Finite Group Databasebeta

← #27171674

Subgroups — #27171674

Order 12928.

380 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 89 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#20265736C211yesmaximal, characteristic
2C2×#15692170C211yesmaximal, characteristic, complemented
3C2×#15692169C212yesmaximal, complemented
4#20264943C212yesmaximal, complemented
5#518×C101C211yesmaximal, characteristic, complemented
6C22×C22⋊D101C2211yescharacteristic
7#15692238—21no—
8#15692236C412yescomplemented
9#15692227—22no—
10#15692224—21no—
11#15692223C412yescomplemented
12C2×C22⋊Dic101C2212yes—
13C2×C22⋊Dic101C2211yescharacteristic
14#15692170—22no—
15#15692169—24no—
16C22×D4×C101C2211yescharacteristic
17C4×D4×C101—24no—
18#117×C101—21no—
19C2×C22⋊C4×C101—21no—
20C2×C22⋊C4×C101C2212yes—
21C23×D101C2×C411yescharacteristic, complemented
22C2×C22⋊D101—22no—
23C2×C22⋊D101—21no—
24C2×C22⋊D101—21no—
25C2×C22⋊D101—22no—
26C2×C22⋊D101—22no—
27C22×Dic101—21no—
28C22×Dic101—22no—
29C22×Dic101C2×C411yescharacteristic, complemented
30#13108053—22no—
31C22⋊Dic101—22no—
32C22⋊Dic101—21no—
33C22⋊Dic101—22no—
34C22⋊Dic101—22no—
35C22⋊Dic101C2×C412yescomplemented
36#13108051—22no—
37C24×C101D411yescharacteristic
38C24×C101C2311yescharacteristic
39C2×D4×C101—22no—
40C2×D4×C101—22no—
41C2×D4×C101C2×C412yes—
42C2×D4×C101D412yes—
43C22×C4×C101—22no—
44C22×C4×C101—22no—
45C22×C4×C101D411yescharacteristic
46C4⋊C4×C101—22no—
47C22⋊C4×C101—22no—
48C22⋊C4×C101—22no—
49C22⋊C4×C101D414yescomplemented
50C42×C101—22no—
51C23×C101—21no—
52C23×C101—21no—
53C23×C101—81no—
54C23×C101—81no—
55C23×C101C22⋊C412yes—
56C23×C101C22×C411yescharacteristic
57C23×C101C2×D411yescharacteristic
58C23×C101C2×D412yes—
59C23×C101C2×D411yescharacteristic
60C23×C101#4112yes—
61C22×D101—21no—
62C22×D101—22no—
63C22×D101—81no—
64D4×C101—24no—
65D4×C101—24no—
66C2×C4×C101—21no—
67C2×C4×C101—21no—
68C2×C4×C101—24no—
69C2×C4×C101—82no—
70C2×C4×C101—82no—
71C2×C4×C101C22⋊C412yes—
72C22⋊D101—41no—
73C22⋊D101—42no—
74C22⋊D101—41no—
75C22⋊D101—82no—
76C2×Dic101—22no—
77C2×Dic101—21no—
78C2×Dic101—22no—
79C2×Dic101—42no—
80C2×Dic101—81no—
81C2×Dic101—82no—
82C22×C101—22no—
83C22×C101—41no—
84C22×C101—42no—
85C22×C101—42no—
86C22×C101—41no—
87C22×C101—41no—
88C22×C101—42no—
89C22×C101—42no—
90C22×C101—81no—
91C22×C101C2×C22⋊C411yescharacteristic
92C22×C101C4×D412yes—
93C22×C101#12011yescharacteristic
94C22×C101#12112yes—
95C22×C101#12311yescharacteristic
96C2×D101—42no—
97C2×D101—41no—
98C2×D101—82no—
99C4×C101—22no—
100C4×C101—44no—
101Dic101—42no—
102Dic101—41no—
103Dic101—82no—
104C2×C101—21no—
105C2×C101—22no—
106C2×C101—21no—
107C2×C101—22no—
108C2×C101—21no—
109C2×C101—42no—
110C2×C101#38611yescharacteristic
111D101—81no—
112#1913—1011nomaximal, Sylow
113C101#191311yescharacteristic, complemented, Sylow
114#534—1011no—
115#518D10111yescharacteristic, complemented
116C2×#99—1011no—
117C2×#99—1012no—
118#342—1012no—
119C22×D4—1011no—
120C22×D4C2×D10111yescharacteristic
121#123—1012no—
122#123—2021no—
123#121—2022no—
124#120—1012no—
125#120—2021no—
126C4×D4—24no—
127#117—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—1011no—
130C2×C22⋊C4—1012no—
131C2×C22⋊C4C2×D10112yes—
132#99—2022no—
133#99—2024no—
134C24—1011no—
135C24C22⋊D10111yescharacteristic
136C24C22×D10111yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—2022no—
140C2×D4—2022no—
141C2×D4—2021no—
142C2×D4—2021no—
143C2×D4—2022no—
144C2×D4C4×D10112yes—
145C2×D4D40412yes—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—1011no—
149C22×C4—2022no—
150C22×C4—2021no—
151C22×C4C22⋊D10111yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—2022no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—1012no—
157C22⋊C4—2022no—
158C22⋊C4—2022no—
159C22⋊C4—2022no—
160C22⋊C4—2021no—
161C22⋊C4—2022no—
162C22⋊C4C22⋊D10114yescomplemented
163C42—22no—
164C23—21no—
165C23—21no—
166C23—81no—
167C23—81no—
168C23—8081no—
169C23—2022no—
170C23—2021no—
171C23#1310805112yes—
172C23C2×C4×D10111yescharacteristic
173C23C2×C22⋊D10111yescharacteristic
174C23C2×C22⋊D10112yes—
175C23C2×D40411yescharacteristic
176C23#1310807412yes—
177D4—24no—
178D4—24no—
179D4—8082no—
180D4—4041no—
181D4—4042no—
182D4—4041no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—8082no—
189C2×C4—8081no—
190C2×C4—4042no—
191C2×C4—2022no—
192C2×C4—2021no—
193C2×C4—2022no—
194C2×C4#1310805112yes—
195C22—22no—
196C22—42no—
197C22—41no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—8082no—
205C22—4042no—
206C22—4041no—
207C22C2×#1310805111yescharacteristic
208C22C4×C22⋊D10112yes—
209C22#1569222411yescharacteristic
210C22#1569222912yes—
211C22#1569223811yescharacteristic
212C4—22no—
213C4—44no—
214C4—8082no—
215C4—4042no—
216C4—4041no—
217C2—21no—
218C2—22no—
219C2—21no—
220C2—21no—
221C2—22no—
222C2—42no—
223C2—8081no—
224C2#2026505611yescharacteristic
subgroupG/H
centerC2#20265056
commutatorC23×C101C22×C4
FrattiniC24C22×D101
Fitting#518×C101C2
socleC2×C101#386
prime radical#20265736C2
cosocleC24×C101C23
hypercenter#518D101
hypocenterC101#1913
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C2×#15692170C2semidirect
3. C2×#15692169C2semidirect
4. #20264943C2semidirect
5. #518×C101C2semidirect
8. #15692236C4semidirect
11. #15692223C4semidirect
21. C23×D101C2×C4semidirect
29. C22×Dic101C2×C4semidirect
35. C22⋊Dic101C2×C4semidirect
49. C22⋊C4×C101D4semidirect
113. C101#1913semidirect
115. #518D101semidirect
162. C22⋊C4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202657362
C2×#156921702
C2×#156921692
#202649432
#518×C1012
#1913101