Subgroups — C₁₃×#10813 #15697152

Order 3328.

674 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#344	C₂	1	2	yes	maximal
3	C₂×C₁₃×#339	C₂	1	2	yes	maximal
4	C₂×C₁₃×#338	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₄×C₁₃×M₁₆	—	2	4	no	—
12	C₂×C₁₃×#97	C₄	1	2	yes	complemented
13	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	complemented
14	C₁₃×#344	—	2	4	no	—
15	C₁₃×#339	—	2	4	no	—
16	C₁₃×#338	C₄	1	4	yes	complemented
17	C₁₃×H(3,ℤ/4)	C₄	1	4	yes	complemented
18	C₂³×C₄×C₁₃	—	2	1	no	—
19	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	—	2	2	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂×C₁₃×M₁₆	—	2	2	no	—
29	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
33	C₂×C₁₃×M₁₆	Q₈	1	2	yes	—
34	C₂²×C₈×C₁₃	—	2	2	no	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
37	C₁₃×#117	—	2	2	no	—
38	C₁₃×#117	—	2	2	no	—
39	C₁₃×#117	—	2	2	no	—
40	C₁₃×#117	C₂×C₄	1	4	yes	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
43	C₂×C₄²×C₁₃	—	2	2	no	—
44	C₂×C₄²×C₁₃	—	2	1	no	—
45	C₂×C₄²×C₁₃	—	2	2	no	—
46	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	D₄	1	4	yes	—
49	C₁₃×#97	—	2	4	no	—
50	C₄×C₈×C₁₃	—	2	4	no	—
51	#10813	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	#41	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	—	4	4	no	—
57	C₂²×C₄×C₁₃	—	4	2	no	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	#41	1	2	yes	—
65	C₂²×C₄×C₁₃	#41	1	2	yes	—
66	C₂²×C₄×C₁₃	#41	1	2	yes	—
67	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
68	C₁₃×M₁₆	—	2	4	no	—
69	C₁₃×M₁₆	—	2	4	no	—
70	C₁₃×M₁₆	—	2	4	no	—
71	C₁₃×M₁₆	—	2	4	no	—
72	C₁₃×M₁₆	—	2	4	no	—
73	C₁₃×M₁₆	—	2	4	no	—
74	C₁₃×M₁₆	—	4	2	no	—
75	C₁₃×M₁₆	—	4	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₂×C₈×C₁₃	—	4	2	no	—
83	C₂×C₈×C₁₃	C₄²	1	4	yes	—
84	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
85	C₁₃×C₄⋊C₄	—	2	4	no	—
86	C₁₃×C₄⋊C₄	—	4	2	no	—
87	C₁₃×C₂²⋊C₄	—	2	4	no	—
88	C₁₃×C₂²⋊C₄	—	4	2	no	—
89	C₄²×C₁₃	—	2	1	no	—
90	C₄²×C₁₃	—	2	1	no	—
91	C₄²×C₁₃	—	2	4	no	—
92	C₄²×C₁₃	—	2	4	no	—
93	C₄²×C₁₃	—	4	2	no	—
94	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
95	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×M₁₆	C₂×C₁₃	1	1	yes	characteristic
97	C₂×#344	C₂×C₁₃	1	2	yes	—
98	C₂×#339	C₂×C₁₃	1	2	yes	—
99	C₂×#338	C₂×C₁₃	1	1	yes	characteristic
100	C₂×H(3,ℤ/4)	C₂×C₁₃	1	1	yes	characteristic
101	C₂³×C₁₃	—	4	2	no	—
102	C₂³×C₁₃	#117	1	1	yes	characteristic
103	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
104	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
105	C₂³×C₁₃	#123	1	2	yes	—
106	C₂³×C₁₃	#126	1	2	yes	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	4	8	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	—	8	4	no	—
114	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#117	1	2	yes	—
119	C₂×C₄×C₁₃	#117	1	2	yes	—
120	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#121	1	2	yes	—
127	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₁₃	#123	1	2	yes	—
129	C₂×C₄×C₁₃	#124	1	2	yes	—
130	C₂×C₄×C₁₃	#125	1	2	yes	—
131	C₂×C₄×C₁₃	#126	1	2	yes	—
132	C₈×C₁₃	—	2	4	no	—
133	C₈×C₁₃	—	2	4	no	—
134	C₈×C₁₃	—	4	4	no	—
135	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
136	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
137	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
138	C₂×#117	C₂²×C₁₃	1	2	yes	—
139	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
140	C₄×M₁₆	—	2	4	no	—
141	C₂×#97	C₄×C₁₃	1	2	yes	complemented
142	C₂×C₄×C₈	C₄×C₁₃	1	2	yes	complemented
143	#344	—	2	4	no	—
144	#339	—	2	4	no	—
145	#338	C₄×C₁₃	1	4	yes	complemented
146	H(3,ℤ/4)	C₄×C₁₃	1	4	yes	complemented
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	2	4	no	—
149	C₂²×C₁₃	—	4	2	no	—
150	C₂²×C₁₃	—	4	1	no	—
151	C₂²×C₁₃	—	4	1	no	—
152	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₁₃	#381	1	1	yes	characteristic
154	C₂²×C₁₃	#383	1	1	yes	characteristic
155	C₂²×C₁₃	#387	1	2	yes	—
156	C₂²×C₁₃	#388	1	2	yes	—
157	C₄×C₁₃	—	2	4	no	—
158	C₄×C₁₃	—	4	2	no	—
159	C₄×C₁₃	—	8	4	no	—
160	C₄×C₁₃	#376	1	1	yes	characteristic
161	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
162	C₄×C₁₃	#380	1	1	yes	characteristic
163	C₄×C₁₃	#382	1	1	yes	characteristic
164	C₄×C₁₃	#387	1	2	yes	—
165	C₄×C₁₃	#388	1	2	yes	—
166	C₂³×C₄	—	2	1	no	—
167	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
178	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
179	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
180	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
181	C₂×M₁₆	Q₈×C₁₃	1	2	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
184	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	C₂×C₄×C₁₃	1	4	yes	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
195	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
196	C₂×C₄²	D₄×C₁₃	1	4	yes	—
197	#97	—	2	4	no	—
198	C₄×C₈	—	2	4	no	—
199	C₂×C₁₃	—	2	1	no	—
200	C₂×C₁₃	—	2	2	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	2	2	no	—
203	C₂×C₁₃	#1434	1	1	yes	characteristic
204	C₂×C₁₃	#1841	1	2	yes	—
205	C₂⁴	C₁₃×#41	1	1	yes	characteristic
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
217	C₂²×C₄	C₁₃×#41	1	2	yes	—
218	C₂²×C₄	C₁₃×#41	1	2	yes	—
219	C₂²×C₄	C₁₃×#41	1	2	yes	—
220	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	C₄²×C₁₃	1	4	yes	—
237	C₂×C₈	C₁₃×C₄⋊C₄	1	4	yes	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	4	2	no	—
247	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
248	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
249	C₁₃	#10813	1	1	yes	characteristic, direct product factor, Sylow
250	C₂³	—	4	2	no	—
251	C₂³	C₁₃×#117	1	1	yes	characteristic
252	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
253	C₂³	C₄×Q₈×C₁₃	1	1	yes	characteristic
254	C₂³	C₁₃×#123	1	2	yes	—
255	C₂³	C₁₃×#126	1	2	yes	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	8	4	no	—
263	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
267	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
268	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
269	C₂×C₄	C₁₃×#117	1	2	yes	—
270	C₂×C₄	C₁₃×#117	1	2	yes	—
271	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
272	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
273	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
274	C₂×C₄	C₄×Q₈×C₁₃	1	1	yes	characteristic
275	C₂×C₄	C₁₃×#121	1	2	yes	—
276	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
277	C₂×C₄	C₁₃×#123	1	2	yes	—
278	C₂×C₄	C₁₃×#124	1	2	yes	—
279	C₂×C₄	C₁₃×#125	1	2	yes	—
280	C₂×C₄	C₁₃×#126	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₈	—	4	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
290	C₂²	C₁₃×#381	1	1	yes	characteristic
291	C₂²	C₁₃×#383	1	1	yes	characteristic
292	C₂²	C₁₃×#387	1	2	yes	—
293	C₂²	C₁₃×#388	1	2	yes	—
294	C₄	—	2	4	no	—
295	C₄	—	4	2	no	—
296	C₄	—	8	4	no	—
297	C₄	C₁₃×#376	1	1	yes	characteristic
298	C₄	C₄×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
299	C₄	C₁₃×#380	1	1	yes	characteristic
300	C₄	C₁₃×#382	1	1	yes	characteristic
301	C₄	C₁₃×#387	1	2	yes	—
302	C₄	C₁₃×#388	1	2	yes	—
303	C₂	—	2	1	no	—
304	C₂	—	2	2	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	C₁₃×#1434	1	1	yes	characteristic
308	C₂	C₁₃×#1841	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	#117
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
socle	C₂²×C₁₃	#383
prime radical	C₂⁴×C₁₃	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂²×C₁₃×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂×C₁₃×#97	C₄	semidirect
13. C₂×C₄×C₈×C₁₃	C₄	semidirect
16. C₁₃×#338	C₄	semidirect
17. C₁₃×H(3,ℤ/4)	C₄	semidirect
51. #10813	C₁₃	direct (N × H)
141. C₂×#97	C₄×C₁₃	semidirect
142. C₂×C₄×C₈	C₄×C₁₃	semidirect
145. #338	C₄×C₁₃	semidirect
146. H(3,ℤ/4)	C₄×C₁₃	semidirect
249. C₁₃	#10813	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×C₄×C₁₃×M₁₆	2
C₂×C₁₃×#344	2
C₂×C₁₃×#339	2
C₂×C₁₃×#338	2
C₂×C₁₃×H(3,ℤ/4)	2
#10813	13

Subgroups — C13×#10813 #15697152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10813 #15697152