Subgroups — C₁₃×#21277 #15707616

Order 3328.

950 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2293	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1972	C₂	1	2	yes	maximal
4	C₁₃×#1943	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1936	C₂	1	4	yes	maximal, complemented
6	C₄×C₁₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₁₃×#102	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
11	C₂×C₁₃×#117	—	2	2	no	—
12	C₄×C₁₃×D₈	C₄	1	8	yes	complemented
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#106	C₂²	1	2	yes	—
15	C₂×C₁₃×#102	C₂²	1	4	yes	—
16	C₂×C₁₃×#102	C₂²	1	2	yes	—
17	C₂×C₁₃×#97	C₂²	1	2	yes	—
18	C₂×C₁₃×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₁₃×#390	C₂²	1	4	yes	—
22	C₁₃×#389	C₂²	1	4	yes	—
23	C₁₃×#388	—	2	4	no	—
24	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄×C₁₃×C₂²⋊C₄	—	2	2	no	—
26	C₄³×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#340	C₂²	1	4	yes	—
28	C₁₃×#340	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₃	—	2	2	no	—
31	C₂×C₁₃×D₈	—	2	4	no	—
32	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₄×D₄×C₁₃	—	2	4	no	—
37	C₄×D₄×C₁₃	C₂×C₄	1	8	yes	—
38	C₁₃×#117	—	4	8	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
44	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₄²×C₁₃	—	2	2	no	—
46	C₂×C₄²×C₁₃	D₄	1	2	yes	—
47	C₂×C₄²×C₁₃	D₄	1	2	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₁₃×#107	C₂×C₄	1	4	yes	—
51	C₁₃×#106	—	2	4	no	—
52	C₁₃×#102	—	2	8	no	—
53	C₁₃×#102	C₂×C₄	1	8	yes	—
54	C₁₃×#97	—	2	4	no	—
55	C₁₃×#97	—	2	4	no	—
56	C₄×C₈×C₁₃	—	2	2	no	—
57	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
58	C₁₃×#95	—	2	4	no	—
59	C₁₃×#95	—	2	2	no	—
60	#21277	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂⁴×C₁₃	—	2	2	no	—
62	C₂×D₄×C₁₃	—	2	4	no	—
63	C₂×D₄×C₁₃	—	2	4	no	—
64	C₂×D₄×C₁₃	—	2	2	no	—
65	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	8	no	—
68	C₂²×C₄×C₁₃	—	2	4	no	—
69	C₂²×C₄×C₁₃	—	4	4	no	—
70	C₂²×C₄×C₁₃	—	4	4	no	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₁₃×D₈	—	2	8	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₃	#41	1	4	yes	—
82	C₂×C₈×C₁₃	#41	1	4	yes	—
83	C₁₃×C₄⋊C₄	—	2	2	no	—
84	C₁₃×C₄⋊C₄	—	2	4	no	—
85	C₁₃×C₄⋊C₄	—	2	4	no	—
86	C₁₃×C₄⋊C₄	—	4	8	no	—
87	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
88	C₁₃×C₂²⋊C₄	—	2	8	no	—
89	C₁₃×C₂²⋊C₄	—	4	8	no	—
90	C₁₃×C₂²⋊C₄	—	4	8	no	—
91	C₄²×C₁₃	—	2	4	no	—
92	C₄²×C₁₃	—	2	2	no	—
93	C₄²×C₁₃	—	2	2	no	—
94	C₄²×C₁₃	—	4	2	no	—
95	C₄²×C₁₃	—	4	4	no	—
96	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
97	C₄²×C₁₃	#41	1	2	yes	—
98	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
99	#2293	C₂×C₁₃	1	2	yes	complemented
100	#1972	C₂×C₁₃	1	2	yes	—
101	#1943	C₂×C₁₃	1	2	yes	complemented
102	#1936	C₂×C₁₃	1	4	yes	complemented
103	C₄×#106	C₂×C₁₃	1	1	yes	characteristic, complemented
104	C₄×#102	C₂×C₁₃	1	2	yes	complemented
105	C₄×#97	C₂×C₁₃	1	1	yes	characteristic, complemented
106	C₂³×C₁₃	—	2	4	no	—
107	C₂³×C₁₃	—	4	4	no	—
108	C₂³×C₁₃	—	4	2	no	—
109	C₂³×C₁₃	—	4	2	no	—
110	C₂³×C₁₃	—	4	4	no	—
111	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	D₄×C₁₃	—	2	4	no	—
113	D₄×C₁₃	—	2	4	no	—
114	D₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	16	no	—
123	C₂×C₄×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	—	4	4	no	—
125	C₂×C₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
129	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
133	C₈×C₁₃	—	2	2	no	—
134	C₈×C₁₃	—	2	4	no	—
135	C₈×C₁₃	#117	1	4	yes	complemented
136	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
137	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
138	C₂×#117	—	2	2	no	—
139	C₄×D₈	C₄×C₁₃	1	8	yes	complemented
140	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
141	C₂×#106	C₂²×C₁₃	1	2	yes	—
142	C₂×#102	C₂²×C₁₃	1	4	yes	—
143	C₂×#102	C₂²×C₁₃	1	2	yes	—
144	C₂×#97	C₂²×C₁₃	1	2	yes	—
145	C₂×#97	C₂²×C₁₃	1	2	yes	—
146	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
148	#390	C₂²×C₁₃	1	4	yes	—
149	#389	C₂²×C₁₃	1	4	yes	—
150	#388	—	2	4	no	—
151	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
152	C₄×C₂²⋊C₄	—	2	2	no	—
153	C₄³	C₂²×C₁₃	1	1	yes	characteristic
154	#340	C₂²×C₁₃	1	2	yes	—
155	#340	C₂²×C₁₃	1	4	yes	—
156	C₂²×C₁₃	—	4	4	no	—
157	C₂²×C₁₃	—	4	4	no	—
158	C₂²×C₁₃	—	4	4	no	—
159	C₂²×C₁₃	—	8	4	no	—
160	C₂²×C₁₃	—	8	4	no	—
161	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂²×C₁₃	#532	1	2	yes	—
163	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
164	C₂²×C₁₃	#575	1	2	yes	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	4	4	no	—
167	C₄×C₁₃	—	4	4	no	—
168	C₄×C₁₃	—	8	4	no	—
169	C₄×C₁₃	#442	1	4	yes	—
170	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
171	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
172	C₄×C₁₃	#529	1	2	yes	—
173	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
174	C₂³×C₄	—	2	2	no	—
175	C₂×D₈	—	2	4	no	—
176	C₂×D₈	C₂×C₄×C₁₃	1	4	yes	—
177	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
179	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
180	C₄×D₄	—	2	4	no	—
181	C₄×D₄	C₂×C₄×C₁₃	1	8	yes	—
182	#117	—	4	8	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
185	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₂²⋊C₄	—	2	4	no	—
188	C₂×C₂²⋊C₄	—	2	4	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	D₄×C₁₃	1	2	yes	—
191	C₂×C₄²	D₄×C₁₃	1	2	yes	—
192	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
193	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
194	#107	C₂×C₄×C₁₃	1	4	yes	—
195	#106	—	2	4	no	—
196	#102	—	2	8	no	—
197	#102	C₂×C₄×C₁₃	1	8	yes	—
198	#97	—	2	4	no	—
199	#97	—	2	4	no	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
202	#95	—	2	4	no	—
203	#95	—	2	2	no	—
204	C₂×C₁₃	—	8	4	no	—
205	C₂×C₁₃	#2293	1	1	yes	characteristic
206	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
207	C₂×C₁₃	#2943	1	1	yes	characteristic
208	C₂×C₁₃	#3171	1	4	yes	—
209	C₂⁴	—	2	2	no	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	C₂²×C₄×C₁₃	1	4	yes	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	8	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	4	4	no	—
218	C₂²×C₄	—	4	4	no	—
219	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
220	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
223	D₈	—	2	8	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
228	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
229	C₂×C₈	C₁₃×#41	1	4	yes	—
230	C₂×C₈	C₁₃×#41	1	4	yes	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	4	8	no	—
235	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
236	C₂²⋊C₄	—	2	8	no	—
237	C₂²⋊C₄	—	4	8	no	—
238	C₂²⋊C₄	—	4	8	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	2	no	—
243	C₄²	—	4	4	no	—
244	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
245	C₄²	C₁₃×#41	1	2	yes	—
246	C₁₃	#21277	1	1	yes	characteristic, direct product factor, Sylow
247	C₂³	—	2	4	no	—
248	C₂³	—	4	4	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	2	no	—
251	C₂³	—	4	4	no	—
252	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
253	D₄	—	2	4	no	—
254	D₄	—	2	4	no	—
255	D₄	—	4	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	16	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
268	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
269	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
270	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
271	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
272	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
273	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	2	4	no	—
276	C₈	C₁₃×#117	1	4	yes	complemented
277	C₂²	—	4	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	8	4	no	—
281	C₂²	—	8	4	no	—
282	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
283	C₂²	C₁₃×#532	1	2	yes	—
284	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	2	yes	—
285	C₂²	C₁₃×#575	1	2	yes	—
286	C₄	—	2	2	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	4	no	—
289	C₄	—	8	4	no	—
290	C₄	C₁₃×#442	1	4	yes	—
291	C₄	C₂×C₁₃×#117	1	1	yes	characteristic
292	C₄	C₄×C₁₃×#41	1	1	yes	characteristic
293	C₄	C₁₃×#529	1	2	yes	—
294	C₂	—	8	4	no	—
295	C₂	C₁₃×#2293	1	1	yes	characteristic
296	C₂	C₂×C₁₃×#442	1	1	yes	characteristic
297	C₂	C₁₃×#2943	1	1	yes	characteristic
298	C₂	C₁₃×#3171	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2293	C₂	semidirect
4. C₁₃×#1943	C₂	semidirect
5. C₁₃×#1936	C₂	semidirect
6. C₄×C₁₃×#106	C₂	semidirect
7. C₄×C₁₃×#102	C₂	semidirect
8. C₄×C₁₃×#97	C₂	semidirect
12. C₄×C₁₃×D₈	C₄	semidirect
60. #21277	C₁₃	direct (N × H)
99. #2293	C₂×C₁₃	semidirect
101. #1943	C₂×C₁₃	semidirect
102. #1936	C₂×C₁₃	semidirect
103. C₄×#106	C₂×C₁₃	semidirect
104. C₄×#102	C₂×C₁₃	semidirect
105. C₄×#97	C₂×C₁₃	semidirect
135. C₈×C₁₃	#117	semidirect
139. C₄×D₈	C₄×C₁₃	semidirect
246. C₁₃	#21277	direct (N × H)
276. C₈	C₁₃×#117	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×C₁₃×D₈	2
C₁₃×#2293	2
C₁₃×#1972	2
C₁₃×#1943	2
C₁₃×#1936	2
C₄×C₁₃×#106	2
C₄×C₁₃×#102	2
C₄×C₁₃×#97	2
#21277	13

Subgroups — C13×#21277 #15707616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#21277 #15707616