Subgroups — C₁₃×#32870 #15719209

Order 3328.

502 conjugacy classes of subgroups.

The subgroup lattice has 292 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×#2098	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2081	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2069	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2048	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2045	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2018	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#1903	C₂	1	2	yes	maximal, complemented
9	C₁₃×#1858	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×AGL(1,ℤ/8)	C₂²	1	2	yes	—
11	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
12	C₁₃×#568	C₂²	1	1	yes	characteristic
13	C₁₃×#430	C₂²	1	2	yes	—
14	C₁₃×#430	C₂²	1	1	yes	characteristic
15	C₁₃×#430	C₂²	1	2	yes	—
16	C₁₃×#424	C₂²	1	2	yes	—
17	C₁₃×#421	C₂²	1	2	yes	—
18	C₁₃×#418	C₂²	1	2	yes	—
19	C₁₃×#418	C₂²	1	2	yes	—
20	C₁₃×#413	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×#100	C₂²	1	2	yes	—
22	C₂×C₁₃×#100	C₂²	1	2	yes	—
23	C₂×C₁₃×#100	C₂²	1	2	yes	—
24	C₁₃×#408	C₂²	1	1	yes	characteristic
25	C₁₃×#344	C₂²	1	2	yes	—
26	C₁₃×#344	C₂²	1	2	yes	—
27	C₁₃×#341	C₂²	1	1	yes	characteristic
28	C₁₃×#341	C₂²	1	2	yes	—
29	C₁₃×#338	C₂²	1	2	yes	—
30	C₁₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
31	C₂×C₁₃×#41	C₂³	1	1	yes	characteristic
32	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
33	C₁₃×AGL(1,ℤ/8)	—	4	2	no	—
34	C₁₃×AGL(1,ℤ/8)	—	4	2	no	—
35	C₂×C₁₃×SD₁₆	—	2	2	no	—
36	C₂×C₁₃×D₈	—	2	2	no	—
37	C₂×C₁₃×D₈	—	4	1	no	—
38	C₂×C₁₃×D₈	—	4	2	no	—
39	C₁₃×#131	—	4	2	no	—
40	C₂×C₁₃×M₁₆	—	4	1	no	—
41	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
42	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
43	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
44	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
45	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×#117	C₂³	1	2	yes	—
48	C₁₃×#108	—	4	1	no	—
49	C₁₃×#108	—	4	2	no	—
50	C₁₃×#108	—	4	2	no	—
51	C₁₃×C₄⋊C₈	—	4	2	no	—
52	C₁₃×#104	—	4	2	no	—
53	C₁₃×#102	—	4	2	no	—
54	C₁₃×#102	—	4	2	no	—
55	C₁₃×#101	—	2	1	no	—
56	C₁₃×#100	—	2	1	no	—
57	C₁₃×#100	—	4	2	no	—
58	C₁₃×#100	—	4	2	no	—
59	C₁₃×#100	—	4	2	no	—
60	C₁₃×C₂²⋊C₈	—	4	1	no	—
61	#32870	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂⁴×C₁₃	—	2	2	no	—
63	C₁₃×#41	—	2	2	no	—
64	C₁₃×#41	—	2	2	no	—
65	C₂×Q₈×C₁₃	#41	1	1	yes	characteristic
66	C₂×D₄×C₁₃	—	2	1	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	2	2	no	—
69	C₂×D₄×C₁₃	—	2	2	no	—
70	C₂×D₄×C₁₃	—	2	2	no	—
71	C₂×D₄×C₁₃	—	2	2	no	—
72	C₂×D₄×C₁₃	—	2	2	no	—
73	C₂×D₄×C₁₃	#41	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	—	2	1	no	—
75	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
76	C₁₃×SD₁₆	—	4	4	no	—
77	C₁₃×D₈	—	4	4	no	—
78	C₁₃×D₈	—	4	4	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	—	2	2	no	—
84	C₁₃×M₁₆	—	2	2	no	—
85	C₁₃×M₁₆	—	2	2	no	—
86	C₁₃×M₁₆	—	2	2	no	—
87	C₁₃×M₁₆	—	2	2	no	—
88	C₁₃×M₁₆	—	4	2	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	—	2	1	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	—	4	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₂²⋊C₄	—	2	2	no	—
99	C₄²×C₁₃	—	2	2	no	—
100	#2103	C₂×C₁₃	1	2	yes	complemented
101	#2098	C₂×C₁₃	1	1	yes	characteristic, complemented
102	#2081	C₂×C₁₃	1	2	yes	complemented
103	#2069	C₂×C₁₃	1	2	yes	complemented
104	#2048	C₂×C₁₃	1	2	yes	complemented
105	#2045	C₂×C₁₃	1	2	yes	complemented
106	#2018	C₂×C₁₃	1	1	yes	characteristic
107	#1903	C₂×C₁₃	1	2	yes	complemented
108	#1858	C₂×C₁₃	1	1	yes	characteristic, complemented
109	C₂³×C₁₃	—	2	1	no	—
110	C₂³×C₁₃	—	4	2	no	—
111	C₂³×C₁₃	—	4	2	no	—
112	C₂³×C₁₃	—	4	2	no	—
113	C₂³×C₁₃	—	16	2	no	—
114	C₂³×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
115	Q₈×C₁₃	—	2	2	no	—
116	D₄×C₁₃	—	2	2	no	—
117	D₄×C₁₃	—	4	2	no	—
118	D₄×C₁₃	—	4	2	no	—
119	D₄×C₁₃	—	4	2	no	—
120	D₄×C₁₃	—	4	2	no	—
121	D₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	4	1	no	—
124	C₂×C₄×C₁₃	—	4	1	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
128	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
129	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
130	C₈×C₁₃	—	4	2	no	—
131	C₈×C₁₃	—	4	2	no	—
132	C₈×C₁₃	—	4	1	no	—
133	C₈×C₁₃	—	4	2	no	—
134	C₈×C₁₃	—	4	1	no	—
135	C₈×C₁₃	—	4	2	no	—
136	C₂×AGL(1,ℤ/8)	C₂²×C₁₃	1	2	yes	—
137	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
138	#568	C₂²×C₁₃	1	1	yes	characteristic
139	#430	C₂²×C₁₃	1	2	yes	—
140	#430	C₂²×C₁₃	1	1	yes	characteristic
141	#430	C₂²×C₁₃	1	2	yes	—
142	#424	C₂²×C₁₃	1	2	yes	—
143	#421	C₂²×C₁₃	1	2	yes	—
144	#418	C₂²×C₁₃	1	2	yes	—
145	#418	C₂²×C₁₃	1	2	yes	—
146	#413	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×#100	C₂²×C₁₃	1	2	yes	—
148	C₂×#100	C₂²×C₁₃	1	2	yes	—
149	C₂×#100	C₂²×C₁₃	1	2	yes	—
150	#408	C₂²×C₁₃	1	1	yes	characteristic
151	#344	C₂²×C₁₃	1	2	yes	—
152	#344	C₂²×C₁₃	1	2	yes	—
153	#341	C₂²×C₁₃	1	1	yes	characteristic
154	#341	C₂²×C₁₃	1	2	yes	—
155	#338	C₂²×C₁₃	1	2	yes	—
156	H(3,ℤ/4)	C₂²×C₁₃	1	1	yes	characteristic
157	C₂²×C₁₃	—	4	1	no	—
158	C₂²×C₁₃	—	4	1	no	—
159	C₂²×C₁₃	—	8	1	no	—
160	C₂²×C₁₃	—	8	2	no	—
161	C₂²×C₁₃	—	16	2	no	—
162	C₂²×C₁₃	—	16	2	no	—
163	C₂²×C₁₃	—	16	2	no	—
164	C₂²×C₁₃	#539	1	1	yes	characteristic
165	C₂²×C₁₃	#562	1	2	yes	—
166	C₄×C₁₃	—	4	1	no	—
167	C₄×C₁₃	—	8	2	no	—
168	C₄×C₁₃	#538	1	2	yes	—
169	C₄×C₁₃	#560	1	1	yes	characteristic
170	C₄×C₁₃	#560	1	1	yes	characteristic
171	C₂×#41	C₂³×C₁₃	1	1	yes	characteristic
172	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
173	AGL(1,ℤ/8)	—	4	2	no	—
174	AGL(1,ℤ/8)	—	4	2	no	—
175	C₂×SD₁₆	—	2	2	no	—
176	C₂×D₈	—	2	2	no	—
177	C₂×D₈	—	4	1	no	—
178	C₂×D₈	—	4	2	no	—
179	#131	—	4	2	no	—
180	C₂×M₁₆	—	4	1	no	—
181	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
182	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
183	C₂×M₁₆	C₂³×C₁₃	1	1	yes	characteristic
184	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
185	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
186	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
187	#117	C₂³×C₁₃	1	2	yes	—
188	#108	—	4	1	no	—
189	#108	—	4	2	no	—
190	#108	—	4	2	no	—
191	C₄⋊C₈	—	4	2	no	—
192	#104	—	4	2	no	—
193	#102	—	4	2	no	—
194	#102	—	4	2	no	—
195	#101	—	2	1	no	—
196	#100	—	2	1	no	—
197	#100	—	4	2	no	—
198	#100	—	4	2	no	—
199	#100	—	4	2	no	—
200	C₂²⋊C₈	—	4	1	no	—
201	C₂×C₁₃	—	2	2	no	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	—	8	1	no	—
204	C₂×C₁₃	—	16	2	no	—
205	C₂×C₁₃	#2826	1	1	yes	characteristic
206	C₂⁴	—	2	2	no	—
207	#41	—	2	2	no	—
208	#41	—	2	2	no	—
209	C₂×Q₈	C₁₃×#41	1	1	yes	characteristic
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	C₁₃×#41	1	1	yes	characteristic
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
220	SD₁₆	—	4	4	no	—
221	D₈	—	4	4	no	—
222	D₈	—	4	4	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	M₁₆	—	2	2	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	2	2	no	—
230	M₁₆	—	2	2	no	—
231	M₁₆	—	2	2	no	—
232	M₁₆	—	4	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₁₃	#32870	1	1	yes	characteristic, direct product factor, Sylow
245	C₂³	—	2	1	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	16	2	no	—
250	C₂³	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
251	Q₈	—	2	2	no	—
252	D₄	—	2	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	D₄	—	4	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
263	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
264	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
265	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
266	C₈	—	4	2	no	—
267	C₈	—	4	2	no	—
268	C₈	—	4	1	no	—
269	C₈	—	4	2	no	—
270	C₈	—	4	1	no	—
271	C₈	—	4	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	8	1	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	16	2	no	—
277	C₂²	—	16	2	no	—
278	C₂²	—	16	2	no	—
279	C₂²	C₁₃×#539	1	1	yes	characteristic
280	C₂²	C₁₃×#562	1	2	yes	—
281	C₄	—	4	1	no	—
282	C₄	—	8	2	no	—
283	C₄	C₁₃×#538	1	2	yes	—
284	C₄	C₁₃×#560	1	1	yes	characteristic
285	C₄	C₁₃×#560	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	2	1	no	—
288	C₂	—	8	1	no	—
289	C₂	—	16	2	no	—
290	C₂	C₁₃×#2826	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#2826
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#2018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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×#2098	C₂	semidirect
3. C₁₃×#2081	C₂	semidirect
4. C₁₃×#2069	C₂	semidirect
5. C₁₃×#2048	C₂	semidirect
6. C₁₃×#2045	C₂	semidirect
8. C₁₃×#1903	C₂	semidirect
9. C₁₃×#1858	C₂	semidirect
61. #32870	C₁₃	direct (N × H)
100. #2103	C₂×C₁₃	semidirect
101. #2098	C₂×C₁₃	semidirect
102. #2081	C₂×C₁₃	semidirect
103. #2069	C₂×C₁₃	semidirect
104. #2048	C₂×C₁₃	semidirect
105. #2045	C₂×C₁₃	semidirect
107. #1903	C₂×C₁₃	semidirect
108. #1858	C₂×C₁₃	semidirect
244. C₁₃	#32870	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×#2098	2
C₁₃×#2081	2
C₁₃×#2069	2
C₁₃×#2048	2
C₁₃×#2045	2
C₁₃×#2018	2
C₁₃×#1903	2
C₁₃×#1858	2
#32870	13

Subgroups — C13×#32870 #15719209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32870 #15719209