Finite Group Databasebeta

← #28372828

Subgroups — #28372828

Order 13952.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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#20295579C211yesmaximal, characteristic, complemented
2C2×#16812661C211yesmaximal, characteristic
3#20295329C212yesmaximal, complemented
4#20295325C212yesmaximal, complemented
5#20295315C212yesmaximal, complemented
6#20295305C212yesmaximal, complemented
7#20295066C211yesmaximal, characteristic, complemented
8#20295049C211yesmaximal, characteristic, complemented
9#20295032C211yesmaximal, characteristic
10#20295030C211yesmaximal, characteristic
11#428×C109C211yesmaximal, characteristic, complemented
12C22×D436C2211yescharacteristic
13C22×C4×D109—21no—
14C2×#13182013C2211yescharacteristic
15#16812698C2211yescharacteristic, complemented
16#16812698C2211yescharacteristic, complemented
17#16812690C2211yescharacteristic, complemented
18#16812690C2211yescharacteristic, complemented
19#16812676—21no—
20#16812676—21no—
21C4×Dic218—21no—
22C4×Dic218—21no—
23#16812661—21no—
24#16812661—21no—
25#16812661C2212yes—
26#16812661C2212yes—
27#16812658C2211yescharacteristic
28C2×#13182006—21no—
29C2×C4×Dic109C2211yescharacteristic
30#16812632C2212yescomplemented
31#16812630C2212yescomplemented
32#16812628C2212yescomplemented
33#16812628C2212yescomplemented
34#16812621C2212yes—
35#16812621C2212yes—
36#16812619C2212yes—
37#16812619C2212yes—
38C2×M16×C109C2211yescharacteristic
39#117×C109C2211yescharacteristic
40C2×C4⋊C4×C109C2211yescharacteristic
41#107×C109C2212yescomplemented
42#106×C109C2212yescomplemented
43C23×D109—21no—
44C2×Dic218—21no—
45C2×Dic218—21no—
46C2×D436—21no—
47C2×D436—22no—
48C2×D436Q814yescomplemented
49C2×D436C2312yes—
50C22×Dic109D411yescharacteristic
51C22×Dic109D411yescharacteristic
52C2×C4×D109—22no—
53C2×C4×D109—22no—
54C2×C4×D109—44no—
55#13182013—22no—
56C2×#11724784C2312yes—
57C4⋊Dic109—21no—
58C4⋊Dic109—21no—
59C4⋊Dic109C2311yescharacteristic
60C4⋊Dic109C2311yescharacteristic
61#13182008—21no—
62#13182008—21no—
63#13182008—21no—
64#13182008—21no—
65C22⋊Dic109—21no—
66#13182006—22no—
67#13182006—22no—
68C4×Dic109—21no—
69C4×Dic109D411yescharacteristic
70C4×Dic109D411yescharacteristic
71C4×Dic109C2311yescharacteristic
72C4×Dic109C2311yescharacteristic
73C22×C4×C109—21no—
74C22×C4×C109C2311yescharacteristic
75M16×C109—22no—
76C2×C8×C109C2312yes—
77C4⋊C4×C109—21no—
78C4⋊C4×C109C2311yescharacteristic
79C4⋊C4×C109C2312yes—
80C4⋊C4×C109C2311yescharacteristic
81C22⋊C4×C109—21no—
82C42×C109—21no—
83C23×C109C2×D411yescharacteristic
84C22×D109—24no—
85C22×D109—22no—
86C22×D109—42no—
87C22×D109—42no—
88C2×C4×C109—21no—
89C2×C4×C109—21no—
90C2×C4×C109—22no—
91C2×C4×C109—42no—
92C2×C4×C109C2×D411yescharacteristic
93C2×C4×C109C2×Q812yes—
94C2×C4×C109#4112yes—
95C2×C4×C109C2411yescharacteristic
96C2×Dic109—22no—
97C2×Dic109—21no—
98C2×Dic109—21no—
99C2×Dic109—41no—
100C2×Dic109C2×D411yescharacteristic
101C2×Dic109C2×D411yescharacteristic
102C2×Dic109C2×D411yescharacteristic
103C2×Dic109C2×D411yescharacteristic
104D436—24no—
105D436—22no—
106D436C2×Q814yes—
107C4×D109—44no—
108C4×D109—44no—
109Dic218—42no—
110Dic218—42no—
111C8×C109—22no—
112#11724784—22no—
113C22×C109—21no—
114C22×C109—21no—
115C22×C109C22⋊Q812yes—
116C22×C109C22×D411yescharacteristic
117C2×D109—24no—
118C2×D109—48no—
119C2×D109—42no—
120C2×D109—42no—
121C4×C109—41no—
122C4×C109—42no—
123C4×C109—41no—
124C4×C109C22⋊Q812yes—
125C4×C109C22×Q811yescharacteristic
126C4×C109C2×#4111yescharacteristic
127Dic109—21no—
128Dic109—21no—
129Dic109—21no—
130Dic109—41no—
131Dic109—41no—
132Dic109AGL(1,ℤ/8)12yes—
133C2×C109—22no—
134C2×C109C2×C22⋊Q811yescharacteristic
135C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C109#57511yescharacteristic
137D109—44no—
138#3077—1091nomaximal, Sylow
139C109#307711yescharacteristic, complemented, Sylow
140#533—1091no—
141C2×C4×D4—1091no—
142#478—1092no—
143#478—1092no—
144#476—1092no—
145#474—1092no—
146#428—1091no—
147#428D10911yescharacteristic, complemented
148#423—1091no—
149#418—1091no—
150#418—1091no—
151C22×D4—1091no—
152C23×C4—2181no—
153C2×M16—1091no—
154C2×M16C2×D10911yescharacteristic
155C4⋊Q8—1091no—
156C4⋊Q8—1091no—
157#125—1091no—
158#125—1091no—
159C22⋊Q8—2181no—
160C22⋊Q8—2181no—
161C4×Q8—2181no—
162C4×Q8—2181no—
163C4×D4—1092no—
164C4×D4—1092no—
165C4×D4—2181no—
166C4×D4—2181no—
167#117—1091no—
168#117C2×D10911yescharacteristic
169C2×C4⋊C4C2×D10911yescharacteristic
170C2×C22⋊C4—2181no—
171C2×C42—1091no—
172#107—1092no—
173#107C2×D10912yescomplemented
174#106—1092no—
175#106C2×D10912yescomplemented
176C4⋊C8—1092no—
177C4⋊C8—1092no—
178#102—1092no—
179#102—1092no—
180#102—1092no—
181#102—1092no—
182C24—2181no—
183C2×Q8—2181no—
184C2×Q8—2181no—
185C2×D4—1094no—
186C2×D4—1092no—
187C2×D4—2181no—
188C2×D4—2182no—
189C22×C4—21no—
190C22×C4—1091no—
191C22×C4—1091no—
192C22×C4—4364no—
193C22×C4—2182no—
194C22×C4—2182no—
195C22×C4C22×D10911yescharacteristic
196M16—22no—
197M16—2182no—
198C2×C8—1092no—
199C2×C8C22×D10912yes—
200C4⋊C4—21no—
201C4⋊C4—1091no—
202C4⋊C4—1091no—
203C4⋊C4—2181no—
204C4⋊C4—2181no—
205C4⋊C4—2181no—
206C4⋊C4—2181no—
207C4⋊C4—2181no—
208C4⋊C4—2181no—
209C4⋊C4C22×D10911yescharacteristic
210C4⋊C4C22×D10911yescharacteristic
211C4⋊C4C22×D10912yes—
212C22⋊C4—21no—
213C22⋊C4—2182no—
214C22⋊C4—2182no—
215C22⋊C4—2181no—
216C42—21no—
217C42—1091no—
218C42—1091no—
219C42—1091no—
220C42—1091no—
221C42—2181no—
222C23—4362no—
223C23—4362no—
224C23—2184no—
225C23—2182no—
226C23D4×D10911yescharacteristic
227Q8—4362no—
228Q8—4362no—
229D4—1094no—
230D4—2184no—
231D4—2182no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—1091no—
237C2×C4—1091no—
238C2×C4—1091no—
239C2×C4—1091no—
240C2×C4—4364no—
241C2×C4—4364no—
242C2×C4—4361no—
243C2×C4—2182no—
244C2×C4—2181no—
245C2×C4—2181no—
246C2×C4D4×D10911yescharacteristic
247C2×C4Q8×D10912yes—
248C2×C4#1318202912yes—
249C2×C4C23×D10911yescharacteristic
250C8—22no—
251C8—2182no—
252C22—21no—
253C22—21no—
254C22—4368no—
255C22—4362no—
256C22—4362no—
257C22—2184no—
258C22#1681267712yes—
259C22C2×D4×D10911yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—1092no—
264C4—4361no—
265C4—4361no—
266C4—2181no—
267C4—2181no—
268C4—2181no—
269C4#1681267712yes—
270C4C2×Q8×D10911yescharacteristic
271C4C2×#1318202911yescharacteristic
272C2—22no—
273C2—4364no—
274C2C2×#1681267711yescharacteristic
275C2AGL(1,ℤ/8)×D10911yescharacteristic
276C2#2029582511yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#428×C109C2
socleC22×C109C22×D4
prime radicalC22×D436C22
hypercenter#428D109
hypocenterC109#3077
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #20295579C2semidirect
3. #20295329C2semidirect
4. #20295325C2semidirect
5. #20295315C2semidirect
6. #20295305C2semidirect
7. #20295066C2semidirect
8. #20295049C2semidirect
11. #428×C109C2semidirect
15. #16812698C22semidirect
16. #16812698C22semidirect
17. #16812690C22semidirect
18. #16812690C22semidirect
30. #16812632C22semidirect
31. #16812630C22semidirect
32. #16812628C22semidirect
33. #16812628C22semidirect
41. #107×C109C22semidirect
42. #106×C109C22semidirect
48. C2×D436Q8semidirect
139. C109#3077semidirect
147. #428D109semidirect
173. #107C2×D109semidirect
175. #106C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202955792
C2×#168126612
#202953292
#202953252
#202953152
#202953052
#202950662
#202950492
#202950322
#202950302
#428×C1092
#3077109