Subgroups — C₁₃×#3356 #13112414

Order 1664.

342 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#556	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#554	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#485	C₂	1	2	yes	maximal, complemented
5	C₁₃×#483	C₂	1	2	yes	maximal, complemented
6	C₁₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#470	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#436	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#428	C₂	1	2	yes	maximal, complemented
12	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
14	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₃×#126	—	2	2	no	—
17	C₁₃×#125	—	2	1	no	—
18	C₁₃×#124	—	2	1	no	—
19	C₁₃×C₂²⋊Q₈	—	2	2	no	—
20	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₄×Q₈×C₁₃	—	2	1	no	—
23	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₃	—	2	1	no	—
26	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#117	C₂²	1	2	yes	—
28	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
29	C₁₃×#107	C₂²	1	1	yes	characteristic
30	C₁₃×#107	C₂²	1	2	yes	—
31	C₁₃×#106	C₂²	1	2	yes	—
32	C₁₃×#106	C₂²	1	2	yes	—
33	C₁₃×#106	C₂²	1	2	yes	—
34	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₁₃×#103	C₂²	1	1	yes	characteristic
36	C₁₃×#103	C₂²	1	1	yes	characteristic
37	C₁₃×#103	C₂²	1	2	yes	—
38	C₁₃×#103	C₂²	1	2	yes	—
39	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₃	—	2	1	no	—
42	C₂×Q₈×C₁₃	—	2	1	no	—
43	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	—	2	2	no	—
47	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
48	C₁₃×Q₁₆	—	2	2	no	—
49	C₁₃×M₁₆	—	2	4	no	—
50	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂³	1	2	yes	—
52	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×C₄⋊C₄	—	2	2	no	—
54	C₁₃×C₄⋊C₄	—	2	1	no	—
55	C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₁₃×C₄⋊C₄	—	2	1	no	—
57	C₁₃×C₄⋊C₄	—	2	1	no	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	2	1	no	—
60	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
61	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
62	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
64	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
65	C₁₃×C₂²⋊C₄	—	2	2	no	—
66	C₁₃×C₂²⋊C₄	—	2	2	no	—
67	C₁₃×C₂²⋊C₄	—	2	2	no	—
68	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
69	C₄²×C₁₃	—	2	2	no	—
70	C₄²×C₁₃	—	2	1	no	—
71	C₄²×C₁₃	—	2	1	no	—
72	C₄²×C₁₃	C₂³	1	1	yes	characteristic
73	#3356	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₁₃	C₂×D₄	1	2	yes	—
75	Q₈×C₁₃	—	2	1	no	—
76	Q₈×C₁₃	—	2	1	no	—
77	Q₈×C₁₃	—	2	1	no	—
78	Q₈×C₁₃	—	2	1	no	—
79	Q₈×C₁₃	—	4	2	no	—
80	Q₈×C₁₃	—	8	1	no	—
81	D₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	8	2	no	—
91	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₈×C₁₃	—	2	2	no	—
96	C₈×C₁₃	—	2	1	no	—
97	C₈×C₁₃	#41	1	2	yes	—
98	#556	C₂×C₁₃	1	1	yes	characteristic
99	#554	C₂×C₁₃	1	1	yes	characteristic
100	#498	C₂×C₁₃	1	1	yes	characteristic, complemented
101	#485	C₂×C₁₃	1	2	yes	complemented
102	#483	C₂×C₁₃	1	2	yes	complemented
103	#477	C₂×C₁₃	1	1	yes	characteristic, complemented
104	#475	C₂×C₁₃	1	1	yes	characteristic, complemented
105	#470	C₂×C₁₃	1	2	yes	complemented
106	C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
107	#436	C₂×C₁₃	1	1	yes	characteristic
108	#428	C₂×C₁₃	1	2	yes	complemented
109	C₂²×C₁₃	—	2	2	no	—
110	C₂²×C₁₃	—	2	2	no	—
111	C₂²×C₁₃	—	2	2	no	—
112	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₄×C₁₃	—	2	2	no	—
114	C₄×C₁₃	—	2	1	no	—
115	C₄×C₁₃	—	4	1	no	—
116	C₄×C₁₃	—	4	1	no	—
117	C₄×C₁₃	—	4	2	no	—
118	C₄×C₁₃	—	4	1	no	—
119	C₄×C₁₃	—	4	1	no	—
120	C₄×C₁₃	—	4	2	no	—
121	C₄×C₁₃	#137	1	2	yes	—
122	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
123	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
125	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
126	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
127	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
128	#126	—	2	2	no	—
129	#125	—	2	1	no	—
130	#124	—	2	1	no	—
131	C₂²⋊Q₈	—	2	2	no	—
132	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
133	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
134	C₄×Q₈	—	2	1	no	—
135	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
136	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
137	C₄×D₄	—	2	1	no	—
138	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
139	#117	C₂²×C₁₃	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
141	#107	C₂²×C₁₃	1	1	yes	characteristic
142	#107	C₂²×C₁₃	1	2	yes	—
143	#106	C₂²×C₁₃	1	2	yes	—
144	#106	C₂²×C₁₃	1	2	yes	—
145	#106	C₂²×C₁₃	1	2	yes	—
146	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
147	#103	C₂²×C₁₃	1	1	yes	characteristic
148	#103	C₂²×C₁₃	1	1	yes	characteristic
149	#103	C₂²×C₁₃	1	2	yes	—
150	#103	C₂²×C₁₃	1	2	yes	—
151	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
152	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×C₁₃	—	4	2	no	—
154	C₂×C₁₃	#547	1	1	yes	characteristic
155	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
156	C₂×C₁₃	#577	1	1	yes	characteristic
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
160	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
161	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
164	Q₁₆	—	2	2	no	—
165	M₁₆	—	2	4	no	—
166	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
167	C₂×C₈	C₂³×C₁₃	1	2	yes	—
168	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
177	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
178	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
179	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
180	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	C₂³×C₁₃	1	1	yes	characteristic
189	C₁₃	#3356	1	1	yes	characteristic, direct product factor, Sylow
190	C₂³	C₂×D₄×C₁₃	1	2	yes	—
191	Q₈	—	2	1	no	—
192	Q₈	—	2	1	no	—
193	Q₈	—	2	1	no	—
194	Q₈	—	2	1	no	—
195	Q₈	—	4	2	no	—
196	Q₈	—	8	1	no	—
197	D₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	8	2	no	—
207	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
208	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
209	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	2	1	no	—
213	C₈	C₁₃×#41	1	2	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	4	1	no	—
221	C₄	—	4	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	1	no	—
225	C₄	—	4	2	no	—
226	C₄	C₁₃×#137	1	2	yes	—
227	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
228	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
229	C₂	—	4	2	no	—
230	C₂	C₁₃×#547	1	1	yes	characteristic
231	C₂	C₂×C₁₃×#137	1	1	yes	characteristic
232	C₂	C₁₃×#577	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₁₃×#498	C₂	semidirect
4. C₁₃×#485	C₂	semidirect
5. C₁₃×#483	C₂	semidirect
6. C₁₃×#477	C₂	semidirect
7. C₁₃×#475	C₂	semidirect
8. C₁₃×#470	C₂	semidirect
9. C₄×C₁₃×Q₁₆	C₂	semidirect
11. C₁₃×#428	C₂	semidirect
73. #3356	C₁₃	direct (N × H)
100. #498	C₂×C₁₃	semidirect
101. #485	C₂×C₁₃	semidirect
102. #483	C₂×C₁₃	semidirect
103. #477	C₂×C₁₃	semidirect
104. #475	C₂×C₁₃	semidirect
105. #470	C₂×C₁₃	semidirect
106. C₄×Q₁₆	C₂×C₁₃	semidirect
108. #428	C₂×C₁₃	semidirect
189. C₁₃	#3356	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×#556	2
C₁₃×#554	2
C₁₃×#498	2
C₁₃×#485	2
C₁₃×#483	2
C₁₃×#477	2
C₁₃×#475	2
C₁₃×#470	2
C₄×C₁₃×Q₁₆	2
C₁₃×#436	2
C₁₃×#428	2
#3356	13

Subgroups — C13×#3356 #13112414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3356 #13112414