Finite Group Databasebeta

← #28366389

Subgroups — #28366389

Order 13952.

1232 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 189 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
1C23×D436C211yesmaximal, characteristic, complemented
2C2×#16812674C212yesmaximal, complemented
3C2×#16812669C212yesmaximal, complemented
4C2×C4×C22⋊D109C211yesmaximal, characteristic, complemented
5#20294944C212yesmaximal, complemented
6#20294936C212yesmaximal, complemented
7#20294927C212yesmaximal, complemented
8#20294881C212yesmaximal, complemented
9#380×C109C211yesmaximal, characteristic, complemented
10C24×D109C2212yes—
11C22×D436—28no—
12C22×D436—24no—
13C22×D436C2212yes—
14C22×D436C2212yes—
15C22×D436C2214yescomplemented
16C22×C22⋊D109C2211yescharacteristic
17C22×C4×D109C2211yescharacteristic, complemented
18#16812674—28no—
19#16812669—28no—
20C4×C22⋊D109—24no—
21C2×#13182008C2211yescharacteristic, complemented
22C2×C22⋊Dic109C2211yescharacteristic
23C2×#13182006C2214yescomplemented
24C2×#13182006C2211yescharacteristic, complemented
25C2×#13182006C2214yescomplemented
26C2×#13182006C2214yescomplemented
27C2×C4×Dic109C2211yescharacteristic, complemented
28C23×C4×C109C2211yescharacteristic
29C2×C4⋊C4×C109C2212yescomplemented
30C2×C22⋊C4×C109C2212yes—
31#95×C109C2212yescomplemented
32C23×D109—24no—
33C23×D109—28no—
34C23×D109—24no—
35C23×D109—24no—
36C23×D109—44no—
37C23×D109C2314yes—
38C23×D109C2311yescharacteristic
39C2×D436—232no—
40C2×D436—28no—
41C2×D436—24no—
42C2×D436—24no—
43C2×D436—24no—
44C2×D436—28no—
45C2×D436—28no—
46C2×D436—44no—
47C2×D436D418yescomplemented
48C2×C22⋊D109—22no—
49C2×C22⋊D109—22no—
50C2×C22⋊D109D414yescomplemented
51C22×Dic109C2311yescharacteristic
52C22×Dic109C2311yescharacteristic
53C22×Dic109C2311yescharacteristic
54C2×C4×D109—22no—
55C2×C4×D109—24no—
56#13182008—22no—
57C22⋊Dic109—22no—
58#13182006—28no—
59#13182006—22no—
60#13182006—28no—
61#13182006—44no—
62C4×Dic109—22no—
63C24×C109C2311yescharacteristic
64C22×C4×C109—24no—
65C22×C4×C109D414yescomplemented
66C22×C4×C109C2314yes—
67C22×C4×C109C2311yescharacteristic
68C22×C4×C109C2311yescharacteristic
69C4⋊C4×C109—24no—
70C22⋊C4×C109—24no—
71C23×C109—22no—
72C23×C109—22no—
73C23×C109C2×D412yes—
74C23×C109C2×D412yes—
75C23×C109#4112yes—
76C23×C109C2411yescharacteristic
77C22×D109—216no—
78C22×D109—21no—
79C22×D109—24no—
80C22×D109—22no—
81C22×D109—24no—
82C22×D109—24no—
83C22×D109—21no—
84C22×D109—24no—
85C22×D109—24no—
86C22×D109—22no—
87C22×D109—24no—
88C22×D109—416no—
89C22×D109—416no—
90C22×D109—48no—
91C22×D109—48no—
92C22×D109C2×D418yes—
93C22×D109C2×D412yescomplemented
94C2×C4×C109—22no—
95C2×C4×C109—28no—
96C2×C4×C109—24no—
97C2×C4×C109—24no—
98C2×C4×C109—24no—
99C2×C4×C109C2×D414yes—
100C2×C4×C109C2×D412yes—
101C2×C4×C109C2×D412yes—
102C22⋊D109—28no—
103C2×Dic109—21no—
104C2×Dic109—21no—
105C2×Dic109—21no—
106C2×Dic109—22no—
107C2×Dic109—21no—
108C2×Dic109—22no—
109C2×Dic109C2×D412yescomplemented
110D436—216no—
111D436—216no—
112D436—416no—
113C4×D109—44no—
114C22×C109—22no—
115C22×C109—22no—
116C22×C109—24no—
117C22×C109—24no—
118C22×C109#12114yescomplemented
119C22×C109C22×D412yes—
120C22×C109C22×D411yescharacteristic
121C22×C109C22×D411yescharacteristic
122C22×C109C2×#4111yescharacteristic
123C22×C109H(5,𝔽2)12yes—
124C2×D109—28no—
125C2×D109—22no—
126C2×D109—24no—
127C2×D109—28no—
128C2×D109—28no—
129C2×D109—432no—
130C2×D109—41no—
131C2×D109—44no—
132C2×D109—41no—
133C2×D109—44no—
134C2×D109—44no—
135C2×D109—42no—
136C2×D109—44no—
137C4×C109—22no—
138C4×C109—44no—
139C4×C109#12014yescomplemented
140Dic109—22no—
141Dic109—41no—
142Dic109—41no—
143C2×C109—24no—
144C2×C109C2×#12011yescharacteristic
145C2×C109C2×#12111yescharacteristic
146C2×C109#53511yescharacteristic
147C2×C109D4212yes—
148C2×C109#54612yes—
149D109—42no—
150D109—48no—
151#2408—1091nomaximal, Sylow
152C109#240811yescharacteristic, complemented, Sylow
153C23×D4—1091no—
154C2×#121—1092no—
155C2×#120—1092no—
156C2×C4×D4—1091no—
157#394—1092no—
158#392—1092no—
159#390—1092no—
160#380D10911yescharacteristic, complemented
161#379—1092no—
162C25—1092no—
163C22×D4—1091no—
164C22×D4—1092no—
165C22×D4—1094no—
166C22×D4—1092no—
167C22×D4—2188no—
168C22×D4—2184no—
169C23×C4—1091no—
170C23×C4C2×D10911yescharacteristic
171#121—2188no—
172#120—2188no—
173C4×D4—2184no—
174C2×C4⋊C4—1091no—
175C2×C4⋊C4C2×D10912yescomplemented
176C2×C22⋊C4—1094no—
177C2×C22⋊C4—1091no—
178C2×C22⋊C4—1091no—
179C2×C22⋊C4—1094no—
180C2×C22⋊C4—1094no—
181C2×C22⋊C4C2×D10912yes—
182C2×C42—1091no—
183#95C2×D10912yescomplemented
184C24—1094no—
185C24—1091no—
186C24—4364no—
187C24—2188no—
188C24—2184no—
189C24—2184no—
190C24—2184no—
191C24C22×D10911yescharacteristic
192C2×D4—1094no—
193C2×D4—1098no—
194C2×D4—4364no—
195C2×D4—2188no—
196C2×D4—2188no—
197C2×D4—2188no—
198C2×D4—21832no—
199C2×D4—2182no—
200C2×D4—2182no—
201C2×D4—2184no—
202C2×D4—2184no—
203C2×D4—2184no—
204C22×C4—24no—
205C22×C4—1091no—
206C22×C4—1091no—
207C22×C4—1091no—
208C22×C4—2184no—
209C22×C4—2182no—
210C22×C4D43614yescomplemented
211C22×C4C22×D10914yes—
212C22×C4C22×D10911yescharacteristic
213C22×C4C22×D10911yescharacteristic
214C4⋊C4—24no—
215C4⋊C4—2182no—
216C22⋊C4—24no—
217C22⋊C4—4364no—
218C22⋊C4—2188no—
219C22⋊C4—2188no—
220C22⋊C4—2182no—
221C22⋊C4—2182no—
222C42—2182no—
223C23—22no—
224C23—22no—
225C23—1098no—
226C23—1092no—
227C23—4368no—
228C23—43616no—
229C23—43616no—
230C23—4368no—
231C23—21816no—
232C23—2184no—
233C23—2182no—
234C23—2184no—
235C23—2181no—
236C23—2184no—
237C23—2181no—
238C23—2184no—
239C23—2184no—
240C23—2182no—
241C23—2184no—
242C23D4×D10912yes—
243C23C2×D43612yes—
244C23#1318203012yes—
245C23C23×D10911yescharacteristic
246D4—43616no—
247D4—21816no—
248D4—2188no—
249D4—21816no—
250C2×C4—22no—
251C2×C4—28no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—1092no—
256C2×C4—4364no—
257C2×C4—2182no—
258C2×C4—2181no—
259C2×C4—2181no—
260C2×C4—2182no—
261C2×C4—2181no—
262C2×C4—2181no—
263C2×C4D4×D10914yes—
264C2×C4C2×D43612yes—
265C2×C4C2×D43612yes—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—24no—
270C22—43632no—
271C22—4361no—
272C22—4364no—
273C22—4364no—
274C22—4361no—
275C22—4364no—
276C22—4364no—
277C22—4362no—
278C22—2188no—
279C22—2182no—
280C22—2184no—
281C22—2188no—
282C22—2188no—
283C22#1681267414yescomplemented
284C22C2×D4×D10912yes—
285C22C2×D4×D10911yescharacteristic
286C22C22×D43611yescharacteristic
287C22C2×#1318203011yescharacteristic
288C22#1681274512yes—
289C4—22no—
290C4—44no—
291C4—4361no—
292C4—4361no—
293C4—2182no—
294C4#1681266914yescomplemented
295C2—24no—
296C2—4368no—
297C2—4362no—
298C2C2×#1681266911yescharacteristic
299C2C2×#1681267411yescharacteristic
300C2#2029559511yescharacteristic
301C2#2029566012yes—
302C2#2029566712yes—
subgroupG/H
center, FrattiniC23C23×D109
commutator, socle, cosocleC23×C109C24
Fitting#380×C109C2
hypercenter#380D109
hypocenterC109#2408
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×D436C2semidirect
2. C2×#16812674C2semidirect
3. C2×#16812669C2semidirect
4. C2×C4×C22⋊D109C2semidirect
5. #20294944C2semidirect
6. #20294936C2semidirect
7. #20294927C2semidirect
8. #20294881C2semidirect
9. #380×C109C2semidirect
15. C22×D436C22semidirect
17. C22×C4×D109C22semidirect
21. C2×#13182008C22semidirect
23. C2×#13182006C22semidirect
24. C2×#13182006C22semidirect
25. C2×#13182006C22semidirect
26. C2×#13182006C22semidirect
27. C2×C4×Dic109C22semidirect
29. C2×C4⋊C4×C109C22semidirect
31. #95×C109C22semidirect
47. C2×D436D4semidirect
50. C2×C22⋊D109D4semidirect
65. C22×C4×C109D4semidirect
93. C22×D109C2×D4semidirect
109. C2×Dic109C2×D4semidirect
118. C22×C109#121semidirect
139. C4×C109#120semidirect
152. C109#2408semidirect
160. #380D109semidirect
175. C2×C4⋊C4C2×D109semidirect
183. #95C2×D109semidirect
210. C22×C4D436semidirect
283. C22#16812674semidirect
294. C4#16812669semidirect

Maximal subgroups

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

subgroupindex
C23×D4362
C2×#168126742
C2×#168126692
C2×C4×C22⋊D1092
#202949442
#202949362
#202949272
#202948812
#380×C1092
#2408109