Finite Group Databasebeta

← #28502053

Subgroups — #28502053

Order 15104.

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

normal subgroups only

This group has 262 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#21426313C212yesmaximal
2#21425676C211yesmaximal, characteristic
3#21422090C212yesmaximal
4#21422049C211yesmaximal, characteristic
5C59×#3261C211yesmaximal, characteristic
6#16837451C2211yescharacteristic
7#16837224—22no—
8#16837218—22no—
9#16837184C2212yes—
10#16837172—21no—
11#16837167—21no—
12C59⋊#338C2211yescharacteristic
13C59×#541—21no—
14C59×#479—22no—
15C59×#470C2212yes—
16C59×#467—21no—
17C59×#462—22no—
18C59×#424C2211yescharacteristic
19C2×#11745664—21no—
20C2×#11745664—21no—
21#14273667—22no—
22#14273663—22no—
23C2×Q16×C59—21no—
24C2×Q16×C59D412yes—
25C2×M16×C59C2311yescharacteristic
26C22×C8×C59D411yescharacteristic
27C4⋊Q8×C59—21no—
28#126×C59—42no—
29#125×C59—21no—
30#125×C59—42no—
31C22⋊Q8×C59—22no—
32C4×Q8×C59—22no—
33#117×C59D411yescharacteristic
34#107×C59—21no—
35#106×C59D412yes—
36C4⋊C8×C59—22no—
37#103×C59—22no—
38#103×C59—22no—
39C2×Q8×C59—22no—
40C22×C4×C59C2×D411yescharacteristic
41Q16×C59—42no—
42Q16×C59—81no—
43M16×C59—41no—
44C2×C8×C59—21no—
45C2×C8×C59—41no—
46C2×C8×C59C2×D412yes—
47C4⋊C4×C59—21no—
48C4⋊C4×C59—22no—
49C4⋊C4×C59—42no—
50C4⋊C4×C59—42no—
51C4⋊C4×C59—42no—
52C4⋊C4×C59—42no—
53C22⋊C4×C59—42no—
54C22⋊C4×C59SD1612yes—
55C42×C59—21no—
56C42×C59—42no—
57#11745664—41no—
58#11745664—41no—
59#11745664—41no—
60#11745664—41no—
61C2×C59⋊C8—41no—
62C2×C59⋊C8—41no—
63C59⋊C16—42no—
64C23×C59C2×SD1611yescharacteristic
65Q8×C59—42no—
66Q8×C59—42no—
67C2×C4×C59—21no—
68C2×C4×C59—21no—
69C2×C4×C59—22no—
70C2×C4×C59—42no—
71C2×C4×C59—42no—
72C2×C4×C59#12011yescharacteristic
73C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
74C8×C59—22no—
75C8×C59—41no—
76C59⋊C8—81no—
77C59⋊C8—81no—
78#12305—591nomaximal, Sylow
79C22×C59—21no—
80C22×C59—41no—
81C22×C59#45011yescharacteristic
82C4×C59—21no—
83C4×C59—42no—
84C4×C59—82no—
85C4×C59—82no—
86C4×C59#45411yescharacteristic
87C4×C59#45811yescharacteristic
88#3261D5911yescharacteristic
89#2239—592no—
90#2103—591no—
91#1342—592no—
92#1328—591no—
93C2×C59—21no—
94C2×C59—41no—
95C2×C59#161611yescharacteristic
96#541—21no—
97#479—22no—
98#470C2×D5912yes—
99#467—21no—
100#462—22no—
101#430—591no—
102#424C2×D5911yescharacteristic
103#365—1182no—
104#362—1182no—
105#349—592no—
106#344—1181no—
107#341—1181no—
108#338—591no—
109C59#1230511yescharacteristic, complemented, Sylow
110C2×Q16—21no—
111C2×Q16C22⋊D5912yes—
112C2×M16—1181no—
113C2×M16—1181no—
114C2×M16C22×D5911yescharacteristic
115C22×C8C22⋊D5911yescharacteristic
116C4⋊Q8—21no—
117#126—42no—
118#125—21no—
119#125—42no—
120C22⋊Q8—22no—
121C4×Q8—22no—
122#117C22⋊D5911yescharacteristic
123M32—1182no—
124#108—1182no—
125#107—21no—
126#106C22⋊D5912yes—
127C4⋊C8—22no—
128#103—22no—
129#103—22no—
130C2×Q8—22no—
131C22×C4C2×C22⋊D5911yescharacteristic
132Q16—42no—
133Q16—81no—
134M16—41no—
135M16—2361no—
136M16—2361no—
137M16—2361no—
138M16—2361no—
139C2×C8—21no—
140C2×C8—41no—
141C2×C8—2361no—
142C2×C8—2361no—
143C2×C8C2×C22⋊D5912yes—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4#1174567012yes—
152C42—21no—
153C42—42no—
154C16—2362no—
155C23C2×#1174567011yescharacteristic
156Q8—42no—
157Q8—42no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4#1427369611yescharacteristic
164C2×C4#1427375311yescharacteristic
165C8—22no—
166C8—41no—
167C8—4721no—
168C8—4721no—
169C22—21no—
170C22—41no—
171C22#1683755311yescharacteristic
172C4—21no—
173C4—42no—
174C4—82no—
175C4—82no—
176C4#1683757511yescharacteristic
177C4#1683759211yescharacteristic
178C2—21no—
179C2—41no—
180C2#2142313311yescharacteristic
subgroupG/H
centerC2#21423133
commutator, cosocleC2×M16×C59C23
FrattiniC2×M16C22×D59
FittingC59×#3261C2
socleC2×C59#1616
prime radicalC23×C59C2×SD16
hypercenter#3261D59
hypocenterC59#12305
LCSG ▷ C2×M16×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3261

Complements

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

NH ≅ G/Ntype
109. C59#12305semidirect

Maximal subgroups

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

subgroupindex
#214263132
#214256762
#214220902
#214220492
C59×#32612
#1230559