Subgroups — C₁₃×#9858 #15696197

Order 3328.

428 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3131	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1492	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1492	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#533	C₂²	1	1	yes	characteristic
9	C₁₃×#532	C₂²	1	1	yes	characteristic
10	C₁₃×#488	—	2	2	no	—
11	C₁₃×#487	C₄	1	1	yes	characteristic
12	C₁₃×#487	C₄	1	1	yes	characteristic
13	C₁₃×#486	C₄	1	1	yes	characteristic
14	C₁₃×#486	C₄	1	1	yes	characteristic
15	C₂×C₁₃×#108	—	2	2	no	—
16	C₁₃×#423	C₂²	1	1	yes	characteristic
17	C₁₃×#423	C₂²	1	1	yes	characteristic
18	C₁₃×#416	—	2	2	no	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₁₃×Q₈⋊C₈	—	2	2	no	—
23	C₁₃×Q₈⋊C₈	—	2	2	no	—
24	C₁₃×D₄⋊C₈	—	2	2	no	—
25	C₁₃×D₄⋊C₈	—	2	2	no	—
26	C₁₃×#322	—	2	2	no	—
27	C₂×C₁₃×#41	—	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	C₂²×C₈×C₁₃	—	2	1	no	—
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
34	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
35	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
36	C₁₃×#127	C₂×C₄	1	1	yes	characteristic
37	C₁₃×#125	—	2	1	no	—
38	C₁₃×#124	—	2	1	no	—
39	C₁₃×C₂²⋊Q₈	—	2	1	no	—
40	C₁₃×C₂²⋊Q₈	—	2	1	no	—
41	C₁₃×#121	—	2	1	no	—
42	C₁₃×#121	—	2	1	no	—
43	C₄×Q₈×C₁₃	—	2	1	no	—
44	C₄×Q₈×C₁₃	—	2	1	no	—
45	C₄×D₄×C₁₃	—	2	1	no	—
46	C₄×D₄×C₁₃	—	2	1	no	—
47	C₁₃×#117	—	2	1	no	—
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×#108	—	4	4	no	—
50	C₁₃×C₄⋊C₈	—	2	1	no	—
51	C₁₃×C₄⋊C₈	—	2	1	no	—
52	C₁₃×C₄⋊C₈	—	2	1	no	—
53	C₁₃×C₄⋊C₈	—	2	1	no	—
54	C₁₃×#103	—	2	2	no	—
55	C₁₃×#103	—	2	2	no	—
56	C₁₃×#102	—	2	2	no	—
57	C₁₃×#102	—	2	2	no	—
58	C₄×C₈×C₁₃	—	2	1	no	—
59	C₄×C₈×C₁₃	—	2	1	no	—
60	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	D₄	1	2	yes	—
63	C₄×C₈×C₁₃	D₄	1	2	yes	—
64	C₄×C₈×C₁₃	D₄	1	2	yes	—
65	#9858	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₁₃×#41	—	4	2	no	—
67	C₁₃×#41	—	4	2	no	—
68	C₂×Q₈×C₁₃	—	2	1	no	—
69	C₂×Q₈×C₁₃	—	2	1	no	—
70	C₂×Q₈×C₁₃	—	2	1	no	—
71	C₂×D₄×C₁₃	—	2	1	no	—
72	C₂×D₄×C₁₃	—	2	1	no	—
73	C₂×D₄×C₁₃	—	2	1	no	—
74	C₂×D₄×C₁₃	—	4	1	no	—
75	C₂²×C₄×C₁₃	—	4	1	no	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
79	C₁₃×M₁₆	—	4	2	no	—
80	C₁₃×M₁₆	—	4	2	no	—
81	C₂×C₈×C₁₃	—	2	4	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	4	2	no	—
87	C₂×C₈×C₁₃	—	4	1	no	—
88	C₂×C₈×C₁₃	—	4	1	no	—
89	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
91	C₁₃×C₄⋊C₄	—	2	1	no	—
92	C₁₃×C₄⋊C₄	—	2	1	no	—
93	C₁₃×C₄⋊C₄	—	2	1	no	—
94	C₁₃×C₄⋊C₄	—	2	1	no	—
95	C₁₃×C₄⋊C₄	—	2	1	no	—
96	C₁₃×C₄⋊C₄	—	2	1	no	—
97	C₁₃×C₄⋊C₄	—	4	1	no	—
98	C₁₃×C₄⋊C₄	—	4	1	no	—
99	C₁₃×C₂²⋊C₄	—	4	1	no	—
100	C₁₃×C₂²⋊C₄	—	4	1	no	—
101	C₁₃×C₂²⋊C₄	—	4	1	no	—
102	C₄²×C₁₃	—	4	1	no	—
103	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	#3131	C₂×C₁₃	1	1	yes	characteristic
108	#1848	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#1493	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#1493	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#1492	C₂×C₁₃	1	1	yes	characteristic
112	#1492	C₂×C₁₃	1	1	yes	characteristic
113	C₂×#322	C₂×C₁₃	1	1	yes	characteristic, complemented
114	C₂³×C₁₃	—	4	1	no	—
115	C₂³×C₁₃	#124	1	1	yes	characteristic
116	Q₈×C₁₃	—	4	2	no	—
117	Q₈×C₁₃	—	4	2	no	—
118	Q₈×C₁₃	—	4	2	no	—
119	D₄×C₁₃	—	4	2	no	—
120	D₄×C₁₃	—	4	2	no	—
121	D₄×C₁₃	—	4	2	no	—
122	D₄×C₁₃	—	8	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	4	1	no	—
126	C₂×C₄×C₁₃	—	4	1	no	—
127	C₂×C₄×C₁₃	—	4	1	no	—
128	C₂×C₄×C₁₃	—	8	1	no	—
129	C₂×C₄×C₁₃	—	8	1	no	—
130	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
136	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
137	C₂×C₄×C₁₃	#137	1	2	yes	—
138	C₈×C₁₃	—	2	4	no	—
139	C₈×C₁₃	—	2	2	no	—
140	C₈×C₁₃	—	2	2	no	—
141	C₈×C₁₃	—	4	2	no	—
142	C₈×C₁₃	—	8	1	no	—
143	C₈×C₁₃	—	8	1	no	—
144	#533	C₂²×C₁₃	1	1	yes	characteristic
145	#532	C₂²×C₁₃	1	1	yes	characteristic
146	#488	—	2	2	no	—
147	#487	C₄×C₁₃	1	1	yes	characteristic
148	#487	C₄×C₁₃	1	1	yes	characteristic
149	#486	C₄×C₁₃	1	1	yes	characteristic
150	#486	C₄×C₁₃	1	1	yes	characteristic
151	C₂×#108	—	2	2	no	—
152	#423	C₂²×C₁₃	1	1	yes	characteristic
153	#423	C₂²×C₁₃	1	1	yes	characteristic
154	#416	—	2	2	no	—
155	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
158	Q₈⋊C₈	—	2	2	no	—
159	Q₈⋊C₈	—	2	2	no	—
160	D₄⋊C₈	—	2	2	no	—
161	D₄⋊C₈	—	2	2	no	—
162	#322	—	2	2	no	—
163	C₂²×C₁₃	—	2	1	no	—
164	C₂²×C₁₃	—	2	1	no	—
165	C₂²×C₁₃	—	8	1	no	—
166	C₂²×C₁₃	—	8	1	no	—
167	C₂²×C₁₃	—	8	1	no	—
168	C₂²×C₁₃	#390	1	1	yes	characteristic
169	C₂²×C₁₃	#489	1	2	yes	—
170	C₄×C₁₃	—	2	1	no	—
171	C₄×C₁₃	—	2	1	no	—
172	C₄×C₁₃	—	8	1	no	—
173	C₄×C₁₃	—	8	1	no	—
174	C₄×C₁₃	—	8	1	no	—
175	C₄×C₁₃	#417	1	2	yes	—
176	C₄×C₁₃	#443	1	2	yes	—
177	C₄×C₁₃	#444	1	2	yes	—
178	C₄×C₁₃	#496	1	1	yes	characteristic
179	C₄×C₁₃	#497	1	1	yes	characteristic
180	C₂×#41	—	2	1	no	—
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	—	2	1	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	2	1	no	—
185	C₂²×C₈	D₄×C₁₃	1	2	yes	—
186	C₄⋊Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
187	C₄⋊Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
188	C₄⋊Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
189	#127	C₂×C₄×C₁₃	1	1	yes	characteristic
190	#125	—	2	1	no	—
191	#124	—	2	1	no	—
192	C₂²⋊Q₈	—	2	1	no	—
193	C₂²⋊Q₈	—	2	1	no	—
194	#121	—	2	1	no	—
195	#121	—	2	1	no	—
196	C₄×Q₈	—	2	1	no	—
197	C₄×Q₈	—	2	1	no	—
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	#117	—	2	1	no	—
201	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
202	#108	—	4	4	no	—
203	C₄⋊C₈	—	2	1	no	—
204	C₄⋊C₈	—	2	1	no	—
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	2	1	no	—
207	#103	—	2	2	no	—
208	#103	—	2	2	no	—
209	#102	—	2	2	no	—
210	#102	—	2	2	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
214	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
215	C₄×C₈	D₄×C₁₃	1	2	yes	—
216	C₄×C₈	D₄×C₁₃	1	2	yes	—
217	C₄×C₈	D₄×C₁₃	1	2	yes	—
218	C₂×C₁₃	—	2	2	no	—
219	C₂×C₁₃	—	16	1	no	—
220	C₂×C₁₃	#1959	1	1	yes	characteristic
221	C₂×C₁₃	#1974	1	1	yes	characteristic
222	C₂×C₁₃	#1975	1	1	yes	characteristic
223	#41	—	4	2	no	—
224	#41	—	4	2	no	—
225	C₂×Q₈	—	2	1	no	—
226	C₂×Q₈	—	2	1	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
234	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
235	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
236	M₁₆	—	4	2	no	—
237	M₁₆	—	4	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
247	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₂²⋊C₄	—	4	1	no	—
257	C₂²⋊C₄	—	4	1	no	—
258	C₂²⋊C₄	—	4	1	no	—
259	C₄²	—	4	1	no	—
260	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
261	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
264	C₁₃	#9858	1	1	yes	characteristic, direct product factor, Sylow
265	C₂³	—	4	1	no	—
266	C₂³	C₁₃×#124	1	1	yes	characteristic
267	Q₈	—	4	2	no	—
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	8	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	8	1	no	—
280	C₂×C₄	—	8	1	no	—
281	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
282	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
283	C₂×C₄	C₄×D₄×C₁₃	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×#127	1	1	yes	characteristic
287	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
288	C₂×C₄	C₁₃×#137	1	2	yes	—
289	C₈	—	2	4	no	—
290	C₈	—	2	2	no	—
291	C₈	—	2	2	no	—
292	C₈	—	4	2	no	—
293	C₈	—	8	1	no	—
294	C₈	—	8	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	8	1	no	—
298	C₂²	—	8	1	no	—
299	C₂²	—	8	1	no	—
300	C₂²	C₁₃×#390	1	1	yes	characteristic
301	C₂²	C₁₃×#489	1	2	yes	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	8	1	no	—
305	C₄	—	8	1	no	—
306	C₄	—	8	1	no	—
307	C₄	C₁₃×#417	1	2	yes	—
308	C₄	C₁₃×#443	1	2	yes	—
309	C₄	C₁₃×#444	1	2	yes	—
310	C₄	C₁₃×#496	1	1	yes	characteristic
311	C₄	C₁₃×#497	1	1	yes	characteristic
312	C₂	—	2	2	no	—
313	C₂	—	16	1	no	—
314	C₂	C₁₃×#1959	1	1	yes	characteristic
315	C₂	C₁₃×#1974	1	1	yes	characteristic
316	C₂	C₁₃×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	#127
commutator	C₄²	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂²×C₁₃	#390
prime radical	C₁₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#1848	C₂	semidirect
3. C₁₃×#1493	C₂	semidirect
4. C₁₃×#1493	C₂	semidirect
7. C₂×C₁₃×#322	C₂	semidirect
65. #9858	C₁₃	direct (N × H)
108. #1848	C₂×C₁₃	semidirect
109. #1493	C₂×C₁₃	semidirect
110. #1493	C₂×C₁₃	semidirect
113. C₂×#322	C₂×C₁₃	semidirect
264. C₁₃	#9858	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×#3131	2
C₁₃×#1848	2
C₁₃×#1493	2
C₁₃×#1492	2
C₂×C₁₃×#322	2
#9858	13

Subgroups — C13×#9858 #15696197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9858 #15696197