Subgroups — #28506700

Order 15104.

298 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837223	C₂	1	1	yes	maximal, characteristic
2	#21425855	C₂	1	2	yes	maximal
3	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#21422185	C₂	1	2	yes	maximal
5	C₅₉×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	2	yes	—
7	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
9	#16837223	—	2	2	no	—
10	C₄⋊(C₅₉⋊C₁₆)	—	2	4	no	—
11	C₄×C₅₉⋊C₁₆	—	2	2	no	—
12	C₂×#108×C₅₉	C₄	1	1	yes	characteristic
13	C₅₉×#427	C₄	1	1	yes	characteristic
14	C₅₉×#405	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#339	C₄	1	2	yes	—
17	#14273667	—	2	2	no	—
18	#14273667	—	2	2	no	—
19	C₂×C₅₉⋊C₁₆	—	2	1	no	—
20	C₂×C₅₉⋊C₁₆	—	2	1	no	—
21	C₂×C₅₉⋊C₁₆	—	2	1	no	—
22	C₂×C₅₉⋊C₁₆	—	2	1	no	—
23	C₂×C₅₉⋊C₁₆	—	2	2	no	—
24	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	—
25	C₂×C₅₉⋊C₁₆	Q₈	1	2	yes	—
26	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	#117×C₅₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
31	#108×C₅₉	—	2	2	no	—
32	#107×C₅₉	C₈	1	2	yes	—
33	#106×C₅₉	C₈	1	2	yes	—
34	#97×C₅₉	—	2	2	no	—
35	C₄×C₈×C₅₉	—	2	2	no	—
36	C₄×C₈×C₅₉	D₄	1	2	yes	—
37	C₄×C₈×C₅₉	Q₈	1	2	yes	—
38	C₂²×C₄×C₅₉	—	2	1	no	—
39	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	M₁₆×C₅₉	—	2	2	no	—
41	M₁₆×C₅₉	—	2	2	no	—
42	C₂×C₈×C₅₉	—	2	2	no	—
43	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
47	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
52	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₅₉	C₂×C₈	1	4	yes	—
54	C₂²⋊C₄×C₅₉	—	2	2	no	—
55	C₄²×C₅₉	—	2	2	no	—
56	C₄²×C₅₉	—	2	1	no	—
57	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
58	C₅₉⋊C₁₆	—	2	2	no	—
59	C₅₉⋊C₁₆	—	2	2	no	—
60	C₅₉⋊C₁₆	—	4	2	no	—
61	C₂³×C₅₉	#131	1	1	yes	characteristic
62	C₂×C₄×C₅₉	—	2	4	no	—
63	C₂×C₄×C₅₉	—	2	2	no	—
64	C₂×C₄×C₅₉	—	2	4	no	—
65	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
71	C₈×C₅₉	—	4	2	no	—
72	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
73	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
74	C₈×C₅₉	#121	1	1	yes	characteristic
75	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₅₉	#125	1	1	yes	characteristic
77	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
78	#13161	—	59	1	no	maximal, Sylow
79	C₂²×C₅₉	—	2	2	no	—
80	C₂²×C₅₉	#424	1	1	yes	characteristic
81	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
82	C₂²×C₅₉	#446	1	1	yes	characteristic
83	C₄×C₅₉	—	2	4	no	—
84	C₄×C₅₉	—	4	4	no	—
85	C₄×C₅₉	#389	1	1	yes	characteristic
86	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₄×C₅₉	#436	1	1	yes	characteristic
88	C₄×C₅₉	C₈×Q₈	1	1	yes	characteristic
89	C₂×#364	—	59	1	no	—
90	#2151	—	59	2	no	—
91	C₂×C₄×C₁₆	—	59	1	no	—
92	#1772	D₅₉	1	1	yes	characteristic
93	#1379	—	59	2	no	—
94	C₂×C₅₉	—	2	1	no	—
95	C₂×C₅₉	—	2	1	no	—
96	C₂×C₅₉	#1939	1	1	yes	characteristic
97	C₂×C₅₉	#2170	1	2	yes	—
98	C₂×M₃₂	—	59	2	no	—
99	C₂²×C₁₆	—	59	1	no	—
100	C₂²×C₁₆	—	59	1	no	—
101	C₂×#108	Dic₅₉	1	1	yes	characteristic
102	#427	Dic₅₉	1	1	yes	characteristic
103	#405	C₂×D₅₉	1	2	yes	—
104	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
105	#364	—	118	2	no	—
106	C₄⋊C₁₆	—	118	4	no	—
107	C₄×C₁₆	—	118	2	no	—
108	#339	Dic₅₉	1	2	yes	—
109	C₅₉	#13161	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₁₆	C₂×Dic₅₉	1	2	yes	—
111	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
113	#117	C₂×Dic₅₉	1	2	yes	—
114	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
115	M₃₂	—	118	2	no	—
116	M₃₂	—	118	2	no	—
117	C₂×C₁₆	—	59	2	no	—
118	C₂×C₁₆	—	59	2	no	—
119	C₂×C₁₆	—	118	1	no	—
120	C₂×C₁₆	—	118	1	no	—
121	C₂×C₁₆	—	118	1	no	—
122	C₂×C₁₆	—	118	1	no	—
123	C₂×C₁₆	—	118	2	no	—
124	#108	—	2	2	no	—
125	#107	C₅₉⋊C₈	1	2	yes	—
126	#106	C₅₉⋊C₈	1	2	yes	—
127	#97	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	Dic₁₁₈	1	2	yes	—
130	C₄×C₈	D₂₃₆	1	2	yes	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
133	M₁₆	—	2	2	no	—
134	M₁₆	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
137	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
138	C₂×C₈	#11745664	1	2	yes	—
139	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
140	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
141	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
142	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
143	C₂×C₈	#11745687	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
145	C₂×C₈	#11745689	1	1	yes	characteristic
146	C₄⋊C₄	C₂×C₅₉⋊C₈	1	4	yes	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	2	1	no	—
150	C₄²	C₄⋊Dic₅₉	1	2	yes	—
151	C₁₆	—	118	2	no	—
152	C₁₆	—	118	2	no	—
153	C₁₆	—	236	2	no	—
154	C₂³	#14273734	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
159	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
160	C₂×C₄	Q₈×Dic₅₉	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
162	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
163	C₂×C₄	#14273734	1	1	yes	characteristic
164	C₈	—	4	2	no	—
165	C₈	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
166	C₈	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
167	C₈	#14273702	1	1	yes	characteristic
168	C₈	#14273707	1	1	yes	characteristic
169	C₈	#14273718	1	1	yes	characteristic
170	C₈	#14273726	1	1	yes	characteristic
171	C₂²	—	2	2	no	—
172	C₂²	#16837429	1	1	yes	characteristic
173	C₂²	D₄×C₅₉⋊C₈	1	1	yes	characteristic
174	C₂²	#16837533	1	1	yes	characteristic
175	C₄	—	2	4	no	—
176	C₄	—	4	4	no	—
177	C₄	#16837301	1	1	yes	characteristic
178	C₄	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
179	C₄	#16837480	1	1	yes	characteristic
180	C₄	Q₈×C₅₉⋊C₈	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#21424815	1	1	yes	characteristic
184	C₂	#21425933	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1772	C₂
socle	C₂²×C₅₉	C₈×D₄
prime radical	C₂³×C₅₉	#131
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1772	D₅₉
hypocenter	C₅₉	#13161
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
109. C₅₉	#13161	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837223	2
#21425855	2
C₂×C₄×C₅₉⋊C₁₆	2
#21422185	2
C₅₉×#1772	2
#13161	59