Subgroups — C₁₃×#13363 #15699702

Order 3328.

450 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#357	C₂	1	2	yes	maximal
3	C₁₃×#1970	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1380	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	C₁₃×#500	—	2	4	no	—
10	C₂×C₁₃×#107	—	2	1	no	—
11	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#102	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#390	—	2	1	no	—
16	C₁₃×#357	—	2	4	no	—
17	C₂²×Q₈×C₁₃	—	4	1	no	—
18	C₂²×D₄×C₁₃	—	2	1	no	—
19	C₂×C₁₃×D₈	—	2	2	no	—
20	C₂×C₁₃×D₈	—	2	2	no	—
21	C₂×C₁₃×D₈	—	2	1	no	—
22	C₂×C₁₃×D₈	Q₈	1	2	yes	complemented
23	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	C₁₃×C₄⋊Q₈	—	2	4	no	—
26	C₁₃×C₄⋊Q₈	—	4	2	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
30	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
31	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×C₁₃×C₁₆	—	2	2	no	—
34	C₂×C₁₃×C₁₆	—	2	2	no	—
35	C₂×C₁₃×C₁₆	—	2	2	no	—
36	C₁₃×#107	—	2	4	no	—
37	C₁₃×#107	—	2	2	no	—
38	C₁₃×#107	—	2	2	no	—
39	C₁₃×#107	D₄	1	4	yes	—
40	C₁₃×#102	—	4	2	no	—
41	C₄×C₈×C₁₃	—	2	2	no	—
42	#13363	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂⁴×C₁₃	—	4	1	no	—
44	C₂×Q₈×C₁₃	—	4	4	no	—
45	C₂×Q₈×C₁₃	—	8	2	no	—
46	C₂×Q₈×C₁₃	—	8	2	no	—
47	C₂×D₄×C₁₃	—	2	2	no	—
48	C₂×D₄×C₁₃	—	4	1	no	—
49	C₂×D₄×C₁₃	—	4	1	no	—
50	C₂×D₄×C₁₃	—	4	1	no	—
51	C₂×D₄×C₁₃	—	4	1	no	—
52	C₂×D₄×C₁₃	—	4	1	no	—
53	C₂²×C₄×C₁₃	—	4	2	no	—
54	C₂²×C₄×C₁₃	—	4	1	no	—
55	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₁₃×D₈	—	2	4	no	—
59	C₁₃×D₈	—	4	2	no	—
60	C₂×C₈×C₁₃	—	2	1	no	—
61	C₂×C₈×C₁₃	—	2	1	no	—
62	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
63	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₁₃	#41	1	2	yes	—
66	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
67	C₁₃×C₄⋊C₄	—	2	4	no	—
68	C₁₃×C₄⋊C₄	—	4	4	no	—
69	C₁₃×C₄⋊C₄	—	4	2	no	—
70	C₁₃×C₄⋊C₄	—	4	1	no	—
71	C₁₃×C₄⋊C₄	—	4	1	no	—
72	C₁₃×C₂²⋊C₄	—	8	2	no	—
73	C₁₃×C₂²⋊C₄	—	8	2	no	—
74	C₄²×C₁₃	—	2	1	no	—
75	C₄²×C₁₃	—	2	1	no	—
76	C₁₃×C₁₆	—	4	2	no	—
77	C₂×#500	C₂×C₁₃	1	1	yes	characteristic, complemented
78	C₂×#357	C₂×C₁₃	1	2	yes	—
79	#1970	C₂×C₁₃	1	1	yes	characteristic
80	#1380	C₂×C₁₃	1	2	yes	complemented
81	#1374	C₂×C₁₃	1	1	yes	characteristic, complemented
82	C₂³×C₁₃	—	4	2	no	—
83	C₂³×C₁₃	—	8	1	no	—
84	C₂³×C₁₃	—	8	1	no	—
85	C₂³×C₁₃	—	8	1	no	—
86	C₂³×C₁₃	—	8	1	no	—
87	C₂³×C₁₃	—	8	1	no	—
88	C₂³×C₁₃	—	8	1	no	—
89	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
90	Q₈×C₁₃	—	8	8	no	—
91	D₄×C₁₃	—	4	2	no	—
92	D₄×C₁₃	—	4	2	no	—
93	D₄×C₁₃	—	8	1	no	—
94	D₄×C₁₃	—	8	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	4	4	no	—
100	C₂×C₄×C₁₃	—	8	2	no	—
101	C₂×C₄×C₁₃	—	8	2	no	—
102	C₂×C₄×C₁₃	—	8	1	no	—
103	C₂×C₄×C₁₃	—	8	1	no	—
104	C₂×C₄×C₁₃	—	8	1	no	—
105	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
106	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
107	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
113	C₈×C₁₃	—	2	2	no	—
114	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
115	C₈×C₁₃	#124	1	2	yes	—
116	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
117	C₂×C₄⋊Q₈	—	2	1	no	—
118	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
119	#500	—	2	4	no	—
120	C₂×#107	—	2	1	no	—
121	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
122	C₂×#107	C₂²×C₁₃	1	2	yes	—
123	C₂×#102	—	2	1	no	—
124	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
125	#390	—	2	1	no	—
126	#357	—	2	4	no	—
127	C₂²×C₁₃	—	8	2	no	—
128	C₂²×C₁₃	—	8	2	no	—
129	C₂²×C₁₃	—	8	2	no	—
130	C₂²×C₁₃	—	16	1	no	—
131	C₂²×C₁₃	—	16	1	no	—
132	C₂²×C₁₃	—	16	1	no	—
133	C₂²×C₁₃	—	16	1	no	—
134	C₂²×C₁₃	#447	1	1	yes	characteristic
135	C₂²×C₁₃	#474	1	1	yes	characteristic
136	C₂²×C₁₃	#486	1	1	yes	characteristic
137	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
138	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
139	C₂²×C₁₃	#509	1	1	yes	characteristic
140	C₂²×C₁₃	#510	1	1	yes	characteristic
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	2	2	no	—
143	C₄×C₁₃	—	8	4	no	—
144	C₄×C₁₃	—	16	1	no	—
145	C₄×C₁₃	#393	1	1	yes	characteristic
146	C₄×C₁₃	#448	1	1	yes	characteristic
147	C₄×C₁₃	#476	1	1	yes	characteristic
148	C₄×C₁₃	#489	1	1	yes	characteristic
149	C₂²×Q₈	—	4	1	no	—
150	C₂²×D₄	—	2	1	no	—
151	C₂×D₈	—	2	2	no	—
152	C₂×D₈	—	2	2	no	—
153	C₂×D₈	—	2	1	no	—
154	C₂×D₈	Q₈×C₁₃	1	2	yes	complemented
155	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
156	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
157	C₄⋊Q₈	—	2	4	no	—
158	C₄⋊Q₈	—	4	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	4	1	no	—
162	C₂×C₂²⋊C₄	—	4	1	no	—
163	C₂×C₂²⋊C₄	—	4	1	no	—
164	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	#107	—	2	4	no	—
169	#107	—	2	2	no	—
170	#107	—	2	2	no	—
171	#107	D₄×C₁₃	1	4	yes	—
172	#102	—	4	2	no	—
173	C₄×C₈	—	2	2	no	—
174	C₂×C₁₃	—	16	2	no	—
175	C₂×C₁₃	#2023	1	1	yes	characteristic
176	C₂×C₁₃	#2184	1	1	yes	characteristic
177	C₂×C₁₃	#2185	1	1	yes	characteristic
178	C₂×C₁₃	#2224	1	1	yes	characteristic
179	C₂×C₁₃	#2226	1	1	yes	characteristic
180	C₂×C₁₃	#2240	1	1	yes	characteristic
181	C₂×C₁₃	#2243	1	1	yes	characteristic
182	C₂⁴	—	4	1	no	—
183	C₂×Q₈	—	4	4	no	—
184	C₂×Q₈	—	8	2	no	—
185	C₂×Q₈	—	8	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂×D₄	—	4	1	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
195	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
197	D₈	—	2	4	no	—
198	D₈	—	4	2	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
202	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
203	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
204	C₂×C₈	C₁₃×#41	1	2	yes	—
205	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₂²⋊C₄	—	8	2	no	—
212	C₂²⋊C₄	—	8	2	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₁₆	—	4	2	no	—
216	C₁₃	#13363	1	1	yes	characteristic, direct product factor, Sylow
217	C₂³	—	4	2	no	—
218	C₂³	—	8	1	no	—
219	C₂³	—	8	1	no	—
220	C₂³	—	8	1	no	—
221	C₂³	—	8	1	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	8	1	no	—
224	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
225	Q₈	—	8	8	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	2	no	—
228	D₄	—	8	1	no	—
229	D₄	—	8	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	1	no	—
238	C₂×C₄	—	8	1	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	C₁₃×D₁₆	1	2	yes	—
241	C₂×C₄	C₁₃×SD₃₂	1	2	yes	—
242	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
243	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
244	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
245	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
246	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	C₁₃×C₂²⋊Q₈	1	2	yes	—
250	C₈	C₁₃×#124	1	2	yes	—
251	C₂²	—	8	2	no	—
252	C₂²	—	8	2	no	—
253	C₂²	—	8	2	no	—
254	C₂²	—	16	1	no	—
255	C₂²	—	16	1	no	—
256	C₂²	—	16	1	no	—
257	C₂²	—	16	1	no	—
258	C₂²	C₁₃×#447	1	1	yes	characteristic
259	C₂²	C₁₃×#474	1	1	yes	characteristic
260	C₂²	C₁₃×#486	1	1	yes	characteristic
261	C₂²	C₂×C₁₃×D₁₆	1	1	yes	characteristic
262	C₂²	C₂×C₁₃×SD₃₂	1	1	yes	characteristic
263	C₂²	C₁₃×#509	1	1	yes	characteristic
264	C₂²	C₁₃×#510	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	8	4	no	—
268	C₄	—	16	1	no	—
269	C₄	C₁₃×#393	1	1	yes	characteristic
270	C₄	C₁₃×#448	1	1	yes	characteristic
271	C₄	C₁₃×#476	1	1	yes	characteristic
272	C₄	C₁₃×#489	1	1	yes	characteristic
273	C₂	—	16	2	no	—
274	C₂	C₁₃×#2023	1	1	yes	characteristic
275	C₂	C₁₃×#2184	1	1	yes	characteristic
276	C₂	C₁₃×#2185	1	1	yes	characteristic
277	C₂	C₁₃×#2224	1	1	yes	characteristic
278	C₂	C₁₃×#2226	1	1	yes	characteristic
279	C₂	C₁₃×#2240	1	1	yes	characteristic
280	C₂	C₁₃×#2243	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
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
1. C₂×C₁₃×#500	C₂	semidirect
4. C₁₃×#1380	C₂	semidirect
5. C₁₃×#1374	C₂	semidirect
22. C₂×C₁₃×D₈	Q₈	semidirect
42. #13363	C₁₃	direct (N × H)
77. C₂×#500	C₂×C₁₃	semidirect
80. #1380	C₂×C₁₃	semidirect
81. #1374	C₂×C₁₃	semidirect
154. C₂×D₈	Q₈×C₁₃	semidirect
216. C₁₃	#13363	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×#500	2
C₂×C₁₃×#357	2
C₁₃×#1970	2
C₁₃×#1380	2
C₁₃×#1374	2
#13363	13

Subgroups — C13×#13363 #15699702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13363 #15699702