Subgroups — C₁₃×#2454 #13111512

Order 1664.

548 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#399	C₂	1	2	yes	maximal, complemented
4	C₁₃×#396	C₂	1	2	yes	maximal, complemented
5	C₁₃×#390	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#390	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#385	C₂	1	2	yes	maximal, complemented
10	C₁₃×#380	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
14	C₁₃×#125	—	2	4	no	—
15	C₄×D₄×C₁₃	—	2	4	no	—
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
33	C₁₃×#95	C₂²	1	1	yes	characteristic
34	C₁₃×#95	C₂²	1	1	yes	characteristic
35	C₁₃×#95	C₂²	1	2	yes	—
36	C₁₃×#95	C₂²	1	2	yes	—
37	C₂⁴×C₁₃	C₂³	1	2	yes	—
38	C₂×D₄×C₁₃	—	2	2	no	—
39	C₂×D₄×C₁₃	—	2	2	no	—
40	C₂×D₄×C₁₃	Q₈	1	4	yes	—
41	C₂²×C₄×C₁₃	—	2	4	no	—
42	C₂²×C₄×C₁₃	—	2	4	no	—
43	C₂²×C₄×C₁₃	—	4	2	no	—
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×C₄⋊C₄	—	2	4	no	—
55	C₁₃×C₄⋊C₄	—	2	4	no	—
56	C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×C₄⋊C₄	—	2	2	no	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	2	2	no	—
60	C₁₃×C₄⋊C₄	—	4	2	no	—
61	C₁₃×C₄⋊C₄	—	4	2	no	—
62	C₁₃×C₄⋊C₄	D₄	1	4	yes	—
63	C₁₃×C₂²⋊C₄	—	2	4	no	—
64	C₁₃×C₂²⋊C₄	—	2	4	no	—
65	C₁₃×C₂²⋊C₄	—	4	2	no	—
66	C₁₃×C₂²⋊C₄	—	4	2	no	—
67	C₁₃×C₂²⋊C₄	—	4	2	no	—
68	C₄²×C₁₃	—	2	2	no	—
69	C₄²×C₁₃	—	2	2	no	—
70	C₄²×C₁₃	—	2	2	no	—
71	#2454	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₁₃	—	2	2	no	—
73	C₂³×C₁₃	—	2	2	no	—
74	C₂³×C₁₃	—	2	2	no	—
75	C₂³×C₁₃	—	2	2	no	—
76	C₂³×C₁₃	—	2	2	no	—
77	C₂³×C₁₃	—	2	2	no	—
78	C₂³×C₁₃	C₂×Q₈	1	4	yes	—
79	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
80	D₄×C₁₃	—	2	8	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	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	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	4	8	no	—
106	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
107	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
109	C₂×C₄×C₁₃	#41	1	2	yes	—
110	C₂×C₄×C₁₃	#41	1	2	yes	—
111	C₂×C₄×C₁₃	#41	1	2	yes	—
112	C₂×#125	C₂×C₁₃	1	1	yes	characteristic, complemented
113	C₂×C₄×D₄	C₂×C₁₃	1	1	yes	characteristic
114	#399	C₂×C₁₃	1	2	yes	complemented
115	#396	C₂×C₁₃	1	2	yes	complemented
116	#390	C₂×C₁₃	1	1	yes	characteristic
117	#390	C₂×C₁₃	1	1	yes	characteristic
118	#389	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#389	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#385	C₂×C₁₃	1	2	yes	complemented
121	#380	C₂×C₁₃	1	2	yes	complemented
122	C₄×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic, complemented
123	C₂²×C₁₃	—	2	4	no	—
124	C₂²×C₁₃	—	2	4	no	—
125	C₂²×C₁₃	—	2	4	no	—
126	C₂²×C₁₃	—	4	2	no	—
127	C₂²×C₁₃	—	4	2	no	—
128	C₂²×C₁₃	—	4	2	no	—
129	C₂²×C₁₃	—	4	2	no	—
130	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
131	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
132	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
133	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
134	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
135	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	2	4	no	—
140	C₄×C₁₃	—	4	1	no	—
141	C₄×C₁₃	—	4	2	no	—
142	C₄×C₁₃	—	4	1	no	—
143	C₄×C₁₃	—	4	1	no	—
144	C₄×C₁₃	—	4	1	no	—
145	C₄×C₁₃	—	4	2	no	—
146	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
147	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
148	C₂³×C₄	C₂²×C₁₃	1	2	yes	—
149	#125	—	2	4	no	—
150	C₄×D₄	—	2	4	no	—
151	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
152	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
153	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
158	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
159	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
161	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
162	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
163	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
164	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
165	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
166	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
167	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
168	#95	C₂²×C₁₃	1	1	yes	characteristic
169	#95	C₂²×C₁₃	1	1	yes	characteristic
170	#95	C₂²×C₁₃	1	2	yes	—
171	#95	C₂²×C₁₃	1	2	yes	—
172	C₂×C₁₃	—	4	4	no	—
173	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
174	C₂×C₁₃	#529	1	1	yes	characteristic
175	C₂×C₁₃	#540	1	1	yes	characteristic
176	C₂×C₁₃	#546	1	1	yes	characteristic
177	C₂×C₁₃	#550	1	1	yes	characteristic
178	C₂×C₁₃	#554	1	1	yes	characteristic
179	C₂×C₁₃	#554	1	1	yes	characteristic
180	C₂⁴	C₂³×C₁₃	1	2	yes	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	Q₈×C₁₃	1	4	yes	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
189	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
193	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
194	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	D₄×C₁₃	1	4	yes	—
206	C₂²⋊C₄	—	2	4	no	—
207	C₂²⋊C₄	—	2	4	no	—
208	C₂²⋊C₄	—	4	2	no	—
209	C₂²⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₁₃	#2454	1	1	yes	characteristic, direct product factor, Sylow
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	C₂×Q₈×C₁₃	1	4	yes	—
222	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
223	D₄	—	2	8	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	8	no	—
249	C₂×C₄	C₂×D₄×C₁₃	1	4	yes	—
250	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
251	C₂×C₄	C₂×Q₈×C₁₃	1	2	yes	—
252	C₂×C₄	C₁₃×#41	1	2	yes	—
253	C₂×C₄	C₁₃×#41	1	2	yes	—
254	C₂×C₄	C₁₃×#41	1	2	yes	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
263	C₂²	C₂²×Q₈×C₁₃	1	1	yes	characteristic
264	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
265	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
266	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
267	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
268	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	4	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	2	no	—
278	C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
279	C₂	—	4	4	no	—
280	C₂	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
281	C₂	C₁₃×#529	1	1	yes	characteristic
282	C₂	C₁₃×#540	1	1	yes	characteristic
283	C₂	C₁₃×#546	1	1	yes	characteristic
284	C₂	C₁₃×#550	1	1	yes	characteristic
285	C₂	C₁₃×#554	1	1	yes	characteristic
286	C₂	C₁₃×#554	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂²×D₄×C₁₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#125	C₂	semidirect
3. C₁₃×#399	C₂	semidirect
4. C₁₃×#396	C₂	semidirect
7. C₁₃×#389	C₂	semidirect
8. C₁₃×#389	C₂	semidirect
9. C₁₃×#385	C₂	semidirect
10. C₁₃×#380	C₂	semidirect
11. C₄×C₁₃×C₄⋊C₄	C₂	semidirect
71. #2454	C₁₃	direct (N × H)
112. C₂×#125	C₂×C₁₃	semidirect
114. #399	C₂×C₁₃	semidirect
115. #396	C₂×C₁₃	semidirect
118. #389	C₂×C₁₃	semidirect
119. #389	C₂×C₁₃	semidirect
120. #385	C₂×C₁₃	semidirect
121. #380	C₂×C₁₃	semidirect
122. C₄×C₄⋊C₄	C₂×C₁₃	semidirect
214. C₁₃	#2454	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×#125	2
C₂×C₄×D₄×C₁₃	2
C₁₃×#399	2
C₁₃×#396	2
C₁₃×#390	2
C₁₃×#389	2
C₁₃×#385	2
C₁₃×#380	2
C₄×C₁₃×C₄⋊C₄	2
#2454	13

Subgroups — C13×#2454 #13111512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2454 #13111512