Subgroups — #15777918

Order 3328.

414 conjugacy classes of subgroups.

The subgroup lattice has 144 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	2	yes	maximal
2	#13112914	C₂	1	4	yes	maximal
3	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
5	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₄³×C₁₃	C₄	1	1	yes	characteristic
9	C₂²×#6090	C₂²	1	4	yes	—
10	C₄×#6090	C₄	1	8	yes	complemented
11	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
12	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	C₈	1	2	yes	—
15	C₂×C₄²×C₁₃	C₈	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₄×C₈×C₁₃	—	2	4	no	—
19	C₄×C₈×C₁₃	—	2	4	no	—
20	C₄×C₈×C₁₃	C₈	1	4	yes	—
21	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
22	C₂×#6090	—	2	8	no	—
23	C₂×#6090	C₂×C₄	1	8	yes	—
24	#12590	—	13	1	no	maximal, Sylow
25	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
26	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₃	—	2	2	no	—
30	C₂×C₈×C₁₃	—	2	2	no	—
31	C₂×C₈×C₁₃	C₄²	1	2	yes	—
32	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
34	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
35	C₂×C₈×C₁₃	#41	1	4	yes	—
36	C₄²×C₁₃	—	2	4	no	—
37	C₄²×C₁₃	—	2	2	no	—
38	C₄²×C₁₃	—	2	2	no	—
39	C₄²×C₁₃	C₁₆	1	8	yes	complemented
40	C₄²×C₁₃	C₄²	1	2	yes	—
41	C₄²×C₁₃	C₂×C₈	1	2	yes	—
42	#6090	—	2	8	no	—
43	C₂×C₄×C₁₆	—	13	2	no	—
44	C₄²×C₈	D₁₃	1	1	yes	characteristic
45	#1374	—	13	4	no	—
46	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
47	C₂×C₄×C₁₃	—	2	4	no	—
48	C₂×C₄×C₁₃	—	2	4	no	—
49	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
50	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
51	C₂×C₄×C₁₃	C₂×C₁₆	1	8	yes	—
52	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
53	C₂×C₄×C₁₃	M₃₂	1	4	yes	—
54	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₁₃	#117	1	2	yes	—
56	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
58	C₈×C₁₃	—	2	4	no	—
59	C₈×C₁₃	#97	1	4	yes	—
60	C₈×C₁₃	#117	1	4	yes	—
61	C₂²×C₁₆	—	13	4	no	—
62	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
63	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
65	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
66	C₄³	Dic₁₃	1	1	yes	characteristic
67	C₄×C₁₆	—	13	8	no	—
68	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
69	C₂²×C₁₃	#431	1	2	yes	—
70	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
71	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
72	C₄×C₁₃	—	2	8	no	—
73	C₄×C₁₃	C₄×C₁₆	1	4	yes	—
74	C₄×C₁₃	#346	1	4	yes	—
75	C₄×C₁₃	#376	1	1	yes	characteristic
76	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
77	C₄×C₁₃	#431	1	2	yes	—
78	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
79	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
80	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
81	C₂×C₄²	#1055	1	2	yes	—
82	C₂×C₄²	#1055	1	2	yes	—
83	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
84	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
85	C₂×C₁₆	—	13	8	no	—
86	C₂×C₁₆	—	26	8	no	—
87	C₄×C₈	—	2	4	no	—
88	C₄×C₈	—	2	4	no	—
89	C₄×C₈	#1055	1	4	yes	—
90	C₄×C₈	C₄×D₁₃	1	4	yes	—
91	C₂×C₁₃	#1744	1	1	yes	characteristic
92	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
93	C₂×C₁₃	C₂×#346	1	1	yes	characteristic
94	C₂×C₁₃	#2162	1	4	yes	—
95	C₂²×C₄	C₂×#1055	1	4	yes	—
96	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
97	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
98	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	C₂×#1055	1	4	yes	—
102	C₂×C₈	#6099	1	4	yes	—
103	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
104	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
105	C₂×C₈	#6127	1	4	yes	—
106	C₄²	—	2	2	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	4	no	—
109	C₄²	#6090	1	8	yes	complemented
110	C₄²	C₂×#1055	1	2	yes	—
111	C₄²	C₄×Dic₁₃	1	2	yes	—
112	C₁₆	—	26	8	no	—
113	C₁₃	#12590	1	1	yes	characteristic, complemented, Sylow
114	C₂³	C₂²×#1055	1	1	yes	characteristic
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	C₄×#1055	1	2	yes	—
118	C₂×C₄	C₄×#1055	1	2	yes	—
119	C₂×C₄	C₂×#6090	1	8	yes	—
120	C₂×C₄	C₂×#6090	1	4	yes	—
121	C₂×C₄	#88052	1	4	yes	—
122	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
123	C₂×C₄	C₂×#6099	1	2	yes	—
124	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
125	C₂×C₄	#88180	1	2	yes	—
126	C₈	—	2	4	no	—
127	C₈	#88043	1	4	yes	—
128	C₈	#88126	1	4	yes	—
129	C₂²	C₂×C₄×#1055	1	1	yes	characteristic
130	C₂²	#11718586	1	2	yes	—
131	C₂²	C₂²×#6090	1	2	yes	—
132	C₂²	C₂×#88052	1	2	yes	—
133	C₄	—	2	8	no	—
134	C₄	C₄×#6090	1	4	yes	—
135	C₄	#11718053	1	4	yes	—
136	C₄	C₂×#88043	1	1	yes	characteristic
137	C₄	#11718586	1	2	yes	—
138	C₄	#11718593	1	1	yes	characteristic
139	C₂	#13114582	1	1	yes	characteristic
140	C₂	C₂×C₄×#6090	1	1	yes	characteristic
141	C₂	C₂×#11718053	1	1	yes	characteristic
142	C₂	#13116635	1	4	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₁₃
commutator	C₂×C₁₃	C₂×C₄×C₁₆
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄²×C₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄²×C₈	D₁₃
hypocenter	C₁₃	#12590
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
10. C₄×#6090	C₄	semidirect
39. C₄²×C₁₃	C₁₆	semidirect
109. C₄²	#6090	semidirect
113. 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
#13112914	2
C₄²×C₈×C₁₃	2
#12590	13