Subgroups — C₁₃×#32947 #15719286

Order 3328.

1306 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3573	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3057	C₂	1	3	yes	maximal, complemented
3	C₁₃×#3028	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#3027	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1910	C₂	1	3	yes	maximal, complemented
6	C₁₃×#1909	C₂	1	3	yes	maximal, complemented
7	C₁₃×#1812	C₂	1	3	yes	maximal, complemented
8	C₁₃×#585	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic, complemented
10	C₁₃×#568	—	2	6	no	—
11	C₁₃×#568	C₂²	1	3	yes	—
12	C₂×C₁₃×#131	—	2	6	no	—
13	C₂×C₁₃×#131	C₂²	1	3	yes	—
14	C₁₃×#536	C₂²	1	3	yes	—
15	C₁₃×#535	C₂²	1	3	yes	—
16	C₁₃×#468	—	2	6	no	—
17	C₁₃×#465	—	2	6	no	—
18	C₁₃×#458	—	2	4	no	—
19	C₁₃×B(4,𝔽₂)	—	2	4	no	—
20	C₁₃×#427	C₂²	1	3	yes	complemented
21	C₁₃×#419	C₂²	1	3	yes	complemented
22	C₁₃×#418	C₂²	1	3	yes	complemented
23	C₁₃×#410	C₂²	1	3	yes	—
24	C₁₃×#410	C₂²	1	3	yes	—
25	C₁₃×H(3,ℤ/4)	C₂²	1	6	yes	complemented
26	C₁₃×2₋¹⁺⁴	—	2	3	no	—
27	C₁₃×H(5,𝔽₂)	—	2	3	no	—
28	C₁₃×H(5,𝔽₂)	D₄	1	4	yes	—
29	C₂×C₁₃×#41	—	2	6	no	—
30	C₂×C₁₃×#41	C₂³	1	3	yes	—
31	C₂²×Q₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×D₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₁₃×SD₁₆	—	2	3	no	—
34	C₂×C₁₃×SD₁₆	—	2	3	no	—
35	C₁₃×#131	—	2	4	no	—
36	C₁₃×#131	—	2	12	no	—
37	C₁₃×#131	—	2	6	no	—
38	C₁₃×#131	—	2	12	no	—
39	C₁₃×#131	—	2	6	no	—
40	C₂×C₁₃×M₁₆	—	2	12	no	—
41	C₂×C₁₃×M₁₆	—	2	6	no	—
42	C₂×C₁₃×M₁₆	D₄	1	6	yes	complemented
43	C₂×C₁₃×M₁₆	C₂³	1	3	yes	—
44	C₂²×C₈×C₁₃	—	2	4	no	—
45	C₂²×C₈×C₁₃	D₄	1	6	yes	complemented
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊Q₈	—	2	3	no	—
48	C₁₃×#127	—	2	3	no	—
49	C₁₃×#124	—	2	3	no	—
50	C₁₃×#124	—	2	3	no	—
51	C₁₃×#123	—	2	6	no	—
52	C₁₃×C₂²⋊Q₈	—	2	6	no	—
53	C₁₃×#121	—	2	6	no	—
54	C₁₃×#120	—	2	6	no	—
55	C₁₃×#117	C₂³	1	3	yes	—
56	C₁₃×#117	C₂³	1	3	yes	—
57	C₁₃×#107	—	2	3	no	—
58	C₁₃×#106	—	2	3	no	—
59	C₁₃×#103	—	2	6	no	—
60	C₁₃×#102	—	2	6	no	—
61	C₁₃×#99	—	2	6	no	—
62	C₁₃×#99	—	2	6	no	—
63	#32947	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂⁴×C₁₃	—	2	1	no	—
65	C₁₃×#41	—	2	4	no	—
66	C₁₃×#41	—	2	12	no	—
67	C₁₃×#41	—	2	12	no	—
68	C₁₃×#41	—	2	6	no	—
69	C₁₃×#41	—	2	6	no	—
70	C₂×Q₈×C₁₃	—	2	6	no	—
71	C₂×Q₈×C₁₃	—	2	3	no	—
72	C₂×Q₈×C₁₃	—	2	3	no	—
73	C₂×Q₈×C₁₃	#41	1	3	yes	—
74	C₂×D₄×C₁₃	—	2	6	no	—
75	C₂×D₄×C₁₃	—	2	12	no	—
76	C₂×D₄×C₁₃	—	2	3	no	—
77	C₂×D₄×C₁₃	—	2	3	no	—
78	C₂×D₄×C₁₃	—	4	6	no	—
79	C₂×D₄×C₁₃	C₂×D₄	1	6	yes	—
80	C₂×D₄×C₁₃	#41	1	3	yes	—
81	C₂²×C₄×C₁₃	—	2	4	no	—
82	C₂²×C₄×C₁₃	—	2	1	no	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
84	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₁₃×SD₁₆	—	4	1	no	—
86	C₁₃×SD₁₆	—	4	3	no	—
87	C₁₃×M₁₆	—	2	12	no	—
88	C₁₃×M₁₆	—	2	6	no	—
89	C₁₃×M₁₆	—	2	12	no	—
90	C₂×C₈×C₁₃	—	2	12	no	—
91	C₂×C₈×C₁₃	—	2	6	no	—
92	C₂×C₈×C₁₃	—	2	6	no	—
93	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
94	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
95	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
96	C₁₃×C₄⋊C₄	—	2	3	no	—
97	C₁₃×C₄⋊C₄	—	2	3	no	—
98	C₁₃×C₄⋊C₄	—	4	6	no	—
99	C₁₃×C₂²⋊C₄	—	2	3	no	—
100	C₁₃×C₂²⋊C₄	—	2	3	no	—
101	C₁₃×C₂²⋊C₄	—	4	6	no	—
102	C₁₃×C₂²⋊C₄	—	4	6	no	—
103	C₄²×C₁₃	—	2	3	no	—
104	C₄²×C₁₃	—	2	3	no	—
105	#3573	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#3057	C₂×C₁₃	1	3	yes	complemented
107	#3028	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#3027	C₂×C₁₃	1	1	yes	characteristic
109	#1910	C₂×C₁₃	1	3	yes	complemented
110	#1909	C₂×C₁₃	1	3	yes	complemented
111	#1812	C₂×C₁₃	1	3	yes	complemented
112	C₂³×C₁₃	—	2	4	no	—
113	C₂³×C₁₃	—	4	3	no	—
114	C₂³×C₁₃	—	16	1	no	—
115	C₂³×C₁₃	#121	1	6	yes	—
116	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	Q₈×C₁₃	—	2	4	no	—
118	Q₈×C₁₃	—	2	6	no	—
119	Q₈×C₁₃	—	4	1	no	—
120	Q₈×C₁₃	—	4	3	no	—
121	D₄×C₁₃	—	2	12	no	—
122	D₄×C₁₃	—	2	12	no	—
123	D₄×C₁₃	—	2	6	no	—
124	D₄×C₁₃	—	4	1	no	—
125	D₄×C₁₃	—	4	3	no	—
126	D₄×C₁₃	—	16	6	no	—
127	C₂×C₄×C₁₃	—	2	12	no	—
128	C₂×C₄×C₁₃	—	2	6	no	—
129	C₂×C₄×C₁₃	—	2	6	no	—
130	C₂×C₄×C₁₃	—	4	3	no	—
131	C₂×C₄×C₁₃	—	4	3	no	—
132	C₂×C₄×C₁₃	—	4	3	no	—
133	C₂×C₄×C₁₃	#121	1	6	yes	—
134	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
135	C₂×C₄×C₁₃	C₂×#41	1	3	yes	—
136	C₈×C₁₃	—	2	6	no	—
137	C₈×C₁₃	#120	1	1	yes	characteristic
138	C₈×C₁₃	#120	1	3	yes	—
139	#585	C₂²×C₁₃	1	1	yes	characteristic
140	C₂²×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic, complemented
141	#568	—	2	6	no	—
142	#568	C₂²×C₁₃	1	3	yes	—
143	C₂×#131	—	2	6	no	—
144	C₂×#131	C₂²×C₁₃	1	3	yes	—
145	#536	C₂²×C₁₃	1	3	yes	—
146	#535	C₂²×C₁₃	1	3	yes	—
147	#468	—	2	6	no	—
148	#465	—	2	6	no	—
149	#458	—	2	4	no	—
150	B(4,𝔽₂)	—	2	4	no	—
151	#427	C₂²×C₁₃	1	3	yes	complemented
152	#419	C₂²×C₁₃	1	3	yes	complemented
153	#418	C₂²×C₁₃	1	3	yes	complemented
154	#410	C₂²×C₁₃	1	3	yes	—
155	#410	C₂²×C₁₃	1	3	yes	—
156	H(3,ℤ/4)	C₂²×C₁₃	1	6	yes	complemented
157	C₂²×C₁₃	—	2	6	no	—
158	C₂²×C₁₃	—	4	1	no	—
159	C₂²×C₁₃	—	4	6	no	—
160	C₂²×C₁₃	—	8	6	no	—
161	C₂²×C₁₃	—	8	4	no	—
162	C₂²×C₁₃	—	8	1	no	—
163	C₂²×C₁₃	C₂×#121	1	3	yes	—
164	C₄×C₁₃	—	2	6	no	—
165	C₄×C₁₃	—	8	3	no	—
166	C₄×C₁₃	—	8	3	no	—
167	C₄×C₁₃	—	8	1	no	—
168	C₄×C₁₃	C₂×#120	1	1	yes	characteristic
169	C₄×C₁₃	#525	1	3	yes	—
170	2₋¹⁺⁴	—	2	3	no	—
171	H(5,𝔽₂)	—	2	3	no	—
172	H(5,𝔽₂)	D₄×C₁₃	1	4	yes	—
173	C₂×#41	—	2	6	no	—
174	C₂×#41	C₂³×C₁₃	1	3	yes	—
175	C₂²×Q₈	C₂³×C₁₃	1	1	yes	characteristic
176	C₂²×D₄	C₂³×C₁₃	1	1	yes	characteristic
177	C₂×SD₁₆	—	2	3	no	—
178	C₂×SD₁₆	—	2	3	no	—
179	#131	—	2	4	no	—
180	#131	—	2	12	no	—
181	#131	—	2	6	no	—
182	#131	—	2	12	no	—
183	#131	—	2	6	no	—
184	C₂×M₁₆	—	2	12	no	—
185	C₂×M₁₆	—	2	6	no	—
186	C₂×M₁₆	D₄×C₁₃	1	6	yes	complemented
187	C₂×M₁₆	C₂³×C₁₃	1	3	yes	—
188	C₂²×C₈	—	2	4	no	—
189	C₂²×C₈	D₄×C₁₃	1	6	yes	complemented
190	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
191	C₄⋊Q₈	—	2	3	no	—
192	#127	—	2	3	no	—
193	#124	—	2	3	no	—
194	#124	—	2	3	no	—
195	#123	—	2	6	no	—
196	C₂²⋊Q₈	—	2	6	no	—
197	#121	—	2	6	no	—
198	#120	—	2	6	no	—
199	#117	C₂³×C₁₃	1	3	yes	—
200	#117	C₂³×C₁₃	1	3	yes	—
201	#107	—	2	3	no	—
202	#106	—	2	3	no	—
203	#103	—	2	6	no	—
204	#102	—	2	6	no	—
205	#99	—	2	6	no	—
206	#99	—	2	6	no	—
207	C₂×C₁₃	—	2	3	no	—
208	C₂×C₁₃	—	4	6	no	—
209	C₂×C₁₃	—	16	1	no	—
210	C₂×C₁₃	#2847	1	1	yes	characteristic
211	C₂⁴	—	2	1	no	—
212	#41	—	2	4	no	—
213	#41	—	2	12	no	—
214	#41	—	2	12	no	—
215	#41	—	2	6	no	—
216	#41	—	2	6	no	—
217	C₂×Q₈	—	2	6	no	—
218	C₂×Q₈	—	2	3	no	—
219	C₂×Q₈	—	2	3	no	—
220	C₂×Q₈	C₁₃×#41	1	3	yes	—
221	C₂×D₄	—	2	6	no	—
222	C₂×D₄	—	2	12	no	—
223	C₂×D₄	—	2	3	no	—
224	C₂×D₄	—	2	3	no	—
225	C₂×D₄	—	4	6	no	—
226	C₂×D₄	C₂×D₄×C₁₃	1	6	yes	—
227	C₂×D₄	C₁₃×#41	1	3	yes	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	C₂×D₄×C₁₃	1	6	yes	—
231	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
232	SD₁₆	—	4	1	no	—
233	SD₁₆	—	4	3	no	—
234	M₁₆	—	2	12	no	—
235	M₁₆	—	2	6	no	—
236	M₁₆	—	2	12	no	—
237	C₂×C₈	—	2	12	no	—
238	C₂×C₈	—	2	6	no	—
239	C₂×C₈	—	2	6	no	—
240	C₂×C₈	C₂×D₄×C₁₃	1	6	yes	—
241	C₂×C₈	C₂×D₄×C₁₃	1	3	yes	—
242	C₂×C₈	C₂×D₄×C₁₃	1	3	yes	—
243	C₄⋊C₄	—	2	3	no	—
244	C₄⋊C₄	—	2	3	no	—
245	C₄⋊C₄	—	4	6	no	—
246	C₂²⋊C₄	—	2	3	no	—
247	C₂²⋊C₄	—	2	3	no	—
248	C₂²⋊C₄	—	4	6	no	—
249	C₂²⋊C₄	—	4	6	no	—
250	C₄²	—	2	3	no	—
251	C₄²	—	2	3	no	—
252	C₁₃	#32947	1	1	yes	characteristic, direct product factor, Sylow
253	C₂³	—	2	4	no	—
254	C₂³	—	4	3	no	—
255	C₂³	—	16	1	no	—
256	C₂³	C₁₃×#121	1	6	yes	—
257	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
258	Q₈	—	2	4	no	—
259	Q₈	—	2	6	no	—
260	Q₈	—	4	1	no	—
261	Q₈	—	4	3	no	—
262	D₄	—	2	12	no	—
263	D₄	—	2	12	no	—
264	D₄	—	2	6	no	—
265	D₄	—	4	1	no	—
266	D₄	—	4	3	no	—
267	D₄	—	16	6	no	—
268	C₂×C₄	—	2	12	no	—
269	C₂×C₄	—	2	6	no	—
270	C₂×C₄	—	2	6	no	—
271	C₂×C₄	—	4	3	no	—
272	C₂×C₄	—	4	3	no	—
273	C₂×C₄	—	4	3	no	—
274	C₂×C₄	C₁₃×#121	1	6	yes	—
275	C₂×C₄	C₂²×D₄×C₁₃	1	3	yes	—
276	C₂×C₄	C₂×C₁₃×#41	1	3	yes	—
277	C₈	—	2	6	no	—
278	C₈	C₁₃×#120	1	1	yes	characteristic
279	C₈	C₁₃×#120	1	3	yes	—
280	C₂²	—	2	6	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	6	no	—
283	C₂²	—	8	6	no	—
284	C₂²	—	8	4	no	—
285	C₂²	—	8	1	no	—
286	C₂²	C₂×C₁₃×#121	1	3	yes	—
287	C₄	—	2	6	no	—
288	C₄	—	8	3	no	—
289	C₄	—	8	3	no	—
290	C₄	—	8	1	no	—
291	C₄	C₂×C₁₃×#120	1	1	yes	characteristic
292	C₄	C₁₃×#525	1	3	yes	—
293	C₂	—	2	3	no	—
294	C₂	—	4	6	no	—
295	C₂	—	16	1	no	—
296	C₂	C₁₃×#2847	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#2847
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#3027	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3573	C₂	semidirect
2. C₁₃×#3057	C₂	semidirect
3. C₁₃×#3028	C₂	semidirect
5. C₁₃×#1910	C₂	semidirect
6. C₁₃×#1909	C₂	semidirect
7. C₁₃×#1812	C₂	semidirect
9. C₂²×C₁₃×SD₁₆	C₂²	semidirect
20. C₁₃×#427	C₂²	semidirect
21. C₁₃×#419	C₂²	semidirect
22. C₁₃×#418	C₂²	semidirect
25. C₁₃×H(3,ℤ/4)	C₂²	semidirect
42. C₂×C₁₃×M₁₆	D₄	semidirect
45. C₂²×C₈×C₁₃	D₄	semidirect
63. #32947	C₁₃	direct (N × H)
105. #3573	C₂×C₁₃	semidirect
106. #3057	C₂×C₁₃	semidirect
107. #3028	C₂×C₁₃	semidirect
109. #1910	C₂×C₁₃	semidirect
110. #1909	C₂×C₁₃	semidirect
111. #1812	C₂×C₁₃	semidirect
140. C₂²×SD₁₆	C₂²×C₁₃	semidirect
151. #427	C₂²×C₁₃	semidirect
152. #419	C₂²×C₁₃	semidirect
153. #418	C₂²×C₁₃	semidirect
156. H(3,ℤ/4)	C₂²×C₁₃	semidirect
186. C₂×M₁₆	D₄×C₁₃	semidirect
189. C₂²×C₈	D₄×C₁₃	semidirect
252. C₁₃	#32947	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₃×#3573	2
C₁₃×#3057	2
C₁₃×#3028	2
C₁₃×#3027	2
C₁₃×#1910	2
C₁₃×#1909	2
C₁₃×#1812	2
#32947	13

Subgroups — C13×#32947 #15719286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32947 #15719286