Subgroups — C₁₃×#17962 #15704301

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2931	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1594	C₂	1	4	yes	maximal, complemented
4	C₁₃×#1580	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1492	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#131	—	2	2	no	—
9	C₁₃×#532	C₂²	1	2	yes	—
10	C₁₃×#500	C₄	1	1	yes	characteristic
11	C₁₃×#493	C₄	1	1	yes	characteristic
12	C₁₃×#486	C₄	1	2	yes	—
13	C₁₃×#466	C₄	1	4	yes	—
14	C₄×C₁₃×D₈	C₂²	1	4	yes	—
15	C₁₃×#436	—	2	2	no	—
16	C₁₃×#436	C₂²	1	4	yes	—
17	C₈×D₄×C₁₃	—	2	2	no	—
18	C₈×D₄×C₁₃	C₂²	1	4	yes	—
19	C₁₃×#423	C₂²	1	2	yes	—
20	C₁₃×#408	—	2	2	no	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#334	C₂²	1	4	yes	—
25	C₁₃×D₄⋊C₈	C₂²	1	8	yes	—
26	C₁₃×#322	C₂²	1	2	yes	—
27	C₁₃×#322	C₂²	1	2	yes	—
28	C₂×C₁₃×#41	—	2	2	no	—
29	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
30	C₁₃×#131	—	4	8	no	—
31	C₂×C₁₃×M₁₆	—	2	2	no	—
32	C₂×C₁₃×M₁₆	—	2	4	no	—
33	C₂²×C₈×C₁₃	—	2	1	no	—
34	C₂²×C₈×C₁₃	—	2	4	no	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₃	D₄	1	2	yes	—
37	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	C₁₃×#127	C₂×C₄	1	2	yes	—
39	C₁₃×#124	—	2	2	no	—
40	C₁₃×#121	—	2	2	no	—
41	C₁₃×#121	C₂×C₄	1	4	yes	—
42	C₄×D₄×C₁₃	—	2	2	no	—
43	C₄×D₄×C₁₃	C₂³	1	4	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×#107	C₂×C₄	1	4	yes	—
46	C₁₃×C₄⋊C₈	—	2	2	no	—
47	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
48	C₁₃×#102	C₂×C₄	1	8	yes	—
49	C₁₃×C₂²⋊C₈	—	2	4	no	—
50	C₁₃×C₂²⋊C₈	—	2	4	no	—
51	C₄×C₈×C₁₃	—	2	1	no	—
52	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	D₄	1	2	yes	—
55	C₄×C₈×C₁₃	C₂³	1	2	yes	—
56	C₄×C₈×C₁₃	C₂³	1	2	yes	—
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	#17962	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₁₃×#41	—	4	4	no	—
61	C₁₃×#41	—	4	4	no	—
62	C₂×Q₈×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	—	2	4	no	—
64	C₂×D₄×C₁₃	—	2	2	no	—
65	C₂×D₄×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₁₃	—	2	4	no	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
71	C₁₃×D₈	—	2	8	no	—
72	C₁₃×M₁₆	—	4	8	no	—
73	C₁₃×M₁₆	—	4	4	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	4	8	no	—
79	C₂×C₈×C₁₃	—	4	2	no	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
84	C₁₃×C₄⋊C₄	—	2	2	no	—
85	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
86	C₁₃×C₂²⋊C₄	—	2	4	no	—
87	C₁₃×C₂²⋊C₄	—	2	4	no	—
88	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	#41	1	1	yes	characteristic
91	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
92	#3169	C₂×C₁₃	1	1	yes	characteristic
93	#2931	C₂×C₁₃	1	2	yes	complemented
94	#1594	C₂×C₁₃	1	4	yes	complemented
95	#1580	C₂×C₁₃	1	4	yes	complemented
96	#1566	C₂×C₁₃	1	1	yes	characteristic, complemented
97	#1492	C₂×C₁₃	1	2	yes	complemented
98	C₂×#322	C₂×C₁₃	1	1	yes	characteristic, complemented
99	C₂³×C₁₃	—	2	4	no	—
100	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
101	Q₈×C₁₃	—	4	4	no	—
102	D₄×C₁₃	—	2	8	no	—
103	D₄×C₁₃	—	4	8	no	—
104	D₄×C₁₃	—	4	4	no	—
105	D₄×C₁₃	—	4	4	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	4	4	no	—
110	C₂×C₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
115	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
118	C₈×C₁₃	—	2	2	no	—
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	—	2	2	no	—
121	C₈×C₁₃	—	4	2	no	—
122	C₈×C₁₃	—	4	4	no	—
123	C₈×C₁₃	#131	1	4	yes	complemented
124	C₂×#131	—	2	2	no	—
125	#532	C₂²×C₁₃	1	2	yes	—
126	#500	C₄×C₁₃	1	1	yes	characteristic
127	#493	C₄×C₁₃	1	1	yes	characteristic
128	#486	C₄×C₁₃	1	2	yes	—
129	#466	C₄×C₁₃	1	4	yes	—
130	C₄×D₈	C₂²×C₁₃	1	4	yes	—
131	#436	—	2	2	no	—
132	#436	C₂²×C₁₃	1	4	yes	—
133	C₈×D₄	—	2	2	no	—
134	C₈×D₄	C₂²×C₁₃	1	4	yes	—
135	#423	C₂²×C₁₃	1	2	yes	—
136	#408	—	2	2	no	—
137	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
140	#334	C₂²×C₁₃	1	4	yes	—
141	D₄⋊C₈	C₂²×C₁₃	1	8	yes	—
142	#322	C₂²×C₁₃	1	2	yes	—
143	#322	C₂²×C₁₃	1	2	yes	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	—	2	1	no	—
146	C₂²×C₁₃	—	4	4	no	—
147	C₂²×C₁₃	—	4	4	no	—
148	C₂²×C₁₃	—	4	4	no	—
149	C₂²×C₁₃	#442	1	2	yes	—
150	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
151	C₄×C₁₃	—	2	1	no	—
152	C₄×C₁₃	—	2	1	no	—
153	C₄×C₁₃	—	4	4	no	—
154	C₄×C₁₃	#442	1	2	yes	—
155	C₄×C₁₃	#444	1	2	yes	—
156	C₄×C₁₃	C₂×#131	1	2	yes	—
157	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
158	C₂×#41	—	2	2	no	—
159	C₂×D₈	C₂×C₄×C₁₃	1	4	yes	—
160	#131	—	4	8	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	4	no	—
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	—	2	4	no	—
165	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
166	C₂²×C₈	D₄×C₁₃	1	2	yes	—
167	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
168	#127	C₂×C₄×C₁₃	1	2	yes	—
169	#124	—	2	2	no	—
170	#121	—	2	2	no	—
171	#121	C₂×C₄×C₁₃	1	4	yes	—
172	C₄×D₄	—	2	2	no	—
173	C₄×D₄	C₂³×C₁₃	1	4	yes	—
174	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
175	#107	C₂×C₄×C₁₃	1	4	yes	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
178	#102	C₂×C₄×C₁₃	1	8	yes	—
179	C₂²⋊C₈	—	2	4	no	—
180	C₂²⋊C₈	—	2	4	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
183	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
184	C₄×C₈	D₄×C₁₃	1	2	yes	—
185	C₄×C₈	C₂³×C₁₃	1	2	yes	—
186	C₄×C₈	C₂³×C₁₃	1	2	yes	—
187	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
189	C₂×C₁₃	—	2	2	no	—
190	C₂×C₁₃	—	8	4	no	—
191	C₂×C₁₃	#2931	1	1	yes	characteristic
192	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
193	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
194	#41	—	4	4	no	—
195	#41	—	4	4	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	C₂²×C₄×C₁₃	1	4	yes	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
205	D₈	—	2	8	no	—
206	M₁₆	—	4	8	no	—
207	M₁₆	—	4	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	4	8	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
215	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
216	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
217	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
220	C₂²⋊C₄	—	2	4	no	—
221	C₂²⋊C₄	—	2	4	no	—
222	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
223	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
224	C₄²	C₁₃×#41	1	1	yes	characteristic
225	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
226	C₁₃	#17962	1	1	yes	characteristic, direct product factor, Sylow
227	C₂³	—	2	4	no	—
228	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
229	Q₈	—	4	4	no	—
230	D₄	—	2	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
240	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
241	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
242	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
243	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
244	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
245	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	4	2	no	—
250	C₈	—	4	4	no	—
251	C₈	C₁₃×#131	1	4	yes	complemented
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	C₁₃×#442	1	2	yes	—
258	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	4	no	—
262	C₄	C₁₃×#442	1	2	yes	—
263	C₄	C₁₃×#444	1	2	yes	—
264	C₄	C₂×C₁₃×#131	1	2	yes	—
265	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	8	4	no	—
268	C₂	C₁₃×#2931	1	1	yes	characteristic
269	C₂	C₂×C₁₃×#442	1	1	yes	characteristic
270	C₂	C₂×C₁₃×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₁₃×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2931	C₂	semidirect
3. C₁₃×#1594	C₂	semidirect
4. C₁₃×#1580	C₂	semidirect
5. C₁₃×#1566	C₂	semidirect
6. C₁₃×#1492	C₂	semidirect
7. C₂×C₁₃×#322	C₂	semidirect
59. #17962	C₁₃	direct (N × H)
93. #2931	C₂×C₁₃	semidirect
94. #1594	C₂×C₁₃	semidirect
95. #1580	C₂×C₁₃	semidirect
96. #1566	C₂×C₁₃	semidirect
97. #1492	C₂×C₁₃	semidirect
98. C₂×#322	C₂×C₁₃	semidirect
123. C₈×C₁₃	#131	semidirect
226. C₁₃	#17962	direct (N × H)
251. C₈	C₁₃×#131	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#3169	2
C₁₃×#2931	2
C₁₃×#1594	2
C₁₃×#1580	2
C₁₃×#1566	2
C₁₃×#1492	2
C₂×C₁₃×#322	2
#17962	13

Subgroups — C13×#17962 #15704301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#17962 #15704301