Subgroups — #15777917

Order 3328.

414 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13112913	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
8	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
9	C₂²×#6090	C₂²	1	2	yes	—
10	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
11	C₂×C₄×#1055	C₄	1	1	yes	characteristic
12	C₄×C₁₃×C₁₆	C₄	1	4	yes	complemented
13	C₄×#6090	C₄	1	4	yes	complemented
14	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
18	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
19	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
20	C₂²×#1055	C₂×C₄	1	2	yes	—
21	C₂×C₁₃×C₁₆	—	2	4	no	—
22	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
23	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₈×Dic₁₃	—	2	4	no	—
26	C₈×Dic₁₃	—	2	4	no	—
27	C₂×#6090	—	2	4	no	—
28	C₂×#6090	C₂×C₄	1	4	yes	—
29	C₄×#1055	C₈	1	4	yes	—
30	#12590	—	13	1	no	maximal, Sylow
31	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₁₃	C₂×C₈	1	4	yes	—
35	C₂×C₈×C₁₃	C₄²	1	2	yes	—
36	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
37	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃	#41	1	4	yes	—
40	C₄²×C₁₃	C₄²	1	2	yes	—
41	C₄²×C₁₃	C₂×C₈	1	2	yes	—
42	C₄×Dic₁₃	—	2	4	no	—
43	C₄×Dic₁₃	—	2	2	no	—
44	C₄×Dic₁₃	—	2	2	no	—
45	C₄×Dic₁₃	C₁₆	1	8	yes	complemented
46	C₂×#1055	—	2	2	no	—
47	C₂×#1055	—	2	2	no	—
48	C₂×#1055	C₂×C₈	1	4	yes	—
49	C₁₃×C₁₆	—	2	4	no	—
50	#6090	—	2	4	no	—
51	C₂×C₄×C₁₆	—	13	1	no	—
52	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
53	C₄²×C₈	—	13	1	no	—
54	#1374	—	13	4	no	—
55	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
57	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
58	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
59	C₂×C₄×C₁₃	M₃₂	1	4	yes	—
60	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#117	1	2	yes	—
62	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
64	C₂×Dic₁₃	—	2	4	no	—
65	C₂×Dic₁₃	—	2	4	no	—
66	C₂×Dic₁₃	C₂×C₁₆	1	8	yes	—
67	C₈×C₁₃	#97	1	4	yes	—
68	C₈×C₁₃	#117	1	2	yes	—
69	C₈×C₁₃	#117	1	2	yes	—
70	#1055	—	2	4	no	—
71	C₂²×C₁₆	—	13	2	no	—
72	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
73	C₂×C₄×C₈	—	13	1	no	—
74	C₂×C₄×C₈	—	13	2	no	—
75	C₂×C₄×C₈	—	13	2	no	—
76	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
77	C₄³	—	13	1	no	—
78	C₄×C₁₆	—	13	4	no	—
79	C₄×C₁₆	Dic₁₃	1	4	yes	complemented
80	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
81	C₂²×C₁₃	#431	1	2	yes	—
82	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
83	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
84	C₄×C₁₃	C₄×C₁₆	1	4	yes	—
85	C₄×C₁₃	#346	1	4	yes	—
86	C₄×C₁₃	#376	1	1	yes	characteristic
87	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
88	C₄×C₁₃	#431	1	2	yes	—
89	Dic₁₃	—	2	8	no	—
90	C₂²×C₈	—	13	2	no	—
91	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
93	C₂×C₄²	—	13	2	no	—
94	C₂×C₄²	—	13	2	no	—
95	C₂×C₄²	—	13	2	no	—
96	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
97	C₂×C₁₆	—	2	4	no	—
98	C₂×C₁₆	—	13	4	no	—
99	C₂×C₁₆	—	26	4	no	—
100	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
101	C₄×C₈	—	13	4	no	—
102	C₄×C₈	—	26	4	no	—
103	C₄×C₈	—	26	4	no	—
104	C₄×C₈	C₄×D₁₃	1	2	yes	—
105	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
106	C₂×C₁₃	#1744	1	1	yes	characteristic
107	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
108	C₂×C₁₃	C₂×#346	1	1	yes	characteristic
109	C₂×C₁₃	#2162	1	4	yes	—
110	C₂²×C₄	—	13	4	no	—
111	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
112	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
113	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
114	C₂×C₈	—	13	4	no	—
115	C₂×C₈	—	26	2	no	—
116	C₂×C₈	—	26	2	no	—
117	C₂×C₈	#6094	1	4	yes	—
118	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
119	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
120	C₂×C₈	#6127	1	4	yes	—
121	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
122	C₄²	—	13	8	no	—
123	C₄²	—	26	2	no	—
124	C₄²	—	26	2	no	—
125	C₄²	—	26	4	no	—
126	C₄²	C₈×D₁₃	1	2	yes	—
127	C₄²	C₄×Dic₁₃	1	2	yes	—
128	C₁₆	—	2	4	no	—
129	C₁₆	—	26	4	no	—
130	C₁₃	#12590	1	1	yes	characteristic, complemented, Sylow
131	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
132	C₂×C₄	—	13	8	no	—
133	C₂×C₄	—	26	4	no	—
134	C₂×C₄	—	26	4	no	—
135	C₂×C₄	C₁₆×D₁₃	1	4	yes	—
136	C₂×C₄	#88038	1	4	yes	—
137	C₂×C₄	C₈×Dic₁₃	1	2	yes	—
138	C₂×C₄	C₈×Dic₁₃	1	2	yes	—
139	C₂×C₄	#88126	1	2	yes	—
140	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
141	C₂×C₄	C₂×#6094	1	2	yes	—
142	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
143	C₈	—	26	4	no	—
144	C₈	#88055	1	4	yes	—
145	C₈	#88126	1	2	yes	—
146	C₈	#88180	1	2	yes	—
147	C₂²	#11718346	1	2	yes	—
148	C₂²	C₂×C₁₆×D₁₃	1	2	yes	—
149	C₂²	C₂×#88038	1	2	yes	—
150	C₂²	C₂×C₈×Dic₁₃	1	1	yes	characteristic
151	C₄	—	26	8	no	—
152	C₄	C₁₆×Dic₁₃	1	4	yes	—
153	C₄	#11718094	1	4	yes	—
154	C₄	#11718346	1	2	yes	—
155	C₄	#11718593	1	1	yes	characteristic
156	C₄	C₂×#88055	1	1	yes	characteristic
157	C₂	#13114581	1	1	yes	characteristic
158	C₂	C₂×C₁₆×Dic₁₃	1	1	yes	characteristic
159	C₂	C₂×#11718094	1	1	yes	characteristic
160	C₂	#13116633	1	4	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₁₃
commutator	C₂×C₁₃	C₂×C₄×C₁₆
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	#12590
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₄×C₁₃×C₁₆	C₄	semidirect
13. C₄×#6090	C₄	semidirect
45. C₄×Dic₁₃	C₁₆	semidirect
79. C₄×C₁₆	Dic₁₃	semidirect
130. C₁₃	#12590	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#6090	2
C₄×C₈×Dic₁₃	2
#13112913	2
C₂×C₄×C₁₃×C₁₆	2
#12590	13