Subgroups — C₁₃×#20570 #15706909

Order 3328.

878 conjugacy classes of subgroups.

The subgroup lattice has 272 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	C₁₃×#1770	C₂	1	2	yes	maximal
5	C₁₃×#1767	C₂	1	2	yes	maximal
6	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1745	C₂	1	2	yes	maximal
8	C₁₃×#1744	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	4	yes	—
15	C₂×C₁₃×#97	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
18	C₄×C₁₃×C₄⋊C₄	C₄	1	4	yes	—
19	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
20	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₄³×C₁₃	C₂²	1	1	yes	characteristic
23	C₄³×C₁₃	C₂²	1	2	yes	—
24	C₁₃×#336	C₂²	1	1	yes	characteristic
25	C₁₃×#336	C₂²	1	1	yes	characteristic
26	C₁₃×#336	C₂²	1	2	yes	—
27	C₁₃×#336	C₂²	1	2	yes	—
28	C₁₃×#336	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	4	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	2	yes	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₁₃×C₄⋊C₄	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	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
48	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
49	C₁₃×C₄⋊C₈	C₂×C₄	1	16	yes	—
50	C₁₃×#97	—	4	4	no	—
51	C₁₃×#97	C₂×C₄	1	16	yes	—
52	C₄×C₈×C₁₃	—	4	2	no	—
53	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
54	C₁₃×#95	C₂×C₄	1	2	yes	—
55	C₁₃×#95	C₂×C₄	1	4	yes	—
56	#20570	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂×Q₈×C₁₃	—	2	4	no	—
58	C₂×Q₈×C₁₃	C₄²	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₁₃	—	2	4	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	4	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
76	C₁₃×C₄⋊C₄	—	2	8	no	—
77	C₁₃×C₄⋊C₄	—	2	4	no	—
78	C₁₃×C₄⋊C₄	C₄²	1	8	yes	—
79	C₁₃×C₄⋊C₄	C₄²	1	4	yes	—
80	C₄²×C₁₃	—	2	8	no	—
81	C₄²×C₁₃	—	2	1	no	—
82	C₄²×C₁₃	—	2	2	no	—
83	C₄²×C₁₃	—	2	2	no	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	—	2	2	no	—
86	C₄²×C₁₃	—	4	2	no	—
87	C₄²×C₁₃	—	4	4	no	—
88	C₄²×C₁₃	M₁₆	1	8	yes	—
89	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
90	C₄²×C₁₃	C₂²×C₄	1	8	yes	—
91	C₂×#446	C₂×C₁₃	1	2	yes	—
92	C₄²×Q₈	C₂×C₁₃	1	1	yes	characteristic
93	#1771	C₂×C₁₃	1	4	yes	—
94	#1770	C₂×C₁₃	1	2	yes	—
95	#1767	C₂×C₁₃	1	2	yes	—
96	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic
97	#1745	C₂×C₁₃	1	2	yes	—
98	#1744	C₂×C₁₃	1	1	yes	characteristic
99	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
100	Q₈×C₁₃	—	2	8	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	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
113	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
114	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
116	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
117	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
118	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
119	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
121	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
122	C₈×C₁₃	—	2	4	no	—
123	C₈×C₁₃	—	2	8	no	—
124	C₈×C₁₃	—	2	4	no	—
125	C₂×C₄×Q₈	C₄×C₁₃	1	2	yes	—
126	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
127	#446	C₄×C₁₃	1	16	yes	complemented
128	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
129	C₂×C₄⋊C₈	C₂²×C₁₃	1	4	yes	—
130	C₂×#97	C₂²×C₁₃	1	4	yes	—
131	C₂×#97	C₂²×C₁₃	1	4	yes	—
132	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
133	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
134	C₄×C₄⋊C₄	C₄×C₁₃	1	4	yes	—
135	C₄×C₄⋊C₄	C₄×C₁₃	1	2	yes	—
136	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
137	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
138	C₄³	C₂²×C₁₃	1	1	yes	characteristic
139	C₄³	C₂²×C₁₃	1	2	yes	—
140	#336	C₂²×C₁₃	1	1	yes	characteristic
141	#336	C₂²×C₁₃	1	1	yes	characteristic
142	#336	C₂²×C₁₃	1	2	yes	—
143	#336	C₂²×C₁₃	1	2	yes	—
144	#336	C₂²×C₁₃	1	4	yes	—
145	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
146	C₂²×C₁₃	#519	1	1	yes	characteristic
147	C₂²×C₁₃	#520	1	1	yes	characteristic
148	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
149	C₂²×C₁₃	#568	1	2	yes	—
150	C₄×C₁₃	—	2	4	no	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	—	2	4	no	—
153	C₄×C₁₃	—	2	4	no	—
154	C₄×C₁₃	—	2	4	no	—
155	C₄×C₁₃	—	4	4	no	—
156	C₄×C₁₃	C₄×M₁₆	1	4	yes	—
157	C₄×C₁₃	#519	1	2	yes	—
158	C₄×C₁₃	#520	1	2	yes	—
159	C₂²×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
161	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
162	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
163	C₄×Q₈	—	2	8	no	—
164	C₄×Q₈	C₂×C₄×C₁₃	1	8	yes	—
165	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
166	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
167	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
168	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
169	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
170	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
171	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
172	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
173	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
176	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
177	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
178	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
179	C₄⋊C₈	C₂×C₄×C₁₃	1	16	yes	—
180	#97	—	4	4	no	—
181	#97	C₂×C₄×C₁₃	1	16	yes	—
182	C₄×C₈	—	4	2	no	—
183	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	—
184	#95	C₂×C₄×C₁₃	1	2	yes	—
185	#95	C₂×C₄×C₁₃	1	4	yes	—
186	C₂×C₁₃	#2276	1	1	yes	characteristic
187	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
188	C₂×C₁₃	#2875	1	1	yes	characteristic
189	C₂×C₁₃	#2983	1	2	yes	—
190	C₂×C₁₃	#2989	1	2	yes	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	C₄²×C₁₃	1	4	yes	—
193	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
194	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
196	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
197	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
199	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
208	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
209	C₂×C₈	C₂²×C₄×C₁₃	1	8	yes	—
210	C₄⋊C₄	—	2	8	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	C₄²×C₁₃	1	8	yes	—
213	C₄⋊C₄	C₄²×C₁₃	1	4	yes	—
214	C₄²	—	2	8	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	4	2	no	—
221	C₄²	—	4	4	no	—
222	C₄²	C₁₃×M₁₆	1	8	yes	—
223	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
224	C₄²	C₂²×C₄×C₁₃	1	8	yes	—
225	C₁₃	#20570	1	1	yes	characteristic, direct product factor, Sylow
226	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
227	Q₈	—	2	8	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
239	C₂×C₄	C₂×C₄²×C₁₃	1	4	yes	—
240	C₂×C₄	C₂×C₄²×C₁₃	1	4	yes	—
241	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
242	C₂×C₄	C₂×C₁₃×M₁₆	1	4	yes	—
243	C₂×C₄	C₂×C₁₃×M₁₆	1	4	yes	—
244	C₂×C₄	C₂×C₁₃×M₁₆	1	4	yes	—
245	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
246	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
247	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
248	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
249	C₈	—	2	4	no	—
250	C₈	—	2	8	no	—
251	C₈	—	2	4	no	—
252	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
253	C₂²	C₁₃×#519	1	1	yes	characteristic
254	C₂²	C₁₃×#520	1	1	yes	characteristic
255	C₂²	C₂²×C₁₃×M₁₆	1	2	yes	—
256	C₂²	C₁₃×#568	1	2	yes	—
257	C₄	—	2	4	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	4	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	4	no	—
263	C₄	C₄×C₁₃×M₁₆	1	4	yes	—
264	C₄	C₁₃×#519	1	2	yes	—
265	C₄	C₁₃×#520	1	2	yes	—
266	C₂	C₁₃×#2276	1	1	yes	characteristic
267	C₂	C₂×C₄×C₁₃×M₁₆	1	1	yes	characteristic
268	C₂	C₁₃×#2875	1	1	yes	characteristic
269	C₂	C₁₃×#2983	1	2	yes	—
270	C₂	C₁₃×#2989	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
56. #20570	C₁₃	direct (N × H)
127. #446	C₄×C₁₃	semidirect
225. C₁₃	#20570	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×#446	2
C₄²×Q₈×C₁₃	2
C₁₃×#1771	2
C₁₃×#1770	2
C₁₃×#1767	2
C₄×C₁₃×C₄⋊C₈	2
C₁₃×#1745	2
C₁₃×#1744	2
#20570	13

Subgroups — C13×#20570 #15706909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20570 #15706909