Finite Group Databasebeta

← #28372524

Subgroups — #28372524

Order 13952.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812729C211yesmaximal, characteristic, complemented
2C2×#16812686C211yesmaximal, characteristic, complemented
3#41×Dic109C211yesmaximal, characteristic, complemented
4#20295246C214yesmaximal, complemented
5#20295239C214yesmaximal, complemented
6#20295049C211yesmaximal, characteristic, complemented
7#20295036C211yesmaximal, characteristic, complemented
8#20295031C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C109C211yesmaximal, characteristic
10C22×Dic218C2211yescharacteristic
11#16812729—24no—
12C2×#13182019C2212yescomplemented
13C2×#13182017C2212yescomplemented
14C2×#13182013C2211yescharacteristic
15#16812686—21no—
16#16812686—21no—
17#16812686C2214yes—
18Q8×Dic109C2212yescomplemented
19D4×Dic109—22no—
20D4×Dic109C2212yescomplemented
21#16812658—21no—
22#16812658C2211yescharacteristic
23C2×C22⋊Dic109—22no—
24C2×C4×Dic109—21no—
25C2×C4×Dic109C2211yescharacteristic
26#16812628C2214yescomplemented
27#16812624C2214yescomplemented
28#16812620C2214yes—
29C2×#41×C109C2211yescharacteristic
30C22×D4×C109C2211yescharacteristic
31AGL(1,ℤ/8)×C109—24no—
32C2×SD16×C109C2212yes—
33C2×D8×C109C2212yes—
34C2×M16×C109C2211yescharacteristic
35C2×Dic218—22no—
36C2×Dic218—22no—
37C2×Dic218—21no—
38C2×Dic218C2312yes—
39C22×Dic109—21no—
40C22×Dic109—22no—
41C22×Dic109D412yes—
42#13182019—24no—
43#13182017—24no—
44#13182013—22no—
45C2×#11724784C2312yes—
46C4⋊Dic109—22no—
47C4⋊Dic109C2312yes—
48C22⋊Dic109—21no—
49C22⋊Dic109—24no—
50C22⋊Dic109—24no—
51C22⋊Dic109—22no—
52C4×Dic109—21no—
53C4×Dic109—22no—
54C4×Dic109D411yescharacteristic
55C4×Dic109D411yescharacteristic
56C4×Dic109C2312yes—
57C24×C109—21no—
58#41×C109—22no—
59#41×C109D414yes—
60C2×Q8×C109C2311yescharacteristic
61C2×D4×C109—21no—
62C2×D4×C109—22no—
63C2×D4×C109—21no—
64C2×D4×C109—22no—
65C2×D4×C109C2312yes—
66C2×D4×C109C2311yescharacteristic
67C22×C4×C109—21no—
68C22×C4×C109C2311yescharacteristic
69SD16×C109—24no—
70D8×C109—24no—
71M16×C109—22no—
72C2×C8×C109C2312yes—
73C23×C109—21no—
74C23×C109—22no—
75C23×C109—44no—
76C23×C109—42no—
77C23×C109C2×D411yescharacteristic
78Q8×C109—21no—
79Q8×C109C2×D412yes—
80D4×C109—24no—
81D4×C109—22no—
82D4×C109—21no—
83D4×C109—22no—
84D4×C109—42no—
85D4×C109C2×D412yes—
86C2×C4×C109—21no—
87C2×C4×C109—24no—
88C2×C4×C109C2×D411yescharacteristic
89C2×C4×C109C2×D412yes—
90C2×C4×C109#4112yes—
91C2×C4×C109C2411yescharacteristic
92C2×Dic109—22no—
93C2×Dic109—22no—
94C2×Dic109—42no—
95C2×Dic109—44no—
96C2×Dic109—42no—
97C2×Dic109C2×D412yes—
98C2×Dic109C2×D412yes—
99Dic218—22no—
100Dic218—22no—
101Dic218—42no—
102C8×C109—22no—
103#11724784—22no—
104C22×C109—22no—
105C22×C109—22no—
106C22×C109—42no—
107C22×C109—42no—
108C22×C109—44no—
109C22×C109—84no—
110C22×C109#12112yes—
111C22×C109C22×D411yescharacteristic
112C4×C109—22no—
113C4×C109#12112yes—
114C4×C109C22×D411yescharacteristic
115C4×C109C2×#4111yescharacteristic
116Dic109—22no—
117Dic109—22no—
118Dic109—42no—
119Dic109—42no—
120C2×C109—22no—
121C2×C109—42no—
122C2×C109—82no—
123C2×C109C2×#12111yescharacteristic
124C2×C109#57512yes—
125#3040—1091nomaximal, Sylow
126C109#304011yescharacteristic, complemented, Sylow
127C2×#137—1091no—
128C2×AGL(1,ℤ/8)D10911yescharacteristic
129C2×#124—1091no—
130C4×#41—1091no—
131#464—1094no—
132#463—1094no—
133#423—1091no—
134#419—1091no—
135#418—1091no—
136C2×#41C2×D10911yescharacteristic
137C22×Q8—1091no—
138C22×D4C2×D10911yescharacteristic
139#137—2184no—
140AGL(1,ℤ/8)—24no—
141C2×Q16—1092no—
142C2×SD16—1092no—
143C2×SD16C2×D10912yes—
144C2×D8C2×D10912yes—
145C2×M16—1091no—
146C2×M16C2×D10911yescharacteristic
147#124—1094no—
148#124—2181no—
149#124—2181no—
150C4×Q8—1092no—
151C4×D4—1092no—
152C4×D4—2182no—
153#117—1091no—
154#117—2181no—
155C2×C22⋊C4—2182no—
156C2×C42—1091no—
157C2×C42—2181no—
158C4⋊C8—1094no—
159#103—1094no—
160#102—1094no—
161C24—21no—
162#41—22no—
163#41C22⋊D10914yes—
164C2×Q8—1092no—
165C2×Q8—2182no—
166C2×Q8—2182no—
167C2×Q8—2181no—
168C2×Q8C22×D10911yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4C22×D10912yes—
174C2×D4C22×D10911yescharacteristic
175C22×C4—21no—
176C22×C4—1092no—
177C22×C4—2182no—
178C22×C4—2181no—
179C22×C4C22×D10911yescharacteristic
180Q16—2184no—
181SD16—24no—
182SD16—2184no—
183D8—24no—
184M16—22no—
185M16—2182no—
186C2×C8—1092no—
187C2×C8C22×D10912yes—
188C4⋊C4—1092no—
189C4⋊C4—2182no—
190C22⋊C4—2184no—
191C22⋊C4—2184no—
192C22⋊C4—2182no—
193C22⋊C4—2181no—
194C42—1091no—
195C42—1091no—
196C42—1092no—
197C42—2182no—
198C42—2181no—
199C23—21no—
200C23—22no—
201C23—44no—
202C23—42no—
203C23D4×D10911yescharacteristic
204Q8—21no—
205Q8—4362no—
206Q8—2182no—
207Q8—2182no—
208Q8C2×C22⋊D10912yes—
209D4—24no—
210D4—22no—
211D4—21no—
212D4—22no—
213D4—42no—
214D4C2×C22⋊D10912yes—
215C2×C4—21no—
216C2×C4—24no—
217C2×C4—1092no—
218C2×C4—1092no—
219C2×C4—4362no—
220C2×C4—4364no—
221C2×C4—4362no—
222C2×C4—2182no—
223C2×C4—2182no—
224C2×C4D4×D10911yescharacteristic
225C2×C4C2×C22⋊D10912yes—
226C2×C4#1318202812yes—
227C2×C4C23×D10911yescharacteristic
228C8—22no—
229C8—2182no—
230C22—22no—
231C22—22no—
232C22—42no—
233C22—42no—
234C22—44no—
235C22—84no—
236C22#1681267112yes—
237C22C2×D4×D10911yescharacteristic
238C4—22no—
239C4—4362no—
240C4—4362no—
241C4—2182no—
242C4—2182no—
243C4#1681267112yes—
244C4C22×C22⋊D10911yescharacteristic
245C4C2×#1318202811yescharacteristic
246C2—22no—
247C2—42no—
248C2—82no—
249C2C2×#1681267111yescharacteristic
250C2#2029582212yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting, prime radicalC2×AGL(1,ℤ/8)×C109C2
socleC22×C109C22×D4
hypercenterC2×AGL(1,ℤ/8)D109
hypocenterC109#3040
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16812729C2semidirect
2. C2×#16812686C2semidirect
3. #41×Dic109C2semidirect
4. #20295246C2semidirect
5. #20295239C2semidirect
6. #20295049C2semidirect
7. #20295036C2semidirect
8. #20295031C2semidirect
12. C2×#13182019C22semidirect
13. C2×#13182017C22semidirect
18. Q8×Dic109C22semidirect
20. D4×Dic109C22semidirect
26. #16812628C22semidirect
27. #16812624C22semidirect
126. C109#3040semidirect

Maximal subgroups

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

subgroupindex
C2×#168127292
C2×#168126862
#41×Dic1092
#202952462
#202952392
#202950492
#202950362
#202950312
C2×AGL(1,ℤ/8)×C1092
#3040109