Subgroups — C₁₃×#10452 #15696791

Order 3328.

408 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#476	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1800	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1570	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1478	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#476	—	2	2	no	—
11	C₁₃×#476	—	2	2	no	—
12	C₁₃×#431	C₂²	1	1	yes	characteristic
13	C₁₃×#431	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#106	—	2	1	no	—
15	C₁₃×#423	C₂²	1	1	yes	characteristic
16	C₁₃×#423	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#102	—	2	1	no	—
19	C₂×C₁₃×#100	—	2	1	no	—
20	C₁₃×#343	—	2	1	no	—
21	C₁₃×#334	—	2	2	no	—
22	C₁₃×#333	—	2	2	no	—
23	C₁₃×#332	—	2	2	no	—
24	C₁₃×D₄⋊C₈	—	2	2	no	—
25	C₂²×Q₈×C₁₃	—	2	1	no	—
26	C₂²×D₄×C₁₃	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	—	2	1	no	—
28	C₂×C₁₃×M₁₆	—	2	1	no	—
29	C₂×C₁₃×M₁₆	—	2	1	no	—
30	C₂²×C₈×C₁₃	—	2	1	no	—
31	C₁₃×C₄⋊Q₈	—	2	1	no	—
32	C₁₃×C₄⋊Q₈	—	2	1	no	—
33	C₁₃×C₄⋊Q₈	—	2	1	no	—
34	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
35	C₄×D₄×C₁₃	—	2	1	no	—
36	C₄×D₄×C₁₃	—	2	1	no	—
37	C₄×D₄×C₁₃	—	2	1	no	—
38	C₄×D₄×C₁₃	—	2	1	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
41	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
42	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×#106	—	2	4	no	—
45	C₁₃×C₄⋊C₈	—	2	1	no	—
46	C₁₃×C₄⋊C₈	—	2	1	no	—
47	C₁₃×C₄⋊C₈	—	2	1	no	—
48	C₁₃×C₄⋊C₈	—	2	1	no	—
49	C₁₃×C₄⋊C₈	—	2	1	no	—
50	C₁₃×C₄⋊C₈	—	2	1	no	—
51	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
52	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
53	C₁₃×#102	—	2	4	no	—
54	C₁₃×#100	—	4	2	no	—
55	C₁₃×C₂²⋊C₈	—	2	1	no	—
56	C₁₃×C₂²⋊C₈	—	2	1	no	—
57	C₄×C₈×C₁₃	—	2	1	no	—
58	C₄×C₈×C₁₃	—	2	1	no	—
59	#10452	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂⁴×C₁₃	—	2	1	no	—
61	C₂×Q₈×C₁₃	—	2	2	no	—
62	C₂×Q₈×C₁₃	—	4	1	no	—
63	C₂×Q₈×C₁₃	—	8	2	no	—
64	C₂×D₄×C₁₃	—	2	1	no	—
65	C₂×D₄×C₁₃	—	2	1	no	—
66	C₂×D₄×C₁₃	—	2	1	no	—
67	C₂×D₄×C₁₃	—	2	1	no	—
68	C₂×D₄×C₁₃	—	2	1	no	—
69	C₂×D₄×C₁₃	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₁₃	—	2	1	no	—
71	C₂²×C₄×C₁₃	—	4	1	no	—
72	C₂²×C₄×C₁₃	—	4	1	no	—
73	C₂²×C₄×C₁₃	—	4	1	no	—
74	C₂²×C₄×C₁₃	—	8	2	no	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
78	C₁₃×M₁₆	—	4	1	no	—
79	C₁₃×M₁₆	—	4	1	no	—
80	C₁₃×M₁₆	—	4	1	no	—
81	C₁₃×M₁₆	—	4	1	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	4	1	no	—
85	C₂×C₈×C₁₃	—	4	1	no	—
86	C₂×C₈×C₁₃	—	4	1	no	—
87	C₂×C₈×C₁₃	—	4	1	no	—
88	C₂×C₈×C₁₃	—	4	1	no	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	2	1	no	—
91	C₁₃×C₄⋊C₄	—	2	1	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	4	1	no	—
94	C₁₃×C₄⋊C₄	—	4	1	no	—
95	C₁₃×C₂²⋊C₄	—	4	1	no	—
96	C₁₃×C₂²⋊C₄	—	4	1	no	—
97	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₁₃	#41	1	1	yes	characteristic
101	C₂×#476	C₂×C₁₃	1	1	yes	characteristic
102	#1800	C₂×C₁₃	1	1	yes	characteristic
103	#1570	C₂×C₁₃	1	1	yes	characteristic, complemented
104	#1549	C₂×C₁₃	1	1	yes	characteristic, complemented
105	#1548	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#1478	C₂×C₁₃	1	1	yes	characteristic
107	#1301	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₂³×C₁₃	—	2	2	no	—
109	C₂³×C₁₃	—	4	1	no	—
110	C₂³×C₁₃	—	4	1	no	—
111	C₂³×C₁₃	—	8	1	no	—
112	C₂³×C₁₃	—	8	1	no	—
113	C₂³×C₁₃	#123	1	1	yes	characteristic
114	Q₈×C₁₃	—	8	2	no	—
115	Q₈×C₁₃	—	8	2	no	—
116	D₄×C₁₃	—	2	2	no	—
117	D₄×C₁₃	—	2	2	no	—
118	D₄×C₁₃	—	4	1	no	—
119	D₄×C₁₃	—	4	1	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	4	1	no	—
122	C₂×C₄×C₁₃	—	4	1	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	8	2	no	—
125	C₂×C₄×C₁₃	—	8	2	no	—
126	C₂×C₄×C₁₃	—	8	2	no	—
127	C₂×C₄×C₁₃	—	8	1	no	—
128	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
138	C₈×C₁₃	—	4	2	no	—
139	C₈×C₁₃	—	4	2	no	—
140	C₈×C₁₃	—	8	1	no	—
141	C₈×C₁₃	—	8	1	no	—
142	C₈×C₁₃	—	8	1	no	—
143	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
145	#476	—	2	2	no	—
146	#476	—	2	2	no	—
147	#431	C₂²×C₁₃	1	1	yes	characteristic
148	#431	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×#106	—	2	1	no	—
150	#423	C₂²×C₁₃	1	1	yes	characteristic
151	#423	C₂²×C₁₃	1	1	yes	characteristic
152	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×#102	—	2	1	no	—
154	C₂×#100	—	2	1	no	—
155	#343	—	2	1	no	—
156	#334	—	2	2	no	—
157	#333	—	2	2	no	—
158	#332	—	2	2	no	—
159	D₄⋊C₈	—	2	2	no	—
160	C₂²×C₁₃	—	2	1	no	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	—	4	2	no	—
163	C₂²×C₁₃	—	8	2	no	—
164	C₂²×C₁₃	—	8	2	no	—
165	C₂²×C₁₃	—	8	1	no	—
166	C₂²×C₁₃	—	8	1	no	—
167	C₂²×C₁₃	#394	1	1	yes	characteristic
168	C₂²×C₁₃	#483	1	1	yes	characteristic
169	C₂²×C₁₃	#485	1	1	yes	characteristic
170	C₄×C₁₃	—	2	1	no	—
171	C₄×C₁₃	—	2	1	no	—
172	C₄×C₁₃	—	2	1	no	—
173	C₄×C₁₃	—	8	2	no	—
174	C₄×C₁₃	—	8	1	no	—
175	C₄×C₁₃	#448	1	1	yes	characteristic
176	C₄×C₁₃	#452	1	1	yes	characteristic
177	C₄×C₁₃	#465	1	1	yes	characteristic
178	C₄×C₁₃	#470	1	1	yes	characteristic
179	C₄×C₁₃	#471	1	2	yes	—
180	C₂²×Q₈	—	2	1	no	—
181	C₂²×D₄	D₄×C₁₃	1	1	yes	characteristic
182	C₂³×C₄	—	2	1	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂²×C₈	—	2	1	no	—
186	C₄⋊Q₈	—	2	1	no	—
187	C₄⋊Q₈	—	2	1	no	—
188	C₄⋊Q₈	—	2	1	no	—
189	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	—	2	1	no	—
192	C₄×D₄	—	2	1	no	—
193	C₄×D₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
197	C₂×C₂²⋊C₄	—	2	1	no	—
198	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
199	#106	—	2	4	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	—	2	1	no	—
203	C₄⋊C₈	—	2	1	no	—
204	C₄⋊C₈	—	2	1	no	—
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
207	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
208	#102	—	2	4	no	—
209	#100	—	4	2	no	—
210	C₂²⋊C₈	—	2	1	no	—
211	C₂²⋊C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₂×C₁₃	—	2	1	no	—
215	C₂×C₁₃	—	2	1	no	—
216	C₂×C₁₃	—	8	2	no	—
217	C₂×C₁₃	#2035	1	1	yes	characteristic
218	C₂×C₁₃	#2046	1	1	yes	characteristic
219	C₂×C₁₃	#2048	1	1	yes	characteristic
220	C₂⁴	—	2	1	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	4	1	no	—
223	C₂×Q₈	—	8	2	no	—
224	C₂×D₄	—	2	1	no	—
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	C₁₃×SD₁₆	1	2	yes	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	8	2	no	—
235	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
236	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
237	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
238	M₁₆	—	4	1	no	—
239	M₁₆	—	4	1	no	—
240	M₁₆	—	4	1	no	—
241	M₁₆	—	4	1	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₂²⋊C₄	—	4	1	no	—
256	C₂²⋊C₄	—	4	1	no	—
257	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₄²	C₁₃×#41	1	1	yes	characteristic
261	C₁₃	#10452	1	1	yes	characteristic, direct product factor, Sylow
262	C₂³	—	2	2	no	—
263	C₂³	—	4	1	no	—
264	C₂³	—	4	1	no	—
265	C₂³	—	8	1	no	—
266	C₂³	—	8	1	no	—
267	C₂³	C₁₃×#123	1	1	yes	characteristic
268	Q₈	—	8	2	no	—
269	Q₈	—	8	2	no	—
270	D₄	—	2	2	no	—
271	D₄	—	2	2	no	—
272	D₄	—	4	1	no	—
273	D₄	—	4	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	—	8	1	no	—
282	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
283	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
284	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
285	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
286	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
287	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
288	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
289	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
290	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
291	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
292	C₈	—	4	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	8	1	no	—
295	C₈	—	8	1	no	—
296	C₈	—	8	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	1	no	—
303	C₂²	—	8	1	no	—
304	C₂²	C₁₃×#394	1	1	yes	characteristic
305	C₂²	C₁₃×#483	1	1	yes	characteristic
306	C₂²	C₁₃×#485	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	8	1	no	—
311	C₄	—	8	2	no	—
312	C₄	C₁₃×#448	1	1	yes	characteristic
313	C₄	C₁₃×#452	1	1	yes	characteristic
314	C₄	C₁₃×#465	1	1	yes	characteristic
315	C₄	C₁₃×#470	1	1	yes	characteristic
316	C₄	C₁₃×#471	1	2	yes	—
317	C₂	—	2	1	no	—
318	C₂	—	2	1	no	—
319	C₂	—	8	2	no	—
320	C₂	C₁₃×#2035	1	1	yes	characteristic
321	C₂	C₁₃×#2046	1	1	yes	characteristic
322	C₂	C₁₃×#2048	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂²×D₄×C₁₃	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#1570	C₂	semidirect
4. C₁₃×#1549	C₂	semidirect
5. C₁₃×#1548	C₂	semidirect
7. C₁₃×#1301	C₂	semidirect
59. #10452	C₁₃	direct (N × H)
103. #1570	C₂×C₁₃	semidirect
104. #1549	C₂×C₁₃	semidirect
105. #1548	C₂×C₁₃	semidirect
107. #1301	C₂×C₁₃	semidirect
261. C₁₃	#10452	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×#476	2
C₁₃×#1800	2
C₁₃×#1570	2
C₁₃×#1549	2
C₁₃×#1548	2
C₁₃×#1478	2
C₁₃×#1301	2
#10452	13

Subgroups — C13×#10452 #15696791

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10452 #15696791