Subgroups — C₁₃×#17989 #15704328

Order 3328.

624 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2981	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1586	C₂	1	4	yes	maximal, complemented
4	C₁₃×#1581	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1496	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#532	C₂²	1	2	yes	—
9	C₁₃×#500	C₄	1	1	yes	characteristic
10	C₁₃×#493	C₄	1	1	yes	characteristic
11	C₁₃×#486	C₄	1	2	yes	—
12	C₁₃×#466	C₄	1	2	yes	—
13	C₁₃×#466	C₄	1	2	yes	—
14	C₄×C₁₃×D₈	C₂²	1	2	yes	—
15	C₄×C₁₃×D₈	C₂²	1	2	yes	—
16	C₁₃×#436	C₂²	1	4	yes	—
17	C₁₃×#435	—	2	4	no	—
18	C₈×D₄×C₁₃	C₂²	1	4	yes	—
19	C₁₃×#432	C₂²	1	2	yes	—
20	C₁₃×#431	C₂²	1	1	yes	characteristic
21	C₁₃×#431	C₂²	1	1	yes	characteristic
22	C₁₃×#408	—	2	4	no	—
23	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#335	C₂²	1	2	yes	—
25	C₁₃×#334	C₂²	1	2	yes	—
26	C₁₃×D₄⋊C₈	C₂²	1	8	yes	—
27	C₁₃×#322	C₂²	1	2	yes	—
28	C₁₃×#322	C₂²	1	2	yes	—
29	C₂×C₁₃×#41	—	2	2	no	—
30	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
31	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
32	C₂×C₁₃×M₁₆	—	2	4	no	—
33	C₂²×C₈×C₁₃	—	2	4	no	—
34	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
37	C₁₃×#127	C₂×C₄	1	2	yes	—
38	C₁₃×#124	—	2	2	no	—
39	C₁₃×#121	—	2	2	no	—
40	C₁₃×#121	C₂×C₄	1	4	yes	—
41	C₄×D₄×C₁₃	—	2	2	no	—
42	C₄×D₄×C₁₃	C₂³	1	4	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×#107	C₂×C₄	1	2	yes	—
45	C₁₃×#107	C₂×C₄	1	2	yes	—
46	C₁₃×C₄⋊C₈	—	2	1	no	—
47	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
48	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
49	C₁₃×#102	C₂×C₄	1	4	yes	—
50	C₁₃×#102	C₂×C₄	1	4	yes	—
51	C₁₃×C₂²⋊C₈	—	2	2	no	—
52	C₁₃×C₂²⋊C₈	—	2	1	no	—
53	C₁₃×C₂²⋊C₈	—	2	4	no	—
54	C₁₃×C₂²⋊C₈	—	2	4	no	—
55	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
56	C₁₃×#97	—	2	2	no	—
57	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂³	1	2	yes	—
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	2	yes	—
63	#17989	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₁₃×#41	—	8	4	no	—
65	C₂×Q₈×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	2	4	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	2	2	no	—
69	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₁₃	—	2	4	no	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
74	C₁₃×D₈	—	4	2	no	—
75	C₁₃×D₈	—	4	2	no	—
76	C₁₃×M₁₆	—	8	4	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	4	1	no	—
80	C₂×C₈×C₁₃	—	4	1	no	—
81	C₂×C₈×C₁₃	—	8	4	no	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
89	C₁₃×C₂²⋊C₄	—	2	4	no	—
90	C₁₃×C₂²⋊C₄	—	2	4	no	—
91	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₃	#41	1	1	yes	characteristic
94	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
95	#3169	C₂×C₁₃	1	1	yes	characteristic
96	#2981	C₂×C₁₃	1	2	yes	complemented
97	#1586	C₂×C₁₃	1	4	yes	complemented
98	#1581	C₂×C₁₃	1	4	yes	complemented
99	#1568	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#1496	C₂×C₁₃	1	2	yes	complemented
101	#1452	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₂³×C₁₃	—	2	4	no	—
103	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
104	Q₈×C₁₃	—	8	2	no	—
105	D₄×C₁₃	—	2	4	no	—
106	D₄×C₁₃	—	2	4	no	—
107	D₄×C₁₃	—	8	2	no	—
108	D₄×C₁₃	—	8	2	no	—
109	D₄×C₁₃	—	8	4	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	4	1	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	4	4	no	—
117	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
123	C₈×C₁₃	—	2	1	no	—
124	C₈×C₁₃	—	2	1	no	—
125	C₈×C₁₃	—	2	1	no	—
126	C₈×C₁₃	—	2	1	no	—
127	C₈×C₁₃	—	2	4	no	—
128	C₈×C₁₃	—	4	2	no	—
129	C₈×C₁₃	—	4	4	no	—
130	#532	C₂²×C₁₃	1	2	yes	—
131	#500	C₄×C₁₃	1	1	yes	characteristic
132	#493	C₄×C₁₃	1	1	yes	characteristic
133	#486	C₄×C₁₃	1	2	yes	—
134	#466	C₄×C₁₃	1	2	yes	—
135	#466	C₄×C₁₃	1	2	yes	—
136	C₄×D₈	C₂²×C₁₃	1	2	yes	—
137	C₄×D₈	C₂²×C₁₃	1	2	yes	—
138	#436	C₂²×C₁₃	1	4	yes	—
139	#435	—	2	4	no	—
140	C₈×D₄	C₂²×C₁₃	1	4	yes	—
141	#432	C₂²×C₁₃	1	2	yes	—
142	#431	C₂²×C₁₃	1	1	yes	characteristic
143	#431	C₂²×C₁₃	1	1	yes	characteristic
144	#408	—	2	4	no	—
145	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
146	#335	C₂²×C₁₃	1	2	yes	—
147	#334	C₂²×C₁₃	1	2	yes	—
148	D₄⋊C₈	C₂²×C₁₃	1	8	yes	—
149	#322	C₂²×C₁₃	1	2	yes	—
150	#322	C₂²×C₁₃	1	2	yes	—
151	C₂²×C₁₃	—	2	1	no	—
152	C₂²×C₁₃	—	2	1	no	—
153	C₂²×C₁₃	—	2	1	no	—
154	C₂²×C₁₃	—	4	4	no	—
155	C₂²×C₁₃	—	4	4	no	—
156	C₂²×C₁₃	—	4	4	no	—
157	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄×C₁₃	—	2	1	no	—
159	C₄×C₁₃	—	2	1	no	—
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	4	4	no	—
163	C₄×C₁₃	#568	1	1	yes	characteristic
164	C₄×C₁₃	#568	1	1	yes	characteristic
165	C₄×C₁₃	#576	1	2	yes	—
166	C₂×#41	—	2	2	no	—
167	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
168	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
172	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
173	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
174	#127	C₂×C₄×C₁₃	1	2	yes	—
175	#124	—	2	2	no	—
176	#121	—	2	2	no	—
177	#121	C₂×C₄×C₁₃	1	4	yes	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	C₂³×C₁₃	1	4	yes	—
180	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
181	#107	C₂×C₄×C₁₃	1	2	yes	—
182	#107	C₂×C₄×C₁₃	1	2	yes	—
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
185	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
186	#102	C₂×C₄×C₁₃	1	4	yes	—
187	#102	C₂×C₄×C₁₃	1	4	yes	—
188	C₂²⋊C₈	—	2	2	no	—
189	C₂²⋊C₈	—	2	1	no	—
190	C₂²⋊C₈	—	2	4	no	—
191	C₂²⋊C₈	—	2	4	no	—
192	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
193	#97	—	2	2	no	—
194	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
195	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
196	C₄×C₈	C₂³×C₁₃	1	2	yes	—
197	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
198	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
199	C₄×C₈	C₂³×C₁₃	1	2	yes	—
200	C₂×C₁₃	—	4	1	no	—
201	C₂×C₁₃	—	8	4	no	—
202	C₂×C₁₃	#2932	1	1	yes	characteristic
203	C₂×C₁₃	#2948	1	1	yes	characteristic
204	C₂×C₁₃	#2957	1	1	yes	characteristic
205	#41	—	8	4	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₂²×C₄×C₁₃	1	4	yes	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
214	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
215	D₈	—	4	2	no	—
216	D₈	—	4	2	no	—
217	M₁₆	—	8	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	8	4	no	—
223	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
224	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
226	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
233	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
234	C₄²	C₁₃×#41	1	1	yes	characteristic
235	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
236	C₁₃	#17989	1	1	yes	characteristic, direct product factor, Sylow
237	C₂³	—	2	4	no	—
238	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
239	Q₈	—	8	2	no	—
240	D₄	—	2	4	no	—
241	D₄	—	2	4	no	—
242	D₄	—	8	2	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
253	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
255	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
256	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	4	no	—
263	C₈	—	4	2	no	—
264	C₈	—	4	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	4	no	—
271	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	4	no	—
277	C₄	C₁₃×#568	1	1	yes	characteristic
278	C₄	C₁₃×#568	1	1	yes	characteristic
279	C₄	C₁₃×#576	1	2	yes	—
280	C₂	—	4	1	no	—
281	C₂	—	8	4	no	—
282	C₂	C₁₃×#2932	1	1	yes	characteristic
283	C₂	C₁₃×#2948	1	1	yes	characteristic
284	C₂	C₁₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₁₃×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₁₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2981	C₂	semidirect
3. C₁₃×#1586	C₂	semidirect
4. C₁₃×#1581	C₂	semidirect
5. C₁₃×#1568	C₂	semidirect
6. C₁₃×#1496	C₂	semidirect
7. C₁₃×#1452	C₂	semidirect
63. #17989	C₁₃	direct (N × H)
96. #2981	C₂×C₁₃	semidirect
97. #1586	C₂×C₁₃	semidirect
98. #1581	C₂×C₁₃	semidirect
99. #1568	C₂×C₁₃	semidirect
100. #1496	C₂×C₁₃	semidirect
101. #1452	C₂×C₁₃	semidirect
236. C₁₃	#17989	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#3169	2
C₁₃×#2981	2
C₁₃×#1586	2
C₁₃×#1581	2
C₁₃×#1568	2
C₁₃×#1496	2
C₁₃×#1452	2
#17989	13

Subgroups — C13×#17989 #15704328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#17989 #15704328