Subgroups — C₁₃×#8601 #15694940

Order 3328.

462 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1564	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1458	C₂	1	2	yes	maximal
4	C₁₃×#1275	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
8	C₁₃×#431	C₂²	1	4	yes	—
9	C₁₃×#405	—	2	2	no	—
10	C₁₃×#405	—	2	2	no	—
11	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
12	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#335	—	2	2	no	—
16	C₁₃×#334	—	2	2	no	—
17	C₁₃×#324	—	2	2	no	—
18	C₁₃×#323	—	2	2	no	—
19	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	—	2	2	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂²×C₈×C₁₃	—	2	1	no	—
24	C₂²×C₈×C₁₃	—	2	1	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	C₈	1	4	yes	complemented
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₁₃×#117	—	2	2	no	—
32	C₁₃×#117	—	2	2	no	—
33	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
34	C₂×C₁₃×C₄⋊C₄	Q₈	1	1	yes	characteristic
35	C₂×C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
36	C₂×C₁₃×C₂²⋊C₄	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×C₄⋊C₈	—	2	4	no	—
40	C₁₃×C₂²⋊C₈	—	2	4	no	—
41	C₁₃×#97	—	2	2	no	—
42	C₁₃×#97	—	2	2	no	—
43	C₄×C₈×C₁₃	—	2	1	no	—
44	C₄×C₈×C₁₃	—	2	1	no	—
45	C₄×C₈×C₁₃	—	2	4	no	—
46	C₄×C₈×C₁₃	D₄	1	2	yes	—
47	C₄×C₈×C₁₃	D₄	1	2	yes	—
48	#8601	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	—	2	1	no	—
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₁₃×M₁₆	—	2	8	no	—
63	C₂×C₈×C₁₃	—	2	1	no	—
64	C₂×C₈×C₁₃	—	2	1	no	—
65	C₂×C₈×C₁₃	—	2	1	no	—
66	C₂×C₈×C₁₃	—	2	1	no	—
67	C₂×C₈×C₁₃	—	2	1	no	—
68	C₂×C₈×C₁₃	—	2	1	no	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	2	4	no	—
71	C₂×C₈×C₁₃	—	4	4	no	—
72	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
77	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
78	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₂²⋊C₄	—	2	2	no	—
82	C₁₃×C₂²⋊C₄	—	2	2	no	—
83	C₄²×C₁₃	—	2	1	no	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	C₂×D₄	1	2	yes	—
86	C₄²×C₁₃	#41	1	2	yes	—
87	C₂×#405	C₂×C₁₃	1	1	yes	characteristic
88	#1564	C₂×C₁₃	1	2	yes	complemented
89	#1458	C₂×C₁₃	1	2	yes	—
90	#1275	C₂×C₁₃	1	2	yes	complemented
91	C₂³×C₁₃	—	2	1	no	—
92	C₂³×C₁₃	—	2	1	no	—
93	C₂³×C₁₃	—	4	1	no	—
94	C₂³×C₁₃	—	4	1	no	—
95	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
96	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	4	1	no	—
106	C₂×C₄×C₁₃	—	4	1	no	—
107	C₂×C₄×C₁₃	—	4	1	no	—
108	C₂×C₄×C₁₃	—	4	1	no	—
109	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#121	1	2	yes	—
113	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
118	C₈×C₁₃	—	2	2	no	—
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	—	2	2	no	—
121	C₈×C₁₃	—	8	4	no	—
122	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
123	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
124	C₂²×M₁₆	C₄×C₁₃	1	1	yes	characteristic
125	C₂³×C₈	C₄×C₁₃	1	1	yes	characteristic
126	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
127	#431	C₂²×C₁₃	1	4	yes	—
128	#405	—	2	2	no	—
129	#405	—	2	2	no	—
130	C₂×#97	C₄×C₁₃	1	1	yes	characteristic
131	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×C₄×C₈	C₄×C₁₃	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
134	#335	—	2	2	no	—
135	#334	—	2	2	no	—
136	#324	—	2	2	no	—
137	#323	—	2	2	no	—
138	C₂²×C₁₃	—	2	1	no	—
139	C₂²×C₁₃	—	2	1	no	—
140	C₂²×C₁₃	—	2	2	no	—
141	C₂²×C₁₃	—	4	2	no	—
142	C₂²×C₁₃	—	4	2	no	—
143	C₂²×C₁₃	—	4	1	no	—
144	C₂²×C₁₃	—	4	1	no	—
145	C₂²×C₁₃	#380	1	1	yes	characteristic
146	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
147	C₂²×C₁₃	#423	1	1	yes	characteristic
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	4	1	no	—
152	C₄×C₁₃	—	4	1	no	—
153	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
154	C₄×C₁₃	#407	1	1	yes	characteristic
155	C₄×C₁₃	#431	1	1	yes	characteristic
156	C₄×C₁₃	#432	1	1	yes	characteristic
157	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	C₈×C₁₃	1	4	yes	complemented
165	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
166	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
167	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
168	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
169	#117	—	2	2	no	—
170	#117	—	2	2	no	—
171	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
172	C₂×C₄⋊C₄	Q₈×C₁₃	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	D₄×C₁₃	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	Q₈×C₁₃	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
176	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
177	C₄⋊C₈	—	2	4	no	—
178	C₂²⋊C₈	—	2	4	no	—
179	#97	—	2	2	no	—
180	#97	—	2	2	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	2	4	no	—
184	C₄×C₈	D₄×C₁₃	1	2	yes	—
185	C₄×C₈	D₄×C₁₃	1	2	yes	—
186	C₂×C₁₃	—	2	1	no	—
187	C₂×C₁₃	—	2	1	no	—
188	C₂×C₁₃	—	4	2	no	—
189	C₂×C₁₃	#1797	1	1	yes	characteristic
190	C₂×C₁₃	#1812	1	1	yes	characteristic
191	C₂×C₁₃	#1813	1	1	yes	characteristic
192	C₂⁴	C₁₃×C₄⋊C₄	1	1	yes	characteristic
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	C₁₃×C₄⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	C₁₃×C₄⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	C₁₃×C₄⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
201	C₂²×C₄	C₁₃×M₁₆	1	2	yes	—
202	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
205	M₁₆	—	2	8	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	4	no	—
215	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
216	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
218	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
219	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
220	C₂×C₈	C₁₃×M₁₆	1	2	yes	—
221	C₂×C₈	C₁₃×M₁₆	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	2	1	no	—
228	C₄²	C₂×D₄×C₁₃	1	2	yes	—
229	C₄²	C₁₃×#41	1	2	yes	—
230	C₁₃	#8601	1	1	yes	characteristic, direct product factor, Sylow
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	4	1	no	—
234	C₂³	—	4	1	no	—
235	C₂³	C₁₃×C₄⋊C₈	1	2	yes	—
236	C₂³	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₁₃×C₄⋊C₈	1	2	yes	—
250	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
252	C₂×C₄	C₁₃×#121	1	2	yes	—
253	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
254	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
255	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
256	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
257	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₈	—	2	2	no	—
261	C₈	—	8	4	no	—
262	C₈	C₁₃×C₂²⋊C₈	1	2	yes	—
263	C₈	C₁₃×C₂²⋊C₈	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	C₁₃×#380	1	1	yes	characteristic
272	C₂²	C₂×C₁₃×C₄⋊C₈	1	1	yes	characteristic
273	C₂²	C₁₃×#423	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
280	C₄	C₁₃×#407	1	1	yes	characteristic
281	C₄	C₁₃×#431	1	1	yes	characteristic
282	C₄	C₁₃×#432	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	4	2	no	—
286	C₂	C₁₃×#1797	1	1	yes	characteristic
287	C₂	C₁₃×#1812	1	1	yes	characteristic
288	C₂	C₁₃×#1813	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	#117
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂²×C₁₃	#380
prime radical	C₂⁴×C₁₃	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#1564	C₂	semidirect
4. C₁₃×#1275	C₂	semidirect
26. C₂²×C₈×C₁₃	C₈	semidirect
48. #8601	C₁₃	direct (N × H)
88. #1564	C₂×C₁₃	semidirect
90. #1275	C₂×C₁₃	semidirect
164. C₂²×C₈	C₈×C₁₃	semidirect
230. C₁₃	#8601	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×#405	2
C₁₃×#1564	2
C₁₃×#1458	2
C₁₃×#1275	2
#8601	13

Subgroups — C13×#8601 #15694940

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#8601 #15694940