Subgroups — D₁₃×#2252 #16018393

Order 3328.

558 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃×#500	C₂	1	2	yes	maximal
2	#13117111	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117110	C₂	1	4	yes	maximal, complemented
4	#13117109	C₂	1	2	yes	maximal, complemented
5	D₁₃×#366	C₂	1	4	yes	maximal
6	C₄×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₃×C₄⋊Q₈	—	2	2	no	—
9	C₁₃×#500	C₂²	1	2	yes	—
10	#11718739	—	2	2	no	—
11	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
12	D₁₃×#107	—	2	2	no	—
13	D₁₃×#107	C₂²	1	4	yes	—
14	#11718535	—	2	2	no	—
15	#11718535	C₂²	1	4	yes	—
16	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
17	#11718343	C₂²	1	2	yes	—
18	C₁₃×#366	C₂²	1	4	yes	—
19	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
20	#11718096	C₂²	1	4	yes	—
21	C₁₆×Dic₁₃	C₂²	1	2	yes	—
22	#11718081	C₂²	1	8	yes	—
23	C₄×#6090	C₂²	1	1	yes	characteristic
24	C₂×Q₈×D₁₃	—	4	2	no	—
25	C₁₃×C₄⋊Q₈	—	2	2	no	—
26	#88198	—	4	2	no	—
27	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
28	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×D₁₃	—	2	4	no	—
30	C₄⋊C₄×D₁₃	—	4	2	no	—
31	#88142	—	2	4	no	—
32	#88142	—	4	2	no	—
33	C₄²×D₁₃	D₄	1	1	yes	characteristic
34	#88123	—	2	2	no	—
35	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
36	C₁₃×#107	—	2	2	no	—
37	C₁₃×#107	C₂³	1	4	yes	—
38	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	#88058	—	2	2	no	—
40	#88058	C₂³	1	4	yes	—
41	C₈×Dic₁₃	D₄	1	1	yes	characteristic
42	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
43	C₂×#6090	C₂³	1	2	yes	—
44	#88047	—	2	4	no	—
45	#88047	—	2	4	no	—
46	C₄×#1055	D₄	1	1	yes	characteristic
47	C₁₆×D₁₃	Q₈	1	8	yes	complemented
48	C₂×#2252	—	13	1	no	maximal, Sylow
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	Q₈×D₁₃	—	8	8	no	—
51	C₂×C₄×D₁₃	—	4	4	no	—
52	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
53	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
54	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₂₆	—	4	2	no	—
56	C₂×Dic₂₆	—	4	4	no	—
57	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
59	C₁₃×C₄⋊C₄	—	2	4	no	—
60	C₁₃×C₄⋊C₄	—	4	2	no	—
61	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₄⋊Dic₁₃	—	2	4	no	—
63	C₄⋊Dic₁₃	—	4	2	no	—
64	#6101	—	4	4	no	—
65	#6101	—	4	4	no	—
66	C₄×Dic₁₃	D₈	1	1	yes	characteristic
67	C₄×Dic₁₃	D₈	1	1	yes	characteristic
68	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂×#1055	C₂×D₄	1	1	yes	characteristic
70	C₂×#1055	C₂×D₄	1	1	yes	characteristic
71	C₈×D₁₃	Q₁₆	1	4	yes	—
72	C₈×D₁₃	C₂×Q₈	1	4	yes	—
73	C₁₃×C₁₆	C₂×Q₈	1	4	yes	complemented
74	#6090	C₂×Q₈	1	4	yes	complemented
75	C₂×#500	—	13	2	no	—
76	#2252	—	13	2	no	—
77	#2252	—	13	4	no	—
78	#2252	—	13	1	no	—
79	#2252	D₁₃	1	1	yes	characteristic, direct product factor
80	C₂×#366	—	13	4	no	—
81	C₂×C₄×C₁₆	—	13	1	no	—
82	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
83	Q₈×C₁₃	—	8	4	no	—
84	C₂×C₄×C₁₃	—	4	4	no	—
85	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
88	C₂×Dic₁₃	—	4	4	no	—
89	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
90	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
91	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
92	C₄×D₁₃	—	8	8	no	—
93	C₄×D₁₃	D₁₆	1	4	yes	—
94	C₄×D₁₃	Q₃₂	1	4	yes	—
95	C₄×D₁₃	C₄⋊Q₈	1	2	yes	—
96	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
97	Dic₂₆	—	8	4	no	—
98	Dic₂₆	—	8	8	no	—
99	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
100	C₈×C₁₃	C₂²×Q₈	1	2	yes	—
101	#1055	C₄⋊Q₈	1	2	yes	—
102	#1055	C₂×Q₁₆	1	2	yes	—
103	C₂×C₄⋊Q₈	—	26	2	no	—
104	C₂²×C₁₆	—	13	2	no	—
105	#500	—	13	4	no	—
106	#500	—	13	2	no	—
107	#500	—	26	2	no	—
108	#500	—	26	2	no	—
109	#500	C₂×D₁₃	1	2	yes	—
110	C₂×#107	—	13	4	no	—
111	C₂×#107	—	26	2	no	—
112	C₂×C₄×C₈	—	13	1	no	—
113	#366	—	13	4	no	—
114	#366	—	13	8	no	—
115	#366	C₂×D₁₃	1	4	yes	—
116	C₄×C₁₆	—	13	2	no	—
117	C₄×C₁₆	—	13	1	no	—
118	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
119	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
120	C₂×D₁₃	#500	1	2	yes	—
121	C₂×D₁₃	C₂×D₁₆	1	2	yes	—
122	C₂×D₁₃	C₂×Q₃₂	1	2	yes	—
123	C₄×C₁₃	—	8	4	no	—
124	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
125	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
126	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
127	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
128	Dic₁₃	—	8	4	no	—
129	Dic₁₃	#500	1	2	yes	—
130	Dic₁₃	C₂×D₁₆	1	2	yes	—
131	Dic₁₃	C₂×Q₃₂	1	2	yes	—
132	C₂²×Q₈	—	52	2	no	—
133	C₂²×C₈	—	13	1	no	—
134	C₂²×C₈	—	13	1	no	—
135	C₄⋊Q₈	—	2	2	no	—
136	C₄⋊Q₈	—	26	4	no	—
137	C₄⋊Q₈	—	26	2	no	—
138	C₄⋊Q₈	—	52	2	no	—
139	C₄⋊Q₈	—	52	2	no	—
140	C₂×C₄⋊C₄	—	26	4	no	—
141	C₂×C₄⋊C₄	—	52	2	no	—
142	C₂×C₄²	—	13	1	no	—
143	C₂×C₁₆	—	13	2	no	—
144	C₂×C₁₆	—	13	8	no	—
145	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
146	#107	—	2	2	no	—
147	#107	—	13	4	no	—
148	#107	—	26	4	no	—
149	#107	—	26	4	no	—
150	#107	—	26	2	no	—
151	#107	C₂²×D₁₃	1	4	yes	—
152	C₄×C₈	—	13	1	no	—
153	C₄×C₈	—	13	1	no	—
154	C₄×C₈	—	13	1	no	—
155	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
156	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
157	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
158	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
159	D₁₃	#2252	1	4	yes	direct product factor
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	52	4	no	—
162	C₂×Q₈	—	52	2	no	—
163	C₂×Q₈	—	104	8	no	—
164	C₂²×C₄	—	13	1	no	—
165	C₂²×C₄	—	13	1	no	—
166	C₂²×C₄	—	13	1	no	—
167	C₂²×C₄	—	52	4	no	—
168	C₂×C₈	—	13	1	no	—
169	C₂×C₈	—	13	1	no	—
170	C₂×C₈	—	13	4	no	—
171	C₂×C₈	—	13	4	no	—
172	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
173	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	26	4	no	—
177	C₄⋊C₄	—	52	4	no	—
178	C₄⋊C₄	—	52	4	no	—
179	C₄⋊C₄	—	52	2	no	—
180	C₄²	—	13	1	no	—
181	C₄²	—	13	1	no	—
182	C₄²	—	13	1	no	—
183	C₄²	D₄×D₁₃	1	1	yes	characteristic
184	C₁₆	—	13	4	no	—
185	C₁₆	Q₈×D₁₃	1	4	yes	complemented
186	C₁₃	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
187	C₂³	—	13	1	no	—
188	Q₈	—	8	4	no	—
189	Q₈	—	104	8	no	—
190	Q₈	—	104	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	13	1	no	—
193	C₂×C₄	—	13	1	no	—
194	C₂×C₄	—	13	1	no	—
195	C₂×C₄	—	13	2	no	—
196	C₂×C₄	—	13	2	no	—
197	C₂×C₄	—	13	4	no	—
198	C₂×C₄	—	13	4	no	—
199	C₂×C₄	—	52	4	no	—
200	C₂×C₄	—	104	8	no	—
201	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
202	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
204	C₈	—	13	2	no	—
205	C₈	—	13	2	no	—
206	C₈	Q₁₆×D₁₃	1	2	yes	—
207	C₈	C₂×Q₈×D₁₃	1	2	yes	—
208	C₂²	—	13	2	no	—
209	C₂²	—	13	2	no	—
210	C₂²	—	13	2	no	—
211	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
212	C₄	—	8	4	no	—
213	C₄	—	13	2	no	—
214	C₄	—	13	2	no	—
215	C₄	—	13	2	no	—
216	C₄	—	104	4	no	—
217	C₄	D₁₃×D₁₆	1	2	yes	—
218	C₄	D₁₃×Q₃₂	1	2	yes	—
219	C₄	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
220	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
221	C₂	—	13	4	no	—
222	C₂	D₁₃×#500	1	1	yes	characteristic
223	C₂	C₂×D₁₃×D₁₆	1	1	yes	characteristic
224	C₂	C₂×D₁₃×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2252	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₃	C₂×D₈
hypercenter	#2252	D₁₃
hypocenter	C₁₃	C₂×#2252
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
2. #13117111	C₂	semidirect
3. #13117110	C₂	semidirect
4. #13117109	C₂	semidirect
7. C₁₃×#2252	C₂	semidirect
47. C₁₆×D₁₃	Q₈	semidirect
73. C₁₃×C₁₆	C₂×Q₈	semidirect
74. #6090	C₂×Q₈	semidirect
79. #2252	D₁₃	direct (N × H)
159. D₁₃	#2252	direct (N × H)
185. C₁₆	Q₈×D₁₃	semidirect
186. C₁₃	C₂×#2252	semidirect

Maximal subgroups

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

subgroup	index
D₁₃×#500	2
#13117111	2
#13117110	2
#13117109	2
D₁₃×#366	2
C₄×C₁₆×D₁₃	2
C₁₃×#2252	2
C₂×#2252	13

Subgroups — D13×#2252 #16018393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₃×#2252 #16018393