Subgroups — C₁₃×#13251 #15699590

Order 3328.

478 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#359	C₂	1	4	yes	maximal, complemented
4	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#135	—	2	1	no	—
6	C₂×C₁₃×#135	C₂²	1	2	yes	—
7	C₁₃×#532	—	2	1	no	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	C₁₃×#495	—	2	1	no	—
10	C₁₃×#495	C₄	1	2	yes	—
11	C₁₃×#494	C₄	1	1	yes	characteristic
12	C₁₃×#493	C₄	1	1	yes	characteristic
13	C₁₃×#492	—	2	1	no	—
14	C₂×C₁₃×#108	C₂²	1	2	yes	—
15	C₄×C₁₃×M₁₆	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×C₄⋊C₁₆	—	2	1	no	—
18	C₁₃×C₄⋊C₁₆	—	2	1	no	—
19	C₁₃×#359	—	2	8	no	—
20	C₁₃×#341	—	2	1	no	—
21	C₂×C₁₃×#41	—	2	2	no	—
22	C₁₃×#135	—	2	4	no	—
23	C₁₃×#135	—	2	4	no	—
24	C₁₃×#135	—	4	4	no	—
25	C₂×C₁₃×Q₁₆	—	2	1	no	—
26	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₃×SD₁₆	—	2	2	no	—
28	C₂×C₁₃×SD₁₆	—	2	2	no	—
29	C₂×C₁₃×D₈	—	2	1	no	—
30	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
31	C₂×C₁₃×M₁₆	—	2	2	no	—
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×C₄⋊Q₈	—	2	1	no	—
35	C₁₃×#127	—	2	1	no	—
36	C₁₃×#124	—	2	2	no	—
37	C₁₃×#121	—	4	1	no	—
38	C₁₃×#121	—	4	1	no	—
39	C₄×D₄×C₁₃	—	4	1	no	—
40	C₄×D₄×C₁₃	—	4	1	no	—
41	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
42	C₂×C₁₃×C₁₆	—	2	4	no	—
43	C₂×C₁₃×C₁₆	—	2	2	no	—
44	C₁₃×#108	—	2	2	no	—
45	C₁₃×#108	D₄	1	4	yes	complemented
46	C₁₃×#97	—	4	1	no	—
47	C₄×C₈×C₁₃	—	4	1	no	—
48	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
52	#13251	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₁₃×#41	—	4	4	no	—
54	C₁₃×#41	—	4	4	no	—
55	C₂×Q₈×C₁₃	—	2	2	no	—
56	C₂×D₄×C₁₃	—	2	2	no	—
57	C₂×D₄×C₁₃	—	4	2	no	—
58	C₂×D₄×C₁₃	—	4	1	no	—
59	C₂×D₄×C₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₁₃×Q₁₆	—	2	4	no	—
65	C₁₃×Q₁₆	—	4	2	no	—
66	C₁₃×SD₁₆	—	4	4	no	—
67	C₁₃×SD₁₆	—	4	2	no	—
68	C₁₃×SD₁₆	—	4	2	no	—
69	C₁₃×D₈	—	2	4	no	—
70	C₁₃×D₈	—	4	2	no	—
71	C₁₃×M₁₆	—	2	4	no	—
72	C₁₃×M₁₆	—	4	2	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	4	2	no	—
75	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	#41	1	2	yes	—
81	C₁₃×C₄⋊C₄	—	4	1	no	—
82	C₁₃×C₄⋊C₄	—	4	1	no	—
83	C₁₃×C₂²⋊C₄	—	4	2	no	—
84	C₁₃×C₂²⋊C₄	—	4	2	no	—
85	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₁₃	D₈	1	1	yes	characteristic
88	C₄²×C₁₃	D₈	1	1	yes	characteristic
89	C₁₃×C₁₆	—	4	2	no	—
90	#3147	C₂×C₁₃	1	1	yes	characteristic
91	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
92	C₂×#359	C₂×C₁₃	1	4	yes	complemented
93	C₄×#108	C₂×C₁₃	1	1	yes	characteristic, complemented
94	C₂³×C₁₃	—	4	2	no	—
95	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
96	Q₈×C₁₃	—	4	2	no	—
97	Q₈×C₁₃	—	4	2	no	—
98	D₄×C₁₃	—	4	2	no	—
99	D₄×C₁₃	—	4	2	no	—
100	D₄×C₁₃	—	8	2	no	—
101	D₄×C₁₃	—	8	4	no	—
102	D₄×C₁₃	—	8	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	4	1	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	8	4	no	—
107	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#121	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
115	C₈×C₁₃	—	2	1	no	—
116	C₈×C₁₃	—	2	1	no	—
117	C₈×C₁₃	—	4	4	no	—
118	C₈×C₁₃	C₄×D₄	1	2	yes	—
119	C₈×C₁₃	#124	1	1	yes	characteristic
120	C₈×C₁₃	#127	1	1	yes	characteristic
121	C₂×#135	—	2	1	no	—
122	C₂×#135	C₂²×C₁₃	1	2	yes	—
123	#532	—	2	1	no	—
124	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
125	#495	—	2	1	no	—
126	#495	C₄×C₁₃	1	2	yes	—
127	#494	C₄×C₁₃	1	1	yes	characteristic
128	#493	C₄×C₁₃	1	1	yes	characteristic
129	#492	—	2	1	no	—
130	C₂×#108	C₂²×C₁₃	1	2	yes	—
131	C₄×M₁₆	—	2	1	no	—
132	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
133	C₄⋊C₁₆	—	2	1	no	—
134	C₄⋊C₁₆	—	2	1	no	—
135	#359	—	2	8	no	—
136	#341	—	2	1	no	—
137	C₂²×C₁₃	—	2	1	no	—
138	C₂²×C₁₃	—	2	1	no	—
139	C₂²×C₁₃	—	8	2	no	—
140	C₂²×C₁₃	—	8	2	no	—
141	C₂²×C₁₃	—	8	2	no	—
142	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
143	C₂²×C₁₃	#460	1	2	yes	—
144	C₄×C₁₃	—	2	1	no	—
145	C₄×C₁₃	—	2	1	no	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	8	2	no	—
148	C₄×C₁₃	#359	1	2	yes	complemented
149	C₄×C₁₃	#390	1	1	yes	characteristic
150	C₄×C₁₃	#418	1	1	yes	characteristic
151	C₄×C₁₃	#459	1	2	yes	—
152	C₂×#41	—	2	2	no	—
153	#135	—	2	4	no	—
154	#135	—	2	4	no	—
155	#135	—	4	4	no	—
156	C₂×Q₁₆	—	2	1	no	—
157	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
158	C₂×SD₁₆	—	2	2	no	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
165	C₄⋊Q₈	—	2	1	no	—
166	#127	—	2	1	no	—
167	#124	—	2	2	no	—
168	#121	—	4	1	no	—
169	#121	—	4	1	no	—
170	C₄×D₄	—	4	1	no	—
171	C₄×D₄	—	4	1	no	—
172	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
173	C₂×C₁₆	—	2	4	no	—
174	C₂×C₁₆	—	2	2	no	—
175	#108	—	2	2	no	—
176	#108	D₄×C₁₃	1	4	yes	complemented
177	#97	—	4	1	no	—
178	C₄×C₈	—	4	1	no	—
179	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
180	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
181	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
182	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
183	C₂×C₁₃	—	2	2	no	—
184	C₂×C₁₃	—	16	2	no	—
185	C₂×C₁₃	#1982	1	1	yes	characteristic
186	C₂×C₁₃	C₂×#359	1	1	yes	characteristic
187	C₂×C₁₃	#2143	1	1	yes	characteristic
188	#41	—	4	4	no	—
189	#41	—	4	4	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
197	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
199	Q₁₆	—	2	4	no	—
200	Q₁₆	—	4	2	no	—
201	SD₁₆	—	4	4	no	—
202	SD₁₆	—	4	2	no	—
203	SD₁₆	—	4	2	no	—
204	D₈	—	2	4	no	—
205	D₈	—	4	2	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	4	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
211	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
213	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
214	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
215	C₂×C₈	C₁₃×#41	1	2	yes	—
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
221	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
222	C₄²	C₁₃×D₈	1	1	yes	characteristic
223	C₄²	C₁₃×D₈	1	1	yes	characteristic
224	C₁₆	—	4	2	no	—
225	C₁₃	#13251	1	1	yes	characteristic, direct product factor, Sylow
226	C₂³	—	4	2	no	—
227	C₂³	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
228	Q₈	—	4	2	no	—
229	Q₈	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	8	2	no	—
233	D₄	—	8	4	no	—
234	D₄	—	8	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	8	4	no	—
239	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
240	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
241	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
242	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
243	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄	C₁₃×#121	1	2	yes	—
245	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
246	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	4	4	no	—
250	C₈	C₄×D₄×C₁₃	1	2	yes	—
251	C₈	C₁₃×#124	1	1	yes	characteristic
252	C₈	C₁₃×#127	1	1	yes	characteristic
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	8	2	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
259	C₂²	C₁₃×#460	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	8	2	no	—
264	C₄	C₁₃×#359	1	2	yes	complemented
265	C₄	C₁₃×#390	1	1	yes	characteristic
266	C₄	C₁₃×#418	1	1	yes	characteristic
267	C₄	C₁₃×#459	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	16	2	no	—
270	C₂	C₁₃×#1982	1	1	yes	characteristic
271	C₂	C₂×C₁₃×#359	1	1	yes	characteristic
272	C₂	C₁₃×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×#102
prime radical	C₁₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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×#359	C₂	semidirect
4. C₄×C₁₃×#108	C₂	semidirect
45. C₁₃×#108	D₄	semidirect
52. #13251	C₁₃	direct (N × H)
91. C₂×C₄⋊C₁₆	C₂×C₁₃	semidirect
92. C₂×#359	C₂×C₁₃	semidirect
93. C₄×#108	C₂×C₁₃	semidirect
148. C₄×C₁₃	#359	semidirect
176. #108	D₄×C₁₃	semidirect
225. C₁₃	#13251	direct (N × H)
264. C₄	C₁₃×#359	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₃×#3147	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×C₁₃×#359	2
C₄×C₁₃×#108	2
#13251	13

Subgroups — C13×#13251 #15699590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13251 #15699590