Subgroups — C₁₃×#13407 #15699746

Order 3328.

566 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×C₁₃×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#357	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₃×#357	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1942	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×D₈	—	2	1	no	—
7	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
8	C₂×C₁₃×#127	—	2	1	no	—
9	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
10	C₁₃×#493	—	2	4	no	—
11	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#389	—	2	1	no	—
15	C₁₃×#357	—	2	4	no	—
16	C₁₃×#357	—	2	4	no	—
17	C₁₃×#340	—	2	1	no	—
18	C₂²×D₄×C₁₃	—	2	2	no	—
19	C₂²×D₄×C₁₃	—	4	1	no	—
20	C₂²×D₄×C₁₃	—	4	1	no	—
21	C₂×C₁₃×D₈	—	2	4	no	—
22	C₂×C₁₃×D₈	—	2	2	no	—
23	C₂×C₁₃×D₈	—	2	2	no	—
24	C₂×C₁₃×D₈	—	2	2	no	—
25	C₂×C₁₃×D₈	—	2	2	no	—
26	C₂×C₁₃×D₈	—	2	2	no	—
27	C₂×C₁₃×D₈	—	4	2	no	—
28	C₂×C₁₃×D₈	—	4	2	no	—
29	C₂×C₁₃×D₈	D₄	1	4	yes	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×#127	—	2	4	no	—
33	C₁₃×#127	—	4	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
36	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₁₃×C₁₆	—	2	2	no	—
39	C₂×C₁₃×C₁₆	—	2	2	no	—
40	C₂×C₁₃×C₁₆	—	2	2	no	—
41	C₁₃×#107	—	2	1	no	—
42	C₁₃×#107	—	2	2	no	—
43	C₁₃×#107	D₄	1	2	yes	complemented
44	C₄×C₈×C₁₃	—	2	2	no	—
45	C₁₃×#95	—	4	1	no	—
46	#13407	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂⁴×C₁₃	—	4	2	no	—
48	C₂×D₄×C₁₃	—	2	4	no	—
49	C₂×D₄×C₁₃	—	4	4	no	—
50	C₂×D₄×C₁₃	—	4	4	no	—
51	C₂×D₄×C₁₃	—	4	2	no	—
52	C₂×D₄×C₁₃	—	4	2	no	—
53	C₂×D₄×C₁₃	—	4	2	no	—
54	C₂×D₄×C₁₃	—	4	2	no	—
55	C₂×D₄×C₁₃	—	4	2	no	—
56	C₂×D₄×C₁₃	—	8	2	no	—
57	C₂×D₄×C₁₃	—	8	2	no	—
58	C₂×D₄×C₁₃	—	8	2	no	—
59	C₂×D₄×C₁₃	—	8	2	no	—
60	C₂²×C₄×C₁₃	—	4	1	no	—
61	C₂²×C₄×C₁₃	—	4	1	no	—
62	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₁₃×D₈	—	2	4	no	—
66	C₁₃×D₈	—	2	4	no	—
67	C₁₃×D₈	—	4	8	no	—
68	C₁₃×D₈	—	4	2	no	—
69	C₁₃×D₈	—	4	2	no	—
70	C₂×C₈×C₁₃	—	2	1	no	—
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
73	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₃	#41	1	2	yes	—
76	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
77	C₁₃×C₄⋊C₄	—	2	2	no	—
78	C₁₃×C₄⋊C₄	—	4	1	no	—
79	C₁₃×C₄⋊C₄	—	4	1	no	—
80	C₁₃×C₄⋊C₄	—	4	1	no	—
81	C₁₃×C₄⋊C₄	—	8	2	no	—
82	C₄²×C₁₃	—	2	1	no	—
83	C₄²×C₁₃	—	2	1	no	—
84	C₁₃×C₁₆	—	4	2	no	—
85	C₂×#493	C₂×C₁₃	1	1	yes	characteristic
86	C₂×#357	C₂×C₁₃	1	2	yes	complemented
87	C₂×#357	C₂×C₁₃	1	2	yes	complemented
88	#1942	C₂×C₁₃	1	1	yes	characteristic, complemented
89	#1374	C₂×C₁₃	1	1	yes	characteristic, complemented
90	C₂³×C₁₃	—	4	4	no	—
91	C₂³×C₁₃	—	8	2	no	—
92	C₂³×C₁₃	—	8	2	no	—
93	C₂³×C₁₃	—	8	2	no	—
94	C₂³×C₁₃	—	8	2	no	—
95	C₂³×C₁₃	—	8	2	no	—
96	C₂³×C₁₃	—	8	2	no	—
97	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
98	D₄×C₁₃	—	4	4	no	—
99	D₄×C₁₃	—	4	4	no	—
100	D₄×C₁₃	—	8	8	no	—
101	D₄×C₁₃	—	8	8	no	—
102	D₄×C₁₃	—	8	2	no	—
103	D₄×C₁₃	—	8	2	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	8	1	no	—
110	C₂×C₄×C₁₃	—	8	1	no	—
111	C₂×C₄×C₁₃	—	8	1	no	—
112	C₂×C₄×C₁₃	—	8	1	no	—
113	C₂×C₄×C₁₃	—	8	1	no	—
114	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
115	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
116	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₈×C₁₃	—	2	2	no	—
123	C₈×C₁₃	#121	1	2	yes	—
124	C₈×C₁₃	#123	1	2	yes	—
125	C₂²×D₈	—	2	1	no	—
126	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
127	C₂×#127	—	2	1	no	—
128	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
129	#493	—	2	4	no	—
130	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
133	#389	—	2	1	no	—
134	#357	—	2	4	no	—
135	#357	—	2	4	no	—
136	#340	—	2	1	no	—
137	C₂²×C₁₃	—	8	4	no	—
138	C₂²×C₁₃	—	8	4	no	—
139	C₂²×C₁₃	—	8	4	no	—
140	C₂²×C₁₃	—	16	2	no	—
141	C₂²×C₁₃	—	16	2	no	—
142	C₂²×C₁₃	—	16	2	no	—
143	C₂²×C₁₃	—	16	2	no	—
144	C₂²×C₁₃	#447	1	1	yes	characteristic
145	C₂²×C₁₃	#459	1	1	yes	characteristic
146	C₂²×C₁₃	#486	1	1	yes	characteristic
147	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
149	C₂²×C₁₃	#509	1	1	yes	characteristic
150	C₂²×C₁₃	#509	1	1	yes	characteristic
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	8	2	no	—
154	C₄×C₁₃	—	16	1	no	—
155	C₄×C₁₃	#394	1	1	yes	characteristic
156	C₄×C₁₃	#450	1	1	yes	characteristic
157	C₄×C₁₃	#460	1	1	yes	characteristic
158	C₄×C₁₃	#490	1	1	yes	characteristic
159	C₂²×D₄	—	2	2	no	—
160	C₂²×D₄	—	4	1	no	—
161	C₂²×D₄	—	4	1	no	—
162	C₂×D₈	—	2	4	no	—
163	C₂×D₈	—	2	2	no	—
164	C₂×D₈	—	2	2	no	—
165	C₂×D₈	—	2	2	no	—
166	C₂×D₈	—	2	2	no	—
167	C₂×D₈	—	2	2	no	—
168	C₂×D₈	—	4	2	no	—
169	C₂×D₈	—	4	2	no	—
170	C₂×D₈	D₄×C₁₃	1	4	yes	—
171	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
173	#127	—	2	4	no	—
174	#127	—	4	2	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	4	1	no	—
178	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	2	2	no	—
181	C₂×C₁₆	—	2	2	no	—
182	#107	—	2	1	no	—
183	#107	—	2	2	no	—
184	#107	D₄×C₁₃	1	2	yes	complemented
185	C₄×C₈	—	2	2	no	—
186	#95	—	4	1	no	—
187	C₂×C₁₃	—	16	4	no	—
188	C₂×C₁₃	#2041	1	1	yes	characteristic
189	C₂×C₁₃	#2184	1	1	yes	characteristic
190	C₂×C₁₃	#2187	1	1	yes	characteristic
191	C₂×C₁₃	#2206	1	1	yes	characteristic
192	C₂×C₁₃	#2207	1	1	yes	characteristic
193	C₂×C₁₃	#2240	1	1	yes	characteristic
194	C₂×C₁₃	#2244	1	1	yes	characteristic
195	C₂⁴	—	4	2	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	4	4	no	—
198	C₂×D₄	—	4	4	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	8	2	no	—
205	C₂×D₄	—	8	2	no	—
206	C₂×D₄	—	8	2	no	—
207	C₂×D₄	—	8	2	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
211	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
213	D₈	—	2	4	no	—
214	D₈	—	2	4	no	—
215	D₈	—	4	8	no	—
216	D₈	—	4	2	no	—
217	D₈	—	4	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
221	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
222	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
223	C₂×C₈	C₁₃×#41	1	2	yes	—
224	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	8	2	no	—
230	C₄²	—	2	1	no	—
231	C₄²	—	2	1	no	—
232	C₁₆	—	4	2	no	—
233	C₁₃	#13407	1	1	yes	characteristic, direct product factor, Sylow
234	C₂³	—	4	4	no	—
235	C₂³	—	8	2	no	—
236	C₂³	—	8	2	no	—
237	C₂³	—	8	2	no	—
238	C₂³	—	8	2	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
242	D₄	—	4	4	no	—
243	D₄	—	4	4	no	—
244	D₄	—	8	8	no	—
245	D₄	—	8	8	no	—
246	D₄	—	8	2	no	—
247	D₄	—	8	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	8	1	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	8	1	no	—
256	C₂×C₄	—	8	1	no	—
257	C₂×C₄	—	8	1	no	—
258	C₂×C₄	C₁₃×D₁₆	1	2	yes	—
259	C₂×C₄	C₁₃×SD₃₂	1	2	yes	—
260	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
261	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
262	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
263	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
264	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
265	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	C₁₃×#121	1	2	yes	—
268	C₈	C₁₃×#123	1	2	yes	—
269	C₂²	—	8	4	no	—
270	C₂²	—	8	4	no	—
271	C₂²	—	8	4	no	—
272	C₂²	—	16	2	no	—
273	C₂²	—	16	2	no	—
274	C₂²	—	16	2	no	—
275	C₂²	—	16	2	no	—
276	C₂²	C₁₃×#447	1	1	yes	characteristic
277	C₂²	C₁₃×#459	1	1	yes	characteristic
278	C₂²	C₁₃×#486	1	1	yes	characteristic
279	C₂²	C₂×C₁₃×D₁₆	1	1	yes	characteristic
280	C₂²	C₂×C₁₃×SD₃₂	1	1	yes	characteristic
281	C₂²	C₁₃×#509	1	1	yes	characteristic
282	C₂²	C₁₃×#509	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	8	2	no	—
286	C₄	—	16	1	no	—
287	C₄	C₁₃×#394	1	1	yes	characteristic
288	C₄	C₁₃×#450	1	1	yes	characteristic
289	C₄	C₁₃×#460	1	1	yes	characteristic
290	C₄	C₁₃×#490	1	1	yes	characteristic
291	C₂	—	16	4	no	—
292	C₂	C₁₃×#2041	1	1	yes	characteristic
293	C₂	C₁₃×#2184	1	1	yes	characteristic
294	C₂	C₁₃×#2187	1	1	yes	characteristic
295	C₂	C₁₃×#2206	1	1	yes	characteristic
296	C₂	C₁₃×#2207	1	1	yes	characteristic
297	C₂	C₁₃×#2240	1	1	yes	characteristic
298	C₂	C₁₃×#2244	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#493	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#357	C₂	semidirect
3. C₂×C₁₃×#357	C₂	semidirect
4. C₁₃×#1942	C₂	semidirect
5. C₁₃×#1374	C₂	semidirect
43. C₁₃×#107	D₄	semidirect
46. #13407	C₁₃	direct (N × H)
86. C₂×#357	C₂×C₁₃	semidirect
87. C₂×#357	C₂×C₁₃	semidirect
88. #1942	C₂×C₁₃	semidirect
89. #1374	C₂×C₁₃	semidirect
184. #107	D₄×C₁₃	semidirect
233. C₁₃	#13407	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×#493	2
C₂×C₁₃×#357	2
C₁₃×#1942	2
C₁₃×#1374	2
#13407	13

Subgroups — C13×#13407 #15699746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13407 #15699746