Subgroups — C₁₃×#33465 #15719804

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#508	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2957	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	C₂×C₁₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2143	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	C₂²	1	2	yes	—
9	C₁₃×#568	—	2	2	no	—
10	C₁₃×#508	C₄	1	2	yes	complemented
11	C₁₃×#508	C₄	1	2	yes	complemented
12	C₁₃×#508	C₂²	1	2	yes	—
13	C₁₃×#508	C₂²	1	2	yes	—
14	C₂×C₁₃×Q₃₂	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₃×SD₃₂	C₄	1	2	yes	complemented
16	C₂×C₁₃×D₁₆	C₄	1	1	yes	characteristic, complemented
17	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
19	C₁₃×#444	—	2	4	no	—
20	C₁₃×#443	C₂²	1	8	yes	complemented
21	C₂×C₁₃×#108	C₂²	1	2	yes	—
22	C₁₃×#421	—	2	2	no	—
23	C₁₃×#405	C₂²	1	1	yes	characteristic
24	C₁₃×#368	C₂²	1	2	yes	complemented
25	C₁₃×#368	C₂²	1	2	yes	complemented
26	C₁₃×#359	C₂²	1	8	yes	complemented
27	C₁₃×#346	C₂²	1	2	yes	complemented
28	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
29	C₂×C₁₃×#41	—	2	2	no	—
30	C₁₃×#135	—	2	2	no	—
31	C₁₃×#135	—	2	2	no	—
32	C₁₃×#135	C₂×C₄	1	4	yes	—
33	C₁₃×#135	C₂³	1	4	yes	—
34	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×SD₁₆	—	2	2	no	—
36	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
37	C₁₃×#131	—	2	8	no	—
38	C₁₃×#131	—	4	4	no	—
39	C₂×C₁₃×M₁₆	—	2	2	no	—
40	C₂×C₁₃×M₁₆	—	4	4	no	—
41	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×#117	D₄	1	1	yes	characteristic
44	C₁₃×Q₃₂	C₂×C₄	1	2	yes	—
45	C₁₃×Q₃₂	C₂×C₄	1	2	yes	—
46	C₁₃×SD₃₂	C₂×C₄	1	4	yes	—
47	C₁₃×SD₃₂	C₂×C₄	1	4	yes	—
48	C₁₃×D₁₆	C₂×C₄	1	2	yes	—
49	C₁₃×D₁₆	C₂×C₄	1	2	yes	—
50	C₁₃×M₃₂	—	2	1	no	—
51	C₁₃×M₃₂	—	2	1	no	—
52	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
54	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
57	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₁₃×#108	—	2	2	no	—
60	C₁₃×#108	C₂³	1	4	yes	—
61	C₁₃×#104	—	2	8	no	—
62	C₁₃×#97	D₄	1	2	yes	—
63	C₄×C₈×C₁₃	C₂³	1	2	yes	—
64	#33465	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₁₃×#41	—	2	4	no	—
66	C₁₃×#41	—	2	4	no	—
67	C₁₃×#41	—	4	2	no	—
68	C₁₃×#41	—	4	2	no	—
69	C₂×Q₈×C₁₃	—	2	2	no	—
70	C₂×D₄×C₁₃	—	2	2	no	—
71	C₂×D₄×C₁₃	—	4	2	no	—
72	C₂²×C₄×C₁₃	—	4	2	no	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₁₃×Q₁₆	—	2	2	no	—
75	C₁₃×Q₁₆	C₂²×C₄	1	4	yes	—
76	C₁₃×SD₁₆	—	2	4	no	—
77	C₁₃×SD₁₆	—	2	4	no	—
78	C₁₃×D₈	—	2	2	no	—
79	C₁₃×D₈	C₂²×C₄	1	4	yes	—
80	C₁₃×M₁₆	—	2	4	no	—
81	C₁₃×M₁₆	—	4	2	no	—
82	C₁₃×M₁₆	—	4	8	no	—
83	C₁₃×M₁₆	D₈	1	2	yes	—
84	C₁₃×M₁₆	D₈	1	2	yes	—
85	C₂×C₈×C₁₃	—	4	2	no	—
86	C₂×C₈×C₁₃	—	4	8	no	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
92	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	—	2	1	no	—
95	C₁₃×C₂²⋊C₄	—	2	1	no	—
96	C₄²×C₁₃	C₂×D₄	1	2	yes	—
97	C₁₃×C₁₆	—	2	2	no	—
98	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
99	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
100	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
101	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
102	C₂×#508	C₂×C₁₃	1	1	yes	characteristic
103	#2957	C₂×C₁₃	1	2	yes	complemented
104	#2179	C₂×C₁₃	1	4	yes	complemented
105	#2178	C₂×C₁₃	1	4	yes	complemented
106	C₂×#368	C₂×C₁₃	1	1	yes	characteristic, complemented
107	#2143	C₂×C₁₃	1	2	yes	complemented
108	#2108	C₂×C₁₃	1	1	yes	characteristic, complemented
109	C₂³×C₁₃	—	4	2	no	—
110	C₂³×C₁₃	#135	1	1	yes	characteristic
111	Q₈×C₁₃	—	2	4	no	—
112	Q₈×C₁₃	—	4	2	no	—
113	D₄×C₁₃	—	2	4	no	—
114	D₄×C₁₃	—	4	4	no	—
115	D₄×C₁₃	—	4	4	no	—
116	D₄×C₁₃	—	4	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	—	4	4	no	—
121	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₈×C₁₃	—	4	4	no	—
128	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
129	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
130	C₈×C₁₃	C₂×D₈	1	4	yes	—
131	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
133	C₂×#135	C₂²×C₁₃	1	2	yes	—
134	#568	—	2	2	no	—
135	#508	C₄×C₁₃	1	2	yes	complemented
136	#508	C₄×C₁₃	1	2	yes	complemented
137	#508	C₂²×C₁₃	1	2	yes	—
138	#508	C₂²×C₁₃	1	2	yes	—
139	C₂×Q₃₂	C₄×C₁₃	1	1	yes	characteristic, complemented
140	C₂×SD₃₂	C₄×C₁₃	1	2	yes	complemented
141	C₂×D₁₆	C₄×C₁₃	1	1	yes	characteristic, complemented
142	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
143	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
144	#444	—	2	4	no	—
145	#443	C₂²×C₁₃	1	8	yes	complemented
146	C₂×#108	C₂²×C₁₃	1	2	yes	—
147	#421	—	2	2	no	—
148	#405	C₂²×C₁₃	1	1	yes	characteristic
149	#368	C₂²×C₁₃	1	2	yes	complemented
150	#368	C₂²×C₁₃	1	2	yes	complemented
151	#359	C₂²×C₁₃	1	8	yes	complemented
152	#346	C₂²×C₁₃	1	2	yes	complemented
153	C₄×C₁₆	C₂²×C₁₃	1	2	yes	complemented
154	C₂²×C₁₃	—	4	1	no	—
155	C₂²×C₁₃	—	4	4	no	—
156	C₂²×C₁₃	—	16	2	no	—
157	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
158	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
160	C₄×C₁₃	—	4	2	no	—
161	C₄×C₁₃	—	4	4	no	—
162	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
163	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
164	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
166	C₂×#41	—	2	2	no	—
167	#135	—	2	2	no	—
168	#135	—	2	2	no	—
169	#135	C₂×C₄×C₁₃	1	4	yes	—
170	#135	C₂³×C₁₃	1	4	yes	—
171	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
172	C₂×SD₁₆	—	2	2	no	—
173	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
174	#131	—	2	8	no	—
175	#131	—	4	4	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	4	4	no	—
178	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
180	#117	D₄×C₁₃	1	1	yes	characteristic
181	Q₃₂	C₂×C₄×C₁₃	1	2	yes	—
182	Q₃₂	C₂×C₄×C₁₃	1	2	yes	—
183	SD₃₂	C₂×C₄×C₁₃	1	4	yes	—
184	SD₃₂	C₂×C₄×C₁₃	1	4	yes	—
185	D₁₆	C₂×C₄×C₁₃	1	2	yes	—
186	D₁₆	C₂×C₄×C₁₃	1	2	yes	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
190	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
191	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
192	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
193	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
194	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
195	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
196	#108	—	2	2	no	—
197	#108	C₂³×C₁₃	1	4	yes	—
198	#104	—	2	8	no	—
199	#97	D₄×C₁₃	1	2	yes	—
200	C₄×C₈	C₂³×C₁₃	1	2	yes	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	—	2	1	no	—
204	C₂×C₁₃	—	8	4	no	—
205	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
206	#41	—	2	4	no	—
207	#41	—	2	4	no	—
208	#41	—	4	2	no	—
209	#41	—	4	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
215	Q₁₆	—	2	2	no	—
216	Q₁₆	C₂²×C₄×C₁₃	1	4	yes	—
217	SD₁₆	—	2	4	no	—
218	SD₁₆	—	2	4	no	—
219	D₈	—	2	2	no	—
220	D₈	C₂²×C₄×C₁₃	1	4	yes	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	8	no	—
224	M₁₆	C₁₃×D₈	1	2	yes	—
225	M₁₆	C₁₃×D₈	1	2	yes	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	8	no	—
228	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
229	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
230	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
232	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
233	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
234	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₂²⋊C₄	—	2	1	no	—
237	C₄²	C₂×D₄×C₁₃	1	2	yes	—
238	C₁₆	—	2	2	no	—
239	C₁₆	C₂²×C₄×C₁₃	1	1	yes	characteristic
240	C₁₆	C₂²×C₄×C₁₃	1	1	yes	characteristic
241	C₁₆	C₂²×C₄×C₁₃	1	1	yes	characteristic
242	C₁₆	C₂²×C₄×C₁₃	1	1	yes	characteristic
243	C₁₃	#33465	1	1	yes	characteristic, direct product factor, Sylow
244	C₂³	—	4	2	no	—
245	C₂³	C₁₃×#135	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	4	2	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
257	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
258	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
259	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
260	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
261	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
262	C₈	—	4	4	no	—
263	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
264	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
265	C₈	C₂×C₁₃×D₈	1	4	yes	—
266	C₈	C₂³×C₄×C₁₃	1	1	yes	characteristic
267	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
268	C₂²	—	4	1	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	16	2	no	—
271	C₂²	C₄×C₁₃×D₈	1	1	yes	characteristic
272	C₂²	C₄×C₁₃×D₈	1	1	yes	characteristic
273	C₂²	C₂×C₁₃×#135	1	1	yes	characteristic
274	C₄	—	4	2	no	—
275	C₄	—	4	4	no	—
276	C₄	C₄×C₁₃×D₈	1	1	yes	characteristic
277	C₄	C₄×C₁₃×D₈	1	1	yes	characteristic
278	C₄	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
279	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	8	4	no	—
284	C₂	C₂×C₄×C₁₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	C₂²×D₈
commutator	C₈	C₂³×C₄×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂×C₁₃	C₂×C₄×D₈
prime radical	C₂×C₁₃×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₁₃×M₁₆ ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2957	C₂	semidirect
3. C₁₃×#2179	C₂	semidirect
4. C₁₃×#2178	C₂	semidirect
5. C₂×C₁₃×#368	C₂	semidirect
6. C₁₃×#2143	C₂	semidirect
7. C₁₃×#2108	C₂	semidirect
10. C₁₃×#508	C₄	semidirect
11. C₁₃×#508	C₄	semidirect
14. C₂×C₁₃×Q₃₂	C₄	semidirect
15. C₂×C₁₃×SD₃₂	C₄	semidirect
16. C₂×C₁₃×D₁₆	C₄	semidirect
20. C₁₃×#443	C₂²	semidirect
24. C₁₃×#368	C₂²	semidirect
25. C₁₃×#368	C₂²	semidirect
26. C₁₃×#359	C₂²	semidirect
27. C₁₃×#346	C₂²	semidirect
28. C₄×C₁₃×C₁₆	C₂²	semidirect
64. #33465	C₁₃	direct (N × H)
103. #2957	C₂×C₁₃	semidirect
104. #2179	C₂×C₁₃	semidirect
105. #2178	C₂×C₁₃	semidirect
106. C₂×#368	C₂×C₁₃	semidirect
107. #2143	C₂×C₁₃	semidirect
108. #2108	C₂×C₁₃	semidirect
135. #508	C₄×C₁₃	semidirect
136. #508	C₄×C₁₃	semidirect
139. C₂×Q₃₂	C₄×C₁₃	semidirect
140. C₂×SD₃₂	C₄×C₁₃	semidirect
141. C₂×D₁₆	C₄×C₁₃	semidirect
145. #443	C₂²×C₁₃	semidirect
149. #368	C₂²×C₁₃	semidirect
150. #368	C₂²×C₁₃	semidirect
151. #359	C₂²×C₁₃	semidirect
152. #346	C₂²×C₁₃	semidirect
153. C₄×C₁₆	C₂²×C₁₃	semidirect
243. C₁₃	#33465	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×#508	2
C₁₃×#2957	2
C₁₃×#2179	2
C₁₃×#2178	2
C₂×C₁₃×#368	2
C₁₃×#2143	2
C₁₃×#2108	2
#33465	13

Subgroups — C13×#33465 #15719804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33465 #15719804