	C₂²×C₄	C₂×#261	1	1	yes	characteristic, complemented
164	C₄⋊C₄	—	13	16	no	—
165	C₂²⋊C₄	—	13	16	no	—
166	C₂²⋊C₄	—	13	32	no	—
167	C₄²	—	13	16	no	—
168	C₁₃	C₂×#518	1	1	yes	characteristic, complemented, Sylow
169	C₂³	—	4	4	no	—
170	C₂³	—	13	1	no	—
171	C₂³	—	13	2	no	—
172	C₂³	—	13	8	no	—
173	C₂³	—	13	2	no	—
174	C₂³	—	13	8	no	—
175	C₂³	—	13	1	no	—
176	C₂³	—	13	4	no	—
177	C₂³	—	13	16	no	—
178	C₂³	—	13	8	no	—
179	C₂³	—	26	16	no	—
180	C₂³	—	26	4	no	—
181	C₂³	—	52	4	no	—
182	C₂³	—	52	8	no	—
183	C₂³	—	52	16	no	—
184	C₂³	—	52	16	no	—
185	C₂³	C₂²×#261	1	1	yes	characteristic
186	C₂³	C₂²×#261	1	4	yes	—
187	C₂³	C₂²×#261	1	8	yes	complemented
188	D₄	—	13	32	no	—
189	D₄	—	13	16	no	—
190	D₄	C₂²×#261	1	16	yes	complemented
191	C₂×C₄	—	13	4	no	—
192	C₂×C₄	—	13	2	no	—
193	C₂×C₄	—	13	4	no	—
194	C₂×C₄	—	13	4	no	—
195	C₂×C₄	—	13	8	no	—
196	C₂×C₄	—	13	16	no	—
197	C₂×C₄	—	26	16	no	—
198	C₂×C₄	—	52	32	no	—
199	C₂×C₄	C₂²×#261	1	2	yes	—
200	C₂×C₄	C₂²×#261	1	4	yes	complemented
201	C₂²	—	2	2	no	—
202	C₂²	—	2	8	no	—
203	C₂²	—	4	8	no	—
204	C₂²	—	13	4	no	—
205	C₂²	—	13	1	no	—
206	C₂²	—	13	1	no	—
207	C₂²	—	13	2	no	—
208	C₂²	—	13	8	no	—
209	C₂²	—	26	16	no	—
210	C₂²	—	26	16	no	—
211	C₂²	—	26	4	no	—
212	C₂²	—	26	4	no	—
213	C₂²	—	26	8	no	—
214	C₂²	—	26	2	no	—
215	C₂²	—	52	8	no	—
216	C₂²	—	52	16	no	—
217	C₂²	C₂³×#261	1	2	yes	—
218	C₂²	C₂³×#261	1	8	yes	—
219	C₂²	C₂³×#261	1	1	yes	characteristic
220	C₄	—	13	4	no	—
221	C₄	—	26	16	no	—
222	C₄	C₂³×#261	1	4	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	2	8	no	—
225	C₂	—	13	2	no	—
226	C₂	—	13	2	no	—
227	C₂	—	26	2	no	—
228	C₂	—	26	8	no	—
229	C₂	#11719623	1	2	yes	direct product factor
230	C₂	C₂⁴×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×#261
commutator	C₂×C₁₃	C₂⁴×C₄
Frattini	C₂	C₂⁴×#261
Fitting	C₂²×D₄×C₁₃	C₄
socle	C₂²×C₁₃	C₂³×C₄
prime radical	C₂²×D₄×D₁₃	C₂
cosocle	C₂×D₁₃	C₂⁵
hypercenter	C₂²×D₄	#261
hypocenter	C₁₃	C₂×#518
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂²⋊#261	C₂	semidirect
3. #11719623	C₂	direct (N × H)
4. C₂×D₄×#261	C₂	semidirect
5. C₂×#88237	C₂	semidirect
7. C₂³×#261	C₂²	semidirect
8. C₂²×D₄×C₁₃	C₄	semidirect
9. C₂²×C₂²⋊D₁₃	C₄	semidirect
10. C₂×D₄×D₁₃	C₄	semidirect
11. C₂×D₄×D₁₃	C₄	semidirect
14. C₂²×D₅₂	C₄	semidirect
17. C₂×C₂²⋊#261	C₂²	semidirect
18. C₂×C₂²⋊#261	C₂²	semidirect
19. D₄×#261	C₂²	semidirect
20. #88237	C₂²	semidirect
21. C₂×C₄⋊#261	C₂²	semidirect
22. C₂×C₄×#261	C₂²	semidirect
23. C₂⁴×C₁₃	C₂×C₄	semidirect
25. C₂³×D₁₃	C₂×C₄	semidirect
26. C₂³×D₁₃	C₂×C₄	semidirect
27. C₂³×D₁₃	C₂×C₄	semidirect
33. C₂²×#261	C₂³	semidirect
34. C₂×D₄×C₁₃	C₂×C₄	semidirect
35. C₂×D₄×C₁₃	C₂×C₄	semidirect
36. C₂²×C₄×C₁₃	C₂×C₄	semidirect
37. C₂×C₂²⋊D₁₃	C₂×C₄	semidirect
38. C₂×C₂²⋊D₁₃	C₂×C₄	semidirect
39. C₂²×Dic₁₃	C₂×C₄	semidirect
40. D₄×D₁₃	C₂×C₄	semidirect
41. D₄×D₁₃	C₂×C₄	semidirect
43. C₂×D₅₂	C₂×C₄	semidirect
44. C₂×D₅₂	C₂×C₄	semidirect
45. C₂×C₄×D₁₃	C₂×C₄	semidirect
46. C₂×C₄×D₁₃	C₂×C₄	semidirect
50. C₂²⋊#261	C₂³	semidirect
51. C₄⋊#261	C₂³	semidirect
52. C₄×#261	C₂³	semidirect
57. C₂³×C₁₃	C₂²×C₄	semidirect
68. C₂²×D₁₃	C₂²×C₄	semidirect
69. C₂²×D₁₃	C₂²×C₄	semidirect
76. D₄×C₁₃	C₂²×C₄	semidirect
78. C₂×C₄×C₁₃	C₂²×C₄	semidirect
79. C₂²⋊D₁₃	C₂²×C₄	semidirect
81. C₂×Dic₁₃	C₂²×C₄	semidirect
82. D₅₂	C₂²×C₄	semidirect
84. C₄×D₁₃	C₂²×C₄	semidirect
120. C₂²×D₄	#261	semidirect
132. C₂×C₁₃	#518	semidirect
136. D₁₃	#518	semidirect
145. C₂⁴	C₂×#261	semidirect
153. C₂×D₄	C₂×#261	semidirect
154. C₂×D₄	C₂×#261	semidirect
163. C₂²×C₄	C₂×#261	semidirect
168. C₁₃	C₂×#518	semidirect
187. C₂³	C₂²×#261	semidirect
190. D₄	C₂²×#261	semidirect
200. C₂×C₄	C₂²×#261	semidirect
229. C₂	#11719623	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×D₄×D₁₃	2
C₂²×C₂²⋊#261	2
#11719623	2
C₂×D₄×#261	2
C₂×#88237	2
C₂×#518	13

Subgroups — C2×#11719623 #13131813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719623 #13131813