Subgroups — C₁₃×#9663 #15696002

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 274 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×#1954	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#322	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	2	no	—
9	C₄×C₁₃×D₈	—	2	2	no	—
10	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₁₃×#102	C₄	1	2	yes	—
13	C₂×C₁₃×C₂²⋊C₈	—	2	4	no	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#335	—	2	2	no	—
18	C₁₃×D₄⋊C₈	—	2	4	no	—
19	C₁₃×#322	—	2	2	no	—
20	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₃	—	2	2	no	—
22	C₂×C₁₃×D₈	—	2	4	no	—
23	C₂×C₁₃×D₈	—	2	2	no	—
24	C₂²×C₈×C₁₃	—	2	1	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	D₄	1	2	yes	—
29	C₄×D₄×C₁₃	—	2	2	no	—
30	C₄×D₄×C₁₃	—	2	2	no	—
31	C₄×D₄×C₁₃	—	2	2	no	—
32	C₄×D₄×C₁₃	—	2	2	no	—
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×#107	—	2	2	no	—
37	C₁₃×C₄⋊C₈	—	2	2	no	—
38	C₁₃×C₄⋊C₈	—	2	2	no	—
39	C₁₃×#102	—	2	4	no	—
40	C₁₃×C₂²⋊C₈	—	4	8	no	—
41	C₄×C₈×C₁₃	—	2	1	no	—
42	C₄×C₈×C₁₃	—	2	1	no	—
43	C₄×C₈×C₁₃	—	2	1	no	—
44	C₄×C₈×C₁₃	D₄	1	2	yes	—
45	C₄×C₈×C₁₃	D₄	1	2	yes	—
46	C₄×C₈×C₁₃	D₄	1	2	yes	—
47	#9663	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂⁴×C₁₃	—	2	2	no	—
49	C₂×D₄×C₁₃	—	2	2	no	—
50	C₂×D₄×C₁₃	—	2	2	no	—
51	C₂×D₄×C₁₃	—	2	2	no	—
52	C₂×D₄×C₁₃	—	2	2	no	—
53	C₂×D₄×C₁₃	—	2	2	no	—
54	C₂×D₄×C₁₃	M₁₆	1	4	yes	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	4	2	no	—
57	C₂²×C₄×C₁₃	—	4	2	no	—
58	C₂²×C₄×C₁₃	—	4	4	no	—
59	C₂²×C₄×C₁₃	—	4	4	no	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
63	C₁₃×D₈	—	2	8	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	4	2	no	—
70	C₂×C₈×C₁₃	—	4	2	no	—
71	C₂×C₈×C₁₃	—	4	4	no	—
72	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₃	D₈	1	4	yes	complemented
75	C₁₃×C₄⋊C₄	—	2	2	no	—
76	C₁₃×C₄⋊C₄	—	2	2	no	—
77	C₁₃×C₂²⋊C₄	—	4	2	no	—
78	C₁₃×C₂²⋊C₄	—	4	2	no	—
79	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₃	#41	1	1	yes	characteristic
83	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
84	#1954	C₂×C₁₃	1	2	yes	complemented
85	C₂×#335	C₂×C₁₃	1	1	yes	characteristic, complemented
86	C₂×D₄⋊C₈	C₂×C₁₃	1	2	yes	complemented
87	C₂×#322	C₂×C₁₃	1	1	yes	characteristic, complemented
88	C₂³×C₁₃	—	2	4	no	—
89	C₂³×C₁₃	—	4	2	no	—
90	C₂³×C₁₃	—	4	2	no	—
91	C₂³×C₁₃	—	4	4	no	—
92	C₂³×C₁₃	—	4	4	no	—
93	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
94	D₄×C₁₃	—	2	4	no	—
95	D₄×C₁₃	—	2	4	no	—
96	D₄×C₁₃	—	4	2	no	—
97	D₄×C₁₃	—	4	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	4	4	no	—
101	C₂×C₄×C₁₃	—	8	4	no	—
102	C₂×C₄×C₁₃	—	8	2	no	—
103	C₂×C₄×C₁₃	—	8	2	no	—
104	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₁₃	#131	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₁₃	#135	1	2	yes	—
114	C₈×C₁₃	—	2	2	no	—
115	C₈×C₁₃	—	2	2	no	—
116	C₈×C₁₃	—	2	4	no	—
117	C₈×C₁₃	—	4	2	no	—
118	C₈×C₁₃	—	8	2	no	—
119	C₂²×D₈	C₄×C₁₃	1	1	yes	characteristic
120	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
121	C₄×D₈	—	2	2	no	—
122	C₄×D₈	—	2	2	no	—
123	C₂×#107	C₄×C₁₃	1	1	yes	characteristic
124	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
125	C₂×#102	C₄×C₁₃	1	2	yes	—
126	C₂×C₂²⋊C₈	—	2	4	no	—
127	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
130	#335	—	2	2	no	—
131	D₄⋊C₈	—	2	4	no	—
132	#322	—	2	2	no	—
133	C₂²×C₁₃	—	4	8	no	—
134	C₂²×C₁₃	—	4	4	no	—
135	C₂²×C₁₃	—	8	4	no	—
136	C₂²×C₁₃	—	8	2	no	—
137	C₂²×C₁₃	—	8	2	no	—
138	C₂²×C₁₃	#390	1	1	yes	characteristic
139	C₂²×C₁₃	#435	1	2	yes	—
140	C₂²×C₁₃	C₄×D₈	1	2	yes	—
141	C₂²×C₁₃	#444	1	2	yes	—
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	—	2	1	no	—
144	C₄×C₁₃	—	8	2	no	—
145	C₄×C₁₃	#407	1	1	yes	characteristic
146	C₄×C₁₃	#408	1	1	yes	characteristic
147	C₄×C₁₃	#435	1	2	yes	—
148	C₄×C₁₃	#466	1	2	yes	—
149	C₄×C₁₃	#493	1	1	yes	characteristic
150	C₄×C₁₃	#495	1	1	yes	characteristic
151	C₂²×D₄	C₂×C₄×C₁₃	1	2	yes	—
152	C₂³×C₄	—	2	2	no	—
153	C₂×D₈	—	2	4	no	—
154	C₂×D₈	—	2	2	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
158	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
159	C₂²×C₈	D₄×C₁₃	1	2	yes	—
160	C₄×D₄	—	2	2	no	—
161	C₄×D₄	—	2	2	no	—
162	C₄×D₄	—	2	2	no	—
163	C₄×D₄	—	2	2	no	—
164	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
165	C₂×C₂²⋊C₄	—	2	2	no	—
166	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
167	#107	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	#102	—	2	4	no	—
171	C₂²⋊C₈	—	4	8	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	—	2	1	no	—
174	C₄×C₈	—	2	1	no	—
175	C₄×C₈	D₄×C₁₃	1	2	yes	—
176	C₄×C₈	D₄×C₁₃	1	2	yes	—
177	C₄×C₈	D₄×C₁₃	1	2	yes	—
178	C₂×C₁₃	—	8	4	no	—
179	C₂×C₁₃	#1581	1	4	yes	—
180	C₂×C₁₃	#1955	1	1	yes	characteristic
181	C₂×C₁₃	#1970	1	1	yes	characteristic
182	C₂×C₁₃	#1975	1	1	yes	characteristic
183	C₂⁴	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	C₁₃×M₁₆	1	4	yes	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	4	no	—
194	C₂²×C₄	—	4	4	no	—
195	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
197	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
198	D₈	—	2	8	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
208	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
209	C₂×C₈	C₁₃×D₈	1	4	yes	complemented
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	4	2	no	—
213	C₂²⋊C₄	—	4	2	no	—
214	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
215	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
216	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
217	C₄²	C₁₃×#41	1	1	yes	characteristic
218	C₁₃	#9663	1	1	yes	characteristic, direct product factor, Sylow
219	C₂³	—	2	4	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	2	no	—
222	C₂³	—	4	4	no	—
223	C₂³	—	4	4	no	—
224	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
225	D₄	—	2	4	no	—
226	D₄	—	2	4	no	—
227	D₄	—	4	2	no	—
228	D₄	—	4	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
237	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
238	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
239	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
240	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
241	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
242	C₂×C₄	C₁₃×#131	1	2	yes	—
243	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
244	C₂×C₄	C₁₃×#135	1	2	yes	—
245	C₈	—	2	2	no	—
246	C₈	—	2	2	no	—
247	C₈	—	2	4	no	—
248	C₈	—	4	2	no	—
249	C₈	—	8	2	no	—
250	C₂²	—	4	8	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	8	4	no	—
253	C₂²	—	8	2	no	—
254	C₂²	—	8	2	no	—
255	C₂²	C₁₃×#390	1	1	yes	characteristic
256	C₂²	C₁₃×#435	1	2	yes	—
257	C₂²	C₄×C₁₃×D₈	1	2	yes	—
258	C₂²	C₁₃×#444	1	2	yes	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	8	2	no	—
262	C₄	C₁₃×#407	1	1	yes	characteristic
263	C₄	C₁₃×#408	1	1	yes	characteristic
264	C₄	C₁₃×#435	1	2	yes	—
265	C₄	C₁₃×#466	1	2	yes	—
266	C₄	C₁₃×#493	1	1	yes	characteristic
267	C₄	C₁₃×#495	1	1	yes	characteristic
268	C₂	—	8	4	no	—
269	C₂	C₁₃×#1581	1	4	yes	—
270	C₂	C₁₃×#1955	1	1	yes	characteristic
271	C₂	C₁₃×#1970	1	1	yes	characteristic
272	C₂	C₁₃×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂²×C₁₃×D₈	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#1954	C₂	semidirect
3. C₂×C₁₃×#335	C₂	semidirect
4. C₂×C₁₃×D₄⋊C₈	C₂	semidirect
5. C₂×C₁₃×#322	C₂	semidirect
47. #9663	C₁₃	direct (N × H)
74. C₂×C₈×C₁₃	D₈	semidirect
84. #1954	C₂×C₁₃	semidirect
85. C₂×#335	C₂×C₁₃	semidirect
86. C₂×D₄⋊C₈	C₂×C₁₃	semidirect
87. C₂×#322	C₂×C₁₃	semidirect
209. C₂×C₈	C₁₃×D₈	semidirect
218. C₁₃	#9663	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×#1954	2
C₂×C₁₃×#335	2
C₂×C₁₃×D₄⋊C₈	2
C₂×C₁₃×#322	2
#9663	13

Subgroups — C13×#9663 #15696002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9663 #15696002