Finite Group Databasebeta

← #28372955

Subgroups — #28372955

Order 13952.

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

normal subgroups only

This group has 458 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
1#20295537C211yesmaximal, characteristic
2#20295361C212yesmaximal, complemented
3#20295356C212yesmaximal, complemented
4#20295344C212yesmaximal, complemented
5#20295339C212yesmaximal, complemented
6#20295066C212yesmaximal, complemented
7#20295025C212yesmaximal, complemented
8#20294990C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C109C211yesmaximal, characteristic, complemented
10C2×#13182029C2211yescharacteristic
11C2×#13182013C2212yes—
12#16812682—21no—
13#16812678C2211yescharacteristic
14#16812678C2211yescharacteristic
15#16812673C2211yescharacteristic
16#16812673C2211yescharacteristic
17#16812668—21no—
18C4×C22⋊D109—21no—
19C4×C22⋊D109—21no—
20#16812658C2211yescharacteristic
21#16812632C2214yescomplemented
22#16812630C2214yescomplemented
23#16812622C2214yescomplemented
24#16812619C2214yescomplemented
25C22⋊#11724784C2212yes—
26C22⋊#11724784C2212yes—
27C23×C4×C109—21no—
28C23×C4×C109C2211yescharacteristic
29C2×C4⋊C4×C109—21no—
30C2×C4⋊C4×C109—21no—
31C2×C4⋊C4×C109—21no—
32C2×C4⋊C4×C109C2212yes—
33C2×C4⋊C4×C109C2212yes—
34C2×C4⋊C4×C109C2212yes—
35#13182029—24no—
36C2×Dic218C2311yescharacteristic
37C2×D436C2311yescharacteristic
38C2×C22⋊D109—21no—
39C2×C22⋊D109—21no—
40C2×C22⋊D109—21no—
41C2×C4×D109—21no—
42#13182013—24no—
43C2×#11724784C2314yes—
44C4⋊Dic109C2311yescharacteristic
45C4⋊Dic109C2311yescharacteristic
46#13182008—21no—
47#13182008—21no—
48C22⋊Dic109—21no—
49C22⋊Dic109—21no—
50C22⋊Dic109—21no—
51#13182006—21no—
52#13182006—21no—
53C4×Dic109—21no—
54C24×C109D411yescharacteristic
55C22×C4×C109—24no—
56C22×C4×C109—22no—
57C22×C4×C109—22no—
58C22×C4×C109—22no—
59C22×C4×C109—44no—
60C22×C4×C109D411yescharacteristic
61C22×C4×C109D411yescharacteristic
62C22×C4×C109D411yescharacteristic
63C22×C4×C109C2311yescharacteristic
64C22×C4×C109C2311yescharacteristic
65C22×C4×C109C2311yescharacteristic
66C4⋊C4×C109—22no—
67C4⋊C4×C109—22no—
68C4⋊C4×C109—22no—
69C4⋊C4×C109C2314yes—
70C23×C109—22no—
71C23×C109—22no—
72C23×C109—22no—
73C23×C109C2×D411yescharacteristic
74C23×C109C2×D411yescharacteristic
75C23×C109C2×D411yescharacteristic
76C22×D109—21no—
77C2×C4×C109—22no—
78C2×C4×C109—22no—
79C2×C4×C109—22no—
80C2×C4×C109—22no—
81C2×C4×C109—22no—
82C2×C4×C109—24no—
83C2×C4×C109—44no—
84C2×C4×C109—44no—
85C2×C4×C109—44no—
86C2×C4×C109C2×D411yescharacteristic
87C2×C4×C109C2×D411yescharacteristic
88C2×C4×C109C2×D411yescharacteristic
89C2×C4×C109#4114yes—
90C2×C4×C109C2411yescharacteristic
91C22⋊D109—42no—
92C22⋊D109—42no—
93C22⋊D109—42no—
94C2×Dic109—21no—
95C2×Dic109—21no—
96C2×Dic109—21no—
97D436—22no—
98C4×D109—42no—
99Dic218—22no—
100#11724784—24no—
101C22×C109—28no—
102C22×C109—21no—
103C22×C109—21no—
104C22×C109—21no—
105C22×C109—21no—
106C22×C109—21no—
107C22×C109—21no—
108C22×C109#12312yes—
109C22×C109AGL(1,ℤ/8)12yes—
110C22×C109#13712yes—
111C22×C109C22×D411yescharacteristic
112C2×D109—41no—
113C2×D109—41no—
114C2×D109—41no—
115C4×C109—21no—
116C4×C109—21no—
117C4×C109—44no—
118C4×C109#12312yes—
119C4×C109C2×#4112yes—
120Dic109—41no—
121Dic109—41no—
122Dic109—41no—
123C2×C109—22no—
124C2×C109—22no—
125C2×C109—22no—
126C2×C109C2×#12311yescharacteristic
127C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C109C2×#13711yescharacteristic
129D109—81no—
130#3092—1091nomaximal, Sylow
131C109#309211yescharacteristic, complemented, Sylow
132#525—1091no—
133C22×C4⋊C4D10911yescharacteristic, complemented
134#484—1092no—
135#483—1092no—
136#481—1092no—
137#480—1092no—
138#428—1092no—
139#417—1092no—
140#407—1091no—
141C2×#41—1091no—
142C23×C4—21no—
143C23×C4C2×D10911yescharacteristic
144C2×M16—1092no—
145#123—2181no—
146C22⋊Q8—1091no—
147C22⋊Q8—1091no—
148#121—1091no—
149#121—1091no—
150#120—2181no—
151C4×D4—2181no—
152C4×D4—2181no—
153#117—1091no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×D10912yes—
158C2×C4⋊C4C2×D10912yes—
159C2×C4⋊C4C2×D10912yes—
160#107—1094no—
161#106—1094no—
162#103—1094no—
163#102—1094no—
164C22⋊C8—1092no—
165C22⋊C8—1092no—
166C24C22⋊D10911yescharacteristic
167#41—2184no—
168C2×Q8—1091no—
169C2×D4—1091no—
170C2×D4—2181no—
171C2×D4—2181no—
172C2×D4—2181no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—44no—
178C22×C4—2181no—
179C22×C4C22⋊D10911yescharacteristic
180C22×C4C22⋊D10911yescharacteristic
181C22×C4C22⋊D10911yescharacteristic
182C22×C4C22×D10911yescharacteristic
183C22×C4C22×D10911yescharacteristic
184C22×C4C22×D10911yescharacteristic
185M16—2184no—
186C2×C8—1094no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—1091no—
191C4⋊C4—1091no—
192C4⋊C4—2181no—
193C4⋊C4—2181no—
194C4⋊C4C22×D10914yes—
195C22⋊C4—2181no—
196C22⋊C4—2181no—
197C22⋊C4—2181no—
198C22⋊C4—2181no—
199C22⋊C4—2181no—
200C42—2181no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—2181no—
205C23C2×C22⋊D10911yescharacteristic
206C23C2×C22⋊D10911yescharacteristic
207C23C2×C22⋊D10911yescharacteristic
208Q8—2182no—
209D4—4362no—
210D4—4362no—
211D4—4362no—
212D4—2182no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—4362no—
223C2×C4—2181no—
224C2×C4—2181no—
225C2×C4—2181no—
226C2×C4C2×C22⋊D10911yescharacteristic
227C2×C4C2×C22⋊D10911yescharacteristic
228C2×C4C2×C22⋊D10911yescharacteristic
229C2×C4#1318202914yes—
230C2×C4C23×D10911yescharacteristic
231C8—2184no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—4361no—
240C22—4361no—
241C22—4361no—
242C22#1681268212yes—
243C22#1681272512yes—
244C22#1681273012yes—
245C22C22×C22⋊D10911yescharacteristic
246C4—21no—
247C4—21no—
248C4—44no—
249C4—4361no—
250C4—4361no—
251C4—4361no—
252C4#1681268212yes—
253C4C2×#1318202912yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—8721no—
258C2C2×#1681268211yescharacteristic
259C2C2×#1681272511yescharacteristic
260C2C2×#1681273011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC22×C4⋊C4×C109C2
socleC22×C109C22×D4
prime radical#20295537C2
hypercenterC22×C4⋊C4D109
hypocenterC109#3092
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20295361C2semidirect
3. #20295356C2semidirect
4. #20295344C2semidirect
5. #20295339C2semidirect
6. #20295066C2semidirect
7. #20295025C2semidirect
8. #20294990C2semidirect
9. C22×C4⋊C4×C109C2semidirect
21. #16812632C22semidirect
22. #16812630C22semidirect
23. #16812622C22semidirect
24. #16812619C22semidirect
131. C109#3092semidirect
133. C22×C4⋊C4D109semidirect

Maximal subgroups

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

subgroupindex
#202955372
#202953612
#202953562
#202953442
#202953392
#202950662
#202950252
#202949902
C22×C4⋊C4×C1092
#3092109