Subgroups — C₁₃×#17981 #15704320

Order 3328.

726 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	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2931	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1586	C₂	1	8	yes	maximal, complemented
4	C₁₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1492	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#131	—	2	2	no	—
8	C₁₃×#532	C₂²	1	2	yes	—
9	C₁₃×#500	C₄	1	1	yes	characteristic
10	C₁₃×#493	C₄	1	1	yes	characteristic
11	C₁₃×#486	C₄	1	2	yes	—
12	C₁₃×#466	C₄	1	4	yes	—
13	C₄×C₁₃×D₈	C₂²	1	4	yes	—
14	C₁₃×#436	—	2	2	no	—
15	C₁₃×#436	C₂²	1	4	yes	—
16	C₈×D₄×C₁₃	—	2	2	no	—
17	C₈×D₄×C₁₃	C₂²	1	4	yes	—
18	C₁₃×#423	C₂²	1	2	yes	—
19	C₁₃×#408	—	2	2	no	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
23	C₁₃×#334	C₂²	1	4	yes	—
24	C₁₃×D₄⋊C₈	C₂²	1	8	yes	—
25	C₁₃×#322	C₂²	1	4	yes	—
26	C₂×C₁₃×#41	—	2	2	no	—
27	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
28	C₁₃×#131	—	4	8	no	—
29	C₂×C₁₃×M₁₆	—	2	2	no	—
30	C₂×C₁₃×M₁₆	—	2	4	no	—
31	C₂²×C₈×C₁₃	—	2	1	no	—
32	C₂²×C₈×C₁₃	—	2	4	no	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₁₃	D₄	1	2	yes	—
35	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₁₃×#127	C₂×C₄	1	2	yes	—
37	C₁₃×#124	—	2	2	no	—
38	C₁₃×#121	—	2	2	no	—
39	C₁₃×#121	C₂×C₄	1	4	yes	—
40	C₄×D₄×C₁₃	—	2	2	no	—
41	C₄×D₄×C₁₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×#107	C₂×C₄	1	4	yes	—
44	C₁₃×C₄⋊C₈	—	2	2	no	—
45	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
46	C₁₃×#102	C₂×C₄	1	8	yes	—
47	C₁₃×C₂²⋊C₈	—	2	4	no	—
48	C₁₃×C₂²⋊C₈	—	2	4	no	—
49	C₄×C₈×C₁₃	—	2	1	no	—
50	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂³	1	2	yes	—
55	C₄×C₈×C₁₃	C₂³	1	2	yes	—
56	C₄×C₈×C₁₃	C₂³	1	2	yes	—
57	#17981	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₁₃×#41	—	4	4	no	—
59	C₁₃×#41	—	4	4	no	—
60	C₂×Q₈×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	—	2	4	no	—
62	C₂×D₄×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	—	2	2	no	—
64	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₃	—	2	4	no	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
69	C₁₃×D₈	—	4	4	no	—
70	C₁₃×M₁₆	—	4	8	no	—
71	C₁₃×M₁₆	—	4	4	no	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	4	2	no	—
77	C₂×C₈×C₁₃	—	4	8	no	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
82	C₁₃×C₄⋊C₄	—	2	2	no	—
83	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
84	C₁₃×C₂²⋊C₄	—	2	4	no	—
85	C₁₃×C₂²⋊C₄	—	2	4	no	—
86	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₃	#41	1	1	yes	characteristic
89	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
90	#3169	C₂×C₁₃	1	1	yes	characteristic
91	#2931	C₂×C₁₃	1	2	yes	complemented
92	#1586	C₂×C₁₃	1	8	yes	complemented
93	#1566	C₂×C₁₃	1	1	yes	characteristic, complemented
94	#1492	C₂×C₁₃	1	2	yes	complemented
95	C₂×#322	C₂×C₁₃	1	1	yes	characteristic, complemented
96	C₂³×C₁₃	—	2	4	no	—
97	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
98	Q₈×C₁₃	—	4	4	no	—
99	D₄×C₁₃	—	2	8	no	—
100	D₄×C₁₃	—	4	8	no	—
101	D₄×C₁₃	—	4	4	no	—
102	D₄×C₁₃	—	4	4	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
112	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
115	C₈×C₁₃	—	2	4	no	—
116	C₈×C₁₃	—	2	2	no	—
117	C₈×C₁₃	—	2	2	no	—
118	C₈×C₁₃	—	2	2	no	—
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	—	4	4	no	—
121	C₂×#131	—	2	2	no	—
122	#532	C₂²×C₁₃	1	2	yes	—
123	#500	C₄×C₁₃	1	1	yes	characteristic
124	#493	C₄×C₁₃	1	1	yes	characteristic
125	#486	C₄×C₁₃	1	2	yes	—
126	#466	C₄×C₁₃	1	4	yes	—
127	C₄×D₈	C₂²×C₁₃	1	4	yes	—
128	#436	—	2	2	no	—
129	#436	C₂²×C₁₃	1	4	yes	—
130	C₈×D₄	—	2	2	no	—
131	C₈×D₄	C₂²×C₁₃	1	4	yes	—
132	#423	C₂²×C₁₃	1	2	yes	—
133	#408	—	2	2	no	—
134	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
137	#334	C₂²×C₁₃	1	4	yes	—
138	D₄⋊C₈	C₂²×C₁₃	1	8	yes	—
139	#322	C₂²×C₁₃	1	4	yes	—
140	C₂²×C₁₃	—	2	1	no	—
141	C₂²×C₁₃	—	2	1	no	—
142	C₂²×C₁₃	—	4	4	no	—
143	C₂²×C₁₃	—	4	4	no	—
144	C₂²×C₁₃	—	4	4	no	—
145	C₂²×C₁₃	#442	1	2	yes	—
146	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	—	4	4	no	—
150	C₄×C₁₃	#442	1	2	yes	—
151	C₄×C₁₃	#443	1	2	yes	—
152	C₄×C₁₃	#568	1	2	yes	—
153	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
154	C₂×#41	—	2	2	no	—
155	C₂×D₈	C₂×C₄×C₁₃	1	4	yes	—
156	#131	—	4	8	no	—
157	C₂×M₁₆	—	2	2	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂²×C₈	—	2	1	no	—
160	C₂²×C₈	—	2	4	no	—
161	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
162	C₂²×C₈	D₄×C₁₃	1	2	yes	—
163	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
164	#127	C₂×C₄×C₁₃	1	2	yes	—
165	#124	—	2	2	no	—
166	#121	—	2	2	no	—
167	#121	C₂×C₄×C₁₃	1	4	yes	—
168	C₄×D₄	—	2	2	no	—
169	C₄×D₄	C₂³×C₁₃	1	4	yes	—
170	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
171	#107	C₂×C₄×C₁₃	1	4	yes	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
174	#102	C₂×C₄×C₁₃	1	8	yes	—
175	C₂²⋊C₈	—	2	4	no	—
176	C₂²⋊C₈	—	2	4	no	—
177	C₄×C₈	—	2	1	no	—
178	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
179	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
180	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
181	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
182	C₄×C₈	C₂³×C₁₃	1	2	yes	—
183	C₄×C₈	C₂³×C₁₃	1	2	yes	—
184	C₄×C₈	C₂³×C₁₃	1	2	yes	—
185	C₂×C₁₃	—	2	2	no	—
186	C₂×C₁₃	—	8	4	no	—
187	C₂×C₁₃	#2932	1	1	yes	characteristic
188	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
189	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
190	#41	—	4	4	no	—
191	#41	—	4	4	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	C₂²×C₄×C₁₃	1	4	yes	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
201	D₈	—	4	4	no	—
202	M₁₆	—	4	8	no	—
203	M₁₆	—	4	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	8	no	—
210	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
211	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
212	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
213	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
219	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
220	C₄²	C₁₃×#41	1	1	yes	characteristic
221	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
222	C₁₃	#17981	1	1	yes	characteristic, direct product factor, Sylow
223	C₂³	—	2	4	no	—
224	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
225	Q₈	—	4	4	no	—
226	D₄	—	2	8	no	—
227	D₄	—	4	8	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
236	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
238	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
239	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
240	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₈	—	2	2	no	—
247	C₈	—	4	4	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	C₁₃×#442	1	2	yes	—
254	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	4	no	—
258	C₄	C₁₃×#442	1	2	yes	—
259	C₄	C₁₃×#443	1	2	yes	—
260	C₄	C₁₃×#568	1	2	yes	—
261	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	8	4	no	—
264	C₂	C₁₃×#2932	1	1	yes	characteristic
265	C₂	C₂×C₁₃×#442	1	1	yes	characteristic
266	C₂	C₂×C₁₃×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₁₃×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2931	C₂	semidirect
3. C₁₃×#1586	C₂	semidirect
4. C₁₃×#1566	C₂	semidirect
5. C₁₃×#1492	C₂	semidirect
6. C₂×C₁₃×#322	C₂	semidirect
57. #17981	C₁₃	direct (N × H)
91. #2931	C₂×C₁₃	semidirect
92. #1586	C₂×C₁₃	semidirect
93. #1566	C₂×C₁₃	semidirect
94. #1492	C₂×C₁₃	semidirect
95. C₂×#322	C₂×C₁₃	semidirect
222. C₁₃	#17981	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×#3169	2
C₁₃×#2931	2
C₁₃×#1586	2
C₁₃×#1566	2
C₁₃×#1492	2
C₂×C₁₃×#322	2
#17981	13

Subgroups — C13×#17981 #15704320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#17981 #15704320