Subgroups — C₁₃×#3324 #13112382

Order 1664.

448 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×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄²×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#482	C₂	1	2	yes	maximal, complemented
5	C₁₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#468	C₂	1	2	yes	maximal, complemented
7	C₁₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#450	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#418	C₂	1	2	yes	maximal, complemented
12	C₂²×D₄×C₁₃	—	2	1	no	—
13	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
14	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	complemented
16	C₂×C₁₃×M₁₆	C₂²	1	2	yes	complemented
17	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₁₃×#127	C₂²	1	1	yes	characteristic
19	C₁₃×#126	—	2	2	no	—
20	C₁₃×#124	—	2	1	no	—
21	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
22	C₁₃×#121	—	2	1	no	—
23	C₁₃×#121	C₂²	1	2	yes	—
24	C₁₃×#120	—	2	2	no	—
25	C₄×Q₈×C₁₃	—	2	1	no	—
26	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
27	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×#117	C₂²	1	2	yes	complemented
30	C₁₃×#107	C₂²	1	2	yes	complemented
31	C₁₃×#106	C₂²	1	1	yes	characteristic, complemented
32	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
33	C₁₃×#103	C₂²	1	1	yes	characteristic, complemented
34	C₁₃×#102	C₂²	1	1	yes	characteristic
35	C₁₃×#102	C₂²	1	2	yes	—
36	C₁₃×#102	C₂²	1	2	yes	complemented
37	C₁₃×#102	C₂²	1	2	yes	complemented
38	C₁₃×#102	C₂²	1	2	yes	complemented
39	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
41	C₂⁴×C₁₃	—	2	2	no	—
42	C₂×Q₈×C₁₃	—	2	1	no	—
43	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₁₃	—	2	4	no	—
45	C₂×D₄×C₁₃	—	2	2	no	—
46	C₂×D₄×C₁₃	—	2	2	no	—
47	C₂×D₄×C₁₃	—	2	1	no	—
48	C₂×D₄×C₁₃	—	2	1	no	—
49	C₂×D₄×C₁₃	—	4	2	no	—
50	C₂×D₄×C₁₃	—	4	4	no	—
51	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
52	C₂×D₄×C₁₃	C₂³	1	2	yes	—
53	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
56	C₁₃×SD₁₆	—	2	2	no	—
57	C₁₃×SD₁₆	—	4	2	no	—
58	C₁₃×M₁₆	—	4	2	no	—
59	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂³	1	2	yes	—
62	C₁₃×C₄⋊C₄	—	2	2	no	—
63	C₁₃×C₄⋊C₄	—	2	1	no	—
64	C₁₃×C₄⋊C₄	—	2	1	no	—
65	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
66	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
68	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₁₃×C₂²⋊C₄	—	2	2	no	—
70	C₁₃×C₂²⋊C₄	—	2	2	no	—
71	C₁₃×C₂²⋊C₄	—	2	1	no	—
72	C₁₃×C₂²⋊C₄	—	2	2	no	—
73	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	—	2	1	no	—
76	C₄²×C₁₃	C₂³	1	1	yes	characteristic
77	#3324	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₁₃	—	2	1	no	—
79	C₂³×C₁₃	—	2	2	no	—
80	C₂³×C₁₃	—	4	4	no	—
81	C₂³×C₁₃	—	4	4	no	—
82	C₂³×C₁₃	—	8	2	no	—
83	C₂³×C₁₃	C₂×D₄	1	2	yes	—
84	Q₈×C₁₃	—	2	1	no	—
85	Q₈×C₁₃	—	2	1	no	—
86	Q₈×C₁₃	—	4	2	no	—
87	D₄×C₁₃	—	2	2	no	—
88	D₄×C₁₃	—	2	2	no	—
89	D₄×C₁₃	—	2	1	no	—
90	D₄×C₁₃	—	2	2	no	—
91	D₄×C₁₃	—	4	4	no	—
92	D₄×C₁₃	—	4	4	no	—
93	D₄×C₁₃	—	4	2	no	—
94	D₄×C₁₃	—	8	1	no	—
95	D₄×C₁₃	#41	1	2	yes	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₈×C₁₃	—	2	1	no	—
108	C₈×C₁₃	—	2	1	no	—
109	C₈×C₁₃	—	2	2	no	—
110	#556	C₂×C₁₃	1	1	yes	characteristic, complemented
111	D₄²	C₂×C₁₃	1	1	yes	characteristic
112	#486	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#482	C₂×C₁₃	1	2	yes	complemented
114	#474	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#468	C₂×C₁₃	1	2	yes	complemented
116	#460	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#450	C₂×C₁₃	1	2	yes	complemented
118	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#436	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#418	C₂×C₁₃	1	2	yes	complemented
121	C₂²×C₁₃	—	2	2	no	—
122	C₂²×C₁₃	—	2	2	no	—
123	C₂²×C₁₃	—	2	2	no	—
124	C₂²×C₁₃	—	2	2	no	—
125	C₂²×C₁₃	—	4	1	no	—
126	C₂²×C₁₃	—	4	1	no	—
127	C₂²×C₁₃	—	4	2	no	—
128	C₂²×C₁₃	—	4	2	no	—
129	C₂²×C₁₃	—	4	2	no	—
130	C₂²×C₁₃	—	8	8	no	—
131	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	—	2	1	no	—
134	C₄×C₁₃	—	4	1	no	—
135	C₄×C₁₃	—	4	1	no	—
136	C₄×C₁₃	—	4	2	no	—
137	C₄×C₁₃	—	4	1	no	—
138	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
139	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
140	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂²×D₄	—	2	1	no	—
142	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
143	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	complemented
145	C₂×M₁₆	C₂²×C₁₃	1	2	yes	complemented
146	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
147	#127	C₂²×C₁₃	1	1	yes	characteristic
148	#126	—	2	2	no	—
149	#124	—	2	1	no	—
150	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	complemented
151	#121	—	2	1	no	—
152	#121	C₂²×C₁₃	1	2	yes	—
153	#120	—	2	2	no	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
156	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
157	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
158	#117	C₂²×C₁₃	1	2	yes	complemented
159	#107	C₂²×C₁₃	1	2	yes	complemented
160	#106	C₂²×C₁₃	1	1	yes	characteristic, complemented
161	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
162	#103	C₂²×C₁₃	1	1	yes	characteristic, complemented
163	#102	C₂²×C₁₃	1	1	yes	characteristic
164	#102	C₂²×C₁₃	1	2	yes	—
165	#102	C₂²×C₁₃	1	2	yes	complemented
166	#102	C₂²×C₁₃	1	2	yes	complemented
167	#102	C₂²×C₁₃	1	2	yes	complemented
168	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	complemented
169	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
170	C₂×C₁₃	—	4	2	no	—
171	C₂×C₁₃	—	4	2	no	—
172	C₂×C₁₃	—	8	2	no	—
173	C₂×C₁₃	#546	1	1	yes	characteristic
174	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂×C₁₃	#577	1	1	yes	characteristic
176	C₂⁴	—	2	2	no	—
177	C₂×Q₈	—	2	1	no	—
178	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	2	1	no	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	4	2	no	—
185	C₂×D₄	—	4	4	no	—
186	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
187	C₂×D₄	C₂³×C₁₃	1	2	yes	—
188	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
191	SD₁₆	—	2	2	no	—
192	SD₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
195	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
196	C₂×C₈	C₂³×C₁₃	1	2	yes	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
201	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
202	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
203	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	C₂³×C₁₃	1	1	yes	characteristic
212	C₁₃	#3324	1	1	yes	characteristic, direct product factor, Sylow
213	C₂³	—	2	1	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	4	4	no	—
216	C₂³	—	4	4	no	—
217	C₂³	—	8	2	no	—
218	C₂³	C₂×D₄×C₁₃	1	2	yes	—
219	Q₈	—	2	1	no	—
220	Q₈	—	2	1	no	—
221	Q₈	—	4	2	no	—
222	D₄	—	2	2	no	—
223	D₄	—	2	2	no	—
224	D₄	—	2	1	no	—
225	D₄	—	2	2	no	—
226	D₄	—	4	4	no	—
227	D₄	—	4	4	no	—
228	D₄	—	4	2	no	—
229	D₄	—	8	1	no	—
230	D₄	C₁₃×#41	1	2	yes	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
239	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
240	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
241	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	1	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	8	8	no	—
255	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	1	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	1	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	1	no	—
262	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
263	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
264	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
265	C₂	—	4	2	no	—
266	C₂	—	4	2	no	—
267	C₂	—	8	2	no	—
268	C₂	C₁₃×#546	1	1	yes	characteristic
269	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
270	C₂	C₁₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	D₄²×C₁₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#556	C₂	semidirect
3. C₁₃×#486	C₂	semidirect
4. C₁₃×#482	C₂	semidirect
5. C₁₃×#474	C₂	semidirect
6. C₁₃×#468	C₂	semidirect
7. C₁₃×#460	C₂	semidirect
8. C₁₃×#450	C₂	semidirect
9. C₄×C₁₃×SD₁₆	C₂	semidirect
10. C₁₃×#436	C₂	semidirect
11. C₁₃×#418	C₂	semidirect
15. C₂×C₁₃×SD₁₆	C₂²	semidirect
16. C₂×C₁₃×M₁₆	C₂²	semidirect
17. C₁₃×C₄⋊Q₈	C₂²	semidirect
21. C₁₃×C₂²⋊Q₈	C₂²	semidirect
26. C₄×Q₈×C₁₃	C₂²	semidirect
29. C₁₃×#117	C₂²	semidirect
30. C₁₃×#107	C₂²	semidirect
31. C₁₃×#106	C₂²	semidirect
32. C₁₃×C₄⋊C₈	C₂²	semidirect
33. C₁₃×#103	C₂²	semidirect
36. C₁₃×#102	C₂²	semidirect
37. C₁₃×#102	C₂²	semidirect
38. C₁₃×#102	C₂²	semidirect
39. C₁₃×C₂²⋊C₈	C₂²	semidirect
40. C₄×C₈×C₁₃	C₂²	semidirect
77. #3324	C₁₃	direct (N × H)
110. #556	C₂×C₁₃	semidirect
112. #486	C₂×C₁₃	semidirect
113. #482	C₂×C₁₃	semidirect
114. #474	C₂×C₁₃	semidirect
115. #468	C₂×C₁₃	semidirect
116. #460	C₂×C₁₃	semidirect
117. #450	C₂×C₁₃	semidirect
118. C₄×SD₁₆	C₂×C₁₃	semidirect
119. #436	C₂×C₁₃	semidirect
120. #418	C₂×C₁₃	semidirect
144. C₂×SD₁₆	C₂²×C₁₃	semidirect
145. C₂×M₁₆	C₂²×C₁₃	semidirect
146. C₄⋊Q₈	C₂²×C₁₃	semidirect
150. C₂²⋊Q₈	C₂²×C₁₃	semidirect
155. C₄×Q₈	C₂²×C₁₃	semidirect
158. #117	C₂²×C₁₃	semidirect
159. #107	C₂²×C₁₃	semidirect
160. #106	C₂²×C₁₃	semidirect
161. C₄⋊C₈	C₂²×C₁₃	semidirect
162. #103	C₂²×C₁₃	semidirect
165. #102	C₂²×C₁₃	semidirect
166. #102	C₂²×C₁₃	semidirect
167. #102	C₂²×C₁₃	semidirect
168. C₂²⋊C₈	C₂²×C₁₃	semidirect
169. C₄×C₈	C₂²×C₁₃	semidirect
212. C₁₃	#3324	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×#556	2
D₄²×C₁₃	2
C₁₃×#486	2
C₁₃×#482	2
C₁₃×#474	2
C₁₃×#468	2
C₁₃×#460	2
C₁₃×#450	2
C₄×C₁₃×SD₁₆	2
C₁₃×#436	2
C₁₃×#418	2
#3324	13

Subgroups — C13×#3324 #13112382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3324 #13112382