Finite Group Databasebeta

← #28372667

Subgroups — #28372667

Order 13952.

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

normal subgroups only

This group has 492 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×#16812705C211yesmaximal, characteristic, complemented
2#20295600C211yesmaximal, characteristic, complemented
3#20295593C211yesmaximal, characteristic
4#20295269C214yesmaximal, complemented
5#20295252C214yesmaximal, complemented
6#20295064C211yesmaximal, characteristic, complemented
7#20295035C211yesmaximal, characteristic, complemented
8#20295031C211yesmaximal, characteristic, complemented
9C22×SD16×C109C211yesmaximal, characteristic, complemented
10C2×#13182029C2211yescharacteristic
11#16812705—24no—
12C2×#13182013C2211yescharacteristic
13C2×#13182012C2212yescomplemented
14C2×C8×D109C2212yescomplemented
15#16812697—21no—
16#16812695—21no—
17#16812688—21no—
18#16812686—21no—
19#16812682—22no—
20#16812675C2214yescomplemented
21#16812670C2214yes—
22#16812668—22no—
23#16812658C2211yescharacteristic
24#16812658C2211yescharacteristic
25#16812633C2212yescomplemented
26#16812631C2212yescomplemented
27#16812623C2214yescomplemented
28#16812620C2214yes—
29C22×Q8×C109C2211yescharacteristic
30C22×D4×C109C2211yescharacteristic
31C2×SD16×C109—24no—
32C2×SD16×C109C2212yes—
33C2×SD16×C109C2212yes—
34C22×C8×C109C2211yescharacteristic
35#13182029—22no—
36#13182029—22no—
37C2×Dic218D411yescharacteristic
38C2×D436D411yescharacteristic
39C2×C22⋊D109—24no—
40C2×C22⋊D109D412yes—
41C2×C4×D109C2312yes—
42#13182013—22no—
43#13182012—24no—
44C2×#11724784C2312yes—
45C8×D109—24no—
46C4⋊Dic109C2312yes—
47C4⋊Dic109C2312yes—
48#13182008—24no—
49C22⋊Dic109—21no—
50C22⋊Dic109—21no—
51C22⋊Dic109—24no—
52#13182006—24no—
53C4×Dic109—21no—
54C4×Dic109—21no—
55C24×C109—21no—
56C2×Q8×C109—21no—
57C2×Q8×C109—22no—
58C2×Q8×C109—22no—
59C2×Q8×C109C2312yes—
60C2×D4×C109—22no—
61C2×D4×C109—21no—
62C2×D4×C109—22no—
63C2×D4×C109C2312yes—
64C22×C4×C109—21no—
65C22×C4×C109C2311yescharacteristic
66SD16×C109—24no—
67SD16×C109—24no—
68C2×C8×C109D414yescomplemented
69C2×C8×C109C2311yescharacteristic
70C2×C8×C109C2311yescharacteristic
71C23×C109—22no—
72C23×C109—42no—
73C23×C109—82no—
74C23×C109C2×D411yescharacteristic
75C22×D109C2×D412yes—
76Q8×C109—22no—
77Q8×C109—22no—
78Q8×C109—42no—
79D4×C109—22no—
80D4×C109—22no—
81D4×C109—42no—
82C2×C4×C109—22no—
83C2×C4×C109—42no—
84C2×C4×C109C2×D411yescharacteristic
85C2×C4×C109C2×D412yes—
86C2×C4×C109#4112yes—
87C2×C4×C109C2411yescharacteristic
88C22⋊D109—24no—
89C22⋊D109—84no—
90C2×Dic109—22no—
91C2×Dic109—22no—
92C2×Dic109C2×D412yes—
93D436—22no—
94C4×D109—22no—
95C4×D109—22no—
96Dic218—22no—
97C8×C109C2×D412yes—
98C8×C109C2×D412yes—
99#11724784—22no—
100C22×C109—22no—
101C22×C109—42no—
102C22×C109—44no—
103C22×C109—84no—
104C22×C109#12112yes—
105C22×C109C22×D411yescharacteristic
106C2×D109—22no—
107C2×D109—24no—
108C4×C109—42no—
109C4×C109#12112yes—
110C4×C109C22×D411yescharacteristic
111C4×C109C2×#4111yescharacteristic
112Dic109—22no—
113Dic109—42no—
114Dic109—42no—
115C2×C109—22no—
116C2×C109—82no—
117C2×C109C2×#12111yescharacteristic
118C2×C109#57712yes—
119D109—42no—
120#3057—1091nomaximal, Sylow
121C109#305711yescharacteristic, complemented, Sylow
122C22×SD16D10911yescharacteristic, complemented
123C2×#131—1091no—
124#536—1091no—
125#535—1091no—
126#468—1094no—
127#465—1094no—
128#427—1091no—
129#419—1091no—
130#418—1091no—
131C2×#41—1091no—
132C22×Q8C2×D10911yescharacteristic
133C22×D4C2×D10911yescharacteristic
134C2×SD16—24no—
135C2×SD16C2×D10912yes—
136C2×SD16C2×D10912yes—
137#131—2184no—
138C2×M16—1091no—
139C2×M16—1092no—
140C22×C8—1092no—
141C22×C8C2×D10911yescharacteristic
142C4⋊Q8—2181no—
143#127—2181no—
144#124—2181no—
145#124—2181no—
146#123—2182no—
147C22⋊Q8—1094no—
148#121—1094no—
149#120—2182no—
150#117—1091no—
151#117—1091no—
152#107—1092no—
153#106—1092no—
154#103—1094no—
155#102—1094no—
156C24—21no—
157#41—2182no—
158#41—2182no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—1091no—
163C2×Q8C22×D10912yes—
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—1092no—
168C2×D4—1091no—
169C2×D4—2184no—
170C2×D4C22×D10912yes—
171C22×C4—21no—
172C22×C4—1092no—
173C22×C4C22×D10911yescharacteristic
174SD16—24no—
175SD16—24no—
176M16—2182no—
177M16—2184no—
178C2×C8—1092no—
179C2×C8—2184no—
180C2×C8C22⋊D10914yescomplemented
181C2×C8C22×D10911yescharacteristic
182C2×C8C22×D10911yescharacteristic
183C4⋊C4—1092no—
184C4⋊C4—1092no—
185C4⋊C4—2184no—
186C22⋊C4—2184no—
187C22⋊C4—2184no—
188C22⋊C4—2181no—
189C22⋊C4—2181no—
190C42—2181no—
191C42—2181no—
192C23—22no—
193C23—42no—
194C23—82no—
195C23—1092no—
196C23D4×D10911yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—42no—
200Q8—2182no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—8724no—
205D4—2184no—
206D4—2182no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—1092no—
210C2×C4—2182no—
211C2×C4—2182no—
212C2×C4—2182no—
213C2×C4—2182no—
214C2×C4D4×D10911yescharacteristic
215C2×C4C2×C22⋊D10912yes—
216C2×C4#1318202812yes—
217C2×C4C23×D10911yescharacteristic
218C8—2182no—
219C8C2×C22⋊D10912yes—
220C8C2×C22⋊D10912yes—
221C22—22no—
222C22—42no—
223C22—44no—
224C22—84no—
225C22—2182no—
226C22—2184no—
227C22#1681267012yes—
228C22C2×D4×D10911yescharacteristic
229C4—42no—
230C4—4362no—
231C4—4362no—
232C4—2182no—
233C4#1681267012yes—
234C4C22×C22⋊D10911yescharacteristic
235C4C2×#1318202811yescharacteristic
236C2—22no—
237C2—82no—
238C2—4362no—
239C2C2×#1681267011yescharacteristic
240C2#2029583212yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC22×SD16×C109C2
socleC22×C109C22×D4
prime radical#20295593C2
hypercenterC22×SD16D109
hypocenterC109#3057
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16812705C2semidirect
2. #20295600C2semidirect
4. #20295269C2semidirect
5. #20295252C2semidirect
6. #20295064C2semidirect
7. #20295035C2semidirect
8. #20295031C2semidirect
9. C22×SD16×C109C2semidirect
13. C2×#13182012C22semidirect
14. C2×C8×D109C22semidirect
20. #16812675C22semidirect
25. #16812633C22semidirect
26. #16812631C22semidirect
27. #16812623C22semidirect
68. C2×C8×C109D4semidirect
121. C109#3057semidirect
122. C22×SD16D109semidirect
180. C2×C8C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127052
#202956002
#202955932
#202952692
#202952522
#202950642
#202950352
#202950312
C22×SD16×C1092
#3057109