Subgroups — #28505078

Order 15104.

282 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#21424268	C₂	1	1	yes	maximal, characteristic
3	#21422189	C₂	1	4	yes	maximal
4	C₅₉×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14273663	—	2	1	no	—
7	C₂×#14273663	C₂²	1	2	yes	—
8	#16837426	—	2	1	no	—
9	C₄×C₅₉⋊C₁₆	C₄	1	2	yes	—
10	C₄×C₅₉⋊C₁₆	C₄	1	2	yes	—
11	C₂×#108×C₅₉	—	2	1	no	—
12	C₂×#108×C₅₉	C₂²	1	2	yes	—
13	C₅₉×#423	—	2	1	no	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×#11745664	—	2	2	no	—
16	C₂×C₅₉⋊C₁₆	—	2	2	no	—
17	C₂×C₅₉⋊C₁₆	—	2	2	no	—
18	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
19	#14273663	—	2	4	no	—
20	#14273663	—	4	2	no	—
21	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
22	C₂×M₁₆×C₅₉	—	2	2	no	—
23	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
26	#108×C₅₉	—	2	4	no	—
27	#108×C₅₉	—	4	2	no	—
28	C₄⋊C₈×C₅₉	—	4	2	no	—
29	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	Q₁₆	1	2	yes	—
34	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	M₁₆×C₅₉	—	4	2	no	—
36	M₁₆×C₅₉	—	4	2	no	—
37	C₂×C₈×C₅₉	—	4	2	no	—
38	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
41	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
42	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₅₉	#41	1	2	yes	—
45	C₂×C₈×C₅₉	#41	1	2	yes	—
46	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₅₉	D₈	1	2	yes	—
49	#11745664	—	4	2	no	—
50	#11745664	—	4	2	no	—
51	C₂×C₅₉⋊C₈	—	4	2	no	—
52	C₅₉⋊C₁₆	—	2	4	no	—
53	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
54	C₂×C₄×C₅₉	—	2	4	no	—
55	C₂×C₄×C₅₉	#107	1	2	yes	—
56	C₂×C₄×C₅₉	#107	1	2	yes	—
57	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
59	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₅₉	—	8	2	no	—
64	C₈×C₅₉	#106	1	2	yes	—
65	C₈×C₅₉	#106	1	2	yes	—
66	C₈×C₅₉	#117	1	2	yes	—
67	C₈×C₅₉	#125	1	2	yes	—
68	C₅₉⋊C₈	—	8	2	no	—
69	#12891	—	59	1	no	maximal, Sylow
70	C₂²×C₅₉	—	2	2	no	—
71	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
72	C₂²×C₅₉	#487	1	1	yes	characteristic
73	C₂²×C₅₉	#487	1	1	yes	characteristic
74	C₄×C₅₉	—	2	2	no	—
75	C₄×C₅₉	#368	1	4	yes	—
76	C₄×C₅₉	#382	1	1	yes	characteristic
77	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
78	C₄×C₅₉	#486	1	1	yes	characteristic
79	C₄×C₅₉	#486	1	1	yes	characteristic
80	C₂×C₄×C₁₆	—	59	1	no	—
81	#1848	—	59	1	no	—
82	#1848	D₅₉	1	1	yes	characteristic
83	#1381	—	59	4	no	—
84	C₂×C₅₉	—	2	1	no	—
85	C₂×C₅₉	—	2	1	no	—
86	C₂×C₅₉	#1834	1	1	yes	characteristic
87	C₂×C₅₉	C₂×#368	1	2	yes	—
88	C₂²×C₁₆	—	59	2	no	—
89	C₂×#108	—	2	1	no	—
90	C₂×#108	—	59	2	no	—
91	C₂×#108	—	118	1	no	—
92	C₂×#108	C₂×D₅₉	1	2	yes	—
93	#423	—	2	1	no	—
94	#423	—	118	1	no	—
95	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
96	C₄×C₁₆	—	59	2	no	—
97	C₄×C₁₆	—	59	2	no	—
98	C₅₉	#12891	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	118	2	no	—
101	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
103	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
104	C₂×C₁₆	—	59	4	no	—
105	C₂×C₁₆	—	118	2	no	—
106	C₂×C₁₆	—	118	2	no	—
107	#108	—	2	4	no	—
108	#108	—	4	2	no	—
109	#108	—	118	4	no	—
110	#108	—	236	2	no	—
111	C₄⋊C₈	—	4	2	no	—
112	C₄⋊C₈	—	236	2	no	—
113	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
114	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
115	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
116	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
117	C₂²×C₄	#11745672	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
119	M₁₆	—	4	2	no	—
120	M₁₆	—	4	2	no	—
121	M₁₆	—	236	2	no	—
122	M₁₆	—	236	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	236	2	no	—
125	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
126	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
127	C₂×C₈	#11745670	1	2	yes	—
128	C₂×C₈	#11745671	1	2	yes	—
129	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
131	C₂×C₈	#11745687	1	2	yes	—
132	C₂×C₈	#11745689	1	2	yes	—
133	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
134	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
135	C₄²	#11745669	1	2	yes	—
136	C₁₆	—	118	4	no	—
137	C₂³	C₂×#11745672	1	1	yes	characteristic
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	#14273660	1	2	yes	—
140	C₂×C₄	#14273660	1	2	yes	—
141	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
142	C₂×C₄	#14273714	1	1	yes	characteristic
143	C₂×C₄	#14273716	1	1	yes	characteristic
144	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
145	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
146	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
147	C₈	—	8	2	no	—
148	C₈	—	472	2	no	—
149	C₈	#14273658	1	2	yes	—
150	C₈	#14273658	1	2	yes	—
151	C₈	#14273685	1	2	yes	—
152	C₈	#14273718	1	2	yes	—
153	C₂²	—	2	2	no	—
154	C₂²	C₂×#14273660	1	1	yes	characteristic
155	C₂²	#16837752	1	1	yes	characteristic
156	C₂²	#16837754	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	#16837231	1	4	yes	—
159	C₄	#16837267	1	1	yes	characteristic
160	C₄	C₂×#14273658	1	1	yes	characteristic
161	C₄	#16837748	1	1	yes	characteristic
162	C₄	#16837750	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	#21424223	1	1	yes	characteristic
166	C₂	C₂×#16837231	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1848	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂³×C₅₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1848	D₅₉
hypocenter	C₅₉	#12891
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
98. C₅₉	#12891	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅₉⋊C₁₆	2
#21424268	2
#21422189	2
C₅₉×#1848	2
#12891	59