Subgroups — C₁₃×#2439 #13111497

Order 1664.

508 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#396	C₂	1	2	yes	maximal
4	C₁₃×#393	C₂	1	2	yes	maximal
5	C₁₃×#391	C₂	1	2	yes	maximal, complemented
6	C₁₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#387	C₂	1	2	yes	maximal, complemented
9	C₁₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#380	C₂	1	1	yes	maximal, characteristic
11	C₄×C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊Q₈	—	2	4	no	—
15	C₁₃×#117	—	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	1	yes	characteristic
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₁₃×C₂²⋊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×C₄²×C₁₃	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
30	C₁₃×#95	C₂²	1	1	yes	characteristic
31	C₁₃×#95	C₂²	1	1	yes	characteristic
32	C₁₃×#95	C₂²	1	2	yes	—
33	C₁₃×#95	C₂²	1	2	yes	—
34	C₁₃×#95	C₂²	1	2	yes	—
35	C₁₃×#95	C₂²	1	2	yes	—
36	C₁₃×#95	C₂²	1	2	yes	—
37	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₁₃	—	2	4	no	—
39	C₂×Q₈×C₁₃	—	2	4	no	—
40	C₂×Q₈×C₁₃	—	4	2	no	—
41	C₂²×C₄×C₁₃	—	2	2	no	—
42	C₂²×C₄×C₁₃	—	2	2	no	—
43	C₂²×C₄×C₁₃	Q₈	1	4	yes	complemented
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	2	yes	—
48	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
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	2	no	—
55	C₁₃×C₄⋊C₄	—	2	4	no	—
56	C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×C₄⋊C₄	—	2	4	no	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	4	2	no	—
60	C₁₃×C₄⋊C₄	D₄	1	4	yes	—
61	C₁₃×C₂²⋊C₄	—	2	4	no	—
62	C₁₃×C₂²⋊C₄	—	2	2	no	—
63	C₁₃×C₂²⋊C₄	—	2	2	no	—
64	C₁₃×C₂²⋊C₄	—	4	2	no	—
65	C₄²×C₁₃	—	2	4	no	—
66	C₄²×C₁₃	—	2	2	no	—
67	C₄²×C₁₃	—	2	2	no	—
68	#2439	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₁₃	—	2	1	no	—
70	C₂³×C₁₃	—	2	1	no	—
71	C₂³×C₁₃	—	2	1	no	—
72	C₂³×C₁₃	—	2	1	no	—
73	C₂³×C₁₃	—	2	1	no	—
74	C₂³×C₁₃	—	2	1	no	—
75	C₂³×C₁₃	C₂×Q₈	1	2	yes	—
76	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
77	Q₈×C₁₃	—	4	8	no	—
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	—	2	1	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	8	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	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	C₂×D₄	1	4	yes	—
103	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
106	C₂×C₄×C₁₃	#41	1	2	yes	—
107	C₂×C₄×C₁₃	#41	1	2	yes	—
108	C₂×C₄×C₁₃	#41	1	2	yes	—
109	C₂×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
110	C₂×#117	C₂×C₁₃	1	1	yes	characteristic
111	#396	C₂×C₁₃	1	2	yes	—
112	#393	C₂×C₁₃	1	2	yes	—
113	#391	C₂×C₁₃	1	2	yes	complemented
114	#389	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#389	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#387	C₂×C₁₃	1	2	yes	complemented
117	#382	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#380	C₂×C₁₃	1	1	yes	characteristic
119	C₄×C₂²⋊C₄	C₂×C₁₃	1	1	yes	characteristic
120	C₂²×C₁₃	—	2	2	no	—
121	C₂²×C₁₃	—	2	2	no	—
122	C₂²×C₁₃	—	2	2	no	—
123	C₂²×C₁₃	—	4	1	no	—
124	C₂²×C₁₃	—	4	1	no	—
125	C₂²×C₁₃	—	4	1	no	—
126	C₂²×C₁₃	—	4	1	no	—
127	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
129	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
130	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
131	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
132	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	4	no	—
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	—	4	2	no	—
139	C₄×C₁₃	—	4	2	no	—
140	C₄×C₁₃	—	4	1	no	—
141	C₄×C₁₃	—	4	2	no	—
142	C₄×C₁₃	—	4	1	no	—
143	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
144	C₂²×Q₈	C₂²×C₁₃	1	2	yes	—
145	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
146	C₄⋊Q₈	—	2	4	no	—
147	#117	—	2	4	no	—
148	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
153	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
154	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
156	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
159	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
160	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
162	#95	C₂²×C₁₃	1	1	yes	characteristic
163	#95	C₂²×C₁₃	1	1	yes	characteristic
164	#95	C₂²×C₁₃	1	2	yes	—
165	#95	C₂²×C₁₃	1	2	yes	—
166	#95	C₂²×C₁₃	1	2	yes	—
167	#95	C₂²×C₁₃	1	2	yes	—
168	#95	C₂²×C₁₃	1	2	yes	—
169	C₂×C₁₃	—	4	2	no	—
170	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
171	C₂×C₁₃	#533	1	1	yes	characteristic
172	C₂×C₁₃	#544	1	1	yes	characteristic
173	C₂×C₁₃	#546	1	1	yes	characteristic
174	C₂×C₁₃	#548	1	1	yes	characteristic
175	C₂×C₁₃	#555	1	1	yes	characteristic
176	C₂×C₁₃	#556	1	1	yes	characteristic
177	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	Q₈×C₁₃	1	4	yes	complemented
184	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
186	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
187	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
188	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
189	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
190	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	D₄×C₁₃	1	4	yes	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₂²⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	—	4	2	no	—
205	C₄²	—	2	4	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₁₃	#2439	1	1	yes	characteristic, direct product factor, Sylow
209	C₂³	—	2	1	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	C₂×Q₈×C₁₃	1	2	yes	—
216	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
217	Q₈	—	4	8	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	8	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	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	C₂×D₄×C₁₃	1	4	yes	—
243	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
244	C₂×C₄	C₂×Q₈×C₁₃	1	2	yes	—
245	C₂×C₄	C₂×Q₈×C₁₃	1	2	yes	—
246	C₂×C₄	C₁₃×#41	1	2	yes	—
247	C₂×C₄	C₁₃×#41	1	2	yes	—
248	C₂×C₄	C₁₃×#41	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
257	C₂²	C₂²×Q₈×C₁₃	1	1	yes	characteristic
258	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
259	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
260	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
261	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
262	C₂²	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	2	no	—
271	C₄	—	4	1	no	—
272	C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
273	C₂	—	4	2	no	—
274	C₂	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
275	C₂	C₁₃×#533	1	1	yes	characteristic
276	C₂	C₁₃×#544	1	1	yes	characteristic
277	C₂	C₁₃×#546	1	1	yes	characteristic
278	C₂	C₁₃×#548	1	1	yes	characteristic
279	C₂	C₁₃×#555	1	1	yes	characteristic
280	C₂	C₁₃×#556	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂³
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×C₄⋊Q₈	C₂	semidirect
5. C₁₃×#391	C₂	semidirect
6. C₁₃×#389	C₂	semidirect
7. C₁₃×#389	C₂	semidirect
8. C₁₃×#387	C₂	semidirect
9. C₁₃×#382	C₂	semidirect
43. C₂²×C₄×C₁₃	Q₈	semidirect
68. #2439	C₁₃	direct (N × H)
109. C₂×C₄⋊Q₈	C₂×C₁₃	semidirect
113. #391	C₂×C₁₃	semidirect
114. #389	C₂×C₁₃	semidirect
115. #389	C₂×C₁₃	semidirect
116. #387	C₂×C₁₃	semidirect
117. #382	C₂×C₁₃	semidirect
183. C₂²×C₄	Q₈×C₁₃	semidirect
208. C₁₃	#2439	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×C₄⋊Q₈	2
C₂×C₁₃×#117	2
C₁₃×#396	2
C₁₃×#393	2
C₁₃×#391	2
C₁₃×#389	2
C₁₃×#387	2
C₁₃×#382	2
C₁₃×#380	2
C₄×C₁₃×C₂²⋊C₄	2
#2439	13

Subgroups — C13×#2439 #13111497

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2439 #13111497