Subgroups — C₁₃×#13248 #15699587

Order 3328.

394 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#510	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2155	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2143	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2141	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1979	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1976	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₁₃×#510	—	2	2	no	—
11	C₁₃×#510	—	2	2	no	—
12	C₂×C₁₃×Q₃₂	C₄	1	2	yes	complemented
13	C₂×C₁₃×SD₃₂	C₄	1	2	yes	complemented
14	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	C₁₃×#421	—	2	1	no	—
19	C₁₃×#419	—	2	1	no	—
20	C₂×C₁₃×#101	—	2	1	no	—
21	C₁₃×#410	—	2	1	no	—
22	C₁₃×#405	C₂²	1	1	yes	characteristic
23	C₁₃×#365	—	2	2	no	—
24	C₁₃×#359	—	2	2	no	—
25	C₁₃×#358	—	2	2	no	—
26	C₁₃×#347	—	2	2	no	—
27	C₂×C₁₃×#41	—	2	1	no	—
28	C₂²×Q₈×C₁₃	—	2	1	no	—
29	C₁₃×#135	—	2	1	no	—
30	C₁₃×#135	—	2	1	no	—
31	C₁₃×#135	—	2	1	no	—
32	C₁₃×#135	—	2	1	no	—
33	C₂×C₁₃×Q₁₆	—	2	1	no	—
34	C₂×C₁₃×Q₁₆	—	2	1	no	—
35	C₂×C₁₃×Q₁₆	—	2	1	no	—
36	C₂×C₁₃×Q₁₆	—	2	1	no	—
37	C₂×C₁₃×Q₁₆	—	2	1	no	—
38	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₃×SD₁₆	—	2	1	no	—
41	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₃×M₁₆	—	2	1	no	—
43	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×#117	—	2	1	no	—
46	C₁₃×#117	D₄	1	1	yes	characteristic
47	C₁₃×Q₃₂	—	2	4	no	—
48	C₁₃×SD₃₂	—	2	4	no	—
49	C₁₃×M₃₂	—	2	1	no	—
50	C₁₃×M₃₂	—	2	1	no	—
51	C₁₃×M₃₂	D₄	1	2	yes	—
52	C₁₃×M₃₂	D₄	1	2	yes	—
53	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
54	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
55	C₁₃×#108	—	2	1	no	—
56	C₁₃×#108	—	2	1	no	—
57	C₁₃×#107	—	2	1	no	—
58	C₁₃×#106	—	2	1	no	—
59	C₁₃×#104	—	4	2	no	—
60	C₁₃×#103	—	4	2	no	—
61	C₁₃×#101	—	4	2	no	—
62	C₁₃×#99	—	4	2	no	—
63	C₁₃×#97	—	2	1	no	—
64	C₄×C₈×C₁₃	—	2	1	no	—
65	#13248	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₁₃×#41	—	4	1	no	—
67	C₁₃×#41	—	4	1	no	—
68	C₁₃×#41	—	4	1	no	—
69	C₁₃×#41	—	4	1	no	—
70	C₂×Q₈×C₁₃	—	2	1	no	—
71	C₂×Q₈×C₁₃	—	2	2	no	—
72	C₂×Q₈×C₁₃	—	4	1	no	—
73	C₂×Q₈×C₁₃	—	4	1	no	—
74	C₂×Q₈×C₁₃	—	4	1	no	—
75	C₂×Q₈×C₁₃	—	4	1	no	—
76	C₂×Q₈×C₁₃	—	4	1	no	—
77	C₂×D₄×C₁₃	—	2	1	no	—
78	C₂×D₄×C₁₃	—	4	1	no	—
79	C₂²×C₄×C₁₃	—	4	1	no	—
80	C₂²×C₄×C₁₃	—	4	1	no	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₁₃×Q₁₆	—	2	2	no	—
83	C₁₃×Q₁₆	—	2	1	no	—
84	C₁₃×Q₁₆	—	2	2	no	—
85	C₁₃×Q₁₆	—	2	1	no	—
86	C₁₃×Q₁₆	—	4	1	no	—
87	C₁₃×Q₁₆	—	4	1	no	—
88	C₁₃×SD₁₆	—	4	1	no	—
89	C₁₃×SD₁₆	—	4	1	no	—
90	C₁₃×D₈	—	2	1	no	—
91	C₁₃×D₈	—	2	1	no	—
92	C₁₃×M₁₆	—	2	1	no	—
93	C₁₃×M₁₆	—	2	1	no	—
94	C₁₃×M₁₆	—	4	1	no	—
95	C₁₃×M₁₆	—	4	1	no	—
96	C₂×C₈×C₁₃	—	2	1	no	—
97	C₂×C₈×C₁₃	—	4	1	no	—
98	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
103	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
104	C₁₃×C₄⋊C₄	—	2	1	no	—
105	C₁₃×C₄⋊C₄	—	4	1	no	—
106	C₁₃×C₂²⋊C₄	—	2	1	no	—
107	C₁₃×C₂²⋊C₄	—	4	1	no	—
108	C₄²×C₁₃	—	2	1	no	—
109	C₄²×C₁₃	—	4	1	no	—
110	C₁₃×C₁₆	—	2	2	no	—
111	C₁₃×C₁₆	C₂²⋊C₄	1	4	yes	—
112	C₂×#510	C₂×C₁₃	1	1	yes	characteristic
113	#2155	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#2143	C₂×C₁₃	1	1	yes	characteristic
115	#2141	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#2110	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#1979	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#1976	C₂×C₁₃	1	1	yes	characteristic
119	C₂³×C₁₃	—	4	1	no	—
120	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
121	Q₈×C₁₃	—	4	1	no	—
122	Q₈×C₁₃	—	4	2	no	—
123	Q₈×C₁₃	—	4	2	no	—
124	Q₈×C₁₃	—	4	1	no	—
125	Q₈×C₁₃	—	8	1	no	—
126	Q₈×C₁₃	—	8	1	no	—
127	D₄×C₁₃	—	4	1	no	—
128	D₄×C₁₃	—	4	1	no	—
129	D₄×C₁₃	—	8	1	no	—
130	D₄×C₁₃	—	8	1	no	—
131	C₂×C₄×C₁₃	—	4	1	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	1	no	—
134	C₂×C₄×C₁₃	—	8	1	no	—
135	C₂×C₄×C₁₃	—	8	1	no	—
136	C₂×C₄×C₁₃	—	8	1	no	—
137	C₂×C₄×C₁₃	—	8	1	no	—
138	C₂×C₄×C₁₃	—	8	1	no	—
139	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
142	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₈×C₁₃	—	4	1	no	—
146	C₈×C₁₃	—	8	1	no	—
147	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
149	C₈×C₁₃	#121	1	1	yes	characteristic
150	C₈×C₁₃	#124	1	1	yes	characteristic
151	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
152	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
153	#510	—	2	2	no	—
154	#510	—	2	2	no	—
155	C₂×Q₃₂	C₄×C₁₃	1	2	yes	complemented
156	C₂×SD₃₂	C₄×C₁₃	1	2	yes	complemented
157	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
159	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
160	#427	C₂²×C₁₃	1	1	yes	characteristic
161	#421	—	2	1	no	—
162	#419	—	2	1	no	—
163	C₂×#101	—	2	1	no	—
164	#410	—	2	1	no	—
165	#405	C₂²×C₁₃	1	1	yes	characteristic
166	#365	—	2	2	no	—
167	#359	—	2	2	no	—
168	#358	—	2	2	no	—
169	#347	—	2	2	no	—
170	C₂²×C₁₃	—	4	1	no	—
171	C₂²×C₁₃	—	8	1	no	—
172	C₂²×C₁₃	—	16	1	no	—
173	C₂²×C₁₃	#442	1	1	yes	characteristic
174	C₂²×C₁₃	#469	1	1	yes	characteristic
175	C₂²×C₁₃	#496	1	1	yes	characteristic
176	C₄×C₁₃	—	8	1	no	—
177	C₄×C₁₃	—	8	1	no	—
178	C₄×C₁₃	—	8	2	no	—
179	C₄×C₁₃	—	16	1	no	—
180	C₄×C₁₃	#390	1	1	yes	characteristic
181	C₄×C₁₃	#442	1	1	yes	characteristic
182	C₄×C₁₃	#469	1	1	yes	characteristic
183	C₄×C₁₃	#496	1	1	yes	characteristic
184	C₂×#41	—	2	1	no	—
185	C₂²×Q₈	—	2	1	no	—
186	#135	—	2	1	no	—
187	#135	—	2	1	no	—
188	#135	—	2	1	no	—
189	#135	—	2	1	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	—	2	1	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	—	2	1	no	—
195	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
196	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
197	C₂×SD₁₆	—	2	1	no	—
198	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
202	#117	—	2	1	no	—
203	#117	D₄×C₁₃	1	1	yes	characteristic
204	Q₃₂	—	2	4	no	—
205	SD₃₂	—	2	4	no	—
206	M₃₂	—	2	1	no	—
207	M₃₂	—	2	1	no	—
208	M₃₂	D₄×C₁₃	1	2	yes	—
209	M₃₂	D₄×C₁₃	1	2	yes	—
210	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
211	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
212	#108	—	2	1	no	—
213	#108	—	2	1	no	—
214	#107	—	2	1	no	—
215	#106	—	2	1	no	—
216	#104	—	4	2	no	—
217	#103	—	4	2	no	—
218	#101	—	4	2	no	—
219	#99	—	4	2	no	—
220	#97	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	C₂×C₁₃	—	2	1	no	—
223	C₂×C₁₃	—	2	1	no	—
224	C₂×C₁₃	—	2	1	no	—
225	C₂×C₁₃	—	16	1	no	—
226	C₂×C₁₃	#1972	1	1	yes	characteristic
227	#41	—	4	1	no	—
228	#41	—	4	1	no	—
229	#41	—	4	1	no	—
230	#41	—	4	1	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
243	Q₁₆	—	2	2	no	—
244	Q₁₆	—	2	1	no	—
245	Q₁₆	—	2	2	no	—
246	Q₁₆	—	2	1	no	—
247	Q₁₆	—	4	1	no	—
248	Q₁₆	—	4	1	no	—
249	SD₁₆	—	4	1	no	—
250	SD₁₆	—	4	1	no	—
251	D₈	—	2	1	no	—
252	D₈	—	2	1	no	—
253	M₁₆	—	2	1	no	—
254	M₁₆	—	2	1	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	—	4	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
263	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
264	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	4	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	4	1	no	—
271	C₁₆	—	2	2	no	—
272	C₁₆	C₁₃×C₂²⋊C₄	1	4	yes	—
273	C₁₃	#13248	1	1	yes	characteristic, direct product factor, Sylow
274	C₂³	—	4	1	no	—
275	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
276	Q₈	—	4	1	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	4	1	no	—
280	Q₈	—	8	1	no	—
281	Q₈	—	8	1	no	—
282	D₄	—	4	1	no	—
283	D₄	—	4	1	no	—
284	D₄	—	8	1	no	—
285	D₄	—	8	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
295	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
296	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
297	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
298	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
299	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
300	C₈	—	4	1	no	—
301	C₈	—	8	1	no	—
302	C₈	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
303	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
304	C₈	C₁₃×#121	1	1	yes	characteristic
305	C₈	C₁₃×#124	1	1	yes	characteristic
306	C₂²	—	4	1	no	—
307	C₂²	—	8	1	no	—
308	C₂²	—	16	1	no	—
309	C₂²	C₁₃×#442	1	1	yes	characteristic
310	C₂²	C₁₃×#469	1	1	yes	characteristic
311	C₂²	C₁₃×#496	1	1	yes	characteristic
312	C₄	—	8	1	no	—
313	C₄	—	8	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	16	1	no	—
316	C₄	C₁₃×#390	1	1	yes	characteristic
317	C₄	C₁₃×#442	1	1	yes	characteristic
318	C₄	C₁₃×#469	1	1	yes	characteristic
319	C₄	C₁₃×#496	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	16	1	no	—
324	C₂	C₁₃×#1972	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#1972
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2155	C₂	semidirect
4. C₁₃×#2141	C₂	semidirect
5. C₁₃×#2110	C₂	semidirect
6. C₁₃×#1979	C₂	semidirect
12. C₂×C₁₃×Q₃₂	C₄	semidirect
13. C₂×C₁₃×SD₃₂	C₄	semidirect
65. #13248	C₁₃	direct (N × H)
113. #2155	C₂×C₁₃	semidirect
115. #2141	C₂×C₁₃	semidirect
116. #2110	C₂×C₁₃	semidirect
117. #1979	C₂×C₁₃	semidirect
155. C₂×Q₃₂	C₄×C₁₃	semidirect
156. C₂×SD₃₂	C₄×C₁₃	semidirect
273. C₁₃	#13248	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×#510	2
C₁₃×#2155	2
C₁₃×#2143	2
C₁₃×#2141	2
C₁₃×#2110	2
C₁₃×#1979	2
C₁₃×#1976	2
#13248	13

Subgroups — C13×#13248 #15699587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13248 #15699587