Subgroups — C₁₃×#32702 #15719041

Order 3328.

480 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2103	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2102	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2080	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2050	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1937	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1881	C₂	1	2	yes	maximal
7	C₁₃×#1861	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1858	C₂	1	2	yes	maximal, complemented
9	C₁₃×#1837	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#575	C₂²	1	1	yes	characteristic
11	C₁₃×#568	C₂²	1	2	yes	—
12	C₁₃×#430	C₂²	1	1	yes	characteristic
13	C₁₃×#430	C₂²	1	2	yes	—
14	C₁₃×#424	C₂²	1	2	yes	—
15	C₁₃×#423	C₂²	1	1	yes	characteristic
16	C₁₃×#423	C₂²	1	1	yes	characteristic
17	C₁₃×#421	C₂²	1	2	yes	—
18	C₂×C₁₃×#104	C₂²	1	2	yes	—
19	C₁₃×#416	C₂²	1	2	yes	—
20	C₁₃×#413	C₂²	1	2	yes	—
21	C₁₃×#408	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
23	C₁₃×#344	C₂²	1	2	yes	—
24	C₁₃×#344	C₂²	1	2	yes	—
25	C₁₃×#341	C₂²	1	2	yes	—
26	C₁₃×#341	C₂²	1	2	yes	—
27	C₁₃×#341	C₂²	1	2	yes	—
28	C₁₃×#339	C₂²	1	2	yes	—
29	C₁₃×#338	C₂²	1	1	yes	characteristic
30	C₁₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
31	C₂×C₁₃×#41	C₂³	1	2	yes	—
32	C₁₃×#137	—	2	2	no	—
33	C₁₃×AGL(1,ℤ/8)	—	2	2	no	—
34	C₁₃×#135	—	4	1	no	—
35	C₁₃×#131	—	4	2	no	—
36	C₁₃×#131	—	4	2	no	—
37	C₂×C₁₃×M₁₆	—	4	2	no	—
38	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
39	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
40	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
42	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
44	C₁₃×#117	C₂³	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×#108	—	4	2	no	—
47	C₁₃×#108	—	4	1	no	—
48	C₁₃×C₄⋊C₈	—	4	1	no	—
49	C₁₃×C₄⋊C₈	—	4	1	no	—
50	C₁₃×C₄⋊C₈	—	4	2	no	—
51	C₁₃×#104	—	2	2	no	—
52	C₁₃×#104	—	2	2	no	—
53	C₁₃×#104	—	4	2	no	—
54	C₁₃×#103	—	2	2	no	—
55	C₁₃×#102	—	2	2	no	—
56	C₁₃×#101	—	2	2	no	—
57	C₁₃×#100	—	2	2	no	—
58	C₁₃×C₂²⋊C₈	—	4	2	no	—
59	C₄×C₈×C₁₃	—	2	1	no	—
60	C₄×C₈×C₁₃	—	2	1	no	—
61	#32702	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₁₃×#41	—	2	2	no	—
63	C₁₃×#41	—	2	2	no	—
64	C₁₃×#41	—	2	2	no	—
65	C₁₃×#41	—	2	2	no	—
66	C₂×Q₈×C₁₃	#41	1	2	yes	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	#41	1	2	yes	—
69	C₂²×C₄×C₁₃	—	2	1	no	—
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₁₃×Q₁₆	—	4	1	no	—
73	C₁₃×SD₁₆	—	4	2	no	—
74	C₁₃×D₈	—	4	1	no	—
75	C₁₃×M₁₆	—	2	2	no	—
76	C₁₃×M₁₆	—	2	2	no	—
77	C₁₃×M₁₆	—	2	2	no	—
78	C₁₃×M₁₆	—	2	2	no	—
79	C₁₃×M₁₆	—	2	2	no	—
80	C₁₃×M₁₆	—	2	2	no	—
81	C₁₃×M₁₆	—	2	2	no	—
82	C₁₃×M₁₆	—	2	2	no	—
83	C₁₃×M₁₆	—	4	4	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	1	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	2	1	no	—
92	C₂×C₈×C₁₃	—	4	4	no	—
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₂²⋊C₄	—	2	2	no	—
95	C₄²×C₁₃	—	2	1	no	—
96	C₄²×C₁₃	—	2	2	no	—
97	C₄²×C₁₃	#41	1	1	yes	characteristic
98	C₄²×C₁₃	#41	1	1	yes	characteristic
99	#2103	C₂×C₁₃	1	2	yes	complemented
100	#2102	C₂×C₁₃	1	1	yes	characteristic, complemented
101	#2080	C₂×C₁₃	1	2	yes	complemented
102	#2050	C₂×C₁₃	1	1	yes	characteristic
103	#1937	C₂×C₁₃	1	2	yes	complemented
104	#1881	C₂×C₁₃	1	2	yes	—
105	#1861	C₂×C₁₃	1	2	yes	complemented
106	#1858	C₂×C₁₃	1	2	yes	complemented
107	#1837	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₂³×C₁₃	—	2	2	no	—
109	C₂³×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
110	Q₈×C₁₃	—	2	2	no	—
111	Q₈×C₁₃	—	2	2	no	—
112	D₄×C₁₃	—	2	2	no	—
113	D₄×C₁₃	—	2	2	no	—
114	D₄×C₁₃	—	4	2	no	—
115	D₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
127	C₈×C₁₃	—	4	1	no	—
128	C₈×C₁₃	—	4	1	no	—
129	C₈×C₁₃	—	4	2	no	—
130	C₈×C₁₃	—	4	2	no	—
131	C₈×C₁₃	—	4	2	no	—
132	C₈×C₁₃	—	4	2	no	—
133	#575	C₂²×C₁₃	1	1	yes	characteristic
134	#568	C₂²×C₁₃	1	2	yes	—
135	#430	C₂²×C₁₃	1	1	yes	characteristic
136	#430	C₂²×C₁₃	1	2	yes	—
137	#424	C₂²×C₁₃	1	2	yes	—
138	#423	C₂²×C₁₃	1	1	yes	characteristic
139	#423	C₂²×C₁₃	1	1	yes	characteristic
140	#421	C₂²×C₁₃	1	2	yes	—
141	C₂×#104	C₂²×C₁₃	1	2	yes	—
142	#416	C₂²×C₁₃	1	2	yes	—
143	#413	C₂²×C₁₃	1	2	yes	—
144	#408	C₂²×C₁₃	1	2	yes	—
145	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
146	#344	C₂²×C₁₃	1	2	yes	—
147	#344	C₂²×C₁₃	1	2	yes	—
148	#341	C₂²×C₁₃	1	2	yes	—
149	#341	C₂²×C₁₃	1	2	yes	—
150	#341	C₂²×C₁₃	1	2	yes	—
151	#339	C₂²×C₁₃	1	2	yes	—
152	#338	C₂²×C₁₃	1	1	yes	characteristic
153	H(3,ℤ/4)	C₂²×C₁₃	1	1	yes	characteristic
154	C₂²×C₁₃	—	4	1	no	—
155	C₂²×C₁₃	—	4	2	no	—
156	C₂²×C₁₃	—	8	2	no	—
157	C₂²×C₁₃	#539	1	1	yes	characteristic
158	C₂²×C₁₃	#539	1	2	yes	—
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	—	4	1	no	—
161	C₄×C₁₃	—	4	2	no	—
162	C₄×C₁₃	—	8	2	no	—
163	C₄×C₁₃	#551	1	1	yes	characteristic
164	C₄×C₁₃	#553	1	2	yes	—
165	C₄×C₁₃	H(3,𝔽₄)	1	1	yes	characteristic
166	C₂×#41	C₂³×C₁₃	1	2	yes	—
167	#137	—	2	2	no	—
168	AGL(1,ℤ/8)	—	2	2	no	—
169	#135	—	4	1	no	—
170	#131	—	4	2	no	—
171	#131	—	4	2	no	—
172	C₂×M₁₆	—	4	2	no	—
173	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
174	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
175	C₂×M₁₆	C₂³×C₁₃	1	1	yes	characteristic
176	C₂×M₁₆	C₂³×C₁₃	1	1	yes	characteristic
177	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
178	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
179	#117	C₂³×C₁₃	1	2	yes	—
180	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
181	#108	—	4	2	no	—
182	#108	—	4	1	no	—
183	C₄⋊C₈	—	4	1	no	—
184	C₄⋊C₈	—	4	1	no	—
185	C₄⋊C₈	—	4	2	no	—
186	#104	—	2	2	no	—
187	#104	—	2	2	no	—
188	#104	—	4	2	no	—
189	#103	—	2	2	no	—
190	#102	—	2	2	no	—
191	#101	—	2	2	no	—
192	#100	—	2	2	no	—
193	C₂²⋊C₈	—	4	2	no	—
194	C₄×C₈	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₂×C₁₃	—	2	1	no	—
197	C₂×C₁₃	—	2	2	no	—
198	C₂×C₁₃	—	8	2	no	—
199	C₂×C₁₃	#2782	1	1	yes	characteristic
200	#41	—	2	2	no	—
201	#41	—	2	2	no	—
202	#41	—	2	2	no	—
203	#41	—	2	2	no	—
204	C₂×Q₈	C₁₃×#41	1	2	yes	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	C₁₃×#41	1	2	yes	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
210	Q₁₆	—	4	1	no	—
211	SD₁₆	—	4	2	no	—
212	D₈	—	4	1	no	—
213	M₁₆	—	2	2	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	2	2	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	—	2	2	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	4	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	4	4	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	2	no	—
235	C₄²	C₁₃×#41	1	1	yes	characteristic
236	C₄²	C₁₃×#41	1	1	yes	characteristic
237	C₁₃	#32702	1	1	yes	characteristic, direct product factor, Sylow
238	C₂³	—	2	2	no	—
239	C₂³	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	Q₈	—	2	2	no	—
242	D₄	—	2	2	no	—
243	D₄	—	2	2	no	—
244	D₄	—	4	2	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
254	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
255	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
256	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₈	—	4	2	no	—
260	C₈	—	4	2	no	—
261	C₈	—	4	2	no	—
262	C₈	—	4	2	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	C₁₃×#539	1	1	yes	characteristic
267	C₂²	C₁₃×#539	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	2	no	—
271	C₄	—	8	2	no	—
272	C₄	C₁₃×#551	1	1	yes	characteristic
273	C₄	C₁₃×#553	1	2	yes	—
274	C₄	C₁₃×H(3,𝔽₄)	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	2	no	—
277	C₂	—	8	2	no	—
278	C₂	C₁₃×#2782	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂×C₁₃	#2782
prime radical	C₁₃×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2103	C₂	semidirect
2. C₁₃×#2102	C₂	semidirect
3. C₁₃×#2080	C₂	semidirect
5. C₁₃×#1937	C₂	semidirect
7. C₁₃×#1861	C₂	semidirect
8. C₁₃×#1858	C₂	semidirect
9. C₁₃×#1837	C₂	semidirect
61. #32702	C₁₃	direct (N × H)
99. #2103	C₂×C₁₃	semidirect
100. #2102	C₂×C₁₃	semidirect
101. #2080	C₂×C₁₃	semidirect
103. #1937	C₂×C₁₃	semidirect
105. #1861	C₂×C₁₃	semidirect
106. #1858	C₂×C₁₃	semidirect
107. #1837	C₂×C₁₃	semidirect
237. C₁₃	#32702	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×#2103	2
C₁₃×#2102	2
C₁₃×#2080	2
C₁₃×#2050	2
C₁₃×#1937	2
C₁₃×#1881	2
C₁₃×#1861	2
C₁₃×#1858	2
C₁₃×#1837	2
#32702	13

Subgroups — C13×#32702 #15719041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32702 #15719041