Subgroups — C₁₃×#12869 #15699208

Order 3328.

404 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1810	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1382	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×C₁₃×#117	—	2	1	no	—
7	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
8	C₂×C₁₃×#108	C₂²	1	2	yes	—
9	C₁₃×#427	—	2	1	no	—
10	C₁₃×#427	—	2	1	no	—
11	C₁₃×#427	C₄	1	2	yes	—
12	C₁₃×#427	C₂²	1	2	yes	—
13	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
14	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
15	C₂×C₁₃×#106	—	2	1	no	—
16	C₁₃×#407	—	2	1	no	—
17	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
18	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
19	C₁₃×#341	—	2	1	no	—
20	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂²×C₈×C₁₃	—	2	4	no	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×#117	—	2	2	no	—
30	C₁₃×#117	—	2	2	no	—
31	C₁₃×#117	—	4	1	no	—
32	C₁₃×#117	—	4	1	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
35	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
36	C₂×C₄²×C₁₃	—	4	1	no	—
37	C₂×C₁₃×C₁₆	—	2	2	no	—
38	C₂×C₁₃×C₁₆	—	2	4	no	—
39	C₁₃×#108	—	2	4	no	—
40	C₁₃×#107	—	2	1	no	—
41	C₁₃×#107	—	2	1	no	—
42	C₁₃×#107	C₂×C₄	1	2	yes	—
43	C₁₃×#107	C₂×C₄	1	2	yes	—
44	C₁₃×#106	—	2	2	no	—
45	C₁₃×#106	—	2	2	no	—
46	C₁₃×C₂²⋊C₈	—	4	1	no	—
47	C₁₃×C₂²⋊C₈	—	4	1	no	—
48	#12869	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	4	1	no	—
53	C₂²×C₄×C₁₃	—	4	1	no	—
54	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₁₃×M₁₆	—	4	2	no	—
61	C₁₃×M₁₆	—	4	2	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	4	2	no	—
68	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	#41	1	4	yes	—
73	C₁₃×C₄⋊C₄	—	2	2	no	—
74	C₁₃×C₄⋊C₄	—	2	2	no	—
75	C₁₃×C₄⋊C₄	—	4	1	no	—
76	C₁₃×C₄⋊C₄	—	4	1	no	—
77	C₁₃×C₂²⋊C₄	—	4	2	no	—
78	C₁₃×C₂²⋊C₄	—	4	2	no	—
79	C₄²×C₁₃	—	4	2	no	—
80	C₄²×C₁₃	—	4	2	no	—
81	C₁₃×C₁₆	—	4	2	no	—
82	C₂×#427	C₂×C₁₃	1	1	yes	characteristic
83	C₂×C₂²⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic
84	#1810	C₂×C₁₃	1	1	yes	characteristic
85	#1382	C₂×C₁₃	1	4	yes	complemented
86	C₂³×C₁₃	—	2	2	no	—
87	C₂³×C₁₃	—	2	2	no	—
88	C₂³×C₁₃	—	4	1	no	—
89	C₂³×C₁₃	#102	1	1	yes	characteristic
90	C₂³×C₁₃	#102	1	1	yes	characteristic
91	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	8	2	no	—
102	C₂×C₄×C₁₃	—	8	2	no	—
103	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#123	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#137	1	2	yes	—
109	C₈×C₁₃	—	2	1	no	—
110	C₈×C₁₃	—	2	1	no	—
111	C₈×C₁₃	—	8	2	no	—
112	C₈×C₁₃	#117	1	2	yes	—
113	C₈×C₁₃	#123	1	2	yes	—
114	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
115	C₂×#117	—	2	1	no	—
116	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
117	C₂×#108	C₂²×C₁₃	1	2	yes	—
118	#427	—	2	1	no	—
119	#427	—	2	1	no	—
120	#427	C₄×C₁₃	1	2	yes	—
121	#427	C₂²×C₁₃	1	2	yes	—
122	C₂×#107	C₄×C₁₃	1	1	yes	characteristic
123	C₂×#107	C₄×C₁₃	1	1	yes	characteristic
124	C₂×#106	—	2	1	no	—
125	#407	—	2	1	no	—
126	C₂²⋊C₁₆	—	2	1	no	—
127	C₂²⋊C₁₆	—	2	1	no	—
128	#341	—	2	1	no	—
129	C₂²×C₁₃	—	2	1	no	—
130	C₂²×C₁₃	—	2	1	no	—
131	C₂²×C₁₃	—	2	1	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	—	2	1	no	—
135	C₂²×C₁₃	—	2	1	no	—
136	C₂²×C₁₃	—	4	4	no	—
137	C₂²×C₁₃	#359	1	2	yes	—
138	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
139	C₂²×C₁₃	#483	1	2	yes	—
140	C₄×C₁₃	—	2	1	no	—
141	C₄×C₁₃	—	2	1	no	—
142	C₄×C₁₃	—	8	2	no	—
143	C₄×C₁₃	—	8	2	no	—
144	C₄×C₁₃	#381	1	1	yes	characteristic
145	C₄×C₁₃	#419	1	1	yes	characteristic
146	C₄×C₁₃	#480	1	2	yes	—
147	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
148	C₂×M₁₆	—	2	2	no	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
151	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
152	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
153	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
154	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
156	#117	—	2	2	no	—
157	#117	—	2	2	no	—
158	#117	—	4	1	no	—
159	#117	—	4	1	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₂²⋊C₄	—	4	1	no	—
163	C₂×C₄²	—	4	1	no	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	—	2	4	no	—
166	#108	—	2	4	no	—
167	#107	—	2	1	no	—
168	#107	—	2	1	no	—
169	#107	C₂×C₄×C₁₃	1	2	yes	—
170	#107	C₂×C₄×C₁₃	1	2	yes	—
171	#106	—	2	2	no	—
172	#106	—	2	2	no	—
173	C₂²⋊C₈	—	4	1	no	—
174	C₂²⋊C₈	—	4	1	no	—
175	C₂×C₁₃	—	2	2	no	—
176	C₂×C₁₃	—	2	2	no	—
177	C₂×C₁₃	—	4	1	no	—
178	C₂×C₁₃	#1823	1	1	yes	characteristic
179	C₂×C₁₃	C₂×#359	1	1	yes	characteristic
180	C₂×C₁₃	#2143	1	1	yes	characteristic
181	C₂⁴	C₁₃×SD₁₆	1	1	yes	characteristic
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
189	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
190	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
203	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
204	C₂×C₈	C₁₃×#41	1	4	yes	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	4	1	no	—
208	C₄⋊C₄	—	4	1	no	—
209	C₂²⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₄²	—	4	2	no	—
212	C₄²	—	4	2	no	—
213	C₁₆	—	4	2	no	—
214	C₁₃	#12869	1	1	yes	characteristic, direct product factor, Sylow
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	4	1	no	—
218	C₂³	C₁₃×#102	1	1	yes	characteristic
219	C₂³	C₁₃×#102	1	1	yes	characteristic
220	C₂³	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
233	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
234	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
235	C₂×C₄	C₁₃×#123	1	2	yes	—
236	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
237	C₂×C₄	C₁₃×#137	1	2	yes	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	8	2	no	—
241	C₈	C₁₃×#117	1	2	yes	—
242	C₈	C₁₃×#123	1	2	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	4	no	—
251	C₂²	C₁₃×#359	1	2	yes	—
252	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
253	C₂²	C₁₃×#483	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	2	no	—
258	C₄	C₁₃×#381	1	1	yes	characteristic
259	C₄	C₁₃×#419	1	1	yes	characteristic
260	C₄	C₁₃×#480	1	2	yes	—
261	C₂	—	2	2	no	—
262	C₂	—	2	2	no	—
263	C₂	—	4	1	no	—
264	C₂	C₁₃×#1823	1	1	yes	characteristic
265	C₂	C₂×C₁₃×#359	1	1	yes	characteristic
266	C₂	C₁₃×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂⁴×C₁₃	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₃×#1382	C₂	semidirect
48. #12869	C₁₃	direct (N × H)
85. #1382	C₂×C₁₃	semidirect
214. C₁₃	#12869	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×#427	2
C₂×C₁₃×C₂²⋊C₁₆	2
C₁₃×#1810	2
C₁₃×#1382	2
#12869	13

Subgroups — C13×#12869 #15699208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12869 #15699208