Subgroups — #15771903

Order 3328.

330 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	1	yes	maximal, characteristic
2	#13113119	C₂	1	1	yes	maximal, characteristic
3	#13112752	C₂	1	2	yes	maximal, complemented
4	C₁₆×#1055	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×#6090	C₄	1	2	yes	—
9	#11718586	C₄	1	2	yes	—
10	C₁₃×#346	C₄	1	2	yes	—
11	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
12	C₄×#6090	C₄	1	2	yes	—
13	C₄×#6090	C₂²	1	1	yes	characteristic
14	C₄×#6090	C₂²	1	1	yes	characteristic
15	#11718047	C₄	1	2	yes	—
16	C₈×#1055	C₂²	1	2	yes	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₁₃×M₃₂	C₈	1	8	yes	complemented
21	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
22	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²⋊#1055	C₈	1	4	yes	—
27	C₂×#6090	C₈	1	8	yes	complemented
28	C₂×#6090	C₂×C₄	1	2	yes	—
29	C₂×#6090	C₂×C₄	1	2	yes	—
30	C₄⋊#1055	C₈	1	4	yes	—
31	C₄×#1055	C₂×C₄	1	4	yes	—
32	#11399	—	13	1	no	maximal, Sylow
33	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
38	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₁₃	C₄²	1	1	yes	characteristic
42	C₄²×C₁₃	C₂×C₈	1	2	yes	—
43	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×#1055	C₂×C₈	1	8	yes	—
45	C₁₃×C₁₆	C₂×C₈	1	8	yes	—
46	#6090	C₂×C₈	1	4	yes	—
47	#6090	C₂×C₈	1	4	yes	—
48	C₄×M₃₂	D₁₃	1	1	yes	characteristic
49	C₂×C₄×C₁₆	—	13	1	no	—
50	#1450	—	13	1	no	—
51	#1311	—	13	2	no	—
52	C₈×C₁₆	—	13	2	no	—
53	C₂³×C₁₃	C₄×C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	—	2	4	no	—
55	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
56	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
58	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
59	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
61	C₈×C₁₃	C₄×C₈	1	2	yes	—
62	C₈×C₁₃	C₄×C₈	1	2	yes	—
63	C₈×C₁₃	C₂²×C₈	1	2	yes	—
64	C₈×C₁₃	C₂²×C₈	1	2	yes	—
65	#1055	—	2	8	no	—
66	C₂×M₃₂	Dic₁₃	1	2	yes	—
67	C₂²×C₁₆	—	13	2	no	—
68	#431	—	13	2	no	—
69	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
70	#346	Dic₁₃	1	2	yes	—
71	C₄×C₁₆	—	13	2	no	—
72	C₄×C₁₆	—	13	1	no	—
73	C₄×C₁₆	—	13	1	no	—
74	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
75	#322	—	13	2	no	—
76	C₈²	—	13	2	no	—
77	C₂²×C₁₃	—	2	2	no	—
78	C₂²×C₁₃	C₈²	1	2	yes	—
79	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
80	C₄×C₁₃	—	2	2	no	—
81	C₄×C₁₃	C₈²	1	2	yes	—
82	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
83	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
84	C₄×C₁₃	#504	1	4	yes	—
85	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
86	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
87	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
88	M₃₂	#1055	1	8	yes	complemented
89	C₂×C₁₆	—	13	8	no	—
90	C₂×C₁₆	—	13	2	no	—
91	C₂×C₁₆	—	13	2	no	—
92	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
93	C₄⋊C₈	—	13	4	no	—
94	C₂²⋊C₈	—	13	4	no	—
95	C₄×C₈	—	13	4	no	—
96	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
97	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
98	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
99	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
100	C₂×C₁₃	—	2	2	no	—
101	C₂×C₁₃	C₂×C₈²	1	1	yes	characteristic
102	C₂×C₁₃	#2108	1	2	yes	—
103	C₂²×C₄	C₈×D₁₃	1	2	yes	—
104	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
105	C₂×C₈	—	13	8	no	—
106	C₂×C₈	C₈×D₁₃	1	4	yes	—
107	C₂×C₈	C₂×#1055	1	4	yes	—
108	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
109	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
110	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
111	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
112	C₄²	C₈×D₁₃	1	2	yes	—
113	C₄²	C₄×Dic₁₃	1	1	yes	characteristic
114	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
115	C₁₆	—	13	4	no	—
116	C₁₆	—	13	4	no	—
117	C₁₆	C₂×#1055	1	8	yes	—
118	C₁₃	#11399	1	1	yes	characteristic, complemented, Sylow
119	C₂³	C₈×Dic₁₃	1	1	yes	characteristic
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	C₄×#1055	1	2	yes	—
122	C₂×C₄	C₈×Dic₁₃	1	2	yes	—
123	C₂×C₄	C₈×Dic₁₃	1	2	yes	—
124	C₂×C₄	C₈×Dic₁₃	1	1	yes	characteristic
125	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
126	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
127	C₈	—	26	8	no	—
128	C₈	C₄×#1055	1	2	yes	—
129	C₈	C₈×Dic₁₃	1	2	yes	—
130	C₈	C₂×C₈×D₁₃	1	2	yes	—
131	C₈	C₂²×#1055	1	2	yes	—
132	C₂²	—	2	2	no	—
133	C₂²	C₈×#1055	1	2	yes	—
134	C₂²	C₂×C₈×Dic₁₃	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	C₈×#1055	1	2	yes	—
137	C₄	#11718568	1	4	yes	—
138	C₄	C₂×C₄×#1055	1	1	yes	characteristic
139	C₄	C₂×C₈×Dic₁₃	1	1	yes	characteristic
140	C₂	—	2	2	no	—
141	C₂	C₂×C₈×#1055	1	1	yes	characteristic
142	C₂	#13116435	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₁₃
commutator	C₂×C₁₃	C₂×C₈²
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂×C₄×C₈
prime radical	C₂³×C₁₃	C₄×C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#11399
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #13112752	C₂	semidirect
4. C₁₆×#1055	C₂	semidirect
20. C₁₃×M₃₂	C₈	semidirect
27. C₂×#6090	C₈	semidirect
88. M₃₂	#1055	semidirect
118. C₁₃	#11399	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
#13113119	2
#13112752	2
C₁₆×#1055	2
C₄×C₁₃×M₃₂	2
#11399	13