Subgroups — C₁₃×#13155 #15699494

Order 3328.

350 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2140	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#347	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1972	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1946	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1386	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
12	C₁₃×#427	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
14	C₁₃×#424	—	2	1	no	—
15	C₂×C₁₃×#102	—	2	1	no	—
16	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
17	C₁₃×#390	—	2	1	no	—
18	C₁₃×#361	—	2	2	no	—
19	C₁₃×#357	C₄	1	4	yes	complemented
20	C₁₃×#347	—	2	2	no	—
21	C₁₃×#339	—	2	1	no	—
22	C₂²×D₄×C₁₃	—	2	1	no	—
23	C₂×C₁₃×D₈	—	2	1	no	—
24	C₂×C₁₃×D₈	—	2	1	no	—
25	C₂×C₁₃×D₈	—	2	1	no	—
26	C₂×C₁₃×D₈	—	2	1	no	—
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
29	C₂×C₁₃×M₁₆	—	2	1	no	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×#117	—	2	1	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
35	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₁₃×M₃₂	—	2	1	no	—
37	C₁₃×M₃₂	—	2	1	no	—
38	C₁₃×M₃₂	—	2	1	no	—
39	C₁₃×M₃₂	D₄	1	2	yes	—
40	C₂×C₁₃×C₁₆	—	2	1	no	—
41	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
42	C₁₃×#108	—	2	1	no	—
43	C₁₃×#108	—	2	1	no	—
44	C₁₃×#107	C₂×C₄	1	2	yes	—
45	C₁₃×#106	—	2	1	no	—
46	C₁₃×#106	—	2	1	no	—
47	C₁₃×#106	—	2	1	no	—
48	C₁₃×C₄⋊C₈	—	4	2	no	—
49	C₁₃×#102	—	4	2	no	—
50	C₁₃×#97	—	2	1	no	—
51	C₁₃×#97	—	2	1	no	—
52	#13155	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂⁴×C₁₃	—	4	1	no	—
54	C₂×D₄×C₁₃	—	2	2	no	—
55	C₂×D₄×C₁₃	—	4	1	no	—
56	C₂×D₄×C₁₃	—	4	1	no	—
57	C₂×D₄×C₁₃	—	4	1	no	—
58	C₂×D₄×C₁₃	—	4	1	no	—
59	C₂×D₄×C₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	—	2	1	no	—
61	C₂²×C₄×C₁₃	—	4	1	no	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₁₃×D₈	—	2	2	no	—
64	C₁₃×D₈	—	2	2	no	—
65	C₁₃×D₈	—	4	1	no	—
66	C₁₃×D₈	—	4	1	no	—
67	C₁₃×M₁₆	—	4	1	no	—
68	C₁₃×M₁₆	—	4	1	no	—
69	C₂×C₈×C₁₃	—	2	1	no	—
70	C₂×C₈×C₁₃	—	2	1	no	—
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	—	4	1	no	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
76	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
77	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
78	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₄⋊C₄	—	4	1	no	—
81	C₁₃×C₄⋊C₄	—	4	1	no	—
82	C₁₃×C₂²⋊C₄	—	4	1	no	—
83	C₁₃×C₂²⋊C₄	—	8	4	no	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	C₄²×C₁₃	—	4	1	no	—
87	C₁₃×C₁₆	—	2	2	no	—
88	C₁₃×C₁₆	—	2	2	no	—
89	C₂×#361	C₂×C₁₃	1	1	yes	characteristic
90	#2140	C₂×C₁₃	1	1	yes	characteristic
91	C₂×#347	C₂×C₁₃	1	1	yes	characteristic, complemented
92	#1972	C₂×C₁₃	1	1	yes	characteristic
93	#1946	C₂×C₁₃	1	1	yes	characteristic, complemented
94	#1386	C₂×C₁₃	1	1	yes	characteristic, complemented
95	#1383	C₂×C₁₃	1	1	yes	characteristic, complemented
96	C₂³×C₁₃	—	4	2	no	—
97	C₂³×C₁₃	—	8	2	no	—
98	C₂³×C₁₃	—	8	2	no	—
99	C₂³×C₁₃	—	8	1	no	—
100	C₂³×C₁₃	—	8	1	no	—
101	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
102	D₄×C₁₃	—	4	2	no	—
103	D₄×C₁₃	—	4	2	no	—
104	D₄×C₁₃	—	8	1	no	—
105	D₄×C₁₃	—	8	1	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	4	1	no	—
109	C₂×C₄×C₁₃	—	8	2	no	—
110	C₂×C₄×C₁₃	—	8	1	no	—
111	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
114	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
117	C₈×C₁₃	—	4	1	no	—
118	C₈×C₁₃	—	8	1	no	—
119	C₈×C₁₃	#117	1	1	yes	characteristic
120	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
121	C₈×C₁₃	#123	1	1	yes	characteristic
122	C₈×C₁₃	#126	1	1	yes	characteristic
123	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
124	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
125	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
126	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
127	#427	C₂²×C₁₃	1	1	yes	characteristic
128	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
129	#424	—	2	1	no	—
130	C₂×#102	—	2	1	no	—
131	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
132	#390	—	2	1	no	—
133	#361	—	2	2	no	—
134	#357	C₄×C₁₃	1	4	yes	complemented
135	#347	—	2	2	no	—
136	#339	—	2	1	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	8	4	no	—
139	C₂²×C₁₃	—	8	2	no	—
140	C₂²×C₁₃	—	16	2	no	—
141	C₂²×C₁₃	—	16	1	no	—
142	C₂²×C₁₃	—	16	1	no	—
143	C₂²×C₁₃	#442	1	1	yes	characteristic
144	C₂²×C₁₃	#468	1	1	yes	characteristic
145	C₂²×C₁₃	#490	1	1	yes	characteristic
146	C₄×C₁₃	—	8	1	no	—
147	C₄×C₁₃	—	8	2	no	—
148	C₄×C₁₃	—	16	1	no	—
149	C₄×C₁₃	#388	1	1	yes	characteristic
150	C₄×C₁₃	#440	1	1	yes	characteristic
151	C₄×C₁₃	#469	1	1	yes	characteristic
152	C₄×C₁₃	#489	1	1	yes	characteristic
153	C₂²×D₄	—	2	1	no	—
154	C₂×D₈	—	2	1	no	—
155	C₂×D₈	—	2	1	no	—
156	C₂×D₈	—	2	1	no	—
157	C₂×D₈	—	2	1	no	—
158	C₂×D₈	—	2	1	no	—
159	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
160	C₂×M₁₆	—	2	1	no	—
161	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
163	#117	—	2	1	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₂²⋊C₄	—	4	2	no	—
166	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
167	M₃₂	—	2	1	no	—
168	M₃₂	—	2	1	no	—
169	M₃₂	—	2	1	no	—
170	M₃₂	D₄×C₁₃	1	2	yes	—
171	C₂×C₁₆	—	2	1	no	—
172	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
173	#108	—	2	1	no	—
174	#108	—	2	1	no	—
175	#107	C₂×C₄×C₁₃	1	2	yes	—
176	#106	—	2	1	no	—
177	#106	—	2	1	no	—
178	#106	—	2	1	no	—
179	C₄⋊C₈	—	4	2	no	—
180	#102	—	4	2	no	—
181	#97	—	2	1	no	—
182	#97	—	2	1	no	—
183	C₂×C₁₃	—	2	1	no	—
184	C₂×C₁₃	—	2	1	no	—
185	C₂×C₁₃	—	16	2	no	—
186	C₂×C₁₃	#1936	1	1	yes	characteristic
187	C₂×C₁₃	AGL(1,ℤ/16)	1	2	yes	—
188	C₂⁴	—	4	1	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
198	D₈	—	2	2	no	—
199	D₈	—	2	2	no	—
200	D₈	—	4	1	no	—
201	D₈	—	4	1	no	—
202	M₁₆	—	4	1	no	—
203	M₁₆	—	4	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
210	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
211	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
212	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
213	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₂²⋊C₄	—	4	1	no	—
218	C₂²⋊C₄	—	8	4	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	4	1	no	—
222	C₁₆	—	2	2	no	—
223	C₁₆	—	2	2	no	—
224	C₁₃	#13155	1	1	yes	characteristic, direct product factor, Sylow
225	C₂³	—	4	2	no	—
226	C₂³	—	8	2	no	—
227	C₂³	—	8	2	no	—
228	C₂³	—	8	1	no	—
229	C₂³	—	8	1	no	—
230	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
231	D₄	—	4	2	no	—
232	D₄	—	4	2	no	—
233	D₄	—	8	1	no	—
234	D₄	—	8	1	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
241	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
242	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
243	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
244	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
245	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
246	C₈	—	4	1	no	—
247	C₈	—	8	1	no	—
248	C₈	C₁₃×#117	1	1	yes	characteristic
249	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
250	C₈	C₁₃×#123	1	1	yes	characteristic
251	C₈	C₁₃×#126	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	8	4	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	16	2	no	—
256	C₂²	—	16	1	no	—
257	C₂²	—	16	1	no	—
258	C₂²	C₁₃×#442	1	1	yes	characteristic
259	C₂²	C₁₃×#468	1	1	yes	characteristic
260	C₂²	C₁₃×#490	1	1	yes	characteristic
261	C₄	—	8	1	no	—
262	C₄	—	8	2	no	—
263	C₄	—	16	1	no	—
264	C₄	C₁₃×#388	1	1	yes	characteristic
265	C₄	C₁₃×#440	1	1	yes	characteristic
266	C₄	C₁₃×#469	1	1	yes	characteristic
267	C₄	C₁₃×#489	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	16	2	no	—
271	C₂	C₁₃×#1936	1	1	yes	characteristic
272	C₂	C₁₃×AGL(1,ℤ/16)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	#442
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₁₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₃×#347	C₂	semidirect
5. C₁₃×#1946	C₂	semidirect
6. C₁₃×#1386	C₂	semidirect
7. C₁₃×#1383	C₂	semidirect
19. C₁₃×#357	C₄	semidirect
52. #13155	C₁₃	direct (N × H)
91. C₂×#347	C₂×C₁₃	semidirect
93. #1946	C₂×C₁₃	semidirect
94. #1386	C₂×C₁₃	semidirect
95. #1383	C₂×C₁₃	semidirect
134. #357	C₄×C₁₃	semidirect
224. C₁₃	#13155	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×#361	2
C₁₃×#2140	2
C₂×C₁₃×#347	2
C₁₃×#1972	2
C₁₃×#1946	2
C₁₃×#1386	2
C₁₃×#1383	2
#13155	13

Subgroups — C13×#13155 #15699494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13155 #15699494