Subgroups — C₁₃×#9659 #15695998

Order 3328.

590 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1954	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×D₈	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
8	C₄×C₁₃×D₈	—	2	2	no	—
9	C₄×C₁₃×D₈	—	2	2	no	—
10	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₁₃×#102	C₄	1	2	yes	—
13	C₂×C₁₃×C₂²⋊C₈	—	2	4	no	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#334	—	2	2	no	—
18	C₁₃×D₄⋊C₈	—	2	4	no	—
19	C₁₃×#322	—	2	2	no	—
20	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₃	—	2	2	no	—
22	C₂×C₁₃×D₈	—	2	2	no	—
23	C₂×C₁₃×D₈	—	2	2	no	—
24	C₂×C₁₃×D₈	C₈	1	4	yes	complemented
25	C₂²×C₈×C₁₃	—	2	1	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	2	yes	—
30	C₄×D₄×C₁₃	—	2	2	no	—
31	C₄×D₄×C₁₃	—	2	2	no	—
32	C₄×D₄×C₁₃	—	2	2	no	—
33	C₄×D₄×C₁₃	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₁₃×#107	—	2	2	no	—
38	C₁₃×C₄⋊C₈	—	2	2	no	—
39	C₁₃×C₄⋊C₈	—	2	2	no	—
40	C₁₃×#102	—	2	4	no	—
41	C₁₃×C₂²⋊C₈	—	4	8	no	—
42	C₄×C₈×C₁₃	—	2	1	no	—
43	C₄×C₈×C₁₃	—	2	1	no	—
44	C₄×C₈×C₁₃	—	2	1	no	—
45	C₄×C₈×C₁₃	D₄	1	2	yes	—
46	C₄×C₈×C₁₃	D₄	1	2	yes	—
47	C₄×C₈×C₁₃	D₄	1	2	yes	—
48	#9659	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	—	2	2	no	—
50	C₂×D₄×C₁₃	—	2	2	no	—
51	C₂×D₄×C₁₃	—	2	2	no	—
52	C₂×D₄×C₁₃	—	2	2	no	—
53	C₂×D₄×C₁₃	—	2	2	no	—
54	C₂×D₄×C₁₃	—	2	2	no	—
55	C₂×D₄×C₁₃	C₂×C₈	1	4	yes	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	4	2	no	—
58	C₂²×C₄×C₁₃	—	4	2	no	—
59	C₂²×C₄×C₁₃	—	4	4	no	—
60	C₂²×C₄×C₁₃	—	4	4	no	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
64	C₁₃×D₈	—	2	8	no	—
65	C₂×C₈×C₁₃	—	2	1	no	—
66	C₂×C₈×C₁₃	—	2	1	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	4	2	no	—
71	C₂×C₈×C₁₃	—	4	2	no	—
72	C₂×C₈×C₁₃	—	4	4	no	—
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
77	C₁₃×C₄⋊C₄	—	2	2	no	—
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₂²⋊C₄	—	4	2	no	—
80	C₁₃×C₂²⋊C₄	—	4	2	no	—
81	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₃	#41	1	1	yes	characteristic
85	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
86	#1954	C₂×C₁₃	1	2	yes	complemented
87	C₂×#334	C₂×C₁₃	1	1	yes	characteristic, complemented
88	C₂×D₄⋊C₈	C₂×C₁₃	1	2	yes	complemented
89	C₂×#322	C₂×C₁₃	1	1	yes	characteristic, complemented
90	C₂³×C₁₃	—	2	4	no	—
91	C₂³×C₁₃	—	4	2	no	—
92	C₂³×C₁₃	—	4	2	no	—
93	C₂³×C₁₃	—	4	4	no	—
94	C₂³×C₁₃	—	4	4	no	—
95	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
96	D₄×C₁₃	—	2	4	no	—
97	D₄×C₁₃	—	2	4	no	—
98	D₄×C₁₃	—	4	2	no	—
99	D₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	—	4	4	no	—
103	C₂×C₄×C₁₃	—	8	4	no	—
104	C₂×C₄×C₁₃	—	8	2	no	—
105	C₂×C₄×C₁₃	—	8	2	no	—
106	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
116	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
117	C₈×C₁₃	—	2	2	no	—
118	C₈×C₁₃	—	4	2	no	—
119	C₈×C₁₃	—	4	2	no	—
120	C₈×C₁₃	—	8	2	no	—
121	C₈×C₁₃	C₂²⋊C₈	1	4	yes	complemented
122	C₂²×D₈	C₄×C₁₃	1	1	yes	characteristic
123	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
124	C₄×D₈	—	2	2	no	—
125	C₄×D₈	—	2	2	no	—
126	C₂×#107	C₄×C₁₃	1	1	yes	characteristic
127	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
128	C₂×#102	C₄×C₁₃	1	2	yes	—
129	C₂×C₂²⋊C₈	—	2	4	no	—
130	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
133	#334	—	2	2	no	—
134	D₄⋊C₈	—	2	4	no	—
135	#322	—	2	2	no	—
136	C₂²×C₁₃	—	4	8	no	—
137	C₂²×C₁₃	—	4	4	no	—
138	C₂²×C₁₃	—	8	4	no	—
139	C₂²×C₁₃	—	8	2	no	—
140	C₂²×C₁₃	—	8	2	no	—
141	C₂²×C₁₃	#390	1	1	yes	characteristic
142	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
143	C₂²×C₁₃	#436	1	1	yes	characteristic
144	C₂²×C₁₃	#442	1	2	yes	—
145	C₂²×C₁₃	#444	1	2	yes	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	8	2	no	—
149	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₄×C₁₃	#408	1	1	yes	characteristic
151	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
152	C₄×C₁₃	#436	1	1	yes	characteristic
153	C₄×C₁₃	#469	1	2	yes	—
154	C₄×C₁₃	#496	1	2	yes	—
155	C₂²×D₄	C₂×C₄×C₁₃	1	2	yes	—
156	C₂³×C₄	—	2	2	no	—
157	C₂×D₈	—	2	2	no	—
158	C₂×D₈	—	2	2	no	—
159	C₂×D₈	C₈×C₁₃	1	4	yes	complemented
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
163	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
164	C₂²×C₈	D₄×C₁₃	1	2	yes	—
165	C₄×D₄	—	2	2	no	—
166	C₄×D₄	—	2	2	no	—
167	C₄×D₄	—	2	2	no	—
168	C₄×D₄	—	2	2	no	—
169	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
170	C₂×C₂²⋊C₄	—	2	2	no	—
171	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
172	#107	—	2	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	#102	—	2	4	no	—
176	C₂²⋊C₈	—	4	8	no	—
177	C₄×C₈	—	2	1	no	—
178	C₄×C₈	—	2	1	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	D₄×C₁₃	1	2	yes	—
181	C₄×C₈	D₄×C₁₃	1	2	yes	—
182	C₄×C₈	D₄×C₁₃	1	2	yes	—
183	C₂×C₁₃	—	8	4	no	—
184	C₂×C₁₃	#1580	1	2	yes	—
185	C₂×C₁₃	#1594	1	2	yes	—
186	C₂×C₁₃	#1954	1	1	yes	characteristic
187	C₂×C₁₃	#1972	1	1	yes	characteristic
188	C₂×C₁₃	#1975	1	1	yes	characteristic
189	C₂⁴	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	C₂×C₈×C₁₃	1	4	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
204	D₈	—	2	8	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	4	no	—
213	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
214	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
215	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
216	C₂×C₈	C₁₃×M₁₆	1	2	yes	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
222	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
223	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
224	C₄²	C₁₃×#41	1	1	yes	characteristic
225	C₁₃	#9659	1	1	yes	characteristic, direct product factor, Sylow
226	C₂³	—	2	4	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	4	no	—
230	C₂³	—	4	4	no	—
231	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
232	D₄	—	2	4	no	—
233	D₄	—	2	4	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
244	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
245	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
246	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
247	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
249	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
250	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
251	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
252	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
253	C₈	—	2	2	no	—
254	C₈	—	4	2	no	—
255	C₈	—	4	2	no	—
256	C₈	—	8	2	no	—
257	C₈	C₁₃×C₂²⋊C₈	1	4	yes	complemented
258	C₂²	—	4	8	no	—
259	C₂²	—	4	4	no	—
260	C₂²	—	8	4	no	—
261	C₂²	—	8	2	no	—
262	C₂²	—	8	2	no	—
263	C₂²	C₁₃×#390	1	1	yes	characteristic
264	C₂²	C₈×D₄×C₁₃	1	1	yes	characteristic
265	C₂²	C₁₃×#436	1	1	yes	characteristic
266	C₂²	C₁₃×#442	1	2	yes	—
267	C₂²	C₁₃×#444	1	2	yes	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	8	2	no	—
271	C₄	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
272	C₄	C₁₃×#408	1	1	yes	characteristic
273	C₄	C₈×D₄×C₁₃	1	1	yes	characteristic
274	C₄	C₁₃×#436	1	1	yes	characteristic
275	C₄	C₁₃×#469	1	2	yes	—
276	C₄	C₁₃×#496	1	2	yes	—
277	C₂	—	8	4	no	—
278	C₂	C₁₃×#1580	1	2	yes	—
279	C₂	C₁₃×#1594	1	2	yes	—
280	C₂	C₁₃×#1954	1	1	yes	characteristic
281	C₂	C₁₃×#1972	1	1	yes	characteristic
282	C₂	C₁₃×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂²×C₁₃×D₈	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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×#1954	C₂	semidirect
3. C₂×C₁₃×#334	C₂	semidirect
4. C₂×C₁₃×D₄⋊C₈	C₂	semidirect
5. C₂×C₁₃×#322	C₂	semidirect
24. C₂×C₁₃×D₈	C₈	semidirect
48. #9659	C₁₃	direct (N × H)
86. #1954	C₂×C₁₃	semidirect
87. C₂×#334	C₂×C₁₃	semidirect
88. C₂×D₄⋊C₈	C₂×C₁₃	semidirect
89. C₂×#322	C₂×C₁₃	semidirect
121. C₈×C₁₃	C₂²⋊C₈	semidirect
159. C₂×D₈	C₈×C₁₃	semidirect
225. C₁₃	#9659	direct (N × H)
257. C₈	C₁₃×C₂²⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₃×D₈	2
C₁₃×#1954	2
C₂×C₁₃×#334	2
C₂×C₁₃×D₄⋊C₈	2
C₂×C₁₃×#322	2
#9659	13

Subgroups — C13×#9659 #15695998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9659 #15695998