Subgroups — C₁₃×#3273 #13112331

Order 1664.

520 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₁₃×#546	C₂	1	2	yes	maximal, complemented
3	C₁₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#468	C₂	1	2	yes	maximal, complemented
5	C₁₃×#463	C₂	1	2	yes	maximal, complemented
6	C₁₃×#452	C₂	1	2	yes	maximal, complemented
7	C₁₃×#447	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#41	C₂²	1	2	yes	—
11	C₂²×D₄×C₁₃	C₂²	1	2	yes	complemented
12	C₁₃×AGL(1,ℤ/8)	—	2	8	no	—
13	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	complemented
15	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	complemented
16	C₂×C₁₃×D₈	C₂²	1	2	yes	—
17	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×M₁₆	C₂²	1	2	yes	complemented
20	C₁₃×#124	C₂²	1	2	yes	complemented
21	C₁₃×#123	—	2	2	no	—
22	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
23	C₁₃×#121	—	2	2	no	—
24	C₁₃×#121	C₂²	1	2	yes	—
25	C₁₃×#120	—	2	2	no	—
26	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₃	C₂²	1	2	yes	complemented
28	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
29	C₁₃×#107	C₂²	1	1	yes	characteristic, complemented
30	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
31	C₁₃×#103	C₂²	1	2	yes	complemented
32	C₁₃×#102	C₂²	1	2	yes	complemented
33	C₁₃×#102	C₂²	1	2	yes	complemented
34	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
35	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
36	C₂⁴×C₁₃	—	2	2	no	—
37	C₁₃×#41	—	2	4	no	—
38	C₁₃×#41	—	4	2	no	—
39	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
40	C₂×D₄×C₁₃	—	2	4	no	—
41	C₂×D₄×C₁₃	—	2	2	no	—
42	C₂×D₄×C₁₃	—	2	2	no	—
43	C₂×D₄×C₁₃	—	2	2	no	—
44	C₂×D₄×C₁₃	—	4	2	no	—
45	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
46	C₂×D₄×C₁₃	C₂³	1	2	yes	—
47	C₂×D₄×C₁₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
51	C₁₃×Q₁₆	—	4	1	no	—
52	C₁₃×SD₁₆	—	2	4	no	—
53	C₁₃×SD₁₆	—	2	4	no	—
54	C₁₃×D₈	—	2	4	no	—
55	C₁₃×D₈	—	4	1	no	—
56	C₁₃×D₈	D₄	1	4	yes	complemented
57	C₁₃×M₁₆	—	2	4	no	—
58	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂³	1	2	yes	—
60	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₁₃×C₄⋊C₄	—	2	2	no	—
62	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
64	C₁₃×C₂²⋊C₄	—	2	2	no	—
65	C₁₃×C₂²⋊C₄	—	2	2	no	—
66	C₁₃×C₂²⋊C₄	—	2	2	no	—
67	C₁₃×C₂²⋊C₄	—	2	2	no	—
68	C₁₃×C₂²⋊C₄	—	2	2	no	—
69	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
70	C₄²×C₁₃	C₂³	1	1	yes	characteristic
71	#3273	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₁₃	—	2	2	no	—
73	C₂³×C₁₃	—	2	2	no	—
74	C₂³×C₁₃	—	4	4	no	—
75	C₂³×C₁₃	—	4	4	no	—
76	C₂³×C₁₃	—	8	2	no	—
77	C₂³×C₁₃	C₂×D₄	1	2	yes	—
78	Q₈×C₁₃	—	2	2	no	—
79	Q₈×C₁₃	—	2	2	no	—
80	D₄×C₁₃	—	2	2	no	—
81	D₄×C₁₃	—	2	2	no	—
82	D₄×C₁₃	—	2	2	no	—
83	D₄×C₁₃	—	4	1	no	—
84	D₄×C₁₃	—	4	4	no	—
85	D₄×C₁₃	—	4	4	no	—
86	D₄×C₁₃	—	8	2	no	—
87	D₄×C₁₃	C₂×D₄	1	4	yes	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	4	4	no	—
93	C₂×C₄×C₁₃	—	8	2	no	—
94	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₈×C₁₃	—	2	2	no	—
99	C₈×C₁₃	—	2	1	no	—
100	C₈×C₁₃	C₂×D₄	1	2	yes	—
101	C₂×AGL(1,ℤ/8)	C₂×C₁₃	1	2	yes	complemented
102	#546	C₂×C₁₃	1	2	yes	complemented
103	#495	C₂×C₁₃	1	1	yes	characteristic, complemented
104	#468	C₂×C₁₃	1	2	yes	complemented
105	#463	C₂×C₁₃	1	2	yes	complemented
106	#452	C₂×C₁₃	1	2	yes	complemented
107	#447	C₂×C₁₃	1	2	yes	complemented
108	C₄×D₈	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#436	C₂×C₁₃	1	1	yes	characteristic, complemented
110	C₂²×C₁₃	—	2	2	no	—
111	C₂²×C₁₃	—	2	4	no	—
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	2	2	no	—
114	C₂²×C₁₃	—	4	2	no	—
115	C₂²×C₁₃	—	4	2	no	—
116	C₂²×C₁₃	—	4	2	no	—
117	C₂²×C₁₃	—	4	2	no	—
118	C₂²×C₁₃	—	4	2	no	—
119	C₂²×C₁₃	—	8	8	no	—
120	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	—	2	1	no	—
123	C₄×C₁₃	—	2	1	no	—
124	C₄×C₁₃	—	4	2	no	—
125	C₄×C₁₃	—	4	2	no	—
126	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂×#41	C₂²×C₁₃	1	2	yes	—
129	C₂²×D₄	C₂²×C₁₃	1	2	yes	complemented
130	AGL(1,ℤ/8)	—	2	8	no	—
131	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic, complemented
132	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	complemented
133	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	complemented
134	C₂×D₈	C₂²×C₁₃	1	2	yes	—
135	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
137	C₂×M₁₆	C₂²×C₁₃	1	2	yes	complemented
138	#124	C₂²×C₁₃	1	2	yes	complemented
139	#123	—	2	2	no	—
140	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	complemented
141	#121	—	2	2	no	—
142	#121	C₂²×C₁₃	1	2	yes	—
143	#120	—	2	2	no	—
144	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
145	C₄×D₄	C₂²×C₁₃	1	2	yes	complemented
146	C₂×C₂²⋊C₄	—	2	2	no	—
147	#107	C₂²×C₁₃	1	1	yes	characteristic, complemented
148	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
149	#103	C₂²×C₁₃	1	2	yes	complemented
150	#102	C₂²×C₁₃	1	2	yes	complemented
151	#102	C₂²×C₁₃	1	2	yes	complemented
152	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	complemented
153	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
154	C₂×C₁₃	—	4	2	no	—
155	C₂×C₁₃	—	4	4	no	—
156	C₂×C₁₃	—	8	2	no	—
157	C₂×C₁₃	D₄²	1	1	yes	characteristic
158	C₂×C₁₃	#575	1	1	yes	characteristic
159	C₂×C₁₃	#576	1	1	yes	characteristic
160	C₂⁴	—	2	2	no	—
161	#41	—	2	4	no	—
162	#41	—	4	2	no	—
163	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	2	2	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
170	C₂×D₄	C₂³×C₁₃	1	2	yes	—
171	C₂×D₄	C₂³×C₁₃	1	2	yes	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
175	Q₁₆	—	4	1	no	—
176	SD₁₆	—	2	4	no	—
177	SD₁₆	—	2	4	no	—
178	D₈	—	2	4	no	—
179	D₈	—	4	1	no	—
180	D₈	D₄×C₁₃	1	4	yes	complemented
181	M₁₆	—	2	4	no	—
182	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
183	C₂×C₈	C₂³×C₁₃	1	2	yes	—
184	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
187	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
194	C₄²	C₂³×C₁₃	1	1	yes	characteristic
195	C₁₃	#3273	1	1	yes	characteristic, direct product factor, Sylow
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	4	4	no	—
199	C₂³	—	4	4	no	—
200	C₂³	—	8	2	no	—
201	C₂³	C₂×D₄×C₁₃	1	2	yes	—
202	Q₈	—	2	2	no	—
203	Q₈	—	2	2	no	—
204	D₄	—	2	2	no	—
205	D₄	—	2	2	no	—
206	D₄	—	2	2	no	—
207	D₄	—	4	1	no	—
208	D₄	—	4	4	no	—
209	D₄	—	4	4	no	—
210	D₄	—	8	2	no	—
211	D₄	C₂×D₄×C₁₃	1	4	yes	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
219	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
220	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
221	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	C₂×D₄×C₁₃	1	2	yes	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	8	8	no	—
235	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
242	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂	—	4	2	no	—
244	C₂	—	4	4	no	—
245	C₂	—	8	2	no	—
246	C₂	D₄²×C₁₃	1	1	yes	characteristic
247	C₂	C₁₃×#575	1	1	yes	characteristic
248	C₂	C₁₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×AGL(1,ℤ/8)	C₂	semidirect
2. C₁₃×#546	C₂	semidirect
3. C₁₃×#495	C₂	semidirect
4. C₁₃×#468	C₂	semidirect
5. C₁₃×#463	C₂	semidirect
6. C₁₃×#452	C₂	semidirect
7. C₁₃×#447	C₂	semidirect
8. C₄×C₁₃×D₈	C₂	semidirect
9. C₁₃×#436	C₂	semidirect
11. C₂²×D₄×C₁₃	C₂²	semidirect
13. C₂×C₁₃×Q₁₆	C₂²	semidirect
14. C₂×C₁₃×SD₁₆	C₂²	semidirect
15. C₂×C₁₃×SD₁₆	C₂²	semidirect
19. C₂×C₁₃×M₁₆	C₂²	semidirect
20. C₁₃×#124	C₂²	semidirect
22. C₁₃×C₂²⋊Q₈	C₂²	semidirect
27. C₄×D₄×C₁₃	C₂²	semidirect
29. C₁₃×#107	C₂²	semidirect
30. C₁₃×C₄⋊C₈	C₂²	semidirect
31. C₁₃×#103	C₂²	semidirect
32. C₁₃×#102	C₂²	semidirect
33. C₁₃×#102	C₂²	semidirect
34. C₁₃×C₂²⋊C₈	C₂²	semidirect
35. C₄×C₈×C₁₃	C₂²	semidirect
56. C₁₃×D₈	D₄	semidirect
71. #3273	C₁₃	direct (N × H)
101. C₂×AGL(1,ℤ/8)	C₂×C₁₃	semidirect
102. #546	C₂×C₁₃	semidirect
103. #495	C₂×C₁₃	semidirect
104. #468	C₂×C₁₃	semidirect
105. #463	C₂×C₁₃	semidirect
106. #452	C₂×C₁₃	semidirect
107. #447	C₂×C₁₃	semidirect
108. C₄×D₈	C₂×C₁₃	semidirect
109. #436	C₂×C₁₃	semidirect
129. C₂²×D₄	C₂²×C₁₃	semidirect
131. C₂×Q₁₆	C₂²×C₁₃	semidirect
132. C₂×SD₁₆	C₂²×C₁₃	semidirect
133. C₂×SD₁₆	C₂²×C₁₃	semidirect
137. C₂×M₁₆	C₂²×C₁₃	semidirect
138. #124	C₂²×C₁₃	semidirect
140. C₂²⋊Q₈	C₂²×C₁₃	semidirect
145. C₄×D₄	C₂²×C₁₃	semidirect
147. #107	C₂²×C₁₃	semidirect
148. C₄⋊C₈	C₂²×C₁₃	semidirect
149. #103	C₂²×C₁₃	semidirect
150. #102	C₂²×C₁₃	semidirect
151. #102	C₂²×C₁₃	semidirect
152. C₂²⋊C₈	C₂²×C₁₃	semidirect
153. C₄×C₈	C₂²×C₁₃	semidirect
180. D₈	D₄×C₁₃	semidirect
195. C₁₃	#3273	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×AGL(1,ℤ/8)	2
C₁₃×#546	2
C₁₃×#495	2
C₁₃×#468	2
C₁₃×#463	2
C₁₃×#452	2
C₁₃×#447	2
C₄×C₁₃×D₈	2
C₁₃×#436	2
#3273	13

Subgroups — C13×#3273 #13112331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3273 #13112331