Finite Group Databasebeta

← #27172190

Subgroups — #27172190

Order 12928.

348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 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#20265975C211yesmaximal, characteristic
2#20265751C211yesmaximal, characteristic, complemented
3#20265193C211yesmaximal, characteristic, complemented
4#20265186C211yesmaximal, characteristic, complemented
5#20265151C211yesmaximal, characteristic, complemented
6#20264949C211yesmaximal, characteristic, complemented
7C2×#137×C101C211yesmaximal, characteristic, complemented
8C2×#13108074C2211yescharacteristic
9#15692286—21no—
10#15692285—21no—
11#15692281—21no—
12#15692280—21no—
13#15692275—22no—
14C2×#13108058C2211yescharacteristic
15#15692253—21no—
16#15692244—21no—
17#15692238—22no—
18#15692231—22no—
19#15692214C2211yescharacteristic
20#15692214C2211yescharacteristic
21#15692181—22no—
22#15692179—22no—
23#15692170—22no—
24C2×#41×C101C2211yescharacteristic
25C22×Q8×C101C2211yescharacteristic
26#137×C101—22no—
27#137×C101—22no—
28C2×Q16×C101—21no—
29C2×SD16×C101—21no—
30C2×M16×C101C2211yescharacteristic
31#13108074—21no—
32#13108074—21no—
33#13108074—21no—
34#13108074—21no—
35C2×Dic202D411yescharacteristic
36C2×D404D411yescharacteristic
37C2×C22⋊D101—21no—
38C2×C4×D101—21no—
39#13108064—22no—
40#13108062—22no—
41#13108061—22no—
42#13108059—22no—
43#13108058—21no—
44#13108058—21no—
45C2×#11717579D412yes—
46C4⋊Dic101—21no—
47C4⋊Dic101—21no—
48#13108053—24no—
49C22⋊Dic101—21no—
50C22⋊Dic101D412yes—
51#13108051—24no—
52C4×Dic101—21no—
53C4×Dic101—21no—
54#41×C101—21no—
55#41×C101—21no—
56#41×C101—21no—
57#41×C101—21no—
58C2×Q8×C101—21no—
59C2×Q8×C101—21no—
60C2×Q8×C101—21no—
61C2×Q8×C101—21no—
62C2×Q8×C101—21no—
63C2×Q8×C101D411yescharacteristic
64C2×Q8×C101D412yes—
65C2×D4×C101—21no—
66C2×D4×C101D411yescharacteristic
67C22×C4×C101—21no—
68C22×C4×C101—21no—
69C22×C4×C101C2311yescharacteristic
70Q16×C101—24no—
71SD16×C101—24no—
72M16×C101—21no—
73M16×C101D412yes—
74C2×C8×C101—21no—
75C23×C101—21no—
76C23×C101C2×D411yescharacteristic
77C22×D101—21no—
78Q8×C101—22no—
79Q8×C101—21no—
80Q8×C101—21no—
81Q8×C101—22no—
82Q8×C101—41no—
83Q8×C101—41no—
84D4×C101—21no—
85D4×C101—21no—
86D4×C101—41no—
87D4×C101—41no—
88C2×C4×C101—22no—
89C2×C4×C101—21no—
90C2×C4×C101—41no—
91C2×C4×C101—41no—
92C2×C4×C101—41no—
93C2×C4×C101—41no—
94C2×C4×C101C2×D411yescharacteristic
95C2×C4×C101C2×D411yescharacteristic
96C2×C4×C101C2×D411yescharacteristic
97C2×C4×C101C2×D411yescharacteristic
98C2×C4×C101C2×D411yescharacteristic
99C2×C4×C101#4111yescharacteristic
100C22⋊D101—41no—
101C22⋊D101—41no—
102C2×Dic101—22no—
103C2×Dic101—21no—
104C2×Dic101—41no—
105D404—21no—
106D404—21no—
107C4×D101—41no—
108C4×D101—41no—
109Dic202—21no—
110Dic202—21no—
111C8×C101—22no—
112#11717579—22no—
113C22×C101—41no—
114C22×C101—41no—
115C22×C101—81no—
116C22×C101#12011yescharacteristic
117C22×C101#12111yescharacteristic
118C22×C101#12711yescharacteristic
119C2×D101—42no—
120C2×D101—41no—
121C4×C101—41no—
122C4×C101—42no—
123C4×C101#12011yescharacteristic
124C4×C101#12011yescharacteristic
125C4×C101#12111yescharacteristic
126C4×C101#12111yescharacteristic
127Dic101—42no—
128Dic101—41no—
129Dic101—81no—
130C2×C101—21no—
131C2×C101—21no—
132C2×C101—21no—
133C2×C101—81no—
134C2×C101#39211yescharacteristic
135D101—81no—
136#2009—1011nomaximal, Sylow
137C101#200911yescharacteristic, complemented, Sylow
138#575—1011no—
139C2×#137D10111yescharacteristic, complemented
140#536—1011no—
141#421—1011no—
142#419—1011no—
143#410—1011no—
144#344—1011no—
145C2×#41—1011no—
146C2×#41C2×D10111yescharacteristic
147C22×Q8C2×D10111yescharacteristic
148#137—22no—
149#137—22no—
150#137—2021no—
151#137—2021no—
152AGL(1,ℤ/8)—2021no—
153AGL(1,ℤ/8)—2021no—
154#135—2022no—
155C2×Q16—21no—
156C2×SD16—21no—
157C2×M16—1011no—
158C2×M16C2×D10111yescharacteristic
159C4⋊Q8—2021no—
160#124—2021no—
161#123—2022no—
162C22⋊Q8—2022no—
163#117—1011no—
164#117—1011no—
165#104—2022no—
166#103—2022no—
167#99—2022no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—2021no—
173#41—2021no—
174#41—2021no—
175#41—2021no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—1011no—
182C2×Q8C22⋊D10112yes—
183C2×Q8C22⋊D10111yescharacteristic
184C2×D4—21no—
185C2×D4—1011no—
186C2×D4—2021no—
187C2×D4C22⋊D10111yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—2021no—
191C22×C4C22×D10111yescharacteristic
192Q16—24no—
193Q16—2022no—
194SD16—24no—
195SD16—2022no—
196SD16—2022no—
197D8—2022no—
198M16—21no—
199M16—2021no—
200M16—2021no—
201M16C22⋊D10112yes—
202C2×C8—21no—
203C2×C8—1012no—
204C4⋊C4—2024no—
205C4⋊C4—2021no—
206C4⋊C4—2021no—
207C22⋊C4—1012no—
208C22⋊C4—2024no—
209C22⋊C4—2021no—
210C42—2021no—
211C42—2021no—
212C23—21no—
213C23—2021no—
214C23D4×D10111yescharacteristic
215Q8—21no—
216Q8—22no—
217Q8—21no—
218Q8—22no—
219Q8—41no—
220Q8—41no—
221Q8—2021no—
222Q8—2021no—
223D4—21no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—4041no—
228D4—4041no—
229D4—2021no—
230D4—2021no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—4041no—
238C2×C4—4041no—
239C2×C4—4041no—
240C2×C4—2022no—
241C2×C4—2021no—
242C2×C4D4×D10111yescharacteristic
243C2×C4D4×D10111yescharacteristic
244C2×C4C2×C22⋊D10111yescharacteristic
245C2×C4C2×C22⋊D10111yescharacteristic
246C2×C4C2×C22⋊D10111yescharacteristic
247C2×C4#1310807311yescharacteristic
248C8—22no—
249C8—2022no—
250C22—41no—
251C22—41no—
252C22—81no—
253C22—4042no—
254C22—4041no—
255C22#1569222311yescharacteristic
256C22#1569222711yescharacteristic
257C22#1569225111yescharacteristic
258C4—41no—
259C4—42no—
260C4—8081no—
261C4—4042no—
262C4—4041no—
263C4#1569222311yescharacteristic
264C4#1569222411yescharacteristic
265C4#1569222611yescharacteristic
266C4#1569222711yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—8081no—
272C2#2026508511yescharacteristic
subgroupG/H
centerC2#20265085
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
FittingC2×#137×C101C2
socleC2×C101#392
prime radical#20265975C2
hypercenterC2×#137D101
hypocenterC101#2009
LCSG ▷ C22×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. #20265751C2semidirect
3. #20265193C2semidirect
4. #20265186C2semidirect
5. #20265151C2semidirect
6. #20264949C2semidirect
7. C2×#137×C101C2semidirect
137. C101#2009semidirect
139. C2×#137D101semidirect

Maximal subgroups

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

subgroupindex
#202659752
#202657512
#202651932
#202651862
#202651512
#202649492
C2×#137×C1012
#2009101