Subgroups — C₄×C₁₃×#443 #15706848

Order 3328.

1046 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	C₂×C₁₃×#443	C₂	1	2	yes	maximal
2	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×#131	C₂	1	2	yes	maximal, complemented
4	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1772	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1764	C₂	1	4	yes	maximal, complemented
7	C₄×C₁₃×#104	C₂	1	2	yes	maximal, complemented
8	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#131	C₂²	1	4	yes	—
11	C₄×C₁₃×#41	C₂²	1	2	yes	—
12	C₁₃×#443	C₄	1	16	yes	direct product factor
13	C₄×C₁₃×Q₁₆	C₄	1	2	yes	complemented
14	C₄×C₁₃×SD₁₆	C₄	1	4	yes	complemented
15	C₄×C₁₃×D₈	C₄	1	2	yes	complemented
16	C₂×C₁₃×#108	C₂²	1	2	yes	—
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#104	C₂²	1	4	yes	—
19	C₁₃×#416	C₂²	1	2	yes	—
20	C₁₃×#405	—	2	4	no	—
21	C₁₃×#405	C₂²	1	4	yes	—
22	C₄×C₁₃×M₁₆	—	2	2	no	—
23	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₃	—	2	2	no	—
25	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
29	C₄³×C₁₃	C₂²	1	1	yes	characteristic
30	C₁₃×#341	C₂²	1	2	yes	—
31	C₁₃×#339	C₂²	1	4	yes	—
32	C₂×C₁₃×#41	C₂³	1	2	yes	—
33	C₁₃×#135	C₂×C₄	1	8	yes	—
34	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
37	C₁₃×#131	—	2	8	no	—
38	C₁₃×#131	C₂×C₄	1	16	yes	complemented
39	C₂×C₁₃×M₁₆	—	2	4	no	—
40	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
41	C₂²×C₈×C₁₃	—	2	4	no	—
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
45	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
46	C₄×D₄×C₁₃	—	2	4	no	—
47	C₄×D₄×C₁₃	C₂×C₄	1	4	yes	—
48	C₁₃×#117	—	2	2	no	—
49	C₁₃×#117	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	—	2	2	no	—
51	C₂×C₄²×C₁₃	—	2	2	no	—
52	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×#108	C₂×C₄	1	8	yes	complemented
55	C₁₃×#107	C₂×C₄	1	2	yes	—
56	C₁₃×#106	C₂×C₄	1	2	yes	—
57	C₁₃×#104	C₂×C₄	1	16	yes	complemented
58	C₁₃×#103	C₂×C₄	1	4	yes	—
59	C₁₃×#102	C₂×C₄	1	4	yes	—
60	C₁₃×#97	—	2	4	no	—
61	C₁₃×#97	—	2	2	no	—
62	C₄×C₈×C₁₃	—	2	4	no	—
63	C₄×C₈×C₁₃	—	2	4	no	—
64	C₄×C₈×C₁₃	—	2	2	no	—
65	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
68	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	D₄	1	2	yes	—
70	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
71	C₄×#443	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₁₃×#41	—	2	4	no	—
73	C₁₃×#41	C₂²×C₄	1	8	yes	—
74	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×D₄×C₁₃	—	2	2	no	—
76	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	—	2	2	no	—
79	C₂²×C₄×C₁₃	—	2	4	no	—
80	C₂²×C₄×C₁₃	#41	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₁₃×Q₁₆	C₄²	1	4	yes	complemented
83	C₁₃×SD₁₆	C₄²	1	8	yes	complemented
84	C₁₃×D₈	C₄²	1	4	yes	complemented
85	C₁₃×M₁₆	—	2	4	no	—
86	C₁₃×M₁₆	—	2	16	no	—
87	C₁₃×M₁₆	C₂²×C₄	1	8	yes	—
88	C₂×C₈×C₁₃	—	2	4	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	16	no	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
96	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₁₃	#41	1	4	yes	—
99	C₁₃×C₄⋊C₄	—	2	4	no	—
100	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
101	C₁₃×C₂²⋊C₄	—	2	2	no	—
102	C₁₃×C₂²⋊C₄	—	2	4	no	—
103	C₄²×C₁₃	—	2	8	no	—
104	C₄²×C₁₃	—	2	4	no	—
105	C₄²×C₁₃	—	2	2	no	—
106	C₄²×C₁₃	—	2	2	no	—
107	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
108	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
109	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×#443	C₂×C₁₃	1	2	yes	—
112	C₄×#135	C₂×C₁₃	1	1	yes	characteristic
113	C₄×#131	C₂×C₁₃	1	2	yes	complemented
114	C₄×#108	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#1772	C₂×C₁₃	1	2	yes	complemented
116	#1764	C₂×C₁₃	1	4	yes	complemented
117	C₄×#104	C₂×C₁₃	1	2	yes	complemented
118	C₄²×C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
119	C₂³×C₁₃	—	2	2	no	—
120	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
121	Q₈×C₁₃	—	2	2	no	—
122	Q₈×C₁₃	C₂×C₄²	1	4	yes	—
123	D₄×C₁₃	—	2	8	no	—
124	D₄×C₁₃	—	2	2	no	—
125	D₄×C₁₃	C₂×C₄²	1	4	yes	—
126	C₂×C₄×C₁₃	—	2	8	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	2	8	no	—
130	C₂×C₄×C₁₃	—	2	4	no	—
131	C₂×C₄×C₁₃	—	2	4	no	—
132	C₂×C₄×C₁₃	—	4	8	no	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
136	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
139	C₈×C₁₃	—	2	8	no	—
140	C₈×C₁₃	—	2	2	no	—
141	C₈×C₁₃	C₂×C₄²	1	2	yes	—
142	C₈×C₁₃	C₂×C₄²	1	2	yes	—
143	C₈×C₁₃	C₄×D₄	1	4	yes	—
144	C₈×C₁₃	C₄×D₄	1	4	yes	—
145	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
146	C₂×#131	C₂²×C₁₃	1	4	yes	—
147	C₄×#41	C₂²×C₁₃	1	2	yes	—
148	#443	C₄×C₁₃	1	16	yes	direct product factor
149	C₄×Q₁₆	C₄×C₁₃	1	2	yes	complemented
150	C₄×SD₁₆	C₄×C₁₃	1	4	yes	complemented
151	C₄×D₈	C₄×C₁₃	1	2	yes	complemented
152	C₂×#108	C₂²×C₁₃	1	2	yes	—
153	#427	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×#104	C₂²×C₁₃	1	4	yes	—
155	#416	C₂²×C₁₃	1	2	yes	—
156	#405	—	2	4	no	—
157	#405	C₂²×C₁₃	1	4	yes	—
158	C₄×M₁₆	—	2	2	no	—
159	C₄×M₁₆	C₂²×C₁₃	1	2	yes	—
160	C₂×C₄×C₈	—	2	2	no	—
161	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
162	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
164	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
165	C₄³	C₂²×C₁₃	1	1	yes	characteristic
166	#341	C₂²×C₁₃	1	2	yes	—
167	#339	C₂²×C₁₃	1	4	yes	—
168	C₂²×C₁₃	—	2	2	no	—
169	C₂²×C₁₃	—	2	4	no	—
170	C₂²×C₁₃	—	4	4	no	—
171	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂²×C₁₃	C₄×#41	1	2	yes	—
173	C₄×C₁₃	—	2	8	no	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	2	4	no	—
176	C₄×C₁₃	—	4	4	no	—
177	C₄×C₁₃	#443	1	4	yes	direct product factor
178	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
179	C₄×C₁₃	C₂×C₄×D₄	1	2	yes	—
180	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
181	C₂×#41	C₂³×C₁₃	1	2	yes	—
182	#135	C₂×C₄×C₁₃	1	8	yes	—
183	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
184	C₂×SD₁₆	C₂×C₄×C₁₃	1	2	yes	—
185	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic
186	#131	—	2	8	no	—
187	#131	C₂×C₄×C₁₃	1	16	yes	complemented
188	C₂×M₁₆	—	2	4	no	—
189	C₂×M₁₆	C₂³×C₁₃	1	4	yes	—
190	C₂²×C₈	—	2	4	no	—
191	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
194	C₄×Q₈	C₂×C₄×C₁₃	1	4	yes	—
195	C₄×D₄	—	2	4	no	—
196	C₄×D₄	C₂×C₄×C₁₃	1	4	yes	—
197	#117	—	2	2	no	—
198	#117	C₂³×C₁₃	1	2	yes	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
202	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
203	#108	C₂×C₄×C₁₃	1	8	yes	complemented
204	#107	C₂×C₄×C₁₃	1	2	yes	—
205	#106	C₂×C₄×C₁₃	1	2	yes	—
206	#104	C₂×C₄×C₁₃	1	16	yes	complemented
207	#103	C₂×C₄×C₁₃	1	4	yes	—
208	#102	C₂×C₄×C₁₃	1	4	yes	—
209	#97	—	2	4	no	—
210	#97	—	2	2	no	—
211	C₄×C₈	—	2	4	no	—
212	C₄×C₈	—	2	4	no	—
213	C₄×C₈	—	2	2	no	—
214	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
215	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
216	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	complemented
217	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
218	C₄×C₈	D₄×C₁₃	1	2	yes	—
219	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
220	C₂×C₁₃	—	2	2	no	—
221	C₂×C₁₃	—	4	4	no	—
222	C₂×C₁₃	C₄²×D₄	1	1	yes	characteristic
223	C₂×C₁₃	C₂×#443	1	2	yes	—
224	#41	—	2	4	no	—
225	#41	C₂²×C₄×C₁₃	1	8	yes	—
226	C₂×Q₈	C₂²×C₄×C₁₃	1	2	yes	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	C₂²×C₄×C₁₃	1	2	yes	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	C₁₃×#41	1	2	yes	—
233	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
234	Q₁₆	C₄²×C₁₃	1	4	yes	complemented
235	SD₁₆	C₄²×C₁₃	1	8	yes	complemented
236	D₈	C₄²×C₁₃	1	4	yes	complemented
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	16	no	—
239	M₁₆	C₂²×C₄×C₁₃	1	8	yes	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	16	no	—
243	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
244	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
245	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
246	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
247	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
248	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
249	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
250	C₂×C₈	C₁₃×#41	1	4	yes	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₄²	—	2	8	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
260	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
261	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
263	C₁₃	C₄×#443	1	1	yes	characteristic, direct product factor, Sylow
264	C₂³	—	2	2	no	—
265	C₂³	C₂×C₁₃×#41	1	1	yes	characteristic
266	Q₈	—	2	2	no	—
267	Q₈	C₂×C₄²×C₁₃	1	4	yes	—
268	D₄	—	2	8	no	—
269	D₄	—	2	2	no	—
270	D₄	C₂×C₄²×C₁₃	1	4	yes	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
279	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
280	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
281	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
283	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
284	C₈	—	2	8	no	—
285	C₈	—	2	2	no	—
286	C₈	C₂×C₄²×C₁₃	1	2	yes	—
287	C₈	C₂×C₄²×C₁₃	1	2	yes	—
288	C₈	C₄×D₄×C₁₃	1	4	yes	—
289	C₈	C₄×D₄×C₁₃	1	4	yes	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
294	C₂²	C₄×C₁₃×#41	1	2	yes	—
295	C₄	—	2	8	no	—
296	C₄	—	2	4	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	4	no	—
299	C₄	C₁₃×#443	1	4	yes	direct product factor
300	C₄	C₂²×C₄²×C₁₃	1	1	yes	characteristic
301	C₄	C₂×C₄×D₄×C₁₃	1	2	yes	—
302	C₄	C₄×C₁₃×#41	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	4	4	no	—
305	C₂	C₄²×D₄×C₁₃	1	1	yes	characteristic
306	C₂	C₂×C₁₃×#443	1	2	yes	—

	subgroup	G/H
center	C₄×C₈×C₁₃	D₄
commutator	C₄	C₂²×C₄²×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₁₃×#131	C₂	semidirect
4. C₄×C₁₃×#108	C₂	semidirect
5. C₁₃×#1772	C₂	semidirect
6. C₁₃×#1764	C₂	semidirect
7. C₄×C₁₃×#104	C₂	semidirect
8. C₄²×C₈×C₁₃	C₂	semidirect
12. C₁₃×#443	C₄	direct (N × H)
13. C₄×C₁₃×Q₁₆	C₄	semidirect
14. C₄×C₁₃×SD₁₆	C₄	semidirect
15. C₄×C₁₃×D₈	C₄	semidirect
38. C₁₃×#131	C₂×C₄	semidirect
54. C₁₃×#108	C₂×C₄	semidirect
57. C₁₃×#104	C₂×C₄	semidirect
67. C₄×C₈×C₁₃	C₂×C₄	semidirect
71. C₄×#443	C₁₃	direct (N × H)
82. C₁₃×Q₁₆	C₄²	semidirect
83. C₁₃×SD₁₆	C₄²	semidirect
84. C₁₃×D₈	C₄²	semidirect
113. C₄×#131	C₂×C₁₃	semidirect
114. C₄×#108	C₂×C₁₃	semidirect
115. #1772	C₂×C₁₃	semidirect
116. #1764	C₂×C₁₃	semidirect
117. C₄×#104	C₂×C₁₃	semidirect
118. C₄²×C₈	C₂×C₁₃	semidirect
148. #443	C₄×C₁₃	direct (N × H)
149. C₄×Q₁₆	C₄×C₁₃	semidirect
150. C₄×SD₁₆	C₄×C₁₃	semidirect
151. C₄×D₈	C₄×C₁₃	semidirect
177. C₄×C₁₃	#443	direct (N × H)
187. #131	C₂×C₄×C₁₃	semidirect
203. #108	C₂×C₄×C₁₃	semidirect
206. #104	C₂×C₄×C₁₃	semidirect
216. C₄×C₈	C₂×C₄×C₁₃	semidirect
234. Q₁₆	C₄²×C₁₃	semidirect
235. SD₁₆	C₄²×C₁₃	semidirect
236. D₈	C₄²×C₁₃	semidirect
263. C₁₃	C₄×#443	direct (N × H)
299. C₄	C₁₃×#443	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×#443	2
C₄×C₁₃×#135	2
C₄×C₁₃×#131	2
C₄×C₁₃×#108	2
C₁₃×#1772	2
C₁₃×#1764	2
C₄×C₁₃×#104	2
C₄²×C₈×C₁₃	2
C₄×#443	13

Subgroups — C4×C13×#443 #15706848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₁₃×#443 #15706848