Subgroups — C₂²×#16837819 #29495153

Order 15104.

2118 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×#16837819	C₂	1	24	yes	maximal, direct product factor
2	C₂²×#14273667	C₂	1	3	yes	maximal, complemented
3	C₂³×C₅₉⋊C₁₆	C₂	1	3	yes	maximal, complemented
4	C₂²×#131×C₅₉	C₂	1	1	yes	maximal, characteristic
5	#16837819	C₂²	1	64	yes	direct product factor
6	C₂×#14273667	C₂²	1	36	yes	complemented
7	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₅₉⋊C₁₆	C₂²	1	3	yes	—
9	C₂²×C₅₉⋊C₁₆	C₂²	1	36	yes	complemented
10	C₂²×#41×C₅₉	C₄	1	1	yes	characteristic
11	C₂×#131×C₅₉	C₄	1	12	yes	—
12	C₂×#131×C₅₉	C₂²	1	12	yes	—
13	C₂²×M₁₆×C₅₉	C₄	1	3	yes	—
14	C₂³×C₈×C₅₉	C₂²	1	3	yes	—
15	#14273667	C₂³	1	48	yes	complemented
16	C₂×C₅₉⋊C₁₆	C₂³	1	6	yes	—
17	C₂×C₅₉⋊C₁₆	C₂³	1	18	yes	—
18	C₂×C₅₉⋊C₁₆	C₂³	1	48	yes	complemented
19	C₂×#41×C₅₉	C₈	1	12	yes	—
20	C₂×#41×C₅₉	C₂×C₄	1	12	yes	—
21	C₂²×Q₈×C₅₉	C₈	1	1	yes	characteristic
22	C₂²×D₄×C₅₉	C₈	1	3	yes	—
23	C₂³×C₄×C₅₉	C₂×C₄	1	3	yes	—
24	#131×C₅₉	C₂×C₄	1	48	yes	—
25	#131×C₅₉	C₂³	1	16	yes	—
26	C₂×M₁₆×C₅₉	C₂×C₄	1	36	yes	—
27	C₂²×C₈×C₅₉	C₂×C₄	1	3	yes	—
28	C₂²×C₈×C₅₉	C₂×C₄	1	18	yes	—
29	C₂²×C₈×C₅₉	C₂³	1	18	yes	—
30	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	C₂⁴×C₅₉	C₂×C₈	1	3	yes	—
32	#41×C₅₉	C₂×C₈	1	48	yes	—
33	#41×C₅₉	C₂²×C₄	1	16	yes	—
34	C₂×Q₈×C₅₉	C₂×C₈	1	12	yes	—
35	C₂×D₄×C₅₉	C₂×C₈	1	36	yes	—
36	C₂²×C₄×C₅₉	C₂×C₈	1	3	yes	—
37	C₂²×C₄×C₅₉	C₂×C₈	1	18	yes	—
38	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	C₂²×C₄	1	18	yes	—
40	M₁₆×C₅₉	C₂²×C₄	1	48	yes	—
41	C₂×C₈×C₅₉	C₂²×C₄	1	3	yes	—
42	C₂×C₈×C₅₉	C₂²×C₄	1	18	yes	—
43	C₂×C₈×C₅₉	C₂²×C₄	1	36	yes	—
44	C₂×C₈×C₅₉	C₂⁴	1	3	yes	—
45	C₂×C₈×C₅₉	C₂⁴	1	12	yes	—
46	C₅₉⋊C₁₆	C₂⁴	1	4	yes	—
47	C₅₉⋊C₁₆	C₂⁴	1	12	yes	—
48	C₂³×C₅₉	—	2	12	no	—
49	C₂³×C₅₉	C₂²×C₈	1	18	yes	—
50	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
51	Q₈×C₅₉	C₂²×C₈	1	16	yes	—
52	D₄×C₅₉	C₂²×C₈	1	48	yes	—
53	C₂×C₄×C₅₉	C₂²×C₈	1	3	yes	—
54	C₂×C₄×C₅₉	C₂²×C₈	1	18	yes	—
55	C₂×C₄×C₅₉	C₂²×C₈	1	36	yes	—
56	C₂×C₄×C₅₉	C₂³×C₄	1	3	yes	—
57	C₂×C₄×C₅₉	C₂³×C₄	1	12	yes	—
58	C₈×C₅₉	C₂³×C₄	1	3	yes	—
59	C₈×C₅₉	C₂³×C₄	1	12	yes	—
60	C₈×C₅₉	C₂⁵	1	1	yes	characteristic
61	C₂²×#504	—	59	1	no	maximal, Sylow
62	C₂²×C₅₉	—	2	36	no	—
63	C₂²×C₅₉	#504	1	4	yes	complemented
64	C₂²×C₅₉	C₂³×C₈	1	12	yes	—
65	C₂²×C₅₉	C₂³×C₈	1	3	yes	—
66	C₄×C₅₉	C₂³×C₈	1	3	yes	—
67	C₄×C₅₉	C₂³×C₈	1	12	yes	—
68	C₄×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
69	C₂²×#131	D₅₉	1	1	yes	characteristic
70	C₂×#504	—	59	24	no	—
71	C₂²×M₃₂	—	59	3	no	—
72	C₂³×C₁₆	—	59	3	no	—
73	C₂×C₅₉	—	2	12	no	—
74	C₂×C₅₉	C₂×#504	1	6	yes	complemented
75	C₂×C₅₉	C₂⁴×C₈	1	1	yes	characteristic
76	C₂²×#41	Dic₅₉	1	1	yes	characteristic
77	C₂×#131	Dic₅₉	1	12	yes	—
78	C₂×#131	C₂×D₅₉	1	12	yes	—
79	C₂²×M₁₆	Dic₅₉	1	3	yes	—
80	C₂³×C₈	C₂×D₅₉	1	3	yes	—
81	#504	—	59	64	no	—
82	C₂×M₃₂	—	59	36	no	—
83	C₂²×C₁₆	—	59	36	no	—
84	C₂²×C₁₆	—	59	1	no	—
85	C₂²×C₁₆	—	59	3	no	—
86	C₅₉	C₂²×#504	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	C₅₉⋊C₈	1	12	yes	—
88	C₂×#41	C₂×Dic₅₉	1	12	yes	—
89	C₂²×Q₈	C₅₉⋊C₈	1	1	yes	characteristic
90	C₂²×D₄	C₅₉⋊C₈	1	3	yes	—
91	C₂³×C₄	C₂×Dic₅₉	1	3	yes	—
92	#131	C₂×Dic₅₉	1	48	yes	—
93	#131	C₂²×D₅₉	1	16	yes	—
94	C₂×M₁₆	C₂×Dic₅₉	1	36	yes	—
95	C₂²×C₈	C₂×Dic₅₉	1	3	yes	—
96	C₂²×C₈	C₂×Dic₅₉	1	18	yes	—
97	C₂²×C₈	C₂²×D₅₉	1	18	yes	—
98	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
99	M₃₂	—	59	48	no	—
100	C₂×C₁₆	—	59	6	no	—
101	C₂×C₁₆	—	59	18	no	—
102	C₂×C₁₆	—	59	48	no	—
103	C₂⁴	C₂×C₅₉⋊C₈	1	3	yes	—
104	#41	C₂×C₅₉⋊C₈	1	48	yes	—
105	#41	C₂²×Dic₅₉	1	16	yes	—
106	C₂×Q₈	C₂×C₅₉⋊C₈	1	12	yes	—
107	C₂×D₄	C₂×C₅₉⋊C₈	1	36	yes	—
108	C₂²×C₄	C₂×C₅₉⋊C₈	1	3	yes	—
109	C₂²×C₄	C₂×C₅₉⋊C₈	1	18	yes	—
110	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
111	C₂²×C₄	C₂²×Dic₅₉	1	18	yes	—
112	M₁₆	C₂²×Dic₅₉	1	48	yes	—
113	C₂×C₈	C₂²×Dic₅₉	1	3	yes	—
114	C₂×C₈	C₂²×Dic₅₉	1	18	yes	—
115	C₂×C₈	C₂²×Dic₅₉	1	36	yes	—
116	C₂×C₈	C₂³×D₅₉	1	3	yes	—
117	C₂×C₈	C₂³×D₅₉	1	12	yes	—
118	C₁₆	—	59	4	no	—
119	C₁₆	—	59	12	no	—
120	C₂³	—	2	12	no	—
121	C₂³	C₂²×C₅₉⋊C₈	1	18	yes	—
122	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
123	Q₈	C₂²×C₅₉⋊C₈	1	16	yes	—
124	D₄	C₂²×C₅₉⋊C₈	1	48	yes	—
125	C₂×C₄	C₂²×C₅₉⋊C₈	1	3	yes	—
126	C₂×C₄	C₂²×C₅₉⋊C₈	1	18	yes	—
127	C₂×C₄	C₂²×C₅₉⋊C₈	1	36	yes	—
128	C₂×C₄	C₂³×Dic₅₉	1	3	yes	—
129	C₂×C₄	C₂³×Dic₅₉	1	12	yes	—
130	C₈	C₂³×Dic₅₉	1	3	yes	—
131	C₈	C₂³×Dic₅₉	1	12	yes	—
132	C₈	C₂⁴×D₅₉	1	1	yes	characteristic
133	C₂²	—	2	36	no	—
134	C₂²	#16837819	1	4	yes	direct product factor
135	C₂²	C₂³×C₅₉⋊C₈	1	12	yes	—
136	C₂²	C₂³×C₅₉⋊C₈	1	3	yes	—
137	C₄	C₂³×C₅₉⋊C₈	1	3	yes	—
138	C₄	C₂³×C₅₉⋊C₈	1	12	yes	—
139	C₄	C₂⁴×Dic₅₉	1	1	yes	characteristic
140	C₂	—	2	12	no	—
141	C₂	C₂×#16837819	1	6	yes	direct product factor
142	C₂	C₂⁴×C₅₉⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₉
commutator	C₂×C₅₉	C₂⁴×C₈
Frattini	C₈	C₂⁴×D₅₉
Fitting	C₂²×#131×C₅₉	C₂
socle	C₂³×C₅₉	C₂²×C₈
prime radical	C₂²×#41×C₅₉	C₄
cosocle	C₈×C₅₉	C₂⁵
hypercenter	C₂²×#131	D₅₉
hypocenter	C₅₉	C₂²×#504
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#16837819	C₂	direct (N × H)
2. C₂²×#14273667	C₂	semidirect
3. C₂³×C₅₉⋊C₁₆	C₂	semidirect
5. #16837819	C₂²	direct (N × H)
6. C₂×#14273667	C₂²	semidirect
9. C₂²×C₅₉⋊C₁₆	C₂²	semidirect
15. #14273667	C₂³	semidirect
18. C₂×C₅₉⋊C₁₆	C₂³	semidirect
63. C₂²×C₅₉	#504	semidirect
74. C₂×C₅₉	C₂×#504	semidirect
86. C₅₉	C₂²×#504	semidirect
134. C₂²	#16837819	direct (N × H)
141. C₂	C₂×#16837819	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16837819	2
C₂²×#14273667	2
C₂³×C₅₉⋊C₁₆	2
C₂²×#131×C₅₉	2
C₂²×#504	59

Subgroups — C22×#16837819 #29495153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16837819 #29495153