Subgroups — C₁₃×#32458 #15718797

Order 3328.

760 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2486	C₂	1	2	yes	maximal
2	C₁₃×#2086	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2075	C₂	1	4	yes	maximal, complemented
4	C₁₃×#1954	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1942	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1852	C₂	1	4	yes	maximal, complemented
7	C₂×C₁₃×#123	—	2	2	no	—
8	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₁₃×#107	C₂²	1	2	yes	—
11	C₂×C₁₃×#107	C₂²	1	2	yes	—
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₂²⋊C₈	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#398	—	2	2	no	—
17	C₁₃×#397	—	2	2	no	—
18	C₁₃×#389	C₂²	1	4	yes	—
19	C₁₃×#381	—	2	2	no	—
20	C₁₃×#380	C₂²	1	2	yes	—
21	C₁₃×#380	C₂²	1	4	yes	—
22	C₁₃×#340	C₂²	1	2	yes	—
23	C₁₃×#340	C₂²	1	4	yes	—
24	C₁₃×#340	C₂²	1	4	yes	—
25	C₂²×D₄×C₁₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₃	—	2	2	no	—
27	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
30	C₁₃×#123	—	4	8	no	—
31	C₄×D₄×C₁₃	—	4	1	no	—
32	C₄×D₄×C₁₃	—	4	1	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
36	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
37	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
40	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
42	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×#107	—	2	4	no	—
45	C₁₃×#107	—	4	2	no	—
46	C₁₃×#107	—	4	2	no	—
47	C₁₃×C₄⋊C₈	—	4	1	no	—
48	C₁₃×C₂²⋊C₈	—	2	8	no	—
49	C₄×C₈×C₁₃	—	4	1	no	—
50	C₁₃×#95	—	2	4	no	—
51	C₁₃×#95	—	2	2	no	—
52	C₁₃×#95	—	2	4	no	—
53	#32458	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂⁴×C₁₃	C₂×D₄	1	2	yes	—
55	C₂×D₄×C₁₃	—	4	1	no	—
56	C₂×D₄×C₁₃	—	4	1	no	—
57	C₂×D₄×C₁₃	D₈	1	4	yes	—
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	4	no	—
62	C₂²×C₄×C₁₃	—	2	4	no	—
63	C₂²×C₄×C₁₃	—	2	4	no	—
64	C₂²×C₄×C₁₃	—	4	4	no	—
65	C₂²×C₄×C₁₃	—	4	4	no	—
66	C₂²×C₄×C₁₃	—	4	4	no	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	#41	1	4	yes	—
76	C₁₃×C₄⋊C₄	—	2	4	no	—
77	C₁₃×C₄⋊C₄	—	2	4	no	—
78	C₁₃×C₄⋊C₄	—	2	4	no	—
79	C₁₃×C₄⋊C₄	—	2	4	no	—
80	C₁₃×C₄⋊C₄	—	4	1	no	—
81	C₁₃×C₄⋊C₄	—	4	4	no	—
82	C₁₃×C₄⋊C₄	—	4	8	no	—
83	C₁₃×C₂²⋊C₄	—	4	2	no	—
84	C₁₃×C₂²⋊C₄	—	4	4	no	—
85	C₁₃×C₂²⋊C₄	—	4	8	no	—
86	C₄²×C₁₃	—	2	1	no	—
87	C₄²×C₁₃	—	2	1	no	—
88	#2486	C₂×C₁₃	1	2	yes	—
89	#2086	C₂×C₁₃	1	2	yes	complemented
90	#2075	C₂×C₁₃	1	4	yes	complemented
91	#1954	C₂×C₁₃	1	1	yes	characteristic
92	#1942	C₂×C₁₃	1	2	yes	complemented
93	#1852	C₂×C₁₃	1	4	yes	complemented
94	C₂³×C₁₃	—	2	2	no	—
95	C₂³×C₁₃	—	2	4	no	—
96	C₂³×C₁₃	—	2	2	no	—
97	C₂³×C₁₃	—	2	4	no	—
98	C₂³×C₁₃	C₂×D₈	1	4	yes	—
99	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
100	D₄×C₁₃	—	4	4	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	—	4	4	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	—	4	4	no	—
114	C₂×C₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	8	8	no	—
117	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
119	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
122	C₈×C₁₃	—	2	4	no	—
123	C₈×C₁₃	—	4	2	no	—
124	C₂×#123	—	2	2	no	—
125	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
126	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
127	C₂×#107	C₂²×C₁₃	1	2	yes	—
128	C₂×#107	C₂²×C₁₃	1	2	yes	—
129	C₂×#107	C₂²×C₁₃	1	2	yes	—
130	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×C₂²⋊C₈	C₂²×C₁₃	1	4	yes	—
132	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
133	#398	—	2	2	no	—
134	#397	—	2	2	no	—
135	#389	C₂²×C₁₃	1	4	yes	—
136	#381	—	2	2	no	—
137	#380	C₂²×C₁₃	1	2	yes	—
138	#380	C₂²×C₁₃	1	4	yes	—
139	#340	C₂²×C₁₃	1	2	yes	—
140	#340	C₂²×C₁₃	1	4	yes	—
141	#340	C₂²×C₁₃	1	4	yes	—
142	C₂²×C₁₃	—	2	8	no	—
143	C₂²×C₁₃	—	2	4	no	—
144	C₂²×C₁₃	—	4	4	no	—
145	C₂²×C₁₃	—	4	2	no	—
146	C₂²×C₁₃	—	4	2	no	—
147	C₂²×C₁₃	#534	1	1	yes	characteristic
148	C₂²×C₁₃	#547	1	2	yes	—
149	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
150	C₂²×C₁₃	#575	1	1	yes	characteristic
151	C₂²×C₁₃	#578	1	2	yes	—
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	4	1	no	—
154	C₄×C₁₃	—	4	2	no	—
155	C₄×C₁₃	—	8	4	no	—
156	C₄×C₁₃	—	8	4	no	—
157	C₄×C₁₃	#539	1	2	yes	—
158	C₄×C₁₃	#559	1	1	yes	characteristic
159	C₄×C₁₃	#563	1	1	yes	characteristic
160	C₂²×D₄	D₄×C₁₃	1	1	yes	characteristic
161	C₂³×C₄	—	2	2	no	—
162	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
163	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
164	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
165	#123	—	4	8	no	—
166	C₄×D₄	—	4	1	no	—
167	C₄×D₄	—	4	1	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₄⋊C₄	—	4	2	no	—
171	C₂×C₄⋊C₄	—	4	2	no	—
172	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
174	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
175	C₂×C₂²⋊C₄	—	2	2	no	—
176	C₂×C₂²⋊C₄	—	2	4	no	—
177	C₂×C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
178	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
179	#107	—	2	4	no	—
180	#107	—	4	2	no	—
181	#107	—	4	2	no	—
182	C₄⋊C₈	—	4	1	no	—
183	C₂²⋊C₈	—	2	8	no	—
184	C₄×C₈	—	4	1	no	—
185	#95	—	2	4	no	—
186	#95	—	2	2	no	—
187	#95	—	2	4	no	—
188	C₂×C₁₃	—	4	4	no	—
189	C₂×C₁₃	#2724	1	1	yes	characteristic
190	C₂×C₁₃	#3186	1	1	yes	characteristic
191	C₂×C₁₃	#3202	1	1	yes	characteristic
192	C₂×C₁₃	#3334	1	2	yes	—
193	C₂×C₁₃	#3352	1	2	yes	—
194	C₂⁴	C₂×D₄×C₁₃	1	2	yes	—
195	C₂×D₄	—	4	1	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂×D₄	C₁₃×D₈	1	4	yes	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
210	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	C₁₃×#41	1	4	yes	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	8	no	—
223	C₂²⋊C₄	—	4	2	no	—
224	C₂²⋊C₄	—	4	4	no	—
225	C₂²⋊C₄	—	4	8	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	2	1	no	—
228	C₁₃	#32458	1	1	yes	characteristic, direct product factor, Sylow
229	C₂³	—	2	2	no	—
230	C₂³	—	2	4	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	4	no	—
233	C₂³	C₂×C₁₃×D₈	1	4	yes	—
234	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
235	D₄	—	4	4	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	8	8	no	—
252	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
253	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
254	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
255	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
256	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
257	C₈	—	2	4	no	—
258	C₈	—	4	2	no	—
259	C₂²	—	2	8	no	—
260	C₂²	—	2	4	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	C₁₃×#534	1	1	yes	characteristic
265	C₂²	C₁₃×#547	1	2	yes	—
266	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
267	C₂²	C₁₃×#575	1	1	yes	characteristic
268	C₂²	C₁₃×#578	1	2	yes	—
269	C₄	—	2	2	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	8	4	no	—
273	C₄	—	8	4	no	—
274	C₄	C₁₃×#539	1	2	yes	—
275	C₄	C₁₃×#559	1	1	yes	characteristic
276	C₄	C₁₃×#563	1	1	yes	characteristic
277	C₂	—	4	4	no	—
278	C₂	C₁₃×#2724	1	1	yes	characteristic
279	C₂	C₁₃×#3186	1	1	yes	characteristic
280	C₂	C₁₃×#3202	1	1	yes	characteristic
281	C₂	C₁₃×#3334	1	2	yes	—
282	C₂	C₁₃×#3352	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂²×D₄×C₁₃	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2086	C₂	semidirect
3. C₁₃×#2075	C₂	semidirect
5. C₁₃×#1942	C₂	semidirect
6. C₁₃×#1852	C₂	semidirect
53. #32458	C₁₃	direct (N × H)
89. #2086	C₂×C₁₃	semidirect
90. #2075	C₂×C₁₃	semidirect
92. #1942	C₂×C₁₃	semidirect
93. #1852	C₂×C₁₃	semidirect
228. C₁₃	#32458	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×#2486	2
C₁₃×#2086	2
C₁₃×#2075	2
C₁₃×#1954	2
C₁₃×#1942	2
C₁₃×#1852	2
#32458	13

Subgroups — C13×#32458 #15718797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32458 #15718797