Finite Group Databasebeta

Subgroups — #28509193

Order 15104.

194 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837235	C₂	1	1	yes	maximal, characteristic
2	#21425877	C₂	1	1	yes	maximal, characteristic
3	#366×D₅₉	C₂	1	1	yes	maximal, characteristic
4	#21422298	C₂	1	1	yes	maximal, characteristic, complemented
5	#21422297	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422295	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×D₅₉	C₂²	1	1	yes	characteristic
9	#16837439	—	2	1	no	—
10	#107×D₅₉	—	2	1	no	—
11	#16837235	C₄	1	4	yes	complemented
12	C₂×C₅₉⋊C₃₂	C₂²	1	1	yes	characteristic
13	#16837229	—	2	1	no	—
14	#16837228	C₂²	1	1	yes	characteristic
15	#16837228	C₂²	1	1	yes	characteristic
16	#16837227	—	2	1	no	—
17	C₂×C₃₂×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#366	C₂²	1	1	yes	characteristic
19	C₅₉×#366	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₅₉	D₄	1	1	yes	characteristic
21	#14273685	—	4	1	no	—
22	C₄⋊C₄×D₅₉	—	4	1	no	—
23	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
24	C₁₆×D₅₉	C₂×C₄	1	2	yes	—
25	C₁₆×D₅₉	Q₈	1	2	yes	—
26	#14273661	—	2	1	no	—
27	#14273661	—	2	1	no	—
28	#14273660	—	4	1	no	—
29	#14273658	—	4	1	no	—
30	C₅₉⋊C₃₂	C₂×C₄	1	2	yes	complemented
31	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
32	#107×C₅₉	—	2	1	no	—
33	#107×C₅₉	—	2	1	no	—
34	C₃₂×C₅₉	C₂×C₄	1	2	yes	complemented
35	C₂×C₄×D₅₉	—	8	1	no	—
36	C₂×C₄×D₅₉	D₈	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₅₉	—	4	1	no	—
39	C₄⋊C₄×C₅₉	—	4	1	no	—
40	(C₂×D₅₉)⋊C₄	—	8	1	no	—
41	C₄⋊Dic₅₉	—	4	1	no	—
42	C₄⋊Dic₅₉	—	4	1	no	—
43	Dic₅₉⋊C₄	—	8	1	no	—
44	C₄×Dic₅₉	—	8	1	no	—
45	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
46	C₈×D₅₉	C₄⋊C₄	1	2	yes	—
47	C₈×D₅₉	Q₁₆	1	2	yes	—
48	C₁₆×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₁₆×C₅₉	C₂×Q₈	1	1	yes	characteristic
50	C₅₉⋊C₁₆	C₄⋊C₄	1	2	yes	—
51	C₂²×D₅₉	D₁₆	1	1	yes	characteristic
52	C₂×C₄×C₅₉	—	8	1	no	—
53	C₂×C₄×C₅₉	—	8	1	no	—
54	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₅₉	—	8	1	no	—
56	C₂×Dic₅₉	—	8	1	no	—
57	C₂×Dic₅₉	D₁₆	1	1	yes	characteristic
58	C₄×D₅₉	—	16	2	no	—
59	C₄×D₅₉	#107	1	2	yes	—
60	C₄×D₅₉	Q₃₂	1	2	yes	—
61	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₅₉⋊C₈	#107	1	2	yes	—
64	#13634	—	59	1	no	maximal, Sylow
65	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
66	C₂×D₅₉	—	2	1	no	—
67	C₂×D₅₉	—	2	1	no	—
68	C₂×D₅₉	#366	1	2	yes	—
69	C₄×C₅₉	—	16	1	no	—
70	C₄×C₅₉	—	16	1	no	—
71	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
72	C₄×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
73	Dic₅₉	—	16	1	no	—
74	Dic₅₉	—	16	1	no	—
75	Dic₅₉	#366	1	2	yes	—
76	C₂×M₆₄	—	59	1	no	—
77	#2158	—	59	1	no	—
78	C₂×#366	—	59	1	no	—
79	#1424	—	59	1	no	—
80	#1424	—	59	1	no	—
81	#1423	—	59	1	no	—
82	#1423	D₅₉	1	1	yes	characteristic, complemented
83	C₂×C₅₉	C₂×#366	1	1	yes	characteristic
84	C₂×C₅₉	#2263	1	1	yes	characteristic
85	C₂×C₅₉	#2264	1	1	yes	characteristic
86	D₅₉	—	2	2	no	—
87	C₂²×C₁₆	—	59	1	no	—
88	#427	—	118	1	no	—
89	C₂×#107	—	118	1	no	—
90	M₆₄	—	59	4	no	—
91	C₂×C₃₂	—	59	1	no	—
92	C₂×C₃₂	C₂×D₅₉	1	1	yes	characteristic
93	#367	—	118	1	no	—
94	#366	—	59	1	no	—
95	#366	—	59	1	no	—
96	#366	—	118	1	no	—
97	#366	C₂×D₅₉	1	1	yes	characteristic
98	#366	C₂×D₅₉	1	1	yes	characteristic
99	C₅₉	#13634	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₈	—	59	1	no	—
101	#117	—	236	1	no	—
102	C₂×C₄⋊C₄	—	236	1	no	—
103	C₂×C₁₆	—	59	1	no	—
104	C₂×C₁₆	—	59	2	no	—
105	C₂×C₁₆	—	59	2	no	—
106	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
107	#107	—	2	1	no	—
108	#107	—	2	1	no	—
109	#107	—	118	1	no	—
110	#107	—	118	1	no	—
111	#107	—	236	1	no	—
112	#106	—	236	1	no	—
113	C₃₂	—	59	2	no	—
114	C₃₂	C₄×D₅₉	1	2	yes	complemented
115	C₂²×C₄	—	59	1	no	—
116	C₂²×C₄	—	472	1	no	—
117	C₂×C₈	—	59	1	no	—
118	C₂×C₈	—	59	2	no	—
119	C₂×C₈	—	59	2	no	—
120	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	4	1	no	—
123	C₄⋊C₄	—	472	1	no	—
124	C₄⋊C₄	—	236	1	no	—
125	C₄⋊C₄	—	236	1	no	—
126	C₂²⋊C₄	—	472	1	no	—
127	C₄²	—	472	1	no	—
128	C₁₆	—	59	2	no	—
129	C₁₆	C₂×C₄×D₅₉	1	1	yes	characteristic
130	C₁₆	Q₈×D₅₉	1	1	yes	characteristic
131	C₂³	—	59	1	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	8	1	no	—
134	C₂×C₄	—	59	1	no	—
135	C₂×C₄	—	59	2	no	—
136	C₂×C₄	—	59	2	no	—
137	C₂×C₄	—	944	2	no	—
138	C₂×C₄	—	472	1	no	—
139	C₂×C₄	—	472	1	no	—
140	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
141	C₈	—	59	2	no	—
142	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
143	C₈	Q₁₆×D₅₉	1	1	yes	characteristic
144	C₂²	—	59	2	no	—
145	C₂²	—	118	1	no	—
146	C₂²	—	118	1	no	—
147	C₂²	D₁₆×D₅₉	1	1	yes	characteristic
148	C₄	—	16	1	no	—
149	C₄	—	16	1	no	—
150	C₄	—	59	2	no	—
151	C₄	—	944	1	no	—
152	C₄	—	944	1	no	—
153	C₄	#107×D₅₉	1	1	yes	characteristic
154	C₄	Q₃₂×D₅₉	1	1	yes	characteristic
155	C₂	—	118	2	no	—
156	C₂	#366×D₅₉	1	1	yes	characteristic
157	C₂	#21426407	1	1	yes	characteristic
158	C₂	#21426415	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₅₉
commutator	C₁₆×C₅₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₅₉×#1423	C₂
socle	C₂²×C₅₉	C₂×D₁₆
prime radical	C₂²×D₅₉	D₁₆
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	#1423	D₅₉
hypocenter	C₅₉	#13634
LCS	G ▷ C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1423

Complements

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

N	H ≅ G/N	type
4. #21422298	C₂	semidirect
5. #21422297	C₂	semidirect
6. #21422295	C₂	semidirect
7. C₅₉×#1423	C₂	semidirect
11. #16837235	C₄	semidirect
30. C₅₉⋊C₃₂	C₂×C₄	semidirect
34. C₃₂×C₅₉	C₂×C₄	semidirect
82. #1423	D₅₉	semidirect
99. C₅₉	#13634	semidirect
114. C₃₂	C₄×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837235	2
#21425877	2
#366×D₅₉	2
#21422298	2
#21422297	2
#21422295	2
C₅₉×#1423	2
#13634	59