Subgroups — C₂×#41×#261 #13131812

Order 1664.

1728 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₁₃	C₂	1	1	yes	maximal, characteristic
2	#41×#261	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×#261	C₂	1	2	yes	maximal, complemented
4	C₂×D₄×#261	C₂	1	6	yes	maximal, complemented
5	C₂×#88237	C₂	1	3	yes	maximal, complemented
6	C₂²×C₄×#261	C₂	1	3	yes	maximal, complemented
7	C₂³×#261	C₂²	1	6	yes	—
8	C₂×C₁₃×#41	C₄	1	1	yes	characteristic, complemented
9	#41×D₁₃	C₄	1	8	yes	complemented
10	#41×D₁₃	C₂²	1	8	yes	—
11	C₂×#6131	C₄	1	1	yes	characteristic, complemented
12	C₂×Q₈×D₁₃	C₂²	1	1	yes	characteristic
13	C₂×#6129	C₄	1	3	yes	complemented
14	C₂×D₄×D₁₃	C₂²	1	3	yes	—
15	C₂×#6127	C₄	1	3	yes	complemented
16	C₂²×C₄×D₁₃	C₂²	1	3	yes	—
17	C₂×C₂²⋊#261	C₂²	1	6	yes	—
18	Q₈×#261	C₂²	1	16	yes	complemented
19	D₄×#261	C₂²	1	48	yes	complemented
20	#88237	C₂²	1	24	yes	complemented
21	C₂×C₄⋊#261	C₂²	1	6	yes	—
22	C₂×C₄×#261	C₂²	1	1	yes	characteristic
23	C₂×C₄×#261	C₂²	1	3	yes	—
24	C₂×C₄×#261	C₂²	1	6	yes	—
25	C₂×C₄×#261	C₂²	1	24	yes	complemented
26	C₂³×D₁₃	C₂³	1	3	yes	—
27	C₂²×#261	—	2	24	no	—
28	C₂²×#261	C₂³	1	6	yes	—
29	C₂²×#261	C₂³	1	24	yes	—
30	C₂²×#261	C₂³	1	2	yes	—
31	C₁₃×#41	C₂×C₄	1	8	yes	complemented
32	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic, complemented
33	C₂×D₄×C₁₃	C₂×C₄	1	3	yes	complemented
34	C₂²×C₄×C₁₃	C₂×C₄	1	3	yes	complemented
35	C₂×C₂²⋊D₁₃	C₂×C₄	1	6	yes	complemented
36	C₂²×Dic₁₃	C₂×C₄	1	3	yes	complemented
37	#6131	C₂×C₄	1	8	yes	complemented
38	Q₈×D₁₃	C₂×C₄	1	4	yes	complemented
39	Q₈×D₁₃	C₂³	1	4	yes	—
40	#6129	C₂×C₄	1	24	yes	complemented
41	D₄×D₁₃	C₂×C₄	1	12	yes	complemented
42	D₄×D₁₃	C₂³	1	12	yes	—
43	#6127	C₂×C₄	1	24	yes	complemented
44	C₂×D₅₂	C₂×C₄	1	3	yes	complemented
45	C₂×C₄×D₁₃	C₂×C₄	1	12	yes	complemented
46	C₂×C₄×D₁₃	C₂×C₄	1	3	yes	complemented
47	C₂×C₄×D₁₃	C₂×C₄	1	3	yes	complemented
48	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₁₃	C₂³	1	3	yes	—
50	C₂×C₄×D₁₃	C₂³	1	12	yes	—
51	C₂×Dic₂₆	C₂×C₄	1	3	yes	complemented
52	C₂²⋊#261	C₂³	1	24	yes	—
53	C₄⋊#261	C₂³	1	24	yes	—
54	C₄×#261	C₂³	1	4	yes	—
55	C₄×#261	C₂³	1	12	yes	—
56	C₄×#261	C₂³	1	24	yes	—
57	C₂×C₄×#41	—	13	1	no	maximal, Sylow
58	C₂³×C₁₃	C₂²×C₄	1	3	yes	—
59	C₂²×D₁₃	—	2	12	no	—
60	C₂²×D₁₃	C₂²×C₄	1	3	yes	—
61	C₂²×D₁₃	C₂²×C₄	1	6	yes	—
62	C₂²×D₁₃	C₂⁴	1	6	yes	—
63	C₂²×D₁₃	C₂⁴	1	1	yes	characteristic
64	C₂×#261	—	2	12	no	—
65	C₂×#261	—	2	48	no	—
66	C₂×#261	#41	1	8	yes	direct product factor
67	C₂×#261	C₂⁴	1	12	yes	—
68	C₂×#261	C₂⁴	1	4	yes	—
69	Q₈×C₁₃	C₂²×C₄	1	4	yes	complemented
70	D₄×C₁₃	C₂²×C₄	1	12	yes	complemented
71	C₂×C₄×C₁₃	C₂²×C₄	1	3	yes	—
72	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂²×C₄	1	12	yes	complemented
74	C₂²⋊D₁₃	C₂²×C₄	1	24	yes	complemented
75	C₂×Dic₁₃	C₂²×C₄	1	3	yes	—
76	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₁₃	C₂²×C₄	1	12	yes	complemented
78	D₅₂	C₂²×C₄	1	12	yes	complemented
79	C₄×D₁₃	C₂²×C₄	1	2	yes	—
80	C₄×D₁₃	C₂²×C₄	1	6	yes	—
81	C₄×D₁₃	C₂²×C₄	1	12	yes	complemented
82	C₄×D₁₃	C₂²×C₄	1	12	yes	complemented
83	C₄×D₁₃	C₂⁴	1	2	yes	—
84	C₄×D₁₃	C₂⁴	1	6	yes	—
85	Dic₂₆	C₂²×C₄	1	12	yes	complemented
86	C₂²×#41	—	13	1	no	—
87	C₄×#41	—	13	16	no	—
88	C₂×C₄×Q₈	—	13	2	no	—
89	C₂×C₄×D₄	—	13	6	no	—
90	C₂×#117	—	13	3	no	—
91	C₂²×C₄²	—	13	3	no	—
92	C₂²×C₁₃	—	2	6	no	—
93	C₂²×C₁₃	C₂³×C₄	1	6	yes	—
94	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₁₃	—	2	6	no	—
96	C₂×D₁₃	—	2	12	no	—
97	C₂×D₁₃	—	2	12	no	—
98	C₂×D₁₃	C₂³×C₄	1	6	yes	—
99	C₂×D₁₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×D₁₃	C₂×#41	1	4	yes	—
101	C₂×D₁₃	C₂⁵	1	1	yes	characteristic
102	#261	—	2	12	no	—
103	#261	C₂×#41	1	8	yes	direct product factor
104	C₄×C₁₃	C₂³×C₄	1	6	yes	—
105	C₄×C₁₃	C₂³×C₄	1	2	yes	—
106	Dic₁₃	C₂³×C₄	1	6	yes	—
107	Dic₁₃	C₂³×C₄	1	2	yes	—
108	C₂×#41	—	13	8	no	—
109	C₂×#41	—	13	8	no	—
110	C₂×#41	—	13	1	no	—
111	C₂×#41	—	13	3	no	—
112	C₂×#41	—	13	3	no	—
113	C₂×#41	#261	1	1	yes	characteristic, direct product factor
114	C₂²×Q₈	—	13	1	no	—
115	C₂²×D₄	—	13	3	no	—
116	C₂³×C₄	—	13	3	no	—
117	C₂³×C₄	—	13	6	no	—
118	C₄×Q₈	—	13	16	no	—
119	C₄×D₄	—	13	48	no	—
120	#117	—	13	24	no	—
121	C₂×C₄⋊C₄	—	13	6	no	—
122	C₂×C₂²⋊C₄	—	13	6	no	—
123	C₂×C₄²	—	13	3	no	—
124	C₂×C₄²	—	13	6	no	—
125	C₂×C₄²	—	13	24	no	—
126	C₂×C₄²	—	13	1	no	—
127	C₂×C₁₃	—	2	6	no	—
128	C₂×C₁₃	C₄×#41	1	2	yes	complemented
129	C₂×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
130	D₁₃	—	2	6	no	—
131	D₁₃	C₄×#41	1	2	yes	complemented
132	D₁₃	C₂²×#41	1	2	yes	—
133	C₂⁴	—	13	3	no	—
134	#41	—	13	8	no	—
135	#41	—	13	24	no	—
136	#41	—	13	24	no	—
137	#41	C₂×#261	1	8	yes	direct product factor
138	C₂×Q₈	—	13	4	no	—
139	C₂×Q₈	—	13	4	no	—
140	C₂×Q₈	—	13	3	no	—
141	C₂×Q₈	C₂×#261	1	1	yes	characteristic, complemented
142	C₂×D₄	—	13	6	no	—
143	C₂×D₄	—	13	12	no	—
144	C₂×D₄	—	13	12	no	—
145	C₂×D₄	—	13	3	no	—
146	C₂×D₄	C₂×#261	1	3	yes	complemented
147	C₂²×C₄	—	13	2	no	—
148	C₂²×C₄	—	13	3	no	—
149	C₂²×C₄	—	13	3	no	—
150	C₂²×C₄	—	13	3	no	—
151	C₂²×C₄	—	13	1	no	—
152	C₂²×C₄	—	13	3	no	—
153	C₂²×C₄	—	13	12	no	—
154	C₂²×C₄	—	13	12	no	—
155	C₂²×C₄	—	13	6	no	—
156	C₂²×C₄	—	13	24	no	—
157	C₂²×C₄	—	26	24	no	—
158	C₂²×C₄	C₂×#261	1	3	yes	complemented
159	C₄⋊C₄	—	13	24	no	—
160	C₂²⋊C₄	—	13	24	no	—
161	C₄²	—	13	4	no	—
162	C₄²	—	13	12	no	—
163	C₄²	—	13	24	no	—
164	C₁₃	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
165	C₂³	—	13	1	no	—
166	C₂³	—	13	6	no	—
167	C₂³	—	13	6	no	—
168	C₂³	—	13	3	no	—
169	C₂³	—	26	12	no	—
170	C₂³	C₂²×#261	1	3	yes	—
171	Q₈	—	13	12	no	—
172	Q₈	C₂²×#261	1	4	yes	complemented
173	D₄	—	13	24	no	—
174	D₄	—	13	12	no	—
175	D₄	C₂²×#261	1	12	yes	complemented
176	C₂×C₄	—	13	4	no	—
177	C₂×C₄	—	13	8	no	—
178	C₂×C₄	—	13	1	no	—
179	C₂×C₄	—	13	3	no	—
180	C₂×C₄	—	13	12	no	—
181	C₂×C₄	—	13	2	no	—
182	C₂×C₄	—	13	6	no	—
183	C₂×C₄	—	13	2	no	—
184	C₂×C₄	—	13	6	no	—
185	C₂×C₄	—	13	12	no	—
186	C₂×C₄	—	13	12	no	—
187	C₂×C₄	—	13	12	no	—
188	C₂×C₄	—	26	48	no	—
189	C₂×C₄	—	26	12	no	—
190	C₂×C₄	C₂²×#261	1	3	yes	—
191	C₂×C₄	C₂²×#261	1	1	yes	characteristic
192	C₂×C₄	C₂²×#261	1	12	yes	complemented
193	C₂²	—	2	6	no	—
194	C₂²	—	13	4	no	—
195	C₂²	—	13	1	no	—
196	C₂²	—	13	1	no	—
197	C₂²	—	13	6	no	—
198	C₂²	—	26	12	no	—
199	C₂²	—	26	12	no	—
200	C₂²	—	26	6	no	—
201	C₂²	C₂³×#261	1	6	yes	—
202	C₂²	C₂³×#261	1	1	yes	characteristic
203	C₄	—	13	8	no	—
204	C₄	—	13	2	no	—
205	C₄	—	13	6	no	—
206	C₄	—	26	12	no	—
207	C₄	C₂³×#261	1	2	yes	—
208	C₄	C₂³×#261	1	6	yes	—
209	C₂	—	2	6	no	—
210	C₂	—	13	2	no	—
211	C₂	—	13	2	no	—
212	C₂	—	26	6	no	—
213	C₂	#41×#261	1	2	yes	direct product factor
214	C₂	C₂⁴×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#261
commutator	C₂×C₁₃	C₂⁴×C₄
Frattini	C₂	C₂⁴×#261
Fitting	C₂×C₁₃×#41	C₄
socle	C₂²×C₁₃	C₂³×C₄
prime radical	C₂×#41×D₁₃	C₂
cosocle	C₂×D₁₃	C₂⁵
hypercenter	C₂×#41	#261
hypocenter	C₁₃	C₂×C₄×#41
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. #41×#261	C₂	direct (N × H)
3. C₂×Q₈×#261	C₂	semidirect
4. C₂×D₄×#261	C₂	semidirect
5. C₂×#88237	C₂	semidirect
6. C₂²×C₄×#261	C₂	semidirect
8. C₂×C₁₃×#41	C₄	semidirect
9. #41×D₁₃	C₄	semidirect
11. C₂×#6131	C₄	semidirect
13. C₂×#6129	C₄	semidirect
15. C₂×#6127	C₄	semidirect
18. Q₈×#261	C₂²	semidirect
19. D₄×#261	C₂²	semidirect
20. #88237	C₂²	semidirect
25. C₂×C₄×#261	C₂²	semidirect
31. C₁₃×#41	C₂×C₄	semidirect
32. C₂×Q₈×C₁₃	C₂×C₄	semidirect
33. C₂×D₄×C₁₃	C₂×C₄	semidirect
34. C₂²×C₄×C₁₃	C₂×C₄	semidirect
35. C₂×C₂²⋊D₁₃	C₂×C₄	semidirect
36. C₂²×Dic₁₃	C₂×C₄	semidirect
37. #6131	C₂×C₄	semidirect
38. Q₈×D₁₃	C₂×C₄	semidirect
40. #6129	C₂×C₄	semidirect
41. D₄×D₁₃	C₂×C₄	semidirect
43. #6127	C₂×C₄	semidirect
44. C₂×D₅₂	C₂×C₄	semidirect
45. C₂×C₄×D₁₃	C₂×C₄	semidirect
46. C₂×C₄×D₁₃	C₂×C₄	semidirect
47. C₂×C₄×D₁₃	C₂×C₄	semidirect
51. C₂×Dic₂₆	C₂×C₄	semidirect
66. C₂×#261	#41	direct (N × H)
69. Q₈×C₁₃	C₂²×C₄	semidirect
70. D₄×C₁₃	C₂²×C₄	semidirect
73. C₂×C₄×C₁₃	C₂²×C₄	semidirect
74. C₂²⋊D₁₃	C₂²×C₄	semidirect
77. C₂×Dic₁₃	C₂²×C₄	semidirect
78. D₅₂	C₂²×C₄	semidirect
81. C₄×D₁₃	C₂²×C₄	semidirect
82. C₄×D₁₃	C₂²×C₄	semidirect
85. Dic₂₆	C₂²×C₄	semidirect
103. #261	C₂×#41	direct (N × H)
113. C₂×#41	#261	direct (N × H)
128. C₂×C₁₃	C₄×#41	semidirect
131. D₁₃	C₄×#41	semidirect
137. #41	C₂×#261	direct (N × H)
141. C₂×Q₈	C₂×#261	semidirect
146. C₂×D₄	C₂×#261	semidirect
158. C₂²×C₄	C₂×#261	semidirect
164. C₁₃	C₂×C₄×#41	semidirect
172. Q₈	C₂²×#261	semidirect
175. D₄	C₂²×#261	semidirect
192. C₂×C₄	C₂²×#261	semidirect
213. C₂	#41×#261	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×#41×D₁₃	2
#41×#261	2
C₂×Q₈×#261	2
C₂×D₄×#261	2
C₂×#88237	2
C₂²×C₄×#261	2
C₂×C₄×#41	13

Subgroups — C2×#41×#261 #13131812

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#41×#261 #13131812