Subgroups — C₁₃×#3585 #13112643

Order 1664.

1464 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#585	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#577	C₂	1	8	yes	maximal, complemented
4	C₁₃×#576	C₂	1	8	yes	maximal, complemented
5	C₁₃×#575	C₂	1	6	yes	maximal, complemented
6	C₂×C₁₃×AGL(1,ℤ/8)	C₂	1	6	yes	maximal, complemented
7	C₁₃×#568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×2₋¹⁺⁴	—	2	1	no	—
9	C₁₃×2₋¹⁺⁴	C₂²	1	4	yes	complemented
10	C₁₃×H(5,𝔽₂)	—	2	4	no	—
11	C₁₃×H(5,𝔽₂)	—	2	3	no	—
12	C₁₃×H(5,𝔽₂)	C₂²	1	8	yes	complemented
13	C₁₃×H(5,𝔽₂)	C₂²	1	4	yes	complemented
14	C₂×C₁₃×#41	—	2	4	no	—
15	C₂×C₁₃×#41	—	2	1	no	—
16	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#41	C₂²	1	3	yes	complemented
18	C₂×C₁₃×#41	C₂²	1	6	yes	complemented
19	C₂²×D₄×C₁₃	—	2	3	no	—
20	C₂²×D₄×C₁₃	C₂²	1	3	yes	complemented
21	C₁₃×#137	C₂²	1	4	yes	complemented
22	C₁₃×#137	C₂²	1	12	yes	complemented
23	C₁₃×AGL(1,ℤ/8)	C₂²	1	12	yes	complemented
24	C₁₃×AGL(1,ℤ/8)	C₂²	1	24	yes	complemented
25	C₁₃×AGL(1,ℤ/8)	C₂²	1	12	yes	complemented
26	C₁₃×#135	C₂²	1	24	yes	complemented
27	C₂×C₁₃×SD₁₆	C₂²	1	12	yes	complemented
28	C₂×C₁₃×D₈	C₂²	1	12	yes	complemented
29	C₁₃×#131	C₂²	1	8	yes	complemented
30	C₂×C₁₃×M₁₆	C₂²	1	6	yes	complemented
31	C₂⁴×C₁₃	—	2	3	no	—
32	C₁₃×#41	—	2	8	no	—
33	C₁₃×#41	—	2	4	no	—
34	C₁₃×#41	—	2	12	no	—
35	C₁₃×#41	—	2	6	no	—
36	C₁₃×#41	—	2	6	no	—
37	C₁₃×#41	—	2	6	no	—
38	C₁₃×#41	D₄	1	4	yes	—
39	C₁₃×#41	C₂³	1	12	yes	—
40	C₁₃×#41	C₂³	1	4	yes	—
41	C₁₃×#41	C₂³	1	4	yes	—
42	C₁₃×#41	C₂³	1	12	yes	complemented
43	C₂×Q₈×C₁₃	—	2	4	no	—
44	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
45	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₁₃	C₂³	1	6	yes	complemented
47	C₂×D₄×C₁₃	—	2	3	no	—
48	C₂×D₄×C₁₃	—	2	6	no	—
49	C₂×D₄×C₁₃	—	2	24	no	—
50	C₂×D₄×C₁₃	—	2	12	no	—
51	C₂×D₄×C₁₃	—	2	3	no	—
52	C₂×D₄×C₁₃	—	2	6	no	—
53	C₂×D₄×C₁₃	D₄	1	3	yes	—
54	C₂×D₄×C₁₃	C₂³	1	12	yes	—
55	C₂×D₄×C₁₃	C₂³	1	3	yes	—
56	C₂×D₄×C₁₃	C₂³	1	6	yes	complemented
57	C₂²×C₄×C₁₃	—	2	3	no	—
58	C₂²×C₄×C₁₃	—	2	6	no	—
59	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
60	C₁₃×Q₁₆	C₂³	1	8	yes	complemented
61	C₁₃×SD₁₆	C₂³	1	24	yes	complemented
62	C₁₃×SD₁₆	C₂³	1	8	yes	complemented
63	C₁₃×D₈	C₂³	1	24	yes	complemented
64	C₁₃×M₁₆	C₂³	1	4	yes	—
65	C₁₃×M₁₆	C₂³	1	12	yes	complemented
66	C₂×C₈×C₁₃	C₂³	1	12	yes	complemented
67	#3585	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₁₃	—	2	3	no	—
69	C₂³×C₁₃	—	2	6	no	—
70	C₂³×C₁₃	—	2	12	no	—
71	C₂³×C₁₃	—	8	6	no	—
72	C₂³×C₁₃	C₂×D₄	1	3	yes	—
73	Q₈×C₁₃	—	2	6	no	—
74	Q₈×C₁₃	—	2	1	no	—
75	Q₈×C₁₃	C₂×D₄	1	4	yes	—
76	Q₈×C₁₃	C₂⁴	1	4	yes	—
77	Q₈×C₁₃	C₂⁴	1	2	yes	—
78	D₄×C₁₃	—	2	12	no	—
79	D₄×C₁₃	—	2	4	no	—
80	D₄×C₁₃	—	2	12	no	—
81	D₄×C₁₃	—	2	3	no	—
82	D₄×C₁₃	—	2	12	no	—
83	D₄×C₁₃	—	2	6	no	—
84	D₄×C₁₃	C₂×D₄	1	12	yes	—
85	D₄×C₁₃	C₂⁴	1	4	yes	—
86	D₄×C₁₃	C₂⁴	1	6	yes	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	6	no	—
89	C₂×C₄×C₁₃	—	2	4	no	—
90	C₂×C₄×C₁₃	—	2	12	no	—
91	C₂×C₄×C₁₃	—	2	12	no	—
92	C₂×C₄×C₁₃	C₂×D₄	1	3	yes	—
93	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
94	C₂×C₄×C₁₃	C₂⁴	1	6	yes	—
95	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₈×C₁₃	C₂⁴	1	8	yes	—
97	#585	C₂×C₁₃	1	1	yes	characteristic, complemented
98	C₂×H(5,𝔽₂)	C₂×C₁₃	1	1	yes	characteristic, complemented
99	#577	C₂×C₁₃	1	8	yes	complemented
100	#576	C₂×C₁₃	1	8	yes	complemented
101	#575	C₂×C₁₃	1	6	yes	complemented
102	C₂×AGL(1,ℤ/8)	C₂×C₁₃	1	6	yes	complemented
103	#568	C₂×C₁₃	1	1	yes	characteristic, complemented
104	C₂²×C₁₃	—	2	4	no	—
105	C₂²×C₁₃	—	2	6	no	—
106	C₂²×C₁₃	—	4	3	no	—
107	C₂²×C₁₃	—	4	6	no	—
108	C₂²×C₁₃	—	8	6	no	—
109	C₂²×C₁₃	—	8	12	no	—
110	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
111	C₂²×C₁₃	C₂²×D₄	1	6	yes	—
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	—	2	4	no	—
114	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₁₃	C₂²×D₄	1	6	yes	—
116	C₄×C₁₃	C₂⁵	1	1	yes	characteristic
117	2₋¹⁺⁴	—	2	1	no	—
118	2₋¹⁺⁴	C₂²×C₁₃	1	4	yes	complemented
119	H(5,𝔽₂)	—	2	4	no	—
120	H(5,𝔽₂)	—	2	3	no	—
121	H(5,𝔽₂)	C₂²×C₁₃	1	8	yes	complemented
122	H(5,𝔽₂)	C₂²×C₁₃	1	4	yes	complemented
123	C₂×#41	—	2	4	no	—
124	C₂×#41	—	2	1	no	—
125	C₂×#41	C₂²×C₁₃	1	1	yes	characteristic
126	C₂×#41	C₂²×C₁₃	1	3	yes	complemented
127	C₂×#41	C₂²×C₁₃	1	6	yes	complemented
128	C₂²×D₄	—	2	3	no	—
129	C₂²×D₄	C₂²×C₁₃	1	3	yes	complemented
130	#137	C₂²×C₁₃	1	4	yes	complemented
131	#137	C₂²×C₁₃	1	12	yes	complemented
132	AGL(1,ℤ/8)	C₂²×C₁₃	1	12	yes	complemented
133	AGL(1,ℤ/8)	C₂²×C₁₃	1	24	yes	complemented
134	AGL(1,ℤ/8)	C₂²×C₁₃	1	12	yes	complemented
135	#135	C₂²×C₁₃	1	24	yes	complemented
136	C₂×SD₁₆	C₂²×C₁₃	1	12	yes	complemented
137	C₂×D₈	C₂²×C₁₃	1	12	yes	complemented
138	#131	C₂²×C₁₃	1	8	yes	complemented
139	C₂×M₁₆	C₂²×C₁₃	1	6	yes	complemented
140	C₂×C₁₃	—	2	1	no	—
141	C₂×C₁₃	—	2	6	no	—
142	C₂×C₁₃	—	4	6	no	—
143	C₂×C₁₃	—	4	4	no	—
144	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
145	C₂⁴	—	2	3	no	—
146	#41	—	2	8	no	—
147	#41	—	2	4	no	—
148	#41	—	2	12	no	—
149	#41	—	2	6	no	—
150	#41	—	2	6	no	—
151	#41	—	2	6	no	—
152	#41	D₄×C₁₃	1	4	yes	—
153	#41	C₂³×C₁₃	1	12	yes	—
154	#41	C₂³×C₁₃	1	4	yes	—
155	#41	C₂³×C₁₃	1	4	yes	—
156	#41	C₂³×C₁₃	1	12	yes	complemented
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
159	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
160	C₂×Q₈	C₂³×C₁₃	1	6	yes	complemented
161	C₂×D₄	—	2	3	no	—
162	C₂×D₄	—	2	6	no	—
163	C₂×D₄	—	2	24	no	—
164	C₂×D₄	—	2	12	no	—
165	C₂×D₄	—	2	3	no	—
166	C₂×D₄	—	2	6	no	—
167	C₂×D₄	D₄×C₁₃	1	3	yes	—
168	C₂×D₄	C₂³×C₁₃	1	12	yes	—
169	C₂×D₄	C₂³×C₁₃	1	3	yes	—
170	C₂×D₄	C₂³×C₁₃	1	6	yes	complemented
171	C₂²×C₄	—	2	3	no	—
172	C₂²×C₄	—	2	6	no	—
173	C₂²×C₄	C₂³×C₁₃	1	3	yes	—
174	Q₁₆	C₂³×C₁₃	1	8	yes	complemented
175	SD₁₆	C₂³×C₁₃	1	24	yes	complemented
176	SD₁₆	C₂³×C₁₃	1	8	yes	complemented
177	D₈	C₂³×C₁₃	1	24	yes	complemented
178	M₁₆	C₂³×C₁₃	1	4	yes	—
179	M₁₆	C₂³×C₁₃	1	12	yes	complemented
180	C₂×C₈	C₂³×C₁₃	1	12	yes	complemented
181	C₁₃	#3585	1	1	yes	characteristic, direct product factor, Sylow
182	C₂³	—	2	3	no	—
183	C₂³	—	2	6	no	—
184	C₂³	—	2	12	no	—
185	C₂³	—	8	6	no	—
186	C₂³	C₂×D₄×C₁₃	1	3	yes	—
187	Q₈	—	2	6	no	—
188	Q₈	—	2	1	no	—
189	Q₈	C₂×D₄×C₁₃	1	4	yes	—
190	Q₈	C₂⁴×C₁₃	1	4	yes	—
191	Q₈	C₂⁴×C₁₃	1	2	yes	—
192	D₄	—	2	12	no	—
193	D₄	—	2	4	no	—
194	D₄	—	2	12	no	—
195	D₄	—	2	3	no	—
196	D₄	—	2	12	no	—
197	D₄	—	2	6	no	—
198	D₄	C₂×D₄×C₁₃	1	12	yes	—
199	D₄	C₂⁴×C₁₃	1	4	yes	—
200	D₄	C₂⁴×C₁₃	1	6	yes	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	6	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	12	no	—
205	C₂×C₄	—	2	12	no	—
206	C₂×C₄	C₂×D₄×C₁₃	1	3	yes	—
207	C₂×C₄	C₂×D₄×C₁₃	1	6	yes	—
208	C₂×C₄	C₂⁴×C₁₃	1	6	yes	—
209	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
210	C₈	C₂⁴×C₁₃	1	8	yes	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	6	no	—
213	C₂²	—	4	3	no	—
214	C₂²	—	4	6	no	—
215	C₂²	—	8	6	no	—
216	C₂²	—	8	12	no	—
217	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
218	C₂²	C₂²×D₄×C₁₃	1	6	yes	—
219	C₄	—	2	2	no	—
220	C₄	—	2	4	no	—
221	C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
222	C₄	C₂²×D₄×C₁₃	1	6	yes	—
223	C₄	C₂⁵×C₁₃	1	1	yes	characteristic
224	C₂	—	2	1	no	—
225	C₂	—	2	6	no	—
226	C₂	—	4	6	no	—
227	C₂	—	4	4	no	—
228	C₂	C₂³×D₄×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	C₂³×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₁₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₁₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#585	C₂	semidirect
2. C₂×C₁₃×H(5,𝔽₂)	C₂	semidirect
3. C₁₃×#577	C₂	semidirect
4. C₁₃×#576	C₂	semidirect
5. C₁₃×#575	C₂	semidirect
6. C₂×C₁₃×AGL(1,ℤ/8)	C₂	semidirect
7. C₁₃×#568	C₂	semidirect
9. C₁₃×2₋¹⁺⁴	C₂²	semidirect
12. C₁₃×H(5,𝔽₂)	C₂²	semidirect
13. C₁₃×H(5,𝔽₂)	C₂²	semidirect
17. C₂×C₁₃×#41	C₂²	semidirect
18. C₂×C₁₃×#41	C₂²	semidirect
20. C₂²×D₄×C₁₃	C₂²	semidirect
21. C₁₃×#137	C₂²	semidirect
22. C₁₃×#137	C₂²	semidirect
23. C₁₃×AGL(1,ℤ/8)	C₂²	semidirect
24. C₁₃×AGL(1,ℤ/8)	C₂²	semidirect
25. C₁₃×AGL(1,ℤ/8)	C₂²	semidirect
26. C₁₃×#135	C₂²	semidirect
27. C₂×C₁₃×SD₁₆	C₂²	semidirect
28. C₂×C₁₃×D₈	C₂²	semidirect
29. C₁₃×#131	C₂²	semidirect
30. C₂×C₁₃×M₁₆	C₂²	semidirect
42. C₁₃×#41	C₂³	semidirect
46. C₂×Q₈×C₁₃	C₂³	semidirect
56. C₂×D₄×C₁₃	C₂³	semidirect
60. C₁₃×Q₁₆	C₂³	semidirect
61. C₁₃×SD₁₆	C₂³	semidirect
62. C₁₃×SD₁₆	C₂³	semidirect
63. C₁₃×D₈	C₂³	semidirect
65. C₁₃×M₁₆	C₂³	semidirect
66. C₂×C₈×C₁₃	C₂³	semidirect
67. #3585	C₁₃	direct (N × H)
97. #585	C₂×C₁₃	semidirect
98. C₂×H(5,𝔽₂)	C₂×C₁₃	semidirect
99. #577	C₂×C₁₃	semidirect
100. #576	C₂×C₁₃	semidirect
101. #575	C₂×C₁₃	semidirect
102. C₂×AGL(1,ℤ/8)	C₂×C₁₃	semidirect
103. #568	C₂×C₁₃	semidirect
118. 2₋¹⁺⁴	C₂²×C₁₃	semidirect
121. H(5,𝔽₂)	C₂²×C₁₃	semidirect
122. H(5,𝔽₂)	C₂²×C₁₃	semidirect
126. C₂×#41	C₂²×C₁₃	semidirect
127. C₂×#41	C₂²×C₁₃	semidirect
129. C₂²×D₄	C₂²×C₁₃	semidirect
130. #137	C₂²×C₁₃	semidirect
131. #137	C₂²×C₁₃	semidirect
132. AGL(1,ℤ/8)	C₂²×C₁₃	semidirect
133. AGL(1,ℤ/8)	C₂²×C₁₃	semidirect
134. AGL(1,ℤ/8)	C₂²×C₁₃	semidirect
135. #135	C₂²×C₁₃	semidirect
136. C₂×SD₁₆	C₂²×C₁₃	semidirect
137. C₂×D₈	C₂²×C₁₃	semidirect
138. #131	C₂²×C₁₃	semidirect
139. C₂×M₁₆	C₂²×C₁₃	semidirect
156. #41	C₂³×C₁₃	semidirect
160. C₂×Q₈	C₂³×C₁₃	semidirect
170. C₂×D₄	C₂³×C₁₃	semidirect
174. Q₁₆	C₂³×C₁₃	semidirect
175. SD₁₆	C₂³×C₁₃	semidirect
176. SD₁₆	C₂³×C₁₃	semidirect
177. D₈	C₂³×C₁₃	semidirect
179. M₁₆	C₂³×C₁₃	semidirect
180. C₂×C₈	C₂³×C₁₃	semidirect
181. C₁₃	#3585	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#585	2
C₂×C₁₃×H(5,𝔽₂)	2
C₁₃×#577	2
C₁₃×#576	2
C₁₃×#575	2
C₂×C₁₃×AGL(1,ℤ/8)	2
C₁₃×#568	2
#3585	13

Subgroups — C13×#3585 #13112643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3585 #13112643