Subgroups — C₁₃×#7728 #15694067

Order 3328.

494 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2857	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1817	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1797	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1271	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₁₃×C₂²⋊Q₈	—	2	1	no	—
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
9	C₂×C₁₃×C₂²⋊C₈	—	2	1	no	—
10	C₂×C₁₃×C₂²⋊C₈	—	2	1	no	—
11	C₁₃×#391	—	2	1	no	—
12	C₁₃×#389	—	2	1	no	—
13	C₁₃×#389	C₂²	1	2	yes	—
14	C₁₃×#384	C₄	1	1	yes	characteristic
15	C₁₃×#380	—	2	1	no	—
16	C₁₃×#380	C₄	1	2	yes	—
17	C₁₃×#336	—	2	1	no	—
18	C₁₃×#336	C₂²	1	2	yes	—
19	C₁₃×#336	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₁₃×C₄⋊Q₈	—	4	2	no	—
27	C₁₃×C₂²⋊Q₈	—	4	2	no	—
28	C₁₃×C₂²⋊Q₈	—	4	2	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
36	C₂×C₄²×C₁₃	—	4	2	no	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₄⋊C₈	—	4	2	no	—
42	C₁₃×C₂²⋊C₈	—	4	2	no	—
43	C₁₃×C₂²⋊C₈	—	4	2	no	—
44	C₁₃×#95	—	2	2	no	—
45	C₁₃×#95	—	2	2	no	—
46	#7728	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Q₈×C₁₃	—	4	2	no	—
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	C₂×Q₈×C₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	4	1	no	—
58	C₂²×C₄×C₁₃	—	4	1	no	—
59	C₂²×C₄×C₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	—	4	1	no	—
61	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	D₈	1	2	yes	—
65	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	—	4	2	no	—
70	C₂×C₈×C₁₃	—	4	2	no	—
71	C₂×C₈×C₁₃	—	4	2	no	—
72	C₂×C₈×C₁₃	—	4	2	no	—
73	C₂×C₈×C₁₃	—	4	2	no	—
74	C₂×C₈×C₁₃	—	4	2	no	—
75	C₁₃×C₄⋊C₄	—	2	4	no	—
76	C₁₃×C₄⋊C₄	—	4	2	no	—
77	C₁₃×C₄⋊C₄	—	4	2	no	—
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₁₃×C₄⋊C₄	—	4	2	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₄⋊C₄	—	4	2	no	—
82	C₁₃×C₂²⋊C₄	—	4	4	no	—
83	C₄²×C₁₃	—	2	2	no	—
84	C₄²×C₁₃	—	2	2	no	—
85	C₄²×C₁₃	—	4	1	no	—
86	C₄²×C₁₃	—	4	1	no	—
87	#2857	C₂×C₁₃	1	1	yes	characteristic
88	#1817	C₂×C₁₃	1	1	yes	characteristic
89	#1797	C₂×C₁₃	1	1	yes	characteristic
90	#1271	C₂×C₁₃	1	4	yes	complemented
91	C₂³×C₁₃	—	2	1	no	—
92	C₂³×C₁₃	—	2	1	no	—
93	C₂³×C₁₃	—	2	1	no	—
94	C₂³×C₁₃	—	2	1	no	—
95	C₂³×C₁₃	—	2	1	no	—
96	C₂³×C₁₃	—	2	1	no	—
97	C₂³×C₁₃	#102	1	2	yes	—
98	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	Q₈×C₁₃	—	8	4	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	4	2	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	4	2	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	#99	1	2	yes	—
123	C₂×C₄×C₁₃	#101	1	2	yes	—
124	C₂×C₄×C₁₃	#102	1	2	yes	—
125	C₂×C₄×C₁₃	#131	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#137	1	2	yes	—
129	C₈×C₁₃	—	8	2	no	—
130	C₈×C₁₃	—	8	2	no	—
131	C₂×C₄⋊Q₈	C₄×C₁₃	1	1	yes	characteristic
132	C₂×C₂²⋊Q₈	—	2	1	no	—
133	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×C₄⋊C₈	—	2	1	no	—
135	C₂×C₂²⋊C₈	—	2	1	no	—
136	C₂×C₂²⋊C₈	—	2	1	no	—
137	#391	—	2	1	no	—
138	#389	—	2	1	no	—
139	#389	C₂²×C₁₃	1	2	yes	—
140	#384	C₄×C₁₃	1	1	yes	characteristic
141	#380	—	2	1	no	—
142	#380	C₄×C₁₃	1	2	yes	—
143	#336	—	2	1	no	—
144	#336	C₂²×C₁₃	1	2	yes	—
145	#336	C₂²×C₁₃	1	2	yes	—
146	C₂²×C₁₃	—	2	2	no	—
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	—	4	1	no	—
150	C₂²×C₁₃	—	4	1	no	—
151	C₂²×C₁₃	—	4	1	no	—
152	C₂²×C₁₃	—	4	1	no	—
153	C₂²×C₁₃	#408	1	1	yes	characteristic
154	C₂²×C₁₃	C₂×#99	1	1	yes	characteristic
155	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
157	C₂²×C₁₃	#419	1	1	yes	characteristic
158	C₂²×C₁₃	#421	1	1	yes	characteristic
159	C₂²×C₁₃	#421	1	1	yes	characteristic
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	—	2	2	no	—
162	C₄×C₁₃	—	2	2	no	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	8	2	no	—
168	C₄×C₁₃	—	8	2	no	—
169	C₂²×Q₈	—	2	1	no	—
170	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
171	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
172	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₄⋊Q₈	—	4	2	no	—
176	C₂²⋊Q₈	—	4	2	no	—
177	C₂²⋊Q₈	—	4	2	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
183	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₄²	—	4	2	no	—
186	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
187	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
188	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
190	C₄⋊C₈	—	4	2	no	—
191	C₂²⋊C₈	—	4	2	no	—
192	C₂²⋊C₈	—	4	2	no	—
193	#95	—	2	2	no	—
194	#95	—	2	2	no	—
195	C₂×C₁₃	—	4	2	no	—
196	C₂×C₁₃	#1472	1	1	yes	characteristic
197	C₂×C₁₃	#1480	1	1	yes	characteristic
198	C₂×C₁₃	#1497	1	1	yes	characteristic
199	C₂×C₁₃	#1511	1	1	yes	characteristic
200	C₂×C₁₃	#1519	1	1	yes	characteristic
201	C₂×C₁₃	#1537	1	1	yes	characteristic
202	C₂×C₁₃	#1546	1	1	yes	characteristic
203	C₂⁴	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
204	C₂×Q₈	—	4	2	no	—
205	C₂×Q₈	—	4	2	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	C₁₃×D₈	1	2	yes	—
221	C₂²×C₄	C₁₃×SD₁₆	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₁₃	#7728	1	1	yes	characteristic, direct product factor, Sylow
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	C₁₃×#102	1	2	yes	—
251	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
252	Q₈	—	8	4	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	C₁₃×#99	1	2	yes	—
276	C₂×C₄	C₁₃×#101	1	2	yes	—
277	C₂×C₄	C₁₃×#102	1	2	yes	—
278	C₂×C₄	C₁₃×#131	1	2	yes	—
279	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
280	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
281	C₂×C₄	C₁₃×#137	1	2	yes	—
282	C₈	—	8	2	no	—
283	C₈	—	8	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	C₁₃×#408	1	1	yes	characteristic
292	C₂²	C₂×C₁₃×#99	1	1	yes	characteristic
293	C₂²	C₂×C₁₃×#101	1	1	yes	characteristic
294	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
295	C₂²	C₁₃×#419	1	1	yes	characteristic
296	C₂²	C₁₃×#421	1	1	yes	characteristic
297	C₂²	C₁₃×#421	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	1	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₂	—	4	2	no	—
308	C₂	C₁₃×#1472	1	1	yes	characteristic
309	C₂	C₁₃×#1480	1	1	yes	characteristic
310	C₂	C₁₃×#1497	1	1	yes	characteristic
311	C₂	C₁₃×#1511	1	1	yes	characteristic
312	C₂	C₁₃×#1519	1	1	yes	characteristic
313	C₂	C₁₃×#1537	1	1	yes	characteristic
314	C₂	C₁₃×#1546	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₁₃×#1271	C₂	semidirect
46. #7728	C₁₃	direct (N × H)
90. #1271	C₂×C₁₃	semidirect
243. C₁₃	#7728	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×#2857	2
C₁₃×#1817	2
C₁₃×#1797	2
C₁₃×#1271	2
#7728	13

Subgroups — C13×#7728 #15694067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#7728 #15694067