Subgroups — C₁₃×#13397 #15699736

Order 3328.

390 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	C₁₃×#3059	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2155	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2141	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2117	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1814	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
10	C₁₃×#536	—	2	1	no	—
11	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₁₃×#470	—	2	2	no	—
14	C₁₃×#467	—	2	2	no	—
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	C₁₃×#427	C₂²	1	1	yes	characteristic
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	C₁₃×#421	—	2	1	no	—
19	C₁₃×#419	—	2	1	no	—
20	C₁₃×#365	—	2	2	no	—
21	C₁₃×#362	—	2	2	no	—
22	C₁₃×#358	—	2	2	no	—
23	C₁₃×#350	—	2	2	no	—
24	C₁₃×#344	—	2	1	no	—
25	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
26	C₂²×Q₈×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	C₁₃×#131	—	2	1	no	—
34	C₁₃×#131	—	2	1	no	—
35	C₁₃×#131	—	2	1	no	—
36	C₁₃×#131	—	2	1	no	—
37	C₂×C₁₃×M₁₆	—	2	1	no	—
38	C₂×C₁₃×M₁₆	—	2	1	no	—
39	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	—	2	1	no	—
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×C₄⋊Q₈	—	4	1	no	—
43	C₁₃×#124	—	4	1	no	—
44	C₁₃×#123	—	4	2	no	—
45	C₁₃×C₂²⋊Q₈	—	2	4	no	—
46	C₁₃×#117	—	2	1	no	—
47	C₁₃×#117	—	2	1	no	—
48	C₁₃×M₃₂	—	2	1	no	—
49	C₁₃×M₃₂	—	2	1	no	—
50	C₁₃×M₃₂	—	2	1	no	—
51	C₁₃×M₃₂	—	2	1	no	—
52	C₂×C₁₃×C₁₆	—	2	1	no	—
53	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
54	C₁₃×#108	—	2	1	no	—
55	C₁₃×#108	—	2	1	no	—
56	C₁₃×#107	—	2	1	no	—
57	C₁₃×#107	—	2	1	no	—
58	C₁₃×#106	—	2	1	no	—
59	C₁₃×#106	D₄	1	2	yes	—
60	C₁₃×#104	—	4	2	no	—
61	C₁₃×#103	—	2	4	no	—
62	#13397	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₁₃×#41	—	2	1	no	—
64	C₁₃×#41	—	2	1	no	—
65	C₁₃×#41	—	2	1	no	—
66	C₁₃×#41	—	2	1	no	—
67	C₂×Q₈×C₁₃	—	2	2	no	—
68	C₂×Q₈×C₁₃	—	4	1	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	SD₁₆	1	1	yes	characteristic
74	C₂×D₄×C₁₃	—	2	1	no	—
75	C₂×D₄×C₁₃	SD₁₆	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	—	2	1	no	—
77	C₂²×C₄×C₁₃	—	4	1	no	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₁₃×Q₁₆	—	2	2	no	—
80	C₁₃×Q₁₆	—	2	2	no	—
81	C₁₃×Q₁₆	—	4	1	no	—
82	C₁₃×Q₁₆	—	4	1	no	—
83	C₁₃×M₁₆	—	2	1	no	—
84	C₁₃×M₁₆	—	2	1	no	—
85	C₁₃×M₁₆	—	4	1	no	—
86	C₁₃×M₁₆	—	4	1	no	—
87	C₁₃×M₁₆	—	4	1	no	—
88	C₁₃×M₁₆	—	4	1	no	—
89	C₂×C₈×C₁₃	—	2	1	no	—
90	C₂×C₈×C₁₃	—	4	1	no	—
91	C₂×C₈×C₁₃	—	4	1	no	—
92	C₂×C₈×C₁₃	—	4	1	no	—
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
97	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
98	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	4	4	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₁₃×C₂²⋊C₄	—	4	4	no	—
103	C₁₃×C₂²⋊C₄	—	4	1	no	—
104	C₁₃×C₂²⋊C₄	—	4	1	no	—
105	C₄²×C₁₃	—	4	1	no	—
106	C₄²×C₁₃	—	4	1	no	—
107	C₁₃×C₁₆	—	2	2	no	—
108	C₁₃×C₁₆	—	4	1	no	—
109	#3059	C₂×C₁₃	1	1	yes	characteristic
110	#2155	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₂×#362	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#2141	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#2117	C₂×C₁₃	1	1	yes	characteristic
114	#1814	C₂×C₁₃	1	1	yes	characteristic
115	#1384	C₂×C₁₃	1	1	yes	characteristic, complemented
116	C₂³×C₁₃	—	2	1	no	—
117	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
118	Q₈×C₁₃	—	2	1	no	—
119	Q₈×C₁₃	—	2	1	no	—
120	Q₈×C₁₃	—	4	2	no	—
121	Q₈×C₁₃	—	4	2	no	—
122	Q₈×C₁₃	—	8	1	no	—
123	Q₈×C₁₃	—	8	1	no	—
124	D₄×C₁₃	—	2	1	no	—
125	D₄×C₁₃	—	2	1	no	—
126	D₄×C₁₃	—	4	1	no	—
127	D₄×C₁₃	—	4	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	4	1	no	—
130	C₂×C₄×C₁₃	—	4	1	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	—	8	1	no	—
136	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
141	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₈×C₁₃	—	4	1	no	—
143	C₈×C₁₃	—	8	1	no	—
144	C₈×C₁₃	#121	1	1	yes	characteristic
145	C₈×C₁₃	#121	1	1	yes	characteristic
146	C₈×C₁₃	#123	1	1	yes	characteristic
147	C₈×C₁₃	#124	1	1	yes	characteristic
148	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
150	#536	—	2	1	no	—
151	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
152	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
153	#470	—	2	2	no	—
154	#467	—	2	2	no	—
155	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
156	#427	C₂²×C₁₃	1	1	yes	characteristic
157	#427	C₂²×C₁₃	1	1	yes	characteristic
158	#421	—	2	1	no	—
159	#419	—	2	1	no	—
160	#365	—	2	2	no	—
161	#362	—	2	2	no	—
162	#358	—	2	2	no	—
163	#350	—	2	2	no	—
164	#344	—	2	1	no	—
165	C₂²×C₁₃	—	4	1	no	—
166	C₂²×C₁₃	—	4	2	no	—
167	C₂²×C₁₃	—	4	1	no	—
168	C₂²×C₁₃	#450	1	1	yes	characteristic
169	C₂²×C₁₃	#469	1	1	yes	characteristic
170	C₂²×C₁₃	#482	1	1	yes	characteristic
171	C₄×C₁₃	—	4	1	no	—
172	C₄×C₁₃	—	8	2	no	—
173	C₄×C₁₃	—	8	2	no	—
174	C₄×C₁₃	—	16	1	no	—
175	C₄×C₁₃	#394	1	1	yes	characteristic
176	C₄×C₁₃	#449	1	1	yes	characteristic
177	C₄×C₁₃	#465	1	1	yes	characteristic
178	C₄×C₁₃	#481	1	1	yes	characteristic
179	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
180	C₂²×Q₈	—	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	#131	—	2	1	no	—
188	#131	—	2	1	no	—
189	#131	—	2	1	no	—
190	#131	—	2	1	no	—
191	C₂×M₁₆	—	2	1	no	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
194	C₂²×C₈	—	2	1	no	—
195	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
196	C₄⋊Q₈	—	4	1	no	—
197	#124	—	4	1	no	—
198	#123	—	4	2	no	—
199	C₂²⋊Q₈	—	2	4	no	—
200	#117	—	2	1	no	—
201	#117	—	2	1	no	—
202	M₃₂	—	2	1	no	—
203	M₃₂	—	2	1	no	—
204	M₃₂	—	2	1	no	—
205	M₃₂	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
208	#108	—	2	1	no	—
209	#108	—	2	1	no	—
210	#107	—	2	1	no	—
211	#107	—	2	1	no	—
212	#106	—	2	1	no	—
213	#106	D₄×C₁₃	1	2	yes	—
214	#104	—	4	2	no	—
215	#103	—	2	4	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	#2034	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×SD₁₆	1	1	yes	characteristic
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	C₁₃×SD₁₆	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	1	no	—
240	Q₁₆	—	4	1	no	—
241	M₁₆	—	2	1	no	—
242	M₁₆	—	2	1	no	—
243	M₁₆	—	4	1	no	—
244	M₁₆	—	4	1	no	—
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
253	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
254	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
255	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
256	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	4	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₂²⋊C₄	—	4	4	no	—
261	C₂²⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	1	no	—
263	C₄²	—	4	1	no	—
264	C₄²	—	4	1	no	—
265	C₁₆	—	2	2	no	—
266	C₁₆	—	4	1	no	—
267	C₁₃	#13397	1	1	yes	characteristic, direct product factor, Sylow
268	C₂³	—	2	1	no	—
269	C₂³	C₁₃×AGL(1,ℤ/8)	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	—	4	2	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
289	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
290	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
291	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
292	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
293	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
294	C₈	—	4	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	2	no	—
302	C₂²	—	4	1	no	—
303	C₂²	C₁₃×#450	1	1	yes	characteristic
304	C₂²	C₁₃×#469	1	1	yes	characteristic
305	C₂²	C₁₃×#482	1	1	yes	characteristic
306	C₄	—	4	1	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	16	1	no	—
310	C₄	C₁₃×#394	1	1	yes	characteristic
311	C₄	C₁₃×#449	1	1	yes	characteristic
312	C₄	C₁₃×#465	1	1	yes	characteristic
313	C₄	C₁₃×#481	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	8	1	no	—
318	C₂	C₁₃×#2034	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#2034
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×#2155	C₂	semidirect
3. C₂×C₁₃×#362	C₂	semidirect
4. C₁₃×#2141	C₂	semidirect
7. C₁₃×#1384	C₂	semidirect
62. #13397	C₁₃	direct (N × H)
110. #2155	C₂×C₁₃	semidirect
111. C₂×#362	C₂×C₁₃	semidirect
112. #2141	C₂×C₁₃	semidirect
115. #1384	C₂×C₁₃	semidirect
267. C₁₃	#13397	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#3059	2
C₁₃×#2155	2
C₂×C₁₃×#362	2
C₁₃×#2141	2
C₁₃×#2117	2
C₁₃×#1814	2
C₁₃×#1384	2
#13397	13

Subgroups — C13×#13397 #15699736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13397 #15699736