Subgroups — #28505122

Order 15104.

374 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837227	C₂	1	4	yes	maximal
2	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#21424256	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14273661	—	2	1	no	—
7	C₂×#14273661	C₂²	1	2	yes	—
8	#16837272	—	2	1	no	—
9	#16837227	—	2	8	no	—
10	C₄×C₅₉⋊C₁₆	C₄	1	4	yes	complemented
11	C₂×#107×C₅₉	—	2	1	no	—
12	C₂×#107×C₅₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#384	—	2	1	no	—
15	C₂×C₄⋊Dic₅₉	—	2	2	no	—
16	C₂×C₄⋊Dic₅₉	—	4	2	no	—
17	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
18	C₂×C₅₉⋊C₁₆	D₄	1	4	yes	—
19	C₂×C₅₉⋊C₁₆	Q₈	1	4	yes	—
20	#14273661	—	2	4	no	—
21	#14273661	—	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	#107×C₅₉	—	2	4	no	—
28	#107×C₅₉	—	4	2	no	—
29	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₅₉	Q₈	1	2	yes	—
31	C₂²×C₄×C₅₉	—	4	2	no	—
32	C₂²×C₄×C₅₉	D₈	1	2	yes	—
33	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×Dic₅₉	—	4	2	no	—
35	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₅₉	D₈	1	2	yes	—
37	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
38	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
40	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
41	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
42	C₄⋊C₄×C₅₉	—	4	2	no	—
43	C₄⋊C₄×C₅₉	—	4	2	no	—
44	C₄⋊C₄×C₅₉	—	8	4	no	—
45	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
46	C₄²×C₅₉	Q₁₆	1	2	yes	—
47	C₄⋊Dic₅₉	—	4	2	no	—
48	C₄⋊Dic₅₉	—	4	2	no	—
49	C₄⋊Dic₅₉	—	8	4	no	—
50	C₅₉⋊C₁₆	C₄⋊C₄	1	8	yes	complemented
51	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₅₉	—	8	2	no	—
53	C₂×C₄×C₅₉	—	8	4	no	—
54	C₂×C₄×C₅₉	#107	1	4	yes	—
55	C₂×C₄×C₅₉	D₁₆	1	4	yes	—
56	C₂×C₄×C₅₉	Q₃₂	1	4	yes	—
57	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₅₉	—	8	2	no	—
64	C₂×Dic₅₉	—	8	4	no	—
65	C₈×C₅₉	#107	1	4	yes	—
66	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
67	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
68	#12905	—	59	1	no	maximal, Sylow
69	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
70	C₂²×C₅₉	#493	1	1	yes	characteristic
71	C₂²×C₅₉	#500	1	1	yes	characteristic
72	C₂²×C₅₉	C₂×D₁₆	1	2	yes	—
73	C₂²×C₅₉	C₂×Q₃₂	1	2	yes	—
74	C₄×C₅₉	—	16	2	no	—
75	C₄×C₅₉	#366	1	8	yes	complemented
76	C₄×C₅₉	#384	1	1	yes	characteristic
77	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
78	C₄×C₅₉	#494	1	1	yes	characteristic
79	C₄×C₅₉	#500	1	1	yes	characteristic
80	Dic₅₉	—	16	2	no	—
81	C₂×#366	—	59	4	no	—
82	C₂×C₄×C₁₆	—	59	1	no	—
83	#1845	—	59	1	no	—
84	#1845	D₅₉	1	1	yes	characteristic
85	C₂×C₅₉	#1845	1	1	yes	characteristic
86	C₂×C₅₉	C₂×#366	1	2	yes	—
87	C₂×C₅₉	#2246	1	1	yes	characteristic
88	C₂×C₅₉	#2247	1	1	yes	characteristic
89	C₂×C₅₉	#2252	1	2	yes	—
90	C₂²×C₁₆	—	59	2	no	—
91	C₂×#107	—	2	1	no	—
92	C₂×#107	—	59	2	no	—
93	C₂×#107	—	118	1	no	—
94	C₂×#107	C₂×D₅₉	1	2	yes	—
95	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
96	#384	—	2	1	no	—
97	#384	—	118	1	no	—
98	#366	—	118	8	no	—
99	C₄×C₁₆	—	59	4	no	—
100	C₅₉	#12905	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	2	2	no	—
104	C₂×C₄⋊C₄	—	4	2	no	—
105	C₂×C₄⋊C₄	—	118	2	no	—
106	C₂×C₄⋊C₄	—	236	2	no	—
107	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
108	C₂×C₁₆	—	59	4	no	—
109	C₂×C₁₆	—	59	4	no	—
110	C₂×C₁₆	—	59	4	no	—
111	#107	—	2	4	no	—
112	#107	—	4	2	no	—
113	#107	—	118	4	no	—
114	#107	—	236	2	no	—
115	C₄×C₈	Dic₁₁₈	1	2	yes	—
116	C₄×C₈	C₄×D₅₉	1	2	yes	—
117	C₂²×C₄	—	4	2	no	—
118	C₂²×C₄	—	236	2	no	—
119	C₂²×C₄	#11745669	1	2	yes	—
120	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
121	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
122	C₂×C₈	#11745669	1	2	yes	—
123	C₂×C₈	#11745672	1	2	yes	—
124	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
125	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
126	C₂×C₈	D₄×D₅₉	1	2	yes	—
127	C₂×C₈	Q₈×D₅₉	1	2	yes	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	8	4	no	—
131	C₄⋊C₄	—	472	4	no	—
132	C₄⋊C₄	—	236	2	no	—
133	C₄⋊C₄	—	236	2	no	—
134	C₄²	Dic₅₉⋊C₄	1	2	yes	—
135	C₄²	#11745672	1	2	yes	—
136	C₁₆	—	59	8	no	—
137	C₂³	C₂×#11745669	1	1	yes	characteristic
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	4	no	—
140	C₂×C₄	—	472	2	no	—
141	C₂×C₄	—	472	4	no	—
142	C₂×C₄	#14273660	1	4	yes	—
143	C₂×C₄	#14273668	1	4	yes	—
144	C₂×C₄	#14273673	1	4	yes	—
145	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
146	C₂×C₄	#14273723	1	1	yes	characteristic
147	C₂×C₄	#14273725	1	1	yes	characteristic
148	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
149	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
150	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
151	C₈	#14273660	1	4	yes	—
152	C₈	C₄⋊C₄×D₅₉	1	2	yes	—
153	C₈	#14273726	1	2	yes	—
154	C₂²	C₂×#14273660	1	1	yes	characteristic
155	C₂²	#16837778	1	1	yes	characteristic
156	C₂²	#16837805	1	1	yes	characteristic
157	C₂²	C₂×#14273668	1	2	yes	—
158	C₂²	C₂×#14273673	1	2	yes	—
159	C₄	—	16	2	no	—
160	C₄	—	944	2	no	—
161	C₄	#16837227	1	8	yes	complemented
162	C₄	#16837271	1	1	yes	characteristic
163	C₄	C₂×#14273660	1	1	yes	characteristic
164	C₄	#16837781	1	1	yes	characteristic
165	C₄	#16837805	1	1	yes	characteristic
166	C₂	#21424255	1	1	yes	characteristic
167	C₂	C₂×#16837227	1	2	yes	—
168	C₂	#21426345	1	1	yes	characteristic
169	C₂	#21426348	1	1	yes	characteristic
170	C₂	#21426369	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1845	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1845	D₅₉
hypocenter	C₅₉	#12905
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
10. C₄×C₅₉⋊C₁₆	C₄	semidirect
50. C₅₉⋊C₁₆	C₄⋊C₄	semidirect
75. C₄×C₅₉	#366	semidirect
100. C₅₉	#12905	semidirect
161. C₄	#16837227	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837227	2
C₂×C₄×C₅₉⋊C₁₆	2
#21424256	2
C₅₉×#1845	2
#12905	59