Subgroups — #28505060

Order 15104.

302 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14273667	C₂	1	1	yes	maximal, characteristic
2	#21424268	C₂	1	1	yes	maximal, characteristic
3	#21422194	C₂	1	4	yes	maximal
4	C₅₉×#1844	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	C₂²	1	2	yes	—
6	C₂×#14273663	—	2	1	no	—
7	C₂×#14273663	C₂²	1	2	yes	—
8	#16837426	—	2	1	no	—
9	#16837178	C₄	1	2	yes	complemented
10	C₄×C₅₉⋊C₁₆	C₄	1	2	yes	complemented
11	C₂×#106×C₅₉	—	2	1	no	—
12	C₂×#106×C₅₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#384	—	2	1	no	—
15	C₂×#11745664	—	2	2	no	—
16	#14273667	—	2	2	no	—
17	#14273667	—	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	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₅₉	—	4	2	no	—
26	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
27	#106×C₅₉	—	2	4	no	—
28	#106×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	—	4	2	no	—
34	C₂²×C₄×C₅₉	SD₁₆	1	2	yes	—
35	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	D₈	1	2	yes	—
39	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
40	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₅₉	#41	1	2	yes	—
43	C₂×C₈×C₅₉	#41	1	2	yes	—
44	C₄⋊C₄×C₅₉	—	4	2	no	—
45	C₄⋊C₄×C₅₉	—	4	2	no	—
46	C₄⋊C₄×C₅₉	—	8	4	no	—
47	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₅₉	SD₁₆	1	2	yes	—
50	#11745664	—	4	2	no	—
51	#11745664	—	4	2	no	—
52	C₂×C₅₉⋊C₈	—	4	2	no	—
53	C₅₉⋊C₁₆	—	2	4	no	—
54	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₉	—	2	4	no	—
56	C₂×C₄×C₅₉	—	8	4	no	—
57	C₂×C₄×C₅₉	—	8	2	no	—
58	C₂×C₄×C₅₉	#106	1	2	yes	—
59	C₂×C₄×C₅₉	#106	1	2	yes	—
60	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₈×C₅₉	#107	1	2	yes	—
67	C₈×C₅₉	#107	1	2	yes	—
68	C₈×C₅₉	#117	1	2	yes	—
69	C₈×C₅₉	#125	1	2	yes	—
70	C₅₉⋊C₈	—	8	2	no	—
71	#12886	—	59	1	no	maximal, Sylow
72	C₂²×C₅₉	—	2	2	no	—
73	C₂²×C₅₉	C₂×#106	1	1	yes	characteristic
74	C₂²×C₅₉	#486	1	1	yes	characteristic
75	C₂²×C₅₉	#487	1	1	yes	characteristic
76	C₄×C₅₉	—	2	2	no	—
77	C₄×C₅₉	—	16	2	no	—
78	C₄×C₅₉	#365	1	4	yes	—
79	C₄×C₅₉	#382	1	1	yes	characteristic
80	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
81	C₄×C₅₉	#486	1	1	yes	characteristic
82	C₄×C₅₉	#487	1	1	yes	characteristic
83	C₄×M₃₂	—	59	1	no	—
84	#1848	—	59	1	no	—
85	#1844	D₅₉	1	1	yes	characteristic
86	#1383	—	59	4	no	—
87	C₂×C₅₉	—	2	1	no	—
88	C₂×C₅₉	—	2	1	no	—
89	C₂×C₅₉	#1834	1	1	yes	characteristic
90	C₂×C₅₉	C₂×#365	1	2	yes	—
91	C₂×M₃₂	—	59	2	no	—
92	C₂×#108	—	59	2	no	—
93	C₂×#108	—	118	1	no	—
94	C₂×#106	—	2	1	no	—
95	C₂×#106	C₂×D₅₉	1	2	yes	—
96	#423	—	118	1	no	—
97	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
98	#384	—	2	1	no	—
99	#346	—	59	2	no	—
100	C₄×C₁₆	—	59	2	no	—
101	C₅₉	#12886	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	118	2	no	—
103	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
104	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	4	2	no	—
107	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
108	M₃₂	—	118	2	no	—
109	M₃₂	—	118	2	no	—
110	C₂×C₁₆	—	59	4	no	—
111	#108	—	118	4	no	—
112	#108	—	236	2	no	—
113	#106	—	2	4	no	—
114	#106	—	4	2	no	—
115	C₄⋊C₈	—	236	2	no	—
116	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
117	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
118	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
119	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	#11745670	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
123	M₁₆	—	236	2	no	—
124	M₁₆	—	236	2	no	—
125	C₂×C₈	—	236	2	no	—
126	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
127	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
128	C₂×C₈	#11745669	1	2	yes	—
129	C₂×C₈	#11745672	1	2	yes	—
130	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
131	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
132	C₂×C₈	#11745687	1	2	yes	—
133	C₂×C₈	#11745689	1	2	yes	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	8	4	no	—
137	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
138	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
139	C₄²	#11745671	1	2	yes	—
140	C₁₆	—	118	4	no	—
141	C₂³	C₂×#11745670	1	1	yes	characteristic
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	#14273658	1	2	yes	—
146	C₂×C₄	#14273658	1	2	yes	—
147	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
148	C₂×C₄	#14273714	1	1	yes	characteristic
149	C₂×C₄	#14273716	1	1	yes	characteristic
150	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
151	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
152	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
153	C₈	—	472	2	no	—
154	C₈	#14273660	1	2	yes	—
155	C₈	#14273660	1	2	yes	—
156	C₈	#14273685	1	2	yes	—
157	C₈	#14273718	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	C₂×#14273658	1	1	yes	characteristic
160	C₂²	#16837748	1	1	yes	characteristic
161	C₂²	#16837754	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	16	2	no	—
164	C₄	#16837224	1	4	yes	—
165	C₄	#16837267	1	1	yes	characteristic
166	C₄	C₂×#14273660	1	1	yes	characteristic
167	C₄	#16837750	1	1	yes	characteristic
168	C₄	#16837752	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#21424223	1	1	yes	characteristic
172	C₂	C₂×#16837224	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14273658
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1844	C₂
socle	C₂²×C₅₉	C₂×#106
prime radical	C₂³×C₅₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1844	D₅₉
hypocenter	C₅₉	#12886
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1844

Complements

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

N	H ≅ G/N	type
9. #16837178	C₄	semidirect
10. C₄×C₅₉⋊C₁₆	C₄	semidirect
101. C₅₉	#12886	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14273667	2
#21424268	2
#21422194	2
C₅₉×#1844	2
#12886	59