Subgroups — C₁₃×#33776 #15720115

Order 3328.

592 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2248	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2211	C₂	1	4	yes	maximal, complemented
8	C₁₃×#2189	C₂	1	4	yes	maximal, complemented
9	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#135	—	2	1	no	—
11	C₂×C₁₃×#135	C₂²	1	2	yes	—
12	C₁₃×#533	—	2	1	no	—
13	C₁₃×#532	—	2	1	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	4	yes	—
15	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	—
16	C₁₃×#500	C₂²	1	1	yes	characteristic
17	C₁₃×#495	—	2	1	no	—
18	C₁₃×#495	C₂²	1	2	yes	—
19	C₁₃×#494	C₂²	1	1	yes	characteristic
20	C₁₃×#494	C₂²	1	1	yes	characteristic
21	C₁₃×#493	C₂²	1	1	yes	characteristic
22	C₁₃×#492	—	2	1	no	—
23	C₁₃×#487	—	2	1	no	—
24	C₁₃×#467	—	2	1	no	—
25	C₁₃×#467	C₂²	1	2	yes	—
26	C₄×C₁₃×Q₁₆	—	2	1	no	—
27	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₁₃×#358	C₂²	1	4	yes	—
31	C₁₃×#357	C₂²	1	4	yes	—
32	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
34	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
35	C₂×C₁₃×#41	—	2	2	no	—
36	C₁₃×#135	—	4	4	no	—
37	C₁₃×#135	—	4	4	no	—
38	C₂×C₁₃×Q₁₆	—	2	1	no	—
39	C₂×C₁₃×Q₁₆	—	2	1	no	—
40	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
41	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
42	C₂×C₁₃×SD₁₆	—	2	2	no	—
43	C₂×C₁₃×SD₁₆	—	2	2	no	—
44	C₂×C₁₃×D₈	—	2	1	no	—
45	C₂×C₁₃×D₈	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×C₄⋊Q₈	—	2	1	no	—
49	C₁₃×C₄⋊Q₈	—	2	1	no	—
50	C₁₃×C₄⋊Q₈	—	2	1	no	—
51	C₁₃×#127	—	2	1	no	—
52	C₁₃×#125	—	4	1	no	—
53	C₁₃×#124	—	2	2	no	—
54	C₁₃×C₂²⋊Q₈	—	2	2	no	—
55	C₁₃×C₂²⋊Q₈	—	4	1	no	—
56	C₁₃×#121	—	4	1	no	—
57	C₁₃×#121	—	4	1	no	—
58	C₄×Q₈×C₁₃	—	2	2	no	—
59	C₄×Q₈×C₁₃	—	4	1	no	—
60	C₄×D₄×C₁₃	—	4	1	no	—
61	C₄×D₄×C₁₃	—	4	1	no	—
62	C₁₃×#117	—	4	1	no	—
63	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
64	C₁₃×Q₃₂	—	4	4	no	—
65	C₁₃×SD₃₂	—	4	4	no	—
66	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
67	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
68	C₁₃×#107	—	2	1	no	—
69	C₁₃×#107	C₂³	1	2	yes	—
70	C₁₃×#103	—	2	2	no	—
71	C₁₃×#103	—	2	2	no	—
72	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
73	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	#33776	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₁₃×#41	—	4	4	no	—
78	C₁₃×#41	—	4	4	no	—
79	C₂×Q₈×C₁₃	—	2	2	no	—
80	C₂×Q₈×C₁₃	—	2	2	no	—
81	C₂×Q₈×C₁₃	—	4	1	no	—
82	C₂×D₄×C₁₃	—	2	2	no	—
83	C₂×D₄×C₁₃	—	4	2	no	—
84	C₂×D₄×C₁₃	—	4	1	no	—
85	C₂×D₄×C₁₃	—	4	1	no	—
86	C₂²×C₄×C₁₃	—	4	2	no	—
87	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₁₃×Q₁₆	—	2	4	no	—
91	C₁₃×Q₁₆	—	2	4	no	—
92	C₁₃×Q₁₆	—	4	2	no	—
93	C₁₃×Q₁₆	—	4	2	no	—
94	C₁₃×SD₁₆	—	4	4	no	—
95	C₁₃×SD₁₆	—	4	4	no	—
96	C₁₃×D₈	—	2	4	no	—
97	C₁₃×D₈	—	4	2	no	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	4	1	no	—
106	C₁₃×C₄⋊C₄	—	4	1	no	—
107	C₁₃×C₄⋊C₄	—	4	1	no	—
108	C₁₃×C₄⋊C₄	—	4	1	no	—
109	C₁₃×C₄⋊C₄	—	4	2	no	—
110	C₁₃×C₄⋊C₄	—	4	2	no	—
111	C₁₃×C₄⋊C₄	—	4	1	no	—
112	C₁₃×C₂²⋊C₄	—	4	2	no	—
113	C₁₃×C₂²⋊C₄	—	4	2	no	—
114	C₁₃×C₂²⋊C₄	—	4	2	no	—
115	C₄²×C₁₃	—	4	2	no	—
116	C₄²×C₁₃	D₈	1	1	yes	characteristic
117	C₄²×C₁₃	D₈	1	1	yes	characteristic
118	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₁₃×C₁₆	—	2	2	no	—
121	C₁₃×C₁₆	—	2	2	no	—
122	#3170	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#3147	C₂×C₁₃	1	1	yes	characteristic
124	#2248	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#2247	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#2241	C₂×C₁₃	1	1	yes	characteristic, complemented
127	#2240	C₂×C₁₃	1	1	yes	characteristic, complemented
128	#2211	C₂×C₁₃	1	4	yes	complemented
129	#2189	C₂×C₁₃	1	4	yes	complemented
130	#2162	C₂×C₁₃	1	1	yes	characteristic, complemented
131	C₂³×C₁₃	—	4	2	no	—
132	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
133	Q₈×C₁₃	—	4	2	no	—
134	Q₈×C₁₃	—	4	2	no	—
135	Q₈×C₁₃	—	4	2	no	—
136	Q₈×C₁₃	—	4	2	no	—
137	Q₈×C₁₃	—	8	2	no	—
138	D₄×C₁₃	—	4	2	no	—
139	D₄×C₁₃	—	4	2	no	—
140	D₄×C₁₃	—	8	4	no	—
141	D₄×C₁₃	—	8	2	no	—
142	D₄×C₁₃	—	8	2	no	—
143	C₂×C₄×C₁₃	—	2	1	no	—
144	C₂×C₄×C₁₃	—	2	1	no	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	4	2	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	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
157	C₈×C₁₃	—	2	1	no	—
158	C₈×C₁₃	—	2	1	no	—
159	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
160	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
161	C₂×#135	—	2	1	no	—
162	C₂×#135	C₂²×C₁₃	1	2	yes	—
163	#533	—	2	1	no	—
164	#532	—	2	1	no	—
165	C₂×Q₃₂	C₂²×C₁₃	1	4	yes	—
166	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	—
167	#500	C₂²×C₁₃	1	1	yes	characteristic
168	#495	—	2	1	no	—
169	#495	C₂²×C₁₃	1	2	yes	—
170	#494	C₂²×C₁₃	1	1	yes	characteristic
171	#494	C₂²×C₁₃	1	1	yes	characteristic
172	#493	C₂²×C₁₃	1	1	yes	characteristic
173	#492	—	2	1	no	—
174	#487	—	2	1	no	—
175	#467	—	2	1	no	—
176	#467	C₂²×C₁₃	1	2	yes	—
177	C₄×Q₁₆	—	2	1	no	—
178	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
179	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
180	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
181	#358	C₂²×C₁₃	1	4	yes	—
182	#357	C₂²×C₁₃	1	4	yes	—
183	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
184	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
185	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
186	C₂²×C₁₃	—	2	1	no	—
187	C₂²×C₁₃	—	2	1	no	—
188	C₂²×C₁₃	—	2	1	no	—
189	C₂²×C₁₃	—	8	2	no	—
190	C₂²×C₁₃	—	8	2	no	—
191	C₂²×C₁₃	—	8	2	no	—
192	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
193	C₄×C₁₃	—	2	1	no	—
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	2	1	no	—
196	C₄×C₁₃	—	8	2	no	—
197	C₄×C₁₃	—	8	2	no	—
198	C₄×C₁₃	—	8	2	no	—
199	C₄×C₁₃	#508	1	2	yes	—
200	C₄×C₁₃	#510	1	2	yes	—
201	C₄×C₁₃	#535	1	1	yes	characteristic
202	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
203	C₂×#41	—	2	2	no	—
204	#135	—	4	4	no	—
205	#135	—	4	4	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	—	2	1	no	—
208	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
209	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
210	C₂×SD₁₆	—	2	2	no	—
211	C₂×SD₁₆	—	2	2	no	—
212	C₂×D₈	—	2	1	no	—
213	C₂×D₈	C₂³×C₁₃	1	2	yes	—
214	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
215	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	2	1	no	—
219	#127	—	2	1	no	—
220	#125	—	4	1	no	—
221	#124	—	2	2	no	—
222	C₂²⋊Q₈	—	2	2	no	—
223	C₂²⋊Q₈	—	4	1	no	—
224	#121	—	4	1	no	—
225	#121	—	4	1	no	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×Q₈	—	4	1	no	—
228	C₄×D₄	—	4	1	no	—
229	C₄×D₄	—	4	1	no	—
230	#117	—	4	1	no	—
231	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
232	Q₃₂	—	4	4	no	—
233	SD₃₂	—	4	4	no	—
234	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
235	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
236	#107	—	2	1	no	—
237	#107	C₂³×C₁₃	1	2	yes	—
238	#103	—	2	2	no	—
239	#103	—	2	2	no	—
240	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
241	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
244	C₂×C₁₃	—	4	1	no	—
245	C₂×C₁₃	—	16	2	no	—
246	C₂×C₁₃	#3205	1	1	yes	characteristic
247	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
248	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
249	#41	—	4	4	no	—
250	#41	—	4	4	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	4	2	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
260	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
262	Q₁₆	—	2	4	no	—
263	Q₁₆	—	2	4	no	—
264	Q₁₆	—	4	2	no	—
265	Q₁₆	—	4	2	no	—
266	SD₁₆	—	4	4	no	—
267	SD₁₆	—	4	4	no	—
268	D₈	—	2	4	no	—
269	D₈	—	4	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
273	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
274	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
275	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₄²	—	4	2	no	—
288	C₄²	C₁₃×D₈	1	1	yes	characteristic
289	C₄²	C₁₃×D₈	1	1	yes	characteristic
290	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
291	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
292	C₁₆	—	2	2	no	—
293	C₁₆	—	2	2	no	—
294	C₁₃	#33776	1	1	yes	characteristic, direct product factor, Sylow
295	C₂³	—	4	2	no	—
296	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
297	Q₈	—	4	2	no	—
298	Q₈	—	4	2	no	—
299	Q₈	—	4	2	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	8	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	8	4	no	—
305	D₄	—	8	2	no	—
306	D₄	—	8	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	8	4	no	—
315	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
316	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
317	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
318	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
319	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
320	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
324	C₈	C₁₃×H(5,𝔽₂)	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	C₁₃×#508	1	2	yes	—
339	C₄	C₁₃×#510	1	2	yes	—
340	C₄	C₁₃×#535	1	1	yes	characteristic
341	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
342	C₂	—	4	1	no	—
343	C₂	—	16	2	no	—
344	C₂	C₁₃×#3205	1	1	yes	characteristic
345	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
346	C₂	C₂×C₁₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3170	C₂	semidirect
3. C₁₃×#2248	C₂	semidirect
4. C₁₃×#2247	C₂	semidirect
5. C₁₃×#2241	C₂	semidirect
6. C₁₃×#2240	C₂	semidirect
7. C₁₃×#2211	C₂	semidirect
8. C₁₃×#2189	C₂	semidirect
9. C₁₃×#2162	C₂	semidirect
76. #33776	C₁₃	direct (N × H)
122. #3170	C₂×C₁₃	semidirect
124. #2248	C₂×C₁₃	semidirect
125. #2247	C₂×C₁₃	semidirect
126. #2241	C₂×C₁₃	semidirect
127. #2240	C₂×C₁₃	semidirect
128. #2211	C₂×C₁₃	semidirect
129. #2189	C₂×C₁₃	semidirect
130. #2162	C₂×C₁₃	semidirect
294. C₁₃	#33776	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×#3170	2
C₁₃×#3147	2
C₁₃×#2248	2
C₁₃×#2247	2
C₁₃×#2241	2
C₁₃×#2240	2
C₁₃×#2211	2
C₁₃×#2189	2
C₁₃×#2162	2
#33776	13

Subgroups — C13×#33776 #15720115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33776 #15720115