Subgroups — C₁₃×#20520 #15706859

Order 3328.

938 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#446	C₂	1	2	yes	maximal
2	C₄²×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1771	C₂	1	4	yes	maximal
4	C₈×C₁₃×C₄⋊C₄	C₂	1	2	yes	maximal
5	C₁₃×#1767	C₂	1	2	yes	maximal
6	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic
7	C₄×C₁₃×#97	C₂	1	2	yes	maximal
8	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₃	C₄	1	2	yes	—
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#446	C₄	1	16	yes	complemented
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	4	yes	—
14	C₂×C₁₃×#97	C₂²	1	2	yes	—
15	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#97	C₂²	1	2	yes	—
17	C₂×C₁₃×#97	C₂²	1	4	yes	—
18	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
21	C₄×C₁₃×C₄⋊C₄	C₄	1	4	yes	—
22	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
23	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄³×C₁₃	C₂²	1	2	yes	—
26	C₄³×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#336	C₂²	1	4	yes	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
33	C₄×Q₈×C₁₃	—	2	8	no	—
34	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
49	C₁₃×C₄⋊C₈	C₂×C₄	1	16	yes	—
50	C₁₃×#97	—	2	4	no	—
51	C₁₃×#97	—	2	4	no	—
52	C₁₃×#97	—	2	2	no	—
53	C₁₃×#97	C₂×C₄	1	16	yes	—
54	C₁₃×#97	Q₈	1	4	yes	complemented
55	C₄×C₈×C₁₃	—	2	8	no	—
56	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
57	C₁₃×#95	C₂×C₄	1	4	yes	—
58	C₁₃×#95	C₂×C₄	1	2	yes	—
59	#20520	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×Q₈×C₁₃	—	2	4	no	—
61	C₂×Q₈×C₁₃	C₄²	1	4	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₁₃	—	2	4	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
78	C₂×C₈×C₁₃	#41	1	4	yes	—
79	C₁₃×C₄⋊C₄	—	2	8	no	—
80	C₁₃×C₄⋊C₄	C₄²	1	8	yes	—
81	C₁₃×C₄⋊C₄	C₄²	1	4	yes	—
82	C₁₃×C₄⋊C₄	M₁₆	1	8	yes	—
83	C₄²×C₁₃	—	2	8	no	—
84	C₄²×C₁₃	—	2	4	no	—
85	C₄²×C₁₃	—	2	4	no	—
86	C₄²×C₁₃	—	2	2	no	—
87	C₄²×C₁₃	—	4	4	no	—
88	C₄²×C₁₃	—	4	2	no	—
89	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
90	C₄²×C₁₃	C₂²×C₄	1	8	yes	—
91	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
92	C₄²×C₁₃	#41	1	2	yes	—
93	C₂×#446	C₂×C₁₃	1	2	yes	—
94	C₄²×Q₈	C₂×C₁₃	1	1	yes	characteristic
95	#1771	C₂×C₁₃	1	4	yes	—
96	C₈×C₄⋊C₄	C₂×C₁₃	1	2	yes	—
97	#1767	C₂×C₁₃	1	2	yes	—
98	#1767	C₂×C₁₃	1	1	yes	characteristic
99	C₄×#97	C₂×C₁₃	1	2	yes	—
100	C₄×#97	C₂×C₁₃	1	1	yes	characteristic
101	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
102	Q₈×C₁₃	—	2	8	no	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
115	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
116	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
118	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
119	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
120	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₈×C₁₃	—	2	8	no	—
125	C₈×C₁₃	—	2	4	no	—
126	C₈×C₁₃	C₄×Q₈	1	8	yes	complemented
127	C₂×C₄×Q₈	C₄×C₁₃	1	2	yes	—
128	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
129	#446	C₄×C₁₃	1	16	yes	complemented
130	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
131	C₂×C₄⋊C₈	C₂²×C₁₃	1	4	yes	—
132	C₂×#97	C₂²×C₁₃	1	2	yes	—
133	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×#97	C₂²×C₁₃	1	2	yes	—
135	C₂×#97	C₂²×C₁₃	1	4	yes	—
136	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂²×C₁₃	1	4	yes	—
138	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
139	C₄×C₄⋊C₄	C₄×C₁₃	1	4	yes	—
140	C₄×C₄⋊C₄	C₄×C₁₃	1	2	yes	—
141	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
142	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
143	C₄³	C₂²×C₁₃	1	2	yes	—
144	C₄³	C₂²×C₁₃	1	1	yes	characteristic
145	#336	C₂²×C₁₃	1	4	yes	—
146	#336	C₂²×C₁₃	1	2	yes	—
147	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
150	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
151	C₂²×C₁₃	#568	1	2	yes	—
152	C₄×C₁₃	—	2	4	no	—
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	2	4	no	—
155	C₄×C₁₃	—	2	4	no	—
156	C₄×C₁₃	—	2	8	no	—
157	C₄×C₁₃	—	4	2	no	—
158	C₄×C₁₃	C₄×M₁₆	1	4	yes	—
159	C₄×C₁₃	C₂×C₄×Q₈	1	2	yes	—
160	C₄×C₁₃	C₄×#41	1	2	yes	—
161	C₂²×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
163	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
164	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
165	C₄×Q₈	—	2	8	no	—
166	C₄×Q₈	C₂×C₄×C₁₃	1	8	yes	—
167	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
168	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
169	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
170	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
171	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
172	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
173	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
174	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
175	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
176	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
177	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
179	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
180	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
181	C₄⋊C₈	C₂×C₄×C₁₃	1	16	yes	—
182	#97	—	2	4	no	—
183	#97	—	2	4	no	—
184	#97	—	2	2	no	—
185	#97	C₂×C₄×C₁₃	1	16	yes	—
186	#97	Q₈×C₁₃	1	4	yes	complemented
187	C₄×C₈	—	2	8	no	—
188	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	—
189	#95	C₂×C₄×C₁₃	1	4	yes	—
190	#95	C₂×C₄×C₁₃	1	2	yes	—
191	C₂×C₁₃	C₄²×Q₈	1	1	yes	characteristic
192	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
193	C₂×C₁₃	#2875	1	1	yes	characteristic
194	C₂×C₁₃	Q₈×M₁₆	1	2	yes	—
195	C₂×C₁₃	#2970	1	2	yes	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	C₄²×C₁₃	1	4	yes	—
198	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
199	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
201	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
202	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
203	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
204	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
211	C₂×C₈	C₂²×C₄×C₁₃	1	8	yes	—
212	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
213	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
214	C₂×C₈	C₁₃×#41	1	4	yes	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	C₄²×C₁₃	1	8	yes	—
217	C₄⋊C₄	C₄²×C₁₃	1	4	yes	—
218	C₄⋊C₄	C₁₃×M₁₆	1	8	yes	—
219	C₄²	—	2	8	no	—
220	C₄²	—	2	4	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	4	4	no	—
224	C₄²	—	4	2	no	—
225	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
226	C₄²	C₂²×C₄×C₁₃	1	8	yes	—
227	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
228	C₄²	C₁₃×#41	1	2	yes	—
229	C₁₃	#20520	1	1	yes	characteristic, direct product factor, Sylow
230	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
231	Q₈	—	2	8	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
243	C₂×C₄	C₂×C₄²×C₁₃	1	4	yes	—
244	C₂×C₄	C₂×C₄²×C₁₃	1	4	yes	—
245	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
246	C₂×C₄	C₄×Q₈×C₁₃	1	4	yes	—
247	C₂×C₄	C₂×C₁₃×M₁₆	1	8	yes	—
248	C₂×C₄	C₂×C₁₃×M₁₆	1	4	yes	—
249	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
250	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
252	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
253	C₈	—	2	8	no	—
254	C₈	—	2	4	no	—
255	C₈	C₄×Q₈×C₁₃	1	8	yes	complemented
256	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
257	C₂²	C₂×C₄×Q₈×C₁₃	1	1	yes	characteristic
258	C₂²	C₄×C₁₃×#41	1	1	yes	characteristic
259	C₂²	C₂²×C₁₃×M₁₆	1	2	yes	—
260	C₂²	C₁₃×#568	1	2	yes	—
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	2	8	no	—
266	C₄	—	4	2	no	—
267	C₄	C₄×C₁₃×M₁₆	1	4	yes	—
268	C₄	C₂×C₄×Q₈×C₁₃	1	2	yes	—
269	C₄	C₄×C₁₃×#41	1	2	yes	—
270	C₂	C₄²×Q₈×C₁₃	1	1	yes	characteristic
271	C₂	C₂×C₄×C₁₃×M₁₆	1	1	yes	characteristic
272	C₂	C₁₃×#2875	1	1	yes	characteristic
273	C₂	Q₈×C₁₃×M₁₆	1	2	yes	—
274	C₂	C₁₃×#2970	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₁₃×#446	C₄	semidirect
54. C₁₃×#97	Q₈	semidirect
59. #20520	C₁₃	direct (N × H)
126. C₈×C₁₃	C₄×Q₈	semidirect
129. #446	C₄×C₁₃	semidirect
186. #97	Q₈×C₁₃	semidirect
229. C₁₃	#20520	direct (N × H)
255. C₈	C₄×Q₈×C₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#446	2
C₄²×Q₈×C₁₃	2
C₁₃×#1771	2
C₈×C₁₃×C₄⋊C₄	2
C₁₃×#1767	2
C₄×C₁₃×#97	2
#20520	13

Subgroups — C13×#20520 #15706859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20520 #15706859