Subgroups — C₁₃×#10815 #15697154

Order 3328.

626 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1742	C₂	1	2	yes	maximal
3	C₂×C₁₃×#341	C₂	1	2	yes	maximal
4	C₂×C₁₃×#338	C₂	1	2	yes	maximal
5	C₂²×C₁₃×M₁₆	C₂²	1	4	yes	—
6	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
9	C₁₃×#405	—	2	4	no	—
10	C₂×C₁₃×#97	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
12	C₁₃×#343	—	2	4	no	—
13	C₁₃×#341	—	2	4	no	—
14	C₁₃×#338	C₄	1	8	yes	complemented
15	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
16	C₂×C₁₃×M₁₆	—	2	2	no	—
17	C₂×C₁₃×M₁₆	—	2	2	no	—
18	C₂×C₁₃×M₁₆	—	2	4	no	—
19	C₂×C₁₃×M₁₆	—	2	4	no	—
20	C₂×C₁₃×M₁₆	—	2	4	no	—
21	C₂×C₁₃×M₁₆	—	2	4	no	—
22	C₂×C₁₃×M₁₆	—	2	4	no	—
23	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
24	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂²×C₈×C₁₃	—	2	4	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	D₄	1	4	yes	—
32	C₁₃×#117	—	2	2	no	—
33	C₁₃×#117	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
35	C₂×C₁₃×C₂²⋊C₄	Q₈	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
37	C₁₃×#97	—	2	4	no	—
38	C₄×C₈×C₁₃	—	2	4	no	—
39	#10815	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	#41	1	2	yes	—
47	C₂²×C₄×C₁₃	#41	1	2	yes	—
48	C₂²×C₄×C₁₃	#41	1	2	yes	—
49	C₂²×C₄×C₁₃	#41	1	2	yes	—
50	C₁₃×M₁₆	—	2	8	no	—
51	C₁₃×M₁₆	—	2	8	no	—
52	C₁₃×M₁₆	—	2	8	no	—
53	C₁₃×M₁₆	—	4	4	no	—
54	C₁₃×M₁₆	—	4	4	no	—
55	C₂×C₈×C₁₃	—	2	2	no	—
56	C₂×C₈×C₁₃	—	2	1	no	—
57	C₂×C₈×C₁₃	—	2	1	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	1	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	—	2	8	no	—
62	C₂×C₈×C₁₃	—	2	8	no	—
63	C₂×C₈×C₁₃	—	4	4	no	—
64	C₂×C₈×C₁₃	—	4	4	no	—
65	C₂×C₈×C₁₃	C₄²	1	4	yes	—
66	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
68	C₁₃×C₄⋊C₄	—	2	4	no	—
69	C₁₃×C₂²⋊C₄	—	2	4	no	—
70	C₄²×C₁₃	—	2	2	no	—
71	C₄²×C₁₃	—	2	2	no	—
72	C₂×#405	C₂×C₁₃	1	1	yes	characteristic
73	#1742	C₂×C₁₃	1	2	yes	—
74	C₂×#341	C₂×C₁₃	1	2	yes	—
75	C₂×#338	C₂×C₁₃	1	2	yes	—
76	C₂³×C₁₃	—	4	2	no	—
77	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
79	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
80	C₂³×C₁₃	#125	1	2	yes	—
81	C₂×C₄×C₁₃	—	2	4	no	—
82	C₂×C₄×C₁₃	—	4	4	no	—
83	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#117	1	4	yes	—
87	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
88	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₁₃	#121	1	2	yes	—
93	C₂×C₄×C₁₃	#123	1	4	yes	—
94	C₂×C₄×C₁₃	#124	1	2	yes	—
95	C₂×C₄×C₁₃	#126	1	4	yes	—
96	C₈×C₁₃	—	2	4	no	—
97	C₈×C₁₃	—	2	2	no	—
98	C₈×C₁₃	—	2	2	no	—
99	C₈×C₁₃	—	4	8	no	—
100	C₂²×M₁₆	C₂²×C₁₃	1	4	yes	—
101	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
102	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
103	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
104	#405	—	2	4	no	—
105	C₂×#97	C₄×C₁₃	1	2	yes	—
106	C₂×C₄×C₈	C₄×C₁₃	1	2	yes	—
107	#343	—	2	4	no	—
108	#341	—	2	4	no	—
109	#338	C₄×C₁₃	1	8	yes	complemented
110	C₂²×C₁₃	—	2	2	no	—
111	C₂²×C₁₃	—	2	4	no	—
112	C₂²×C₁₃	—	4	2	no	—
113	C₂²×C₁₃	—	4	2	no	—
114	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₁₃	#380	1	1	yes	characteristic
116	C₂²×C₁₃	#382	1	1	yes	characteristic
117	C₂²×C₁₃	#387	1	4	yes	—
118	C₄×C₁₃	—	4	4	no	—
119	C₄×C₁₃	#376	1	1	yes	characteristic
120	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
121	C₄×C₁₃	#381	1	1	yes	characteristic
122	C₄×C₁₃	#383	1	1	yes	characteristic
123	C₄×C₁₃	#388	1	4	yes	—
124	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
125	C₂×M₁₆	—	2	2	no	—
126	C₂×M₁₆	—	2	2	no	—
127	C₂×M₁₆	—	2	4	no	—
128	C₂×M₁₆	—	2	4	no	—
129	C₂×M₁₆	—	2	4	no	—
130	C₂×M₁₆	—	2	4	no	—
131	C₂×M₁₆	—	2	4	no	—
132	C₂×M₁₆	C₂×C₄×C₁₃	1	8	yes	—
133	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	2	4	no	—
137	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
138	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
139	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
140	C₂²×C₈	D₄×C₁₃	1	4	yes	—
141	#117	—	2	2	no	—
142	#117	—	2	2	no	—
143	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
144	C₂×C₂²⋊C₄	Q₈×C₁₃	1	2	yes	—
145	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
146	#97	—	2	4	no	—
147	C₄×C₈	—	2	4	no	—
148	C₂×C₁₃	—	2	1	no	—
149	C₂×C₁₃	—	2	1	no	—
150	C₂×C₁₃	—	2	4	no	—
151	C₂×C₁₃	#1434	1	1	yes	characteristic
152	C₂×C₁₃	#1813	1	2	yes	—
153	C₂⁴	C₂×Q₈×C₁₃	1	1	yes	characteristic
154	C₂²×C₄	C₁₃×C₄⋊C₄	1	4	yes	—
155	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
156	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
157	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
158	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
159	C₂²×C₄	C₁₃×#41	1	2	yes	—
160	C₂²×C₄	C₁₃×#41	1	2	yes	—
161	C₂²×C₄	C₁₃×#41	1	2	yes	—
162	C₂²×C₄	C₁₃×#41	1	2	yes	—
163	M₁₆	—	2	8	no	—
164	M₁₆	—	2	8	no	—
165	M₁₆	—	2	8	no	—
166	M₁₆	—	4	4	no	—
167	M₁₆	—	4	4	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	4	4	no	—
177	C₂×C₈	—	4	4	no	—
178	C₂×C₈	C₄²×C₁₃	1	4	yes	—
179	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
180	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
181	C₄⋊C₄	—	2	4	no	—
182	C₂²⋊C₄	—	2	4	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₁₃	#10815	1	1	yes	characteristic, direct product factor, Sylow
186	C₂³	—	4	2	no	—
187	C₂³	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
188	C₂³	C₄×Q₈×C₁₃	1	2	yes	—
189	C₂³	C₁₃×C₂²⋊Q₈	1	2	yes	—
190	C₂³	C₁₃×#125	1	2	yes	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
194	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
195	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
196	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
197	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
198	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
199	C₂×C₄	C₁₃×#117	1	4	yes	—
200	C₂×C₄	C₄×D₄×C₁₃	1	4	yes	—
201	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
202	C₂×C₄	C₁₃×#121	1	2	yes	—
203	C₂×C₄	C₁₃×#123	1	4	yes	—
204	C₂×C₄	C₁₃×#124	1	2	yes	—
205	C₂×C₄	C₁₃×#126	1	4	yes	—
206	C₈	—	2	4	no	—
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	4	8	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
215	C₂²	C₁₃×#380	1	1	yes	characteristic
216	C₂²	C₁₃×#382	1	1	yes	characteristic
217	C₂²	C₁₃×#387	1	4	yes	—
218	C₄	—	4	4	no	—
219	C₄	C₁₃×#376	1	1	yes	characteristic
220	C₄	C₄×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
221	C₄	C₁₃×#381	1	1	yes	characteristic
222	C₄	C₁₃×#383	1	1	yes	characteristic
223	C₄	C₁₃×#388	1	4	yes	—
224	C₂	—	2	1	no	—
225	C₂	—	2	1	no	—
226	C₂	—	2	4	no	—
227	C₂	C₁₃×#1434	1	1	yes	characteristic
228	C₂	C₁₃×#1813	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	#117
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
socle	C₂²×C₁₃	#380
prime radical	C₂⁴×C₁₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂²×C₁₃×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₁₃×#338	C₄	semidirect
39. #10815	C₁₃	direct (N × H)
109. #338	C₄×C₁₃	semidirect
185. C₁₃	#10815	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×#405	2
C₁₃×#1742	2
C₂×C₁₃×#341	2
C₂×C₁₃×#338	2
#10815	13

Subgroups — C13×#10815 #15697154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10815 #15697154