Finite Group Databasebeta

← #28502048

Subgroups — #28502048

Order 15104.

286 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426308C212yesmaximal, complemented
2#21425676C211yesmaximal, characteristic, complemented
3#21422087C212yesmaximal, complemented
4#21422049C211yesmaximal, characteristic, complemented
5C59×#3259C211yesmaximal, characteristic
6#16837451C2211yescharacteristic
7#16837224—22no—
8#16837215—22no—
9#16837184C2212yes—
10#16837172—21no—
11#16837167—21no—
12C59⋊#338C2211yescharacteristic
13C59×#540—21no—
14C59×#478—22no—
15C59×#469C2212yes—
16C59×#466—21no—
17C59×#459—22no—
18C59×#424C2211yescharacteristic
19C2×#11745664—21no—
20C2×#11745664—21no—
21#14273667—22no—
22#14273663—22no—
23C2×D8×C59—21no—
24C2×D8×C59D412yes—
25C2×M16×C59C2311yescharacteristic
26C22×C8×C59D411yescharacteristic
27#127×C59—21no—
28#125×C59—21no—
29#123×C59—42no—
30#121×C59—22no—
31#121×C59—42no—
32C4×D4×C59—22no—
33#117×C59D411yescharacteristic
34#107×C59—21no—
35#106×C59D412yes—
36C4⋊C8×C59—22no—
37#102×C59—22no—
38#102×C59—22no—
39C2×D4×C59—22no—
40C2×D4×C59—42no—
41C2×D4×C59—42no—
42C22×C4×C59—42no—
43C22×C4×C59C2×D411yescharacteristic
44D8×C59—42no—
45D8×C59—81no—
46M16×C59—41no—
47C2×C8×C59—21no—
48C2×C8×C59—41no—
49C2×C8×C59C2×D412yes—
50C4⋊C4×C59—21no—
51C4⋊C4×C59—22no—
52C4⋊C4×C59—42no—
53C22⋊C4×C59—42no—
54C22⋊C4×C59—42no—
55C22⋊C4×C59SD1612yes—
56C42×C59—21no—
57#11745664—41no—
58#11745664—41no—
59#11745664—41no—
60#11745664—41no—
61C2×C59⋊C8—41no—
62C2×C59⋊C8—41no—
63C59⋊C16—42no—
64C23×C59—42no—
65C23×C59C2×SD1611yescharacteristic
66D4×C59—42no—
67D4×C59—42no—
68D4×C59—162no—
69D4×C59—162no—
70C2×C4×C59—21no—
71C2×C4×C59—21no—
72C2×C4×C59—22no—
73C2×C4×C59—42no—
74C2×C4×C59—162no—
75C2×C4×C59#12011yescharacteristic
76C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
77C8×C59—22no—
78C8×C59—41no—
79C59⋊C8—81no—
80C59⋊C8—81no—
81#12304—591nomaximal, Sylow
82C22×C59—21no—
83C22×C59—41no—
84C22×C59—82no—
85C22×C59—82no—
86C22×C59—82no—
87C22×C59#45011yescharacteristic
88C4×C59—21no—
89C4×C59—42no—
90C4×C59—82no—
91C4×C59#45411yescharacteristic
92C4×C59#45811yescharacteristic
93#3259D5911yescharacteristic
94#2238—592no—
95#2103—591no—
96#1341—592no—
97#1328—591no—
98C2×C59—21no—
99C2×C59—41no—
100C2×C59—162no—
101C2×C59#161611yescharacteristic
102#540—21no—
103#478—22no—
104#469C2×D5912yes—
105#466—21no—
106#459—22no—
107#430—591no—
108#424C2×D5911yescharacteristic
109#365—1182no—
110#361—1182no—
111#349—592no—
112#344—1181no—
113#341—1181no—
114#338—591no—
115C59#1230411yescharacteristic, complemented, Sylow
116C2×D8—21no—
117C2×D8C22⋊D5912yes—
118C2×M16—1181no—
119C2×M16—1181no—
120C2×M16C22×D5911yescharacteristic
121C22×C8C22⋊D5911yescharacteristic
122#127—21no—
123#125—21no—
124#123—42no—
125#121—22no—
126#121—42no—
127C4×D4—22no—
128#117C22⋊D5911yescharacteristic
129M32—1182no—
130#108—1182no—
131#107—21no—
132#106C22⋊D5912yes—
133C4⋊C8—22no—
134#102—22no—
135#102—22no—
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—42no—
139C22×C4—42no—
140C22×C4C2×C22⋊D5911yescharacteristic
141D8—42no—
142D8—81no—
143M16—41no—
144M16—2361no—
145M16—2361no—
146M16—2361no—
147M16—2361no—
148C2×C8—21no—
149C2×C8—41no—
150C2×C8—2361no—
151C2×C8—2361no—
152C2×C8C2×C22⋊D5912yes—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C22⋊C4#1174567012yes—
159C42—21no—
160C16—2362no—
161C23—42no—
162C23C2×#1174567011yescharacteristic
163D4—42no—
164D4—42no—
165D4—162no—
166D4—162no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—42no—
171C2×C4—162no—
172C2×C4#1427369611yescharacteristic
173C2×C4#1427375311yescharacteristic
174C8—22no—
175C8—41no—
176C8—4721no—
177C8—4721no—
178C22—21no—
179C22—41no—
180C22—82no—
181C22—82no—
182C22—82no—
183C22#1683755311yescharacteristic
184C4—21no—
185C4—42no—
186C4—82no—
187C4#1683757511yescharacteristic
188C4#1683759211yescharacteristic
189C2—21no—
190C2—41no—
191C2—162no—
192C2#2142313311yescharacteristic
subgroupG/H
centerC2#21423133
commutator, cosocleC2×M16×C59C23
FrattiniC2×M16C22×D59
Fitting, prime radicalC59×#3259C2
socleC2×C59#1616
hypercenter#3259D59
hypocenterC59#12304
LCSG ▷ C2×M16×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3259

Complements

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

NH ≅ G/Ntype
1. #21426308C2semidirect
2. #21425676C2semidirect
3. #21422087C2semidirect
4. #21422049C2semidirect
115. C59#12304semidirect

Maximal subgroups

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

subgroupindex
#214263082
#214256762
#214220872
#214220492
C59×#32592
#1230459