Subgroups — C₁₃×#9669 #15696008

Order 3328.

446 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1479	C₂	1	2	yes	maximal
4	C₁₃×#1461	C₂	1	2	yes	maximal
5	C₁₃×#1450	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
8	C₄×C₁₃×Q₁₆	—	2	1	no	—
9	C₄×C₁₃×Q₁₆	—	2	1	no	—
10	C₄×C₁₃×Q₁₆	—	2	1	no	—
11	C₄×C₁₃×Q₁₆	—	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×#103	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×Q₈⋊C₈	—	2	4	no	—
21	C₁₃×#324	—	2	4	no	—
22	C₁₃×#322	—	2	1	no	—
23	C₈²×C₁₃	—	2	1	no	—
24	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×Q₁₆	—	2	1	no	—
26	C₂×C₁₃×Q₁₆	—	2	1	no	—
27	C₂×C₁₃×Q₁₆	—	2	2	no	—
28	C₂×C₁₃×Q₁₆	C₈	1	4	yes	complemented
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₁₃	—	2	2	no	—
32	C₄×Q₈×C₁₃	—	2	2	no	—
33	C₄×Q₈×C₁₃	—	2	2	no	—
34	C₄×Q₈×C₁₃	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₃	—	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×#103	—	2	2	no	—
45	C₁₃×#103	—	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	#9669	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂×Q₈×C₁₃	—	2	2	no	—
58	C₂×Q₈×C₁₃	—	2	2	no	—
59	C₂×Q₈×C₁₃	—	2	2	no	—
60	C₂×Q₈×C₁₃	—	2	2	no	—
61	C₂×Q₈×C₁₃	—	2	2	no	—
62	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
68	C₁₃×Q₁₆	—	2	4	no	—
69	C₁₃×Q₁₆	—	2	4	no	—
70	C₂×C₈×C₁₃	—	2	1	no	—
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	4	2	no	—
75	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
80	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	—	2	2	no	—
83	C₁₃×C₄⋊C₄	—	4	2	no	—
84	C₁₃×C₄⋊C₄	—	4	2	no	—
85	C₄²×C₁₃	—	4	2	no	—
86	C₄²×C₁₃	—	4	2	no	—
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	#41	1	1	yes	characteristic
91	C₂×C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic
92	#1564	C₂×C₁₃	1	1	yes	characteristic
93	#1479	C₂×C₁₃	1	2	yes	—
94	#1461	C₂×C₁₃	1	2	yes	—
95	#1450	C₂×C₁₃	1	1	yes	characteristic
96	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
97	Q₈×C₁₃	—	2	4	no	—
98	Q₈×C₁₃	—	2	4	no	—
99	Q₈×C₁₃	—	4	2	no	—
100	Q₈×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	4	1	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	8	2	no	—
108	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
118	C₈×C₁₃	—	2	1	no	—
119	C₈×C₁₃	—	2	1	no	—
120	C₈×C₁₃	—	4	2	no	—
121	C₈×C₁₃	—	4	2	no	—
122	C₈×C₁₃	—	8	2	no	—
123	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
124	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
125	C₂²×Q₁₆	C₄×C₁₃	1	1	yes	characteristic
126	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
127	C₄×Q₁₆	—	2	1	no	—
128	C₄×Q₁₆	—	2	1	no	—
129	C₄×Q₁₆	—	2	1	no	—
130	C₄×Q₁₆	—	2	1	no	—
131	#431	C₂²×C₁₃	1	2	yes	—
132	#431	C₂²×C₁₃	1	1	yes	characteristic
133	#431	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×#107	C₄×C₁₃	1	1	yes	characteristic
135	C₂×#103	C₄×C₁₃	1	2	yes	—
136	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
137	#335	—	2	1	no	—
138	#334	—	2	1	no	—
139	Q₈⋊C₈	—	2	4	no	—
140	#324	—	2	4	no	—
141	#322	—	2	1	no	—
142	C₈²	—	2	1	no	—
143	C₂²×C₁₃	—	2	1	no	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	#390	1	1	yes	characteristic
146	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
147	C₂²×C₁₃	#436	1	1	yes	characteristic
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	—	4	4	no	—
152	C₄×C₁₃	—	8	2	no	—
153	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
154	C₄×C₁₃	#408	1	1	yes	characteristic
155	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
156	C₄×C₁₃	#436	1	1	yes	characteristic
157	C₄×C₁₃	#473	1	2	yes	—
158	C₂²×Q₈	C₂×C₄×C₁₃	1	2	yes	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×Q₁₆	—	2	2	no	—
162	C₂×Q₁₆	C₈×C₁₃	1	4	yes	complemented
163	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
164	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
165	C₄×Q₈	—	2	2	no	—
166	C₄×Q₈	—	2	2	no	—
167	C₄×Q₈	—	2	2	no	—
168	C₄×Q₈	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
171	C₂×C₄²	—	2	2	no	—
172	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
173	#107	—	2	1	no	—
174	#107	—	2	1	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
178	#103	—	2	2	no	—
179	#103	—	2	2	no	—
180	C₂²⋊C₈	—	2	1	no	—
181	C₂²⋊C₈	—	2	2	no	—
182	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
187	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
188	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
189	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
190	C₂×C₁₃	—	2	1	no	—
191	C₂×C₁₃	—	2	1	no	—
192	C₂×C₁₃	#1954	1	1	yes	characteristic
193	C₂×C₁₃	#1976	1	1	yes	characteristic
194	C₂×C₁₃	#1979	1	1	yes	characteristic
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	C₂×C₈×C₁₃	1	4	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
206	Q₁₆	—	2	4	no	—
207	Q₁₆	—	2	4	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
218	C₂×C₈	C₁₃×M₁₆	1	2	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄²	—	4	2	no	—
224	C₄²	—	4	2	no	—
225	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
226	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
227	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
228	C₄²	C₁₃×#41	1	1	yes	characteristic
229	C₁₃	#9669	1	1	yes	characteristic, direct product factor, Sylow
230	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
231	Q₈	—	2	4	no	—
232	Q₈	—	2	4	no	—
233	Q₈	—	4	2	no	—
234	Q₈	—	4	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
244	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
245	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
246	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
247	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
249	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
250	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
251	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	4	2	no	—
255	C₈	—	4	2	no	—
256	C₈	—	8	2	no	—
257	C₈	C₁₃×C₂²⋊C₈	1	2	yes	—
258	C₈	C₁₃×C₂²⋊C₈	1	2	yes	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	C₁₃×#390	1	1	yes	characteristic
262	C₂²	C₈×D₄×C₁₃	1	1	yes	characteristic
263	C₂²	C₁₃×#436	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	4	no	—
268	C₄	—	8	2	no	—
269	C₄	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
270	C₄	C₁₃×#408	1	1	yes	characteristic
271	C₄	C₈×D₄×C₁₃	1	1	yes	characteristic
272	C₄	C₁₃×#436	1	1	yes	characteristic
273	C₄	C₁₃×#473	1	2	yes	—
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	C₁₃×#1954	1	1	yes	characteristic
277	C₂	C₁₃×#1976	1	1	yes	characteristic
278	C₂	C₁₃×#1979	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	C₂³×C₁₃	C₄×D₄
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
28. C₂×C₁₃×Q₁₆	C₈	semidirect
56. #9669	C₁₃	direct (N × H)
162. C₂×Q₁₆	C₈×C₁₃	semidirect
229. C₁₃	#9669	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×Q₁₆	2
C₁₃×#1564	2
C₁₃×#1479	2
C₁₃×#1461	2
C₁₃×#1450	2
#9669	13

Subgroups — C13×#9669 #15696008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9669 #15696008