Subgroups — #15777749

Order 3328.

402 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×#11718104	C₂	1	4	yes	maximal
2	C₄×#88052	C₂	1	1	yes	maximal, characteristic
3	#13114522	C₂	1	1	yes	maximal, characteristic
4	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×#88055	C₂²	1	2	yes	—
8	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
9	C₂×#88052	C₂²	1	2	yes	—
10	#11718593	C₄	1	1	yes	characteristic
11	C₂×#88043	C₄	1	1	yes	characteristic
12	C₁₃×#346	C₄	1	2	yes	complemented
13	C₄×C₁₃×C₁₆	C₄	1	2	yes	complemented
14	#11718104	C₄	1	16	yes	complemented
15	#11718053	C₄	1	2	yes	complemented
16	C₄×#6090	C₄	1	2	yes	complemented
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
21	C₂²×#1055	C₂×C₄	1	2	yes	—
22	C₁₃×M₃₂	C₂×C₄	1	8	yes	—
23	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	#88071	—	2	2	no	—
29	#88055	C₂×C₄	1	8	yes	—
30	C₈×Dic₁₃	—	2	4	no	—
31	#88052	C₂×C₄	1	8	yes	—
32	C₂×#6090	C₂×C₄	1	4	yes	—
33	#88043	—	2	2	no	—
34	#12541	—	13	1	no	maximal, Sylow
35	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
36	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×Dic₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
41	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
43	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₁₃	C₄²	1	1	yes	characteristic
45	C₄²×C₁₃	C₄²	1	1	yes	characteristic
46	C₄²×C₁₃	M₁₆	1	2	yes	—
47	C₄×Dic₁₃	—	2	2	no	—
48	C₄×Dic₁₃	C₄²	1	4	yes	—
49	C₂×#1055	—	2	2	no	—
50	C₂×#1055	—	2	2	no	—
51	C₂×#1055	C₄²	1	4	yes	—
52	C₁₃×C₁₆	C₄²	1	8	yes	—
53	#6090	C₄²	1	8	yes	—
54	C₂×#347	—	13	4	no	—
55	C₄×M₃₂	—	13	1	no	—
56	C₄×M₃₂	D₁₃	1	1	yes	characteristic
57	#1726	—	13	1	no	—
58	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₄×C₁₃	—	4	2	no	—
60	C₂×C₄×C₁₃	#97	1	2	yes	—
61	C₂×C₄×C₁₃	#97	1	2	yes	—
62	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
64	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
66	C₂×Dic₁₃	—	2	4	no	—
67	C₂×Dic₁₃	—	4	4	no	—
68	C₈×C₁₃	#97	1	2	yes	—
69	C₈×C₁₃	#97	1	2	yes	—
70	C₈×C₁₃	C₂×C₄²	1	2	yes	—
71	C₈×C₁₃	C₂×C₄²	1	2	yes	—
72	#1055	—	2	4	no	—
73	C₂×M₃₂	—	13	2	no	—
74	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
75	C₂×#97	—	13	2	no	—
76	C₂×#97	—	13	1	no	—
77	C₂×C₄×C₈	—	13	2	no	—
78	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
79	#376	—	13	1	no	—
80	#347	—	13	16	no	—
81	#346	—	13	2	no	—
82	#346	Dic₁₃	1	2	yes	complemented
83	C₄×C₁₆	—	13	2	no	—
84	C₄×C₁₆	Dic₁₃	1	2	yes	complemented
85	C₂²×C₁₃	—	2	2	no	—
86	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
87	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
88	C₄×C₁₃	—	2	2	no	—
89	C₄×C₁₃	—	2	2	no	—
90	C₄×C₁₃	C₄³	1	1	yes	characteristic
91	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
92	C₄×C₁₃	C₄×M₁₆	1	2	yes	—
93	Dic₁₃	—	4	4	no	—
94	C₂²×C₈	—	13	2	no	—
95	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
96	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
97	C₂×C₄²	—	13	2	no	—
98	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
99	M₃₂	—	13	8	no	—
100	M₃₂	C₄×D₁₃	1	8	yes	—
101	C₂×C₁₆	—	13	4	no	—
102	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
103	#97	—	13	8	no	—
104	#97	—	26	2	no	—
105	C₄×C₈	—	26	4	no	—
106	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
107	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
108	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
109	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
110	#95	—	26	2	no	—
111	C₂×C₁₃	—	2	2	no	—
112	C₂×C₁₃	C₄×#97	1	1	yes	characteristic
113	C₂×C₁₃	#2110	1	2	yes	—
114	C₂²×C₄	—	26	2	no	—
115	C₂²×C₄	#6094	1	2	yes	—
116	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
117	C₂×C₈	—	13	4	no	—
118	C₂×C₈	—	26	2	no	—
119	C₂×C₈	—	26	2	no	—
120	C₂×C₈	#6094	1	4	yes	—
121	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
122	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
123	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
124	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
125	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
126	C₄²	—	13	4	no	—
127	C₄²	—	26	2	no	—
128	C₄²	#6094	1	2	yes	—
129	C₄²	C₄×Dic₁₃	1	1	yes	characteristic
130	C₄²	C₄×Dic₁₃	1	1	yes	characteristic
131	C₁₆	—	13	8	no	—
132	C₁₆	C₄×Dic₁₃	1	8	yes	—
133	C₁₃	#12541	1	1	yes	characteristic, complemented, Sylow
134	C₂³	C₂×#6094	1	1	yes	characteristic
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	26	4	no	—
137	C₂×C₄	—	52	4	no	—
138	C₂×C₄	#88055	1	2	yes	—
139	C₂×C₄	#88055	1	2	yes	—
140	C₂×C₄	C₄²×D₁₃	1	2	yes	—
141	C₂×C₄	C₂×#6094	1	2	yes	—
142	C₂×C₄	C₂×#6094	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
144	C₈	—	26	4	no	—
145	C₈	#88055	1	2	yes	—
146	C₈	#88055	1	2	yes	—
147	C₈	C₄²×D₁₃	1	2	yes	—
148	C₈	C₂×C₄×Dic₁₃	1	2	yes	—
149	C₂²	—	2	2	no	—
150	C₂²	C₄×#6094	1	2	yes	—
151	C₂²	C₂×#88055	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	2	2	no	—
154	C₄	—	52	4	no	—
155	C₄	C₄×#6094	1	2	yes	—
156	C₄	C₄²×Dic₁₃	1	1	yes	characteristic
157	C₄	C₂×#88055	1	1	yes	characteristic
158	C₂	—	2	2	no	—
159	C₂	C₄×#88055	1	1	yes	characteristic
160	C₂	#13116440	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₁₃
commutator	C₄×C₁₃	C₄³
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂×#97
prime radical	C₂³×C₁₃	C₂×M₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#12541
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
12. C₁₃×#346	C₄	semidirect
13. C₄×C₁₃×C₁₆	C₄	semidirect
14. #11718104	C₄	semidirect
15. #11718053	C₄	semidirect
16. C₄×#6090	C₄	semidirect
82. #346	Dic₁₃	semidirect
84. C₄×C₁₆	Dic₁₃	semidirect
133. C₁₃	#12541	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718104	2
C₄×#88052	2
#13114522	2
C₄×C₁₃×M₃₂	2
#12541	13