Subgroups — C₁₃×#20547 #15706886

Order 3328.

994 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×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1770	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1767	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1752	C₂	1	4	yes	maximal
7	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
8	C₂×C₁₃×#117	C₄	1	2	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#431	C₄	1	8	yes	complemented
11	C₂×C₁₃×C₄⋊C₈	C₄	1	8	yes	complemented
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
16	C₂×C₁₃×#97	C₂²	1	8	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
18	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
19	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₄×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
21	C₄×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₁₃×#376	C₄	1	2	yes	—
23	C₄³×C₁₃	C₂²	1	2	yes	—
24	C₁₃×#336	C₂²	1	8	yes	—
25	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	—	4	4	no	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
32	C₁₃×#117	—	2	8	no	—
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
34	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
45	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
46	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
47	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
48	C₁₃×#97	—	4	8	no	—
49	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
50	C₁₃×#95	C₂×C₄	1	4	yes	—
51	C₁₃×#95	C₂×C₄	1	4	yes	—
52	#20547	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	4	2	no	—
57	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
58	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
59	C₂²×C₄×C₁₃	M₁₆	1	8	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	2	8	no	—
71	C₂×C₈×C₁₃	—	2	4	no	—
72	C₂×C₈×C₁₃	—	4	8	no	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
75	C₁₃×C₄⋊C₄	—	2	8	no	—
76	C₁₃×C₂²⋊C₄	—	2	8	no	—
77	C₄²×C₁₃	—	2	8	no	—
78	C₄²×C₁₃	—	2	4	no	—
79	C₄²×C₁₃	—	2	4	no	—
80	C₄²×C₁₃	—	4	4	no	—
81	C₄²×C₁₃	C₄²	1	4	yes	—
82	C₄²×C₁₃	C₄²	1	4	yes	—
83	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
84	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
85	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
86	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂×#431	C₂×C₁₃	1	1	yes	characteristic
88	C₂²×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic
89	C₄×#117	C₂×C₁₃	1	1	yes	characteristic
90	#1770	C₂×C₁₃	1	4	yes	complemented
91	#1767	C₂×C₁₃	1	4	yes	complemented
92	#1752	C₂×C₁₃	1	4	yes	—
93	C₂³×C₁₃	—	2	2	no	—
94	C₂³×C₁₃	—	2	2	no	—
95	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
96	C₂³×C₁₃	C₂×M₁₆	1	4	yes	—
97	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	2	8	no	—
105	C₂×C₄×C₁₃	—	2	8	no	—
106	C₂×C₄×C₁₃	—	2	8	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	#97	1	16	yes	complemented
109	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
115	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
116	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
119	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
120	C₈×C₁₃	—	2	8	no	—
121	C₈×C₁₃	—	2	8	no	—
122	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
123	C₂×#117	C₄×C₁₃	1	2	yes	—
124	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
125	#431	C₄×C₁₃	1	8	yes	complemented
126	C₂×C₄⋊C₈	C₄×C₁₃	1	8	yes	complemented
127	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
128	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
129	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
130	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
131	C₂×#97	C₂²×C₁₃	1	8	yes	—
132	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
133	C₄×C₄⋊C₄	C₄×C₁₃	1	2	yes	—
134	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
135	C₄×C₂²⋊C₄	C₄×C₁₃	1	2	yes	—
136	C₄×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
137	#376	C₄×C₁₃	1	2	yes	—
138	C₄³	C₂²×C₁₃	1	2	yes	—
139	#336	C₂²×C₁₃	1	8	yes	—
140	C₂²×C₁₃	—	2	2	no	—
141	C₂²×C₁₃	—	2	8	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	C₂×#97	1	4	yes	—
144	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
145	C₂²×C₁₃	#520	1	2	yes	—
146	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
147	C₂²×C₁₃	#568	1	2	yes	—
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	4	8	no	—
152	C₄×C₁₃	C₂×#97	1	4	yes	—
153	C₄×C₁₃	C₂×#97	1	4	yes	—
154	C₄×C₁₃	#518	1	1	yes	characteristic
155	C₄×C₁₃	#519	1	1	yes	characteristic
156	C₄×C₁₃	#520	1	2	yes	—
157	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
158	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
159	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
160	C₂²×C₈	—	4	4	no	—
161	C₂²×C₈	C₂×C₄×C₁₃	1	8	yes	—
162	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
163	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
164	#117	—	2	8	no	—
165	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
166	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
167	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
168	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
169	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
170	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
171	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
172	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
173	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
174	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
176	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
177	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
178	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
179	C₂²⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
180	#97	—	4	8	no	—
181	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	—
182	#95	C₂×C₄×C₁₃	1	4	yes	—
183	#95	C₂×C₄×C₁₃	1	4	yes	—
184	C₂×C₁₃	—	2	4	no	—
185	C₂×C₁₃	#2274	1	1	yes	characteristic
186	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
187	C₂×C₁₃	#2873	1	1	yes	characteristic
188	C₂×C₁₃	#2989	1	4	yes	—
189	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
194	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
195	C₂²×C₄	C₁₃×M₁₆	1	8	yes	—
196	C₂²×C₄	C₂²×C₄×C₁₃	1	8	yes	—
197	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
198	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
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	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	8	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	4	8	no	—
209	C₂×C₈	C₂²×C₄×C₁₃	1	8	yes	—
210	C₂×C₈	C₂²×C₄×C₁₃	1	8	yes	—
211	C₄⋊C₄	—	2	8	no	—
212	C₂²⋊C₄	—	2	8	no	—
213	C₄²	—	2	8	no	—
214	C₄²	—	2	4	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	4	4	no	—
217	C₄²	C₄²×C₁₃	1	4	yes	—
218	C₄²	C₄²×C₁₃	1	4	yes	—
219	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
220	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
221	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
222	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
223	C₁₃	#20547	1	1	yes	characteristic, direct product factor, Sylow
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₂×C₄²×C₁₃	1	2	yes	—
227	C₂³	C₂×C₁₃×M₁₆	1	4	yes	—
228	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₁₃×#97	1	16	yes	complemented
240	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
241	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
242	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
243	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
244	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
245	C₂×C₄	C₂×C₁₃×M₁₆	1	4	yes	—
246	C₂×C₄	C₂×C₁₃×M₁₆	1	4	yes	—
247	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
249	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
250	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
251	C₈	—	2	8	no	—
252	C₈	—	2	8	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	8	no	—
256	C₂²	C₂×C₁₃×#97	1	4	yes	—
257	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
258	C₂²	C₁₃×#520	1	2	yes	—
259	C₂²	C₂²×C₁₃×M₁₆	1	2	yes	—
260	C₂²	C₁₃×#568	1	2	yes	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	8	no	—
265	C₄	C₂×C₁₃×#97	1	4	yes	—
266	C₄	C₂×C₁₃×#97	1	4	yes	—
267	C₄	C₁₃×#518	1	1	yes	characteristic
268	C₄	C₁₃×#519	1	1	yes	characteristic
269	C₄	C₁₃×#520	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	C₁₃×#2274	1	1	yes	characteristic
272	C₂	C₂²×C₁₃×#97	1	1	yes	characteristic
273	C₂	C₁₃×#2873	1	1	yes	characteristic
274	C₂	C₁₃×#2989	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×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
4. C₁₃×#1770	C₂	semidirect
5. C₁₃×#1767	C₂	semidirect
10. C₁₃×#431	C₄	semidirect
11. C₂×C₁₃×C₄⋊C₈	C₄	semidirect
52. #20547	C₁₃	direct (N × H)
90. #1770	C₂×C₁₃	semidirect
91. #1767	C₂×C₁₃	semidirect
108. C₂×C₄×C₁₃	#97	semidirect
125. #431	C₄×C₁₃	semidirect
126. C₂×C₄⋊C₈	C₄×C₁₃	semidirect
223. C₁₃	#20547	direct (N × H)
239. C₂×C₄	C₁₃×#97	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#431	2
C₂²×C₁₃×C₄⋊C₈	2
C₄×C₁₃×#117	2
C₁₃×#1770	2
C₁₃×#1767	2
C₁₃×#1752	2
#20547	13

Subgroups — C13×#20547 #15706886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20547 #15706886