Subgroups — C₁₃×#17970 #15704309

Order 3328.

640 conjugacy classes of subgroups.

The subgroup lattice has 308 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×#1594	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1589	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1580	C₂	1	2	yes	maximal, complemented
6	C₈×C₁₃×D₈	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#1496	C₂	1	2	yes	maximal, complemented
9	C₁₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#532	C₂²	1	2	yes	—
11	C₁₃×#500	C₄	1	1	yes	characteristic
12	C₁₃×#493	C₄	1	1	yes	characteristic
13	C₁₃×#486	C₄	1	2	yes	—
14	C₁₃×#466	C₄	1	2	yes	—
15	C₁₃×#466	C₄	1	2	yes	—
16	C₄×C₁₃×D₈	C₂²	1	2	yes	—
17	C₄×C₁₃×D₈	C₂²	1	2	yes	—
18	C₁₃×#436	C₂²	1	4	yes	—
19	C₁₃×#435	—	2	4	no	—
20	C₈×D₄×C₁₃	C₂²	1	4	yes	—
21	C₁₃×#432	C₂²	1	2	yes	—
22	C₁₃×#431	C₂²	1	1	yes	characteristic
23	C₁₃×#431	C₂²	1	1	yes	characteristic
24	C₁₃×#408	—	2	4	no	—
25	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#335	C₂²	1	2	yes	—
27	C₁₃×#334	C₂²	1	2	yes	—
28	C₁₃×D₄⋊C₈	C₂²	1	4	yes	—
29	C₁₃×D₄⋊C₈	C₂²	1	4	yes	—
30	C₁₃×#322	C₂²	1	1	yes	characteristic
31	C₁₃×#322	C₂²	1	1	yes	characteristic
32	C₈²×C₁₃	C₂²	1	1	yes	characteristic
33	C₈²×C₁₃	C₂²	1	1	yes	characteristic
34	C₂×C₁₃×#41	—	2	2	no	—
35	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
36	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×M₁₆	—	2	4	no	—
38	C₂²×C₈×C₁₃	—	2	4	no	—
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
42	C₁₃×#127	C₂×C₄	1	2	yes	—
43	C₁₃×#124	—	2	2	no	—
44	C₁₃×#121	—	2	2	no	—
45	C₁₃×#121	C₂×C₄	1	4	yes	—
46	C₄×D₄×C₁₃	—	2	2	no	—
47	C₄×D₄×C₁₃	C₂³	1	4	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×#107	C₂×C₄	1	2	yes	—
50	C₁₃×#107	C₂×C₄	1	2	yes	—
51	C₁₃×C₄⋊C₈	—	2	1	no	—
52	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
53	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
54	C₁₃×#102	C₂×C₄	1	4	yes	—
55	C₁₃×#102	C₂×C₄	1	4	yes	—
56	C₁₃×C₂²⋊C₈	—	2	2	no	—
57	C₁₃×C₂²⋊C₈	—	2	1	no	—
58	C₁₃×C₂²⋊C₈	—	2	4	no	—
59	C₁₃×C₂²⋊C₈	—	2	4	no	—
60	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
61	C₁₃×#97	—	2	2	no	—
62	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	#17970	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₁₃×#41	—	8	4	no	—
72	C₂×Q₈×C₁₃	—	2	2	no	—
73	C₂×D₄×C₁₃	—	2	4	no	—
74	C₂×D₄×C₁₃	—	2	2	no	—
75	C₂×D₄×C₁₃	—	2	2	no	—
76	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₁₃	—	2	4	no	—
78	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
81	C₁₃×D₈	—	2	4	no	—
82	C₁₃×D₈	—	2	4	no	—
83	C₁₃×M₁₆	—	8	4	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	4	no	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	—	2	1	no	—
89	C₂×C₈×C₁₃	—	8	4	no	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
98	C₁₃×C₂²⋊C₄	—	2	4	no	—
99	C₁₃×C₂²⋊C₄	—	2	4	no	—
100	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃	#41	1	1	yes	characteristic
103	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
104	#3169	C₂×C₁₃	1	1	yes	characteristic
105	#2981	C₂×C₁₃	1	2	yes	complemented
106	#1594	C₂×C₁₃	1	2	yes	complemented
107	#1589	C₂×C₁₃	1	2	yes	complemented
108	#1580	C₂×C₁₃	1	2	yes	complemented
109	C₈×D₈	C₂×C₁₃	1	2	yes	complemented
110	#1568	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#1496	C₂×C₁₃	1	2	yes	complemented
112	#1450	C₂×C₁₃	1	1	yes	characteristic, complemented
113	C₂³×C₁₃	—	2	4	no	—
114	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
115	Q₈×C₁₃	—	8	2	no	—
116	D₄×C₁₃	—	2	4	no	—
117	D₄×C₁₃	—	2	4	no	—
118	D₄×C₁₃	—	8	2	no	—
119	D₄×C₁₃	—	8	2	no	—
120	D₄×C₁₃	—	8	4	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	4	1	no	—
126	C₂×C₄×C₁₃	—	4	4	no	—
127	C₂×C₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
134	C₈×C₁₃	—	2	1	no	—
135	C₈×C₁₃	—	2	1	no	—
136	C₈×C₁₃	—	2	2	no	—
137	C₈×C₁₃	—	2	2	no	—
138	C₈×C₁₃	—	4	1	no	—
139	C₈×C₁₃	—	4	1	no	—
140	C₈×C₁₃	—	4	4	no	—
141	C₈×C₁₃	#131	1	2	yes	—
142	C₈×C₁₃	#131	1	2	yes	—
143	#532	C₂²×C₁₃	1	2	yes	—
144	#500	C₄×C₁₃	1	1	yes	characteristic
145	#493	C₄×C₁₃	1	1	yes	characteristic
146	#486	C₄×C₁₃	1	2	yes	—
147	#466	C₄×C₁₃	1	2	yes	—
148	#466	C₄×C₁₃	1	2	yes	—
149	C₄×D₈	C₂²×C₁₃	1	2	yes	—
150	C₄×D₈	C₂²×C₁₃	1	2	yes	—
151	#436	C₂²×C₁₃	1	4	yes	—
152	#435	—	2	4	no	—
153	C₈×D₄	C₂²×C₁₃	1	4	yes	—
154	#432	C₂²×C₁₃	1	2	yes	—
155	#431	C₂²×C₁₃	1	1	yes	characteristic
156	#431	C₂²×C₁₃	1	1	yes	characteristic
157	#408	—	2	4	no	—
158	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
159	#335	C₂²×C₁₃	1	2	yes	—
160	#334	C₂²×C₁₃	1	2	yes	—
161	D₄⋊C₈	C₂²×C₁₃	1	4	yes	—
162	D₄⋊C₈	C₂²×C₁₃	1	4	yes	—
163	#322	C₂²×C₁₃	1	1	yes	characteristic
164	#322	C₂²×C₁₃	1	1	yes	characteristic
165	C₈²	C₂²×C₁₃	1	1	yes	characteristic
166	C₈²	C₂²×C₁₃	1	1	yes	characteristic
167	C₂²×C₁₃	—	2	1	no	—
168	C₂²×C₁₃	—	2	1	no	—
169	C₂²×C₁₃	—	2	1	no	—
170	C₂²×C₁₃	—	4	4	no	—
171	C₂²×C₁₃	—	4	4	no	—
172	C₂²×C₁₃	—	4	4	no	—
173	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	1	no	—
178	C₄×C₁₃	—	4	4	no	—
179	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
180	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
181	C₄×C₁₃	#576	1	2	yes	—
182	C₂×#41	—	2	2	no	—
183	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
184	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
185	C₂×M₁₆	—	2	4	no	—
186	C₂²×C₈	—	2	4	no	—
187	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
188	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
189	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
190	#127	C₂×C₄×C₁₃	1	2	yes	—
191	#124	—	2	2	no	—
192	#121	—	2	2	no	—
193	#121	C₂×C₄×C₁₃	1	4	yes	—
194	C₄×D₄	—	2	2	no	—
195	C₄×D₄	C₂³×C₁₃	1	4	yes	—
196	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
197	#107	C₂×C₄×C₁₃	1	2	yes	—
198	#107	C₂×C₄×C₁₃	1	2	yes	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
201	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
202	#102	C₂×C₄×C₁₃	1	4	yes	—
203	#102	C₂×C₄×C₁₃	1	4	yes	—
204	C₂²⋊C₈	—	2	2	no	—
205	C₂²⋊C₈	—	2	1	no	—
206	C₂²⋊C₈	—	2	4	no	—
207	C₂²⋊C₈	—	2	4	no	—
208	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
209	#97	—	2	2	no	—
210	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
211	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
212	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
213	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
214	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
215	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
216	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
217	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
218	C₂×C₁₃	—	4	1	no	—
219	C₂×C₁₃	—	8	4	no	—
220	C₂×C₁₃	#2931	1	1	yes	characteristic
221	C₂×C₁₃	#2948	1	1	yes	characteristic
222	C₂×C₁₃	#2957	1	1	yes	characteristic
223	#41	—	8	4	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	C₂²×C₄×C₁₃	1	4	yes	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
232	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
233	D₈	—	2	4	no	—
234	D₈	—	2	4	no	—
235	M₁₆	—	8	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	8	4	no	—
242	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
243	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
244	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
245	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
246	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
247	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
253	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
254	C₄²	C₁₃×#41	1	1	yes	characteristic
255	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
256	C₁₃	#17970	1	1	yes	characteristic, direct product factor, Sylow
257	C₂³	—	2	4	no	—
258	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
259	Q₈	—	8	2	no	—
260	D₄	—	2	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	8	2	no	—
263	D₄	—	8	2	no	—
264	D₄	—	8	4	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
273	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
274	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
275	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
277	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	2	no	—
281	C₈	—	2	2	no	—
282	C₈	—	4	1	no	—
283	C₈	—	4	1	no	—
284	C₈	—	4	4	no	—
285	C₈	C₁₃×#131	1	2	yes	—
286	C₈	C₁₃×#131	1	2	yes	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	4	no	—
299	C₄	C₂×C₁₃×#131	1	1	yes	characteristic
300	C₄	C₂×C₁₃×#131	1	1	yes	characteristic
301	C₄	C₁₃×#576	1	2	yes	—
302	C₂	—	4	1	no	—
303	C₂	—	8	4	no	—
304	C₂	C₁₃×#2931	1	1	yes	characteristic
305	C₂	C₁₃×#2948	1	1	yes	characteristic
306	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×#1594	C₂	semidirect
4. C₁₃×#1589	C₂	semidirect
5. C₁₃×#1580	C₂	semidirect
6. C₈×C₁₃×D₈	C₂	semidirect
7. C₁₃×#1568	C₂	semidirect
8. C₁₃×#1496	C₂	semidirect
9. C₁₃×#1450	C₂	semidirect
70. #17970	C₁₃	direct (N × H)
105. #2981	C₂×C₁₃	semidirect
106. #1594	C₂×C₁₃	semidirect
107. #1589	C₂×C₁₃	semidirect
108. #1580	C₂×C₁₃	semidirect
109. C₈×D₈	C₂×C₁₃	semidirect
110. #1568	C₂×C₁₃	semidirect
111. #1496	C₂×C₁₃	semidirect
112. #1450	C₂×C₁₃	semidirect
256. C₁₃	#17970	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×#1594	2
C₁₃×#1589	2
C₁₃×#1580	2
C₈×C₁₃×D₈	2
C₁₃×#1568	2
C₁₃×#1496	2
C₁₃×#1450	2
#17970	13

Subgroups — C13×#17970 #15704309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#17970 #15704309