Finite Group Databasebeta

← #28499536

Subgroups — #28499536

Order 15104.

324 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 79 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#21425908C211yesmaximal, characteristic, complemented
2#21424245C211yesmaximal, characteristic, complemented
3#21422087C212yesmaximal, complemented
4#21422081C212yesmaximal, complemented
5C59×#3184C211yesmaximal, characteristic
6#16837223—21no—
7#16837184C2212yes—
8#16837180—21no—
9#16837173—21no—
10#16837162—21no—
11C59⋊#338C2211yescharacteristic
12#16837159C2211yescharacteristic
13C59×#532—21no—
14C59×#498C411yescharacteristic, complemented
15C59×#493C411yescharacteristic, complemented
16C59×#486—21no—
17C59×#469—21no—
18C59×#469C2212yes—
19C59×#442—21no—
20C59×#442C412yescomplemented
21C4×M16×C59C2211yescharacteristic
22C2×#11745664—21no—
23#14273686—21no—
24#14273667—22no—
25C2×#41×C59—41no—
26C2×D8×C59—21no—
27C2×D8×C59C2×C412yes—
28C2×M16×C59D411yescharacteristic
29C2×M16×C59C2311yescharacteristic
30C4⋊Q8×C59—21no—
31#127×C59—21no—
32#124×C59—41no—
33#121×C59—22no—
34#121×C59—41no—
35C4×D4×C59—22no—
36C4×D4×C59—41no—
37C2×C42×C59D411yescharacteristic
38#106×C59—21no—
39#106×C59C2×C412yes—
40#102×C59—22no—
41#102×C59—22no—
42#97×C59C2×C411yescharacteristic
43#97×C59D411yescharacteristic
44C4×C8×C59C2×C411yescharacteristic
45C4×C8×C59D411yescharacteristic
46#41×C59—84no—
47C2×Q8×C59—41no—
48C2×D4×C59—22no—
49C2×D4×C59—41no—
50C2×D4×C59—41no—
51C2×D4×C59—42no—
52C22×C4×C59—21no—
53C22×C4×C59—42no—
54C22×C4×C59C2×D411yescharacteristic
55D8×C59—42no—
56D8×C59—42no—
57M16×C59—41no—
58M16×C59—41no—
59C2×C8×C59C22⋊C411yescharacteristic
60C2×C8×C59C22⋊C411yescharacteristic
61C2×C8×C59C22×C411yescharacteristic
62C2×C8×C59C2×D411yescharacteristic
63C4⋊C4×C59—22no—
64C4⋊C4×C59—41no—
65C22⋊C4×C59—42no—
66C22⋊C4×C59—42no—
67C42×C59—21no—
68C42×C59C22⋊C411yescharacteristic
69C42×C59C22⋊C411yescharacteristic
70C22⋊Dic59—41no—
71C4⋊Dic59—41no—
72C4×Dic59—41no—
73#11745664—41no—
74#11745664—41no—
75C2×C59⋊C8—41no—
76C59⋊C16—42no—
77C23×C59—42no—
78C23×C59AGL(1,ℤ/8)11yescharacteristic
79Q8×C59—82no—
80D4×C59—42no—
81D4×C59—42no—
82D4×C59—82no—
83D4×C59—82no—
84D4×C59—84no—
85C2×C4×C59—21no—
86C2×C4×C59—21no—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—42no—
90C2×C4×C59—42no—
91C2×C4×C59—84no—
92C2×C4×C59C2×C22⋊C411yescharacteristic
93C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
94C2×Dic59—81no—
95C8×C59—21no—
96C8×C59—21no—
97C8×C59—21no—
98C8×C59—21no—
99C59⋊C8—81no—
100#11837—591nomaximal, Sylow
101C22×C59—21no—
102C22×C59—41no—
103C22×C59—82no—
104C22×C59—82no—
105C22×C59—82no—
106C22×C59#41811yescharacteristic
107C4×C59—22no—
108C4×C59—21no—
109C4×C59—41no—
110C4×C59—82no—
111C4×C59#41011yescharacteristic
112C4×C59#42111yescharacteristic
113Dic59—161no—
114#3184D5911yescharacteristic
115#2165—591no—
116#1841—591no—
117#1341—592no—
118#1339—592no—
119C2×C59—21no—
120C2×C59—41no—
121C2×C59—162no—
122C2×C59#151411yescharacteristic
123#532—21no—
124#498Dic5911yescharacteristic, complemented
125#493Dic5911yescharacteristic, complemented
126#486—21no—
127#469—21no—
128#469C2×D5912yes—
129#442—21no—
130#442Dic5912yescomplemented
131C4×M16C2×D5911yescharacteristic
132#364—1181no—
133#349—592no—
134#347—1181no—
135#344—1181no—
136#339—1181no—
137#338—591no—
138H(3,ℤ/4)—591no—
139C59#1183711yescharacteristic, complemented, Sylow
140C2×#41—41no—
141C2×D8—21no—
142C2×D8C2×Dic5912yes—
143C2×M16—1181no—
144C2×M16C22⋊D5911yescharacteristic
145C2×M16C22×D5911yescharacteristic
146C4⋊Q8—21no—
147#127—21no—
148#124—41no—
149#121—22no—
150#121—41no—
151C4×D4—22no—
152C4×D4—41no—
153#117—1181no—
154C2×C42C22⋊D5911yescharacteristic
155M32—1182no—
156#106—21no—
157#106C2×Dic5912yes—
158#102—22no—
159#102—22no—
160#97C2×Dic5911yescharacteristic
161#97C22⋊D5911yescharacteristic
162C4×C8C2×Dic5911yescharacteristic
163C4×C8C22⋊D5911yescharacteristic
164#41—84no—
165C2×Q8—41no—
166C2×D4—22no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—42no—
170C22×C4—21no—
171C22×C4—42no—
172C22×C4C2×C22⋊D5911yescharacteristic
173D8—42no—
174D8—42no—
175M16—41no—
176M16—41no—
177M16—2361no—
178M16—2361no—
179C2×C8—2361no—
180C2×C8C22⋊Dic5911yescharacteristic
181C2×C8C22⋊Dic5911yescharacteristic
182C2×C8C22×Dic5911yescharacteristic
183C2×C8C2×C22⋊D5911yescharacteristic
184C4⋊C4—22no—
185C4⋊C4—41no—
186C4⋊C4—2361no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—2361no—
190C42—21no—
191C42—2361no—
192C42C22⋊Dic5911yescharacteristic
193C42C22⋊Dic5911yescharacteristic
194C16—2362no—
195C23—42no—
196C23#1427375311yescharacteristic
197Q8—82no—
198D4—42no—
199D4—42no—
200D4—82no—
201D4—82no—
202D4—84no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—84no—
210C2×C4—4721no—
211C2×C4C2×C22⋊Dic5911yescharacteristic
212C2×C4#1427375311yescharacteristic
213C8—21no—
214C8—21no—
215C8—21no—
216C8—21no—
217C8—4721no—
218C22—21no—
219C22—41no—
220C22—82no—
221C22—82no—
222C22—82no—
223C22#1683740711yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—82no—
228C4—9441no—
229C4#1683737611yescharacteristic
230C4#1683741911yescharacteristic
231C2—21no—
232C2—41no—
233C2—162no—
234C2#2142264311yescharacteristic
subgroupG/H
centerC2#21422643
commutatorC2×C8×C59C22×C4
FrattiniC2×M16C22×D59
Fitting, prime radicalC59×#3184C2
socleC2×C59#1514
cosocleC2×M16×C59C23
hypercenter#3184D59
hypocenterC59#11837
LCSG ▷ C2×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3184

Complements

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

NH ≅ G/Ntype
1. #21425908C2semidirect
2. #21424245C2semidirect
3. #21422087C2semidirect
4. #21422081C2semidirect
14. C59×#498C4semidirect
15. C59×#493C4semidirect
20. C59×#442C4semidirect
124. #498Dic59semidirect
125. #493Dic59semidirect
130. #442Dic59semidirect
139. C59#11837semidirect

Maximal subgroups

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

subgroupindex
#214259082
#214242452
#214220872
#214220812
C59×#31842
#1183759