Finite Group Databasebeta

← #28502065

Subgroups — #28502065

Order 15104.

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

normal subgroups only

This group has 318 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#21425671C211yesmaximal, characteristic, complemented
3#21422081C212yesmaximal, complemented
4#21422047C211yesmaximal, characteristic, complemented
5C59×#3253C211yesmaximal, characteristic
6#16837451C2211yescharacteristic
7#16837224—22no—
8#16837215—22no—
9#16837184C2212yes—
10#16837174—21no—
11#16837162—21no—
12#16837159C2211yescharacteristic
13C59×#532—21no—
14C59×#476—22no—
15C59×#469—21no—
16C59×#469C2212yes—
17C59×#459—22no—
18C59×#423C2211yescharacteristic
19C2×#11745664—21no—
20#14273686—21no—
21#14273667—22no—
22#14273663—22no—
23C2×#41×C59—41no—
24C2×D8×C59—21no—
25C2×D8×C59D412yes—
26C2×M16×C59D411yescharacteristic
27C2×M16×C59C2311yescharacteristic
28C4⋊Q8×C59—21no—
29#127×C59—21no—
30#124×C59—41no—
31#121×C59—22no—
32#121×C59—41no—
33C4×D4×C59—22no—
34C4×D4×C59—41no—
35C2×C42×C59D411yescharacteristic
36#106×C59—21no—
37#106×C59D412yes—
38C4⋊C8×C59—22no—
39#102×C59—22no—
40#102×C59—22no—
41#41×C59—84no—
42C2×Q8×C59—41no—
43C2×D4×C59—22no—
44C2×D4×C59—41no—
45C2×D4×C59—41no—
46C2×D4×C59—42no—
47C22×C4×C59—21no—
48C22×C4×C59—42no—
49C22×C4×C59C2×D411yescharacteristic
50D8×C59—42no—
51D8×C59—81no—
52M16×C59—41no—
53M16×C59—41no—
54C2×C8×C59—21no—
55C2×C8×C59C2×D412yes—
56C4⋊C4×C59—22no—
57C4⋊C4×C59—41no—
58C22⋊C4×C59—42no—
59C22⋊C4×C59—42no—
60C42×C59—21no—
61C42×C59SD1611yescharacteristic
62C42×C59SD1611yescharacteristic
63C22⋊Dic59—41no—
64C4⋊Dic59—41no—
65C4×Dic59—41no—
66#11745664—41no—
67#11745664—41no—
68C2×C59⋊C8—41no—
69C59⋊C16—42no—
70C23×C59—42no—
71C23×C59AGL(1,ℤ/8)11yescharacteristic
72Q8×C59—82no—
73D4×C59—42no—
74D4×C59—42no—
75D4×C59—82no—
76D4×C59—82no—
77D4×C59—84no—
78C2×C4×C59—21no—
79C2×C4×C59—21no—
80C2×C4×C59—21no—
81C2×C4×C59—21no—
82C2×C4×C59—42no—
83C2×C4×C59—42no—
84C2×C4×C59—84no—
85C2×C4×C59#12011yescharacteristic
86C2×C4×C59C2×SD1611yescharacteristic
87C2×Dic59—81no—
88C8×C59—22no—
89C8×C59—41no—
90C59⋊C8—81no—
91#12307—591nomaximal, Sylow
92C22×C59—21no—
93C22×C59—41no—
94C22×C59—82no—
95C22×C59—82no—
96C22×C59—82no—
97C22×C59#45011yescharacteristic
98C4×C59—22no—
99C4×C59—21no—
100C4×C59—41no—
101C4×C59—82no—
102C4×C59#45411yescharacteristic
103C4×C59#45811yescharacteristic
104Dic59—161no—
105#3253D5911yescharacteristic
106#2238—592no—
107#2102—591no—
108#1339—592no—
109#1327—591no—
110C2×C59—21no—
111C2×C59—41no—
112C2×C59—162no—
113C2×C59#161611yescharacteristic
114#532—21no—
115#476—22no—
116#469—21no—
117#469C2×D5912yes—
118#459—22no—
119#430—591no—
120#423C2×D5911yescharacteristic
121#365—1182no—
122#361—1182no—
123#349—592no—
124#344—1181no—
125#339—1181no—
126H(3,ℤ/4)—591no—
127C59#1230711yescharacteristic, complemented, Sylow
128C2×#41—41no—
129C2×D8—21no—
130C2×D8C22⋊D5912yes—
131C2×M16—1181no—
132C2×M16C22⋊D5911yescharacteristic
133C2×M16C22×D5911yescharacteristic
134C4⋊Q8—21no—
135#127—21no—
136#124—41no—
137#121—22no—
138#121—41no—
139C4×D4—22no—
140C4×D4—41no—
141#117—1181no—
142C2×C42C22⋊D5911yescharacteristic
143M32—1182no—
144#108—1182no—
145#106—21no—
146#106C22⋊D5912yes—
147C4⋊C8—22no—
148#102—22no—
149#102—22no—
150#41—84no—
151C2×Q8—41no—
152C2×D4—22no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—42no—
156C22×C4—21no—
157C22×C4—42no—
158C22×C4C2×C22⋊D5911yescharacteristic
159D8—42no—
160D8—81no—
161M16—41no—
162M16—41no—
163M16—2361no—
164M16—2361no—
165C2×C8—21no—
166C2×C8—2361no—
167C2×C8C2×C22⋊D5912yes—
168C4⋊C4—22no—
169C4⋊C4—41no—
170C4⋊C4—2361no—
171C22⋊C4—42no—
172C22⋊C4—42no—
173C22⋊C4—2361no—
174C42—21no—
175C42—2361no—
176C42#1174567011yescharacteristic
177C42#1174567011yescharacteristic
178C16—2362no—
179C23—42no—
180C23#1427375311yescharacteristic
181Q8—82no—
182D4—42no—
183D4—42no—
184D4—82no—
185D4—82no—
186D4—84no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—84no—
194C2×C4—4721no—
195C2×C4#1427369611yescharacteristic
196C2×C4C2×#1174567011yescharacteristic
197C8—22no—
198C8—41no—
199C8—4721no—
200C22—21no—
201C22—41no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22#1683755311yescharacteristic
206C4—22no—
207C4—21no—
208C4—41no—
209C4—82no—
210C4—9441no—
211C4#1683757511yescharacteristic
212C4#1683759211yescharacteristic
213C2—21no—
214C2—41no—
215C2—162no—
216C2#2142313311yescharacteristic
subgroupG/H
centerC2#21423133
commutator, cosocleC2×M16×C59C23
FrattiniC2×M16C22×D59
Fitting, prime radicalC59×#3253C2
socleC2×C59#1616
hypercenter#3253D59
hypocenterC59#12307
LCSG ▷ C2×M16×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3253

Complements

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

NH ≅ G/Ntype
1. #21426308C2semidirect
2. #21425671C2semidirect
3. #21422081C2semidirect
4. #21422047C2semidirect
127. C59#12307semidirect

Maximal subgroups

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

subgroupindex
#214263082
#214256712
#214220812
#214220472
C59×#32532
#1230759