Subgroups — C₁₃×#3399 #13112457

Order 1664.

418 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#496	C₂	1	2	yes	maximal, complemented
4	C₁₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#463	C₂	1	2	yes	maximal, complemented
6	C₁₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#459	C₂	1	2	yes	maximal, complemented
8	C₁₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#442	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	—	2	1	no	—
13	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
15	C₂×C₁₃×D₈	C₂²	1	2	yes	—
16	C₂×C₁₃×D₈	C₂²	1	2	yes	—
17	C₂×C₁₃×D₈	C₂²	1	2	yes	—
18	C₁₃×#127	C₂²	1	1	yes	characteristic
19	C₁₃×#127	C₂²	1	2	yes	—
20	C₁₃×#127	C₂²	1	1	yes	characteristic
21	C₁₃×#124	—	2	1	no	—
22	C₁₃×#124	C₂²	1	2	yes	—
23	C₁₃×#123	—	2	2	no	—
24	C₁₃×#121	—	2	1	no	—
25	C₁₃×#121	—	2	2	no	—
26	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
27	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₁₃	—	2	1	no	—
29	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
30	C₄×D₄×C₁₃	C₂²	1	2	yes	—
31	C₄×D₄×C₁₃	C₂²	1	2	yes	—
32	C₁₃×#106	C₂²	1	1	yes	characteristic
33	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
35	C₁₃×#103	C₂²	1	1	yes	characteristic
36	C₁₃×#102	C₂²	1	1	yes	characteristic
37	C₁₃×#102	C₂²	1	2	yes	—
38	C₁₃×#102	C₂²	1	2	yes	—
39	C₁₃×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
41	C₁₃×#41	—	4	4	no	—
42	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₁₃	—	2	1	no	—
45	C₂×D₄×C₁₃	—	2	2	no	—
46	C₂×D₄×C₁₃	—	2	1	no	—
47	C₂×D₄×C₁₃	—	2	2	no	—
48	C₂×D₄×C₁₃	—	2	1	no	—
49	C₂×D₄×C₁₃	—	2	1	no	—
50	C₂×D₄×C₁₃	C₂³	1	2	yes	—
51	C₂×D₄×C₁₃	C₂³	1	2	yes	—
52	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₁₃×SD₁₆	—	2	2	no	—
57	C₁₃×SD₁₆	—	4	2	no	—
58	C₁₃×D₈	—	2	4	no	—
59	C₁₃×D₈	—	4	2	no	—
60	C₁₃×D₈	—	4	2	no	—
61	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂³	1	2	yes	—
63	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	—	2	1	no	—
65	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
66	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
67	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₁₃×C₂²⋊C₄	—	2	2	no	—
70	C₁₃×C₂²⋊C₄	—	2	2	no	—
71	C₁₃×C₂²⋊C₄	—	2	1	no	—
72	C₁₃×C₂²⋊C₄	—	2	2	no	—
73	C₁₃×C₂²⋊C₄	—	2	2	no	—
74	C₄²×C₁₃	—	2	1	no	—
75	C₄²×C₁₃	C₂³	1	1	yes	characteristic
76	C₄²×C₁₃	C₂³	1	2	yes	—
77	#3399	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₁₃	—	2	2	no	—
79	C₂³×C₁₃	—	2	2	no	—
80	C₂³×C₁₃	—	2	1	no	—
81	Q₈×C₁₃	—	2	2	no	—
82	Q₈×C₁₃	—	2	1	no	—
83	Q₈×C₁₃	—	2	1	no	—
84	D₄×C₁₃	—	2	2	no	—
85	D₄×C₁₃	—	2	2	no	—
86	D₄×C₁₃	—	2	1	no	—
87	D₄×C₁₃	—	2	2	no	—
88	D₄×C₁₃	—	2	1	no	—
89	D₄×C₁₃	—	2	2	no	—
90	D₄×C₁₃	—	4	4	no	—
91	D₄×C₁₃	—	4	2	no	—
92	D₄×C₁₃	—	4	2	no	—
93	D₄×C₁₃	—	8	2	no	—
94	D₄×C₁₃	—	8	1	no	—
95	D₄×C₁₃	—	8	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	4	4	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	8	2	no	—
102	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₈×C₁₃	—	2	1	no	—
108	C₈×C₁₃	—	2	2	no	—
109	C₈×C₁₃	#41	1	2	yes	complemented
110	#559	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#550	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#496	C₂×C₁₃	1	2	yes	complemented
113	#493	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#463	C₂×C₁₃	1	2	yes	complemented
115	#460	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#459	C₂×C₁₃	1	2	yes	complemented
117	#459	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#446	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#442	C₂×C₁₃	1	2	yes	complemented
120	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
121	C₂²×C₁₃	—	4	2	no	—
122	C₂²×C₁₃	—	4	2	no	—
123	C₂²×C₁₃	—	4	1	no	—
124	C₂²×C₁₃	—	4	2	no	—
125	C₂²×C₁₃	—	4	1	no	—
126	C₂²×C₁₃	—	4	2	no	—
127	C₂²×C₁₃	—	4	2	no	—
128	C₂²×C₁₃	—	4	1	no	—
129	C₂²×C₁₃	—	4	2	no	—
130	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₁₃	—	2	2	no	—
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	—	2	1	no	—
134	C₄×C₁₃	—	4	1	no	—
135	C₄×C₁₃	—	4	1	no	—
136	C₄×C₁₃	—	4	1	no	—
137	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
138	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
139	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
140	C₂×#41	—	2	1	no	—
141	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
142	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
143	C₂×D₈	C₂²×C₁₃	1	2	yes	—
144	C₂×D₈	C₂²×C₁₃	1	2	yes	—
145	C₂×D₈	C₂²×C₁₃	1	2	yes	—
146	#127	C₂²×C₁₃	1	2	yes	—
147	#127	C₂²×C₁₃	1	1	yes	characteristic
148	#127	C₂²×C₁₃	1	1	yes	characteristic
149	#124	—	2	1	no	—
150	#124	C₂²×C₁₃	1	2	yes	—
151	#123	—	2	2	no	—
152	#121	—	2	1	no	—
153	#121	—	2	2	no	—
154	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
155	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
156	C₄×D₄	—	2	1	no	—
157	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
158	C₄×D₄	C₂²×C₁₃	1	2	yes	—
159	C₄×D₄	C₂²×C₁₃	1	2	yes	—
160	#106	C₂²×C₁₃	1	1	yes	characteristic
161	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
162	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
163	#103	C₂²×C₁₃	1	1	yes	characteristic
164	#102	C₂²×C₁₃	1	1	yes	characteristic
165	#102	C₂²×C₁₃	1	2	yes	—
166	#102	C₂²×C₁₃	1	2	yes	—
167	#97	C₂²×C₁₃	1	2	yes	—
168	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
169	C₂×C₁₃	—	8	2	no	—
170	C₂×C₁₃	—	8	1	no	—
171	C₂×C₁₃	—	8	2	no	—
172	C₂×C₁₃	#550	1	1	yes	characteristic
173	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₂×C₁₃	#576	1	1	yes	characteristic
175	#41	—	4	4	no	—
176	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
177	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	2	1	no	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	C₂³×C₁₃	1	2	yes	—
185	C₂×D₄	C₂³×C₁₃	1	2	yes	—
186	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	2	no	—
190	SD₁₆	—	2	2	no	—
191	SD₁₆	—	4	2	no	—
192	D₈	—	2	4	no	—
193	D₈	—	4	2	no	—
194	D₈	—	4	2	no	—
195	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
196	C₂×C₈	C₂³×C₁₃	1	2	yes	—
197	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
200	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
201	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
202	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
203	C₂²⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	C₂³×C₁₃	1	1	yes	characteristic
210	C₄²	C₂³×C₁₃	1	2	yes	—
211	C₁₃	#3399	1	1	yes	characteristic, direct product factor, Sylow
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	1	no	—
215	Q₈	—	2	2	no	—
216	Q₈	—	2	1	no	—
217	Q₈	—	2	1	no	—
218	D₄	—	2	2	no	—
219	D₄	—	2	2	no	—
220	D₄	—	2	1	no	—
221	D₄	—	2	2	no	—
222	D₄	—	2	1	no	—
223	D₄	—	2	2	no	—
224	D₄	—	4	4	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	8	2	no	—
228	D₄	—	8	1	no	—
229	D₄	—	8	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
237	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
238	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
239	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
240	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
241	C₈	—	2	1	no	—
242	C₈	—	2	2	no	—
243	C₈	C₁₃×#41	1	2	yes	complemented
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	2	no	—
253	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	1	no	—
260	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
261	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
262	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
263	C₂	—	8	2	no	—
264	C₂	—	8	1	no	—
265	C₂	—	8	2	no	—
266	C₂	C₁₃×#550	1	1	yes	characteristic
267	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
268	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×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#559	C₂	semidirect
2. C₁₃×#550	C₂	semidirect
3. C₁₃×#496	C₂	semidirect
4. C₁₃×#493	C₂	semidirect
5. C₁₃×#463	C₂	semidirect
6. C₁₃×#460	C₂	semidirect
7. C₁₃×#459	C₂	semidirect
8. C₁₃×#459	C₂	semidirect
9. C₁₃×#446	C₂	semidirect
10. C₁₃×#442	C₂	semidirect
11. C₄×C₁₃×SD₁₆	C₂	semidirect
77. #3399	C₁₃	direct (N × H)
109. C₈×C₁₃	#41	semidirect
110. #559	C₂×C₁₃	semidirect
111. #550	C₂×C₁₃	semidirect
112. #496	C₂×C₁₃	semidirect
113. #493	C₂×C₁₃	semidirect
114. #463	C₂×C₁₃	semidirect
115. #460	C₂×C₁₃	semidirect
116. #459	C₂×C₁₃	semidirect
117. #459	C₂×C₁₃	semidirect
118. #446	C₂×C₁₃	semidirect
119. #442	C₂×C₁₃	semidirect
120. C₄×SD₁₆	C₂×C₁₃	semidirect
211. C₁₃	#3399	direct (N × H)
243. C₈	C₁₃×#41	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#559	2
C₁₃×#550	2
C₁₃×#496	2
C₁₃×#493	2
C₁₃×#463	2
C₁₃×#460	2
C₁₃×#459	2
C₁₃×#446	2
C₁₃×#442	2
C₄×C₁₃×SD₁₆	2
#3399	13

Subgroups — C13×#3399 #13112457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3399 #13112457