Subgroups — C₁₃×#20513 #15706852

Order 3328.

998 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#444	C₂	1	2	yes	maximal
2	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2875	C₂	1	2	yes	maximal, complemented
4	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1773	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1764	C₂	1	4	yes	maximal, complemented
7	C₁₃×#1759	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#131	C₂²	1	4	yes	—
11	C₄×C₁₃×#41	C₂²	1	2	yes	—
12	C₁₃×#444	C₄	1	16	yes	complemented
13	C₄×C₁₃×Q₁₆	C₄	1	2	yes	complemented
14	C₄×C₁₃×SD₁₆	C₄	1	4	yes	complemented
15	C₄×C₁₃×D₈	C₄	1	2	yes	complemented
16	C₂×C₁₃×#108	C₂²	1	2	yes	—
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#104	C₂²	1	4	yes	—
19	C₁₃×#416	C₂²	1	2	yes	—
20	C₁₃×#405	—	2	4	no	—
21	C₁₃×#405	C₂²	1	4	yes	—
22	C₄×C₁₃×M₁₆	—	2	2	no	—
23	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
24	C₂×C₁₃×#97	—	2	2	no	—
25	C₂×C₁₃×#97	C₂²	1	2	yes	—
26	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×#376	C₂²	1	1	yes	characteristic
30	C₁₃×#341	C₂²	1	2	yes	—
31	C₁₃×#339	C₂²	1	4	yes	—
32	C₂×C₁₃×#41	C₂³	1	2	yes	—
33	C₁₃×#135	C₂×C₄	1	8	yes	—
34	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
37	C₁₃×#131	—	2	8	no	—
38	C₁₃×#131	C₂×C₄	1	16	yes	complemented
39	C₂×C₁₃×M₁₆	—	2	4	no	—
40	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
41	C₂²×C₈×C₁₃	—	2	4	no	—
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
45	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
46	C₄×D₄×C₁₃	—	2	4	no	—
47	C₄×D₄×C₁₃	C₂×C₄	1	4	yes	—
48	C₁₃×#117	—	2	2	no	—
49	C₁₃×#117	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	—	2	2	no	—
51	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×#108	C₂×C₄	1	8	yes	complemented
54	C₁₃×#107	C₂×C₄	1	2	yes	—
55	C₁₃×#106	C₂×C₄	1	2	yes	—
56	C₁₃×#104	C₂×C₄	1	16	yes	complemented
57	C₁₃×#103	C₂×C₄	1	4	yes	—
58	C₁₃×#102	C₂×C₄	1	4	yes	—
59	C₁₃×#97	—	2	4	no	—
60	C₁₃×#97	—	2	2	no	—
61	C₁₃×#97	C₂×C₄	1	8	yes	complemented
62	C₁₃×#97	D₄	1	2	yes	—
63	C₁₃×#97	D₄	1	2	yes	—
64	C₄×C₈×C₁₃	—	2	4	no	—
65	C₄×C₈×C₁₃	—	2	4	no	—
66	C₄×C₈×C₁₃	—	2	2	no	—
67	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
69	C₁₃×#95	—	2	2	no	—
70	#20513	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₁₃×#41	—	2	4	no	—
72	C₁₃×#41	C₂²×C₄	1	8	yes	—
73	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂×D₄×C₁₃	—	2	2	no	—
75	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	—	2	4	no	—
79	C₂²×C₄×C₁₃	#41	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₁₃×Q₁₆	C₄²	1	4	yes	—
82	C₁₃×SD₁₆	C₄²	1	8	yes	—
83	C₁₃×D₈	C₄²	1	4	yes	—
84	C₁₃×M₁₆	—	2	4	no	—
85	C₁₃×M₁₆	—	2	16	no	—
86	C₁₃×M₁₆	C₂²×C₄	1	8	yes	—
87	C₂×C₈×C₁₃	—	2	4	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	16	no	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
96	C₂×C₈×C₁₃	#41	1	4	yes	—
97	C₁₃×C₄⋊C₄	—	2	4	no	—
98	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
99	C₁₃×C₂²⋊C₄	—	2	2	no	—
100	C₁₃×C₂²⋊C₄	—	2	4	no	—
101	C₄²×C₁₃	—	2	4	no	—
102	C₄²×C₁₃	—	2	2	no	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
105	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×#444	C₂×C₁₃	1	2	yes	—
109	C₄×#135	C₂×C₁₃	1	1	yes	characteristic
110	#2875	C₂×C₁₃	1	2	yes	complemented
111	C₄×#108	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#1773	C₂×C₁₃	1	2	yes	complemented
113	#1764	C₂×C₁₃	1	4	yes	complemented
114	#1759	C₂×C₁₃	1	2	yes	complemented
115	#1726	C₂×C₁₃	1	1	yes	characteristic, complemented
116	C₂³×C₁₃	—	2	2	no	—
117	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
118	Q₈×C₁₃	—	2	2	no	—
119	Q₈×C₁₃	C₂×C₄²	1	4	yes	—
120	D₄×C₁₃	—	2	8	no	—
121	D₄×C₁₃	—	2	2	no	—
122	D₄×C₁₃	C₂×C₄²	1	4	yes	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	8	no	—
125	C₂×C₄×C₁₃	—	2	4	no	—
126	C₂×C₄×C₁₃	—	2	4	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	4	4	no	—
129	C₂×C₄×C₁₃	—	8	4	no	—
130	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
136	C₈×C₁₃	—	2	8	no	—
137	C₈×C₁₃	—	2	2	no	—
138	C₈×C₁₃	C₂×C₄²	1	2	yes	—
139	C₈×C₁₃	C₂×C₄²	1	2	yes	—
140	C₈×C₁₃	C₄×D₄	1	8	yes	—
141	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
142	C₂×#131	C₂²×C₁₃	1	4	yes	—
143	C₄×#41	C₂²×C₁₃	1	2	yes	—
144	#444	C₄×C₁₃	1	16	yes	complemented
145	C₄×Q₁₆	C₄×C₁₃	1	2	yes	complemented
146	C₄×SD₁₆	C₄×C₁₃	1	4	yes	complemented
147	C₄×D₈	C₄×C₁₃	1	2	yes	complemented
148	C₂×#108	C₂²×C₁₃	1	2	yes	—
149	#427	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×#104	C₂²×C₁₃	1	4	yes	—
151	#416	C₂²×C₁₃	1	2	yes	—
152	#405	—	2	4	no	—
153	#405	C₂²×C₁₃	1	4	yes	—
154	C₄×M₁₆	—	2	2	no	—
155	C₄×M₁₆	C₂²×C₁₃	1	2	yes	—
156	C₂×#97	—	2	2	no	—
157	C₂×#97	C₂²×C₁₃	1	2	yes	—
158	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
160	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
161	#376	C₂²×C₁₃	1	1	yes	characteristic
162	#341	C₂²×C₁₃	1	2	yes	—
163	#339	C₂²×C₁₃	1	4	yes	—
164	C₂²×C₁₃	—	2	2	no	—
165	C₂²×C₁₃	—	2	4	no	—
166	C₂²×C₁₃	—	4	4	no	—
167	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂²×C₁₃	C₄×#41	1	2	yes	—
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	—	2	4	no	—
172	C₄×C₁₃	—	4	4	no	—
173	C₄×C₁₃	—	4	4	no	—
174	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
175	C₄×C₁₃	C₂×C₄×D₄	1	2	yes	—
176	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
177	C₂×#41	C₂³×C₁₃	1	2	yes	—
178	#135	C₂×C₄×C₁₃	1	8	yes	—
179	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
180	C₂×SD₁₆	C₂×C₄×C₁₃	1	2	yes	—
181	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic
182	#131	—	2	8	no	—
183	#131	C₂×C₄×C₁₃	1	16	yes	complemented
184	C₂×M₁₆	—	2	4	no	—
185	C₂×M₁₆	C₂³×C₁₃	1	4	yes	—
186	C₂²×C₈	—	2	4	no	—
187	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
190	C₄×Q₈	C₂×C₄×C₁₃	1	4	yes	—
191	C₄×D₄	—	2	4	no	—
192	C₄×D₄	C₂×C₄×C₁₃	1	4	yes	—
193	#117	—	2	2	no	—
194	#117	C₂³×C₁₃	1	2	yes	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
197	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
198	#108	C₂×C₄×C₁₃	1	8	yes	complemented
199	#107	C₂×C₄×C₁₃	1	2	yes	—
200	#106	C₂×C₄×C₁₃	1	2	yes	—
201	#104	C₂×C₄×C₁₃	1	16	yes	complemented
202	#103	C₂×C₄×C₁₃	1	4	yes	—
203	#102	C₂×C₄×C₁₃	1	4	yes	—
204	#97	—	2	4	no	—
205	#97	—	2	2	no	—
206	#97	C₂×C₄×C₁₃	1	8	yes	complemented
207	#97	D₄×C₁₃	1	2	yes	—
208	#97	D₄×C₁₃	1	2	yes	—
209	C₄×C₈	—	2	4	no	—
210	C₄×C₈	—	2	4	no	—
211	C₄×C₈	—	2	2	no	—
212	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
213	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
214	#95	—	2	2	no	—
215	C₂×C₁₃	—	2	2	no	—
216	C₂×C₁₃	—	4	4	no	—
217	C₂×C₁₃	C₄²×D₄	1	1	yes	characteristic
218	C₂×C₁₃	#2957	1	2	yes	—
219	#41	—	2	4	no	—
220	#41	C₂²×C₄×C₁₃	1	8	yes	—
221	C₂×Q₈	C₂²×C₄×C₁₃	1	2	yes	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	C₂²×C₄×C₁₃	1	2	yes	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	C₁₃×#41	1	2	yes	—
228	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
229	Q₁₆	C₄²×C₁₃	1	4	yes	—
230	SD₁₆	C₄²×C₁₃	1	8	yes	—
231	D₈	C₄²×C₁₃	1	4	yes	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	16	no	—
234	M₁₆	C₂²×C₄×C₁₃	1	8	yes	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	16	no	—
238	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
239	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
240	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
241	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
242	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
243	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
244	C₂×C₈	C₁₃×#41	1	4	yes	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₄²	—	2	4	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
253	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
254	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
255	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
256	C₁₃	#20513	1	1	yes	characteristic, direct product factor, Sylow
257	C₂³	—	2	2	no	—
258	C₂³	C₂×C₁₃×#41	1	1	yes	characteristic
259	Q₈	—	2	2	no	—
260	Q₈	C₂×C₄²×C₁₃	1	4	yes	—
261	D₄	—	2	8	no	—
262	D₄	—	2	2	no	—
263	D₄	C₂×C₄²×C₁₃	1	4	yes	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	8	4	no	—
271	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
272	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
273	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
274	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
277	C₈	—	2	8	no	—
278	C₈	—	2	2	no	—
279	C₈	C₂×C₄²×C₁₃	1	2	yes	—
280	C₈	C₂×C₄²×C₁₃	1	2	yes	—
281	C₈	C₄×D₄×C₁₃	1	8	yes	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	4	4	no	—
285	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
286	C₂²	C₄×C₁₃×#41	1	2	yes	—
287	C₄	—	2	4	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	4	4	no	—
291	C₄	—	4	4	no	—
292	C₄	C₂²×C₄²×C₁₃	1	1	yes	characteristic
293	C₄	C₂×C₄×D₄×C₁₃	1	2	yes	—
294	C₄	C₄×C₁₃×#41	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	4	4	no	—
297	C₂	C₄²×D₄×C₁₃	1	1	yes	characteristic
298	C₂	C₁₃×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2875	C₂	semidirect
4. C₄×C₁₃×#108	C₂	semidirect
5. C₁₃×#1773	C₂	semidirect
6. C₁₃×#1764	C₂	semidirect
7. C₁₃×#1759	C₂	semidirect
8. C₁₃×#1726	C₂	semidirect
12. C₁₃×#444	C₄	semidirect
13. C₄×C₁₃×Q₁₆	C₄	semidirect
14. C₄×C₁₃×SD₁₆	C₄	semidirect
15. C₄×C₁₃×D₈	C₄	semidirect
38. C₁₃×#131	C₂×C₄	semidirect
53. C₁₃×#108	C₂×C₄	semidirect
56. C₁₃×#104	C₂×C₄	semidirect
61. C₁₃×#97	C₂×C₄	semidirect
70. #20513	C₁₃	direct (N × H)
110. #2875	C₂×C₁₃	semidirect
111. C₄×#108	C₂×C₁₃	semidirect
112. #1773	C₂×C₁₃	semidirect
113. #1764	C₂×C₁₃	semidirect
114. #1759	C₂×C₁₃	semidirect
115. #1726	C₂×C₁₃	semidirect
144. #444	C₄×C₁₃	semidirect
145. C₄×Q₁₆	C₄×C₁₃	semidirect
146. C₄×SD₁₆	C₄×C₁₃	semidirect
147. C₄×D₈	C₄×C₁₃	semidirect
183. #131	C₂×C₄×C₁₃	semidirect
198. #108	C₂×C₄×C₁₃	semidirect
201. #104	C₂×C₄×C₁₃	semidirect
206. #97	C₂×C₄×C₁₃	semidirect
256. C₁₃	#20513	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×#444	2
C₄×C₁₃×#135	2
C₁₃×#2875	2
C₄×C₁₃×#108	2
C₁₃×#1773	2
C₁₃×#1764	2
C₁₃×#1759	2
C₁₃×#1726	2
#20513	13

Subgroups — C13×#20513 #15706852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20513 #15706852