Subgroups — C₁₃×#13255 #15699594

Order 3328.

498 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#360	C₂	1	4	yes	maximal, complemented
4	C₄×C₁₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#135	—	2	1	no	—
6	C₂×C₁₃×#135	C₂²	1	2	yes	—
7	C₁₃×#532	—	2	1	no	—
8	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
9	C₁₃×#495	—	2	1	no	—
10	C₁₃×#495	C₄	1	2	yes	complemented
11	C₁₃×#494	C₄	1	1	yes	characteristic, complemented
12	C₁₃×#493	C₄	1	1	yes	characteristic, complemented
13	C₁₃×#492	—	2	1	no	—
14	C₂×C₁₃×#106	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
17	C₁₃×C₄⋊C₁₆	—	2	1	no	—
18	C₁₃×C₄⋊C₁₆	—	2	1	no	—
19	C₁₃×#360	—	2	8	no	—
20	C₁₃×#340	—	2	1	no	—
21	C₂×C₁₃×#41	—	2	2	no	—
22	C₁₃×#135	—	2	4	no	—
23	C₁₃×#135	—	2	4	no	—
24	C₁₃×#135	—	4	4	no	—
25	C₂×C₁₃×Q₁₆	—	2	1	no	—
26	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₃×SD₁₆	—	2	2	no	—
28	C₂×C₁₃×SD₁₆	—	2	2	no	—
29	C₂×C₁₃×D₈	—	2	1	no	—
30	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₁₃×C₄⋊Q₈	—	2	1	no	—
34	C₁₃×#127	—	2	1	no	—
35	C₁₃×#124	—	2	2	no	—
36	C₁₃×#121	—	4	1	no	—
37	C₁₃×#121	—	4	1	no	—
38	C₄×D₄×C₁₃	—	4	1	no	—
39	C₄×D₄×C₁₃	—	4	1	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₄²×C₁₃	—	4	1	no	—
42	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
43	C₁₃×M₃₂	—	2	4	no	—
44	C₂×C₁₃×C₁₆	—	2	2	no	—
45	C₁₃×#106	—	2	2	no	—
46	C₁₃×#106	D₄	1	4	yes	complemented
47	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
51	C₁₃×#95	—	4	1	no	—
52	#13255	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₁₃×#41	—	4	4	no	—
54	C₁₃×#41	—	4	4	no	—
55	C₂×Q₈×C₁₃	—	2	2	no	—
56	C₂×D₄×C₁₃	—	2	2	no	—
57	C₂×D₄×C₁₃	—	4	2	no	—
58	C₂×D₄×C₁₃	—	4	1	no	—
59	C₂×D₄×C₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₁₃×Q₁₆	—	2	4	no	—
66	C₁₃×Q₁₆	—	4	2	no	—
67	C₁₃×SD₁₆	—	4	4	no	—
68	C₁₃×SD₁₆	—	4	2	no	—
69	C₁₃×SD₁₆	—	4	2	no	—
70	C₁₃×D₈	—	2	4	no	—
71	C₁₃×D₈	—	4	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	#41	1	2	yes	—
79	C₁₃×C₄⋊C₄	—	2	4	no	—
80	C₁₃×C₄⋊C₄	—	4	1	no	—
81	C₁₃×C₄⋊C₄	—	4	1	no	—
82	C₁₃×C₄⋊C₄	—	4	2	no	—
83	C₁₃×C₂²⋊C₄	—	4	2	no	—
84	C₁₃×C₂²⋊C₄	—	4	2	no	—
85	C₄²×C₁₃	—	8	2	no	—
86	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
89	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
90	C₁₃×C₁₆	—	4	2	no	—
91	#3147	C₂×C₁₃	1	1	yes	characteristic
92	#2150	C₂×C₁₃	1	1	yes	characteristic, complemented
93	C₂×#360	C₂×C₁₃	1	4	yes	complemented
94	C₄×#106	C₂×C₁₃	1	1	yes	characteristic, complemented
95	C₂³×C₁₃	—	4	2	no	—
96	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
97	Q₈×C₁₃	—	4	2	no	—
98	Q₈×C₁₃	—	4	2	no	—
99	D₄×C₁₃	—	4	2	no	—
100	D₄×C₁₃	—	4	2	no	—
101	D₄×C₁₃	—	8	2	no	—
102	D₄×C₁₃	—	8	4	no	—
103	D₄×C₁₃	—	8	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	4	1	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	8	2	no	—
109	C₂×C₄×C₁₃	—	8	2	no	—
110	C₂×C₄×C₁₃	—	8	4	no	—
111	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#121	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
119	C₈×C₁₃	—	2	1	no	—
120	C₈×C₁₃	—	2	1	no	—
121	C₈×C₁₃	C₄×D₄	1	2	yes	—
122	C₈×C₁₃	#124	1	1	yes	characteristic
123	C₈×C₁₃	#127	1	1	yes	characteristic
124	C₂×#135	—	2	1	no	—
125	C₂×#135	C₂²×C₁₃	1	2	yes	—
126	#532	—	2	1	no	—
127	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
128	#495	—	2	1	no	—
129	#495	C₄×C₁₃	1	2	yes	complemented
130	#494	C₄×C₁₃	1	1	yes	characteristic, complemented
131	#493	C₄×C₁₃	1	1	yes	characteristic, complemented
132	#492	—	2	1	no	—
133	C₂×#106	C₂²×C₁₃	1	2	yes	—
134	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
135	C₄×C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₁₆	—	2	1	no	—
137	C₄⋊C₁₆	—	2	1	no	—
138	#360	—	2	8	no	—
139	#340	—	2	1	no	—
140	C₂²×C₁₃	—	2	1	no	—
141	C₂²×C₁₃	—	2	1	no	—
142	C₂²×C₁₃	—	8	2	no	—
143	C₂²×C₁₃	—	8	2	no	—
144	C₂²×C₁₃	—	8	2	no	—
145	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
146	C₂²×C₁₃	#459	1	2	yes	—
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	8	4	no	—
151	C₄×C₁₃	—	8	2	no	—
152	C₄×C₁₃	#360	1	2	yes	complemented
153	C₄×C₁₃	#390	1	1	yes	characteristic
154	C₄×C₁₃	#418	1	1	yes	characteristic
155	C₄×C₁₃	#460	1	2	yes	—
156	C₂×#41	—	2	2	no	—
157	#135	—	2	4	no	—
158	#135	—	2	4	no	—
159	#135	—	4	4	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂×SD₁₆	—	2	2	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
166	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
168	C₄⋊Q₈	—	2	1	no	—
169	#127	—	2	1	no	—
170	#124	—	2	2	no	—
171	#121	—	4	1	no	—
172	#121	—	4	1	no	—
173	C₄×D₄	—	4	1	no	—
174	C₄×D₄	—	4	1	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄²	—	4	1	no	—
177	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
178	M₃₂	—	2	4	no	—
179	C₂×C₁₆	—	2	2	no	—
180	#106	—	2	2	no	—
181	#106	D₄×C₁₃	1	4	yes	complemented
182	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
183	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
184	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
185	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
186	#95	—	4	1	no	—
187	C₂×C₁₃	—	2	2	no	—
188	C₂×C₁₃	—	16	2	no	—
189	C₂×C₁₃	#1982	1	1	yes	characteristic
190	C₂×C₁₃	C₂×#360	1	1	yes	characteristic
191	C₂×C₁₃	#2145	1	1	yes	characteristic
192	#41	—	4	4	no	—
193	#41	—	4	4	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
202	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
204	Q₁₆	—	2	4	no	—
205	Q₁₆	—	4	2	no	—
206	SD₁₆	—	4	4	no	—
207	SD₁₆	—	4	2	no	—
208	SD₁₆	—	4	2	no	—
209	D₈	—	2	4	no	—
210	D₈	—	4	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
215	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
216	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
217	C₂×C₈	C₁₃×#41	1	2	yes	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₂²⋊C₄	—	4	2	no	—
223	C₂²⋊C₄	—	4	2	no	—
224	C₄²	—	8	2	no	—
225	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
226	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
227	C₄²	C₁₃×SD₁₆	1	1	yes	characteristic
228	C₄²	C₁₃×SD₁₆	1	1	yes	characteristic
229	C₁₆	—	4	2	no	—
230	C₁₃	#13255	1	1	yes	characteristic, direct product factor, Sylow
231	C₂³	—	4	2	no	—
232	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
233	Q₈	—	4	2	no	—
234	Q₈	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	8	2	no	—
238	D₄	—	8	4	no	—
239	D₄	—	8	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
248	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
249	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
250	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
251	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×C₄	C₁₃×#121	1	2	yes	—
253	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
254	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	C₄×D₄×C₁₃	1	2	yes	—
258	C₈	C₁₃×#124	1	1	yes	characteristic
259	C₈	C₁₃×#127	1	1	yes	characteristic
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	8	2	no	—
263	C₂²	—	8	2	no	—
264	C₂²	—	8	2	no	—
265	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
266	C₂²	C₁₃×#459	1	2	yes	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	8	4	no	—
271	C₄	—	8	2	no	—
272	C₄	C₁₃×#360	1	2	yes	complemented
273	C₄	C₁₃×#390	1	1	yes	characteristic
274	C₄	C₁₃×#418	1	1	yes	characteristic
275	C₄	C₁₃×#460	1	2	yes	—
276	C₂	—	2	2	no	—
277	C₂	—	16	2	no	—
278	C₂	C₁₃×#1982	1	1	yes	characteristic
279	C₂	C₂×C₁₃×#360	1	1	yes	characteristic
280	C₂	C₁₃×#2145	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₁₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2150	C₂	semidirect
3. C₂×C₁₃×#360	C₂	semidirect
4. C₄×C₁₃×#106	C₂	semidirect
10. C₁₃×#495	C₄	semidirect
11. C₁₃×#494	C₄	semidirect
12. C₁₃×#493	C₄	semidirect
46. C₁₃×#106	D₄	semidirect
52. #13255	C₁₃	direct (N × H)
92. #2150	C₂×C₁₃	semidirect
93. C₂×#360	C₂×C₁₃	semidirect
94. C₄×#106	C₂×C₁₃	semidirect
129. #495	C₄×C₁₃	semidirect
130. #494	C₄×C₁₃	semidirect
131. #493	C₄×C₁₃	semidirect
152. C₄×C₁₃	#360	semidirect
181. #106	D₄×C₁₃	semidirect
230. C₁₃	#13255	direct (N × H)
272. C₄	C₁₃×#360	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#3147	2
C₁₃×#2150	2
C₂×C₁₃×#360	2
C₄×C₁₃×#106	2
#13255	13

Subgroups — C13×#13255 #15699594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13255 #15699594