Subgroups — C₁₃×#9668 #15696007

Order 3328.

510 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×#1564	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#1478	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1460	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×D₈	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
8	C₄×C₁₃×D₈	—	2	1	no	—
9	C₄×C₁₃×D₈	—	2	1	no	—
10	C₄×C₁₃×D₈	—	2	1	no	—
11	C₄×C₁₃×D₈	—	2	1	no	—
12	C₁₃×#431	C₂²	1	2	yes	—
13	C₁₃×#431	C₂²	1	1	yes	characteristic
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
16	C₂×C₁₃×#102	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#335	—	2	1	no	—
19	C₁₃×#334	—	2	1	no	—
20	C₁₃×D₄⋊C₈	—	2	4	no	—
21	C₁₃×#323	—	2	4	no	—
22	C₁₃×#322	—	2	1	no	—
23	C₈²×C₁₃	—	2	1	no	—
24	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₃	—	2	2	no	—
26	C₂×C₁₃×D₈	—	2	1	no	—
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	—	2	2	no	—
29	C₂×C₁₃×D₈	C₈	1	4	yes	complemented
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₄×D₄×C₁₃	—	2	2	no	—
33	C₄×D₄×C₁₃	—	2	2	no	—
34	C₄×D₄×C₁₃	—	2	2	no	—
35	C₄×D₄×C₁₃	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×#107	—	2	1	no	—
40	C₁₃×#107	—	2	1	no	—
41	C₁₃×C₄⋊C₈	—	2	2	no	—
42	C₁₃×C₄⋊C₈	—	2	1	no	—
43	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
44	C₁₃×#102	—	2	2	no	—
45	C₁₃×#102	—	2	2	no	—
46	C₁₃×C₂²⋊C₈	—	2	1	no	—
47	C₁₃×C₂²⋊C₈	—	2	2	no	—
48	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
49	C₄×C₈×C₁₃	—	2	1	no	—
50	C₄×C₈×C₁₃	—	2	1	no	—
51	C₄×C₈×C₁₃	—	2	2	no	—
52	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
56	#9668	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₃	—	2	2	no	—
58	C₂×D₄×C₁₃	—	2	2	no	—
59	C₂×D₄×C₁₃	—	2	2	no	—
60	C₂×D₄×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	—	2	2	no	—
62	C₂×D₄×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₃	—	4	2	no	—
66	C₂²×C₄×C₁₃	—	4	2	no	—
67	C₂²×C₄×C₁₃	—	8	4	no	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
71	C₁₃×D₈	—	2	4	no	—
72	C₁₃×D₈	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	1	no	—
74	C₂×C₈×C₁₃	—	2	1	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	4	2	no	—
78	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
83	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
84	C₁₃×C₄⋊C₄	—	2	2	no	—
85	C₁₃×C₄⋊C₄	—	2	2	no	—
86	C₁₃×C₂²⋊C₄	—	4	2	no	—
87	C₁₃×C₂²⋊C₄	—	4	2	no	—
88	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₃	#41	1	1	yes	characteristic
92	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
93	#1564	C₂×C₁₃	1	1	yes	characteristic, complemented
94	#1478	C₂×C₁₃	1	2	yes	complemented
95	#1460	C₂×C₁₃	1	2	yes	complemented
96	#1450	C₂×C₁₃	1	1	yes	characteristic, complemented
97	C₂³×C₁₃	—	2	4	no	—
98	C₂³×C₁₃	—	4	2	no	—
99	C₂³×C₁₃	—	4	2	no	—
100	C₂³×C₁₃	—	8	2	no	—
101	C₂³×C₁₃	—	8	2	no	—
102	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
103	D₄×C₁₃	—	2	4	no	—
104	D₄×C₁₃	—	2	4	no	—
105	D₄×C₁₃	—	4	2	no	—
106	D₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	4	1	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	8	4	no	—
111	C₂×C₄×C₁₃	—	8	4	no	—
112	C₂×C₄×C₁₃	—	8	2	no	—
113	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
123	C₈×C₁₃	—	2	1	no	—
124	C₈×C₁₃	—	2	1	no	—
125	C₈×C₁₃	—	4	2	no	—
126	C₈×C₁₃	—	4	2	no	—
127	C₈×C₁₃	—	8	2	no	—
128	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
129	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
130	C₂²×D₈	C₄×C₁₃	1	1	yes	characteristic
131	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
132	C₄×D₈	—	2	1	no	—
133	C₄×D₈	—	2	1	no	—
134	C₄×D₈	—	2	1	no	—
135	C₄×D₈	—	2	1	no	—
136	#431	C₂²×C₁₃	1	2	yes	—
137	#431	C₂²×C₁₃	1	1	yes	characteristic
138	#431	C₂²×C₁₃	1	1	yes	characteristic
139	C₂×#107	C₄×C₁₃	1	1	yes	characteristic
140	C₂×#102	C₄×C₁₃	1	2	yes	—
141	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
142	#335	—	2	1	no	—
143	#334	—	2	1	no	—
144	D₄⋊C₈	—	2	4	no	—
145	#323	—	2	4	no	—
146	#322	—	2	1	no	—
147	C₈²	—	2	1	no	—
148	C₂²×C₁₃	—	2	1	no	—
149	C₂²×C₁₃	—	2	1	no	—
150	C₂²×C₁₃	—	4	4	no	—
151	C₂²×C₁₃	—	8	4	no	—
152	C₂²×C₁₃	—	8	4	no	—
153	C₂²×C₁₃	—	8	2	no	—
154	C₂²×C₁₃	—	8	2	no	—
155	C₂²×C₁₃	#390	1	1	yes	characteristic
156	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
157	C₂²×C₁₃	#436	1	1	yes	characteristic
158	C₄×C₁₃	—	2	1	no	—
159	C₄×C₁₃	—	2	1	no	—
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	8	2	no	—
162	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
163	C₄×C₁₃	#408	1	1	yes	characteristic
164	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
165	C₄×C₁₃	#436	1	1	yes	characteristic
166	C₄×C₁₃	#472	1	2	yes	—
167	C₂²×D₄	C₂×C₄×C₁₃	1	2	yes	—
168	C₂³×C₄	—	2	2	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	—	2	1	no	—
171	C₂×D₈	—	2	2	no	—
172	C₂×D₈	C₈×C₁₃	1	4	yes	complemented
173	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
174	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
175	C₄×D₄	—	2	2	no	—
176	C₄×D₄	—	2	2	no	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
180	C₂×C₂²⋊C₄	—	2	2	no	—
181	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
182	#107	—	2	1	no	—
183	#107	—	2	1	no	—
184	C₄⋊C₈	—	2	2	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
187	#102	—	2	2	no	—
188	#102	—	2	2	no	—
189	C₂²⋊C₈	—	2	1	no	—
190	C₂²⋊C₈	—	2	2	no	—
191	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	2	1	no	—
194	C₄×C₈	—	2	2	no	—
195	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
196	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
197	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
198	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
199	C₂×C₁₃	—	2	1	no	—
200	C₂×C₁₃	—	2	1	no	—
201	C₂×C₁₃	—	8	4	no	—
202	C₂×C₁₃	#1954	1	1	yes	characteristic
203	C₂×C₁₃	#1976	1	1	yes	characteristic
204	C₂×C₁₃	#1978	1	1	yes	characteristic
205	C₂⁴	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₂×C₈×C₁₃	1	4	yes	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	8	4	no	—
216	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
219	D₈	—	2	4	no	—
220	D₈	—	2	4	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
227	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
231	C₂×C₈	C₁₃×M₁₆	1	2	yes	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	4	2	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
237	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
238	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
239	C₄²	C₁₃×#41	1	1	yes	characteristic
240	C₁₃	#9668	1	1	yes	characteristic, direct product factor, Sylow
241	C₂³	—	2	4	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	8	2	no	—
245	C₂³	—	8	2	no	—
246	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
247	D₄	—	2	4	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
259	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
260	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
261	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
262	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
265	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
266	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	4	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	8	2	no	—
272	C₈	C₁₃×C₂²⋊C₈	1	2	yes	—
273	C₈	C₁₃×C₂²⋊C₈	1	2	yes	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	8	4	no	—
278	C₂²	—	8	4	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	C₁₃×#390	1	1	yes	characteristic
282	C₂²	C₈×D₄×C₁₃	1	1	yes	characteristic
283	C₂²	C₁₃×#436	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	8	2	no	—
288	C₄	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
289	C₄	C₁₃×#408	1	1	yes	characteristic
290	C₄	C₈×D₄×C₁₃	1	1	yes	characteristic
291	C₄	C₁₃×#436	1	1	yes	characteristic
292	C₄	C₁₃×#472	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	8	4	no	—
296	C₂	C₁₃×#1954	1	1	yes	characteristic
297	C₂	C₁₃×#1976	1	1	yes	characteristic
298	C₂	C₁₃×#1978	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	#121
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂²×C₁₃	#390
prime radical	C₂²×C₁₃×D₈	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#1564	C₂	semidirect
3. C₁₃×#1478	C₂	semidirect
4. C₁₃×#1460	C₂	semidirect
5. C₁₃×#1450	C₂	semidirect
29. C₂×C₁₃×D₈	C₈	semidirect
56. #9668	C₁₃	direct (N × H)
93. #1564	C₂×C₁₃	semidirect
94. #1478	C₂×C₁₃	semidirect
95. #1460	C₂×C₁₃	semidirect
96. #1450	C₂×C₁₃	semidirect
172. C₂×D₈	C₈×C₁₃	semidirect
240. C₁₃	#9668	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×C₄×C₁₃×D₈	2
C₁₃×#1564	2
C₁₃×#1478	2
C₁₃×#1460	2
C₁₃×#1450	2
#9668	13

Subgroups — C13×#9668 #15696007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9668 #15696007