Subgroups — C₁₃×#33463 #15719802

Order 3328.

652 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3414	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#443	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2179	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2178	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2161	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2143	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	C₂²	1	2	yes	—
9	C₂×C₁₃×#131	—	2	2	no	—
10	C₁₃×#510	C₄	1	4	yes	complemented
11	C₁₃×#509	C₄	1	4	yes	complemented
12	C₁₃×#508	C₂²	1	4	yes	—
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₁₃×#443	—	2	2	no	—
16	C₁₃×#443	—	2	2	no	—
17	C₁₃×#443	C₂²	1	4	yes	complemented
18	C₁₃×#443	C₂²	1	4	yes	complemented
19	C₂×C₁₃×#108	C₂²	1	2	yes	—
20	C₂×C₁₃×#104	—	2	2	no	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₁₃×#368	C₂²	1	4	yes	complemented
23	C₁₃×#359	C₂²	1	8	yes	complemented
24	C₁₃×#346	C₂²	1	2	yes	complemented
25	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
26	C₂×C₁₃×#41	—	2	2	no	—
27	C₁₃×#135	—	2	2	no	—
28	C₁₃×#135	—	2	2	no	—
29	C₁₃×#135	C₂×C₄	1	4	yes	—
30	C₁₃×#135	C₂³	1	4	yes	—
31	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₁₃×SD₁₆	—	2	2	no	—
33	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
34	C₁₃×#131	—	2	4	no	—
35	C₁₃×#131	—	2	4	no	—
36	C₁₃×#131	—	4	2	no	—
37	C₁₃×#131	—	4	2	no	—
38	C₂×C₁₃×M₁₆	—	2	2	no	—
39	C₂×C₁₃×M₁₆	—	4	2	no	—
40	C₂²×C₈×C₁₃	—	4	2	no	—
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
44	C₁₃×Q₃₂	C₂×C₄	1	4	yes	complemented
45	C₁₃×SD₃₂	C₂×C₄	1	8	yes	complemented
46	C₁₃×D₁₆	C₂×C₄	1	4	yes	complemented
47	C₁₃×M₃₂	—	2	2	no	—
48	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
49	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
50	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
51	C₁₃×#108	—	2	2	no	—
52	C₁₃×#108	C₂³	1	4	yes	—
53	C₁₃×#104	—	2	4	no	—
54	C₁₃×#104	—	2	4	no	—
55	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	#33463	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₁₃×#41	—	2	4	no	—
61	C₁₃×#41	—	2	4	no	—
62	C₁₃×#41	—	4	2	no	—
63	C₁₃×#41	—	4	2	no	—
64	C₂×Q₈×C₁₃	—	2	2	no	—
65	C₂×D₄×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	4	2	no	—
67	C₂²×C₄×C₁₃	—	2	1	no	—
68	C₂²×C₄×C₁₃	—	4	2	no	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₁₃×Q₁₆	—	2	2	no	—
71	C₁₃×Q₁₆	C₂²×C₄	1	4	yes	—
72	C₁₃×SD₁₆	—	2	4	no	—
73	C₁₃×SD₁₆	—	2	4	no	—
74	C₁₃×D₈	—	2	2	no	—
75	C₁₃×D₈	C₂²×C₄	1	4	yes	—
76	C₁₃×M₁₆	—	2	4	no	—
77	C₁₃×M₁₆	—	4	2	no	—
78	C₁₃×M₁₆	—	4	4	no	—
79	C₁₃×M₁₆	—	4	4	no	—
80	C₂×C₈×C₁₃	—	4	2	no	—
81	C₂×C₈×C₁₃	—	4	4	no	—
82	C₂×C₈×C₁₃	—	4	4	no	—
83	C₂×C₈×C₁₃	D₈	1	2	yes	—
84	C₂×C₈×C₁₃	D₈	1	2	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	#41	1	2	yes	—
90	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₄²×C₁₃	—	2	1	no	—
92	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₁₃×C₁₆	—	2	2	no	—
95	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
96	#3414	C₂×C₁₃	1	1	yes	characteristic
97	C₂×#443	C₂×C₁₃	1	2	yes	complemented
98	#2179	C₂×C₁₃	1	4	yes	complemented
99	#2178	C₂×C₁₃	1	4	yes	complemented
100	#2161	C₂×C₁₃	1	1	yes	characteristic, complemented
101	#2143	C₂×C₁₃	1	2	yes	complemented
102	C₄×M₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
103	C₂³×C₁₃	—	4	2	no	—
104	C₂³×C₁₃	#135	1	1	yes	characteristic
105	Q₈×C₁₃	—	2	4	no	—
106	Q₈×C₁₃	—	4	2	no	—
107	D₄×C₁₃	—	2	4	no	—
108	D₄×C₁₃	—	4	4	no	—
109	D₄×C₁₃	—	4	4	no	—
110	D₄×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	4	4	no	—
117	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₁₃	—	4	4	no	—
122	C₈×C₁₃	C₄×D₄	1	2	yes	—
123	C₈×C₁₃	C₂×D₈	1	2	yes	—
124	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
125	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
126	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
127	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₂×#135	C₂²×C₁₃	1	2	yes	—
129	C₂×#131	—	2	2	no	—
130	#510	C₄×C₁₃	1	4	yes	complemented
131	#509	C₄×C₁₃	1	4	yes	complemented
132	#508	C₂²×C₁₃	1	4	yes	—
133	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
135	#443	—	2	2	no	—
136	#443	—	2	2	no	—
137	#443	C₂²×C₁₃	1	4	yes	complemented
138	#443	C₂²×C₁₃	1	4	yes	complemented
139	C₂×#108	C₂²×C₁₃	1	2	yes	—
140	C₂×#104	—	2	2	no	—
141	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
142	#368	C₂²×C₁₃	1	4	yes	complemented
143	#359	C₂²×C₁₃	1	8	yes	complemented
144	#346	C₂²×C₁₃	1	2	yes	complemented
145	C₄×C₁₆	C₂²×C₁₃	1	2	yes	complemented
146	C₂²×C₁₃	—	4	1	no	—
147	C₂²×C₁₃	—	4	4	no	—
148	C₂²×C₁₃	—	8	4	no	—
149	C₂²×C₁₃	C₄×D₈	1	2	yes	—
150	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	—	4	1	no	—
153	C₄×C₁₃	—	4	4	no	—
154	C₄×C₁₃	C₄×D₈	1	2	yes	—
155	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
156	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
157	C₂×#41	—	2	2	no	—
158	#135	—	2	2	no	—
159	#135	—	2	2	no	—
160	#135	C₂×C₄×C₁₃	1	4	yes	—
161	#135	C₂³×C₁₃	1	4	yes	—
162	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
163	C₂×SD₁₆	—	2	2	no	—
164	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
165	#131	—	2	4	no	—
166	#131	—	2	4	no	—
167	#131	—	4	2	no	—
168	#131	—	4	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	4	2	no	—
171	C₂²×C₈	—	4	2	no	—
172	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
174	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
175	Q₃₂	C₂×C₄×C₁₃	1	4	yes	complemented
176	SD₃₂	C₂×C₄×C₁₃	1	8	yes	complemented
177	D₁₆	C₂×C₄×C₁₃	1	4	yes	complemented
178	M₃₂	—	2	2	no	—
179	M₃₂	C₂×C₄×C₁₃	1	4	yes	—
180	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
181	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
182	#108	—	2	2	no	—
183	#108	C₂³×C₁₃	1	4	yes	—
184	#104	—	2	4	no	—
185	#104	—	2	4	no	—
186	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
187	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
189	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
190	C₂×C₁₃	—	2	1	no	—
191	C₂×C₁₃	—	2	2	no	—
192	C₂×C₁₃	—	8	4	no	—
193	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
194	#41	—	2	4	no	—
195	#41	—	2	4	no	—
196	#41	—	4	2	no	—
197	#41	—	4	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
204	Q₁₆	—	2	2	no	—
205	Q₁₆	C₂²×C₄×C₁₃	1	4	yes	—
206	SD₁₆	—	2	4	no	—
207	SD₁₆	—	2	4	no	—
208	D₈	—	2	2	no	—
209	D₈	C₂²×C₄×C₁₃	1	4	yes	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	4	2	no	—
212	M₁₆	—	4	4	no	—
213	M₁₆	—	4	4	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	—	4	4	no	—
217	C₂×C₈	C₁₃×D₈	1	2	yes	—
218	C₂×C₈	C₁₃×D₈	1	2	yes	—
219	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
220	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
221	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
222	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
223	C₂×C₈	C₁₃×#41	1	2	yes	—
224	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
225	C₄²	—	2	1	no	—
226	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
227	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
228	C₁₆	—	2	2	no	—
229	C₁₆	C₂²×C₄×C₁₃	1	4	yes	—
230	C₁₃	#33463	1	1	yes	characteristic, direct product factor, Sylow
231	C₂³	—	4	2	no	—
232	C₂³	C₁₃×#135	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	4	2	no	—
235	D₄	—	2	4	no	—
236	D₄	—	4	4	no	—
237	D₄	—	4	4	no	—
238	D₄	—	4	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
246	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
247	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
248	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
249	C₈	—	4	4	no	—
250	C₈	C₄×D₄×C₁₃	1	2	yes	—
251	C₈	C₂×C₁₃×D₈	1	2	yes	—
252	C₈	C₂×C₁₃×D₈	1	1	yes	characteristic
253	C₈	C₂×C₁₃×D₈	1	1	yes	characteristic
254	C₈	C₂³×C₄×C₁₃	1	1	yes	characteristic
255	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
256	C₂²	—	4	1	no	—
257	C₂²	—	4	4	no	—
258	C₂²	—	8	4	no	—
259	C₂²	C₄×C₁₃×D₈	1	2	yes	—
260	C₂²	C₂×C₁₃×#135	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	4	no	—
264	C₄	C₄×C₁₃×D₈	1	2	yes	—
265	C₄	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
266	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
267	C₂	—	2	1	no	—
268	C₂	—	2	2	no	—
269	C₂	—	8	4	no	—
270	C₂	C₂×C₄×C₁₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₃	C₂×D₈
commutator	C₈	C₂³×C₄×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂×C₁₃	C₂×C₄×D₈
prime radical	C₁₃×#3414	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×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
2. C₂×C₁₃×#443	C₂	semidirect
3. C₁₃×#2179	C₂	semidirect
4. C₁₃×#2178	C₂	semidirect
5. C₁₃×#2161	C₂	semidirect
6. C₁₃×#2143	C₂	semidirect
7. C₄×C₁₃×M₃₂	C₂	semidirect
10. C₁₃×#510	C₄	semidirect
11. C₁₃×#509	C₄	semidirect
17. C₁₃×#443	C₂²	semidirect
18. C₁₃×#443	C₂²	semidirect
22. C₁₃×#368	C₂²	semidirect
23. C₁₃×#359	C₂²	semidirect
24. C₁₃×#346	C₂²	semidirect
25. C₄×C₁₃×C₁₆	C₂²	semidirect
44. C₁₃×Q₃₂	C₂×C₄	semidirect
45. C₁₃×SD₃₂	C₂×C₄	semidirect
46. C₁₃×D₁₆	C₂×C₄	semidirect
59. #33463	C₁₃	direct (N × H)
97. C₂×#443	C₂×C₁₃	semidirect
98. #2179	C₂×C₁₃	semidirect
99. #2178	C₂×C₁₃	semidirect
100. #2161	C₂×C₁₃	semidirect
101. #2143	C₂×C₁₃	semidirect
102. C₄×M₃₂	C₂×C₁₃	semidirect
130. #510	C₄×C₁₃	semidirect
131. #509	C₄×C₁₃	semidirect
137. #443	C₂²×C₁₃	semidirect
138. #443	C₂²×C₁₃	semidirect
142. #368	C₂²×C₁₃	semidirect
143. #359	C₂²×C₁₃	semidirect
144. #346	C₂²×C₁₃	semidirect
145. C₄×C₁₆	C₂²×C₁₃	semidirect
175. Q₃₂	C₂×C₄×C₁₃	semidirect
176. SD₃₂	C₂×C₄×C₁₃	semidirect
177. D₁₆	C₂×C₄×C₁₃	semidirect
230. C₁₃	#33463	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×#3414	2
C₂×C₁₃×#443	2
C₁₃×#2179	2
C₁₃×#2178	2
C₁₃×#2161	2
C₁₃×#2143	2
C₄×C₁₃×M₃₂	2
#33463	13

Subgroups — C13×#33463 #15719802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33463 #15719802