Finite Group Databasebeta

← #28372947

Subgroups — #28372947

Order 13952.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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#20295537C211yesmaximal, characteristic
2C2×#16812658C211yesmaximal, characteristic, complemented
3#20295368C212yesmaximal, complemented
4#20295362C211yesmaximal, characteristic, complemented
5#20295357C211yesmaximal, characteristic, complemented
6#20295351C212yesmaximal, complemented
7#20295345C211yesmaximal, characteristic, complemented
8#20295340C211yesmaximal, characteristic, complemented
9#20295064C212yesmaximal, complemented
10#20295023C212yesmaximal, complemented
11C2×C22⋊C8×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13#16812682—21no—
14#16812678C2211yescharacteristic
15#16812678C2211yescharacteristic
16#16812673C2211yescharacteristic
17#16812673C2211yescharacteristic
18#16812668—21no—
19C4×C22⋊D109—21no—
20C4×C22⋊D109—21no—
21#16812658—21no—
22#16812658—21no—
23#16812658C2212yes—
24#16812658C2212yes—
25#16812658C2211yescharacteristic
26C2×C4⋊Dic109C2211yescharacteristic
27C2×C4⋊Dic109C2211yescharacteristic
28C2×C22⋊Dic109—21no—
29C2×C4×Dic109—21no—
30#16812633C2212yescomplemented
31#16812633C2212yescomplemented
32#16812631C2212yescomplemented
33#16812631C2212yescomplemented
34#16812624C2212yescomplemented
35#16812624C2212yescomplemented
36#16812621C2212yescomplemented
37#16812621C2212yescomplemented
38C23×C4×C109C2211yescharacteristic
39C22×C8×C109C2212yes—
40C22⋊C8×C109C2211yescharacteristic
41C22⋊C8×C109C2211yescharacteristic
42C22⋊C8×C109C2212yes—
43#13182029—24no—
44C2×Dic218C2311yescharacteristic
45C2×D436C2311yescharacteristic
46C2×C22⋊D109—21no—
47C2×C22⋊D109—21no—
48C2×C22⋊D109—21no—
49C22×Dic109—21no—
50C22×Dic109—21no—
51C2×C4×D109—21no—
52C4⋊Dic109—21no—
53C4⋊Dic109—21no—
54C4⋊Dic109C2312yes—
55C4⋊Dic109C2312yes—
56C4⋊Dic109C2311yescharacteristic
57C4⋊Dic109C2311yescharacteristic
58#13182008—21no—
59#13182008—21no—
60C22⋊Dic109—22no—
61C22⋊Dic109—22no—
62C22⋊Dic109—21no—
63C22⋊Dic109—21no—
64C22⋊Dic109—21no—
65#13182006—21no—
66#13182006—21no—
67C4×Dic109—22no—
68C4×Dic109—22no—
69C4×Dic109—21no—
70C24×C109D411yescharacteristic
71C22×C4×C109—24no—
72C22×C4×C109D411yescharacteristic
73C22×C4×C109D411yescharacteristic
74C22×C4×C109D411yescharacteristic
75C22×C4×C109C2311yescharacteristic
76C22×C4×C109C2311yescharacteristic
77C22×C4×C109C2311yescharacteristic
78C2×C8×C109—24no—
79C2×C8×C109C2312yes—
80C2×C8×C109C2312yes—
81C23×C109—22no—
82C23×C109—22no—
83C23×C109—41no—
84C23×C109C2×D411yescharacteristic
85C23×C109C2×D411yescharacteristic
86C23×C109C2×D411yescharacteristic
87C22×D109—21no—
88C2×C4×C109—22no—
89C2×C4×C109—22no—
90C2×C4×C109—22no—
91C2×C4×C109—22no—
92C2×C4×C109—22no—
93C2×C4×C109C2×D411yescharacteristic
94C2×C4×C109C2×D411yescharacteristic
95C2×C4×C109C2×D411yescharacteristic
96C2×C4×C109#4114yes—
97C2×C4×C109C2411yescharacteristic
98C22⋊D109—42no—
99C22⋊D109—42no—
100C22⋊D109—42no—
101C2×Dic109—22no—
102C2×Dic109—22no—
103C2×Dic109—21no—
104C2×Dic109—21no—
105C2×Dic109—21no—
106C2×Dic109—42no—
107C2×Dic109—42no—
108D436—22no—
109C4×D109—42no—
110Dic218—22no—
111C8×C109—22no—
112C8×C109—22no—
113C22×C109—21no—
114C22×C109—21no—
115C22×C109—21no—
116C22×C109—21no—
117C22×C109—21no—
118C22×C109—21no—
119C22×C109—21no—
120C22×C109—44no—
121C22×C109#12312yes—
122C22×C109#13512yes—
123C22×C109C22×D411yescharacteristic
124C2×D109—41no—
125C2×D109—41no—
126C2×D109—41no—
127C4×C109—21no—
128C4×C109—21no—
129C4×C109#12312yes—
130C4×C109C2×#4112yes—
131Dic109—41no—
132Dic109—42no—
133Dic109—41no—
134Dic109—42no—
135Dic109—41no—
136C2×C109—22no—
137C2×C109—22no—
138C2×C109—41no—
139C2×C109C2×#12311yescharacteristic
140C2×C109C2×#13511yescharacteristic
141C2×C109#57511yescharacteristic
142D109—81no—
143#3091—1091nomaximal, Sylow
144C109#309111yescharacteristic, complemented, Sylow
145#525—1091no—
146C2×#117—1091no—
147#485—1092no—
148#484—1091no—
149#483—1091no—
150#482—1092no—
151#481—1091no—
152#480—1091no—
153#427—1092no—
154#416—1092no—
155C2×C22⋊C8D10911yescharacteristic, complemented
156C2×#41—1091no—
157C23×C4C2×D10911yescharacteristic
158C22×C8C2×D10912yes—
159#123—2181no—
160C22⋊Q8—1091no—
161C22⋊Q8—1091no—
162#121—1091no—
163#121—1091no—
164#120—2181no—
165C4×D4—2181no—
166C4×D4—2181no—
167#117—1092no—
168#117—1092no—
169#117—1091no—
170#117—2181no—
171#117—2181no—
172C2×C4⋊C4—1091no—
173C2×C4⋊C4—1091no—
174C2×C22⋊C4—2181no—
175C2×C42—2181no—
176#107—1092no—
177#107—1092no—
178#106—1092no—
179#106—1092no—
180#103—1092no—
181#103—1092no—
182#102—1092no—
183#102—1092no—
184C22⋊C8C2×D10911yescharacteristic
185C22⋊C8C2×D10911yescharacteristic
186C22⋊C8C2×D10912yes—
187C24D43611yescharacteristic
188#41—2184no—
189C2×Q8—1091no—
190C2×D4—1091no—
191C2×D4—2181no—
192C2×D4—2181no—
193C2×D4—2181no—
194C22×C4—24no—
195C22×C4—2181no—
196C22×C4—2181no—
197C22×C4—2181no—
198C22×C4D43611yescharacteristic
199C22×C4D43611yescharacteristic
200C22×C4D43611yescharacteristic
201C22×C4C22×D10911yescharacteristic
202C22×C4C22×D10911yescharacteristic
203C22×C4C22×D10911yescharacteristic
204C2×C8—24no—
205C2×C8C22×D10912yes—
206C2×C8C22×D10912yes—
207C4⋊C4—1092no—
208C4⋊C4—1092no—
209C4⋊C4—1091no—
210C4⋊C4—1091no—
211C4⋊C4—2181no—
212C4⋊C4—2181no—
213C4⋊C4—2181no—
214C4⋊C4—2181no—
215C22⋊C4—2182no—
216C22⋊C4—2182no—
217C22⋊C4—2181no—
218C22⋊C4—2181no—
219C22⋊C4—2181no—
220C22⋊C4—2181no—
221C22⋊C4—2181no—
222C42—2182no—
223C42—2182no—
224C42—2181no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—2181no—
229C23C2×D43611yescharacteristic
230C23C2×D43611yescharacteristic
231C23C2×D43611yescharacteristic
232Q8—2182no—
233D4—4362no—
234D4—4362no—
235D4—4362no—
236D4—2182no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—4362no—
243C2×C4—4362no—
244C2×C4—4362no—
245C2×C4—2182no—
246C2×C4—2182no—
247C2×C4—2181no—
248C2×C4—2181no—
249C2×C4—2181no—
250C2×C4C2×D43611yescharacteristic
251C2×C4C2×D43611yescharacteristic
252C2×C4C2×D43611yescharacteristic
253C2×C4#1318202814yes—
254C2×C4C23×D10911yescharacteristic
255C8—22no—
256C8—22no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—4361no—
266C22—4361no—
267C22—4361no—
268C22#1681268312yes—
269C22#1681272112yes—
270C22C22×D43611yescharacteristic
271C4—21no—
272C4—21no—
273C4—4361no—
274C4—4362no—
275C4—4361no—
276C4—4362no—
277C4—4361no—
278C4#1681268312yes—
279C4C2×#1318202812yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—8721no—
284C2C2×#1681268311yescharacteristic
285C2C2×#1681272111yescharacteristic
286C2#2029582611yescharacteristic
subgroupG/H
centerC2×C4C2×D436
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC2×C22⋊C8×C109C2
socleC22×C109C22×D4
prime radical#20295537C2
hypercenterC2×C22⋊C8D109
hypocenterC109#3091
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#16812658C2semidirect
3. #20295368C2semidirect
4. #20295362C2semidirect
5. #20295357C2semidirect
6. #20295351C2semidirect
7. #20295345C2semidirect
8. #20295340C2semidirect
9. #20295064C2semidirect
10. #20295023C2semidirect
11. C2×C22⋊C8×C109C2semidirect
30. #16812633C22semidirect
31. #16812633C22semidirect
32. #16812631C22semidirect
33. #16812631C22semidirect
34. #16812624C22semidirect
35. #16812624C22semidirect
36. #16812621C22semidirect
37. #16812621C22semidirect
144. C109#3091semidirect
155. C2×C22⋊C8D109semidirect

Maximal subgroups

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

subgroupindex
#202955372
C2×#168126582
#202953682
#202953622
#202953572
#202953512
#202953452
#202953402
#202950642
#202950232
C2×C22⋊C8×C1092
#3091109