Finite Group Databasebeta

← #27172158

Subgroups — #27172158

Order 12928.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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#20265741C211yesmaximal, characteristic
2C2×#15692244C211yesmaximal, characteristic, complemented
3C2×#15692182C211yesmaximal, characteristic, complemented
4#20265186C211yesmaximal, characteristic, complemented
5C2×#15692171C211yesmaximal, characteristic, complemented
6#20264938C211yesmaximal, characteristic, complemented
7C2×#137×C101C211yesmaximal, characteristic, complemented
8C22×D404C2211yescharacteristic
9C2×#13108058C2211yescharacteristic
10#15692251—21no—
11#15692244—22no—
12#15692244—21no—
13#15692244—22no—
14#15692229—22no—
15#15692223—22no—
16#15692214C2211yescharacteristic
17C2×#13108051—22no—
18C2×C4×Dic101C2211yescharacteristic
19#15692182—22no—
20#15692179—22no—
21#15692171—22no—
22C2×#41×C101C2211yescharacteristic
23C22×Q8×C101C2211yescharacteristic
24#137×C101—22no—
25#137×C101—22no—
26C2×Q16×C101—21no—
27C2×SD16×C101—21no—
28C2×M16×C101C2211yescharacteristic
29C23×D101D412yes—
30C2×D404—21no—
31C2×D404—21no—
32C2×D404—21no—
33C2×D404—21no—
34C2×D404—21no—
35C2×D404—21no—
36C2×C22⋊D101—24no—
37C22×Dic101—21no—
38#13108058—21no—
39#13108058—21no—
40C2×#11717579—21no—
41C4⋊Dic101D412yes—
42C22⋊Dic101—21no—
43#13108051—28no—
44#13108051—24no—
45C4×Dic101—21no—
46C4×Dic101—21no—
47C4×Dic101D412yescomplemented
48#41×C101—21no—
49#41×C101—21no—
50#41×C101—21no—
51#41×C101—21no—
52C2×Q8×C101—21no—
53C2×Q8×C101—21no—
54C2×Q8×C101—21no—
55C2×Q8×C101—21no—
56C2×Q8×C101—21no—
57C2×Q8×C101D411yescharacteristic
58C2×Q8×C101D412yescomplemented
59C2×D4×C101—21no—
60C2×D4×C101D411yescharacteristic
61C22×C4×C101—21no—
62C22×C4×C101—21no—
63C22×C4×C101C2311yescharacteristic
64Q16×C101—24no—
65SD16×C101—24no—
66M16×C101—21no—
67M16×C101D412yescomplemented
68C2×C8×C101—21no—
69C23×C101—21no—
70C23×C101C2×D411yescharacteristic
71C22×D101—22no—
72C22×D101—22no—
73C22×D101—22no—
74C22×D101—44no—
75Q8×C101—22no—
76Q8×C101—21no—
77Q8×C101—22no—
78Q8×C101—21no—
79Q8×C101—41no—
80Q8×C101—41no—
81D4×C101—21no—
82D4×C101—21no—
83D4×C101—41no—
84D4×C101—41no—
85C2×C4×C101—21no—
86C2×C4×C101—22no—
87C2×C4×C101—41no—
88C2×C4×C101—41no—
89C2×C4×C101—41no—
90C2×C4×C101—41no—
91C2×C4×C101C2×D411yescharacteristic
92C2×C4×C101C2×D411yescharacteristic
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2×D411yescharacteristic
95C2×C4×C101C2×D411yescharacteristic
96C2×C4×C101#4111yescharacteristic
97C22⋊D101—84no—
98C2×Dic101—22no—
99C2×Dic101—22no—
100C2×Dic101—42no—
101D404—41no—
102D404—41no—
103D404—41no—
104D404—41no—
105C8×C101—22no—
106#11717579—41no—
107C22×C101—22no—
108C22×C101—41no—
109C22×C101—42no—
110C22×C101#12011yescharacteristic
111C22×C101#12011yescharacteristic
112C22×C101#12111yescharacteristic
113C2×D101—44no—
114C2×D101—42no—
115C2×D101—44no—
116C2×D101—82no—
117C4×C101—41no—
118C4×C101—42no—
119C4×C101#12011yescharacteristic
120C4×C101#12111yescharacteristic
121C4×C101#12111yescharacteristic
122C4×C101#12711yescharacteristic
123Dic101—42no—
124Dic101—42no—
125C2×C101—21no—
126C2×C101—21no—
127C2×C101—81no—
128C2×C101#39211yescharacteristic
129C2×C101#45512yes—
130D101—82no—
131#2004—1011nomaximal, Sylow
132C101#200411yescharacteristic, complemented, Sylow
133C2×#137D10111yescharacteristic, complemented
134#535—1011no—
135C2×#124—1011no—
136C2×#104—1011no—
137#419—1011no—
138C2×#100—1011no—
139#339—1011no—
140C2×#41C2×D10111yescharacteristic
141C22×Q8C2×D10111yescharacteristic
142C22×D4—1011no—
143#137—22no—
144#137—22no—
145C2×Q16—21no—
146C2×SD16—21no—
147C2×M16—1011no—
148C2×M16C2×D10111yescharacteristic
149#127—2021no—
150#124—2022no—
151#124—2021no—
152#124—2022no—
153#121—2022no—
154#120—2022no—
155#117—1011no—
156C2×C22⋊C4—2022no—
157C2×C42—1011no—
158#104—2022no—
159#103—2022no—
160#100—2022no—
161C24—1012no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22⋊D10111yescharacteristic
172C2×Q8C22⋊D10112yescomplemented
173C2×D4—21no—
174C2×D4—2024no—
175C2×D4—2021no—
176C2×D4—2021no—
177C2×D4—2021no—
178C2×D4—2021no—
179C2×D4—2021no—
180C2×D4—2021no—
181C2×D4C22⋊D10111yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—2021no—
185C22×C4C22×D10111yescharacteristic
186Q16—24no—
187SD16—24no—
188M16—21no—
189M16—2021no—
190M16—2021no—
191M16C22⋊D10112yescomplemented
192C2×C8—21no—
193C2×C8—2021no—
194C4⋊C4—1012no—
195C22⋊C4—2028no—
196C22⋊C4—2024no—
197C22⋊C4—2021no—
198C42—1012no—
199C42—2021no—
200C42—2021no—
201C23—21no—
202C23—4044no—
203C23—2022no—
204C23—2022no—
205C23—2022no—
206C23D4×D10111yescharacteristic
207Q8—22no—
208Q8—21no—
209Q8—22no—
210Q8—21no—
211Q8—41no—
212Q8—41no—
213D4—21no—
214D4—21no—
215D4—41no—
216D4—41no—
217D4—8084no—
218D4—4041no—
219D4—4041no—
220D4—4041no—
221D4—4041no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—4042no—
229C2×C4—2022no—
230C2×C4—2022no—
231C2×C4D4×D10111yescharacteristic
232C2×C4D4×D10111yescharacteristic
233C2×C4C2×C22⋊D10111yescharacteristic
234C2×C4C2×C22⋊D10111yescharacteristic
235C2×C4C2×C22⋊D10111yescharacteristic
236C2×C4#1310807311yescharacteristic
237C8—22no—
238C8—4041no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—8082no—
243C22—4044no—
244C22—4042no—
245C22—4044no—
246C22#1569222311yescharacteristic
247C22#1569222311yescharacteristic
248C22#1569222611yescharacteristic
249C4—41no—
250C4—42no—
251C4—4042no—
252C4—4042no—
253C4#1569222411yescharacteristic
254C4#1569222711yescharacteristic
255C4#1569222711yescharacteristic
256C4#1569225111yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—8082no—
261C2#2026508511yescharacteristic
262C2#2026535312yes—
subgroupG/H
centerC22#15692223
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
FittingC2×#137×C101C2
socleC22×C101#120
prime radical#20265741C2
hypercenterC2×#137D101
hypocenterC101#2004
LCSG ▷ C22×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#15692244C2semidirect
3. C2×#15692182C2semidirect
4. #20265186C2semidirect
5. C2×#15692171C2semidirect
6. #20264938C2semidirect
7. C2×#137×C101C2semidirect
47. C4×Dic101D4semidirect
58. C2×Q8×C101D4semidirect
67. M16×C101D4semidirect
132. C101#2004semidirect
133. C2×#137D101semidirect
172. C2×Q8C22⋊D101semidirect
191. M16C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202657412
C2×#156922442
C2×#156921822
#202651862
C2×#156921712
#202649382
C2×#137×C1012
#2004101