Subgroups — C₁₃×#33443 #15719782

Order 3328.

648 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×#3407	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₃×#443	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2171	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2170	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2117	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	C₄	1	1	yes	characteristic
9	C₂×C₁₃×#131	C₂²	1	2	yes	—
10	C₁₃×#504	—	2	4	no	—
11	C₁₃×#504	C₂²	1	8	yes	complemented
12	C₂×C₁₃×M₃₂	—	2	4	no	—
13	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
14	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₁₃×#443	C₄	1	2	yes	—
17	C₁₃×#443	C₄	1	2	yes	—
18	C₁₃×#443	C₂²	1	2	yes	—
19	C₁₃×#443	C₂²	1	2	yes	—
20	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
21	C₂×C₁₃×#104	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
23	C₁₃×#364	C₂²	1	2	yes	complemented
24	C₁₃×#364	C₂²	1	2	yes	complemented
25	C₁₃×#350	C₂²	1	8	yes	complemented
26	C₁₃×#346	C₂²	1	2	yes	complemented
27	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
28	C₂×C₁₃×#41	C₂×C₄	1	2	yes	—
29	C₁₃×#135	C₈	1	2	yes	—
30	C₁₃×#135	C₈	1	2	yes	—
31	C₁₃×#135	C₂×C₄	1	2	yes	—
32	C₁₃×#135	C₂×C₄	1	2	yes	—
33	C₂×C₁₃×Q₁₆	C₈	1	1	yes	characteristic
34	C₂×C₁₃×SD₁₆	C₈	1	2	yes	—
35	C₂×C₁₃×D₈	C₈	1	1	yes	characteristic
36	C₁₃×#131	—	2	2	no	—
37	C₁₃×#131	—	2	2	no	—
38	C₁₃×#131	C₂×C₄	1	4	yes	—
39	C₁₃×#131	C₂³	1	4	yes	—
40	C₂×C₁₃×M₁₆	—	2	2	no	—
41	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₁₃	—	2	2	no	—
43	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₁₃×M₃₂	—	2	2	no	—
47	C₁₃×M₃₂	—	2	1	no	—
48	C₁₃×M₃₂	—	2	1	no	—
49	C₁₃×M₃₂	—	2	8	no	—
50	C₁₃×M₃₂	D₄	1	2	yes	—
51	C₁₃×M₃₂	D₄	1	2	yes	—
52	C₁₃×M₃₂	C₂³	1	4	yes	—
53	C₂×C₁₃×C₁₆	—	2	2	no	—
54	C₂×C₁₃×C₁₆	—	2	8	no	—
55	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
56	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
57	C₁₃×#108	C₂×C₄	1	2	yes	—
58	C₁₃×#108	C₂×C₄	1	2	yes	—
59	C₁₃×#104	C₂×C₄	1	4	yes	—
60	C₁₃×#104	C₂×C₄	1	4	yes	—
61	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	#33443	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₁₃×#41	—	2	2	no	—
67	C₁₃×#41	—	2	2	no	—
68	C₁₃×#41	C₂×C₈	1	4	yes	—
69	C₁₃×#41	C₂²×C₄	1	4	yes	—
70	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
71	C₂×D₄×C₁₃	—	2	2	no	—
72	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	2	1	no	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₁₃×Q₁₆	C₂×C₈	1	2	yes	—
77	C₁₃×Q₁₆	C₂×C₈	1	2	yes	—
78	C₁₃×SD₁₆	C₂×C₈	1	4	yes	—
79	C₁₃×SD₁₆	C₂×C₈	1	4	yes	—
80	C₁₃×D₈	C₂×C₈	1	2	yes	—
81	C₁₃×D₈	C₂×C₈	1	2	yes	—
82	C₁₃×M₁₆	—	2	2	no	—
83	C₁₃×M₁₆	—	2	4	no	—
84	C₁₃×M₁₆	—	2	4	no	—
85	C₁₃×M₁₆	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	4	no	—
88	C₂×C₈×C₁₃	—	2	4	no	—
89	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
101	C₄²×C₁₃	—	2	1	no	—
102	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₁₃×C₁₆	—	2	4	no	—
105	C₁₃×C₁₆	—	2	2	no	—
106	C₁₃×C₁₆	C₂×D₄	1	4	yes	—
107	#3407	C₂×C₁₃	1	2	yes	complemented
108	C₂×#443	C₂×C₁₃	1	1	yes	characteristic
109	#2171	C₂×C₁₃	1	4	yes	complemented
110	#2170	C₂×C₁₃	1	4	yes	complemented
111	C₂×#364	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#2117	C₂×C₁₃	1	2	yes	complemented
113	C₄×M₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
114	C₂³×C₁₃	—	2	2	no	—
115	C₂³×C₁₃	#131	1	1	yes	characteristic
116	Q₈×C₁₃	—	2	2	no	—
117	Q₈×C₁₃	C₂²×C₈	1	4	yes	—
118	D₄×C₁₃	—	2	4	no	—
119	D₄×C₁₃	—	2	4	no	—
120	D₄×C₁₃	—	2	2	no	—
121	D₄×C₁₃	C₂²×C₈	1	4	yes	—
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	—	2	4	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	4	2	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	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₈×C₁₃	—	2	4	no	—
135	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
136	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
137	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
138	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
139	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
140	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
141	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
143	C₂×#135	C₄×C₁₃	1	1	yes	characteristic
144	C₂×#131	C₂²×C₁₃	1	2	yes	—
145	#504	—	2	4	no	—
146	#504	C₂²×C₁₃	1	8	yes	complemented
147	C₂×M₃₂	—	2	4	no	—
148	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
149	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
151	#443	C₄×C₁₃	1	2	yes	—
152	#443	C₄×C₁₃	1	2	yes	—
153	#443	C₂²×C₁₃	1	2	yes	—
154	#443	C₂²×C₁₃	1	2	yes	—
155	C₂×#108	C₄×C₁₃	1	1	yes	characteristic
156	C₂×#104	C₄×C₁₃	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
158	#364	C₂²×C₁₃	1	2	yes	complemented
159	#364	C₂²×C₁₃	1	2	yes	complemented
160	#350	C₂²×C₁₃	1	8	yes	complemented
161	#346	C₂²×C₁₃	1	2	yes	complemented
162	C₄×C₁₆	C₂²×C₁₃	1	2	yes	complemented
163	C₂²×C₁₃	—	2	4	no	—
164	C₂²×C₁₃	—	4	1	no	—
165	C₂²×C₁₃	—	8	2	no	—
166	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
167	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
168	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	—	2	4	no	—
171	C₄×C₁₃	—	4	1	no	—
172	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
173	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
174	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
175	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
176	C₂×#41	C₂×C₄×C₁₃	1	2	yes	—
177	#135	C₈×C₁₃	1	2	yes	—
178	#135	C₈×C₁₃	1	2	yes	—
179	#135	C₂×C₄×C₁₃	1	2	yes	—
180	#135	C₂×C₄×C₁₃	1	2	yes	—
181	C₂×Q₁₆	C₈×C₁₃	1	1	yes	characteristic
182	C₂×SD₁₆	C₈×C₁₃	1	2	yes	—
183	C₂×D₈	C₈×C₁₃	1	1	yes	characteristic
184	#131	—	2	2	no	—
185	#131	—	2	2	no	—
186	#131	C₂×C₄×C₁₃	1	4	yes	—
187	#131	C₂³×C₁₃	1	4	yes	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
193	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
194	M₃₂	—	2	2	no	—
195	M₃₂	—	2	1	no	—
196	M₃₂	—	2	1	no	—
197	M₃₂	—	2	8	no	—
198	M₃₂	D₄×C₁₃	1	2	yes	—
199	M₃₂	D₄×C₁₃	1	2	yes	—
200	M₃₂	C₂³×C₁₃	1	4	yes	—
201	C₂×C₁₆	—	2	2	no	—
202	C₂×C₁₆	—	2	8	no	—
203	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
204	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
205	#108	C₂×C₄×C₁₃	1	2	yes	—
206	#108	C₂×C₄×C₁₃	1	2	yes	—
207	#104	C₂×C₄×C₁₃	1	4	yes	—
208	#104	C₂×C₄×C₁₃	1	4	yes	—
209	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
210	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
211	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
212	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
213	C₂×C₁₃	—	2	1	no	—
214	C₂×C₁₃	—	2	1	no	—
215	C₂×C₁₃	—	2	1	no	—
216	C₂×C₁₃	—	4	4	no	—
217	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
218	#41	—	2	2	no	—
219	#41	—	2	2	no	—
220	#41	C₂×C₈×C₁₃	1	4	yes	—
221	#41	C₂²×C₄×C₁₃	1	4	yes	—
222	C₂×Q₈	C₂×C₈×C₁₃	1	2	yes	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	C₂×C₈×C₁₃	1	2	yes	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
228	Q₁₆	C₂×C₈×C₁₃	1	2	yes	—
229	Q₁₆	C₂×C₈×C₁₃	1	2	yes	—
230	SD₁₆	C₂×C₈×C₁₃	1	4	yes	—
231	SD₁₆	C₂×C₈×C₁₃	1	4	yes	—
232	D₈	C₂×C₈×C₁₃	1	2	yes	—
233	D₈	C₂×C₈×C₁₃	1	2	yes	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	C₂²×C₄×C₁₃	1	4	yes	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
242	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
243	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
244	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
245	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
246	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
247	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
248	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
250	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
251	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
252	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
253	C₄²	—	2	1	no	—
254	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
255	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
256	C₁₆	—	2	4	no	—
257	C₁₆	—	2	2	no	—
258	C₁₆	C₂×D₄×C₁₃	1	4	yes	—
259	C₁₃	#33443	1	1	yes	characteristic, direct product factor, Sylow
260	C₂³	—	2	2	no	—
261	C₂³	C₁₃×#131	1	1	yes	characteristic
262	Q₈	—	2	2	no	—
263	Q₈	C₂²×C₈×C₁₃	1	4	yes	—
264	D₄	—	2	4	no	—
265	D₄	—	2	4	no	—
266	D₄	—	2	2	no	—
267	D₄	C₂²×C₈×C₁₃	1	4	yes	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
275	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
278	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
279	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
280	C₈	—	2	4	no	—
281	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
282	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
283	C₈	C₂²×C₈×C₁₃	1	1	yes	characteristic
284	C₈	C₂²×C₈×C₁₃	1	1	yes	characteristic
285	C₈	C₂²×C₈×C₁₃	1	1	yes	characteristic
286	C₈	C₂²×C₈×C₁₃	1	1	yes	characteristic
287	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
288	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
289	C₂²	—	2	4	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	C₈×D₄×C₁₃	1	1	yes	characteristic
293	C₂²	C₈×D₄×C₁₃	1	1	yes	characteristic
294	C₂²	C₂×C₁₃×#131	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	4	no	—
297	C₄	—	4	1	no	—
298	C₄	C₈×D₄×C₁₃	1	1	yes	characteristic
299	C₄	C₈×D₄×C₁₃	1	1	yes	characteristic
300	C₄	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
301	C₄	C₂³×C₈×C₁₃	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	4	no	—
306	C₂	C₂×C₈×D₄×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₃	C₂²×D₄
commutator	C₄	C₂³×C₈×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂×C₁₃	C₂×C₈×D₄
prime radical	C₂×C₁₃×#135	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃ ◁ C₂×C₁₃×M₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3407	C₂	semidirect
3. C₁₃×#2171	C₂	semidirect
4. C₁₃×#2170	C₂	semidirect
5. C₂×C₁₃×#364	C₂	semidirect
6. C₁₃×#2117	C₂	semidirect
7. C₄×C₁₃×M₃₂	C₂	semidirect
11. C₁₃×#504	C₂²	semidirect
23. C₁₃×#364	C₂²	semidirect
24. C₁₃×#364	C₂²	semidirect
25. C₁₃×#350	C₂²	semidirect
26. C₁₃×#346	C₂²	semidirect
27. C₄×C₁₃×C₁₆	C₂²	semidirect
65. #33443	C₁₃	direct (N × H)
107. #3407	C₂×C₁₃	semidirect
109. #2171	C₂×C₁₃	semidirect
110. #2170	C₂×C₁₃	semidirect
111. C₂×#364	C₂×C₁₃	semidirect
112. #2117	C₂×C₁₃	semidirect
113. C₄×M₃₂	C₂×C₁₃	semidirect
146. #504	C₂²×C₁₃	semidirect
158. #364	C₂²×C₁₃	semidirect
159. #364	C₂²×C₁₃	semidirect
160. #350	C₂²×C₁₃	semidirect
161. #346	C₂²×C₁₃	semidirect
162. C₄×C₁₆	C₂²×C₁₃	semidirect
259. C₁₃	#33443	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×#3407	2
C₂×C₁₃×#443	2
C₁₃×#2171	2
C₁₃×#2170	2
C₂×C₁₃×#364	2
C₁₃×#2117	2
C₄×C₁₃×M₃₂	2
#33443	13

Subgroups — C13×#33443 #15719782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33443 #15719782