Finite Group Databasebeta

← #28372290

Subgroups — #28372290

Order 13952.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
2C2×#16812730C212yesmaximal, complemented
3#20295186C214yesmaximal, complemented
4#20295181C212yesmaximal, complemented
5#20295176C212yesmaximal, complemented
6#20295025C212yesmaximal, complemented
7#20294990C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C109C211yesmaximal, characteristic, complemented
9C2×#13182029—28no—
10C2×#13182029—21no—
11C2×#13182029—21no—
12C2×#13182029C2212yes—
13C2×#13182029C2212yes—
14C22×Dic218C2211yescharacteristic
15C22×D436C2211yescharacteristic
16C22×C22⋊D109—21no—
17C22×C4×D109—21no—
18#16812730—28no—
19C2×#13182019C2214yescomplemented
20C2×#13182016C2214yescomplemented
21C2×#13182013C2212yescomplemented
22#16812622C2214yescomplemented
23#16812619C2214yescomplemented
24C22⋊#11724784C2212yes—
25C22⋊#11724784C2212yescomplemented
26C22×Q8×C109C2211yescharacteristic, complemented
27C23×C4×C109C2211yescharacteristic
28C22⋊Q8×C109—21no—
29C22⋊Q8×C109—21no—
30C22⋊Q8×C109C2212yes—
31C22⋊Q8×C109C2212yescomplemented
32C2×C4⋊C4×C109—21no—
33C2×C4⋊C4×C109C2211yescharacteristic, complemented
34C2×C22⋊C4×C109—21no—
35C23×D109—21no—
36#13182029—28no—
37#13182029—28no—
38#13182029—24no—
39#13182029—24no—
40#13182029—24no—
41#13182029D418yes—
42C2×Dic218—21no—
43C2×Dic218—24no—
44C2×Dic218C2312yes—
45C2×D436—21no—
46C2×D436—24no—
47C2×D436C2312yes—
48C2×C22⋊D109—28no—
49C2×C22⋊D109—22no—
50C2×C22⋊D109—22no—
51C22×Dic109—21no—
52C2×C4×D109—28no—
53C2×C4×D109—22no—
54C2×C4×D109—22no—
55#13182019—28no—
56#13182016—28no—
57#13182013—24no—
58C2×#11724784C2314yes—
59C24×C109D411yescharacteristic
60C2×Q8×C109—21no—
61C2×Q8×C109—24no—
62C2×Q8×C109C2312yes—
63C22×C4×C109—24no—
64C22×C4×C109—21no—
65C22×C4×C109—21no—
66C22×C4×C109D411yescharacteristic
67C22×C4×C109D411yescharacteristic
68C22×C4×C109D411yescharacteristic
69C22×C4×C109C2311yescharacteristic
70C22×C4×C109C2311yescharacteristic
71C22×C4×C109C2311yescharacteristic
72C4⋊C4×C109—22no—
73C4⋊C4×C109—22no—
74C4⋊C4×C109—21no—
75C4⋊C4×C109C2312yes—
76C22⋊C4×C109—22no—
77C22⋊C4×C109—22no—
78C23×C109—22no—
79C23×C109—22no—
80C23×C109—41no—
81C23×C109C2×D411yescharacteristic
82C23×C109C2×D411yescharacteristic
83C23×C109C2×D411yescharacteristic
84C22×D109—24no—
85C22×D109—22no—
86C22×D109—42no—
87C22×D109—42no—
88Q8×C109—24no—
89Q8×C109—42no—
90C2×C4×C109—22no—
91C2×C4×C109—22no—
92C2×C4×C109—22no—
93C2×C4×C109—22no—
94C2×C4×C109—22no—
95C2×C4×C109—22no—
96C2×C4×C109—22no—
97C2×C4×C109—42no—
98C2×C4×C109—42no—
99C2×C4×C109C2×D414yes—
100C2×C4×C109C2×D411yescharacteristic
101C2×C4×C109C2×D411yescharacteristic
102C2×C4×C109C2×D411yescharacteristic
103C2×C4×C109C2411yescharacteristic
104C22⋊D109—28no—
105C22⋊D109—28no—
106C2×Dic109—24no—
107C2×Dic109—22no—
108D436—24no—
109D436—22no—
110D436C2×D414yes—
111C4×D109—28no—
112C4×D109—28no—
113Dic218—24no—
114Dic218—22no—
115Dic218C2×D414yes—
116#11724784—24no—
117C22×C109—21no—
118C22×C109—21no—
119C22×C109—21no—
120C22×C109—21no—
121C22×C109—21no—
122C22×C109—21no—
123C22×C109—21no—
124C22×C109—44no—
125C22×C109#12012yes—
126C22×C109#13712yescomplemented
127C22×C109C22×D411yescharacteristic
128C2×D109—24no—
129C2×D109—48no—
130C2×D109—42no—
131C2×D109—42no—
132C4×C109—21no—
133C4×C109—21no—
134C4×C109—42no—
135C4×C109—42no—
136C4×C109#12012yes—
137C4×C109C22×D412yes—
138Dic109—24no—
139C2×C109—22no—
140C2×C109—22no—
141C2×C109—41no—
142C2×C109C2×#12011yescharacteristic
143C2×C109C2×#13711yescharacteristic
144C2×C109#57511yescharacteristic
145D109—44no—
146#3007—1091nomaximal, Sylow
147C109#300711yescharacteristic, complemented, Sylow
148C22×#41—1091no—
149C2×#137—1092no—
150C2×C22⋊Q8D10911yescharacteristic, complemented
151#452—1094no—
152#451—1092no—
153#450—1092no—
154#417—1092no—
155#407—1091no—
156C2×#41—1092no—
157C2×#41—1092no—
158C2×#41—2188no—
159C2×#41—2181no—
160C2×#41—2181no—
161C22×Q8—1091no—
162C22×Q8C2×D10911yescharacteristic, complemented
163C22×D4—1091no—
164C22×D4—2181no—
165C23×C4—2181no—
166C23×C4C2×D10911yescharacteristic
167#137—2188no—
168C2×Q16—1094no—
169C2×SD16—1094no—
170C2×M16—1092no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C2×D10912yes—
174C22⋊Q8C2×D10912yescomplemented
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D10911yescharacteristic, complemented
177C2×C22⋊C4—21no—
178#103—1094no—
179#102—1094no—
180C22⋊C8—1092no—
181C22⋊C8—1092no—
182C24—2181no—
183C24C22⋊D10911yescharacteristic
184#41—1098no—
185#41—2184no—
186#41—2184no—
187#41—2188no—
188#41—2188no—
189#41—2184no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—1092no—
193C2×Q8—2184no—
194C2×Q8—2181no—
195C2×Q8C22×D10912yes—
196C2×D4—1092no—
197C2×D4—2184no—
198C2×D4—2188no—
199C2×D4—2182no—
200C2×D4—2182no—
201C2×D4—2181no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—2188no—
206C22×C4—2181no—
207C22×C4—2182no—
208C22×C4—2182no—
209C22×C4C22⋊D10911yescharacteristic
210C22×C4C22⋊D10911yescharacteristic
211C22×C4C22⋊D10911yescharacteristic
212C22×C4C22×D10911yescharacteristic
213C22×C4C22×D10911yescharacteristic
214C22×C4C22×D10911yescharacteristic
215Q16—2188no—
216SD16—2188no—
217M16—2184no—
218C2×C8—1094no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D10912yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—4362no—
229C23—4362no—
230C23—2184no—
231C23—2182no—
232C23C2×C22⋊D10911yescharacteristic
233C23C2×C22⋊D10911yescharacteristic
234C23C2×C22⋊D10911yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—1094no—
238Q8—2184no—
239Q8—2182no—
240D4—1094no—
241D4—2184no—
242D4—2188no—
243D4—2188no—
244D4—2182no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—2188no—
255C2×C4—2184no—
256C2×C4—2182no—
257C2×C4—2188no—
258C2×C4D4×D10914yes—
259C2×C4C2×C22⋊D10911yescharacteristic
260C2×C4C2×C22⋊D10911yescharacteristic
261C2×C4C2×C22⋊D10911yescharacteristic
262C2×C4C23×D10911yescharacteristic
263C8—2184no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—4368no—
273C22—4362no—
274C22—4362no—
275C22—2184no—
276C22#1681266712yes—
277C22#1681273012yescomplemented
278C22C22×C22⋊D10911yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—2184no—
284C4#1681266712yes—
285C4C2×D4×D10912yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—4364no—
290C2C2×#1681266711yescharacteristic
291C2C2×#1681273011yescharacteristic
292C2#2029582411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC2×C22⋊Q8×C109C2
socleC22×C109C22×D4
prime radicalC22×#13182029C2
hypercenterC2×C22⋊Q8D109
hypocenterC109#3007
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16812730C2semidirect
3. #20295186C2semidirect
4. #20295181C2semidirect
5. #20295176C2semidirect
6. #20295025C2semidirect
7. #20294990C2semidirect
8. C2×C22⋊Q8×C109C2semidirect
19. C2×#13182019C22semidirect
20. C2×#13182016C22semidirect
21. C2×#13182013C22semidirect
22. #16812622C22semidirect
23. #16812619C22semidirect
25. C22⋊#11724784C22semidirect
26. C22×Q8×C109C22semidirect
31. C22⋊Q8×C109C22semidirect
33. C2×C4⋊C4×C109C22semidirect
126. C22×C109#137semidirect
147. C109#3007semidirect
150. C2×C22⋊Q8D109semidirect
162. C22×Q8C2×D109semidirect
174. C22⋊Q8C2×D109semidirect
176. C2×C4⋊C4C2×D109semidirect
277. C22#16812730semidirect

Maximal subgroups

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

subgroupindex
C22×#131820292
C2×#168127302
#202951862
#202951812
#202951762
#202950252
#202949902
C2×C22⋊Q8×C1092
#3007109