Subgroups — C₁₃×#20514 #15706853

Order 3328.

998 conjugacy classes of subgroups.

The subgroup lattice has 306 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	C₁₃×#2875	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×#1765	C₂	1	4	yes	maximal, complemented
7	C₁₃×#1759	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1726	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	complemented
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×C₁₃×#97	—	2	2	no	—
25	C₂×C₁₃×#97	C₂²	1	2	yes	—
26	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
29	C₁₃×#376	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	C₂³	1	2	yes	—
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×#108	C₂×C₄	1	8	yes	complemented
54	C₁₃×#107	C₂×C₄	1	2	yes	—
55	C₁₃×#106	C₂×C₄	1	2	yes	—
56	C₁₃×#104	C₂×C₄	1	16	yes	complemented
57	C₁₃×#103	C₂×C₄	1	4	yes	—
58	C₁₃×#102	C₂×C₄	1	4	yes	—
59	C₁₃×#97	—	2	4	no	—
60	C₁₃×#97	—	2	4	no	—
61	C₁₃×#97	—	2	2	no	—
62	C₁₃×#97	D₄	1	1	yes	characteristic
63	C₁₃×#97	D₄	1	1	yes	characteristic
64	C₁₃×#97	D₄	1	2	yes	—
65	C₄×C₈×C₁₃	—	2	4	no	—
66	C₄×C₈×C₁₃	—	2	2	no	—
67	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
70	C₁₃×#95	—	2	2	no	—
71	#20514	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	—
83	C₁₃×SD₁₆	C₄²	1	8	yes	—
84	C₁₃×D₈	C₄²	1	4	yes	—
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	4	no	—
104	C₄²×C₁₃	—	2	2	no	—
105	C₄²×C₁₃	—	2	2	no	—
106	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
107	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×#443	C₂×C₁₃	1	2	yes	—
111	C₄×#135	C₂×C₁₃	1	1	yes	characteristic
112	#2875	C₂×C₁₃	1	2	yes	complemented
113	C₄×#108	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#1772	C₂×C₁₃	1	2	yes	complemented
115	#1765	C₂×C₁₃	1	4	yes	complemented
116	#1759	C₂×C₁₃	1	2	yes	complemented
117	#1726	C₂×C₁₃	1	1	yes	characteristic, complemented
118	C₂³×C₁₃	—	2	2	no	—
119	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
120	Q₈×C₁₃	—	2	2	no	—
121	Q₈×C₁₃	C₂×C₄²	1	4	yes	—
122	D₄×C₁₃	—	2	8	no	—
123	D₄×C₁₃	—	2	2	no	—
124	D₄×C₁₃	C₂×C₄²	1	4	yes	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	8	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	4	no	—
131	C₂×C₄×C₁₃	—	8	4	no	—
132	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
135	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
138	C₈×C₁₃	—	2	8	no	—
139	C₈×C₁₃	—	2	2	no	—
140	C₈×C₁₃	C₂×C₄²	1	2	yes	—
141	C₈×C₁₃	C₂×C₄²	1	2	yes	—
142	C₈×C₁₃	C₄×D₄	1	4	yes	—
143	C₈×C₁₃	C₄×D₄	1	4	yes	—
144	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
145	C₂×#131	C₂²×C₁₃	1	4	yes	—
146	C₄×#41	C₂²×C₁₃	1	2	yes	—
147	#443	C₄×C₁₃	1	16	yes	complemented
148	C₄×Q₁₆	C₄×C₁₃	1	2	yes	complemented
149	C₄×SD₁₆	C₄×C₁₃	1	4	yes	complemented
150	C₄×D₈	C₄×C₁₃	1	2	yes	complemented
151	C₂×#108	C₂²×C₁₃	1	2	yes	—
152	#427	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×#104	C₂²×C₁₃	1	4	yes	—
154	#416	C₂²×C₁₃	1	2	yes	—
155	#405	—	2	4	no	—
156	#405	C₂²×C₁₃	1	4	yes	—
157	C₄×M₁₆	—	2	2	no	—
158	C₄×M₁₆	C₂²×C₁₃	1	2	yes	—
159	C₂×#97	—	2	2	no	—
160	C₂×#97	C₂²×C₁₃	1	2	yes	—
161	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
164	#376	C₂²×C₁₃	1	1	yes	characteristic
165	#341	C₂²×C₁₃	1	2	yes	—
166	#339	C₂²×C₁₃	1	4	yes	—
167	C₂²×C₁₃	—	2	2	no	—
168	C₂²×C₁₃	—	2	4	no	—
169	C₂²×C₁₃	—	4	4	no	—
170	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂²×C₁₃	C₄×#41	1	2	yes	—
172	C₄×C₁₃	—	2	2	no	—
173	C₄×C₁₃	—	2	2	no	—
174	C₄×C₁₃	—	2	4	no	—
175	C₄×C₁₃	—	4	4	no	—
176	C₄×C₁₃	—	4	4	no	—
177	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
178	C₄×C₁₃	C₂×C₄×D₄	1	2	yes	—
179	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
180	C₂×#41	C₂³×C₁₃	1	2	yes	—
181	#135	C₂×C₄×C₁₃	1	8	yes	—
182	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
183	C₂×SD₁₆	C₂×C₄×C₁₃	1	2	yes	—
184	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic
185	#131	—	2	8	no	—
186	#131	C₂×C₄×C₁₃	1	16	yes	complemented
187	C₂×M₁₆	—	2	4	no	—
188	C₂×M₁₆	C₂³×C₁₃	1	4	yes	—
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
193	C₄×Q₈	C₂×C₄×C₁₃	1	4	yes	—
194	C₄×D₄	—	2	4	no	—
195	C₄×D₄	C₂×C₄×C₁₃	1	4	yes	—
196	#117	—	2	2	no	—
197	#117	C₂³×C₁₃	1	2	yes	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
200	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
201	#108	C₂×C₄×C₁₃	1	8	yes	complemented
202	#107	C₂×C₄×C₁₃	1	2	yes	—
203	#106	C₂×C₄×C₁₃	1	2	yes	—
204	#104	C₂×C₄×C₁₃	1	16	yes	complemented
205	#103	C₂×C₄×C₁₃	1	4	yes	—
206	#102	C₂×C₄×C₁₃	1	4	yes	—
207	#97	—	2	4	no	—
208	#97	—	2	4	no	—
209	#97	—	2	2	no	—
210	#97	D₄×C₁₃	1	1	yes	characteristic
211	#97	D₄×C₁₃	1	1	yes	characteristic
212	#97	D₄×C₁₃	1	2	yes	—
213	C₄×C₈	—	2	4	no	—
214	C₄×C₈	—	2	2	no	—
215	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
216	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
217	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	complemented
218	#95	—	2	2	no	—
219	C₂×C₁₃	—	2	2	no	—
220	C₂×C₁₃	—	4	4	no	—
221	C₂×C₁₃	C₄²×D₄	1	1	yes	characteristic
222	C₂×C₁₃	#2957	1	2	yes	—
223	#41	—	2	4	no	—
224	#41	C₂²×C₄×C₁₃	1	8	yes	—
225	C₂×Q₈	C₂²×C₄×C₁₃	1	2	yes	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	C₂²×C₄×C₁₃	1	2	yes	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	C₁₃×#41	1	2	yes	—
232	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
233	Q₁₆	C₄²×C₁₃	1	4	yes	—
234	SD₁₆	C₄²×C₁₃	1	8	yes	—
235	D₈	C₄²×C₁₃	1	4	yes	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	16	no	—
238	M₁₆	C₂²×C₄×C₁₃	1	8	yes	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	16	no	—
242	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
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	4	yes	—
246	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
247	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
248	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
249	C₂×C₈	C₁₃×#41	1	4	yes	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
258	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
259	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₁₃	#20514	1	1	yes	characteristic, direct product factor, Sylow
262	C₂³	—	2	2	no	—
263	C₂³	C₂×C₁₃×#41	1	1	yes	characteristic
264	Q₈	—	2	2	no	—
265	Q₈	C₂×C₄²×C₁₃	1	4	yes	—
266	D₄	—	2	8	no	—
267	D₄	—	2	2	no	—
268	D₄	C₂×C₄²×C₁₃	1	4	yes	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
277	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
278	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
279	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
280	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
281	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
282	C₈	—	2	8	no	—
283	C₈	—	2	2	no	—
284	C₈	C₂×C₄²×C₁₃	1	2	yes	—
285	C₈	C₂×C₄²×C₁₃	1	2	yes	—
286	C₈	C₄×D₄×C₁₃	1	4	yes	—
287	C₈	C₄×D₄×C₁₃	1	4	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
292	C₂²	C₄×C₁₃×#41	1	2	yes	—
293	C₄	—	2	4	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	4	4	no	—
297	C₄	—	4	4	no	—
298	C₄	C₂²×C₄²×C₁₃	1	1	yes	characteristic
299	C₄	C₂×C₄×D₄×C₁₃	1	2	yes	—
300	C₄	C₄×C₁₃×#41	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	4	4	no	—
303	C₂	C₄²×D₄×C₁₃	1	1	yes	characteristic
304	C₂	C₁₃×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂²×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 ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2875	C₂	semidirect
4. C₄×C₁₃×#108	C₂	semidirect
5. C₁₃×#1772	C₂	semidirect
6. C₁₃×#1765	C₂	semidirect
7. C₁₃×#1759	C₂	semidirect
8. C₁₃×#1726	C₂	semidirect
12. C₁₃×#443	C₄	semidirect
13. C₄×C₁₃×Q₁₆	C₄	semidirect
14. C₄×C₁₃×SD₁₆	C₄	semidirect
15. C₄×C₁₃×D₈	C₄	semidirect
38. C₁₃×#131	C₂×C₄	semidirect
53. C₁₃×#108	C₂×C₄	semidirect
56. C₁₃×#104	C₂×C₄	semidirect
69. C₄×C₈×C₁₃	C₂×C₄	semidirect
71. #20514	C₁₃	direct (N × H)
112. #2875	C₂×C₁₃	semidirect
113. C₄×#108	C₂×C₁₃	semidirect
114. #1772	C₂×C₁₃	semidirect
115. #1765	C₂×C₁₃	semidirect
116. #1759	C₂×C₁₃	semidirect
117. #1726	C₂×C₁₃	semidirect
147. #443	C₄×C₁₃	semidirect
148. C₄×Q₁₆	C₄×C₁₃	semidirect
149. C₄×SD₁₆	C₄×C₁₃	semidirect
150. C₄×D₈	C₄×C₁₃	semidirect
186. #131	C₂×C₄×C₁₃	semidirect
201. #108	C₂×C₄×C₁₃	semidirect
204. #104	C₂×C₄×C₁₃	semidirect
217. C₄×C₈	C₂×C₄×C₁₃	semidirect
261. C₁₃	#20514	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
C₁₃×#2875	2
C₄×C₁₃×#108	2
C₁₃×#1772	2
C₁₃×#1765	2
C₁₃×#1759	2
C₁₃×#1726	2
#20514	13

Subgroups — C13×#20514 #15706853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20514 #15706853