Finite Group Databasebeta

← #28366355

Subgroups — #28366355

Order 13952.

1116 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 197 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
1C2×#16812680C212yesmaximal, complemented
2C22×C22⋊Dic109C211yesmaximal, characteristic, complemented
3#20294940C214yesmaximal, complemented
4#20294906C212yesmaximal, complemented
5#20294899C212yesmaximal, complemented
6#20294882C212yesmaximal, complemented
7#20294882C211yesmaximal, characteristic, complemented
8C23×D4×C109C211yesmaximal, characteristic
9C23×Dic109C2211yescharacteristic, complemented
10C23×Dic109C2211yescharacteristic, complemented
11#16812680—28no—
12C2×#13182008C2214yescomplemented
13C2×C22⋊Dic109—24no—
14C2×C22⋊Dic109C2212yes—
15C2×C22⋊Dic109C2212yes—
16C2×C22⋊Dic109C2212yescomplemented
17C2×C22⋊Dic109C2214yescomplemented
18C2×C22⋊Dic109C2212yescomplemented
19C2×C22⋊Dic109C2212yescomplemented
20C2×C22⋊Dic109C2214yescomplemented
21#16812606C2214yescomplemented
22C25×C109C2211yescharacteristic
23C25×C109C2211yescharacteristic
24C22×D4×C109—24no—
25C22×D4×C109—28no—
26C22×D4×C109C2212yes—
27C22×D4×C109C2212yes—
28C23×C4×C109C2211yescharacteristic
29C22×Dic109—22no—
30C22×Dic109—22no—
31C22×Dic109—24no—
32C22×Dic109—42no—
33C22×Dic109C2312yes—
34C22×Dic109C2314yes—
35C22×Dic109C2312yes—
36#13182008—28no—
37C22⋊Dic109—24no—
38C22⋊Dic109—24no—
39C22⋊Dic109—24no—
40C22⋊Dic109—28no—
41C22⋊Dic109—24no—
42C22⋊Dic109—42no—
43C22⋊Dic109—44no—
44C24×C109—24no—
45C24×C109—24no—
46C24×C109—22no—
47C24×C109—24no—
48C24×C109—24no—
49C24×C109—42no—
50C24×C109—42no—
51C24×C109D414yes—
52C24×C109C2311yescharacteristic
53C24×C109C2312yes—
54C24×C109C2311yescharacteristic
55C24×C109C2311yescharacteristic
56C2×D4×C109—232no—
57C2×D4×C109—24no—
58C2×D4×C109—24no—
59C2×D4×C109—24no—
60C2×D4×C109—24no—
61C2×D4×C109—24no—
62C2×D4×C109D418yes—
63C22×C4×C109—22no—
64C22×C4×C109—24no—
65C22×C4×C109C2312yes—
66C23×C109—22no—
67C23×C109—22no—
68C23×C109—22no—
69C23×C109—22no—
70C23×C109—28no—
71C23×C109—28no—
72C23×C109—28no—
73C23×C109—22no—
74C23×C109—22no—
75C23×C109—22no—
76C23×C109—22no—
77C23×C109—24no—
78C23×C109—22no—
79C23×C109—24no—
80C23×C109—44no—
81C23×C109—416no—
82C23×C109—416no—
83C23×C109—44no—
84C23×C109—44no—
85C23×C109C2×D412yes—
86C23×C109C2×D414yes—
87C23×C109C2×D412yes—
88C23×C109C2×D414yes—
89C23×C109C2×D412yes—
90C23×C109#4114yes—
91C23×C109C2411yescharacteristic
92D4×C109—216no—
93D4×C109—216no—
94C2×C4×C109—28no—
95C2×C4×C109—22no—
96C2×C4×C109—22no—
97C2×C4×C109C2×D414yes—
98C2×Dic109—24no—
99C2×Dic109—22no—
100C2×Dic109—22no—
101C2×Dic109—24no—
102C2×Dic109—24no—
103C2×Dic109—24no—
104C2×Dic109—24no—
105C2×Dic109—44no—
106C2×Dic109—44no—
107C22×C109—22no—
108C22×C109—22no—
109C22×C109—22no—
110C22×C109—22no—
111C22×C109—28no—
112C22×C109—28no—
113C22×C109—22no—
114C22×C109—24no—
115C22×C109—22no—
116C22×C109—22no—
117C22×C109—28no—
118C22×C109—416no—
119C22×C109—416no—
120C22×C109—42no—
121C22×C109—42no—
122C22×C109—44no—
123C22×C109—44no—
124C22×C109#12014yes—
125C22×C109#12314yescomplemented
126C22×C109C22×D411yescharacteristic
127C22×C109C22×D412yes—
128C22×C109C2×#4112yes—
129C22×C109H(5,𝔽2)12yes—
130C4×C109—24no—
131Dic109—42no—
132Dic109—42no—
133Dic109—44no—
134C2×C109—24no—
135C2×C109—24no—
136C2×C109—42no—
137C2×C109—44no—
138C2×C109C2×#12011yescharacteristic
139C2×C109C2×#12311yescharacteristic
140C2×C109#53411yescharacteristic
141C2×C109#54614yes—
142#2405—1091nomaximal, Sylow
143C109#240511yescharacteristic, complemented, Sylow
144C23×D4D10911yescharacteristic
145C2×#123—1092no—
146C22×C22⋊C4—1091no—
147#394—1094no—
148#386—1092no—
149#385—1092no—
150#379—1092no—
151#379—1091no—
152C25C2×D10911yescharacteristic
153C25C2×D10911yescharacteristic
154C22×D4—24no—
155C22×D4—28no—
156C22×D4C2×D10912yes—
157C22×D4C2×D10912yes—
158C23×C4—1091no—
159C23×C4—1091no—
160C23×C4C2×D10911yescharacteristic
161#123—2188no—
162C2×C4⋊C4—1094no—
163C2×C22⋊C4—1094no—
164C2×C22⋊C4—1092no—
165C2×C22⋊C4—1092no—
166C2×C22⋊C4—1092no—
167C2×C22⋊C4—1092no—
168C2×C22⋊C4—1094no—
169C2×C22⋊C4—1092no—
170C2×C22⋊C4—2184no—
171#95—1094no—
172C24—24no—
173C24—24no—
174C24—22no—
175C24—24no—
176C24—24no—
177C24—42no—
178C24—42no—
179C24C22⋊D10914yes—
180C24C22×D10911yescharacteristic
181C24C22×D10912yes—
182C24C22×D10911yescharacteristic
183C24C22×D10911yescharacteristic
184C2×D4—232no—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4C22⋊D10918yes—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—1092no—
194C22×C4—1094no—
195C22×C4—1092no—
196C22×C4—4362no—
197C22×C4—2182no—
198C22×C4—2182no—
199C22×C4—2184no—
200C22×C4C22×D10912yes—
201C4⋊C4—2188no—
202C22⋊C4—4362no—
203C22⋊C4—4364no—
204C22⋊C4—2184no—
205C22⋊C4—2184no—
206C22⋊C4—2184no—
207C22⋊C4—2188no—
208C22⋊C4—2184no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—28no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—24no—
221C23—22no—
222C23—24no—
223C23—44no—
224C23—416no—
225C23—416no—
226C23—44no—
227C23—44no—
228C23C2×C22⋊D10912yes—
229C23C2×C22⋊D10914yes—
230C23C2×C22⋊D10914yes—
231C23C2×C22⋊D10912yes—
232C23C2×C22⋊D10912yes—
233C23#1318202814yes—
234C23C23×D10911yescharacteristic
235D4—216no—
236D4—216no—
237C2×C4—28no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—4364no—
241C2×C4—4364no—
242C2×C4—2184no—
243C2×C4—2182no—
244C2×C4—2182no—
245C2×C4—2184no—
246C2×C4—2184no—
247C2×C4—2184no—
248C2×C4—2184no—
249C2×C4C2×C22⋊D10914yes—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—28no—
255C22—28no—
256C22—22no—
257C22—24no—
258C22—22no—
259C22—22no—
260C22—28no—
261C22—416no—
262C22—416no—
263C22—42no—
264C22—42no—
265C22—44no—
266C22—44no—
267C22#1681266814yes—
268C22#1681268014yescomplemented
269C22C22×C22⋊D10911yescharacteristic
270C22C22×C22⋊D10912yes—
271C22C2×#1318202812yes—
272C22#1681274412yes—
273C4—24no—
274C4—4362no—
275C4—4362no—
276C4—4364no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—44no—
281C2C2×#1681266811yescharacteristic
282C2C2×#1681268011yescharacteristic
283C2#2029558511yescharacteristic
284C2#2029567014yes—
subgroupG/H
center, FrattiniC23C23×D109
commutator, socle, cosocleC23×C109C24
Fitting, prime radicalC23×D4×C109C2
hypercenterC23×D4D109
hypocenterC109#2405
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16812680C2semidirect
2. C22×C22⋊Dic109C2semidirect
3. #20294940C2semidirect
4. #20294906C2semidirect
5. #20294899C2semidirect
6. #20294882C2semidirect
7. #20294882C2semidirect
9. C23×Dic109C22semidirect
10. C23×Dic109C22semidirect
12. C2×#13182008C22semidirect
16. C2×C22⋊Dic109C22semidirect
17. C2×C22⋊Dic109C22semidirect
18. C2×C22⋊Dic109C22semidirect
19. C2×C22⋊Dic109C22semidirect
20. C2×C22⋊Dic109C22semidirect
21. #16812606C22semidirect
125. C22×C109#123semidirect
143. C109#2405semidirect
268. C22#16812680semidirect

Maximal subgroups

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

subgroupindex
C2×#168126802
C22×C22⋊Dic1092
#202949402
#202949062
#202948992
#202948822
C23×D4×C1092
#2405109