Subgroups — C₁₃×#20503 #15706842

Order 3328.

1206 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	C₂×C₁₃×#442	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2275	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1776	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1762	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×#102	C₂	1	4	yes	maximal, complemented
8	C₁₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃	C₂²	1	4	yes	—
11	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
12	C₁₃×#442	C₄	1	16	yes	complemented
13	C₄×C₁₃×D₈	C₄	1	8	yes	complemented
14	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#106	C₂²	1	2	yes	—
16	C₂×C₁₃×#102	C₂²	1	2	yes	—
17	C₂×C₁₃×#102	C₂²	1	4	yes	—
18	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
23	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₄×C₁₃×C₂²⋊C₄	—	2	4	no	—
25	C₄×C₁₃×C₂²⋊C₄	—	2	2	no	—
26	C₁₃×#376	C₂²	1	1	yes	characteristic
27	C₂×C₁₃×#95	—	2	2	no	—
28	C₁₃×#340	C₂²	1	2	yes	—
29	C₁₃×#340	C₂²	1	4	yes	—
30	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₁₃	—	2	2	no	—
32	C₂³×C₄×C₁₃	—	2	4	no	—
33	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
34	C₂×C₁₃×D₈	C₂×C₄	1	8	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₄×D₄×C₁₃	—	2	8	no	—
39	C₄×D₄×C₁₃	—	2	4	no	—
40	C₄×D₄×C₁₃	C₂×C₄	1	8	yes	complemented
41	C₄×D₄×C₁₃	C₂×C₄	1	16	yes	complemented
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
44	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
46	C₂×C₄²×C₁₃	—	2	4	no	—
47	C₂×C₄²×C₁₃	—	2	2	no	—
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
50	C₁₃×#107	C₂×C₄	1	4	yes	complemented
51	C₁₃×#106	C₂×C₄	1	8	yes	complemented
52	C₁₃×#102	C₂×C₄	1	8	yes	complemented
53	C₁₃×#102	C₂×C₄	1	16	yes	complemented
54	C₁₃×#97	—	2	2	no	—
55	C₁₃×#97	C₂×C₄	1	8	yes	complemented
56	C₄×C₈×C₁₃	—	2	4	no	—
57	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	complemented
58	C₁₃×#95	—	2	2	no	—
59	C₁₃×#95	—	2	4	no	—
60	C₁₃×#95	D₄	1	2	yes	—
61	C₁₃×#95	D₄	1	2	yes	—
62	#20503	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂⁴×C₁₃	—	2	2	no	—
64	C₂×D₄×C₁₃	—	2	4	no	—
65	C₂×D₄×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
67	C₂×D₄×C₁₃	C₂²×C₄	1	8	yes	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	2	4	no	—
70	C₂²×C₄×C₁₃	—	2	16	no	—
71	C₂²×C₄×C₁₃	—	2	8	no	—
72	C₂²×C₄×C₁₃	—	8	4	no	—
73	C₂²×C₄×C₁₃	—	8	4	no	—
74	C₂²×C₄×C₁₃	—	8	4	no	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₁₃×D₈	C₄²	1	16	yes	complemented
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
87	C₁₃×C₄⋊C₄	—	2	4	no	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
90	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
91	C₁₃×C₂²⋊C₄	—	2	8	no	—
92	C₁₃×C₂²⋊C₄	—	2	16	no	—
93	C₄²×C₁₃	—	8	4	no	—
94	C₄²×C₁₃	—	8	2	no	—
95	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
97	C₄²×C₁₃	#41	1	4	yes	—
98	C₄²×C₁₃	#41	1	2	yes	—
99	C₂×#442	C₂×C₁₃	1	2	yes	—
100	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
101	#2275	C₂×C₁₃	1	2	yes	complemented
102	#1776	C₂×C₁₃	1	2	yes	complemented
103	C₄×#106	C₂×C₁₃	1	1	yes	characteristic, complemented
104	#1762	C₂×C₁₃	1	2	yes	complemented
105	C₄×#102	C₂×C₁₃	1	4	yes	complemented
106	#1726	C₂×C₁₃	1	1	yes	characteristic, complemented
107	C₂³×C₁₃	—	2	8	no	—
108	C₂³×C₁₃	—	2	4	no	—
109	C₂³×C₁₃	—	4	4	no	—
110	C₂³×C₁₃	—	4	2	no	—
111	C₂³×C₁₃	—	4	2	no	—
112	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	D₄×C₁₃	—	2	4	no	—
114	D₄×C₁₃	—	2	8	no	—
115	D₄×C₁₃	C₂×C₄²	1	8	yes	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	—	2	8	no	—
121	C₂×C₄×C₁₃	—	2	4	no	—
122	C₂×C₄×C₁₃	—	4	4	no	—
123	C₂×C₄×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	—	4	4	no	—
125	C₂×C₄×C₁₃	—	8	16	no	—
126	C₂×C₄×C₁₃	—	8	8	no	—
127	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₁₃	C₄×D₄	1	8	yes	—
129	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
131	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
134	C₈×C₁₃	—	2	2	no	—
135	C₈×C₁₃	—	2	4	no	—
136	C₈×C₁₃	C₂×C₄²	1	4	yes	—
137	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
138	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
139	C₂×C₄×D₄	C₂²×C₁₃	1	4	yes	—
140	#442	C₄×C₁₃	1	16	yes	complemented
141	C₄×D₈	C₄×C₁₃	1	8	yes	complemented
142	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×#106	C₂²×C₁₃	1	2	yes	—
144	C₂×#102	C₂²×C₁₃	1	2	yes	—
145	C₂×#102	C₂²×C₁₃	1	4	yes	—
146	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×#97	C₂²×C₁₃	1	2	yes	—
148	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
149	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
150	C₄×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
151	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
152	C₄×C₂²⋊C₄	—	2	2	no	—
153	C₄×C₂²⋊C₄	—	2	4	no	—
154	#376	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×#95	—	2	2	no	—
156	#340	C₂²×C₁₃	1	2	yes	—
157	#340	C₂²×C₁₃	1	4	yes	—
158	C₂²×C₁₃	—	2	8	no	—
159	C₂²×C₁₃	—	4	8	no	—
160	C₂²×C₁₃	—	4	8	no	—
161	C₂²×C₁₃	—	4	4	no	—
162	C₂²×C₁₃	—	4	4	no	—
163	C₂²×C₁₃	C₂×C₄×D₄	1	2	yes	—
164	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂²×C₁₃	#576	1	2	yes	—
166	C₂²×C₁₃	#577	1	2	yes	—
167	C₄×C₁₃	—	2	8	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	—	4	8	no	—
171	C₄×C₁₃	—	4	4	no	—
172	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
173	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
174	C₄×C₁₃	C₄×#41	1	2	yes	—
175	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
176	C₂³×C₄	—	2	2	no	—
177	C₂³×C₄	—	2	4	no	—
178	C₂×D₈	C₂×C₄×C₁₃	1	4	yes	—
179	C₂×D₈	C₂×C₄×C₁₃	1	8	yes	—
180	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
182	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
183	C₄×D₄	—	2	8	no	—
184	C₄×D₄	—	2	4	no	—
185	C₄×D₄	C₂×C₄×C₁₃	1	8	yes	complemented
186	C₄×D₄	C₂×C₄×C₁₃	1	16	yes	complemented
187	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
188	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
189	C₂×C₂²⋊C₄	—	2	4	no	—
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₄²	—	2	4	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
195	#107	C₂×C₄×C₁₃	1	4	yes	complemented
196	#106	C₂×C₄×C₁₃	1	8	yes	complemented
197	#102	C₂×C₄×C₁₃	1	8	yes	complemented
198	#102	C₂×C₄×C₁₃	1	16	yes	complemented
199	#97	—	2	2	no	—
200	#97	C₂×C₄×C₁₃	1	8	yes	complemented
201	C₄×C₈	—	2	4	no	—
202	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	complemented
203	#95	—	2	2	no	—
204	#95	—	2	4	no	—
205	#95	D₄×C₁₃	1	2	yes	—
206	#95	D₄×C₁₃	1	2	yes	—
207	C₂×C₁₃	—	4	8	no	—
208	C₂×C₁₃	C₄²×D₄	1	1	yes	characteristic
209	C₂×C₁₃	#2946	1	4	yes	—
210	C₂×C₁₃	#2948	1	1	yes	characteristic
211	C₂×C₁₃	#2952	1	1	yes	characteristic
212	C₂⁴	—	2	2	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	C₂²×C₄×C₁₃	1	4	yes	—
216	C₂×D₄	C₂²×C₄×C₁₃	1	8	yes	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	16	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	8	4	no	—
222	C₂²×C₄	—	8	4	no	—
223	C₂²×C₄	—	8	4	no	—
224	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
226	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
228	D₈	C₄²×C₁₃	1	16	yes	complemented
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
234	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
235	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	C₂²×C₄×C₁₃	1	8	yes	—
239	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
240	C₂²⋊C₄	—	2	8	no	—
241	C₂²⋊C₄	—	2	16	no	—
242	C₄²	—	8	4	no	—
243	C₄²	—	8	2	no	—
244	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
245	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
246	C₄²	C₁₃×#41	1	4	yes	—
247	C₄²	C₁₃×#41	1	2	yes	—
248	C₁₃	#20503	1	1	yes	characteristic, direct product factor, Sylow
249	C₂³	—	2	8	no	—
250	C₂³	—	2	4	no	—
251	C₂³	—	4	4	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
255	D₄	—	2	4	no	—
256	D₄	—	2	8	no	—
257	D₄	C₂×C₄²×C₁₃	1	8	yes	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	8	16	no	—
268	C₂×C₄	—	8	8	no	—
269	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
270	C₂×C₄	C₄×D₄×C₁₃	1	8	yes	—
271	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
272	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
273	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
274	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
275	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	2	4	no	—
278	C₈	C₂×C₄²×C₁₃	1	4	yes	—
279	C₂²	—	2	8	no	—
280	C₂²	—	4	8	no	—
281	C₂²	—	4	8	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	4	no	—
284	C₂²	C₂×C₄×D₄×C₁₃	1	2	yes	—
285	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
286	C₂²	C₁₃×#576	1	2	yes	—
287	C₂²	C₁₃×#577	1	2	yes	—
288	C₄	—	2	8	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	8	no	—
292	C₄	—	4	4	no	—
293	C₄	C₂²×C₄²×C₁₃	1	1	yes	characteristic
294	C₄	C₄×C₁₃×#41	1	1	yes	characteristic
295	C₄	C₄×C₁₃×#41	1	2	yes	—
296	C₂	—	4	8	no	—
297	C₂	C₄²×D₄×C₁₃	1	1	yes	characteristic
298	C₂	C₁₃×#2946	1	4	yes	—
299	C₂	C₁₃×#2948	1	1	yes	characteristic
300	C₂	C₁₃×#2952	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₁₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2275	C₂	semidirect
4. C₁₃×#1776	C₂	semidirect
5. C₄×C₁₃×#106	C₂	semidirect
6. C₁₃×#1762	C₂	semidirect
7. C₄×C₁₃×#102	C₂	semidirect
8. C₁₃×#1726	C₂	semidirect
12. C₁₃×#442	C₄	semidirect
13. C₄×C₁₃×D₈	C₄	semidirect
40. C₄×D₄×C₁₃	C₂×C₄	semidirect
41. C₄×D₄×C₁₃	C₂×C₄	semidirect
50. C₁₃×#107	C₂×C₄	semidirect
51. C₁₃×#106	C₂×C₄	semidirect
52. C₁₃×#102	C₂×C₄	semidirect
53. C₁₃×#102	C₂×C₄	semidirect
55. C₁₃×#97	C₂×C₄	semidirect
57. C₄×C₈×C₁₃	C₂×C₄	semidirect
62. #20503	C₁₃	direct (N × H)
79. C₁₃×D₈	C₄²	semidirect
101. #2275	C₂×C₁₃	semidirect
102. #1776	C₂×C₁₃	semidirect
103. C₄×#106	C₂×C₁₃	semidirect
104. #1762	C₂×C₁₃	semidirect
105. C₄×#102	C₂×C₁₃	semidirect
106. #1726	C₂×C₁₃	semidirect
140. #442	C₄×C₁₃	semidirect
141. C₄×D₈	C₄×C₁₃	semidirect
185. C₄×D₄	C₂×C₄×C₁₃	semidirect
186. C₄×D₄	C₂×C₄×C₁₃	semidirect
195. #107	C₂×C₄×C₁₃	semidirect
196. #106	C₂×C₄×C₁₃	semidirect
197. #102	C₂×C₄×C₁₃	semidirect
198. #102	C₂×C₄×C₁₃	semidirect
200. #97	C₂×C₄×C₁₃	semidirect
202. C₄×C₈	C₂×C₄×C₁₃	semidirect
228. D₈	C₄²×C₁₃	semidirect
248. C₁₃	#20503	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×#442	2
C₂×C₄×C₁₃×D₈	2
C₁₃×#2275	2
C₁₃×#1776	2
C₄×C₁₃×#106	2
C₁₃×#1762	2
C₄×C₁₃×#102	2
C₁₃×#1726	2
#20503	13

Subgroups — C13×#20503 #15706842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20503 #15706842