Finite Group Databasebeta

← #28372550

Subgroups — #28372550

Order 13952.

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

normal subgroups only

This group has 490 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×#16812730C211yesmaximal, characteristic, complemented
2#20295593C211yesmaximal, characteristic
3#20295504C211yesmaximal, characteristic, complemented
4#20295246C214yesmaximal, complemented
5#20295224C214yesmaximal, complemented
6#20295052C211yesmaximal, characteristic, complemented
7#20295031C211yesmaximal, characteristic, complemented
8#20295023C211yesmaximal, characteristic, complemented
9C22×SD16×C109C211yesmaximal, characteristic, complemented
10C2×#13182029C2211yescharacteristic
11#16812730—24no—
12C2×#13182019C2212yescomplemented
13C2×#13182016C2212yescomplemented
14C2×#13182013C2211yescharacteristic
15#16812695C2212yes—
16#16812686C2212yes—
17#16812670—22no—
18#16812668—22no—
19Q8×Dic109—22no—
20Q8×Dic109C2214yescomplemented
21#16812658—22no—
22#16812658C2211yescharacteristic
23#16812658C2211yescharacteristic
24#16812628C2214yescomplemented
25#16812624C2212yescomplemented
26#16812621C2212yescomplemented
27#16812620C2214yes—
28C22×Q8×C109C2211yescharacteristic
29C22×D4×C109C2211yescharacteristic
30C2×SD16×C109—24no—
31C2×SD16×C109C2214yes—
32C22×C8×C109C2211yescharacteristic
33#13182029—22no—
34#13182029—22no—
35C2×Dic218C2311yescharacteristic
36C2×D436C2311yescharacteristic
37C2×C22⋊D109—21no—
38C2×C22⋊D109—24no—
39C2×C4×D109—21no—
40#13182019—24no—
41#13182016—24no—
42#13182013—22no—
43C2×#11724784C2312yes—
44C4⋊Dic109—24no—
45C4⋊Dic109D412yes—
46C4⋊Dic109C2312yes—
47C22⋊Dic109—21no—
48C22⋊Dic109—24no—
49C22⋊Dic109D412yes—
50C4×Dic109—21no—
51C4×Dic109—24no—
52C4×Dic109C2312yes—
53C24×C109—21no—
54C2×Q8×C109—21no—
55C2×Q8×C109—22no—
56C2×Q8×C109D414yescomplemented
57C2×Q8×C109C2312yes—
58C2×D4×C109—22no—
59C2×D4×C109—21no—
60C2×D4×C109—22no—
61C2×D4×C109C2312yes—
62C22×C4×C109—21no—
63C22×C4×C109C2311yescharacteristic
64SD16×C109—28no—
65C2×C8×C109—22no—
66C2×C8×C109C2312yes—
67C23×C109—22no—
68C23×C109—42no—
69C23×C109—82no—
70C23×C109C2×D411yescharacteristic
71C22×D109—21no—
72Q8×C109—24no—
73Q8×C109—22no—
74Q8×C109C2×D414yes—
75D4×C109—22no—
76D4×C109—22no—
77D4×C109—42no—
78C2×C4×C109—24no—
79C2×C4×C109—22no—
80C2×C4×C109C2×D411yescharacteristic
81C2×C4×C109C2×D412yes—
82C2×C4×C109#4112yes—
83C2×C4×C109C2411yescharacteristic
84C22⋊D109—42no—
85C22⋊D109—84no—
86C2×Dic109—22no—
87C2×Dic109—21no—
88C2×Dic109C2×D414yes—
89D436—21no—
90D436—21no—
91C4×D109—42no—
92Dic218—21no—
93Dic218—21no—
94C8×C109—22no—
95#11724784—22no—
96C22×C109—22no—
97C22×C109—42no—
98C22×C109—44no—
99C22×C109—84no—
100C22×C109#12112yes—
101C22×C109C22×D411yescharacteristic
102C2×D109—41no—
103C2×D109—42no—
104C4×C109—24no—
105C4×C109#12112yes—
106C4×C109C22×D411yescharacteristic
107C4×C109C2×#4111yescharacteristic
108Dic109—24no—
109Dic109—41no—
110Dic109—42no—
111C2×C109—22no—
112C2×C109—82no—
113C2×C109C2×#12111yescharacteristic
114C2×C109#57712yes—
115D109—81no—
116#3043—1091nomaximal, Sylow
117C109#304311yescharacteristic, complemented, Sylow
118C2×#137—1091no—
119C22×SD16D10911yescharacteristic, complemented
120#535—1091no—
121#519—1091no—
122#464—1094no—
123#460—1094no—
124#424—1091no—
125#418—1091no—
126#416—1091no—
127C2×#41—1091no—
128C22×Q8C2×D10911yescharacteristic
129C22×D4C2×D10911yescharacteristic
130#137—2184no—
131C2×Q16—1092no—
132C2×SD16—24no—
133C2×SD16—1092no—
134C2×SD16C2×D10914yes—
135C2×M16—1091no—
136C22×C8C2×D10911yescharacteristic
137#127—1092no—
138#124—1092no—
139#121—2182no—
140#120—2182no—
141C4×Q8—1094no—
142C4×Q8—2182no—
143#117—1091no—
144#117—1091no—
145#117—2182no—
146C4⋊C8—1094no—
147#103—1092no—
148#102—1094no—
149#102—1092no—
150C24—21no—
151#41—2182no—
152#41—2182no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—1091no—
156C2×Q8C22⋊D10914yescomplemented
157C2×Q8C22×D10912yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—1091no—
162C2×D4—2184no—
163C2×D4—2181no—
164C2×D4C22×D10912yes—
165C22×C4—21no—
166C22×C4—2181no—
167C22×C4C22×D10911yescharacteristic
168Q16—2184no—
169SD16—28no—
170SD16—2184no—
171M16—2182no—
172C2×C8—22no—
173C2×C8—1092no—
174C2×C8C22×D10912yes—
175C4⋊C4—1092no—
176C4⋊C4—1092no—
177C4⋊C4—2184no—
178C22⋊C4—1092no—
179C22⋊C4—2184no—
180C22⋊C4—2181no—
181C42—1092no—
182C42—2184no—
183C42—2181no—
184C23—22no—
185C23—42no—
186C23—82no—
187C23—2181no—
188C23D4×D10911yescharacteristic
189Q8—24no—
190Q8—22no—
191Q8—2181no—
192Q8—2181no—
193Q8C2×C22⋊D10914yes—
194D4—22no—
195D4—22no—
196D4—42no—
197D4—8724no—
198D4—4362no—
199D4—2181no—
200D4—2181no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—1094no—
204C2×C4—4362no—
205C2×C4—2182no—
206C2×C4—2181no—
207C2×C4D4×D10911yescharacteristic
208C2×C4C2×C22⋊D10912yes—
209C2×C4#1318202812yes—
210C2×C4C23×D10911yescharacteristic
211C8—22no—
212C8—2182no—
213C22—22no—
214C22—42no—
215C22—44no—
216C22—84no—
217C22—4361no—
218C22—4362no—
219C22#1681267112yes—
220C22C2×D4×D10911yescharacteristic
221C4—24no—
222C4—4361no—
223C4—4362no—
224C4—2184no—
225C4#1681267112yes—
226C4C22×C22⋊D10911yescharacteristic
227C4C2×#1318202811yescharacteristic
228C2—22no—
229C2—82no—
230C2—8721no—
231C2C2×#1681267111yescharacteristic
232C2#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#3043
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×#16812730C2semidirect
3. #20295504C2semidirect
4. #20295246C2semidirect
5. #20295224C2semidirect
6. #20295052C2semidirect
7. #20295031C2semidirect
8. #20295023C2semidirect
9. C22×SD16×C109C2semidirect
12. C2×#13182019C22semidirect
13. C2×#13182016C22semidirect
20. Q8×Dic109C22semidirect
24. #16812628C22semidirect
25. #16812624C22semidirect
26. #16812621C22semidirect
56. C2×Q8×C109D4semidirect
117. C109#3043semidirect
119. C22×SD16D109semidirect
156. C2×Q8C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127302
#202955932
#202955042
#202952462
#202952242
#202950522
#202950312
#202950232
C22×SD16×C1092
#3043109