Finite Group Databasebeta

← #49122287

Subgroups — #49122287

Order 36224.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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#31827029C211yesmaximal, characteristic
2C2×#22583599C211yesmaximal, characteristic, complemented
3C2×#22583537C211yesmaximal, characteristic, complemented
4#31826474C211yesmaximal, characteristic, complemented
5C2×#22583526C211yesmaximal, characteristic, complemented
6#31826226C211yesmaximal, characteristic, complemented
7C2×#137×C283C211yesmaximal, characteristic, complemented
8C22×D1132C2211yescharacteristic
9C2×#17985376C2211yescharacteristic
10#22583606—21no—
11#22583599—22no—
12#22583599—21no—
13#22583599—22no—
14#22583584—22no—
15#22583578—22no—
16#22583569C2211yescharacteristic
17C2×(C2×D283)⋊C4—22no—
18C2×C4×Dic283C2211yescharacteristic
19#22583537—22no—
20#22583534—22no—
21#22583526—22no—
22C2×#41×C283C2211yescharacteristic
23C22×Q8×C283C2211yescharacteristic
24#137×C283—22no—
25#137×C283—22no—
26C2×Q16×C283—21no—
27C2×SD16×C283—21no—
28C2×M16×C283C2211yescharacteristic
29C23×D283D412yes—
30C2×D1132—21no—
31C2×D1132—21no—
32C2×D1132—21no—
33C2×D1132—21no—
34C2×D1132—21no—
35C2×D1132—21no—
36C2×C22⋊D283—24no—
37C22×Dic283—21no—
38#17985376—21no—
39#17985376—21no—
40C2×C283⋊C8—21no—
41C4⋊Dic283D412yes—
42C22⋊Dic283—21no—
43(C2×D283)⋊C4—28no—
44(C2×D283)⋊C4—24no—
45C4×Dic283—21no—
46C4×Dic283—21no—
47C4×Dic283D412yescomplemented
48#41×C283—21no—
49#41×C283—21no—
50#41×C283—21no—
51#41×C283—21no—
52C2×Q8×C283—21no—
53C2×Q8×C283—21no—
54C2×Q8×C283—21no—
55C2×Q8×C283—21no—
56C2×Q8×C283—21no—
57C2×Q8×C283D411yescharacteristic
58C2×Q8×C283D412yescomplemented
59C2×D4×C283—21no—
60C2×D4×C283D411yescharacteristic
61C22×C4×C283—21no—
62C22×C4×C283—21no—
63C22×C4×C283C2311yescharacteristic
64Q16×C283—24no—
65SD16×C283—24no—
66M16×C283—21no—
67M16×C283D412yescomplemented
68C2×C8×C283—21no—
69C22×D283—22no—
70C22×D283—22no—
71C22×D283—22no—
72C22×D283—44no—
73D1132—41no—
74D1132—41no—
75D1132—41no—
76D1132—41no—
77C22⋊D283—84no—
78C2×Dic283—22no—
79C2×Dic283—22no—
80C2×Dic283—42no—
81C283⋊C8—41no—
82C23×C283—21no—
83C23×C283C2×D411yescharacteristic
84Q8×C283—22no—
85Q8×C283—21no—
86Q8×C283—22no—
87Q8×C283—21no—
88Q8×C283—41no—
89Q8×C283—41no—
90D4×C283—21no—
91D4×C283—21no—
92D4×C283—41no—
93D4×C283—41no—
94C2×C4×C283—21no—
95C2×C4×C283—22no—
96C2×C4×C283—41no—
97C2×C4×C283—41no—
98C2×C4×C283—41no—
99C2×C4×C283—41no—
100C2×C4×C283C2×D411yescharacteristic
101C2×C4×C283C2×D411yescharacteristic
102C2×C4×C283C2×D411yescharacteristic
103C2×C4×C283C2×D411yescharacteristic
104C2×C4×C283C2×D411yescharacteristic
105C2×C4×C283#4111yescharacteristic
106C8×C283—22no—
107C22×C283—22no—
108C22×C283—41no—
109C22×C283—42no—
110C22×C283#12011yescharacteristic
111C22×C283#12011yescharacteristic
112C22×C283#12111yescharacteristic
113C2×D283—44no—
114C2×D283—42no—
115C2×D283—44no—
116C2×D283—82no—
117C4×C283—41no—
118C4×C283—42no—
119C4×C283#12011yescharacteristic
120C4×C283#12111yescharacteristic
121C4×C283#12111yescharacteristic
122C4×C283#12711yescharacteristic
123Dic283—42no—
124Dic283—42no—
125C2×C283—21no—
126C2×C283—21no—
127C2×C283—81no—
128C2×C283#39211yescharacteristic
129C2×C283#45512yes—
130D283—82no—
131C283#200411yescharacteristic, complemented, Sylow
132#2004—2831nomaximal, Sylow
133C2×#137D28311yescharacteristic, complemented
134#535—2831no—
135C2×#124—2831no—
136C2×#104—2831no—
137#419—2831no—
138C2×#100—2831no—
139#339—2831no—
140C2×#41C2×D28311yescharacteristic
141C22×Q8C2×D28311yescharacteristic
142C22×D4—2831no—
143#137—22no—
144#137—22no—
145C2×Q16—21no—
146C2×SD16—21no—
147C2×M16—2831no—
148C2×M16C2×D28311yescharacteristic
149#127—5661no—
150#124—5662no—
151#124—5661no—
152#124—5662no—
153#121—5662no—
154#120—5662no—
155#117—2831no—
156C2×C22⋊C4—5662no—
157C2×C42—2831no—
158#104—5662no—
159#103—5662no—
160#100—5662no—
161C24—2832no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22⋊D28311yescharacteristic
172C2×Q8C22⋊D28312yescomplemented
173C2×D4—21no—
174C2×D4—5664no—
175C2×D4—5661no—
176C2×D4—5661no—
177C2×D4—5661no—
178C2×D4—5661no—
179C2×D4—5661no—
180C2×D4—5661no—
181C2×D4C22⋊D28311yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—5661no—
185C22×C4C22×D28311yescharacteristic
186Q16—24no—
187SD16—24no—
188M16—21no—
189M16—5661no—
190M16—5661no—
191M16C22⋊D28312yescomplemented
192C2×C8—21no—
193C2×C8—5661no—
194C4⋊C4—2832no—
195C22⋊C4—5668no—
196C22⋊C4—5664no—
197C22⋊C4—5661no—
198C42—2832no—
199C42—5661no—
200C42—5661no—
201C23—21no—
202C23—5662no—
203C23—5662no—
204C23—5662no—
205C23—11324no—
206C23D4×D28311yescharacteristic
207Q8—22no—
208Q8—21no—
209Q8—22no—
210Q8—21no—
211Q8—41no—
212Q8—41no—
213D4—21no—
214D4—21no—
215D4—41no—
216D4—41no—
217D4—11321no—
218D4—11321no—
219D4—11321no—
220D4—11321no—
221D4—22644no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—5662no—
229C2×C4—5662no—
230C2×C4—11322no—
231C2×C4D4×D28311yescharacteristic
232C2×C4D4×D28311yescharacteristic
233C2×C4C2×C22⋊D28311yescharacteristic
234C2×C4C2×C22⋊D28311yescharacteristic
235C2×C4C2×C22⋊D28311yescharacteristic
236C2×C4#1798539111yescharacteristic
237C8—22no—
238C8—11321no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—11324no—
243C22—11322no—
244C22—11324no—
245C22—22642no—
246C22#2258357811yescharacteristic
247C22#2258357811yescharacteristic
248C22#2258358111yescharacteristic
249C4—41no—
250C4—42no—
251C4—11322no—
252C4—11322no—
253C4#2258357911yescharacteristic
254C4#2258358211yescharacteristic
255C4#2258358211yescharacteristic
256C4#2258360611yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—22642no—
261C2#3182637311yescharacteristic
262C2#3182664112yes—
subgroupG/H
centerC22#22583578
commutator, cosocleC22×C4×C283C23
FrattiniC22×C4C22×D283
FittingC2×#137×C283C2
socleC22×C283#120
prime radical#31827029C2
hypercenterC2×#137D283
hypocenterC283#2004
LCSG ▷ C22×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#22583599C2semidirect
3. C2×#22583537C2semidirect
4. #31826474C2semidirect
5. C2×#22583526C2semidirect
6. #31826226C2semidirect
7. C2×#137×C283C2semidirect
47. C4×Dic283D4semidirect
58. C2×Q8×C283D4semidirect
67. M16×C283D4semidirect
131. C283#2004semidirect
133. C2×#137D283semidirect
172. C2×Q8C22⋊D283semidirect
191. M16C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
#318270292
C2×#225835992
C2×#225835372
#318264742
C2×#225835262
#318262262
C2×#137×C2832
#2004283