Finite Group Databasebeta

Subgroups — #28509608

Order 15104.

160 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 53 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	#21426377	C₂	1	1	yes	maximal, characteristic, complemented
2	#21425977	C₂	1	1	yes	maximal, characteristic
3	#21422302	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422301	C₂	1	1	yes	maximal, characteristic, complemented
5	#21422293	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422283	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1418	C₂	1	1	yes	maximal, characteristic
8	#16837806	—	2	1	no	—
9	#16837519	—	2	1	no	—
10	#16837236	C₂²	1	1	yes	characteristic
11	#16837232	C₂²	1	1	yes	characteristic
12	#16837230	C₂²	1	1	yes	characteristic
13	#16837229	—	2	1	no	—
14	#16837209	—	2	1	no	—
15	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#508	C₂²	1	1	yes	characteristic
17	C₅₉×M₆₄	C₂²	1	1	yes	characteristic
18	C₅₉×#367	C₂²	1	1	yes	characteristic
19	#14273734	—	4	1	no	—
20	#14273726	—	4	1	no	—
21	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
22	#14273663	—	2	1	no	—
23	#14273661	—	2	1	no	—
24	#14273660	—	4	1	no	—
25	#14273654	—	4	1	no	—
26	C₈×Dic₅₉	D₄	1	1	yes	characteristic
27	C₅₉⋊C₃₂	—	2	1	no	—
28	#135×C₅₉	—	2	1	no	—
29	Q₃₂×C₅₉	Q₈	1	1	yes	characteristic, complemented
30	SD₃₂×C₅₉	—	2	1	no	—
31	D₁₆×C₅₉	Q₈	1	1	yes	characteristic, complemented
32	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
33	#107×C₅₉	—	2	1	no	—
34	C₃₂×C₅₉	—	2	1	no	—
35	#41×C₅₉	—	4	1	no	—
36	C₂×Dic₁₁₈	—	8	1	no	—
37	Q₁₆×C₅₉	—	2	1	no	—
38	SD₁₆×C₅₉	—	4	1	no	—
39	D₈×C₅₉	—	2	1	no	—
40	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₅₉	—	4	1	no	—
42	C₄⋊Dic₅₉	—	4	1	no	—
43	Dic₅₉⋊C₄	—	8	1	no	—
44	C₄×Dic₅₉	D₈	1	1	yes	characteristic
45	#11745664	—	4	1	no	—
46	#11745664	—	8	1	no	—
47	C₂×C₅₉⋊C₈	—	8	1	no	—
48	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
49	C₁₆×C₅₉	C₂×Q₈	1	1	yes	characteristic
50	C₁₆×C₅₉	#41	1	1	yes	characteristic
51	C₅₉⋊C₁₆	—	2	1	no	—
52	Q₈×C₅₉	—	4	1	no	—
53	D₄×C₅₉	—	4	1	no	—
54	D₄×C₅₉	—	8	1	no	—
55	C₂×C₄×C₅₉	—	8	1	no	—
56	C₂×C₄×C₅₉	—	8	1	no	—
57	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₅₉	—	2	1	no	—
59	C₂×Dic₅₉	—	8	1	no	—
60	Dic₁₁₈	—	16	2	no	—
61	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
62	C₈×C₅₉	#137	1	1	yes	characteristic
63	C₅₉⋊C₈	—	8	1	no	—
64	C₅₉⋊C₈	SD₃₂	1	1	yes	characteristic
65	C₅₉⋊C₈	SD₃₂	1	1	yes	characteristic
66	#13710	—	59	1	no	maximal, Sylow
67	C₂²×C₅₉	—	8	1	no	—
68	C₂²×C₅₉	#509	1	1	yes	characteristic
69	C₄×C₅₉	—	8	1	no	—
70	C₄×C₅₉	—	16	1	no	—
71	C₄×C₅₉	#474	1	1	yes	characteristic
72	C₄×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
73	Dic₅₉	—	2	2	no	—
74	Dic₅₉	—	16	1	no	—
75	#2254	—	59	1	no	—
76	#2178	—	59	1	no	—
77	#1426	—	59	1	no	—
78	#1426	—	59	1	no	—
79	#1422	—	59	1	no	—
80	#1418	—	59	1	no	—
81	#1418	D₅₉	1	1	yes	characteristic
82	C₂×C₅₉	—	2	1	no	—
83	C₂×C₅₉	—	16	1	no	—
84	C₂×C₅₉	#2226	1	1	yes	characteristic
85	#508	C₂×D₅₉	1	1	yes	characteristic
86	#500	—	118	1	no	—
87	#443	—	118	1	no	—
88	M₆₄	—	59	1	no	—
89	M₆₄	C₂×D₅₉	1	1	yes	characteristic
90	#368	—	59	1	no	—
91	#367	—	59	1	no	—
92	#367	—	118	1	no	—
93	#367	C₂×D₅₉	1	1	yes	characteristic
94	#359	—	118	1	no	—
95	C₄×C₁₆	—	59	1	no	—
96	C₅₉	#13710	1	1	yes	characteristic, complemented, Sylow
97	#135	—	2	1	no	—
98	#131	—	236	1	no	—
99	C₄⋊Q₈	—	236	1	no	—
100	Q₃₂	Dic₁₁₈	1	1	yes	characteristic, complemented
101	SD₃₂	—	2	1	no	—
102	D₁₆	Dic₁₁₈	1	1	yes	characteristic, complemented
103	C₂×C₁₆	—	59	1	no	—
104	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
105	#108	—	118	1	no	—
106	#107	—	2	1	no	—
107	#107	—	118	1	no	—
108	#107	—	236	1	no	—
109	#104	—	236	1	no	—
110	C₄×C₈	—	59	1	no	—
111	C₃₂	—	2	1	no	—
112	C₃₂	—	118	1	no	—
113	#41	—	4	1	no	—
114	C₂×Q₈	—	472	1	no	—
115	Q₁₆	—	2	1	no	—
116	SD₁₆	—	4	1	no	—
117	D₈	—	2	1	no	—
118	M₁₆	—	472	1	no	—
119	M₁₆	—	236	1	no	—
120	C₂×C₈	—	59	1	no	—
121	C₂×C₈	—	472	1	no	—
122	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
123	C₄⋊C₄	—	4	1	no	—
124	C₄⋊C₄	—	472	1	no	—
125	C₄⋊C₄	—	236	1	no	—
126	C₄²	—	59	1	no	—
127	C₁₆	—	118	1	no	—
128	C₁₆	C₂×Dic₁₁₈	1	1	yes	characteristic
129	C₁₆	#11745687	1	1	yes	characteristic
130	Q₈	—	4	1	no	—
131	Q₈	—	944	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	8	1	no	—
136	C₂×C₄	—	118	1	no	—
137	C₂×C₄	—	472	1	no	—
138	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
139	C₈	—	59	1	no	—
140	C₈	—	59	1	no	—
141	C₈	—	472	1	no	—
142	C₈	#14273704	1	1	yes	characteristic
143	C₈	#14273756	1	1	yes	characteristic
144	C₂²	—	8	1	no	—
145	C₂²	#16837836	1	1	yes	characteristic
146	C₄	—	8	1	no	—
147	C₄	—	16	1	no	—
148	C₄	—	118	2	no	—
149	C₄	—	944	1	no	—
150	C₄	#16837680	1	1	yes	characteristic
151	C₄	SD₃₂×D₅₉	1	1	yes	characteristic
152	C₂	—	2	1	no	—
153	C₂	—	16	1	no	—
154	C₂	#21426241	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21426241
commutator, cosocle	C₂×C₁₆×C₅₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₅₉×#1418	C₂
socle	C₂×C₅₉	#2226
prime radical	C₅₉×#508	C₂²
hypercenter	#1418	D₅₉
hypocenter	C₅₉	#13710
LCS	G ▷ C₂×C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1418

Complements

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

N	H ≅ G/N	type
1. #21426377	C₂	semidirect
3. #21422302	C₂	semidirect
4. #21422301	C₂	semidirect
5. #21422293	C₂	semidirect
29. Q₃₂×C₅₉	Q₈	semidirect
31. D₁₆×C₅₉	Q₈	semidirect
96. C₅₉	#13710	semidirect
100. Q₃₂	Dic₁₁₈	semidirect
102. D₁₆	Dic₁₁₈	semidirect

Maximal subgroups

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

subgroup	index
#21426377	2
#21425977	2
#21422302	2
#21422301	2
#21422293	2
#21422283	2
C₅₉×#1418	2
#13710	59