Subgroups — C₁₃×#13151 #15699490

Order 3328.

332 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#359	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2138	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1974	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1945	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
13	C₁₃×#427	C₂²	1	1	yes	characteristic
14	C₁₃×#424	—	2	1	no	—
15	C₂×C₁₃×#104	—	2	1	no	—
16	C₁₃×#408	—	2	1	no	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#359	—	2	2	no	—
19	C₁₃×#358	C₄	1	2	yes	complemented
20	C₁₃×#357	C₄	1	2	yes	complemented
21	C₄×C₁₃×C₁₆	—	2	2	no	—
22	C₁₃×#339	—	2	1	no	—
23	C₂×C₁₃×#41	—	2	1	no	—
24	C₁₃×#135	—	2	1	no	—
25	C₁₃×#135	—	2	1	no	—
26	C₁₃×#135	—	2	1	no	—
27	C₁₃×#135	—	2	1	no	—
28	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₃×SD₁₆	—	2	1	no	—
30	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₃×M₁₆	—	2	1	no	—
32	C₂×C₁₃×M₁₆	—	2	1	no	—
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×#117	—	2	1	no	—
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₂×C₁₃×C₁₆	—	2	1	no	—
38	C₂×C₁₃×C₁₆	—	2	1	no	—
39	C₂×C₁₃×C₁₆	—	2	1	no	—
40	C₂×C₁₃×C₁₆	—	2	1	no	—
41	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
42	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
43	C₁₃×#108	—	2	1	no	—
44	C₁₃×#108	—	2	1	no	—
45	C₁₃×#108	—	2	1	no	—
46	C₁₃×#108	—	2	1	no	—
47	C₁₃×#107	C₂×C₄	1	2	yes	—
48	C₁₃×#106	—	2	1	no	—
49	C₁₃×C₄⋊C₈	—	4	2	no	—
50	C₁₃×#104	—	4	2	no	—
51	C₁₃×C₂²⋊C₈	—	4	2	no	—
52	C₄×C₈×C₁₃	—	2	1	no	—
53	C₄×C₈×C₁₃	—	2	1	no	—
54	#13151	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₁₃×#41	—	4	1	no	—
56	C₁₃×#41	—	4	1	no	—
57	C₁₃×#41	—	4	1	no	—
58	C₁₃×#41	—	4	1	no	—
59	C₂×Q₈×C₁₃	—	2	1	no	—
60	C₂×D₄×C₁₃	—	2	1	no	—
61	C₂×D₄×C₁₃	—	4	1	no	—
62	C₂²×C₄×C₁₃	—	2	1	no	—
63	C₂²×C₄×C₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₁₃×Q₁₆	—	2	1	no	—
66	C₁₃×Q₁₆	—	2	1	no	—
67	C₁₃×SD₁₆	—	4	1	no	—
68	C₁₃×SD₁₆	—	4	1	no	—
69	C₁₃×D₈	—	2	1	no	—
70	C₁₃×D₈	—	2	1	no	—
71	C₁₃×M₁₆	—	4	1	no	—
72	C₁₃×M₁₆	—	4	1	no	—
73	C₁₃×M₁₆	—	4	1	no	—
74	C₁₃×M₁₆	—	4	1	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	—	4	1	no	—
78	C₂×C₈×C₁₃	—	4	1	no	—
79	C₂×C₈×C₁₃	D₈	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₂²⋊C₄	—	4	1	no	—
88	C₄²×C₁₃	—	2	1	no	—
89	C₄²×C₁₃	—	4	1	no	—
90	C₄²×C₁₃	SD₁₆	1	2	yes	—
91	C₁₃×C₁₆	—	2	2	no	—
92	C₁₃×C₁₆	—	2	2	no	—
93	C₂×#359	C₂×C₁₃	1	1	yes	characteristic
94	#2138	C₂×C₁₃	1	1	yes	characteristic
95	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
96	#1974	C₂×C₁₃	1	1	yes	characteristic
97	#1945	C₂×C₁₃	1	1	yes	characteristic, complemented
98	#1382	C₂×C₁₃	1	1	yes	characteristic, complemented
99	#1381	C₂×C₁₃	1	1	yes	characteristic, complemented
100	C₂³×C₁₃	—	4	1	no	—
101	C₂³×C₁₃	#135	1	1	yes	characteristic
102	Q₈×C₁₃	—	4	1	no	—
103	Q₈×C₁₃	—	4	1	no	—
104	D₄×C₁₃	—	4	1	no	—
105	D₄×C₁₃	—	4	1	no	—
106	D₄×C₁₃	—	8	1	no	—
107	D₄×C₁₃	—	8	1	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	—	4	1	no	—
112	C₂×C₄×C₁₃	—	8	1	no	—
113	C₂×C₄×C₁₃	—	8	1	no	—
114	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
120	C₈×C₁₃	—	2	2	no	—
121	C₈×C₁₃	—	8	1	no	—
122	C₈×C₁₃	—	8	1	no	—
123	C₈×C₁₃	#117	1	1	yes	characteristic
124	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
125	C₈×C₁₃	#123	1	1	yes	characteristic
126	C₈×C₁₃	#126	1	1	yes	characteristic
127	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
128	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
129	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
132	#427	C₂²×C₁₃	1	1	yes	characteristic
133	#424	—	2	1	no	—
134	C₂×#104	—	2	1	no	—
135	#408	—	2	1	no	—
136	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
137	#359	—	2	2	no	—
138	#358	C₄×C₁₃	1	2	yes	complemented
139	#357	C₄×C₁₃	1	2	yes	complemented
140	C₄×C₁₆	—	2	2	no	—
141	#339	—	2	1	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	—	8	2	no	—
144	C₂²×C₁₃	—	8	1	no	—
145	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
146	C₂²×C₁₃	#465	1	1	yes	characteristic
147	C₂²×C₁₃	#488	1	1	yes	characteristic
148	C₄×C₁₃	—	2	4	no	—
149	C₄×C₁₃	—	8	2	no	—
150	C₄×C₁₃	—	8	1	no	—
151	C₄×C₁₃	#388	1	1	yes	characteristic
152	C₄×C₁₃	C₄×SD₁₆	1	1	yes	characteristic
153	C₄×C₁₃	#466	1	1	yes	characteristic
154	C₄×C₁₃	#486	1	1	yes	characteristic
155	C₂×#41	—	2	1	no	—
156	#135	—	2	1	no	—
157	#135	—	2	1	no	—
158	#135	—	2	1	no	—
159	#135	—	2	1	no	—
160	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
161	C₂×SD₁₆	—	2	1	no	—
162	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic
163	C₂×M₁₆	—	2	1	no	—
164	C₂×M₁₆	—	2	1	no	—
165	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
167	#117	—	2	1	no	—
168	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
169	C₂×C₁₆	—	2	1	no	—
170	C₂×C₁₆	—	2	1	no	—
171	C₂×C₁₆	—	2	1	no	—
172	C₂×C₁₆	—	2	1	no	—
173	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
174	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
175	#108	—	2	1	no	—
176	#108	—	2	1	no	—
177	#108	—	2	1	no	—
178	#108	—	2	1	no	—
179	#107	C₂×C₄×C₁₃	1	2	yes	—
180	#106	—	2	1	no	—
181	C₄⋊C₈	—	4	2	no	—
182	#104	—	4	2	no	—
183	C₂²⋊C₈	—	4	2	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	2	1	no	—
186	C₂×C₁₃	—	2	1	no	—
187	C₂×C₁₃	—	2	1	no	—
188	C₂×C₁₃	—	16	1	no	—
189	C₂×C₁₃	#1934	1	1	yes	characteristic
190	C₂×C₁₃	#2178	1	2	yes	—
191	#41	—	4	1	no	—
192	#41	—	4	1	no	—
193	#41	—	4	1	no	—
194	#41	—	4	1	no	—
195	C₂×Q₈	—	2	1	no	—
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
201	Q₁₆	—	2	1	no	—
202	Q₁₆	—	2	1	no	—
203	SD₁₆	—	4	1	no	—
204	SD₁₆	—	4	1	no	—
205	D₈	—	2	1	no	—
206	D₈	—	2	1	no	—
207	M₁₆	—	4	1	no	—
208	M₁₆	—	4	1	no	—
209	M₁₆	—	4	1	no	—
210	M₁₆	—	4	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	C₁₃×D₈	1	2	yes	—
216	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
217	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
218	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
219	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
220	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
221	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
222	C₄⋊C₄	—	2	2	no	—
223	C₂²⋊C₄	—	4	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	4	1	no	—
226	C₄²	C₁₃×SD₁₆	1	2	yes	—
227	C₁₆	—	2	2	no	—
228	C₁₆	—	2	2	no	—
229	C₁₃	#13151	1	1	yes	characteristic, direct product factor, Sylow
230	C₂³	—	4	1	no	—
231	C₂³	C₁₃×#135	1	1	yes	characteristic
232	Q₈	—	4	1	no	—
233	Q₈	—	4	1	no	—
234	D₄	—	4	1	no	—
235	D₄	—	4	1	no	—
236	D₄	—	8	1	no	—
237	D₄	—	8	1	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	8	1	no	—
243	C₂×C₄	—	8	1	no	—
244	C₂×C₄	C₄×D₄×C₁₃	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	C₂×C₁₃×D₈	1	1	yes	characteristic
248	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
249	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	8	1	no	—
252	C₈	—	8	1	no	—
253	C₈	C₁₃×#117	1	1	yes	characteristic
254	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
255	C₈	C₁₃×#123	1	1	yes	characteristic
256	C₈	C₁₃×#126	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	1	no	—
260	C₂²	C₄×C₁₃×D₈	1	1	yes	characteristic
261	C₂²	C₁₃×#465	1	1	yes	characteristic
262	C₂²	C₁₃×#488	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	8	2	no	—
265	C₄	—	8	1	no	—
266	C₄	C₁₃×#388	1	1	yes	characteristic
267	C₄	C₄×C₁₃×SD₁₆	1	1	yes	characteristic
268	C₄	C₁₃×#466	1	1	yes	characteristic
269	C₄	C₁₃×#486	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	16	1	no	—
273	C₂	C₁₃×#1934	1	1	yes	characteristic
274	C₂	C₁₃×#2178	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₄×D₈
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
5. C₁₃×#1945	C₂	semidirect
6. C₁₃×#1382	C₂	semidirect
7. C₁₃×#1381	C₂	semidirect
19. C₁₃×#358	C₄	semidirect
20. C₁₃×#357	C₄	semidirect
54. #13151	C₁₃	direct (N × H)
95. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
97. #1945	C₂×C₁₃	semidirect
98. #1382	C₂×C₁₃	semidirect
99. #1381	C₂×C₁₃	semidirect
138. #358	C₄×C₁₃	semidirect
139. #357	C₄×C₁₃	semidirect
229. C₁₃	#13151	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×#359	2
C₁₃×#2138	2
C₂×C₄×C₁₃×C₁₆	2
C₁₃×#1974	2
C₁₃×#1945	2
C₁₃×#1382	2
C₁₃×#1381	2
#13151	13

Subgroups — C13×#13151 #15699490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13151 #15699490