Subgroups — C₂×#13129648 #16776818

Order 3328.

59186 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13129648	C₂	1	64	yes	maximal, direct product factor
2	C₂×#11719652	C₂	1	16	yes	maximal, complemented
3	C₂×#11719648	C₂	1	20	yes	maximal, complemented
4	C₂²×#88254	C₂	1	10	yes	maximal, complemented
5	C₂×D₁₃×2₋¹⁺⁴	C₂	1	16	yes	maximal, complemented
6	C₂²×C₁₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×2₋¹⁺⁴	C₂²	1	16	yes	complemented
8	C₂×C₁₃×2₋¹⁺⁴	C₂²	1	32	yes	complemented
9	C₂²×C₁₃×#41	C₂²	1	10	yes	complemented
10	C₂³×Q₈×C₁₃	C₂²	1	5	yes	complemented
11	#11719652	C₂²	1	512	yes	complemented
12	D₁₃×2₋¹⁺⁴	C₂²	1	512	yes	complemented
13	#11719648	C₂²	1	640	yes	complemented
14	C₂×#88257	C₂²	1	80	yes	complemented
15	C₂×#88256	C₂²	1	80	yes	complemented
16	C₂×#41×D₁₃	C₂²	1	160	yes	complemented
17	C₂×#88254	C₂²	1	80	yes	complemented
18	C₂×#88254	C₂²	1	40	yes	complemented
19	C₂×#88254	C₂²	1	320	yes	complemented
20	C₂²×#6131	C₂²	1	40	yes	complemented
21	C₂²×Q₈×D₁₃	C₂²	1	40	yes	complemented
22	C₂×#88251	C₂²	1	40	yes	complemented
23	C₂²×#6127	C₂²	1	60	yes	complemented
24	C₁₃×2₋¹⁺⁴	C₂³	1	256	yes	complemented
25	C₂×C₁₃×#41	C₂³	1	80	yes	—
26	C₂×C₁₃×#41	C₂³	1	160	yes	complemented
27	C₂²×Q₈×C₁₃	C₂³	1	40	yes	—
28	C₂²×Q₈×C₁₃	C₂³	1	10	yes	—
29	C₂²×Q₈×C₁₃	C₂³	1	80	yes	complemented
30	C₂²×D₄×C₁₃	C₂³	1	10	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	15	yes	—
32	C₂²×C₂²⋊D₁₃	C₂³	1	40	yes	—
33	#88257	C₂³	1	1280	yes	complemented
34	#88256	C₂³	1	1280	yes	complemented
35	#41×D₁₃	C₂³	1	2560	yes	complemented
36	#88254	C₂³	1	640	yes	complemented
37	#88254	C₂³	1	1280	yes	complemented
38	C₂×#6131	C₂³	1	160	yes	—
39	C₂×#6131	C₂³	1	640	yes	complemented
40	C₂×Q₈×D₁₃	C₂³	1	160	yes	—
41	C₂×Q₈×D₁₃	C₂³	1	640	yes	complemented
42	#88251	C₂³	1	640	yes	complemented
43	C₂×#6129	C₂³	1	160	yes	—
44	C₂×D₄×D₁₃	C₂³	1	160	yes	—
45	C₂×#6127	C₂³	1	80	yes	—
46	C₂×#6127	C₂³	1	240	yes	—
47	C₂×#6127	C₂³	1	960	yes	complemented
48	C₂²×D₅₂	C₂³	1	60	yes	—
49	C₂²×C₄×D₁₃	C₂³	1	120	yes	—
50	C₂²×Dic₂₆	C₂³	1	60	yes	—
51	C₂×2₋¹⁺⁶	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	C₂⁴	1	5	yes	—
53	C₂³×D₁₃	C₂⁴	1	40	yes	—
54	C₁₃×#41	C₂⁴	1	640	yes	—
55	C₂×Q₈×C₁₃	C₂⁴	1	40	yes	—
56	C₂×Q₈×C₁₃	C₂⁴	1	320	yes	—
57	C₂×Q₈×C₁₃	C₂⁴	1	80	yes	—
58	C₂×D₄×C₁₃	C₂⁴	1	40	yes	—
59	C₂×D₄×C₁₃	C₂⁴	1	80	yes	—
60	C₂²×C₄×C₁₃	C₂⁴	1	60	yes	—
61	C₂²×C₄×C₁₃	C₂⁴	1	10	yes	—
62	C₂²×C₄×C₁₃	C₂⁴	1	120	yes	—
63	C₂×C₂²⋊D₁₃	C₂⁴	1	16	yes	—
64	C₂×C₂²⋊D₁₃	C₂⁴	1	80	yes	—
65	C₂×C₂²⋊D₁₃	C₂⁴	1	320	yes	—
66	C₂²×Dic₁₃	C₂⁴	1	40	yes	—
67	#6131	C₂⁴	1	1280	yes	—
68	Q₈×D₁₃	C₂⁴	1	1280	yes	—
69	#6129	C₂⁴	1	1280	yes	—
70	D₄×D₁₃	C₂⁴	1	1280	yes	—
71	#6127	C₂⁴	1	640	yes	—
72	#6127	C₂⁴	1	1920	yes	—
73	C₂×D₅₂	C₂⁴	1	80	yes	—
74	C₂×D₅₂	C₂⁴	1	480	yes	—
75	C₂×C₄×D₁₃	C₂⁴	1	160	yes	—
76	C₂×C₄×D₁₃	C₂⁴	1	960	yes	—
77	C₂×Dic₂₆	C₂⁴	1	80	yes	—
78	C₂×Dic₂₆	C₂⁴	1	480	yes	—
79	2₋¹⁺⁶	—	13	64	no	—
80	C₂×#585	—	13	16	no	—
81	C₂×#585	—	13	20	no	—
82	C₂²×2₋¹⁺⁴	—	13	16	no	—
83	C₂²×2₋¹⁺⁴	—	13	10	no	—
84	C₂²×2₋¹⁺⁴	D₁₃	1	1	yes	characteristic, complemented
85	C₂³×C₁₃	—	4	10	no	—
86	C₂³×C₁₃	C₂⁵	1	10	yes	—
87	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
88	C₂³×C₁₃	C₂⁵	1	20	yes	—
89	C₂²×D₁₃	—	4	80	no	—
90	C₂²×D₁₃	C₂⁵	1	16	yes	—
91	C₂²×D₁₃	C₂⁵	1	160	yes	—
92	Q₈×C₁₃	C₂⁵	1	160	yes	—
93	D₄×C₁₃	C₂⁵	1	160	yes	—
94	C₂×C₄×C₁₃	C₂⁵	1	20	yes	—
95	C₂×C₄×C₁₃	C₂⁵	1	240	yes	—
96	C₂×C₄×C₁₃	C₂⁵	1	40	yes	—
97	C₂²⋊D₁₃	C₂⁵	1	64	yes	—
98	C₂²⋊D₁₃	C₂⁵	1	320	yes	—
99	C₂×Dic₁₃	C₂⁵	1	16	yes	—
100	C₂×Dic₁₃	C₂⁵	1	160	yes	—
101	D₅₂	C₂⁵	1	320	yes	—
102	C₄×D₁₃	C₂⁵	1	640	yes	—
103	Dic₂₆	C₂⁵	1	320	yes	—
104	#585	—	13	512	no	—
105	#585	—	13	640	no	—
106	C₂×2₋¹⁺⁴	—	13	512	no	—
107	C₂×2₋¹⁺⁴	—	13	80	no	—
108	C₂×2₋¹⁺⁴	—	13	80	no	—
109	C₂×2₋¹⁺⁴	—	13	40	no	—
110	C₂×2₋¹⁺⁴	—	13	320	no	—
111	C₂×2₋¹⁺⁴	C₂×D₁₃	1	16	yes	complemented
112	C₂×2₋¹⁺⁴	C₂×D₁₃	1	32	yes	complemented
113	C₂×H(5,𝔽₂)	—	13	80	no	—
114	C₂×H(5,𝔽₂)	—	13	40	no	—
115	C₂²×#41	—	13	160	no	—
116	C₂²×#41	—	13	40	no	—
117	C₂²×#41	—	13	60	no	—
118	C₂²×#41	C₂×D₁₃	1	10	yes	complemented
119	C₂³×Q₈	—	13	40	no	—
120	C₂³×Q₈	C₂×D₁₃	1	5	yes	complemented
121	C₂²×C₁₃	—	2	20	no	—
122	C₂²×C₁₃	—	2	2	no	—
123	C₂²×C₁₃	—	4	20	no	—
124	C₂²×C₁₃	C₂⁶	1	20	yes	—
125	C₂²×C₁₃	C₂⁶	1	2	yes	—
126	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
127	C₂×D₁₃	—	2	32	no	—
128	C₂×D₁₃	—	4	160	no	—
129	C₂×D₁₃	C₂⁶	1	32	yes	—
130	C₄×C₁₃	C₂⁶	1	40	yes	—
131	Dic₁₃	C₂⁶	1	32	yes	—
132	2₋¹⁺⁴	—	13	1280	no	—
133	2₋¹⁺⁴	—	13	1280	no	—
134	2₋¹⁺⁴	—	13	640	no	—
135	2₋¹⁺⁴	C₂²×D₁₃	1	256	yes	complemented
136	H(5,𝔽₂)	—	13	1280	no	—
137	H(5,𝔽₂)	—	13	640	no	—
138	C₂×#41	—	13	2560	no	—
139	C₂×#41	—	13	160	no	—
140	C₂×#41	—	13	640	no	—
141	C₂×#41	—	13	160	no	—
142	C₂×#41	—	13	240	no	—
143	C₂×#41	—	13	80	no	—
144	C₂×#41	—	13	960	no	—
145	C₂×#41	C₂²×D₁₃	1	80	yes	—
146	C₂×#41	C₂²×D₁₃	1	160	yes	complemented
147	C₂²×Q₈	—	13	160	no	—
148	C₂²×Q₈	—	13	640	no	—
149	C₂²×Q₈	—	13	60	no	—
150	C₂²×Q₈	C₂²×D₁₃	1	40	yes	—
151	C₂²×Q₈	C₂²×D₁₃	1	10	yes	—
152	C₂²×Q₈	C₂²×D₁₃	1	80	yes	complemented
153	C₂²×D₄	—	13	40	no	—
154	C₂²×D₄	—	13	160	no	—
155	C₂²×D₄	—	13	60	no	—
156	C₂²×D₄	C₂²×D₁₃	1	10	yes	—
157	C₂³×C₄	—	13	120	no	—
158	C₂³×C₄	C₂²×D₁₃	1	15	yes	—
159	C₂×C₁₃	—	2	20	no	—
160	C₂×C₁₃	—	2	2	no	—
161	C₂×C₁₃	2₋¹⁺⁶	1	2	yes	complemented
162	C₂×C₁₃	C₂⁷	1	1	yes	characteristic
163	D₁₃	—	2	32	no	—
164	C₂⁴	—	13	40	no	—
165	C₂⁴	C₂³×D₁₃	1	5	yes	—
166	#41	—	13	1280	no	—
167	#41	—	13	1280	no	—
168	#41	—	13	1920	no	—
169	#41	—	13	640	no	—
170	#41	C₂³×D₁₃	1	640	yes	—
171	C₂×Q₈	—	13	1280	no	—
172	C₂×Q₈	—	13	80	no	—
173	C₂×Q₈	—	13	480	no	—
174	C₂×Q₈	C₂³×D₁₃	1	40	yes	—
175	C₂×Q₈	C₂³×D₁₃	1	320	yes	—
176	C₂×Q₈	C₂³×D₁₃	1	80	yes	—
177	C₂×D₄	—	13	80	no	—
178	C₂×D₄	—	13	16	no	—
179	C₂×D₄	—	13	320	no	—
180	C₂×D₄	—	13	1280	no	—
181	C₂×D₄	—	13	80	no	—
182	C₂×D₄	—	13	480	no	—
183	C₂×D₄	C₂³×D₁₃	1	40	yes	—
184	C₂×D₄	C₂³×D₁₃	1	80	yes	—
185	C₂²×C₄	—	13	40	no	—
186	C₂²×C₄	—	13	160	no	—
187	C₂²×C₄	—	13	960	no	—
188	C₂²×C₄	C₂³×D₁₃	1	60	yes	—
189	C₂²×C₄	C₂³×D₁₃	1	10	yes	—
190	C₂²×C₄	C₂³×D₁₃	1	120	yes	—
191	C₁₃	C₂×2₋¹⁺⁶	1	1	yes	characteristic, complemented, Sylow
192	C₂³	—	4	10	no	—
193	C₂³	—	13	16	no	—
194	C₂³	—	13	160	no	—
195	C₂³	—	52	80	no	—
196	C₂³	C₂⁴×D₁₃	1	10	yes	—
197	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
198	C₂³	C₂⁴×D₁₃	1	20	yes	—
199	Q₈	—	13	320	no	—
200	Q₈	C₂⁴×D₁₃	1	160	yes	—
201	D₄	—	13	320	no	—
202	D₄	—	13	64	no	—
203	D₄	—	13	320	no	—
204	D₄	C₂⁴×D₁₃	1	160	yes	—
205	C₂×C₄	—	13	160	no	—
206	C₂×C₄	—	13	16	no	—
207	C₂×C₄	—	13	640	no	—
208	C₂×C₄	C₂⁴×D₁₃	1	240	yes	—
209	C₂×C₄	C₂⁴×D₁₃	1	40	yes	—
210	C₂×C₄	C₂⁴×D₁₃	1	20	yes	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	20	no	—
213	C₂²	—	4	20	no	—
214	C₂²	—	13	32	no	—
215	C₂²	—	26	32	no	—
216	C₂²	—	52	160	no	—
217	C₂²	C₂⁵×D₁₃	1	20	yes	—
218	C₂²	C₂⁵×D₁₃	1	2	yes	—
219	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
220	C₄	—	13	32	no	—
221	C₄	C₂⁵×D₁₃	1	40	yes	—
222	C₂	—	2	2	no	—
223	C₂	—	2	20	no	—
224	C₂	—	26	32	no	—
225	C₂	#13129648	1	2	yes	direct product factor
226	C₂	C₂⁶×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂⁵×D₁₃
commutator, cosocle	C₂×C₁₃	C₂⁷
Frattini	C₂	C₂⁶×D₁₃
Fitting	C₂²×C₁₃×2₋¹⁺⁴	C₂
socle	C₂²×C₁₃	C₂⁶
hypercenter	C₂²×2₋¹⁺⁴	D₁₃
hypocenter	C₁₃	C₂×2₋¹⁺⁶
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #13129648	C₂	direct (N × H)
2. C₂×#11719652	C₂	semidirect
3. C₂×#11719648	C₂	semidirect
4. C₂²×#88254	C₂	semidirect
5. C₂×D₁₃×2₋¹⁺⁴	C₂	semidirect
6. C₂²×C₁₃×2₋¹⁺⁴	C₂	semidirect
7. C₂×C₁₃×2₋¹⁺⁴	C₂²	semidirect
8. C₂×C₁₃×2₋¹⁺⁴	C₂²	semidirect
9. C₂²×C₁₃×#41	C₂²	semidirect
10. C₂³×Q₈×C₁₃	C₂²	semidirect
11. #11719652	C₂²	semidirect
12. D₁₃×2₋¹⁺⁴	C₂²	semidirect
13. #11719648	C₂²	semidirect
14. C₂×#88257	C₂²	semidirect
15. C₂×#88256	C₂²	semidirect
16. C₂×#41×D₁₃	C₂²	semidirect
17. C₂×#88254	C₂²	semidirect
18. C₂×#88254	C₂²	semidirect
19. C₂×#88254	C₂²	semidirect
20. C₂²×#6131	C₂²	semidirect
21. C₂²×Q₈×D₁₃	C₂²	semidirect
22. C₂×#88251	C₂²	semidirect
23. C₂²×#6127	C₂²	semidirect
24. C₁₃×2₋¹⁺⁴	C₂³	semidirect
26. C₂×C₁₃×#41	C₂³	semidirect
29. C₂²×Q₈×C₁₃	C₂³	semidirect
33. #88257	C₂³	semidirect
34. #88256	C₂³	semidirect
35. #41×D₁₃	C₂³	semidirect
36. #88254	C₂³	semidirect
37. #88254	C₂³	semidirect
39. C₂×#6131	C₂³	semidirect
41. C₂×Q₈×D₁₃	C₂³	semidirect
42. #88251	C₂³	semidirect
47. C₂×#6127	C₂³	semidirect
84. C₂²×2₋¹⁺⁴	D₁₃	semidirect
111. C₂×2₋¹⁺⁴	C₂×D₁₃	semidirect
112. C₂×2₋¹⁺⁴	C₂×D₁₃	semidirect
118. C₂²×#41	C₂×D₁₃	semidirect
120. C₂³×Q₈	C₂×D₁₃	semidirect
135. 2₋¹⁺⁴	C₂²×D₁₃	semidirect
146. C₂×#41	C₂²×D₁₃	semidirect
152. C₂²×Q₈	C₂²×D₁₃	semidirect
161. C₂×C₁₃	2₋¹⁺⁶	semidirect
191. C₁₃	C₂×2₋¹⁺⁶	semidirect
225. C₂	#13129648	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13129648	2
C₂×#11719652	2
C₂×#11719648	2
C₂²×#88254	2
C₂×D₁₃×2₋¹⁺⁴	2
C₂²×C₁₃×2₋¹⁺⁴	2
C₂×2₋¹⁺⁶	13

Subgroups — C2×#13129648 #16776818

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13129648 #16776818