Subgroups — C₁₃×#9657 #15695996

Order 3328.

606 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×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×C₈²×C₁₃	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	C₈²×C₁₃	—	2	2	no	—
20	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₃	—	2	2	no	—
22	C₂×C₁₃×D₈	—	2	2	no	—
23	C₂×C₁₃×D₈	—	2	2	no	—
24	C₂×C₁₃×D₈	C₈	1	4	yes	complemented
25	C₂²×C₈×C₁₃	—	2	1	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	2	yes	—
30	C₄×D₄×C₁₃	—	2	2	no	—
31	C₄×D₄×C₁₃	—	2	2	no	—
32	C₄×D₄×C₁₃	—	2	2	no	—
33	C₄×D₄×C₁₃	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₁₃×#107	—	2	2	no	—
38	C₁₃×C₄⋊C₈	—	2	2	no	—
39	C₁₃×C₄⋊C₈	—	2	2	no	—
40	C₁₃×#102	—	2	4	no	—
41	C₁₃×C₂²⋊C₈	—	4	8	no	—
42	C₄×C₈×C₁₃	—	2	1	no	—
43	C₄×C₈×C₁₃	—	2	1	no	—
44	C₄×C₈×C₁₃	—	2	1	no	—
45	C₄×C₈×C₁₃	D₄	1	2	yes	—
46	C₄×C₈×C₁₃	D₄	1	2	yes	—
47	C₄×C₈×C₁₃	D₄	1	2	yes	—
48	#9657	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×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	—	2	2	no	—
55	C₂×D₄×C₁₃	C₂×C₈	1	4	yes	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	4	2	no	—
58	C₂²×C₄×C₁₃	—	4	2	no	—
59	C₂²×C₄×C₁₃	—	4	4	no	—
60	C₂²×C₄×C₁₃	—	4	4	no	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
64	C₁₃×D₈	—	2	8	no	—
65	C₂×C₈×C₁₃	—	2	4	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	1	no	—
69	C₂×C₈×C₁₃	—	2	1	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	4	2	no	—
72	C₂×C₈×C₁₃	—	4	4	no	—
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
77	C₂×C₈×C₁₃	D₈	1	4	yes	complemented
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₂²⋊C₄	—	4	2	no	—
81	C₁₃×C₂²⋊C₄	—	4	2	no	—
82	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₃	#41	1	1	yes	characteristic
86	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
87	#1954	C₂×C₁₃	1	2	yes	complemented
88	C₂×#335	C₂×C₁₃	1	1	yes	characteristic, complemented
89	C₂×D₄⋊C₈	C₂×C₁₃	1	2	yes	complemented
90	C₂×C₈²	C₂×C₁₃	1	1	yes	characteristic, complemented
91	C₂³×C₁₃	—	2	4	no	—
92	C₂³×C₁₃	—	4	2	no	—
93	C₂³×C₁₃	—	4	2	no	—
94	C₂³×C₁₃	—	4	4	no	—
95	C₂³×C₁₃	—	4	4	no	—
96	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
97	D₄×C₁₃	—	2	4	no	—
98	D₄×C₁₃	—	2	4	no	—
99	D₄×C₁₃	—	4	2	no	—
100	D₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	—	8	4	no	—
105	C₂×C₄×C₁₃	—	8	2	no	—
106	C₂×C₄×C₁₃	—	8	2	no	—
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₁₃	#135	1	2	yes	—
119	C₈×C₁₃	—	2	4	no	—
120	C₈×C₁₃	—	2	2	no	—
121	C₈×C₁₃	—	2	4	no	—
122	C₈×C₁₃	—	8	2	no	—
123	C₈×C₁₃	C₂²⋊C₈	1	4	yes	complemented
124	C₂²×D₈	C₄×C₁₃	1	1	yes	characteristic
125	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
126	C₄×D₈	—	2	2	no	—
127	C₄×D₈	—	2	2	no	—
128	C₂×#107	C₄×C₁₃	1	1	yes	characteristic
129	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
130	C₂×#102	C₄×C₁₃	1	2	yes	—
131	C₂×C₂²⋊C₈	—	2	4	no	—
132	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
135	#335	—	2	2	no	—
136	D₄⋊C₈	—	2	4	no	—
137	C₈²	—	2	2	no	—
138	C₂²×C₁₃	—	4	8	no	—
139	C₂²×C₁₃	—	4	4	no	—
140	C₂²×C₁₃	—	8	4	no	—
141	C₂²×C₁₃	—	8	2	no	—
142	C₂²×C₁₃	—	8	2	no	—
143	C₂²×C₁₃	#390	1	1	yes	characteristic
144	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
145	C₂²×C₁₃	#436	1	1	yes	characteristic
146	C₂²×C₁₃	C₄×D₈	1	2	yes	—
147	C₂²×C₁₃	#443	1	2	yes	—
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	8	2	no	—
151	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
152	C₄×C₁₃	#408	1	1	yes	characteristic
153	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
154	C₄×C₁₃	#436	1	1	yes	characteristic
155	C₄×C₁₃	#466	1	2	yes	—
156	C₄×C₁₃	#493	1	1	yes	characteristic
157	C₄×C₁₃	#495	1	1	yes	characteristic
158	C₂²×D₄	C₂×C₄×C₁₃	1	2	yes	—
159	C₂³×C₄	—	2	2	no	—
160	C₂×D₈	—	2	2	no	—
161	C₂×D₈	—	2	2	no	—
162	C₂×D₈	C₈×C₁₃	1	4	yes	complemented
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
166	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
167	C₂²×C₈	D₄×C₁₃	1	2	yes	—
168	C₄×D₄	—	2	2	no	—
169	C₄×D₄	—	2	2	no	—
170	C₄×D₄	—	2	2	no	—
171	C₄×D₄	—	2	2	no	—
172	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
173	C₂×C₂²⋊C₄	—	2	2	no	—
174	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
175	#107	—	2	2	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	2	2	no	—
178	#102	—	2	4	no	—
179	C₂²⋊C₈	—	4	8	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	D₄×C₁₃	1	2	yes	—
184	C₄×C₈	D₄×C₁₃	1	2	yes	—
185	C₄×C₈	D₄×C₁₃	1	2	yes	—
186	C₂×C₁₃	—	8	4	no	—
187	C₂×C₁₃	C₈×D₈	1	2	yes	—
188	C₂×C₁₃	#1589	1	2	yes	—
189	C₂×C₁₃	#1954	1	1	yes	characteristic
190	C₂×C₁₃	#1970	1	1	yes	characteristic
191	C₂×C₁₃	#1974	1	1	yes	characteristic
192	C₂⁴	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	C₂×C₈×C₁₃	1	4	yes	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	4	4	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
207	D₈	—	2	8	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
217	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
218	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
219	C₂×C₈	C₁₃×M₁₆	1	2	yes	—
220	C₂×C₈	C₁₃×D₈	1	4	yes	complemented
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₂²⋊C₄	—	4	2	no	—
224	C₂²⋊C₄	—	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	#9657	1	1	yes	characteristic, direct product factor, Sylow
230	C₂³	—	2	4	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	4	no	—
235	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
236	D₄	—	2	4	no	—
237	D₄	—	2	4	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
247	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
248	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
249	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
250	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
251	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
253	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
254	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
255	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
256	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
257	C₂×C₄	C₁₃×#135	1	2	yes	—
258	C₈	—	2	4	no	—
259	C₈	—	2	2	no	—
260	C₈	—	2	4	no	—
261	C₈	—	8	2	no	—
262	C₈	C₁₃×C₂²⋊C₈	1	4	yes	complemented
263	C₂²	—	4	8	no	—
264	C₂²	—	4	4	no	—
265	C₂²	—	8	4	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	2	no	—
268	C₂²	C₁₃×#390	1	1	yes	characteristic
269	C₂²	C₈×D₄×C₁₃	1	1	yes	characteristic
270	C₂²	C₁₃×#436	1	1	yes	characteristic
271	C₂²	C₄×C₁₃×D₈	1	2	yes	—
272	C₂²	C₁₃×#443	1	2	yes	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	8	2	no	—
276	C₄	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
277	C₄	C₁₃×#408	1	1	yes	characteristic
278	C₄	C₈×D₄×C₁₃	1	1	yes	characteristic
279	C₄	C₁₃×#436	1	1	yes	characteristic
280	C₄	C₁₃×#466	1	2	yes	—
281	C₄	C₁₃×#493	1	1	yes	characteristic
282	C₄	C₁₃×#495	1	1	yes	characteristic
283	C₂	—	8	4	no	—
284	C₂	C₈×C₁₃×D₈	1	2	yes	—
285	C₂	C₁₃×#1589	1	2	yes	—
286	C₂	C₁₃×#1954	1	1	yes	characteristic
287	C₂	C₁₃×#1970	1	1	yes	characteristic
288	C₂	C₁₃×#1974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂²×C₁₃×D₈	C₄
LCS	G ▷ C₂×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×C₈²×C₁₃	C₂	semidirect
24. C₂×C₁₃×D₈	C₈	semidirect
48. #9657	C₁₃	direct (N × H)
77. C₂×C₈×C₁₃	D₈	semidirect
87. #1954	C₂×C₁₃	semidirect
88. C₂×#335	C₂×C₁₃	semidirect
89. C₂×D₄⋊C₈	C₂×C₁₃	semidirect
90. C₂×C₈²	C₂×C₁₃	semidirect
123. C₈×C₁₃	C₂²⋊C₈	semidirect
162. C₂×D₈	C₈×C₁₃	semidirect
220. C₂×C₈	C₁₃×D₈	semidirect
229. C₁₃	#9657	direct (N × H)
262. C₈	C₁₃×C₂²⋊C₈	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×#1954	2
C₂×C₁₃×#335	2
C₂×C₁₃×D₄⋊C₈	2
C₂×C₈²×C₁₃	2
#9657	13

Subgroups — C13×#9657 #15695996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9657 #15695996