Subgroups — C₁₃×#10878 #15697217

Order 3328.

1062 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2296	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#339	C₂	1	2	yes	maximal
4	C₂×C₁₃×#339	C₂	1	2	yes	maximal
5	C₁₃×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₁₃	—	2	2	no	—
10	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#389	C₄	1	4	yes	complemented
13	C₁₃×#385	—	2	2	no	—
14	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
15	C₄³×C₁₃	—	2	4	no	—
16	C₁₃×#339	—	2	4	no	—
17	C₁₃×#339	C₄	1	8	yes	complemented
18	C₁₃×#336	—	2	2	no	—
19	C₂³×C₄×C₁₃	—	2	2	no	—
20	C₂³×C₄×C₁₃	D₄	1	2	yes	—
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
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	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₁₃×#117	—	2	2	no	—
30	C₁₃×#117	—	2	2	no	—
31	C₁₃×#117	—	2	2	no	—
32	C₁₃×#117	C₂×C₄	1	4	yes	—
33	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
37	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
38	C₂×C₄²×C₁₃	—	2	16	no	—
39	C₂×C₄²×C₁₃	—	2	8	no	—
40	C₂×C₄²×C₁₃	—	2	2	no	—
41	C₂×C₄²×C₁₃	—	2	2	no	—
42	C₂×C₄²×C₁₃	—	2	1	no	—
43	C₂×C₄²×C₁₃	—	2	2	no	—
44	C₂×C₄²×C₁₃	—	2	2	no	—
45	C₂×C₄²×C₁₃	—	2	2	no	—
46	C₂×C₄²×C₁₃	—	2	1	no	—
47	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₁₃	D₄	1	2	yes	—
50	C₂×C₄²×C₁₃	D₄	1	4	yes	—
51	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
52	C₁₃×#95	—	2	4	no	—
53	#10878	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	—	2	16	no	—
56	C₂²×C₄×C₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	8	no	—
62	C₂²×C₄×C₁₃	—	2	2	no	—
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₃	#41	1	2	yes	—
71	C₂²×C₄×C₁₃	#41	1	2	yes	—
72	C₂²×C₄×C₁₃	#41	1	2	yes	—
73	C₁₃×M₁₆	—	2	4	no	—
74	C₁₃×M₁₆	—	2	4	no	—
75	C₁₃×M₁₆	—	4	2	no	—
76	C₁₃×M₁₆	—	4	2	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	4	2	no	—
79	C₂×C₈×C₁₃	—	4	2	no	—
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	2	4	no	—
82	C₁₃×C₄⋊C₄	—	4	8	no	—
83	C₁₃×C₄⋊C₄	C₄²	1	4	yes	complemented
84	C₁₃×C₂²⋊C₄	—	2	4	no	—
85	C₁₃×C₂²⋊C₄	—	4	1	no	—
86	C₁₃×C₂²⋊C₄	—	4	1	no	—
87	C₄²×C₁₃	—	2	32	no	—
88	C₄²×C₁₃	—	2	4	no	—
89	C₄²×C₁₃	—	2	4	no	—
90	C₄²×C₁₃	—	2	4	no	—
91	C₄²×C₁₃	—	2	4	no	—
92	C₄²×C₁₃	—	2	4	no	—
93	C₄²×C₁₃	—	2	4	no	—
94	C₄²×C₁₃	—	4	2	no	—
95	C₄²×C₁₃	C₄⋊C₄	1	8	yes	complemented
96	#2296	C₂×C₁₃	1	1	yes	characteristic
97	C₂×C₄³	C₂×C₁₃	1	1	yes	characteristic
98	C₂×#339	C₂×C₁₃	1	2	yes	—
99	C₂×#339	C₂×C₁₃	1	2	yes	—
100	#1728	C₂×C₁₃	1	1	yes	characteristic
101	C₂³×C₁₃	—	2	4	no	—
102	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₁₃	C₄×D₄	1	2	yes	—
104	C₂³×C₁₃	#120	1	1	yes	characteristic
105	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
106	C₂³×C₁₃	#123	1	1	yes	characteristic
107	C₂×C₄×C₁₃	—	2	16	no	—
108	C₂×C₄×C₁₃	—	2	8	no	—
109	C₂×C₄×C₁₃	—	2	16	no	—
110	C₂×C₄×C₁₃	—	2	16	no	—
111	C₂×C₄×C₁₃	—	2	8	no	—
112	C₂×C₄×C₁₃	—	2	8	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	4	4	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	#104	1	8	yes	—
120	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₁₃	#117	1	2	yes	—
123	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₁₃	#121	1	2	yes	—
130	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₁₃	#123	1	2	yes	—
132	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#125	1	2	yes	—
134	C₂×C₄×C₁₃	#126	1	2	yes	—
135	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
136	C₈×C₁₃	—	4	4	no	—
137	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
138	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
139	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
140	C₂²×C₄²	—	2	2	no	—
141	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
142	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
143	#389	C₄×C₁₃	1	4	yes	complemented
144	#385	—	2	2	no	—
145	C₄×C₄⋊C₄	—	2	2	no	—
146	C₄³	—	2	4	no	—
147	#339	—	2	4	no	—
148	#339	C₄×C₁₃	1	8	yes	complemented
149	#336	—	2	2	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	4	no	—
153	C₂²×C₁₃	—	2	4	no	—
154	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₁₃	#385	1	2	yes	—
156	C₂²×C₁₃	#386	1	1	yes	characteristic
157	C₂²×C₁₃	#387	1	2	yes	—
158	C₂²×C₁₃	#391	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×#104	1	4	yes	—
160	C₄×C₁₃	—	2	16	no	—
161	C₄×C₁₃	—	2	8	no	—
162	C₄×C₁₃	—	4	4	no	—
163	C₄×C₁₃	—	8	2	no	—
164	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
165	C₄×C₁₃	#387	1	2	yes	—
166	C₄×C₁₃	#388	1	2	yes	—
167	C₄×C₁₃	#389	1	2	yes	—
168	C₂³×C₄	—	2	2	no	—
169	C₂³×C₄	D₄×C₁₃	1	2	yes	—
170	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
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	C₂×C₄×C₁₃	1	4	yes	—
177	C₂²×C₈	—	2	2	no	—
178	#117	—	2	2	no	—
179	#117	—	2	2	no	—
180	#117	—	2	2	no	—
181	#117	C₂×C₄×C₁₃	1	4	yes	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
184	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
185	C₂×C₂²⋊C₄	—	2	1	no	—
186	C₂×C₂²⋊C₄	—	2	1	no	—
187	C₂×C₄²	—	2	8	no	—
188	C₂×C₄²	—	2	16	no	—
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	1	no	—
196	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
197	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
198	C₂×C₄²	D₄×C₁₃	1	2	yes	—
199	C₂×C₄²	D₄×C₁₃	1	4	yes	—
200	C₂×C₄²	Q₈×C₁₃	1	4	yes	—
201	#95	—	2	4	no	—
202	C₂×C₁₃	—	2	2	no	—
203	C₂×C₁₃	—	2	2	no	—
204	C₂×C₁₃	#1436	1	1	yes	characteristic
205	C₂×C₁₃	C₄×#104	1	2	yes	—
206	C₂×C₁₃	#1896	1	2	yes	—
207	C₂×C₁₃	#1994	1	2	yes	—
208	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
209	C₂²×C₄	—	2	16	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	8	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
219	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
224	C₂²×C₄	C₁₃×#41	1	2	yes	—
225	C₂²×C₄	C₁₃×#41	1	2	yes	—
226	C₂²×C₄	C₁₃×#41	1	2	yes	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	2	4	no	—
229	M₁₆	—	4	2	no	—
230	M₁₆	—	4	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	8	no	—
237	C₄⋊C₄	C₄²×C₁₃	1	4	yes	complemented
238	C₂²⋊C₄	—	2	4	no	—
239	C₂²⋊C₄	—	4	1	no	—
240	C₂²⋊C₄	—	4	1	no	—
241	C₄²	—	2	32	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	4	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	4	2	no	—
249	C₄²	C₁₃×C₄⋊C₄	1	8	yes	complemented
250	C₁₃	#10878	1	1	yes	characteristic, direct product factor, Sylow
251	C₂³	—	2	4	no	—
252	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
253	C₂³	C₄×D₄×C₁₃	1	2	yes	—
254	C₂³	C₁₃×#120	1	1	yes	characteristic
255	C₂³	C₁₃×C₂²⋊Q₈	1	2	yes	—
256	C₂³	C₁₃×#123	1	1	yes	characteristic
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	16	no	—
260	C₂×C₄	—	2	16	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	C₁₃×#104	1	8	yes	—
270	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
271	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
272	C₂×C₄	C₁₃×#117	1	2	yes	—
273	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
274	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
275	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
276	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
277	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
278	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
279	C₂×C₄	C₁₃×#121	1	2	yes	—
280	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
281	C₂×C₄	C₁₃×#123	1	2	yes	—
282	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
283	C₂×C₄	C₁₃×#125	1	2	yes	—
284	C₂×C₄	C₁₃×#126	1	2	yes	—
285	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
286	C₈	—	4	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	C₄×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
292	C₂²	C₁₃×#385	1	2	yes	—
293	C₂²	C₁₃×#386	1	1	yes	characteristic
294	C₂²	C₁₃×#387	1	2	yes	—
295	C₂²	C₁₃×#391	1	1	yes	characteristic
296	C₂²	C₂×C₁₃×#104	1	4	yes	—
297	C₄	—	2	16	no	—
298	C₄	—	2	8	no	—
299	C₄	—	4	4	no	—
300	C₄	—	8	2	no	—
301	C₄	C₄×C₁₃×C₄⋊C₄	1	2	yes	—
302	C₄	C₁₃×#387	1	2	yes	—
303	C₄	C₁₃×#388	1	2	yes	—
304	C₄	C₁₃×#389	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	2	no	—
307	C₂	C₁₃×#1436	1	1	yes	characteristic
308	C₂	C₄×C₁₃×#104	1	2	yes	—
309	C₂	C₁₃×#1896	1	2	yes	—
310	C₂	C₁₃×#1994	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₁₃×#389	C₄	semidirect
17. C₁₃×#339	C₄	semidirect
53. #10878	C₁₃	direct (N × H)
83. C₁₃×C₄⋊C₄	C₄²	semidirect
95. C₄²×C₁₃	C₄⋊C₄	semidirect
143. #389	C₄×C₁₃	semidirect
148. #339	C₄×C₁₃	semidirect
237. C₄⋊C₄	C₄²×C₁₃	semidirect
249. C₄²	C₁₃×C₄⋊C₄	semidirect
250. C₁₃	#10878	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×#2296	2
C₂×C₄³×C₁₃	2
C₂×C₁₃×#339	2
C₁₃×#1728	2
#10878	13

Subgroups — C13×#10878 #15697217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10878 #15697217