Subgroups — C₁₃×#33586 #15719925

Order 3328.

716 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#509	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#3156	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2948	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2210	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2208	C₂	1	4	yes	maximal, complemented
7	C₁₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2141	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#137	—	2	1	no	—
11	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
14	C₁₃×#536	—	2	1	no	—
15	C₁₃×#518	—	2	1	no	—
16	C₁₃×#509	—	2	4	no	—
17	C₂×C₁₃×SD₃₂	—	2	4	no	—
18	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	—
19	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
20	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
21	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
22	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
23	C₁₃×#497	—	2	2	no	—
24	C₁₃×#495	C₂²	1	2	yes	complemented
25	C₁₃×#494	C₂²	1	2	yes	complemented
26	C₁₃×#442	—	2	2	no	—
27	C₄×C₁₃×D₈	C₂²	1	4	yes	—
28	C₁₃×#427	C₂²	1	1	yes	characteristic
29	C₁₃×#418	—	2	1	no	—
30	C₁₃×#405	C₂²	1	1	yes	characteristic
31	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
32	C₁₃×#358	C₂²	1	2	yes	complemented
33	C₁₃×#358	C₂²	1	4	yes	complemented
34	C₁₃×#357	C₂²	1	2	yes	complemented
35	C₂×C₁₃×#41	—	2	1	no	—
36	C₂²×Q₈×C₁₃	—	2	1	no	—
37	C₂²×D₄×C₁₃	—	2	1	no	—
38	C₁₃×#137	—	4	4	no	—
39	C₁₃×#135	—	2	2	no	—
40	C₁₃×#135	—	2	2	no	—
41	C₂×C₁₃×Q₁₆	—	2	2	no	—
42	C₂×C₁₃×Q₁₆	—	2	2	no	—
43	C₂×C₁₃×Q₁₆	—	2	1	no	—
44	C₂×C₁₃×Q₁₆	—	2	2	no	—
45	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
47	C₂×C₁₃×SD₁₆	—	2	2	no	—
48	C₂×C₁₃×SD₁₆	—	2	1	no	—
49	C₂×C₁₃×D₈	—	2	1	no	—
50	C₂×C₁₃×D₈	—	2	2	no	—
51	C₂×C₁₃×D₈	D₄	1	4	yes	—
52	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
53	C₂×C₁₃×D₈	C₂³	1	2	yes	—
54	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₁₃×C₄⋊Q₈	—	2	2	no	—
57	C₁₃×#124	—	2	2	no	—
58	C₁₃×#123	—	4	2	no	—
59	C₁₃×C₂²⋊Q₈	—	4	2	no	—
60	C₄×D₄×C₁₃	—	2	4	no	—
61	C₄×D₄×C₁₃	—	4	2	no	—
62	C₁₃×#117	—	2	1	no	—
63	C₁₃×#117	D₄	1	1	yes	characteristic
64	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
65	C₁₃×SD₃₂	—	2	8	no	—
66	C₁₃×SD₃₂	—	2	4	no	—
67	C₁₃×D₁₆	—	2	4	no	—
68	C₁₃×M₃₂	—	2	2	no	—
69	C₂×C₁₃×C₁₆	—	2	2	no	—
70	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
71	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
72	C₁₃×#107	C₂³	1	2	yes	—
73	C₁₃×#106	—	2	1	no	—
74	C₁₃×#102	—	2	4	no	—
75	C₁₃×#97	D₄	1	2	yes	—
76	C₄×C₈×C₁₃	C₂³	1	2	yes	—
77	#33586	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂⁴×C₁₃	—	4	1	no	—
79	C₁₃×#41	—	4	2	no	—
80	C₁₃×#41	—	4	2	no	—
81	C₂×Q₈×C₁₃	—	2	2	no	—
82	C₂×Q₈×C₁₃	—	2	1	no	—
83	C₂×Q₈×C₁₃	—	4	1	no	—
84	C₂×Q₈×C₁₃	—	4	2	no	—
85	C₂×Q₈×C₁₃	—	4	2	no	—
86	C₂×D₄×C₁₃	—	2	4	no	—
87	C₂×D₄×C₁₃	—	2	2	no	—
88	C₂×D₄×C₁₃	—	2	1	no	—
89	C₂×D₄×C₁₃	—	4	1	no	—
90	C₂×D₄×C₁₃	—	4	1	no	—
91	C₂×D₄×C₁₃	—	4	2	no	—
92	C₂²×C₄×C₁₃	—	4	1	no	—
93	C₂²×C₄×C₁₃	—	4	1	no	—
94	C₂²×C₄×C₁₃	—	4	4	no	—
95	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₁₃×Q₁₆	—	2	2	no	—
97	C₁₃×Q₁₆	—	2	1	no	—
98	C₁₃×Q₁₆	—	2	2	no	—
99	C₁₃×Q₁₆	—	2	1	no	—
100	C₁₃×Q₁₆	—	4	4	no	—
101	C₁₃×Q₁₆	—	4	2	no	—
102	C₁₃×SD₁₆	—	4	4	no	—
103	C₁₃×SD₁₆	—	4	2	no	—
104	C₁₃×D₈	—	2	4	no	—
105	C₁₃×D₈	—	2	1	no	—
106	C₁₃×D₈	—	2	2	no	—
107	C₁₃×D₈	—	2	1	no	—
108	C₁₃×D₈	C₂×D₄	1	4	yes	—
109	C₁₃×M₁₆	—	2	2	no	—
110	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₁₃	#41	1	2	yes	—
114	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
115	C₁₃×C₄⋊C₄	—	2	2	no	—
116	C₁₃×C₄⋊C₄	—	4	4	no	—
117	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
118	C₁₃×C₂²⋊C₄	—	4	1	no	—
119	C₁₃×C₂²⋊C₄	—	4	4	no	—
120	C₁₃×C₂²⋊C₄	—	4	4	no	—
121	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
122	C₄²×C₁₃	—	4	1	no	—
123	C₄²×C₁₃	C₂×D₄	1	2	yes	—
124	C₁₃×C₁₆	—	2	2	no	—
125	C₁₃×C₁₆	—	2	2	no	—
126	C₂×#509	C₂×C₁₃	1	1	yes	characteristic
127	C₂²×SD₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
128	#3156	C₂×C₁₃	1	1	yes	characteristic, complemented
129	#2948	C₂×C₁₃	1	1	yes	characteristic, complemented
130	#2210	C₂×C₁₃	1	4	yes	complemented
131	#2208	C₂×C₁₃	1	4	yes	complemented
132	#2151	C₂×C₁₃	1	1	yes	characteristic, complemented
133	#2141	C₂×C₁₃	1	1	yes	characteristic, complemented
134	#2138	C₂×C₁₃	1	1	yes	characteristic, complemented
135	C₂³×C₁₃	—	4	2	no	—
136	C₂³×C₁₃	—	4	4	no	—
137	C₂³×C₁₃	—	4	1	no	—
138	C₂³×C₁₃	—	16	2	no	—
139	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
140	Q₈×C₁₃	—	4	1	no	—
141	Q₈×C₁₃	—	4	2	no	—
142	Q₈×C₁₃	—	4	2	no	—
143	Q₈×C₁₃	—	4	1	no	—
144	Q₈×C₁₃	—	8	2	no	—
145	D₄×C₁₃	—	2	4	no	—
146	D₄×C₁₃	—	4	1	no	—
147	D₄×C₁₃	—	4	2	no	—
148	D₄×C₁₃	—	4	1	no	—
149	D₄×C₁₃	—	4	4	no	—
150	D₄×C₁₃	—	8	2	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	4	2	no	—
153	C₂×C₄×C₁₃	—	4	1	no	—
154	C₂×C₄×C₁₃	—	8	2	no	—
155	C₂×C₄×C₁₃	—	8	2	no	—
156	C₂×C₄×C₁₃	—	16	4	no	—
157	C₂×C₄×C₁₃	#121	1	2	yes	—
158	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
159	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
161	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
162	C₈×C₁₃	—	2	2	no	—
163	C₈×C₁₃	#121	1	2	yes	—
164	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
165	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
166	C₂×#137	—	2	1	no	—
167	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
168	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
169	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
170	#536	—	2	1	no	—
171	#518	—	2	1	no	—
172	#509	—	2	4	no	—
173	C₂×SD₃₂	—	2	4	no	—
174	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	—
175	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
176	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
177	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
178	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
179	#497	—	2	2	no	—
180	#495	C₂²×C₁₃	1	2	yes	complemented
181	#494	C₂²×C₁₃	1	2	yes	complemented
182	#442	—	2	2	no	—
183	C₄×D₈	C₂²×C₁₃	1	4	yes	—
184	#427	C₂²×C₁₃	1	1	yes	characteristic
185	#418	—	2	1	no	—
186	#405	C₂²×C₁₃	1	1	yes	characteristic
187	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	complemented
188	#358	C₂²×C₁₃	1	2	yes	complemented
189	#358	C₂²×C₁₃	1	4	yes	complemented
190	#357	C₂²×C₁₃	1	2	yes	complemented
191	C₂²×C₁₃	—	2	2	no	—
192	C₂²×C₁₃	—	4	4	no	—
193	C₂²×C₁₃	—	8	4	no	—
194	C₂²×C₁₃	—	8	2	no	—
195	C₂²×C₁₃	—	8	1	no	—
196	C₂²×C₁₃	—	16	4	no	—
197	C₂²×C₁₃	#459	1	2	yes	—
198	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
199	C₄×C₁₃	—	2	4	no	—
200	C₄×C₁₃	—	8	2	no	—
201	C₄×C₁₃	—	8	1	no	—
202	C₄×C₁₃	—	8	2	no	—
203	C₄×C₁₃	#459	1	2	yes	—
204	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
205	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₂×#41	—	2	1	no	—
207	C₂²×Q₈	—	2	1	no	—
208	C₂²×D₄	—	2	1	no	—
209	#137	—	4	4	no	—
210	#135	—	2	2	no	—
211	#135	—	2	2	no	—
212	C₂×Q₁₆	—	2	2	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	2	2	no	—
215	C₂×Q₁₆	—	2	2	no	—
216	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
217	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
218	C₂×SD₁₆	—	2	1	no	—
219	C₂×SD₁₆	—	2	2	no	—
220	C₂×D₈	—	2	1	no	—
221	C₂×D₈	—	2	2	no	—
222	C₂×D₈	D₄×C₁₃	1	4	yes	—
223	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
224	C₂×D₈	C₂³×C₁₃	1	2	yes	—
225	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
226	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
227	C₄⋊Q₈	—	2	2	no	—
228	#124	—	2	2	no	—
229	#123	—	4	2	no	—
230	C₂²⋊Q₈	—	4	2	no	—
231	C₄×D₄	—	2	4	no	—
232	C₄×D₄	—	4	2	no	—
233	#117	—	2	1	no	—
234	#117	D₄×C₁₃	1	1	yes	characteristic
235	C₂×C₂²⋊C₄	—	4	2	no	—
236	SD₃₂	—	2	8	no	—
237	SD₃₂	—	2	4	no	—
238	D₁₆	—	2	4	no	—
239	M₃₂	—	2	2	no	—
240	C₂×C₁₆	—	2	2	no	—
241	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
242	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
243	#107	C₂³×C₁₃	1	2	yes	—
244	#106	—	2	1	no	—
245	#102	—	2	4	no	—
246	#97	D₄×C₁₃	1	2	yes	—
247	C₄×C₈	C₂³×C₁₃	1	2	yes	—
248	C₂×C₁₃	—	2	2	no	—
249	C₂×C₁₃	—	8	4	no	—
250	C₂×C₁₃	—	16	1	no	—
251	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
252	C₂×C₁₃	#3416	1	2	yes	—
253	C₂⁴	—	4	1	no	—
254	#41	—	4	2	no	—
255	#41	—	4	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	4	2	no	—
260	C₂×Q₈	—	4	2	no	—
261	C₂×D₄	—	2	4	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	4	4	no	—
270	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
271	Q₁₆	—	2	2	no	—
272	Q₁₆	—	2	1	no	—
273	Q₁₆	—	2	2	no	—
274	Q₁₆	—	2	1	no	—
275	Q₁₆	—	4	4	no	—
276	Q₁₆	—	4	2	no	—
277	SD₁₆	—	4	4	no	—
278	SD₁₆	—	4	2	no	—
279	D₈	—	2	4	no	—
280	D₈	—	2	1	no	—
281	D₈	—	2	2	no	—
282	D₈	—	2	1	no	—
283	D₈	C₂×D₄×C₁₃	1	4	yes	—
284	M₁₆	—	2	2	no	—
285	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
286	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
287	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
288	C₂×C₈	C₁₃×#41	1	2	yes	—
289	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	4	4	no	—
292	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
293	C₂²⋊C₄	—	4	1	no	—
294	C₂²⋊C₄	—	4	4	no	—
295	C₂²⋊C₄	—	4	4	no	—
296	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
297	C₄²	—	4	1	no	—
298	C₄²	C₂×D₄×C₁₃	1	2	yes	—
299	C₁₆	—	2	2	no	—
300	C₁₆	—	2	2	no	—
301	C₁₃	#33586	1	1	yes	characteristic, direct product factor, Sylow
302	C₂³	—	4	2	no	—
303	C₂³	—	4	4	no	—
304	C₂³	—	4	1	no	—
305	C₂³	—	16	2	no	—
306	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
307	Q₈	—	4	1	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	8	2	no	—
312	D₄	—	2	4	no	—
313	D₄	—	4	4	no	—
314	D₄	—	4	1	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	1	no	—
317	D₄	—	8	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	8	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	16	4	no	—
324	C₂×C₄	C₁₃×#121	1	2	yes	—
325	C₂×C₄	C₂×C₁₃×D₈	1	4	yes	—
326	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
327	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
328	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	C₁₃×#121	1	2	yes	—
331	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
332	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
333	C₂²	—	2	2	no	—
334	C₂²	—	4	4	no	—
335	C₂²	—	8	4	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	16	4	no	—
339	C₂²	C₁₃×#459	1	2	yes	—
340	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
341	C₄	—	2	4	no	—
342	C₄	—	8	1	no	—
343	C₄	—	8	2	no	—
344	C₄	—	8	2	no	—
345	C₄	C₁₃×#459	1	2	yes	—
346	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
347	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
348	C₂	—	2	2	no	—
349	C₂	—	8	4	no	—
350	C₂	—	16	1	no	—
351	C₂	C₂×C₁₃×#459	1	1	yes	characteristic
352	C₂	C₁₃×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₃×SD₃₂	C₂	semidirect
3. C₁₃×#3156	C₂	semidirect
4. C₁₃×#2948	C₂	semidirect
5. C₁₃×#2210	C₂	semidirect
6. C₁₃×#2208	C₂	semidirect
7. C₁₃×#2151	C₂	semidirect
8. C₁₃×#2141	C₂	semidirect
9. C₁₃×#2138	C₂	semidirect
24. C₁₃×#495	C₂²	semidirect
25. C₁₃×#494	C₂²	semidirect
31. C₁₃×C₄⋊C₁₆	C₂²	semidirect
32. C₁₃×#358	C₂²	semidirect
33. C₁₃×#358	C₂²	semidirect
34. C₁₃×#357	C₂²	semidirect
77. #33586	C₁₃	direct (N × H)
127. C₂²×SD₃₂	C₂×C₁₃	semidirect
128. #3156	C₂×C₁₃	semidirect
129. #2948	C₂×C₁₃	semidirect
130. #2210	C₂×C₁₃	semidirect
131. #2208	C₂×C₁₃	semidirect
132. #2151	C₂×C₁₃	semidirect
133. #2141	C₂×C₁₃	semidirect
134. #2138	C₂×C₁₃	semidirect
180. #495	C₂²×C₁₃	semidirect
181. #494	C₂²×C₁₃	semidirect
187. C₄⋊C₁₆	C₂²×C₁₃	semidirect
188. #358	C₂²×C₁₃	semidirect
189. #358	C₂²×C₁₃	semidirect
190. #357	C₂²×C₁₃	semidirect
301. C₁₃	#33586	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×#509	2
C₂²×C₁₃×SD₃₂	2
C₁₃×#3156	2
C₁₃×#2948	2
C₁₃×#2210	2
C₁₃×#2208	2
C₁₃×#2151	2
C₁₃×#2141	2
C₁₃×#2138	2
#33586	13

Subgroups — C13×#33586 #15719925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33586 #15719925