Finite Group Databasebeta

← #28372525

Subgroups — #28372525

Order 13952.

500 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812732C211yesmaximal, characteristic, complemented
2C2×#16812725C211yesmaximal, characteristic
3C2×#16812686C211yesmaximal, characteristic, complemented
4#20295494C211yesmaximal, characteristic, complemented
5#20295247C214yesmaximal, complemented
6#20295240C214yesmaximal, complemented
7#20295049C211yesmaximal, characteristic, complemented
8#20295034C211yesmaximal, characteristic, complemented
9#418×C109C211yesmaximal, characteristic, complemented
10C2×#13182029C2211yescharacteristic
11C22×Dic218C2211yescharacteristic
12#16812732—24no—
13#16812725—24no—
14C2×Dic436C2212yescomplemented
15C2×#13182018C2212yescomplemented
16C2×#13182017C2212yes—
17C2×#13182014C2212yes—
18C2×#13182013C2211yescharacteristic
19#16812686—21no—
20#16812686—21no—
21#16812686C2214yes—
22#16812664C2212yescomplemented
23#16812661C2212yescomplemented
24#16812659—22no—
25#16812656—21no—
26C2×C22⋊Dic109—22no—
27C2×C4×Dic109C2211yescharacteristic
28C42×D109—21no—
29#16812628C2214yescomplemented
30#16812622C2214yescomplemented
31C22×D4×C109C2211yescharacteristic
32C2×M16×C109C2211yescharacteristic
33#117×C109C2211yescharacteristic
34#102×C109C2214yes—
35#13182029—22no—
36#13182029D414yes—
37C2×Dic218—22no—
38C2×Dic218—22no—
39C2×Dic218—21no—
40C2×Dic218C2312yes—
41C2×Dic218C2311yescharacteristic
42C2×D436C2311yescharacteristic
43C2×C22⋊D109—21no—
44C22×Dic109—21no—
45C22×Dic109D412yes—
46C2×C4×D109—21no—
47C2×C4×D109—22no—
48Dic436—24no—
49#13182018—24no—
50#13182017—24no—
51#13182014—24no—
52#13182013—22no—
53C2×#11724784C2312yes—
54#13182008—22no—
55C22⋊Dic109—24no—
56C22⋊Dic109—24no—
57#13182006—22no—
58C4×Dic109—22no—
59C4×Dic109D411yescharacteristic
60C4×Dic109D411yescharacteristic
61C4×Dic109C2312yes—
62C24×C109—21no—
63C2×D4×C109—22no—
64C2×D4×C109—21no—
65C2×D4×C109—22no—
66C2×D4×C109C2312yes—
67C22×C4×C109C2311yescharacteristic
68M16×C109—22no—
69C2×C8×C109C2312yes—
70C4⋊C4×C109C2312yes—
71C22⋊C4×C109—21no—
72C42×C109—21no—
73C23×C109—22no—
74C23×C109—44no—
75C23×C109—42no—
76C23×C109C2×D411yescharacteristic
77C22×D109—21no—
78D4×C109—22no—
79D4×C109—22no—
80D4×C109—42no—
81C2×C4×C109—22no—
82C2×C4×C109C2×D411yescharacteristic
83C2×C4×C109C2×D412yes—
84C2×C4×C109#4112yes—
85C2×C4×C109C2411yescharacteristic
86C22⋊D109—24no—
87C2×Dic109—21no—
88C2×Dic109—22no—
89C2×Dic109—42no—
90C2×Dic109—42no—
91C2×Dic109C2×D412yes—
92C2×Dic109C2×D412yes—
93D436—21no—
94D436C2×D412yes—
95C4×D109—24no—
96C4×D109—44no—
97Dic218—21no—
98Dic218—22no—
99Dic218—22no—
100Dic218—42no—
101Dic218C2×D412yes—
102C8×C109—22no—
103#11724784—22no—
104C22×C109—22no—
105C22×C109—42no—
106C22×C109—44no—
107C22×C109—84no—
108C22×C109#12112yes—
109C22×C109C22×D411yescharacteristic
110C2×D109—22no—
111C2×D109—42no—
112C4×C109—42no—
113C4×C109#12112yes—
114C4×C109C22×D411yescharacteristic
115C4×C109C2×#4111yescharacteristic
116Dic109—22no—
117Dic109—22no—
118Dic109—22no—
119Dic109—42no—
120C2×C109—22no—
121C2×C109—82no—
122C2×C109C2×#12111yescharacteristic
123C2×C109#57512yes—
124D109—42no—
125#3040—1091nomaximal, Sylow
126C109#304011yescharacteristic, complemented, Sylow
127C2×#137—1091no—
128C2×AGL(1,ℤ/8)—1091no—
129C2×#124—1091no—
130C4×#41—1091no—
131#464—1094no—
132#463—1094no—
133#423—1091no—
134#419—1091no—
135#418D10911yescharacteristic, complemented
136C2×#41—1091no—
137C22×Q8—1091no—
138C22×D4C2×D10911yescharacteristic
139#137—2184no—
140AGL(1,ℤ/8)—2184no—
141C2×Q16—1092no—
142C2×SD16—1092no—
143C2×SD16—1092no—
144C2×D8—1092no—
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—2181no—
154#117C2×D10911yescharacteristic
155C2×C22⋊C4—2182no—
156C2×C42—1091no—
157C2×C42—2181no—
158C4⋊C8—1094no—
159#103—1094no—
160#102C2×D10914yes—
161C24—21no—
162#41—1094no—
163#41—2182no—
164C2×Q8—1091no—
165C2×Q8—1092no—
166C2×Q8—2182no—
167C2×Q8—2182no—
168C2×Q8—2181no—
169C2×D4—22no—
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—1091no—
173C2×D4—2181no—
174C2×D4C22×D10912yes—
175C22×C4—1092no—
176C22×C4—2182no—
177C22×C4—2181no—
178C22×C4—2181no—
179C22×C4C22×D10911yescharacteristic
180Q16—2184no—
181SD16—2184no—
182SD16—2184no—
183D8—2184no—
184M16—22no—
185M16—2182no—
186C2×C8—1092no—
187C2×C8C22×D10912yes—
188C4⋊C4—2182no—
189C4⋊C4C22×D10912yes—
190C22⋊C4—21no—
191C22⋊C4—2182no—
192C22⋊C4—2184no—
193C22⋊C4—2184no—
194C42—21no—
195C42—1091no—
196C42—1091no—
197C42—1092no—
198C42—2182no—
199C23—22no—
200C23—44no—
201C23—42no—
202C23—2181no—
203C23D4×D10911yescharacteristic
204Q8—1092no—
205Q8—4362no—
206Q8—2182no—
207Q8—2182no—
208Q8—2181no—
209D4—22no—
210D4—22no—
211D4—42no—
212D4—1092no—
213D4—2184no—
214D4—2181no—
215C2×C4—22no—
216C2×C4—1092no—
217C2×C4—1092no—
218C2×C4—4364no—
219C2×C4—4362no—
220C2×C4—4362no—
221C2×C4—2182no—
222C2×C4—2181no—
223C2×C4—2184no—
224C2×C4D4×D10911yescharacteristic
225C2×C4D4×D10912yes—
226C2×C4#1318202912yes—
227C2×C4C23×D10911yescharacteristic
228C8—22no—
229C8—2182no—
230C22—22no—
231C22—42no—
232C22—44no—
233C22—84no—
234C22—4362no—
235C22—2182no—
236C22#1681267212yes—
237C22C2×D4×D10911yescharacteristic
238C4—42no—
239C4—4362no—
240C4—2182no—
241C4—2182no—
242C4—2182no—
243C4#1681267212yes—
244C4C2×D4×D10911yescharacteristic
245C4C2×#1318202911yescharacteristic
246C2—22no—
247C2—82no—
248C2—4362no—
249C2C2×#1681267211yescharacteristic
250C2#2029582212yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#418×C109C2
socleC22×C109C22×D4
prime radicalC2×#16812725C2
hypercenter#418D109
hypocenterC109#3040
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#16812732C2semidirect
3. C2×#16812686C2semidirect
4. #20295494C2semidirect
5. #20295247C2semidirect
6. #20295240C2semidirect
7. #20295049C2semidirect
8. #20295034C2semidirect
9. #418×C109C2semidirect
14. C2×Dic436C22semidirect
15. C2×#13182018C22semidirect
22. #16812664C22semidirect
23. #16812661C22semidirect
29. #16812628C22semidirect
30. #16812622C22semidirect
126. C109#3040semidirect
135. #418D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127322
C2×#168127252
C2×#168126862
#202954942
#202952472
#202952402
#202950492
#202950342
#418×C1092
#3040109