Subgroups — C₁₃×#33582 #15719921

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 312 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	2	yes	maximal, complemented
2	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2207	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2206	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2140	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×D₈	—	2	1	no	—
10	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
11	C₂×C₁₃×#127	—	2	1	no	—
12	C₄×C₁₃×#41	—	2	1	no	—
13	C₁₃×#509	—	2	8	no	—
14	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	complemented
15	C₂×C₁₃×D₁₆	C₂²	1	4	yes	complemented
16	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
17	C₁₃×#493	—	2	2	no	—
18	C₁₃×#493	C₂²	1	4	yes	—
19	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	complemented
20	C₄×C₁₃×SD₁₆	—	2	2	no	—
21	C₄×C₁₃×D₈	C₂²	1	2	yes	complemented
22	C₁₃×#427	C₂²	1	1	yes	characteristic
23	C₁₃×#416	—	2	1	no	—
24	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
26	C₁₃×#357	C₂²	1	8	yes	complemented
27	C₂×C₁₃×#41	—	2	1	no	—
28	C₂²×D₄×C₁₃	—	2	2	no	—
29	C₂²×D₄×C₁₃	—	4	2	no	—
30	C₁₃×#135	—	2	2	no	—
31	C₁₃×#135	D₄	1	4	yes	complemented
32	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
33	C₂×C₁₃×SD₁₆	—	2	1	no	—
34	C₂×C₁₃×D₈	—	2	4	no	—
35	C₂×C₁₃×D₈	—	2	2	no	—
36	C₂×C₁₃×D₈	—	2	4	no	—
37	C₂×C₁₃×D₈	—	2	4	no	—
38	C₂×C₁₃×D₈	—	4	4	no	—
39	C₂×C₁₃×D₈	C₂³	1	4	yes	—
40	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×#127	—	2	4	no	—
44	C₁₃×#127	—	4	1	no	—
45	C₁₃×#127	—	4	1	no	—
46	C₄×Q₈×C₁₃	—	2	2	no	—
47	C₄×D₄×C₁₃	—	2	2	no	—
48	C₄×D₄×C₁₃	—	4	2	no	—
49	C₁₃×#117	—	2	1	no	—
50	C₁₃×#117	—	4	1	no	—
51	C₂×C₄²×C₁₃	—	4	1	no	—
52	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
53	C₁₃×SD₃₂	—	2	8	no	—
54	C₁₃×D₁₆	—	2	8	no	—
55	C₁₃×M₃₂	—	2	4	no	—
56	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
57	C₁₃×#107	C₂³	1	2	yes	—
58	C₁₃×#106	—	2	1	no	—
59	C₁₃×#103	—	2	2	no	—
60	C₁₃×#102	—	2	2	no	—
61	C₄×C₈×C₁₃	D₄	1	2	yes	—
62	C₄×C₈×C₁₃	C₂³	1	2	yes	—
63	#33582	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂⁴×C₁₃	—	4	2	no	—
65	C₁₃×#41	—	2	4	no	—
66	C₁₃×#41	—	4	2	no	—
67	C₂×Q₈×C₁₃	—	2	1	no	—
68	C₂×D₄×C₁₃	—	2	1	no	—
69	C₂×D₄×C₁₃	—	2	4	no	—
70	C₂×D₄×C₁₃	—	4	1	no	—
71	C₂×D₄×C₁₃	—	4	4	no	—
72	C₂×D₄×C₁₃	—	4	4	no	—
73	C₂×D₄×C₁₃	—	4	4	no	—
74	C₂×D₄×C₁₃	—	4	2	no	—
75	C₂×D₄×C₁₃	—	4	4	no	—
76	C₂×D₄×C₁₃	—	8	8	no	—
77	C₂²×C₄×C₁₃	—	4	1	no	—
78	C₂²×C₄×C₁₃	—	4	2	no	—
79	C₂²×C₄×C₁₃	D₈	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₁₃×Q₁₆	—	2	1	no	—
82	C₁₃×Q₁₆	C₂×D₄	1	2	yes	complemented
83	C₁₃×SD₁₆	—	2	4	no	—
84	C₁₃×D₈	—	2	4	no	—
85	C₁₃×D₈	—	2	1	no	—
86	C₁₃×D₈	—	2	4	no	—
87	C₁₃×D₈	—	4	8	no	—
88	C₁₃×D₈	—	4	4	no	—
89	C₁₃×D₈	C₂×D₄	1	2	yes	complemented
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₁₃	#41	1	2	yes	—
95	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	4	2	no	—
98	C₁₃×C₂²⋊C₄	—	4	1	no	—
99	C₁₃×C₂²⋊C₄	—	4	2	no	—
100	C₄²×C₁₃	—	4	1	no	—
101	C₄²×C₁₃	—	4	2	no	—
102	C₄²×C₁₃	D₈	1	1	yes	characteristic
103	C₄²×C₁₃	D₈	1	1	yes	characteristic
104	C₄²×C₁₃	C₂×D₄	1	2	yes	—
105	C₁₃×C₁₆	—	2	4	no	—
106	C₂×#509	C₂×C₁₃	1	2	yes	complemented
107	C₂×#493	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₄×#135	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#2207	C₂×C₁₃	1	4	yes	complemented
110	#2206	C₂×C₁₃	1	4	yes	complemented
111	#2150	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#2140	C₂×C₁₃	1	2	yes	complemented
113	C₂³×C₁₃	—	4	1	no	—
114	C₂³×C₁₃	—	4	4	no	—
115	C₂³×C₁₃	—	8	8	no	—
116	C₂³×C₁₃	—	8	4	no	—
117	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
118	Q₈×C₁₃	—	2	2	no	—
119	Q₈×C₁₃	—	4	1	no	—
120	D₄×C₁₃	—	2	2	no	—
121	D₄×C₁₃	—	4	4	no	—
122	D₄×C₁₃	—	4	4	no	—
123	D₄×C₁₃	—	4	1	no	—
124	D₄×C₁₃	—	4	4	no	—
125	D₄×C₁₃	—	8	8	no	—
126	D₄×C₁₃	—	8	4	no	—
127	D₄×C₁₃	—	8	8	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	4	1	no	—
130	C₂×C₄×C₁₃	—	4	4	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	8	4	no	—
133	C₂×C₄×C₁₃	#121	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₁₃	—	2	2	no	—
140	C₈×C₁₃	#121	1	2	yes	—
141	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
143	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
144	C₂²×D₈	—	2	1	no	—
145	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
146	C₂×#127	—	2	1	no	—
147	C₄×#41	—	2	1	no	—
148	#509	—	2	8	no	—
149	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	complemented
150	C₂×D₁₆	C₂²×C₁₃	1	4	yes	complemented
151	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
152	#493	—	2	2	no	—
153	#493	C₂²×C₁₃	1	4	yes	—
154	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	complemented
155	C₄×SD₁₆	—	2	2	no	—
156	C₄×D₈	C₂²×C₁₃	1	2	yes	complemented
157	#427	C₂²×C₁₃	1	1	yes	characteristic
158	#416	—	2	1	no	—
159	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
160	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	complemented
161	#357	C₂²×C₁₃	1	8	yes	complemented
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	4	2	no	—
164	C₂²×C₁₃	—	8	2	no	—
165	C₂²×C₁₃	—	8	8	no	—
166	C₂²×C₁₃	—	8	4	no	—
167	C₂²×C₁₃	—	16	8	no	—
168	C₂²×C₁₃	#459	1	2	yes	—
169	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	—	4	2	no	—
172	C₄×C₁₃	—	8	2	no	—
173	C₄×C₁₃	#459	1	2	yes	—
174	C₄×C₁₃	#509	1	4	yes	complemented
175	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
176	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₂×#41	—	2	1	no	—
178	C₂²×D₄	—	2	2	no	—
179	C₂²×D₄	—	4	2	no	—
180	#135	—	2	2	no	—
181	#135	D₄×C₁₃	1	4	yes	complemented
182	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
183	C₂×SD₁₆	—	2	1	no	—
184	C₂×D₈	—	2	4	no	—
185	C₂×D₈	—	2	4	no	—
186	C₂×D₈	—	2	2	no	—
187	C₂×D₈	—	2	4	no	—
188	C₂×D₈	—	4	4	no	—
189	C₂×D₈	C₂³×C₁₃	1	4	yes	—
190	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
191	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
193	#127	—	2	4	no	—
194	#127	—	4	1	no	—
195	#127	—	4	1	no	—
196	C₄×Q₈	—	2	2	no	—
197	C₄×D₄	—	2	2	no	—
198	C₄×D₄	—	4	2	no	—
199	#117	—	2	1	no	—
200	#117	—	4	1	no	—
201	C₂×C₄²	—	4	1	no	—
202	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
203	SD₃₂	—	2	8	no	—
204	D₁₆	—	2	8	no	—
205	M₃₂	—	2	4	no	—
206	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
207	#107	C₂³×C₁₃	1	2	yes	—
208	#106	—	2	1	no	—
209	#103	—	2	2	no	—
210	#102	—	2	2	no	—
211	C₄×C₈	D₄×C₁₃	1	2	yes	—
212	C₄×C₈	C₂³×C₁₃	1	2	yes	—
213	C₂×C₁₃	—	2	2	no	—
214	C₂×C₁₃	—	8	2	no	—
215	C₂×C₁₃	—	16	4	no	—
216	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
217	C₂×C₁₃	C₂×#509	1	2	yes	—
218	C₂⁴	—	4	2	no	—
219	#41	—	2	4	no	—
220	#41	—	4	2	no	—
221	C₂×Q₈	—	2	1	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂×D₄	—	4	4	no	—
226	C₂×D₄	—	4	4	no	—
227	C₂×D₄	—	4	4	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	8	8	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	C₁₃×D₈	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
235	Q₁₆	—	2	1	no	—
236	Q₁₆	C₂×D₄×C₁₃	1	2	yes	complemented
237	SD₁₆	—	2	4	no	—
238	D₈	—	2	4	no	—
239	D₈	—	2	1	no	—
240	D₈	—	2	4	no	—
241	D₈	—	4	8	no	—
242	D₈	—	4	4	no	—
243	D₈	C₂×D₄×C₁₃	1	2	yes	complemented
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
246	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
247	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
248	C₂×C₈	C₁₃×#41	1	2	yes	—
249	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₂²⋊C₄	—	4	1	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₄²	—	4	1	no	—
255	C₄²	—	4	2	no	—
256	C₄²	C₁₃×D₈	1	1	yes	characteristic
257	C₄²	C₁₃×D₈	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₁₃	1	2	yes	—
259	C₁₆	—	2	4	no	—
260	C₁₃	#33582	1	1	yes	characteristic, direct product factor, Sylow
261	C₂³	—	4	1	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	8	8	no	—
264	C₂³	—	8	4	no	—
265	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
266	Q₈	—	2	2	no	—
267	Q₈	—	4	1	no	—
268	D₄	—	2	2	no	—
269	D₄	—	4	4	no	—
270	D₄	—	4	4	no	—
271	D₄	—	4	1	no	—
272	D₄	—	4	4	no	—
273	D₄	—	8	8	no	—
274	D₄	—	8	4	no	—
275	D₄	—	8	8	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	4	no	—
281	C₂×C₄	C₁₃×#121	1	2	yes	—
282	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
283	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
284	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
285	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
286	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	C₁₃×#121	1	2	yes	—
289	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
290	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
291	C₂²	—	2	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	4	no	—
295	C₂²	—	8	8	no	—
296	C₂²	—	16	8	no	—
297	C₂²	C₁₃×#459	1	2	yes	—
298	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	2	no	—
302	C₄	C₁₃×#459	1	2	yes	—
303	C₄	C₁₃×#509	1	4	yes	complemented
304	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
305	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	8	2	no	—
308	C₂	—	16	4	no	—
309	C₂	C₂×C₁₃×#459	1	1	yes	characteristic
310	C₂	C₂×C₁₃×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
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. C₂×C₁₃×#509	C₂	semidirect
2. C₂×C₁₃×#493	C₂	semidirect
3. C₄×C₁₃×#135	C₂	semidirect
4. C₁₃×#2207	C₂	semidirect
5. C₁₃×#2206	C₂	semidirect
6. C₁₃×#2150	C₂	semidirect
7. C₁₃×#2140	C₂	semidirect
14. C₂×C₁₃×SD₃₂	C₂²	semidirect
15. C₂×C₁₃×D₁₆	C₂²	semidirect
19. C₄×C₁₃×Q₁₆	C₂²	semidirect
21. C₄×C₁₃×D₈	C₂²	semidirect
25. C₁₃×C₄⋊C₁₆	C₂²	semidirect
26. C₁₃×#357	C₂²	semidirect
31. C₁₃×#135	D₄	semidirect
63. #33582	C₁₃	direct (N × H)
82. C₁₃×Q₁₆	C₂×D₄	semidirect
89. C₁₃×D₈	C₂×D₄	semidirect
106. C₂×#509	C₂×C₁₃	semidirect
107. C₂×#493	C₂×C₁₃	semidirect
108. C₄×#135	C₂×C₁₃	semidirect
109. #2207	C₂×C₁₃	semidirect
110. #2206	C₂×C₁₃	semidirect
111. #2150	C₂×C₁₃	semidirect
112. #2140	C₂×C₁₃	semidirect
149. C₂×SD₃₂	C₂²×C₁₃	semidirect
150. C₂×D₁₆	C₂²×C₁₃	semidirect
154. C₄×Q₁₆	C₂²×C₁₃	semidirect
156. C₄×D₈	C₂²×C₁₃	semidirect
160. C₄⋊C₁₆	C₂²×C₁₃	semidirect
161. #357	C₂²×C₁₃	semidirect
174. C₄×C₁₃	#509	semidirect
181. #135	D₄×C₁₃	semidirect
236. Q₁₆	C₂×D₄×C₁₃	semidirect
243. D₈	C₂×D₄×C₁₃	semidirect
260. C₁₃	#33582	direct (N × H)
303. C₄	C₁₃×#509	semidirect

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
C₂×C₁₃×#493	2
C₄×C₁₃×#135	2
C₁₃×#2207	2
C₁₃×#2206	2
C₁₃×#2150	2
C₁₃×#2140	2
#33582	13

Subgroups — C13×#33582 #15719921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33582 #15719921