Subgroups — C₁₃×#13390 #15699729

Order 3328.

382 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#473	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2161	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2117	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1813	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1388	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#137	—	2	1	no	—
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₁₃×#473	—	2	2	no	—
14	C₁₃×#473	—	2	2	no	—
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
18	C₁₃×#413	—	2	1	no	—
19	C₂×C₁₃×#101	—	2	1	no	—
20	C₁₃×#368	—	2	2	no	—
21	C₁₃×#362	—	2	2	no	—
22	C₁₃×#362	—	2	2	no	—
23	C₁₃×#350	—	2	2	no	—
24	C₁₃×#338	—	2	1	no	—
25	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
26	C₂²×Q₈×C₁₃	—	2	1	no	—
27	C₁₃×#137	—	2	4	no	—
28	C₁₃×#137	—	4	2	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	—	2	1	no	—
33	C₂×C₁₃×Q₁₆	—	2	1	no	—
34	C₂×C₁₃×Q₁₆	—	4	1	no	—
35	C₂×C₁₃×Q₁₆	D₄	1	2	yes	—
36	C₂×C₁₃×SD₁₆	—	4	1	no	—
37	C₁₃×#131	—	2	1	no	—
38	C₁₃×#131	—	2	1	no	—
39	C₁₃×#131	—	2	1	no	—
40	C₁₃×#131	—	2	1	no	—
41	C₂×C₁₃×M₁₆	—	2	1	no	—
42	C₂×C₁₃×M₁₆	—	2	1	no	—
43	C₂×C₁₃×M₁₆	—	2	1	no	—
44	C₂×C₁₃×M₁₆	—	2	1	no	—
45	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	—	2	1	no	—
47	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×M₃₂	—	2	1	no	—
49	C₁₃×M₃₂	—	2	1	no	—
50	C₁₃×M₃₂	—	2	1	no	—
51	C₁₃×M₃₂	D₄	1	2	yes	—
52	C₂×C₁₃×C₁₆	—	2	1	no	—
53	C₂×C₁₃×C₁₆	—	2	1	no	—
54	C₁₃×#108	—	2	1	no	—
55	C₁₃×#108	—	2	1	no	—
56	C₁₃×#108	—	2	1	no	—
57	C₁₃×#108	—	2	1	no	—
58	C₁₃×#108	—	2	1	no	—
59	C₁₃×#108	D₄	1	2	yes	—
60	C₁₃×#101	—	2	4	no	—
61	C₁₃×#101	—	4	1	no	—
62	C₁₃×#100	—	4	1	no	—
63	#13390	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₁₃×#41	—	2	1	no	—
65	C₁₃×#41	—	2	1	no	—
66	C₁₃×#41	—	2	1	no	—
67	C₁₃×#41	—	2	1	no	—
68	C₂×Q₈×C₁₃	—	2	2	no	—
69	C₂×Q₈×C₁₃	—	4	1	no	—
70	C₂×Q₈×C₁₃	—	4	1	no	—
71	C₂×Q₈×C₁₃	—	4	1	no	—
72	C₂×Q₈×C₁₃	—	4	1	no	—
73	C₂×Q₈×C₁₃	—	4	1	no	—
74	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
75	C₂×D₄×C₁₃	—	2	1	no	—
76	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	—	2	1	no	—
78	C₂²×C₄×C₁₃	—	4	1	no	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₁₃×Q₁₆	—	2	2	no	—
81	C₁₃×Q₁₆	—	2	2	no	—
82	C₁₃×Q₁₆	—	4	4	no	—
83	C₁₃×Q₁₆	—	4	1	no	—
84	C₁₃×Q₁₆	—	4	1	no	—
85	C₁₃×SD₁₆	—	4	4	no	—
86	C₁₃×M₁₆	—	2	1	no	—
87	C₁₃×M₁₆	—	2	2	no	—
88	C₁₃×M₁₆	—	2	1	no	—
89	C₁₃×M₁₆	—	4	1	no	—
90	C₁₃×M₁₆	—	4	1	no	—
91	C₁₃×M₁₆	—	4	1	no	—
92	C₁₃×M₁₆	—	4	1	no	—
93	C₁₃×M₁₆	—	4	1	no	—
94	C₁₃×M₁₆	—	4	1	no	—
95	C₁₃×M₁₆	—	4	1	no	—
96	C₂×C₈×C₁₃	—	2	1	no	—
97	C₂×C₈×C₁₃	—	4	1	no	—
98	C₂×C₈×C₁₃	—	4	1	no	—
99	C₂×C₈×C₁₃	—	4	1	no	—
100	C₂×C₈×C₁₃	—	4	1	no	—
101	C₂×C₈×C₁₃	—	4	1	no	—
102	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
106	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
107	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
108	C₁₃×C₁₆	—	2	2	no	—
109	C₁₃×C₁₆	—	4	1	no	—
110	C₂×#473	C₂×C₁₃	1	1	yes	characteristic
111	#2161	C₂×C₁₃	1	1	yes	characteristic, complemented
112	C₂×#362	C₂×C₁₃	1	1	yes	characteristic, complemented
113	C₂×#362	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#2117	C₂×C₁₃	1	1	yes	characteristic
115	#1813	C₂×C₁₃	1	1	yes	characteristic
116	#1388	C₂×C₁₃	1	1	yes	characteristic, complemented
117	C₂³×C₁₃	—	2	1	no	—
118	C₂³×C₁₃	#137	1	1	yes	characteristic
119	Q₈×C₁₃	—	2	1	no	—
120	Q₈×C₁₃	—	2	1	no	—
121	Q₈×C₁₃	—	4	2	no	—
122	Q₈×C₁₃	—	4	2	no	—
123	Q₈×C₁₃	—	8	1	no	—
124	Q₈×C₁₃	—	8	1	no	—
125	D₄×C₁₃	—	2	1	no	—
126	D₄×C₁₃	—	2	1	no	—
127	D₄×C₁₃	—	4	1	no	—
128	D₄×C₁₃	—	4	1	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	4	1	no	—
131	C₂×C₄×C₁₃	—	4	1	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	8	1	no	—
134	C₂×C₄×C₁₃	—	8	1	no	—
135	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
140	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₈×C₁₃	—	4	1	no	—
142	C₈×C₁₃	—	4	2	no	—
143	C₈×C₁₃	—	8	1	no	—
144	C₈×C₁₃	—	8	1	no	—
145	C₈×C₁₃	#121	1	1	yes	characteristic
146	C₈×C₁₃	#121	1	1	yes	characteristic
147	C₈×C₁₃	#123	1	1	yes	characteristic
148	C₈×C₁₃	#124	1	1	yes	characteristic
149	C₂×#137	—	2	1	no	—
150	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
152	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
154	#473	—	2	2	no	—
155	#473	—	2	2	no	—
156	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
159	#413	—	2	1	no	—
160	C₂×#101	—	2	1	no	—
161	#368	—	2	2	no	—
162	#362	—	2	2	no	—
163	#362	—	2	2	no	—
164	#350	—	2	2	no	—
165	#338	—	2	1	no	—
166	C₂²×C₁₃	—	4	1	no	—
167	C₂²×C₁₃	—	4	1	no	—
168	C₂²×C₁₃	—	8	1	no	—
169	C₂²×C₁₃	#452	1	1	yes	characteristic
170	C₂²×C₁₃	#468	1	1	yes	characteristic
171	C₂²×C₁₃	#480	1	1	yes	characteristic
172	C₄×C₁₃	—	4	1	no	—
173	C₄×C₁₃	—	8	2	no	—
174	C₄×C₁₃	#394	1	1	yes	characteristic
175	C₄×C₁₃	#447	1	1	yes	characteristic
176	C₄×C₁₃	#466	1	1	yes	characteristic
177	C₄×C₁₃	#482	1	1	yes	characteristic
178	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
179	C₂²×Q₈	—	2	1	no	—
180	#137	—	2	4	no	—
181	#137	—	4	2	no	—
182	C₂×Q₁₆	—	2	1	no	—
183	C₂×Q₁₆	—	2	1	no	—
184	C₂×Q₁₆	—	2	1	no	—
185	C₂×Q₁₆	—	2	1	no	—
186	C₂×Q₁₆	—	2	1	no	—
187	C₂×Q₁₆	—	4	1	no	—
188	C₂×Q₁₆	D₄×C₁₃	1	2	yes	—
189	C₂×SD₁₆	—	4	1	no	—
190	#131	—	2	1	no	—
191	#131	—	2	1	no	—
192	#131	—	2	1	no	—
193	#131	—	2	1	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₂×M₁₆	—	2	1	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
199	C₂²×C₈	—	2	1	no	—
200	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
201	M₃₂	—	2	1	no	—
202	M₃₂	—	2	1	no	—
203	M₃₂	—	2	1	no	—
204	M₃₂	D₄×C₁₃	1	2	yes	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	#108	—	2	1	no	—
208	#108	—	2	1	no	—
209	#108	—	2	1	no	—
210	#108	—	2	1	no	—
211	#108	—	2	1	no	—
212	#108	D₄×C₁₃	1	2	yes	—
213	#101	—	2	4	no	—
214	#101	—	4	1	no	—
215	#100	—	4	1	no	—
216	C₂×C₁₃	—	2	1	no	—
217	C₂×C₁₃	—	2	1	no	—
218	C₂×C₁₃	—	2	1	no	—
219	C₂×C₁₃	—	8	1	no	—
220	C₂×C₁₃	#2033	1	1	yes	characteristic
221	#41	—	2	1	no	—
222	#41	—	2	1	no	—
223	#41	—	2	1	no	—
224	#41	—	2	1	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
237	Q₁₆	—	2	2	no	—
238	Q₁₆	—	2	2	no	—
239	Q₁₆	—	4	4	no	—
240	Q₁₆	—	4	1	no	—
241	Q₁₆	—	4	1	no	—
242	SD₁₆	—	4	4	no	—
243	M₁₆	—	2	1	no	—
244	M₁₆	—	2	2	no	—
245	M₁₆	—	2	1	no	—
246	M₁₆	—	4	1	no	—
247	M₁₆	—	4	1	no	—
248	M₁₆	—	4	1	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
263	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
264	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
265	C₁₆	—	2	2	no	—
266	C₁₆	—	4	1	no	—
267	C₁₃	#13390	1	1	yes	characteristic, direct product factor, Sylow
268	C₂³	—	2	1	no	—
269	C₂³	C₁₃×#137	1	1	yes	characteristic
270	Q₈	—	2	1	no	—
271	Q₈	—	2	1	no	—
272	Q₈	—	4	2	no	—
273	Q₈	—	4	2	no	—
274	Q₈	—	8	1	no	—
275	Q₈	—	8	1	no	—
276	D₄	—	2	1	no	—
277	D₄	—	2	1	no	—
278	D₄	—	4	1	no	—
279	D₄	—	4	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
287	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
288	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
289	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
290	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
291	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
292	C₈	—	4	1	no	—
293	C₈	—	4	2	no	—
294	C₈	—	8	1	no	—
295	C₈	—	8	1	no	—
296	C₈	C₁₃×#121	1	1	yes	characteristic
297	C₈	C₁₃×#121	1	1	yes	characteristic
298	C₈	C₁₃×#123	1	1	yes	characteristic
299	C₈	C₁₃×#124	1	1	yes	characteristic
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	8	1	no	—
303	C₂²	C₁₃×#452	1	1	yes	characteristic
304	C₂²	C₁₃×#468	1	1	yes	characteristic
305	C₂²	C₁₃×#480	1	1	yes	characteristic
306	C₄	—	4	1	no	—
307	C₄	—	8	2	no	—
308	C₄	C₁₃×#394	1	1	yes	characteristic
309	C₄	C₁₃×#447	1	1	yes	characteristic
310	C₄	C₁₃×#466	1	1	yes	characteristic
311	C₄	C₁₃×#482	1	1	yes	characteristic
312	C₂	—	2	1	no	—
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	8	1	no	—
316	C₂	C₁₃×#2033	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#2033
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₁₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2161	C₂	semidirect
3. C₂×C₁₃×#362	C₂	semidirect
4. C₂×C₁₃×#362	C₂	semidirect
7. C₁₃×#1388	C₂	semidirect
63. #13390	C₁₃	direct (N × H)
111. #2161	C₂×C₁₃	semidirect
112. C₂×#362	C₂×C₁₃	semidirect
113. C₂×#362	C₂×C₁₃	semidirect
116. #1388	C₂×C₁₃	semidirect
267. C₁₃	#13390	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×#473	2
C₁₃×#2161	2
C₂×C₁₃×#362	2
C₁₃×#2117	2
C₁₃×#1813	2
C₁₃×#1388	2
#13390	13

Subgroups — C13×#13390 #15699729

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13390 #15699729