Subgroups — C₁₃×#13414 #15699753

Order 3328.

456 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2141	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2140	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1945	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×SD₁₆	—	2	1	no	—
10	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#124	—	2	1	no	—
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₁₃×#495	—	2	2	no	—
15	C₁₃×#495	—	2	2	no	—
16	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	C₁₃×#424	—	2	1	no	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#362	—	2	2	no	—
21	C₁₃×#361	—	2	2	no	—
22	C₁₃×#358	—	2	2	no	—
23	C₁₃×#357	—	2	2	no	—
24	C₁₃×#339	—	2	1	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×SD₁₆	—	2	4	no	—
34	C₂×C₁₃×SD₁₆	—	4	2	no	—
35	C₂×C₁₃×SD₁₆	—	4	2	no	—
36	C₂×C₁₃×D₈	—	2	1	no	—
37	C₂×C₁₃×D₈	—	2	1	no	—
38	C₂×C₁₃×D₈	—	2	1	no	—
39	C₂×C₁₃×D₈	—	2	1	no	—
40	C₂×C₁₃×D₈	—	2	1	no	—
41	C₂×C₁₃×D₈	D₄	1	2	yes	—
42	C₂×C₁₃×M₁₆	—	2	1	no	—
43	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×#124	—	2	4	no	—
46	C₁₃×#124	—	4	2	no	—
47	C₁₃×#117	—	2	1	no	—
48	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
49	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
50	C₁₃×M₃₂	—	2	1	no	—
51	C₁₃×M₃₂	—	2	1	no	—
52	C₁₃×M₃₂	—	2	1	no	—
53	C₁₃×M₃₂	—	2	1	no	—
54	C₂×C₁₃×C₁₆	—	2	1	no	—
55	C₂×C₁₃×C₁₆	—	2	1	no	—
56	C₁₃×#108	—	2	1	no	—
57	C₁₃×#108	—	2	1	no	—
58	C₁₃×#107	D₄	1	2	yes	—
59	C₁₃×#106	—	2	1	no	—
60	C₁₃×C₄⋊C₈	—	4	2	no	—
61	C₄×C₈×C₁₃	—	2	1	no	—
62	C₄×C₈×C₁₃	—	2	1	no	—
63	#13414	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂⁴×C₁₃	—	4	1	no	—
65	C₂×Q₈×C₁₃	—	2	2	no	—
66	C₂×Q₈×C₁₃	—	4	1	no	—
67	C₂×Q₈×C₁₃	—	4	1	no	—
68	C₂×Q₈×C₁₃	—	4	1	no	—
69	C₂×Q₈×C₁₃	—	4	1	no	—
70	C₂×Q₈×C₁₃	—	4	1	no	—
71	C₂×D₄×C₁₃	—	2	2	no	—
72	C₂×D₄×C₁₃	—	4	1	no	—
73	C₂×D₄×C₁₃	—	4	1	no	—
74	C₂×D₄×C₁₃	—	4	1	no	—
75	C₂×D₄×C₁₃	—	4	1	no	—
76	C₂×D₄×C₁₃	—	4	1	no	—
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	1	no	—
83	C₁₃×Q₁₆	—	4	1	no	—
84	C₁₃×SD₁₆	—	4	8	no	—
85	C₁₃×D₈	—	2	2	no	—
86	C₁₃×D₈	—	2	2	no	—
87	C₁₃×D₈	—	4	1	no	—
88	C₁₃×D₈	—	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	—	2	1	no	—
93	C₂×C₈×C₁₃	—	4	1	no	—
94	C₂×C₈×C₁₃	D₈	1	2	yes	—
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
100	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₂²⋊C₄	—	4	8	no	—
103	C₁₃×C₂²⋊C₄	—	4	1	no	—
104	C₄²×C₁₃	—	2	1	no	—
105	C₄²×C₁₃	—	4	1	no	—
106	C₄²×C₁₃	SD₁₆	1	2	yes	—
107	C₁₃×C₁₆	—	4	1	no	—
108	C₁₃×C₁₆	—	4	1	no	—
109	C₂×#495	C₂×C₁₃	1	1	yes	characteristic
110	C₂×#362	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₂×#361	C₂×C₁₃	1	1	yes	characteristic
112	#2141	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#2140	C₂×C₁₃	1	1	yes	characteristic
114	#1945	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#1376	C₂×C₁₃	1	1	yes	characteristic, complemented
116	C₂³×C₁₃	—	4	2	no	—
117	C₂³×C₁₃	—	8	4	no	—
118	C₂³×C₁₃	—	8	1	no	—
119	C₂³×C₁₃	—	8	1	no	—
120	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
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	—	4	2	no	—
126	D₄×C₁₃	—	4	2	no	—
127	D₄×C₁₃	—	8	1	no	—
128	D₄×C₁₃	—	8	1	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	2	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	#124	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₈×C₁₃	—	2	2	no	—
142	C₈×C₁₃	—	8	1	no	—
143	C₈×C₁₃	#121	1	1	yes	characteristic
144	C₈×C₁₃	#121	1	1	yes	characteristic
145	C₈×C₁₃	#123	1	1	yes	characteristic
146	C₈×C₁₃	#123	1	1	yes	characteristic
147	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
148	C₂²×SD₁₆	—	2	1	no	—
149	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×#124	—	2	1	no	—
151	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
152	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
153	#495	—	2	2	no	—
154	#495	—	2	2	no	—
155	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
156	#427	C₂²×C₁₃	1	1	yes	characteristic
157	#424	—	2	1	no	—
158	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
159	#362	—	2	2	no	—
160	#361	—	2	2	no	—
161	#358	—	2	2	no	—
162	#357	—	2	2	no	—
163	#339	—	2	1	no	—
164	C₂²×C₁₃	—	2	2	no	—
165	C₂²×C₁₃	—	8	4	no	—
166	C₂²×C₁₃	—	8	2	no	—
167	C₂²×C₁₃	—	16	2	no	—
168	C₂²×C₁₃	—	16	2	no	—
169	C₂²×C₁₃	#450	1	1	yes	characteristic
170	C₂²×C₁₃	#459	1	1	yes	characteristic
171	C₂²×C₁₃	#490	1	1	yes	characteristic
172	C₄×C₁₃	—	4	2	no	—
173	C₄×C₁₃	—	8	2	no	—
174	C₄×C₁₃	—	8	2	no	—
175	C₄×C₁₃	#394	1	1	yes	characteristic
176	C₄×C₁₃	#447	1	1	yes	characteristic
177	C₄×C₁₃	#460	1	1	yes	characteristic
178	C₄×C₁₃	#486	1	1	yes	characteristic
179	C₂²×Q₈	—	2	1	no	—
180	C₂²×D₄	—	2	1	no	—
181	C₂×Q₁₆	—	2	1	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	D₄×C₁₃	1	2	yes	—
187	C₂×SD₁₆	—	2	4	no	—
188	C₂×SD₁₆	—	4	2	no	—
189	C₂×SD₁₆	—	4	2	no	—
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	—	2	1	no	—
193	C₂×D₈	—	2	1	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	D₄×C₁₃	1	2	yes	—
196	C₂×M₁₆	—	2	1	no	—
197	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
199	#124	—	2	4	no	—
200	#124	—	4	2	no	—
201	#117	—	2	1	no	—
202	C₂×C₂²⋊C₄	—	4	2	no	—
203	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
204	M₃₂	—	2	1	no	—
205	M₃₂	—	2	1	no	—
206	M₃₂	—	2	1	no	—
207	M₃₂	—	2	1	no	—
208	C₂×C₁₆	—	2	1	no	—
209	C₂×C₁₆	—	2	1	no	—
210	#108	—	2	1	no	—
211	#108	—	2	1	no	—
212	#107	D₄×C₁₃	1	2	yes	—
213	#106	—	2	1	no	—
214	C₄⋊C₈	—	4	2	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	—	2	1	no	—
217	C₂×C₁₃	—	2	1	no	—
218	C₂×C₁₃	—	2	1	no	—
219	C₂×C₁₃	—	16	2	no	—
220	C₂×C₁₃	#2041	1	1	yes	characteristic
221	C₂×C₁₃	#2212	1	2	yes	—
222	C₂⁴	—	4	1	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	4	1	no	—
225	C₂×Q₈	—	4	1	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂×D₄	—	4	1	no	—
233	C₂×D₄	—	4	1	no	—
234	C₂×D₄	—	4	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
238	Q₁₆	—	2	2	no	—
239	Q₁₆	—	2	2	no	—
240	Q₁₆	—	4	1	no	—
241	Q₁₆	—	4	1	no	—
242	SD₁₆	—	4	8	no	—
243	D₈	—	2	2	no	—
244	D₈	—	2	2	no	—
245	D₈	—	4	1	no	—
246	D₈	—	4	1	no	—
247	M₁₆	—	4	1	no	—
248	M₁₆	—	4	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	C₁₃×D₈	1	2	yes	—
253	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
254	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
255	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
256	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
257	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
258	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
259	C₄⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	4	8	no	—
261	C₂²⋊C₄	—	4	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	4	1	no	—
264	C₄²	C₁₃×SD₁₆	1	2	yes	—
265	C₁₆	—	4	1	no	—
266	C₁₆	—	4	1	no	—
267	C₁₃	#13414	1	1	yes	characteristic, direct product factor, Sylow
268	C₂³	—	4	2	no	—
269	C₂³	—	8	4	no	—
270	C₂³	—	8	1	no	—
271	C₂³	—	8	1	no	—
272	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	8	1	no	—
276	Q₈	—	8	1	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	8	1	no	—
280	D₄	—	8	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
288	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
289	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
290	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
291	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
292	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	8	1	no	—
295	C₈	C₁₃×#121	1	1	yes	characteristic
296	C₈	C₁₃×#121	1	1	yes	characteristic
297	C₈	C₁₃×#123	1	1	yes	characteristic
298	C₈	C₁₃×#123	1	1	yes	characteristic
299	C₂²	—	2	2	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	16	2	no	—
303	C₂²	—	16	2	no	—
304	C₂²	C₁₃×#450	1	1	yes	characteristic
305	C₂²	C₁₃×#459	1	1	yes	characteristic
306	C₂²	C₁₃×#490	1	1	yes	characteristic
307	C₄	—	4	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	C₁₃×#394	1	1	yes	characteristic
311	C₄	C₁₃×#447	1	1	yes	characteristic
312	C₄	C₁₃×#460	1	1	yes	characteristic
313	C₄	C₁₃×#486	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	16	2	no	—
317	C₂	C₁₃×#2041	1	1	yes	characteristic
318	C₂	C₁₃×#2212	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	#459
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 ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#362	C₂	semidirect
4. C₁₃×#2141	C₂	semidirect
6. C₁₃×#1945	C₂	semidirect
7. C₁₃×#1376	C₂	semidirect
63. #13414	C₁₃	direct (N × H)
110. C₂×#362	C₂×C₁₃	semidirect
112. #2141	C₂×C₁₃	semidirect
114. #1945	C₂×C₁₃	semidirect
115. #1376	C₂×C₁₃	semidirect
267. C₁₃	#13414	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×#495	2
C₂×C₁₃×#362	2
C₂×C₁₃×#361	2
C₁₃×#2141	2
C₁₃×#2140	2
C₁₃×#1945	2
C₁₃×#1376	2
#13414	13

Subgroups — C13×#13414 #15699753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13414 #15699753