Subgroups — C₁₃×#21143 #15707482

Order 3328.

1038 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#424	C₂	1	2	yes	maximal
2	C₂×C₁₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2285	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1939	C₂	1	8	yes	maximal, complemented
6	C₁₃×#1729	C₂	1	2	yes	maximal
7	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#117	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#424	—	2	8	no	—
13	C₁₃×#423	—	2	4	no	—
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	4	yes	—
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	8	yes	—
16	C₂×C₄×C₈×C₁₃	—	2	4	no	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
18	C₁₃×#384	C₄	1	2	yes	—
19	C₁₃×#382	C₄	1	2	yes	—
20	C₁₃×#380	C₄	1	4	yes	—
21	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
22	C₁₃×#336	C₂²	1	8	yes	—
23	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂×C₁₃×M₁₆	—	2	4	no	—
26	C₂×C₁₃×M₁₆	—	2	4	no	—
27	C₂×C₁₃×M₁₆	—	2	4	no	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	—	2	2	no	—
31	C₂²×C₈×C₁₃	—	2	2	no	—
32	C₂²×C₈×C₁₃	D₄	1	4	yes	—
33	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₁₃×#117	—	2	4	no	—
38	C₁₃×#117	—	2	4	no	—
39	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
41	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	D₄	1	4	yes	—
45	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
48	C₁₃×C₄⋊C₈	—	2	16	no	—
49	C₁₃×C₄⋊C₈	—	2	8	no	—
50	C₄×C₈×C₁₃	—	2	8	no	—
51	C₁₃×#95	C₂×C₄	1	8	yes	—
52	#21143	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	8	no	—
55	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₁₃	#41	1	2	yes	—
65	C₂²×C₄×C₁₃	#41	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
67	C₁₃×M₁₆	—	4	8	no	—
68	C₂×C₈×C₁₃	—	2	8	no	—
69	C₂×C₈×C₁₃	—	2	8	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
79	C₁₃×C₄⋊C₄	—	2	8	no	—
80	C₁₃×C₄⋊C₄	—	2	16	no	—
81	C₁₃×C₂²⋊C₄	—	2	8	no	—
82	C₄²×C₁₃	—	2	4	no	—
83	C₄²×C₁₃	—	2	4	no	—
84	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
85	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
86	C₄²×C₁₃	C₂×D₄	1	4	yes	—
87	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
88	C₂×#424	C₂×C₁₃	1	2	yes	—
89	C₂×#423	C₂×C₁₃	1	1	yes	characteristic
90	C₂²×C₄×C₈	C₂×C₁₃	1	1	yes	characteristic
91	#2285	C₂×C₁₃	1	1	yes	characteristic
92	#1939	C₂×C₁₃	1	8	yes	complemented
93	#1729	C₂×C₁₃	1	2	yes	—
94	C₂³×C₁₃	—	2	4	no	—
95	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₁₃	C₄×D₄	1	2	yes	—
97	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
98	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	—	2	16	no	—
101	C₂×C₄×C₁₃	—	2	8	no	—
102	C₂×C₄×C₁₃	—	2	16	no	—
103	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₁₃	#121	1	4	yes	—
109	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
110	C₂×C₄×C₁₃	#125	1	4	yes	—
111	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
112	C₂×C₄×C₁₃	#131	1	8	yes	—
113	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
119	C₈×C₁₃	—	2	4	no	—
120	C₈×C₁₃	—	2	4	no	—
121	C₈×C₁₃	—	4	4	no	—
122	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
123	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
124	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
125	C₂×#117	C₂²×C₁₃	1	2	yes	—
126	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
127	#424	—	2	8	no	—
128	#423	—	2	4	no	—
129	C₂×C₄⋊C₈	C₂²×C₁₃	1	4	yes	—
130	C₂×C₄⋊C₈	C₂²×C₁₃	1	8	yes	—
131	C₂×C₄×C₈	—	2	4	no	—
132	C₂×C₄×C₈	C₂²×C₁₃	1	4	yes	—
133	#384	C₄×C₁₃	1	2	yes	—
134	#382	C₄×C₁₃	1	2	yes	—
135	#380	C₄×C₁₃	1	4	yes	—
136	C₄×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
137	#336	C₂²×C₁₃	1	8	yes	—
138	C₂²×C₁₃	—	2	4	no	—
139	C₂²×C₁₃	—	2	4	no	—
140	C₂²×C₁₃	—	2	4	no	—
141	C₂²×C₁₃	#389	1	4	yes	—
142	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
145	C₂²×C₁₃	C₂×#131	1	4	yes	—
146	C₄×C₁₃	—	2	4	no	—
147	C₄×C₁₃	—	4	8	no	—
148	C₄×C₁₃	#389	1	4	yes	—
149	C₄×C₁₃	#424	1	8	yes	complemented
150	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
151	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
152	C₄×C₁₃	C₂×#125	1	1	yes	characteristic
153	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
154	C₂³×C₄	C₂×C₄×C₁₃	1	2	yes	—
155	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	D₄×C₁₃	1	4	yes	—
164	C₂²×C₈	Q₈×C₁₃	1	4	yes	—
165	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
166	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
167	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
168	#117	—	2	4	no	—
169	#117	—	2	4	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
171	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	8	yes	—
172	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
173	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
175	C₂×C₄²	D₄×C₁₃	1	4	yes	—
176	C₂×C₄²	Q₈×C₁₃	1	4	yes	—
177	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
178	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
179	C₄⋊C₈	—	2	16	no	—
180	C₄⋊C₈	—	2	8	no	—
181	C₄×C₈	—	2	8	no	—
182	#95	C₂×C₄×C₁₃	1	8	yes	—
183	C₂×C₁₃	—	2	4	no	—
184	C₂×C₁₃	C₂×#389	1	1	yes	characteristic
185	C₂×C₁₃	C₂×#424	1	2	yes	—
186	C₂×C₁₃	#2931	1	2	yes	—
187	C₂×C₁₃	#2960	1	2	yes	—
188	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	C₁₃×C₄⋊C₄	1	8	yes	—
191	C₂²×C₄	C₂²×C₄×C₁₃	1	8	yes	—
192	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
193	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
194	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
196	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
197	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
198	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
199	C₂²×C₄	C₁₃×#41	1	2	yes	—
200	C₂²×C₄	C₁₃×#41	1	2	yes	—
201	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
202	M₁₆	—	4	8	no	—
203	C₂×C₈	—	2	8	no	—
204	C₂×C₈	—	2	8	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
213	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄⋊C₄	—	2	16	no	—
216	C₂²⋊C₄	—	2	8	no	—
217	C₄²	—	2	4	no	—
218	C₄²	—	2	4	no	—
219	C₄²	C₁₃×C₄⋊C₄	1	4	yes	—
220	C₄²	C₁₃×C₄⋊C₄	1	4	yes	—
221	C₄²	C₂×D₄×C₁₃	1	4	yes	—
222	C₄²	C₂×Q₈×C₁₃	1	4	yes	—
223	C₁₃	#21143	1	1	yes	characteristic, direct product factor, Sylow
224	C₂³	—	2	4	no	—
225	C₂³	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
226	C₂³	C₄×D₄×C₁₃	1	2	yes	—
227	C₂³	C₄×Q₈×C₁₃	1	2	yes	—
228	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	16	no	—
233	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
234	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
235	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
236	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
237	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
238	C₂×C₄	C₁₃×#121	1	4	yes	—
239	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
240	C₂×C₄	C₁₃×#125	1	4	yes	—
241	C₂×C₄	C₁₃×C₄⋊Q₈	1	4	yes	—
242	C₂×C₄	C₁₃×#131	1	8	yes	—
243	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
244	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
245	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
246	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
247	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
248	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
249	C₈	—	2	4	no	—
250	C₈	—	2	4	no	—
251	C₈	—	4	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	C₁₃×#389	1	4	yes	—
256	C₂²	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
257	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
258	C₂²	C₂×C₄×Q₈×C₁₃	1	1	yes	characteristic
259	C₂²	C₂×C₁₃×#131	1	4	yes	—
260	C₄	—	2	4	no	—
261	C₄	—	4	8	no	—
262	C₄	C₁₃×#389	1	4	yes	—
263	C₄	C₁₃×#424	1	8	yes	complemented
264	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
265	C₄	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
266	C₄	C₂×C₁₃×#125	1	1	yes	characteristic
267	C₄	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
268	C₂	—	2	4	no	—
269	C₂	C₂×C₁₃×#389	1	1	yes	characteristic
270	C₂	C₂×C₁₃×#424	1	2	yes	—
271	C₂	C₁₃×#2931	1	2	yes	—
272	C₂	C₁₃×#2960	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₁₃×#1939	C₂	semidirect
52. #21143	C₁₃	direct (N × H)
92. #1939	C₂×C₁₃	semidirect
149. C₄×C₁₃	#424	semidirect
223. C₁₃	#21143	direct (N × H)
263. C₄	C₁₃×#424	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#424	2
C₂×C₁₃×#423	2
C₂²×C₄×C₈×C₁₃	2
C₁₃×#2285	2
C₁₃×#1939	2
C₁₃×#1729	2
#21143	13

Subgroups — C13×#21143 #15707482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#21143 #15707482