Finite Group Databasebeta

← #27204973

Subgroups — #27204973

Order 13184.

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

normal subgroups only

This group has 398 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
1#20268422C211yesmaximal, characteristic, complemented
2#20268265C211yesmaximal, characteristic
3#20268094C212yesmaximal, complemented
4#20268090C212yesmaximal, complemented
5#20268080C212yesmaximal, complemented
6#20268070C212yesmaximal, complemented
7#20267827C211yesmaximal, characteristic, complemented
8#20267819C211yesmaximal, characteristic, complemented
9#20267797C211yesmaximal, characteristic
10C2×#15692941C211yesmaximal, characteristic
11#428×C103C211yesmaximal, characteristic, complemented
12C22×D412C2211yescharacteristic
13C22×C103⋊C8C2211yescharacteristic
14#15693020—21no—
15#15693020C2212yescomplemented
16#15693012—21no—
17#15693012C2212yescomplemented
18#15693000—22no—
19C4×D412—21no—
20C4×D412—21no—
21#15692983C2212yes—
22#15692983C2212yes—
23#15692980C2211yescharacteristic
24#15692980C2211yescharacteristic
25C22⋊C4×D103—22no—
26#15692954C2211yescharacteristic, complemented
27#15692954C2211yescharacteristic, complemented
28#15692952C2211yescharacteristic, complemented
29#15692952C2211yescharacteristic, complemented
30Dic103⋊C8C2212yescomplemented
31Dic103⋊C8C2212yescomplemented
32#15692943C2212yes—
33#15692943C2212yes—
34#15692941C2212yes—
35#15692941C2212yes—
36C2×M16×C103C2211yescharacteristic
37#117×C103C2211yescharacteristic
38C2×C4⋊C4×C103C2211yescharacteristic
39#107×C103C2212yescomplemented
40#106×C103C2212yescomplemented
41C23×D103—21no—
42C2×Dic206—22no—
43C2×D412—21no—
44C2×D412—22no—
45C2×D412Q814yescomplemented
46C2×D412C2312yes—
47C2×C4×D103—22no—
48C2×C4×D103—22no—
49C2×C103⋊C8—22no—
50C2×C103⋊C8C2311yescharacteristic
51C2×C103⋊C8C2311yescharacteristic
52C4⋊Dic103—22no—
53C4⋊Dic103D411yescharacteristic
54C4⋊Dic103D411yescharacteristic
55C4⋊Dic103C2311yescharacteristic
56C4⋊Dic103C2311yescharacteristic
57Dic103⋊C4—22no—
58Dic103⋊C4—22no—
59C22⋊Dic103—21no—
60C22⋊Dic103D412yes—
61(C2×D103)⋊C4—22no—
62(C2×D103)⋊C4—22no—
63C4×Dic103—21no—
64C4×Dic103C2312yes—
65C22×C4×C103—21no—
66C22×C4×C103C2311yescharacteristic
67M16×C103—22no—
68C2×C8×C103C2312yes—
69C4⋊C4×C103—21no—
70C4⋊C4×C103C2311yescharacteristic
71C4⋊C4×C103C2312yes—
72C4⋊C4×C103C2311yescharacteristic
73C22⋊C4×C103—21no—
74C42×C103—21no—
75C23×C103C2×D411yescharacteristic
76C22×D103—22no—
77C22×D103—24no—
78C22×D103—81no—
79C22×D103—81no—
80C2×C4×C103—21no—
81C2×C4×C103—21no—
82C2×C4×C103—22no—
83C2×C4×C103—81no—
84C2×C4×C103C2×D411yescharacteristic
85C2×C4×C103C2×Q812yes—
86C2×C4×C103#4112yes—
87C2×C4×C103C2411yescharacteristic
88C2×Dic103—21no—
89C2×Dic103—21no—
90C2×Dic103C2×D412yes—
91C2×Dic103C2×D412yes—
92D412—24no—
93D412—22no—
94D412C2×Q814yes—
95C4×D103—82no—
96C4×D103—82no—
97Dic206—82no—
98C8×C103—22no—
99C103⋊C8—21no—
100C103⋊C8—21no—
101C22×C103—21no—
102C22×C103—21no—
103C22×C103C22⋊Q812yes—
104C22×C103C22×D411yescharacteristic
105C2×D103—24no—
106C2×D103—42no—
107C2×D103—42no—
108C2×D103—84no—
109C4×C103—42no—
110C4×C103—41no—
111C4×C103—41no—
112C4×C103C22⋊Q812yes—
113C4×C103C22×Q811yescharacteristic
114C4×C103C2×#4111yescharacteristic
115Dic103—22no—
116Dic103—22no—
117Dic103—41no—
118Dic103—41no—
119C2×C103—22no—
120C2×C103C2×C22⋊Q811yescharacteristic
121C2×C103#57611yescharacteristic
122C2×C103#57711yescharacteristic
123D103—44no—
124#3081—1031nomaximal, Sylow
125C103#308111yescharacteristic, complemented, Sylow
126#544—1031no—
127#518—1031no—
128#478—1032no—
129#478—1032no—
130#476—1032no—
131#474—1032no—
132#428D10311yescharacteristic, complemented
133#427—1031no—
134#424—1031no—
135#418—1031no—
136C2×#102—1031no—
137C22×D4—1031no—
138C2×M16C2×D10311yescharacteristic
139C22×C8—1031no—
140C4⋊Q8—1032no—
141C4⋊Q8—2061no—
142#125—1032no—
143#125—2061no—
144C22⋊Q8—2062no—
145C4×D4—1032no—
146C4×D4—1032no—
147C4×D4—2061no—
148C4×D4—2061no—
149#117—1031no—
150#117—1031no—
151#117C2×D10311yescharacteristic
152C2×C4⋊C4C2×D10311yescharacteristic
153C2×C22⋊C4—2062no—
154#107—1031no—
155#107—1031no—
156#107C2×D10312yescomplemented
157#106—1031no—
158#106—1031no—
159#106C2×D10312yescomplemented
160C4⋊C8—1032no—
161C4⋊C8—1032no—
162#102—1032no—
163#102—1032no—
164#102—1032no—
165#102—1032no—
166C24—2061no—
167C2×Q8—2062no—
168C2×D4—1034no—
169C2×D4—1032no—
170C2×D4—2061no—
171C2×D4—2062no—
172C22×C4—21no—
173C22×C4—2062no—
174C22×C4—2062no—
175C22×C4C22×D10311yescharacteristic
176M16—22no—
177C2×C8—1031no—
178C2×C8—1031no—
179C2×C8—2062no—
180C2×C8C22×D10312yes—
181C4⋊C4—21no—
182C4⋊C4—1031no—
183C4⋊C4—1031no—
184C4⋊C4—1031no—
185C4⋊C4—1031no—
186C4⋊C4—2062no—
187C4⋊C4—2062no—
188C4⋊C4—2062no—
189C4⋊C4C22×D10312yes—
190C4⋊C4C22×D10311yescharacteristic
191C4⋊C4C22×D10311yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—1032no—
194C22⋊C4—2062no—
195C22⋊C4—2062no—
196C22⋊C4—2061no—
197C42—21no—
198C42—1032no—
199C42—2061no—
200C23—8241no—
201C23—8241no—
202C23—2064no—
203C23—2062no—
204C23D4×D10311yescharacteristic
205Q8—8242no—
206D4—1034no—
207D4—2062no—
208D4—2064no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—81no—
213C2×C4—8242no—
214C2×C4—8242no—
215C2×C4—1032no—
216C2×C4—1032no—
217C2×C4—2061no—
218C2×C4—2061no—
219C2×C4D4×D10311yescharacteristic
220C2×C4Q8×D10312yes—
221C2×C4#1311010512yes—
222C2×C4C23×D10311yescharacteristic
223C8—22no—
224C8—2061no—
225C8—2061no—
226C22—21no—
227C22—21no—
228C22—8244no—
229C22—4122no—
230C22—4122no—
231C22—2064no—
232C22#1569299912yes—
233C22C2×D4×D10311yescharacteristic
234C4—42no—
235C4—41no—
236C4—41no—
237C4—4121no—
238C4—4121no—
239C4—2062no—
240C4—2062no—
241C4#1569299912yes—
242C4C2×Q8×D10311yescharacteristic
243C4C2×#1311010511yescharacteristic
244C2—22no—
245C2—4124no—
246C2C2×#1569299911yescharacteristic
247C2#2026859511yescharacteristic
248C2#2026860011yescharacteristic
subgroupG/H
centerC22C2×D4×D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
Fitting#428×C103C2
socleC22×C103C22×D4
prime radicalC22×D412C22
hypercenter#428D103
hypocenterC103#3081
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #20268422C2semidirect
3. #20268094C2semidirect
4. #20268090C2semidirect
5. #20268080C2semidirect
6. #20268070C2semidirect
7. #20267827C2semidirect
8. #20267819C2semidirect
11. #428×C103C2semidirect
15. #15693020C22semidirect
17. #15693012C22semidirect
26. #15692954C22semidirect
27. #15692954C22semidirect
28. #15692952C22semidirect
29. #15692952C22semidirect
30. Dic103⋊C8C22semidirect
31. Dic103⋊C8C22semidirect
39. #107×C103C22semidirect
40. #106×C103C22semidirect
45. C2×D412Q8semidirect
125. C103#3081semidirect
132. #428D103semidirect
156. #107C2×D103semidirect
159. #106C2×D103semidirect

Maximal subgroups

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

subgroupindex
#202684222
#202682652
#202680942
#202680902
#202680802
#202680702
#202678272
#202678192
#202677972
C2×#156929412
#428×C1032
#3081103