Subgroups — C₁₃×#11660 #15697999

Order 3328.

372 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1332	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1331	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#135	C₄	1	1	yes	characteristic
7	C₄×C₁₃×#41	—	2	1	no	—
8	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
9	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₄×C₁₃×SD₁₆	—	2	1	no	—
11	C₄×C₁₃×SD₁₆	—	2	1	no	—
12	C₄×C₁₃×D₈	C₄	1	1	yes	characteristic
13	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
14	C₁₃×#427	C₄	1	1	yes	characteristic
15	C₁₃×#423	—	2	1	no	—
16	C₁₃×#416	—	2	1	no	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×C₄⋊C₁₆	—	2	1	no	—
19	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
20	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
21	C₁₃×#335	—	2	1	no	—
22	C₁₃×#335	C₂²	1	2	yes	—
23	C₂×C₁₃×#41	—	2	1	no	—
24	C₁₃×#135	—	2	2	no	—
25	C₁₃×#135	—	2	2	no	—
26	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₃×SD₁₆	—	2	1	no	—
28	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₃×M₁₆	—	4	1	no	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₁₃	—	2	1	no	—
33	C₄×Q₈×C₁₃	—	2	1	no	—
34	C₄×D₄×C₁₃	—	2	1	no	—
35	C₄×D₄×C₁₃	—	2	1	no	—
36	C₄×D₄×C₁₃	—	4	1	no	—
37	C₄×D₄×C₁₃	—	4	1	no	—
38	C₁₃×#117	—	2	1	no	—
39	C₁₃×#117	—	4	1	no	—
40	C₂×C₄²×C₁₃	—	4	1	no	—
41	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
42	C₂×C₁₃×C₁₆	—	2	2	no	—
43	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
44	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
45	C₁₃×#106	—	2	1	no	—
46	C₁₃×C₄⋊C₈	—	2	2	no	—
47	C₁₃×C₄⋊C₈	—	4	1	no	—
48	C₁₃×#103	—	2	1	no	—
49	C₁₃×#103	—	2	1	no	—
50	C₁₃×#102	—	2	1	no	—
51	C₁₃×#102	—	2	1	no	—
52	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	#11660	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₁₃×#41	—	2	4	no	—
58	C₁₃×#41	—	4	2	no	—
59	C₂×Q₈×C₁₃	—	2	1	no	—
60	C₂×D₄×C₁₃	—	2	1	no	—
61	C₂×D₄×C₁₃	—	4	1	no	—
62	C₂²×C₄×C₁₃	—	4	1	no	—
63	C₂²×C₄×C₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	—	4	1	no	—
65	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
68	C₁₃×Q₁₆	—	2	1	no	—
69	C₁₃×Q₁₆	M₁₆	1	2	yes	complemented
70	C₁₃×SD₁₆	—	2	4	no	—
71	C₁₃×D₈	—	2	1	no	—
72	C₁₃×D₈	M₁₆	1	2	yes	complemented
73	C₁₃×M₁₆	—	8	2	no	—
74	C₂×C₈×C₁₃	—	2	1	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	4	1	no	—
77	C₂×C₈×C₁₃	—	4	2	no	—
78	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₁₃×C₄⋊C₄	—	2	1	no	—
83	C₁₃×C₄⋊C₄	—	2	1	no	—
84	C₁₃×C₄⋊C₄	—	4	1	no	—
85	C₁₃×C₄⋊C₄	—	4	1	no	—
86	C₁₃×C₂²⋊C₄	—	4	1	no	—
87	C₁₃×C₂²⋊C₄	—	4	1	no	—
88	C₁₃×C₂²⋊C₄	—	4	1	no	—
89	C₄²×C₁₃	—	4	1	no	—
90	C₄²×C₁₃	—	4	1	no	—
91	C₄²×C₁₃	—	4	1	no	—
92	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₁₃	D₈	1	1	yes	characteristic
94	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
95	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₁₃×C₁₆	—	4	2	no	—
97	C₄×#135	C₂×C₁₃	1	1	yes	characteristic
98	#2162	C₂×C₁₃	1	1	yes	characteristic, complemented
99	#1567	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#1332	C₂×C₁₃	1	2	yes	complemented
101	#1331	C₂×C₁₃	1	2	yes	complemented
102	C₂³×C₁₃	—	4	1	no	—
103	C₂³×C₁₃	#102	1	1	yes	characteristic
104	Q₈×C₁₃	—	2	2	no	—
105	Q₈×C₁₃	—	4	1	no	—
106	D₄×C₁₃	—	2	2	no	—
107	D₄×C₁₃	—	4	4	no	—
108	D₄×C₁₃	—	4	1	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	—	4	1	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	—	8	2	no	—
119	C₂×C₄×C₁₃	—	8	1	no	—
120	C₂×C₄×C₁₃	—	8	1	no	—
121	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
127	C₈×C₁₃	—	2	1	no	—
128	C₈×C₁₃	—	2	1	no	—
129	C₈×C₁₃	—	2	1	no	—
130	C₈×C₁₃	—	8	2	no	—
131	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
132	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
133	C₂×#135	C₄×C₁₃	1	1	yes	characteristic
134	C₄×#41	—	2	1	no	—
135	C₄×Q₁₆	C₄×C₁₃	1	1	yes	characteristic
136	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
137	C₄×SD₁₆	—	2	1	no	—
138	C₄×SD₁₆	—	2	1	no	—
139	C₄×D₈	C₄×C₁₃	1	1	yes	characteristic
140	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
141	#427	C₄×C₁₃	1	1	yes	characteristic
142	#423	—	2	1	no	—
143	#416	—	2	1	no	—
144	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
145	C₄⋊C₁₆	—	2	1	no	—
146	C₂²⋊C₁₆	—	2	1	no	—
147	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
148	#335	—	2	1	no	—
149	#335	C₂²×C₁₃	1	2	yes	—
150	C₂²×C₁₃	—	2	1	no	—
151	C₂²×C₁₃	—	2	1	no	—
152	C₂²×C₁₃	—	2	1	no	—
153	C₂²×C₁₃	—	4	2	no	—
154	C₂²×C₁₃	—	8	1	no	—
155	C₂²×C₁₃	—	8	1	no	—
156	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
157	C₄×C₁₃	—	2	1	no	—
158	C₄×C₁₃	—	2	1	no	—
159	C₄×C₁₃	—	2	1	no	—
160	C₄×C₁₃	—	4	2	no	—
161	C₄×C₁₃	—	8	1	no	—
162	C₄×C₁₃	—	8	1	no	—
163	C₄×C₁₃	#359	1	2	yes	—
164	C₄×C₁₃	#407	1	1	yes	characteristic
165	C₄×C₁₃	#421	1	1	yes	characteristic
166	C₄×C₁₃	#509	1	1	yes	characteristic
167	C₄×C₁₃	#510	1	1	yes	characteristic
168	C₂×#41	—	2	1	no	—
169	#135	—	2	2	no	—
170	#135	—	2	2	no	—
171	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
172	C₂×SD₁₆	—	2	1	no	—
173	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic
174	C₂×M₁₆	—	4	1	no	—
175	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
176	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
177	C₄×Q₈	—	2	1	no	—
178	C₄×Q₈	—	2	1	no	—
179	C₄×D₄	—	2	1	no	—
180	C₄×D₄	—	2	1	no	—
181	C₄×D₄	—	4	1	no	—
182	C₄×D₄	—	4	1	no	—
183	#117	—	2	1	no	—
184	#117	—	4	1	no	—
185	C₂×C₄²	—	4	1	no	—
186	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
187	C₂×C₁₆	—	2	2	no	—
188	#107	C₂×C₄×C₁₃	1	1	yes	characteristic
189	#107	C₂×C₄×C₁₃	1	1	yes	characteristic
190	#106	—	2	1	no	—
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	4	1	no	—
193	#103	—	2	1	no	—
194	#103	—	2	1	no	—
195	#102	—	2	1	no	—
196	#102	—	2	1	no	—
197	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
198	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
199	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
200	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
201	C₂×C₁₃	—	4	1	no	—
202	C₂×C₁₃	—	8	2	no	—
203	C₂×C₁₃	#1486	1	1	yes	characteristic
204	C₂×C₁₃	#2139	1	1	yes	characteristic
205	C₂×C₁₃	C₂×#359	1	1	yes	characteristic
206	#41	—	2	4	no	—
207	#41	—	4	2	no	—
208	C₂×Q₈	—	2	1	no	—
209	C₂×D₄	—	2	1	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
215	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
216	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
217	Q₁₆	—	2	1	no	—
218	Q₁₆	C₁₃×M₁₆	1	2	yes	complemented
219	SD₁₆	—	2	4	no	—
220	D₈	—	2	1	no	—
221	D₈	C₁₃×M₁₆	1	2	yes	complemented
222	M₁₆	—	8	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
230	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₂²⋊C₄	—	4	1	no	—
236	C₂²⋊C₄	—	4	1	no	—
237	C₂²⋊C₄	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	4	1	no	—
241	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
242	C₄²	C₁₃×D₈	1	1	yes	characteristic
243	C₄²	C₁₃×SD₁₆	1	1	yes	characteristic
244	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
245	C₁₆	—	4	2	no	—
246	C₁₃	#11660	1	1	yes	characteristic, direct product factor, Sylow
247	C₂³	—	4	1	no	—
248	C₂³	C₁₃×#102	1	1	yes	characteristic
249	Q₈	—	2	2	no	—
250	Q₈	—	4	1	no	—
251	D₄	—	2	2	no	—
252	D₄	—	4	4	no	—
253	D₄	—	4	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	8	2	no	—
264	C₂×C₄	—	8	1	no	—
265	C₂×C₄	—	8	1	no	—
266	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
267	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
268	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
269	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
271	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	8	2	no	—
276	C₈	C₂×C₁₃×M₁₆	1	1	yes	characteristic
277	C₈	C₂×C₁₃×M₁₆	1	1	yes	characteristic
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	8	1	no	—
283	C₂²	—	8	1	no	—
284	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	2	no	—
289	C₄	—	8	1	no	—
290	C₄	—	8	1	no	—
291	C₄	C₁₃×#359	1	2	yes	—
292	C₄	C₁₃×#407	1	1	yes	characteristic
293	C₄	C₁₃×#421	1	1	yes	characteristic
294	C₄	C₁₃×#509	1	1	yes	characteristic
295	C₄	C₁₃×#510	1	1	yes	characteristic
296	C₂	—	4	1	no	—
297	C₂	—	8	2	no	—
298	C₂	C₁₃×#1486	1	1	yes	characteristic
299	C₂	C₁₃×#2139	1	1	yes	characteristic
300	C₂	C₂×C₁₃×#359	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₄×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂×C₁₃×#135	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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×#2162	C₂	semidirect
3. C₁₃×#1567	C₂	semidirect
4. C₁₃×#1332	C₂	semidirect
5. C₁₃×#1331	C₂	semidirect
56. #11660	C₁₃	direct (N × H)
69. C₁₃×Q₁₆	M₁₆	semidirect
72. C₁₃×D₈	M₁₆	semidirect
98. #2162	C₂×C₁₃	semidirect
99. #1567	C₂×C₁₃	semidirect
100. #1332	C₂×C₁₃	semidirect
101. #1331	C₂×C₁₃	semidirect
218. Q₁₆	C₁₃×M₁₆	semidirect
221. D₈	C₁₃×M₁₆	semidirect
246. C₁₃	#11660	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₃×#135	2
C₁₃×#2162	2
C₁₃×#1567	2
C₁₃×#1332	2
C₁₃×#1331	2
#11660	13

Subgroups — C13×#11660 #15697999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#11660 #15697999