Subgroups — C₁₃×#18719 #15705058

Order 3328.

928 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3550	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#3230	C₂	1	2	yes	maximal, complemented
3	C₁₃×#3152	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1633	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1630	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1543	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1485	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₃×#137	—	2	2	no	—
9	C₁₃×#563	—	2	2	no	—
10	C₁₃×#559	C₂²	1	2	yes	—
11	Q₈²×C₁₃	C₂²	1	2	yes	—
12	C₁₃×#556	—	2	2	no	—
13	D₄×Q₈×C₁₃	—	2	2	no	—
14	C₁₃×#542	—	2	4	no	—
15	C₁₃×#536	—	2	2	no	—
16	C₁₃×#533	C₂²	1	2	yes	—
17	C₁₃×#532	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₃×#500	—	2	2	no	—
20	C₁₃×#497	—	2	2	no	—
21	C₁₃×#494	C₂²	1	2	yes	—
22	C₁₃×#492	C₂²	1	2	yes	—
23	C₁₃×#487	—	2	2	no	—
24	C₁₃×#484	—	2	4	no	—
25	C₁₃×#477	C₂²	1	8	yes	—
26	C₁₃×#431	C₂²	1	2	yes	—
27	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
28	C₁₃×#333	C₂²	1	2	yes	—
29	C₁₃×#331	C₂²	1	2	yes	—
30	C₁₃×Q₈⋊C₈	C₂²	1	8	yes	—
31	C₂×C₁₃×#41	D₄	1	2	yes	—
32	C₂²×Q₈×C₁₃	—	2	2	no	—
33	C₂²×Q₈×C₁₃	—	2	1	no	—
34	C₁₃×#137	—	4	8	no	—
35	C₂×C₁₃×Q₁₆	—	2	4	no	—
36	C₂×C₁₃×SD₁₆	—	2	4	no	—
37	C₂×C₁₃×M₁₆	—	2	1	no	—
38	C₁₃×C₄⋊Q₈	—	2	4	no	—
39	C₁₃×C₄⋊Q₈	—	2	4	no	—
40	C₁₃×C₄⋊Q₈	—	2	2	no	—
41	C₁₃×C₄⋊Q₈	—	2	1	no	—
42	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
43	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
44	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
45	C₁₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
46	C₁₃×#127	C₂³	1	1	yes	characteristic
47	C₁₃×#126	—	2	8	no	—
48	C₁₃×#125	—	2	2	no	—
49	C₁₃×#124	—	2	4	no	—
50	C₁₃×#124	—	2	4	no	—
51	C₁₃×#124	D₄	1	2	yes	—
52	C₁₃×#123	—	2	4	no	—
53	C₁₃×#123	—	2	4	no	—
54	C₁₃×C₂²⋊Q₈	—	2	4	no	—
55	C₁₃×C₂²⋊Q₈	—	2	8	no	—
56	C₁₃×C₂²⋊Q₈	—	2	2	no	—
57	C₁₃×C₂²⋊Q₈	D₄	1	4	yes	—
58	C₁₃×#121	—	2	2	no	—
59	C₄×Q₈×C₁₃	—	2	4	no	—
60	C₄×Q₈×C₁₃	—	2	2	no	—
61	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
62	C₄×D₄×C₁₃	—	2	4	no	—
63	C₄×D₄×C₁₃	—	2	2	no	—
64	C₁₃×#117	—	2	2	no	—
65	C₁₃×#117	—	2	2	no	—
66	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
68	C₁₃×#107	—	2	8	no	—
69	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
70	C₁₃×#103	—	2	8	no	—
71	C₁₃×C₂²⋊C₈	—	2	2	no	—
72	C₁₃×#97	—	2	1	no	—
73	C₄×C₈×C₁₃	—	2	2	no	—
74	C₄×C₈×C₁₃	C₂³	1	2	yes	—
75	#18719	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₁₃×#41	—	4	4	no	—
77	C₂×Q₈×C₁₃	—	2	4	no	—
78	C₂×Q₈×C₁₃	—	2	2	no	—
79	C₂×Q₈×C₁₃	—	2	2	no	—
80	C₂×Q₈×C₁₃	—	4	4	no	—
81	C₂×Q₈×C₁₃	—	4	1	no	—
82	C₂×Q₈×C₁₃	—	8	4	no	—
83	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
84	C₂×Q₈×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×D₄×C₁₃	—	2	2	no	—
86	C₂×D₄×C₁₃	—	2	2	no	—
87	C₂×D₄×C₁₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₃	—	2	4	no	—
89	C₂²×C₄×C₁₃	—	2	2	no	—
90	C₂²×C₄×C₁₃	—	4	1	no	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₁₃×Q₁₆	—	4	8	no	—
94	C₁₃×SD₁₆	—	4	8	no	—
95	C₁₃×M₁₆	—	2	4	no	—
96	C₂×C₈×C₁₃	—	2	2	no	—
97	C₂×C₈×C₁₃	—	2	4	no	—
98	C₁₃×C₄⋊C₄	—	2	8	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	2	4	no	—
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	2	4	no	—
105	C₁₃×C₄⋊C₄	—	2	4	no	—
106	C₁₃×C₄⋊C₄	—	4	2	no	—
107	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
108	C₁₃×C₂²⋊C₄	—	2	4	no	—
109	C₁₃×C₂²⋊C₄	—	2	8	no	—
110	C₁₃×C₂²⋊C₄	—	2	4	no	—
111	C₁₃×C₂²⋊C₄	—	2	4	no	—
112	C₄²×C₁₃	—	2	4	no	—
113	C₄²×C₁₃	—	2	4	no	—
114	C₄²×C₁₃	C₂×D₄	1	2	yes	—
115	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
117	#3550	C₂×C₁₃	1	1	yes	characteristic
118	#3230	C₂×C₁₃	1	2	yes	complemented
119	#3152	C₂×C₁₃	1	1	yes	characteristic
120	#1633	C₂×C₁₃	1	4	yes	complemented
121	#1630	C₂×C₁₃	1	4	yes	complemented
122	#1543	C₂×C₁₃	1	1	yes	characteristic
123	#1485	C₂×C₁₃	1	2	yes	complemented
124	C₂³×C₁₃	—	2	2	no	—
125	C₂³×C₁₃	#120	1	1	yes	characteristic
126	Q₈×C₁₃	—	2	8	no	—
127	Q₈×C₁₃	—	4	2	no	—
128	Q₈×C₁₃	—	4	4	no	—
129	Q₈×C₁₃	—	4	4	no	—
130	Q₈×C₁₃	—	8	4	no	—
131	Q₈×C₁₃	—	8	2	no	—
132	D₄×C₁₃	—	4	2	no	—
133	D₄×C₁₃	—	4	4	no	—
134	D₄×C₁₃	—	8	2	no	—
135	C₂×C₄×C₁₃	—	2	4	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	4	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	4	no	—
140	C₂×C₄×C₁₃	—	4	4	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	8	4	no	—
143	C₂×C₄×C₁₃	—	8	2	no	—
144	C₂×C₄×C₁₃	#120	1	2	yes	—
145	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
148	C₈×C₁₃	—	2	4	no	—
149	C₈×C₁₃	—	4	4	no	—
150	C₂×#137	—	2	2	no	—
151	#563	—	2	2	no	—
152	#559	C₂²×C₁₃	1	2	yes	—
153	Q₈²	C₂²×C₁₃	1	2	yes	—
154	#556	—	2	2	no	—
155	D₄×Q₈	—	2	2	no	—
156	#542	—	2	4	no	—
157	#536	—	2	2	no	—
158	#533	C₂²×C₁₃	1	2	yes	—
159	#532	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
161	#500	—	2	2	no	—
162	#497	—	2	2	no	—
163	#494	C₂²×C₁₃	1	2	yes	—
164	#492	C₂²×C₁₃	1	2	yes	—
165	#487	—	2	2	no	—
166	#484	—	2	4	no	—
167	#477	C₂²×C₁₃	1	8	yes	—
168	#431	C₂²×C₁₃	1	2	yes	—
169	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
170	#333	C₂²×C₁₃	1	2	yes	—
171	#331	C₂²×C₁₃	1	2	yes	—
172	Q₈⋊C₈	C₂²×C₁₃	1	8	yes	—
173	C₂²×C₁₃	—	2	2	no	—
174	C₂²×C₁₃	—	2	1	no	—
175	C₂²×C₁₃	—	4	4	no	—
176	C₂²×C₁₃	—	4	2	no	—
177	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
178	C₄×C₁₃	—	2	2	no	—
179	C₄×C₁₃	—	2	1	no	—
180	C₄×C₁₃	—	4	4	no	—
181	C₄×C₁₃	—	4	2	no	—
182	C₄×C₁₃	—	4	4	no	—
183	C₄×C₁₃	—	8	2	no	—
184	C₄×C₁₃	#453	1	2	yes	—
185	C₄×C₁₃	#578	1	4	yes	—
186	C₂×#41	D₄×C₁₃	1	2	yes	—
187	C₂²×Q₈	—	2	2	no	—
188	C₂²×Q₈	—	2	1	no	—
189	#137	—	4	8	no	—
190	C₂×Q₁₆	—	2	4	no	—
191	C₂×SD₁₆	—	2	4	no	—
192	C₂×M₁₆	—	2	1	no	—
193	C₄⋊Q₈	—	2	4	no	—
194	C₄⋊Q₈	—	2	4	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	2	1	no	—
197	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
198	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
199	C₄⋊Q₈	C₂³×C₁₃	1	2	yes	—
200	C₄⋊Q₈	C₂³×C₁₃	1	1	yes	characteristic
201	#127	C₂³×C₁₃	1	1	yes	characteristic
202	#126	—	2	8	no	—
203	#125	—	2	2	no	—
204	#124	—	2	4	no	—
205	#124	—	2	4	no	—
206	#124	D₄×C₁₃	1	2	yes	—
207	#123	—	2	4	no	—
208	#123	—	2	4	no	—
209	C₂²⋊Q₈	—	2	4	no	—
210	C₂²⋊Q₈	—	2	8	no	—
211	C₂²⋊Q₈	—	2	2	no	—
212	C₂²⋊Q₈	D₄×C₁₃	1	4	yes	—
213	#121	—	2	2	no	—
214	C₄×Q₈	—	2	4	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×Q₈	C₂³×C₁₃	1	4	yes	—
217	C₄×D₄	—	2	4	no	—
218	C₄×D₄	—	2	2	no	—
219	#117	—	2	2	no	—
220	#117	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
223	#107	—	2	8	no	—
224	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
225	#103	—	2	8	no	—
226	C₂²⋊C₈	—	2	2	no	—
227	#97	—	2	1	no	—
228	C₄×C₈	—	2	2	no	—
229	C₄×C₈	C₂³×C₁₃	1	2	yes	—
230	C₂×C₁₃	—	4	1	no	—
231	C₂×C₁₃	—	8	2	no	—
232	C₂×C₁₃	#3011	1	2	yes	—
233	C₂×C₁₃	C₂×#453	1	1	yes	characteristic
234	#41	—	4	4	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	4	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	8	4	no	—
241	C₂×Q₈	C₂×D₄×C₁₃	1	4	yes	—
242	C₂×Q₈	C₂×D₄×C₁₃	1	2	yes	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	C₂×D₄×C₁₃	1	2	yes	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
251	Q₁₆	—	4	8	no	—
252	SD₁₆	—	4	8	no	—
253	M₁₆	—	2	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	4	no	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
266	C₂²⋊C₄	—	2	4	no	—
267	C₂²⋊C₄	—	2	8	no	—
268	C₂²⋊C₄	—	2	4	no	—
269	C₂²⋊C₄	—	2	4	no	—
270	C₄²	—	2	4	no	—
271	C₄²	—	2	4	no	—
272	C₄²	C₂×D₄×C₁₃	1	2	yes	—
273	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
274	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
275	C₁₃	#18719	1	1	yes	characteristic, direct product factor, Sylow
276	C₂³	—	2	2	no	—
277	C₂³	C₁₃×#120	1	1	yes	characteristic
278	Q₈	—	2	8	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	4	4	no	—
281	Q₈	—	4	4	no	—
282	Q₈	—	8	4	no	—
283	Q₈	—	8	2	no	—
284	D₄	—	4	2	no	—
285	D₄	—	4	4	no	—
286	D₄	—	8	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	8	4	no	—
295	C₂×C₄	—	8	2	no	—
296	C₂×C₄	C₁₃×#120	1	2	yes	—
297	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
298	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
300	C₈	—	2	4	no	—
301	C₈	—	4	4	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	4	2	no	—
306	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	1	no	—
309	C₄	—	4	4	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	4	no	—
312	C₄	—	8	2	no	—
313	C₄	C₁₃×#453	1	2	yes	—
314	C₄	C₁₃×#578	1	4	yes	—
315	C₂	—	4	1	no	—
316	C₂	—	8	2	no	—
317	C₂	C₁₃×#3011	1	2	yes	—
318	C₂	C₂×C₁₃×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃
prime radical	C₁₃×#532	C₂²
LCS	G ▷ 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
2. C₁₃×#3230	C₂	semidirect
4. C₁₃×#1633	C₂	semidirect
5. C₁₃×#1630	C₂	semidirect
7. C₁₃×#1485	C₂	semidirect
75. #18719	C₁₃	direct (N × H)
118. #3230	C₂×C₁₃	semidirect
120. #1633	C₂×C₁₃	semidirect
121. #1630	C₂×C₁₃	semidirect
123. #1485	C₂×C₁₃	semidirect
275. C₁₃	#18719	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×#3550	2
C₁₃×#3230	2
C₁₃×#3152	2
C₁₃×#1633	2
C₁₃×#1630	2
C₁₃×#1543	2
C₁₃×#1485	2
#18719	13

Subgroups — C13×#18719 #15705058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#18719 #15705058