Subgroups — C₁₃×#33688 #15720027

Order 3328.

564 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3184	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2956	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2238	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2230	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2213	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2190	C₂	1	2	yes	maximal, complemented
7	C₁₃×AGL(1,ℤ/16)	C₂	1	4	yes	maximal, complemented
8	C₁₃×#2165	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#575	C₂²	1	1	yes	characteristic
10	C₁₃×#568	—	2	1	no	—
11	C₁₃×#532	—	2	1	no	—
12	C₁₃×#509	C₂²	1	4	yes	—
13	C₁₃×#498	C₂²	1	1	yes	characteristic, complemented
14	C₁₃×#493	C₂²	1	1	yes	characteristic, complemented
15	C₁₃×#486	—	2	1	no	—
16	C₁₃×#469	—	2	1	no	—
17	C₁₃×#469	C₂²	1	2	yes	—
18	C₁₃×#444	—	2	1	no	—
19	C₁₃×#444	C₂²	1	2	yes	—
20	C₁₃×#443	—	2	1	no	—
21	C₁₃×#443	C₂²	1	2	yes	—
22	C₁₃×#442	—	2	1	no	—
23	C₁₃×#442	C₂²	1	2	yes	complemented
24	C₁₃×#430	C₂²	1	1	yes	characteristic
25	C₂×C₁₃×#104	—	2	1	no	—
26	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
27	C₁₃×#365	C₂²	1	4	yes	complemented
28	C₁₃×#364	C₂²	1	2	yes	complemented
29	C₁₃×#361	C₂²	1	4	yes	complemented
30	C₁₃×#360	C₂²	1	4	yes	—
31	C₁₃×#349	C₂²	1	2	yes	—
32	C₁₃×#347	C₂²	1	2	yes	complemented
33	C₂×C₁₃×#41	—	2	1	no	—
34	C₂×C₁₃×#41	—	4	1	no	—
35	C₁₃×#137	—	2	2	no	—
36	C₁₃×AGL(1,ℤ/8)	—	2	2	no	—
37	C₁₃×#135	—	2	1	no	—
38	C₁₃×#135	C₂³	1	2	yes	—
39	C₂×C₁₃×D₈	—	2	1	no	—
40	C₂×C₁₃×D₈	C₂³	1	2	yes	—
41	C₁₃×#131	—	2	2	no	—
42	C₁₃×#131	—	2	2	no	—
43	C₁₃×#131	—	4	2	no	—
44	C₂×C₁₃×M₁₆	—	2	1	no	—
45	C₂×C₁₃×M₁₆	—	4	2	no	—
46	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
47	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
48	C₁₃×C₄⋊Q₈	—	2	1	no	—
49	C₁₃×#127	—	2	1	no	—
50	C₁₃×#124	—	4	1	no	—
51	C₁₃×#121	—	2	2	no	—
52	C₁₃×#121	—	4	1	no	—
53	C₄×D₄×C₁₃	—	2	2	no	—
54	C₄×D₄×C₁₃	—	4	1	no	—
55	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
56	C₁₃×SD₃₂	—	2	4	no	—
57	C₁₃×D₁₆	—	2	4	no	—
58	C₁₃×M₃₂	C₂³	1	4	yes	—
59	C₁₃×#108	—	2	1	no	—
60	C₁₃×#108	C₂³	1	2	yes	—
61	C₁₃×#106	—	2	1	no	—
62	C₁₃×#106	C₂³	1	2	yes	—
63	C₁₃×#104	—	2	2	no	—
64	C₁₃×#104	—	2	2	no	—
65	C₁₃×#102	—	2	2	no	—
66	C₁₃×#102	—	2	2	no	—
67	C₁₃×#97	D₄	1	1	yes	characteristic
68	C₁₃×#97	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	#33688	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₁₃×#41	—	2	2	no	—
73	C₁₃×#41	—	2	2	no	—
74	C₁₃×#41	—	4	1	no	—
75	C₁₃×#41	—	4	1	no	—
76	C₁₃×#41	—	8	4	no	—
77	C₂×Q₈×C₁₃	—	2	1	no	—
78	C₂×Q₈×C₁₃	—	4	1	no	—
79	C₂×D₄×C₁₃	—	2	1	no	—
80	C₂×D₄×C₁₃	—	2	2	no	—
81	C₂×D₄×C₁₃	—	4	1	no	—
82	C₂×D₄×C₁₃	—	4	1	no	—
83	C₂×D₄×C₁₃	—	4	1	no	—
84	C₂×D₄×C₁₃	—	4	2	no	—
85	C₂²×C₄×C₁₃	—	2	1	no	—
86	C₂²×C₄×C₁₃	—	4	1	no	—
87	C₂²×C₄×C₁₃	—	4	2	no	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₁₃×Q₁₆	—	2	1	no	—
90	C₁₃×Q₁₆	#41	1	2	yes	complemented
91	C₁₃×SD₁₆	—	2	2	no	—
92	C₁₃×SD₁₆	—	2	2	no	—
93	C₁₃×D₈	—	2	1	no	—
94	C₁₃×D₈	—	2	4	no	—
95	C₁₃×D₈	—	4	2	no	—
96	C₁₃×D₈	#41	1	2	yes	complemented
97	C₁₃×M₁₆	—	2	2	no	—
98	C₁₃×M₁₆	—	2	2	no	—
99	C₁₃×M₁₆	—	2	2	no	—
100	C₁₃×M₁₆	—	4	2	no	—
101	C₁₃×M₁₆	—	4	1	no	—
102	C₁₃×M₁₆	—	4	2	no	—
103	C₂×C₈×C₁₃	—	4	2	no	—
104	C₂×C₈×C₁₃	—	4	1	no	—
105	C₂×C₈×C₁₃	—	4	2	no	—
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
110	C₁₃×C₄⋊C₄	—	2	2	no	—
111	C₁₃×C₄⋊C₄	—	4	1	no	—
112	C₁₃×C₂²⋊C₄	—	4	2	no	—
113	C₁₃×C₂²⋊C₄	—	4	2	no	—
114	C₄²×C₁₃	—	2	1	no	—
115	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₁₃×C₁₆	—	2	4	no	—
118	#3184	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#2956	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#2238	C₂×C₁₃	1	2	yes	complemented
121	#2230	C₂×C₁₃	1	2	yes	complemented
122	#2213	C₂×C₁₃	1	2	yes	complemented
123	#2190	C₂×C₁₃	1	2	yes	complemented
124	AGL(1,ℤ/16)	C₂×C₁₃	1	4	yes	complemented
125	#2165	C₂×C₁₃	1	1	yes	characteristic, complemented
126	C₂³×C₁₃	—	4	1	no	—
127	C₂³×C₁₃	—	4	2	no	—
128	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
129	Q₈×C₁₃	—	2	2	no	—
130	Q₈×C₁₃	—	4	1	no	—
131	Q₈×C₁₃	—	8	2	no	—
132	D₄×C₁₃	—	2	2	no	—
133	D₄×C₁₃	—	4	1	no	—
134	D₄×C₁₃	—	4	2	no	—
135	D₄×C₁₃	—	4	2	no	—
136	D₄×C₁₃	—	4	2	no	—
137	D₄×C₁₃	—	4	2	no	—
138	D₄×C₁₃	—	8	2	no	—
139	D₄×C₁₃	—	8	2	no	—
140	D₄×C₁₃	—	8	4	no	—
141	C₂×C₄×C₁₃	—	2	1	no	—
142	C₂×C₄×C₁₃	—	2	1	no	—
143	C₂×C₄×C₁₃	—	2	1	no	—
144	C₂×C₄×C₁₃	—	2	1	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	4	1	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	—	8	4	no	—
151	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₁₃	—	2	1	no	—
154	C₈×C₁₃	—	2	1	no	—
155	C₈×C₁₃	—	4	2	no	—
156	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
157	C₈×C₁₃	C₂×#41	1	2	yes	—
158	#575	C₂²×C₁₃	1	1	yes	characteristic
159	#568	—	2	1	no	—
160	#532	—	2	1	no	—
161	#509	C₂²×C₁₃	1	4	yes	—
162	#498	C₂²×C₁₃	1	1	yes	characteristic, complemented
163	#493	C₂²×C₁₃	1	1	yes	characteristic, complemented
164	#486	—	2	1	no	—
165	#469	—	2	1	no	—
166	#469	C₂²×C₁₃	1	2	yes	—
167	#444	—	2	1	no	—
168	#444	C₂²×C₁₃	1	2	yes	—
169	#443	—	2	1	no	—
170	#443	C₂²×C₁₃	1	2	yes	—
171	#442	—	2	1	no	—
172	#442	C₂²×C₁₃	1	2	yes	complemented
173	#430	C₂²×C₁₃	1	1	yes	characteristic
174	C₂×#104	—	2	1	no	—
175	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
176	#365	C₂²×C₁₃	1	4	yes	complemented
177	#364	C₂²×C₁₃	1	2	yes	complemented
178	#361	C₂²×C₁₃	1	4	yes	complemented
179	#360	C₂²×C₁₃	1	4	yes	—
180	#349	C₂²×C₁₃	1	2	yes	—
181	#347	C₂²×C₁₃	1	2	yes	complemented
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	—	4	1	no	—
184	C₂²×C₁₃	—	4	2	no	—
185	C₂²×C₁₃	—	8	2	no	—
186	C₂²×C₁₃	—	8	2	no	—
187	C₂²×C₁₃	—	8	2	no	—
188	C₂²×C₁₃	—	16	1	no	—
189	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	—	2	1	no	—
192	C₄×C₁₃	—	4	1	no	—
193	C₄×C₁₃	—	4	2	no	—
194	C₄×C₁₃	—	8	2	no	—
195	C₄×C₁₃	#525	1	1	yes	characteristic
196	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
197	C₂×#41	—	2	1	no	—
198	C₂×#41	—	4	1	no	—
199	#137	—	2	2	no	—
200	AGL(1,ℤ/8)	—	2	2	no	—
201	#135	—	2	1	no	—
202	#135	C₂³×C₁₃	1	2	yes	—
203	C₂×D₈	—	2	1	no	—
204	C₂×D₈	C₂³×C₁₃	1	2	yes	—
205	#131	—	2	2	no	—
206	#131	—	2	2	no	—
207	#131	—	4	2	no	—
208	C₂×M₁₆	—	2	1	no	—
209	C₂×M₁₆	—	4	2	no	—
210	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
211	C₂×M₁₆	C₂³×C₁₃	1	1	yes	characteristic
212	C₄⋊Q₈	—	2	1	no	—
213	#127	—	2	1	no	—
214	#124	—	4	1	no	—
215	#121	—	2	2	no	—
216	#121	—	4	1	no	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	—	4	1	no	—
219	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
220	SD₃₂	—	2	4	no	—
221	D₁₆	—	2	4	no	—
222	M₃₂	C₂³×C₁₃	1	4	yes	—
223	#108	—	2	1	no	—
224	#108	C₂³×C₁₃	1	2	yes	—
225	#106	—	2	1	no	—
226	#106	C₂³×C₁₃	1	2	yes	—
227	#104	—	2	2	no	—
228	#104	—	2	2	no	—
229	#102	—	2	2	no	—
230	#102	—	2	2	no	—
231	#97	D₄×C₁₃	1	1	yes	characteristic
232	#97	C₂³×C₁₃	1	1	yes	characteristic
233	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
234	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
235	C₂×C₁₃	—	2	1	no	—
236	C₂×C₁₃	—	4	1	no	—
237	C₂×C₁₃	—	8	2	no	—
238	C₂×C₁₃	—	16	2	no	—
239	C₂×C₁₃	#3109	1	1	yes	characteristic
240	#41	—	2	2	no	—
241	#41	—	2	2	no	—
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	#41	—	8	4	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
257	Q₁₆	—	2	1	no	—
258	Q₁₆	C₁₃×#41	1	2	yes	complemented
259	SD₁₆	—	2	2	no	—
260	SD₁₆	—	2	2	no	—
261	D₈	—	2	1	no	—
262	D₈	—	2	4	no	—
263	D₈	—	4	2	no	—
264	D₈	C₁₃×#41	1	2	yes	complemented
265	M₁₆	—	2	2	no	—
266	M₁₆	—	2	2	no	—
267	M₁₆	—	2	2	no	—
268	M₁₆	—	4	2	no	—
269	M₁₆	—	4	1	no	—
270	M₁₆	—	4	2	no	—
271	C₂×C₈	—	4	2	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	2	no	—
274	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
277	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₄²	—	2	1	no	—
283	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
284	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
285	C₁₆	—	2	4	no	—
286	C₁₃	#33688	1	1	yes	characteristic, direct product factor, Sylow
287	C₂³	—	4	1	no	—
288	C₂³	—	4	2	no	—
289	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
290	Q₈	—	2	2	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	8	2	no	—
293	D₄	—	2	2	no	—
294	D₄	—	4	1	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	8	2	no	—
300	D₄	—	8	2	no	—
301	D₄	—	8	4	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	4	2	no	—
317	C₈	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
318	C₈	C₂×C₁₃×#41	1	2	yes	—
319	C₂²	—	2	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	16	1	no	—
326	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	2	no	—
331	C₄	—	8	2	no	—
332	C₄	C₁₃×#525	1	1	yes	characteristic
333	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
334	C₂	—	2	1	no	—
335	C₂	—	4	1	no	—
336	C₂	—	8	2	no	—
337	C₂	—	16	2	no	—
338	C₂	C₁₃×#3109	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₄×C₁₃×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3184	C₂	semidirect
2. C₁₃×#2956	C₂	semidirect
3. C₁₃×#2238	C₂	semidirect
4. C₁₃×#2230	C₂	semidirect
5. C₁₃×#2213	C₂	semidirect
6. C₁₃×#2190	C₂	semidirect
7. C₁₃×AGL(1,ℤ/16)	C₂	semidirect
8. C₁₃×#2165	C₂	semidirect
13. C₁₃×#498	C₂²	semidirect
14. C₁₃×#493	C₂²	semidirect
23. C₁₃×#442	C₂²	semidirect
27. C₁₃×#365	C₂²	semidirect
28. C₁₃×#364	C₂²	semidirect
29. C₁₃×#361	C₂²	semidirect
32. C₁₃×#347	C₂²	semidirect
71. #33688	C₁₃	direct (N × H)
90. C₁₃×Q₁₆	#41	semidirect
96. C₁₃×D₈	#41	semidirect
118. #3184	C₂×C₁₃	semidirect
119. #2956	C₂×C₁₃	semidirect
120. #2238	C₂×C₁₃	semidirect
121. #2230	C₂×C₁₃	semidirect
122. #2213	C₂×C₁₃	semidirect
123. #2190	C₂×C₁₃	semidirect
124. AGL(1,ℤ/16)	C₂×C₁₃	semidirect
125. #2165	C₂×C₁₃	semidirect
162. #498	C₂²×C₁₃	semidirect
163. #493	C₂²×C₁₃	semidirect
172. #442	C₂²×C₁₃	semidirect
176. #365	C₂²×C₁₃	semidirect
177. #364	C₂²×C₁₃	semidirect
178. #361	C₂²×C₁₃	semidirect
181. #347	C₂²×C₁₃	semidirect
258. Q₁₆	C₁₃×#41	semidirect
264. D₈	C₁₃×#41	semidirect
286. C₁₃	#33688	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×#3184	2
C₁₃×#2956	2
C₁₃×#2238	2
C₁₃×#2230	2
C₁₃×#2213	2
C₁₃×#2190	2
C₁₃×AGL(1,ℤ/16)	2
C₁₃×#2165	2
#33688	13

Subgroups — C13×#33688 #15720027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33688 #15720027