Subgroups — C₁₃×#13440 #15699779

Order 3328.

398 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×SD₃₂	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1979	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1978	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×SD₃₂	—	2	2	no	—
11	C₂×C₁₃×SD₃₂	—	2	2	no	—
12	C₂×C₁₃×SD₃₂	—	2	2	no	—
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
17	C₁₃×#419	—	2	1	no	—
18	C₁₃×#418	—	2	1	no	—
19	C₂×C₁₃×#101	—	2	1	no	—
20	C₂×C₁₃×#100	—	2	1	no	—
21	C₁₃×#405	C₂²	1	1	yes	characteristic
22	C₁₃×C₄⋊C₁₆	—	2	2	no	—
23	C₁₃×#362	—	2	2	no	—
24	C₁₃×#361	—	2	2	no	—
25	C₂²×Q₈×C₁₃	—	2	1	no	—
26	C₂²×D₄×C₁₃	—	2	1	no	—
27	C₂×C₁₃×Q₁₆	—	2	1	no	—
28	C₂×C₁₃×Q₁₆	—	2	1	no	—
29	C₂×C₁₃×Q₁₆	—	2	1	no	—
30	C₂×C₁₃×Q₁₆	—	2	1	no	—
31	C₂×C₁₃×Q₁₆	—	2	1	no	—
32	C₂×C₁₃×Q₁₆	D₄	1	2	yes	—
33	C₂×C₁₃×D₈	—	2	1	no	—
34	C₂×C₁₃×D₈	—	2	1	no	—
35	C₂×C₁₃×D₈	—	2	1	no	—
36	C₂×C₁₃×D₈	—	2	1	no	—
37	C₂×C₁₃×D₈	—	2	1	no	—
38	C₂×C₁₃×D₈	D₄	1	2	yes	—
39	C₂×C₁₃×M₁₆	—	2	1	no	—
40	C₂×C₁₃×M₁₆	—	2	1	no	—
41	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×#117	D₄	1	1	yes	characteristic
44	C₁₃×SD₃₂	—	2	8	no	—
45	C₁₃×M₃₂	—	2	1	no	—
46	C₁₃×M₃₂	—	2	1	no	—
47	C₂×C₁₃×C₁₆	—	2	1	no	—
48	C₂×C₁₃×C₁₆	—	2	1	no	—
49	C₂×C₁₃×C₁₆	—	2	1	no	—
50	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
51	C₁₃×#108	—	2	1	no	—
52	C₁₃×#108	—	2	1	no	—
53	C₁₃×#108	—	2	1	no	—
54	C₁₃×#108	—	2	1	no	—
55	C₁₃×#103	—	4	2	no	—
56	C₁₃×#102	—	4	2	no	—
57	C₁₃×#101	—	4	2	no	—
58	C₁₃×#100	—	4	2	no	—
59	C₁₃×#97	—	2	1	no	—
60	C₄×C₈×C₁₃	—	2	1	no	—
61	#13440	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂⁴×C₁₃	—	4	1	no	—
63	C₂×Q₈×C₁₃	—	2	2	no	—
64	C₂×Q₈×C₁₃	—	4	1	no	—
65	C₂×Q₈×C₁₃	—	4	1	no	—
66	C₂×Q₈×C₁₃	—	4	1	no	—
67	C₂×Q₈×C₁₃	—	4	1	no	—
68	C₂×Q₈×C₁₃	—	4	1	no	—
69	C₂×D₄×C₁₃	—	2	2	no	—
70	C₂×D₄×C₁₃	—	4	1	no	—
71	C₂×D₄×C₁₃	—	4	1	no	—
72	C₂×D₄×C₁₃	—	4	1	no	—
73	C₂×D₄×C₁₃	—	4	1	no	—
74	C₂×D₄×C₁₃	—	4	1	no	—
75	C₂²×C₄×C₁₃	—	4	1	no	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₁₃×Q₁₆	—	2	2	no	—
78	C₁₃×Q₁₆	—	2	2	no	—
79	C₁₃×Q₁₆	—	4	1	no	—
80	C₁₃×Q₁₆	—	4	1	no	—
81	C₁₃×D₈	—	2	2	no	—
82	C₁₃×D₈	—	2	2	no	—
83	C₁₃×D₈	—	4	1	no	—
84	C₁₃×D₈	—	4	1	no	—
85	C₁₃×M₁₆	—	2	1	no	—
86	C₁₃×M₁₆	—	2	1	no	—
87	C₁₃×M₁₆	—	4	1	no	—
88	C₁₃×M₁₆	—	4	1	no	—
89	C₁₃×M₁₆	—	4	1	no	—
90	C₁₃×M₁₆	—	4	1	no	—
91	C₂×C₈×C₁₃	—	2	1	no	—
92	C₂×C₈×C₁₃	—	4	1	no	—
93	C₂×C₈×C₁₃	—	4	1	no	—
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
98	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
100	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
101	C₁₃×C₂²⋊C₄	—	2	1	no	—
102	C₄²×C₁₃	—	2	1	no	—
103	C₁₃×C₁₆	—	2	2	no	—
104	C₁₃×C₁₆	—	4	1	no	—
105	C₂²×SD₃₂	C₂×C₁₃	1	1	yes	characteristic
106	#2151	C₂×C₁₃	1	1	yes	characteristic, complemented
107	C₂×#362	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₂×#361	C₂×C₁₃	1	1	yes	characteristic
109	#1979	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#1978	C₂×C₁₃	1	1	yes	characteristic
111	#1381	C₂×C₁₃	1	1	yes	characteristic, complemented
112	C₂³×C₁₃	—	4	2	no	—
113	C₂³×C₁₃	—	8	1	no	—
114	C₂³×C₁₃	—	8	1	no	—
115	C₂³×C₁₃	—	16	2	no	—
116	C₂³×C₁₃	#135	1	1	yes	characteristic
117	Q₈×C₁₃	—	4	2	no	—
118	Q₈×C₁₃	—	4	2	no	—
119	Q₈×C₁₃	—	8	1	no	—
120	Q₈×C₁₃	—	8	1	no	—
121	D₄×C₁₃	—	4	2	no	—
122	D₄×C₁₃	—	4	2	no	—
123	D₄×C₁₃	—	8	1	no	—
124	D₄×C₁₃	—	8	1	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	8	1	no	—
128	C₂×C₄×C₁₃	—	8	1	no	—
129	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₈×C₁₃	—	4	1	no	—
136	C₈×C₁₃	—	8	1	no	—
137	C₈×C₁₃	—	8	1	no	—
138	C₈×C₁₃	#120	1	1	yes	characteristic
139	C₈×C₁₃	#123	1	1	yes	characteristic
140	C₈×C₁₃	#123	1	1	yes	characteristic
141	C₈×C₁₃	#124	1	1	yes	characteristic
142	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
143	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×SD₃₂	—	2	2	no	—
145	C₂×SD₃₂	—	2	2	no	—
146	C₂×SD₃₂	—	2	2	no	—
147	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
148	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
151	#419	—	2	1	no	—
152	#418	—	2	1	no	—
153	C₂×#101	—	2	1	no	—
154	C₂×#100	—	2	1	no	—
155	#405	C₂²×C₁₃	1	1	yes	characteristic
156	C₄⋊C₁₆	—	2	2	no	—
157	#362	—	2	2	no	—
158	#361	—	2	2	no	—
159	C₂²×C₁₃	—	2	2	no	—
160	C₂²×C₁₃	—	8	2	no	—
161	C₂²×C₁₃	—	16	2	no	—
162	C₂²×C₁₃	—	16	2	no	—
163	C₂²×C₁₃	—	16	2	no	—
164	C₂²×C₁₃	#463	1	1	yes	characteristic
165	C₂²×C₁₃	#463	1	1	yes	characteristic
166	C₂²×C₁₃	#466	1	1	yes	characteristic
167	C₄×C₁₃	—	4	2	no	—
168	C₄×C₁₃	—	8	2	no	—
169	C₄×C₁₃	#394	1	1	yes	characteristic
170	C₄×C₁₃	#459	1	1	yes	characteristic
171	C₄×C₁₃	#459	1	1	yes	characteristic
172	C₄×C₁₃	#469	1	1	yes	characteristic
173	C₂²×Q₈	—	2	1	no	—
174	C₂²×D₄	—	2	1	no	—
175	C₂×Q₁₆	—	2	1	no	—
176	C₂×Q₁₆	—	2	1	no	—
177	C₂×Q₁₆	—	2	1	no	—
178	C₂×Q₁₆	—	2	1	no	—
179	C₂×Q₁₆	—	2	1	no	—
180	C₂×Q₁₆	D₄×C₁₃	1	2	yes	—
181	C₂×D₈	—	2	1	no	—
182	C₂×D₈	—	2	1	no	—
183	C₂×D₈	—	2	1	no	—
184	C₂×D₈	—	2	1	no	—
185	C₂×D₈	—	2	1	no	—
186	C₂×D₈	D₄×C₁₃	1	2	yes	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
191	#117	D₄×C₁₃	1	1	yes	characteristic
192	SD₃₂	—	2	8	no	—
193	M₃₂	—	2	1	no	—
194	M₃₂	—	2	1	no	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	—	2	1	no	—
197	C₂×C₁₆	—	2	1	no	—
198	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
199	#108	—	2	1	no	—
200	#108	—	2	1	no	—
201	#108	—	2	1	no	—
202	#108	—	2	1	no	—
203	#103	—	4	2	no	—
204	#102	—	4	2	no	—
205	#101	—	4	2	no	—
206	#100	—	4	2	no	—
207	#97	—	2	1	no	—
208	C₄×C₈	—	2	1	no	—
209	C₂×C₁₃	—	2	1	no	—
210	C₂×C₁₃	—	2	1	no	—
211	C₂×C₁₃	—	16	2	no	—
212	C₂×C₁₃	#2054	1	1	yes	characteristic
213	C₂×C₁₃	#2223	1	2	yes	—
214	C₂⁴	—	4	1	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	4	1	no	—
217	C₂×Q₈	—	4	1	no	—
218	C₂×Q₈	—	4	1	no	—
219	C₂×Q₈	—	4	1	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	4	1	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂×D₄	—	4	1	no	—
226	C₂×D₄	—	4	1	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
229	Q₁₆	—	2	2	no	—
230	Q₁₆	—	2	2	no	—
231	Q₁₆	—	4	1	no	—
232	Q₁₆	—	4	1	no	—
233	D₈	—	2	2	no	—
234	D₈	—	2	2	no	—
235	D₈	—	4	1	no	—
236	D₈	—	4	1	no	—
237	M₁₆	—	2	1	no	—
238	M₁₆	—	2	1	no	—
239	M₁₆	—	4	1	no	—
240	M₁₆	—	4	1	no	—
241	M₁₆	—	4	1	no	—
242	M₁₆	—	4	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
247	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
248	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
249	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
250	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
251	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
252	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₁₆	—	2	2	no	—
256	C₁₆	—	4	1	no	—
257	C₁₃	#13440	1	1	yes	characteristic, direct product factor, Sylow
258	C₂³	—	4	2	no	—
259	C₂³	—	8	1	no	—
260	C₂³	—	8	1	no	—
261	C₂³	—	16	2	no	—
262	C₂³	C₁₃×#135	1	1	yes	characteristic
263	Q₈	—	4	2	no	—
264	Q₈	—	4	2	no	—
265	Q₈	—	8	1	no	—
266	Q₈	—	8	1	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	D₄	—	8	1	no	—
270	D₄	—	8	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
276	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
277	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
278	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
279	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
280	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
281	C₈	—	4	1	no	—
282	C₈	—	8	1	no	—
283	C₈	—	8	1	no	—
284	C₈	C₁₃×#120	1	1	yes	characteristic
285	C₈	C₁₃×#123	1	1	yes	characteristic
286	C₈	C₁₃×#123	1	1	yes	characteristic
287	C₈	C₁₃×#124	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	16	2	no	—
291	C₂²	—	16	2	no	—
292	C₂²	—	16	2	no	—
293	C₂²	C₁₃×#463	1	1	yes	characteristic
294	C₂²	C₁₃×#463	1	1	yes	characteristic
295	C₂²	C₁₃×#466	1	1	yes	characteristic
296	C₄	—	4	2	no	—
297	C₄	—	8	2	no	—
298	C₄	C₁₃×#394	1	1	yes	characteristic
299	C₄	C₁₃×#459	1	1	yes	characteristic
300	C₄	C₁₃×#459	1	1	yes	characteristic
301	C₄	C₁₃×#469	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	16	2	no	—
305	C₂	C₁₃×#2054	1	1	yes	characteristic
306	C₂	C₁₃×#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2151	C₂	semidirect
3. C₂×C₁₃×#362	C₂	semidirect
5. C₁₃×#1979	C₂	semidirect
7. C₁₃×#1381	C₂	semidirect
61. #13440	C₁₃	direct (N × H)
106. #2151	C₂×C₁₃	semidirect
107. C₂×#362	C₂×C₁₃	semidirect
109. #1979	C₂×C₁₃	semidirect
111. #1381	C₂×C₁₃	semidirect
257. C₁₃	#13440	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×SD₃₂	2
C₁₃×#2151	2
C₂×C₁₃×#362	2
C₂×C₁₃×#361	2
C₁₃×#1979	2
C₁₃×#1978	2
C₁₃×#1381	2
#13440	13

Subgroups — C13×#13440 #15699779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13440 #15699779