Subgroups — C₁₃×#13158 #15699497

Order 3328.

316 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1978	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1953	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1388	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₁₃×#430	—	2	1	no	—
12	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
15	C₁₃×#418	—	2	1	no	—
16	C₂×C₁₃×#100	—	2	1	no	—
17	C₁₃×#405	C₂²	1	1	yes	characteristic
18	C₁₃×#361	—	2	2	no	—
19	C₁₃×#361	C₄	1	4	yes	complemented
20	C₁₃×#347	—	2	2	no	—
21	C₁₃×#344	—	2	1	no	—
22	C₂²×D₄×C₁₃	—	2	1	no	—
23	C₂×C₁₃×D₈	—	2	1	no	—
24	C₂×C₁₃×D₈	—	2	1	no	—
25	C₂×C₁₃×D₈	—	2	1	no	—
26	C₂×C₁₃×D₈	—	2	1	no	—
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
29	C₂×C₁₃×M₁₆	—	2	1	no	—
30	C₂×C₁₃×M₁₆	—	2	1	no	—
31	C₂×C₁₃×M₁₆	—	2	1	no	—
32	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×#117	D₄	1	1	yes	characteristic
35	C₁₃×M₃₂	—	2	1	no	—
36	C₁₃×M₃₂	—	2	1	no	—
37	C₁₃×M₃₂	C₂×C₄	1	2	yes	—
38	C₁₃×M₃₂	D₄	1	2	yes	—
39	C₂×C₁₃×C₁₆	—	2	1	no	—
40	C₂×C₁₃×C₁₆	—	2	1	no	—
41	C₁₃×#108	—	2	1	no	—
42	C₁₃×#108	—	2	1	no	—
43	C₁₃×#108	—	2	1	no	—
44	C₁₃×#108	—	2	1	no	—
45	C₁₃×#108	—	2	1	no	—
46	C₁₃×#108	—	4	2	no	—
47	C₁₃×#108	C₂×C₄	1	2	yes	—
48	C₁₃×#102	—	4	2	no	—
49	C₁₃×#100	—	4	2	no	—
50	C₁₃×#97	—	2	1	no	—
51	C₄×C₈×C₁₃	—	2	1	no	—
52	#13158	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂⁴×C₁₃	—	4	1	no	—
54	C₂×D₄×C₁₃	—	2	2	no	—
55	C₂×D₄×C₁₃	—	4	1	no	—
56	C₂×D₄×C₁₃	—	4	1	no	—
57	C₂×D₄×C₁₃	—	4	1	no	—
58	C₂×D₄×C₁₃	—	4	1	no	—
59	C₂×D₄×C₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₁₃×D₈	—	2	2	no	—
62	C₁₃×D₈	—	2	2	no	—
63	C₁₃×D₈	—	4	1	no	—
64	C₁₃×D₈	—	4	1	no	—
65	C₁₃×M₁₆	—	2	2	no	—
66	C₁₃×M₁₆	—	2	1	no	—
67	C₁₃×M₁₆	—	2	1	no	—
68	C₁₃×M₁₆	—	4	1	no	—
69	C₁₃×M₁₆	—	4	1	no	—
70	C₁₃×M₁₆	—	4	1	no	—
71	C₁₃×M₁₆	—	4	1	no	—
72	C₁₃×M₁₆	—	4	1	no	—
73	C₂×C₈×C₁₃	—	2	1	no	—
74	C₂×C₈×C₁₃	—	4	1	no	—
75	C₂×C₈×C₁₃	—	4	1	no	—
76	C₂×C₈×C₁₃	—	4	1	no	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
80	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
81	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
83	C₁₃×C₄⋊C₄	—	2	1	no	—
84	C₁₃×C₂²⋊C₄	—	2	1	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	C₁₃×C₁₆	—	2	2	no	—
87	C₁₃×C₁₆	—	2	2	no	—
88	C₂×#361	C₂×C₁₃	1	1	yes	characteristic
89	C₂×#361	C₂×C₁₃	1	1	yes	characteristic
90	#2110	C₂×C₁₃	1	1	yes	characteristic, complemented
91	#1978	C₂×C₁₃	1	1	yes	characteristic
92	#1953	C₂×C₁₃	1	1	yes	characteristic, complemented
93	#1388	C₂×C₁₃	1	1	yes	characteristic, complemented
94	#1388	C₂×C₁₃	1	1	yes	characteristic, complemented
95	C₂³×C₁₃	—	4	2	no	—
96	C₂³×C₁₃	—	8	1	no	—
97	C₂³×C₁₃	—	8	1	no	—
98	C₂³×C₁₃	—	16	2	no	—
99	C₂³×C₁₃	#137	1	1	yes	characteristic
100	D₄×C₁₃	—	4	2	no	—
101	D₄×C₁₃	—	4	2	no	—
102	D₄×C₁₃	—	8	1	no	—
103	D₄×C₁₃	—	8	1	no	—
104	C₂×C₄×C₁₃	—	4	1	no	—
105	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
108	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₈×C₁₃	—	4	2	no	—
112	C₈×C₁₃	—	4	1	no	—
113	C₈×C₁₃	—	8	1	no	—
114	C₈×C₁₃	—	8	1	no	—
115	C₈×C₁₃	#117	1	1	yes	characteristic
116	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₈×C₁₃	#123	1	1	yes	characteristic
118	C₈×C₁₃	#126	1	1	yes	characteristic
119	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
120	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
121	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
122	#430	—	2	1	no	—
123	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
124	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
125	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
126	#418	—	2	1	no	—
127	C₂×#100	—	2	1	no	—
128	#405	C₂²×C₁₃	1	1	yes	characteristic
129	#361	—	2	2	no	—
130	#361	C₄×C₁₃	1	4	yes	complemented
131	#347	—	2	2	no	—
132	#344	—	2	1	no	—
133	C₂²×C₁₃	—	4	1	no	—
134	C₂²×C₁₃	—	8	2	no	—
135	C₂²×C₁₃	—	16	2	no	—
136	C₂²×C₁₃	—	16	2	no	—
137	C₂²×C₁₃	—	16	2	no	—
138	C₂²×C₁₃	#440	1	1	yes	characteristic
139	C₂²×C₁₃	#468	1	1	yes	characteristic
140	C₂²×C₁₃	#489	1	1	yes	characteristic
141	C₄×C₁₃	—	8	1	no	—
142	C₄×C₁₃	#388	1	1	yes	characteristic
143	C₄×C₁₃	#442	1	1	yes	characteristic
144	C₄×C₁₃	#469	1	1	yes	characteristic
145	C₄×C₁₃	#490	1	1	yes	characteristic
146	C₂²×D₄	—	2	1	no	—
147	C₂×D₈	—	2	1	no	—
148	C₂×D₈	—	2	1	no	—
149	C₂×D₈	—	2	1	no	—
150	C₂×D₈	—	2	1	no	—
151	C₂×D₈	—	2	1	no	—
152	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
153	C₂×M₁₆	—	2	1	no	—
154	C₂×M₁₆	—	2	1	no	—
155	C₂×M₁₆	—	2	1	no	—
156	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
158	#117	D₄×C₁₃	1	1	yes	characteristic
159	M₃₂	—	2	1	no	—
160	M₃₂	—	2	1	no	—
161	M₃₂	C₂×C₄×C₁₃	1	2	yes	—
162	M₃₂	D₄×C₁₃	1	2	yes	—
163	C₂×C₁₆	—	2	1	no	—
164	C₂×C₁₆	—	2	1	no	—
165	#108	—	2	1	no	—
166	#108	—	2	1	no	—
167	#108	—	2	1	no	—
168	#108	—	2	1	no	—
169	#108	—	2	1	no	—
170	#108	—	4	2	no	—
171	#108	C₂×C₄×C₁₃	1	2	yes	—
172	#102	—	4	2	no	—
173	#100	—	4	2	no	—
174	#97	—	2	1	no	—
175	C₄×C₈	—	2	1	no	—
176	C₂×C₁₃	—	2	1	no	—
177	C₂×C₁₃	—	2	1	no	—
178	C₂×C₁₃	—	2	1	no	—
179	C₂×C₁₃	—	16	2	no	—
180	C₂×C₁₃	#1936	1	1	yes	characteristic
181	C₂⁴	—	4	1	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	4	1	no	—
184	C₂×D₄	—	4	1	no	—
185	C₂×D₄	—	4	1	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
189	D₈	—	2	2	no	—
190	D₈	—	2	2	no	—
191	D₈	—	4	1	no	—
192	D₈	—	4	1	no	—
193	M₁₆	—	2	2	no	—
194	M₁₆	—	2	1	no	—
195	M₁₆	—	2	1	no	—
196	M₁₆	—	4	1	no	—
197	M₁₆	—	4	1	no	—
198	M₁₆	—	4	1	no	—
199	M₁₆	—	4	1	no	—
200	M₁₆	—	4	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
206	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
207	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
208	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
209	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
210	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
211	C₄⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₁₆	—	2	2	no	—
215	C₁₆	—	2	2	no	—
216	C₁₃	#13158	1	1	yes	characteristic, direct product factor, Sylow
217	C₂³	—	4	2	no	—
218	C₂³	—	8	1	no	—
219	C₂³	—	8	1	no	—
220	C₂³	—	16	2	no	—
221	C₂³	C₁₃×#137	1	1	yes	characteristic
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	8	1	no	—
225	D₄	—	8	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
228	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
229	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
230	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
231	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
232	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
233	C₈	—	4	2	no	—
234	C₈	—	4	1	no	—
235	C₈	—	8	1	no	—
236	C₈	—	8	1	no	—
237	C₈	C₁₃×#117	1	1	yes	characteristic
238	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
239	C₈	C₁₃×#123	1	1	yes	characteristic
240	C₈	C₁₃×#126	1	1	yes	characteristic
241	C₂²	—	4	1	no	—
242	C₂²	—	8	2	no	—
243	C₂²	—	16	2	no	—
244	C₂²	—	16	2	no	—
245	C₂²	—	16	2	no	—
246	C₂²	C₁₃×#440	1	1	yes	characteristic
247	C₂²	C₁₃×#468	1	1	yes	characteristic
248	C₂²	C₁₃×#489	1	1	yes	characteristic
249	C₄	—	8	1	no	—
250	C₄	C₁₃×#388	1	1	yes	characteristic
251	C₄	C₁₃×#442	1	1	yes	characteristic
252	C₄	C₁₃×#469	1	1	yes	characteristic
253	C₄	C₁₃×#490	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	16	2	no	—
258	C₂	C₁₃×#1936	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#1936
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2110	C₂	semidirect
5. C₁₃×#1953	C₂	semidirect
6. C₁₃×#1388	C₂	semidirect
7. C₁₃×#1388	C₂	semidirect
19. C₁₃×#361	C₄	semidirect
52. #13158	C₁₃	direct (N × H)
90. #2110	C₂×C₁₃	semidirect
92. #1953	C₂×C₁₃	semidirect
93. #1388	C₂×C₁₃	semidirect
94. #1388	C₂×C₁₃	semidirect
130. #361	C₄×C₁₃	semidirect
216. C₁₃	#13158	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×#361	2
C₁₃×#2110	2
C₁₃×#1978	2
C₁₃×#1953	2
C₁₃×#1388	2
#13158	13

Subgroups — C13×#13158 #15699497

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13158 #15699497