Finite Group Databasebeta

← #28509040

Subgroups — #28509040

Order 15104.

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

normal subgroups only

This group has 388 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
1C2×#16837195C212yesmaximal
2#21422267C214yesmaximal, complemented
3C59×#3138C211yesmaximal, characteristic
4C2×#14273642—22no—
5#16837195—22no—
6#16837195C2214yes—
7C59×#533C411yescharacteristic, complemented
8C2×#127×C59C2211yescharacteristic
9C59×#490C412yescomplemented
10C59×#490C2212yes—
11C59×#486—22no—
12C59×#418—22no—
13C4×M16×C59C411yescharacteristic, complemented
14C2×C22⋊Dic59—42no—
15#14273642—24no—
16#14273642—42no—
17C22×D4×C59—41no—
18C22×D4×C59D412yes—
19C2×M16×C59—21no—
20C4⋊Q8×C59—21no—
21#127×C59—41no—
22#127×C59C2×C411yescharacteristic
23#127×C59D412yes—
24#127×C59C2311yescharacteristic
25#125×C59C2×C412yescomplemented
26C22⋊Q8×C59—41no—
27C4×Q8×C59—41no—
28#117×C59—21no—
29C2×C42×C59C2×C411yescharacteristic
30#102×C59—24no—
31#102×C59—24no—
32#97×C59C2×C412yescomplemented
33C4×C8×C59—21no—
34C24×C59—22no—
35C2×Q8×C59—41no—
36C2×D4×C59—22no—
37C2×D4×C59—42no—
38C2×D4×C59—42no—
39C2×D4×C59—41no—
40C2×D4×C59—42no—
41C2×D4×C59—41no—
42C2×D4×C59—84no—
43C2×D4×C59C22⋊C412yes—
44C2×D4×C59C2×D412yes—
45C22×C4×C59—21no—
46C22×C4×C59C22⋊C411yescharacteristic
47C22×Dic59—82no—
48M16×C59—42no—
49C2×C8×C59—22no—
50C4⋊C4×C59—22no—
51C4⋊C4×C59—41no—
52C4⋊C4×C59—42no—
53C22⋊C4×C59—41no—
54C42×C59—21no—
55C42×C59—41no—
56C42×C59C22⋊C411yescharacteristic
57C42×C59C22×C411yescharacteristic
58C22⋊Dic59—44no—
59C22⋊Dic59—82no—
60C23×C59—22no—
61C23×C59—22no—
62C23×C59—42no—
63C23×C59—42no—
64C23×C59—84no—
65C23×C59#9911yescharacteristic
66Q8×C59—82no—
67D4×C59—42no—
68D4×C59—42no—
69D4×C59—82no—
70D4×C59—84no—
71D4×C59—82no—
72D4×C59—82no—
73C2×C4×C59—22no—
74C2×C4×C59—21no—
75C2×C4×C59—21no—
76C2×C4×C59—42no—
77C2×C4×C59—42no—
78C2×C4×C59#9911yescharacteristic
79C2×C4×C59C2×C22⋊C411yescharacteristic
80C2×Dic59—84no—
81C2×Dic59—164no—
82C8×C59—42no—
83#13595—591nomaximal, Sylow
84C22×C59—41no—
85C22×C59—42no—
86C22×C59—42no—
87C22×C59—82no—
88C22×C59—82no—
89C22×C59—84no—
90C22×C59—162no—
91C22×C59#35412yes—
92C22×C59C2×#9911yescharacteristic
93C4×C59—22no—
94C4×C59—21no—
95C4×C59—21no—
96C4×C59—41no—
97C4×C59—82no—
98Dic59—164no—
99#3138D5911yescharacteristic
100C2×#353—592no—
101#1412—594no—
102C2×C59—21no—
103C2×C59—22no—
104C2×C59—82no—
105C2×C59—82no—
106C2×C59C2×#35411yescharacteristic
107#533Dic5911yescharacteristic, complemented
108C2×#127C2×D5911yescharacteristic
109#490Dic5912yescomplemented
110#490C2×D5912yes—
111#486—22no—
112#418—22no—
113C2×#99—1182no—
114C4×M16Dic5911yescharacteristic, complemented
115#353—594no—
116#353—1182no—
117C59#1359511yescharacteristic, complemented, Sylow
118C22×D4—41no—
119C22×D4C22⋊D5912yes—
120C2×M16—21no—
121C4⋊Q8—21no—
122#127—41no—
123#127C2×Dic5911yescharacteristic
124#127C22⋊D5912yes—
125#127C22×D5911yescharacteristic
126#125C2×Dic5912yescomplemented
127C22⋊Q8—41no—
128C4×Q8—41no—
129#117—21no—
130C2×C22⋊C4—2362no—
131C2×C42C2×Dic5911yescharacteristic
132#102—24no—
133#102—24no—
134#99—1184no—
135#99—2362no—
136#97C2×Dic5912yescomplemented
137C4×C8—21no—
138C24—22no—
139C2×Q8—41no—
140C2×D4—22no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—41no—
144C2×D4—42no—
145C2×D4—41no—
146C2×D4—84no—
147C2×D4C22⋊Dic5912yes—
148C2×D4C2×C22⋊D5912yes—
149C22×C4—21no—
150C22×C4—4722no—
151C22×C4C22⋊Dic5911yescharacteristic
152M16—42no—
153C2×C8—22no—
154C4⋊C4—22no—
155C4⋊C4—42no—
156C4⋊C4—41no—
157C22⋊C4—41no—
158C22⋊C4—4722no—
159C22⋊C4—2364no—
160C42—21no—
161C42—41no—
162C42C22⋊Dic5911yescharacteristic
163C42C22×Dic5911yescharacteristic
164C23—22no—
165C23—22no—
166C23—42no—
167C23—42no—
168C23—84no—
169C23#1427364211yescharacteristic
170Q8—82no—
171D4—42no—
172D4—42no—
173D4—82no—
174D4—84no—
175D4—82no—
176D4—82no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—9444no—
183C2×C4—4724no—
184C2×C4#1427364211yescharacteristic
185C2×C4C2×C22⋊Dic5911yescharacteristic
186C8—42no—
187C22—41no—
188C22—42no—
189C22—42no—
190C22—82no—
191C22—82no—
192C22—84no—
193C22—162no—
194C22#1683719712yes—
195C22C2×#1427364211yescharacteristic
196C4—22no—
197C4—21no—
198C4—21no—
199C4—41no—
200C4—82no—
201C4—9444no—
202C2—21no—
203C2—22no—
204C2—82no—
205C2—82no—
206C2C2×#1683719711yescharacteristic
subgroupG/H
centerC2C2×#16837197
commutatorC42×C59C22×C4
Frattini#127C22×D59
FittingC59×#3138C2
socleC2×C59C2×#354
prime radicalC2×#127×C59C22
cosocle#127×C59C23
hypercenter#3138D59
hypocenterC59#13595
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ #3138

Complements

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

NH ≅ G/Ntype
2. #21422267C2semidirect
7. C59×#533C4semidirect
9. C59×#490C4semidirect
13. C4×M16×C59C4semidirect
25. #125×C59C2×C4semidirect
32. #97×C59C2×C4semidirect
107. #533Dic59semidirect
109. #490Dic59semidirect
114. C4×M16Dic59semidirect
117. C59#13595semidirect
126. #125C2×Dic59semidirect
136. #97C2×Dic59semidirect

Maximal subgroups

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

subgroupindex
C2×#168371952
#214222672
C59×#31382
#1359559