Finite Group Databasebeta

← #28366353

Subgroups — #28366353

Order 13952.

1056 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C22×#13182029C211yesmaximal, characteristic, complemented
2C2×#16812669C212yesmaximal, complemented
3C22×#13182006C211yesmaximal, characteristic, complemented
4#20294939C212yesmaximal, complemented
5#20294936C212yesmaximal, complemented
6#20294904C214yesmaximal, complemented
7#20294903C212yesmaximal, complemented
8#381×C109C211yesmaximal, characteristic, complemented
9C24×D109C2211yescharacteristic
10C2×#13182029—28no—
11C2×#13182029—24no—
12C22×Dic218C2211yescharacteristic, complemented
13C22×D436C2214yes—
14C22×D436C2211yescharacteristic, complemented
15C22×C22⋊D109C2212yes—
16C23×Dic109C2211yescharacteristic
17C22×C4×D109C2212yescomplemented
18#16812669—28no—
19C2×#13182006—24no—
20C2×#13182006C2212yes—
21C2×#13182006C2214yes—
22C2×#13182006C2212yes—
23C2×#13182006C2214yescomplemented
24#16812606C2214yescomplemented
25C23×C4×C109C2211yescharacteristic
26C2×C22⋊C4×C109C2212yes—
27#95×C109C2212yescomplemented
28#95×C109C2212yescomplemented
29C23×D109—22no—
30C23×D109—24no—
31C23×D109—24no—
32C23×D109—42no—
33C23×D109C2312yes—
34C23×D109C2312yes—
35#13182029—232no—
36C2×Dic218—22no—
37C2×Dic218—24no—
38C2×D436—24no—
39C2×D436—22no—
40C2×D436—28no—
41C2×D436—28no—
42C2×D436—44no—
43C2×C22⋊D109—24no—
44C2×C22⋊D109—24no—
45C2×C22⋊D109D418yescomplemented
46C22×Dic109—22no—
47C22×Dic109—42no—
48C22×Dic109C2312yes—
49C22×Dic109C2312yes—
50C2×C4×D109—24no—
51C2×C4×D109—24no—
52C2×C4×D109—24no—
53#13182006—24no—
54#13182006—24no—
55#13182006—28no—
56#13182006—44no—
57C24×C109C2311yescharacteristic
58C22×C4×C109—24no—
59C22×C4×C109D414yescomplemented
60C22×C4×C109C2314yes—
61C22×C4×C109C2311yescharacteristic
62C22×C4×C109C2311yescharacteristic
63C22⋊C4×C109—24no—
64C23×C109—22no—
65C23×C109—22no—
66C23×C109C2×D414yes—
67C23×C109C2×D412yes—
68C23×C109C2411yescharacteristic
69C22×D109—28no—
70C22×D109—22no—
71C22×D109—24no—
72C22×D109—22no—
73C22×D109—22no—
74C22×D109—24no—
75C22×D109—22no—
76C22×D109—24no—
77C22×D109—24no—
78C22×D109—416no—
79C22×D109—44no—
80C22×D109—44no—
81C22×D109C2×D414yes—
82C22×D109#4114yescomplemented
83C2×C4×C109—28no—
84C2×C4×C109—22no—
85C2×C4×C109—22no—
86C2×C4×C109—24no—
87C2×C4×C109—24no—
88C2×C4×C109—24no—
89C2×C4×C109C2×D412yes—
90C2×C4×C109C2×D412yes—
91C22⋊D109—216no—
92C2×Dic109—22no—
93C2×Dic109—22no—
94C2×Dic109—22no—
95C2×Dic109—24no—
96C2×Dic109—44no—
97C2×Dic109C2×D414yes—
98D436—28no—
99D436—416no—
100C4×D109—216no—
101Dic218—28no—
102C22×C109—22no—
103C22×C109—22no—
104C22×C109—28no—
105C22×C109#12014yes—
106C22×C109C22×D412yes—
107C22×C109C22×D411yescharacteristic
108C22×C109C2×#4112yes—
109C22×C109H(5,𝔽2)12yes—
110C2×D109—24no—
111C2×D109—24no—
112C2×D109—28no—
113C2×D109—28no—
114C2×D109—416no—
115C2×D109—42no—
116C2×D109—42no—
117C2×D109—42no—
118C2×D109—42no—
119C2×D109—44no—
120C2×D109—44no—
121C4×C109—22no—
122C4×C109—22no—
123C4×C109—44no—
124Dic109—24no—
125Dic109—42no—
126C2×C109—24no—
127C2×C109C2×#12011yescharacteristic
128C2×C109#52511yescharacteristic
129C2×C109#53511yescharacteristic
130C2×C109#54614yes—
131D109—44no—
132D109—44no—
133#2404—1091nomaximal, Sylow
134C109#240411yescharacteristic, complemented, Sylow
135C22×#41—1091no—
136C2×#120—1092no—
137C22×C22⋊C4—1091no—
138#393—1092no—
139#392—1092no—
140#386—1092no—
141#386—1094no—
142#381D10911yescharacteristic, complemented
143C25—1091no—
144C2×#41—2188no—
145C2×#41—2184no—
146C22×Q8—1091no—
147C22×D4—1092no—
148C22×D4—1094no—
149C22×D4—1091no—
150C23×C4—1091no—
151C23×C4—1092no—
152C23×C4C2×D10911yescharacteristic
153#120—2188no—
154C2×C22⋊C4—1094no—
155C2×C22⋊C4—1092no—
156C2×C22⋊C4—1092no—
157C2×C22⋊C4—1094no—
158C2×C22⋊C4—2184no—
159C2×C22⋊C4C2×D10912yes—
160#95—1094no—
161#95C2×D10912yescomplemented
162#95C2×D10912yescomplemented
163C24—1092no—
164C24—1092no—
165C24—4362no—
166C24—2184no—
167C24—2184no—
168C24—2182no—
169C24C22×D10911yescharacteristic
170#41—21832no—
171C2×Q8—2184no—
172C2×Q8—2182no—
173C2×D4—1098no—
174C2×D4—4364no—
175C2×D4—2188no—
176C2×D4—2188no—
177C2×D4—2184no—
178C2×D4—2184no—
179C2×D4—2184no—
180C2×D4—2182no—
181C22×C4—24no—
182C22×C4—1092no—
183C22×C4—1092no—
184C22×C4—4362no—
185C22×C4—2184no—
186C22×C4—2184no—
187C22×C4—2182no—
188C22×C4—2184no—
189C22×C4D43614yescomplemented
190C22×C4C22×D10914yes—
191C22×C4C22×D10911yescharacteristic
192C22×C4C22×D10911yescharacteristic
193C22⋊C4—24no—
194C22⋊C4—4364no—
195C22⋊C4—2188no—
196C22⋊C4—2184no—
197C22⋊C4—2184no—
198C23—22no—
199C23—22no—
200C23—1094no—
201C23—1094no—
202C23—4364no—
203C23—43616no—
204C23—4364no—
205C23—2188no—
206C23—2184no—
207C23—2182no—
208C23—2182no—
209C23—2182no—
210C23—2184no—
211C23—2182no—
212C23—2184no—
213C23—2184no—
214C23D4×D10914yes—
215C23C2×D43612yes—
216C23C23×D10911yescharacteristic
217Q8—2188no—
218D4—43616no—
219D4—2188no—
220D4—21816no—
221C2×C4—28no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—1094no—
228C2×C4—4364no—
229C2×C4—21816no—
230C2×C4—2182no—
231C2×C4—2184no—
232C2×C4—2182no—
233C2×C4—2182no—
234C2×C4C2×D43612yes—
235C2×C4C2×D43612yes—
236C22—22no—
237C22—22no—
238C22—28no—
239C22—43616no—
240C22—4362no—
241C22—4362no—
242C22—4362no—
243C22—4362no—
244C22—4364no—
245C22—4364no—
246C22—2184no—
247C22—2184no—
248C22—2188no—
249C22—2188no—
250C22#1681266914yes—
251C22C2×D4×D10912yes—
252C22C22×D43611yescharacteristic
253C22#41×D10912yes—
254C22#1681274512yes—
255C4—22no—
256C4—22no—
257C4—44no—
258C4—4362no—
259C4—2184no—
260C2—24no—
261C2—4364no—
262C2—4364no—
263C2C2×#1681266911yescharacteristic
264C2#2029553811yescharacteristic
265C2#2029559511yescharacteristic
266C2#2029566814yes—
subgroupG/H
center, FrattiniC23C23×D109
commutator, socle, cosocleC23×C109C24
Fitting#381×C109C2
hypercenter#381D109
hypocenterC109#2404
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C22×#13182029C2semidirect
2. C2×#16812669C2semidirect
3. C22×#13182006C2semidirect
4. #20294939C2semidirect
5. #20294936C2semidirect
6. #20294904C2semidirect
7. #20294903C2semidirect
8. #381×C109C2semidirect
12. C22×Dic218C22semidirect
14. C22×D436C22semidirect
17. C22×C4×D109C22semidirect
23. C2×#13182006C22semidirect
24. #16812606C22semidirect
27. #95×C109C22semidirect
28. #95×C109C22semidirect
45. C2×C22⋊D109D4semidirect
59. C22×C4×C109D4semidirect
82. C22×D109#41semidirect
134. C109#2404semidirect
142. #381D109semidirect
161. #95C2×D109semidirect
162. #95C2×D109semidirect
189. C22×C4D436semidirect

Maximal subgroups

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

subgroupindex
C22×#131820292
C2×#168126692
C22×#131820062
#202949392
#202949362
#202949042
#202949032
#381×C1092
#2404109