Subgroups — C₁₃×#10839 #15697178

Order 3328.

1110 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#423	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#339	C₂	1	4	yes	maximal
5	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
6	C₂×C₁₃×#117	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₃	—	2	2	no	—
8	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#423	—	2	2	no	—
11	C₁₃×#423	C₄	1	4	yes	complemented
12	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
13	C₁₃×#384	C₄	1	2	yes	complemented
14	C₁₃×#382	C₄	1	2	yes	complemented
15	C₁₃×#380	—	2	2	no	—
16	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
17	C₄³×C₁₃	—	2	2	no	—
18	C₄³×C₁₃	C₄	1	4	yes	complemented
19	C₁₃×#339	—	2	8	no	—
20	C₂³×C₄×C₁₃	—	2	2	no	—
21	C₂³×C₄×C₁₃	D₄	1	2	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	4	no	—
25	C₂×C₁₃×M₁₆	—	2	4	no	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₁₃×#117	—	2	4	no	—
29	C₁₃×#117	—	2	4	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
32	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
33	C₂×C₄²×C₁₃	—	2	16	no	—
34	C₂×C₄²×C₁₃	—	2	8	no	—
35	C₂×C₄²×C₁₃	—	2	2	no	—
36	C₂×C₄²×C₁₃	—	2	2	no	—
37	C₂×C₄²×C₁₃	—	2	2	no	—
38	C₂×C₄²×C₁₃	—	2	2	no	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₃	D₄	1	4	yes	—
45	C₂×C₄²×C₁₃	D₄	1	2	yes	—
46	C₂×C₄²×C₁₃	D₄	1	2	yes	—
47	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
48	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
49	C₁₃×C₄⋊C₈	—	2	8	no	—
50	C₁₃×#95	—	2	4	no	—
51	#10839	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	—	2	16	no	—
54	C₂²×C₄×C₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₃	—	2	8	no	—
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
61	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	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	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₃	#41	1	2	yes	—
72	C₂²×C₄×C₁₃	#41	1	2	yes	—
73	C₁₃×M₁₆	—	2	8	no	—
74	C₁₃×M₁₆	—	4	4	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	4	2	no	—
77	C₂×C₈×C₁₃	—	4	2	no	—
78	C₁₃×C₄⋊C₄	—	2	8	no	—
79	C₁₃×C₄⋊C₄	—	4	8	no	—
80	C₁₃×C₂²⋊C₄	—	4	4	no	—
81	C₄²×C₁₃	—	2	32	no	—
82	C₄²×C₁₃	—	2	4	no	—
83	C₄²×C₁₃	—	2	4	no	—
84	C₄²×C₁₃	—	2	4	no	—
85	C₄²×C₁₃	—	4	2	no	—
86	C₄²×C₁₃	—	4	2	no	—
87	C₄²×C₁₃	C₄²	1	2	yes	—
88	C₄²×C₁₃	C₄²	1	2	yes	—
89	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
90	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
91	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
92	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
93	C₄²×C₁₃	C₄⋊C₄	1	8	yes	complemented
94	C₂×#423	C₂×C₁₃	1	1	yes	characteristic
95	#2285	C₂×C₁₃	1	1	yes	characteristic
96	C₂×C₄³	C₂×C₁₃	1	1	yes	characteristic
97	C₂×#339	C₂×C₁₃	1	4	yes	—
98	C₂³×C₁₃	—	2	4	no	—
99	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₁₃	#121	1	2	yes	—
103	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₁₃	—	2	16	no	—
105	C₂×C₄×C₁₃	—	2	16	no	—
106	C₂×C₄×C₁₃	—	2	16	no	—
107	C₂×C₄×C₁₃	—	2	8	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	8	no	—
110	C₂×C₄×C₁₃	—	2	8	no	—
111	C₂×C₄×C₁₃	—	4	4	no	—
112	C₂×C₄×C₁₃	—	4	8	no	—
113	C₂×C₄×C₁₃	#95	1	4	yes	—
114	C₂×C₄×C₁₃	#95	1	4	yes	—
115	C₂×C₄×C₁₃	#104	1	8	yes	—
116	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#117	1	2	yes	—
122	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₁₃	#121	1	2	yes	—
127	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#125	1	2	yes	—
131	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
133	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
134	C₈×C₁₃	—	4	4	no	—
135	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
136	C₂×#117	C₂²×C₁₃	1	2	yes	—
137	C₂²×C₄²	—	2	2	no	—
138	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
139	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
140	#423	—	2	2	no	—
141	#423	C₄×C₁₃	1	4	yes	complemented
142	C₂×C₄⋊C₈	—	2	2	no	—
143	#384	C₄×C₁₃	1	2	yes	complemented
144	#382	C₄×C₁₃	1	2	yes	complemented
145	#380	—	2	2	no	—
146	C₄×C₄⋊C₄	—	2	2	no	—
147	C₄³	—	2	2	no	—
148	C₄³	C₄×C₁₃	1	4	yes	complemented
149	#339	—	2	8	no	—
150	C₂²×C₁₃	—	2	4	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	2	4	no	—
154	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
155	C₂²×C₁₃	#380	1	2	yes	—
156	C₂²×C₁₃	#389	1	2	yes	—
157	C₂²×C₁₃	#390	1	1	yes	characteristic
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	4	no	—
162	C₄×C₁₃	—	8	4	no	—
163	C₄×C₁₃	#339	1	8	yes	complemented
164	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
165	C₄×C₁₃	#382	1	1	yes	characteristic
166	C₄×C₁₃	#384	1	1	yes	characteristic
167	C₄×C₁₃	#389	1	2	yes	—
168	C₄×C₁₃	#390	1	1	yes	characteristic
169	C₄×C₁₃	#391	1	1	yes	characteristic
170	C₂³×C₄	—	2	2	no	—
171	C₂³×C₄	D₄×C₁₃	1	2	yes	—
172	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	4	no	—
175	C₂×M₁₆	—	2	4	no	—
176	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
177	C₂²×C₈	—	2	2	no	—
178	#117	—	2	4	no	—
179	#117	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	4	no	—
181	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
182	C₂×C₂²⋊C₄	—	2	2	no	—
183	C₂×C₄²	—	2	16	no	—
184	C₂×C₄²	—	2	8	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
190	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
191	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
192	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
193	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
194	C₂×C₄²	D₄×C₁₃	1	4	yes	—
195	C₂×C₄²	D₄×C₁₃	1	2	yes	—
196	C₂×C₄²	D₄×C₁₃	1	2	yes	—
197	C₂×C₄²	Q₈×C₁₃	1	4	yes	—
198	C₂×C₄²	Q₈×C₁₃	1	2	yes	—
199	C₄⋊C₈	—	2	8	no	—
200	#95	—	2	4	no	—
201	C₂×C₁₃	—	2	2	no	—
202	C₂×C₁₃	—	2	2	no	—
203	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
204	C₂×C₁₃	C₂×#339	1	2	yes	—
205	C₂×C₁₃	#1962	1	2	yes	—
206	C₂×C₁₃	#1994	1	2	yes	—
207	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
208	C₂²×C₄	—	2	16	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
215	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
216	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
217	C₂²×C₄	C₁₃×C₄⋊C₄	1	4	yes	—
218	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
224	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
226	C₂²×C₄	C₁₃×#41	1	2	yes	—
227	C₂²×C₄	C₁₃×#41	1	2	yes	—
228	M₁₆	—	2	8	no	—
229	M₁₆	—	4	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	4	8	no	—
235	C₂²⋊C₄	—	4	4	no	—
236	C₄²	—	2	32	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	4	2	no	—
241	C₄²	—	4	2	no	—
242	C₄²	C₄²×C₁₃	1	2	yes	—
243	C₄²	C₄²×C₁₃	1	2	yes	—
244	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
245	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
246	C₄²	C₁₃×C₄⋊C₄	1	4	yes	—
247	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
248	C₄²	C₁₃×C₄⋊C₄	1	8	yes	complemented
249	C₁₃	#10839	1	1	yes	characteristic, direct product factor, Sylow
250	C₂³	—	2	4	no	—
251	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
252	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
253	C₂³	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
254	C₂³	C₁₃×#121	1	2	yes	—
255	C₂³	C₁₃×C₂²⋊Q₈	1	2	yes	—
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	16	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	C₁₃×#95	1	4	yes	—
266	C₂×C₄	C₁₃×#95	1	4	yes	—
267	C₂×C₄	C₁₃×#104	1	8	yes	—
268	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
269	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
271	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
272	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₁₃×#117	1	2	yes	—
274	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
275	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
276	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
277	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
278	C₂×C₄	C₁₃×#121	1	2	yes	—
279	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
280	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
281	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
282	C₂×C₄	C₁₃×#125	1	2	yes	—
283	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
284	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
285	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
286	C₈	—	4	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	C₂×C₁₃×#95	1	1	yes	characteristic
292	C₂²	C₁₃×#380	1	2	yes	—
293	C₂²	C₁₃×#389	1	2	yes	—
294	C₂²	C₁₃×#390	1	1	yes	characteristic
295	C₂²	C₁₃×#391	1	1	yes	characteristic
296	C₂²	C₂×C₁₃×#104	1	4	yes	—
297	C₄	—	2	16	no	—
298	C₄	—	2	4	no	—
299	C₄	—	8	4	no	—
300	C₄	C₁₃×#339	1	8	yes	complemented
301	C₄	C₄×C₁₃×C₄⋊C₄	1	2	yes	—
302	C₄	C₁₃×#382	1	1	yes	characteristic
303	C₄	C₁₃×#384	1	1	yes	characteristic
304	C₄	C₁₃×#389	1	2	yes	—
305	C₄	C₁₃×#390	1	1	yes	characteristic
306	C₄	C₁₃×#391	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	C₁₃×C₄⋊#95	1	1	yes	characteristic
310	C₂	C₂×C₁₃×#339	1	2	yes	—
311	C₂	C₁₃×#1962	1	2	yes	—
312	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
11. C₁₃×#423	C₄	semidirect
13. C₁₃×#384	C₄	semidirect
14. C₁₃×#382	C₄	semidirect
18. C₄³×C₁₃	C₄	semidirect
51. #10839	C₁₃	direct (N × H)
93. C₄²×C₁₃	C₄⋊C₄	semidirect
141. #423	C₄×C₁₃	semidirect
143. #384	C₄×C₁₃	semidirect
144. #382	C₄×C₁₃	semidirect
148. C₄³	C₄×C₁₃	semidirect
163. C₄×C₁₃	#339	semidirect
248. C₄²	C₁₃×C₄⋊C₄	semidirect
249. C₁₃	#10839	direct (N × H)
300. C₄	C₁₃×#339	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×#423	2
C₁₃×#2285	2
C₂×C₄³×C₁₃	2
C₂×C₁₃×#339	2
#10839	13

Subgroups — C13×#10839 #15697178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10839 #15697178