Finite Group Databasebeta

← #28498126

Subgroups — #28498126

Order 15104.

280 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 69 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#21432743C211yesmaximal, characteristic
2#21425905C212yesmaximal, complemented
3#21422048C212yesmaximal, complemented
4#21422034C211yesmaximal, characteristic
5C59×#1321C211yesmaximal, characteristic
6#41×Dic59C411yescharacteristic
7#16837460—22no—
8#16837456—22no—
9#16837430C2211yescharacteristic
10#16837428C411yescharacteristic
11#16837369C411yescharacteristic
12M16×Dic59C2211yescharacteristic
13C2×#14273637C411yescharacteristic
14#16837222—22no—
15#16837184C2212yes—
16#16837179—22no—
17C59×#408C2211yescharacteristic
18C59×#349C2212yes—
19C2×#11745664C2×C411yescharacteristic
20C22×C59⋊C8C2×C411yescharacteristic
21Q8×Dic59—21no—
22D4×Dic59—21no—
23D4×Dic59—22no—
24#14273686—21no—
25#14273686C2×C411yescharacteristic
26C2×C4×Dic59—21no—
27C2×C4×Dic59C2×C411yescharacteristic
28#14273667—22no—
29Dic59⋊C8—22no—
30Dic59⋊C8—22no—
31C22⋊(C59⋊C8)—22no—
32#14273637—21no—
33#14273637—21no—
34#14273637D412yes—
35C8×Dic59D412yes—
36C2×#41×C59C2×C411yescharacteristic
37C2×M16×C59C2311yescharacteristic
38C22×C8×C59C2×C411yescharacteristic
39M32×C59—22no—
40C22⋊C8×C59—22no—
41#41×C59—41no—
42#41×C59—41no—
43C2×Q8×C59M1611yescharacteristic
44C2×D4×C59—21no—
45C2×D4×C59M1611yescharacteristic
46C22×C4×C59—21no—
47C22×C4×C59C22×C411yescharacteristic
48C22×Dic59—22no—
49C22×Dic59—21no—
50M16×C59—41no—
51C2×C8×C59—21no—
52C2×C8×C59—41no—
53C2×C8×C59C2×D412yes—
54C22⋊Dic59—22no—
55C22⋊Dic59M1612yes—
56C4⋊Dic59—22no—
57C4⋊Dic59—21no—
58C4×Dic59—22no—
59C4×Dic59—21no—
60C4×Dic59—21no—
61C4×Dic59C22⋊C411yescharacteristic
62C4×Dic59C22⋊C411yescharacteristic
63#11745664—41no—
64C2×C59⋊C8—21no—
65C2×C59⋊C8—41no—
66C2×C59⋊C8C22⋊C412yes—
67C16×C59—42no—
68C59⋊C16—42no—
69C23×C59—21no—
70C23×C59C2×M1611yescharacteristic
71Q8×C59—41no—
72D4×C59—41no—
73D4×C59—42no—
74C2×C4×C59—21no—
75C2×C4×C59—21no—
76C2×C4×C59—21no—
77C2×C4×C59—41no—
78C2×C4×C59—41no—
79C2×C4×C59C2×C22⋊C411yescharacteristic
80C2×C4×C59C2×M1611yescharacteristic
81C2×Dic59—21no—
82C2×Dic59—22no—
83C2×Dic59—21no—
84C2×Dic59—81no—
85C2×Dic59—82no—
86C8×C59—22no—
87C8×C59—41no—
88C59⋊C8—22no—
89C59⋊C8—41no—
90#11523—591nomaximal, Sylow
91C22×C59—21no—
92C22×C59—41no—
93C22×C59—42no—
94C22×C59—41no—
95C22×C59#40711yescharacteristic
96C4×C59—21no—
97C4×C59—41no—
98C4×C59#41011yescharacteristic
99C4×C59#41311yescharacteristic
100Dic59—42no—
101Dic59—41no—
102Dic59—41no—
103#2923—591no—
104#2165—592no—
105#1328—592no—
106#1321—591no—
107#1321D5911yescharacteristic
108C2×C59—21no—
109C2×C59—41no—
110C2×C59—81no—
111C2×C59#146411yescharacteristic
112C4×#41—591no—
113#433—1182no—
114#432—1182no—
115#424—591no—
116#424—591no—
117#408—591no—
118#408C2×D5911yescharacteristic
119C4×M16—591no—
120C2×#97—591no—
121#364—1182no—
122#349—592no—
123#349C2×D5912yes—
124#347—1182no—
125C59#1152311yescharacteristic, complemented, Sylow
126C2×#41C4×D5911yescharacteristic
127C2×M16—591no—
128C2×M16C22×D5911yescharacteristic
129C22×C8—591no—
130C22×C8C4×D5911yescharacteristic
131C4×Q8—1181no—
132C4×D4—1182no—
133C4×D4—1181no—
134#117—591no—
135#117—1181no—
136C2×C42—591no—
137C2×C42—1181no—
138M32—22no—
139M32—1182no—
140C4⋊C8—1182no—
141C4⋊C8—1182no—
142C22⋊C8—22no—
143C22⋊C8—1182no—
144#97—592no—
145#97—1181no—
146#97—1181no—
147C4×C8—592no—
148#41—41no—
149#41—41no—
150C2×Q8#1174565911yescharacteristic
151C2×D4—21no—
152C2×D4#1174565911yescharacteristic
153C22×C4—21no—
154C22×C4—1182no—
155C22×C4—1181no—
156C22×C4C2×C4×D5911yescharacteristic
157M16—41no—
158M16—2361no—
159C2×C8—21no—
160C2×C8—41no—
161C2×C8—592no—
162C2×C8—1181no—
163C2×C8—2361no—
164C2×C8D4×D5912yes—
165C4⋊C4—1182no—
166C4⋊C4—1181no—
167C22⋊C4—592no—
168C22⋊C4—1182no—
169C42—591no—
170C42—591no—
171C42—1182no—
172C42—1181no—
173C42—1181no—
174C16—42no—
175C16—2362no—
176C23—21no—
177C23M16×D5911yescharacteristic
178Q8—41no—
179D4—41no—
180D4—42no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—1181no—
187C2×C4—1181no—
188C2×C4—1182no—
189C2×C4—4721no—
190C2×C4—4722no—
191C2×C4C22⋊C4×D5911yescharacteristic
192C2×C4C2×#1174565911yescharacteristic
193C8—22no—
194C8—41no—
195C8—1182no—
196C8—2361no—
197C22—21no—
198C22—41no—
199C22—42no—
200C22—41no—
201C22#1683736411yescharacteristic
202C4—21no—
203C4—41no—
204C4—2361no—
205C4—2361no—
206C4—2362no—
207C4#1683737411yescharacteristic
208C4#1683738311yescharacteristic
209C2—21no—
210C2—41no—
211C2—81no—
212C2#2142242711yescharacteristic
subgroupG/H
centerC4#16837374
commutatorC22×C4×C59C22×C4
FrattiniC2×M16C22×D59
FittingC59×#1321C2
socleC2×C59#1464
prime radicalC2×#41×C59C2×C4
cosocleC2×M16×C59C23
hypercenter#1321D59
hypocenterC59#11523
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #21425905C2semidirect
3. #21422048C2semidirect
125. C59#11523semidirect

Maximal subgroups

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

subgroupindex
#214327432
#214259052
#214220482
#214220342
C59×#13212
#1152359