Subgroups — C₁₃×#13254 #15699593

Order 3328.

514 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#358	C₂	1	2	yes	maximal, complemented
4	C₂×C₁₃×#357	C₂	1	2	yes	maximal
5	C₄×C₁₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×SD₁₆	—	2	1	no	—
8	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#124	—	2	1	no	—
10	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
11	C₁₃×#495	—	2	2	no	—
12	C₁₃×#495	C₄	1	4	yes	complemented
13	C₂×C₁₃×#107	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
16	C₁₃×C₄⋊C₁₆	—	2	2	no	—
17	C₁₃×#358	—	2	4	no	—
18	C₁₃×#357	—	2	4	no	—
19	C₁₃×#340	—	2	1	no	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂²×D₄×C₁₃	—	2	1	no	—
22	C₂×C₁₃×Q₁₆	—	2	2	no	—
23	C₂×C₁₃×Q₁₆	—	2	1	no	—
24	C₂×C₁₃×Q₁₆	—	2	2	no	—
25	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₁₃×SD₁₆	—	2	4	no	—
27	C₂×C₁₃×SD₁₆	—	4	4	no	—
28	C₂×C₁₃×D₈	—	2	2	no	—
29	C₂×C₁₃×D₈	—	2	1	no	—
30	C₂×C₁₃×D₈	—	2	2	no	—
31	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×#124	—	2	4	no	—
35	C₁₃×#124	—	4	2	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
38	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
39	C₂×C₄²×C₁₃	—	4	1	no	—
40	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
41	C₂×C₁₃×C₁₆	—	2	2	no	—
42	C₂×C₁₃×C₁₆	—	2	2	no	—
43	C₂×C₁₃×C₁₆	—	2	2	no	—
44	C₁₃×#107	—	2	2	no	—
45	C₁₃×#107	D₄	1	4	yes	—
46	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₁₃	D₄	1	2	yes	—
48	C₁₃×#95	—	4	1	no	—
49	#13254	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂⁴×C₁₃	—	4	1	no	—
51	C₂×Q₈×C₁₃	—	2	2	no	—
52	C₂×Q₈×C₁₃	—	4	1	no	—
53	C₂×Q₈×C₁₃	—	4	2	no	—
54	C₂×Q₈×C₁₃	—	4	2	no	—
55	C₂×D₄×C₁₃	—	2	2	no	—
56	C₂×D₄×C₁₃	—	4	2	no	—
57	C₂×D₄×C₁₃	—	4	1	no	—
58	C₂×D₄×C₁₃	—	4	2	no	—
59	C₂²×C₄×C₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₃	—	4	1	no	—
61	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₁₃×Q₁₆	—	2	4	no	—
65	C₁₃×Q₁₆	—	4	1	no	—
66	C₁₃×Q₁₆	—	4	1	no	—
67	C₁₃×SD₁₆	—	4	8	no	—
68	C₁₃×D₈	—	2	4	no	—
69	C₁₃×D₈	—	4	1	no	—
70	C₁₃×D₈	—	4	1	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	#41	1	2	yes	—
77	C₁₃×C₄⋊C₄	—	2	4	no	—
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₁₃×C₂²⋊C₄	—	4	4	no	—
80	C₁₃×C₂²⋊C₄	—	4	4	no	—
81	C₄²×C₁₃	—	8	2	no	—
82	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₁₃	SD₁₆	1	2	yes	—
84	C₁₃×C₁₆	—	4	2	no	—
85	C₂×#495	C₂×C₁₃	1	1	yes	characteristic
86	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
87	C₂×#358	C₂×C₁₃	1	2	yes	complemented
88	C₂×#357	C₂×C₁₃	1	2	yes	—
89	C₄×#107	C₂×C₁₃	1	1	yes	characteristic, complemented
90	C₂³×C₁₃	—	4	2	no	—
91	C₂³×C₁₃	—	8	2	no	—
92	C₂³×C₁₃	—	8	2	no	—
93	C₂³×C₁₃	—	8	1	no	—
94	C₂³×C₁₃	—	8	1	no	—
95	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
96	Q₈×C₁₃	—	4	2	no	—
97	Q₈×C₁₃	—	4	2	no	—
98	Q₈×C₁₃	—	8	2	no	—
99	D₄×C₁₃	—	4	2	no	—
100	D₄×C₁₃	—	4	2	no	—
101	D₄×C₁₃	—	8	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	—	8	2	no	—
108	C₂×C₄×C₁₃	—	8	2	no	—
109	C₂×C₄×C₁₃	—	8	2	no	—
110	C₂×C₄×C₁₃	#102	1	2	yes	—
111	C₂×C₄×C₁₃	#102	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#121	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
116	C₈×C₁₃	—	2	2	no	—
117	C₈×C₁₃	C₄×D₄	1	2	yes	—
118	C₈×C₁₃	#124	1	1	yes	characteristic
119	C₈×C₁₃	#127	1	1	yes	characteristic
120	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
121	C₂²×SD₁₆	—	2	1	no	—
122	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
123	C₂×#124	—	2	1	no	—
124	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
125	#495	—	2	2	no	—
126	#495	C₄×C₁₃	1	4	yes	complemented
127	C₂×#107	C₂²×C₁₃	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
129	C₄×C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₁₆	—	2	2	no	—
131	#358	—	2	4	no	—
132	#357	—	2	4	no	—
133	#340	—	2	1	no	—
134	C₂²×C₁₃	—	8	2	no	—
135	C₂²×C₁₃	—	8	2	no	—
136	C₂²×C₁₃	—	8	2	no	—
137	C₂²×C₁₃	—	16	2	no	—
138	C₂²×C₁₃	—	16	2	no	—
139	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
140	C₂²×C₁₃	#459	1	2	yes	—
141	C₂²×C₁₃	#508	1	2	yes	—
142	C₂²×C₁₃	#509	1	1	yes	characteristic
143	C₂²×C₁₃	#510	1	1	yes	characteristic
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	8	4	no	—
147	C₄×C₁₃	—	8	2	no	—
148	C₄×C₁₃	#390	1	1	yes	characteristic
149	C₄×C₁₃	#418	1	1	yes	characteristic
150	C₄×C₁₃	#460	1	2	yes	—
151	C₂²×Q₈	—	2	1	no	—
152	C₂²×D₄	—	2	1	no	—
153	C₂×Q₁₆	—	2	2	no	—
154	C₂×Q₁₆	—	2	1	no	—
155	C₂×Q₁₆	—	2	2	no	—
156	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
157	C₂×SD₁₆	—	2	4	no	—
158	C₂×SD₁₆	—	4	4	no	—
159	C₂×D₈	—	2	2	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	—	2	2	no	—
162	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
163	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
165	#124	—	2	4	no	—
166	#124	—	4	2	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₂²⋊C₄	—	4	1	no	—
169	C₂×C₂²⋊C₄	—	4	1	no	—
170	C₂×C₄²	—	4	1	no	—
171	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	2	2	no	—
175	#107	—	2	2	no	—
176	#107	D₄×C₁₃	1	4	yes	—
177	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
178	C₄×C₈	D₄×C₁₃	1	2	yes	—
179	#95	—	4	1	no	—
180	C₂×C₁₃	—	16	2	no	—
181	C₂×C₁₃	#1982	1	1	yes	characteristic
182	C₂×C₁₃	#2138	1	1	yes	characteristic
183	C₂×C₁₃	#2139	1	1	yes	characteristic
184	C₂×C₁₃	#2210	1	2	yes	—
185	C₂×C₁₃	#2211	1	2	yes	—
186	C₂⁴	—	4	1	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	4	1	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
198	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
200	Q₁₆	—	2	4	no	—
201	Q₁₆	—	4	1	no	—
202	Q₁₆	—	4	1	no	—
203	SD₁₆	—	4	8	no	—
204	D₈	—	2	4	no	—
205	D₈	—	4	1	no	—
206	D₈	—	4	1	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
209	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
211	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
212	C₂×C₈	C₁₃×#41	1	2	yes	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₂²⋊C₄	—	4	4	no	—
217	C₄²	—	8	2	no	—
218	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
219	C₄²	C₁₃×SD₁₆	1	2	yes	—
220	C₁₆	—	4	2	no	—
221	C₁₃	#13254	1	1	yes	characteristic, direct product factor, Sylow
222	C₂³	—	4	2	no	—
223	C₂³	—	8	2	no	—
224	C₂³	—	8	2	no	—
225	C₂³	—	8	1	no	—
226	C₂³	—	8	1	no	—
227	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
228	Q₈	—	4	2	no	—
229	Q₈	—	4	2	no	—
230	Q₈	—	8	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	4	2	no	—
233	D₄	—	8	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	C₁₃×#102	1	2	yes	—
243	C₂×C₄	C₁₃×#102	1	2	yes	—
244	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
245	C₂×C₄	C₁₃×#121	1	2	yes	—
246	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
247	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
248	C₈	—	2	2	no	—
249	C₈	C₄×D₄×C₁₃	1	2	yes	—
250	C₈	C₁₃×#124	1	1	yes	characteristic
251	C₈	C₁₃×#127	1	1	yes	characteristic
252	C₂²	—	8	2	no	—
253	C₂²	—	8	2	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	16	2	no	—
256	C₂²	—	16	2	no	—
257	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
258	C₂²	C₁₃×#459	1	2	yes	—
259	C₂²	C₁₃×#508	1	2	yes	—
260	C₂²	C₁₃×#509	1	1	yes	characteristic
261	C₂²	C₁₃×#510	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	8	2	no	—
265	C₄	—	8	4	no	—
266	C₄	C₁₃×#390	1	1	yes	characteristic
267	C₄	C₁₃×#418	1	1	yes	characteristic
268	C₄	C₁₃×#460	1	2	yes	—
269	C₂	—	16	2	no	—
270	C₂	C₁₃×#1982	1	1	yes	characteristic
271	C₂	C₁₃×#2138	1	1	yes	characteristic
272	C₂	C₁₃×#2139	1	1	yes	characteristic
273	C₂	C₁₃×#2210	1	2	yes	—
274	C₂	C₁₃×#2211	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
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 ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₁₃×#358	C₂	semidirect
5. C₄×C₁₃×#107	C₂	semidirect
12. C₁₃×#495	C₄	semidirect
49. #13254	C₁₃	direct (N × H)
86. C₂×C₄⋊C₁₆	C₂×C₁₃	semidirect
87. C₂×#358	C₂×C₁₃	semidirect
89. C₄×#107	C₂×C₁₃	semidirect
126. #495	C₄×C₁₃	semidirect
221. C₁₃	#13254	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×#495	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×C₁₃×#358	2
C₂×C₁₃×#357	2
C₄×C₁₃×#107	2
#13254	13

Subgroups — C13×#13254 #15699593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13254 #15699593