Finite Group Databasebeta

← #49124433

Subgroups — #49124433

Order 36224.

520 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 173 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×#22583599C211yesmaximal, characteristic
2#31826372C211yesmaximal, characteristic, complemented
3#31826356C212yesmaximal, complemented
4#31826354C212yesmaximal, complemented
5#31826343C212yesmaximal
6#31826339C212yesmaximal, complemented
7#31826336C212yesmaximal, complemented
8#31826329C212yesmaximal, complemented
9#391×C283C211yesmaximal, characteristic, complemented
10C22×D1132C2211yescharacteristic
11C22×C4×D283C2212yes—
12#22583599C418yescomplemented
13C2×C4⋊Dic283C2211yescharacteristic
14C2×Dic283⋊C4C2212yes—
15C2×Dic283⋊C4C2212yes—
16C2×(C2×D283)⋊C4C2212yes—
17C2×(C2×D283)⋊C4C2212yes—
18C2×(C2×D283)⋊C4C2212yes—
19C2×(C2×D283)⋊C4C2212yes—
20C2×(C2×D283)⋊C4C2212yes—
21C2×C4×Dic283C2211yescharacteristic
22C2×C4×Dic283C2211yescharacteristic
23C283⋊#95C2212yes—
24C283⋊#95C2212yes—
25C283⋊#95C2212yes—
26C283⋊#95C2212yes—
27C22×Q8×C283C2211yescharacteristic
28C2×C4⋊C4×C283C2211yescharacteristic
29C2×C42×C283C2211yescharacteristic
30#95×C283C2212yes—
31#95×C283C2212yes—
32C23×D283C2312yes—
33C2×D1132—41no—
34C2×D1132—41no—
35C2×D1132C2×C414yes—
36C22×Dic283C2312yes—
37C22×Dic283C2312yes—
38C22×Dic283C2312yes—
39C2×C4×D283—42no—
40C2×C4×D283—42no—
41C2×C4×D283—42no—
42C4⋊Dic283—41no—
43Dic283⋊C4—42no—
44Dic283⋊C4—42no—
45(C2×D283)⋊C4—42no—
46(C2×D283)⋊C4—42no—
47(C2×D283)⋊C4—42no—
48(C2×D283)⋊C4C2×C418yescomplemented
49(C2×D283)⋊C4C2×C418yescomplemented
50C4×Dic283—41no—
51C4×Dic283C2×C414yescomplemented
52C2×Q8×C283—41no—
53C2×Q8×C283—41no—
54C2×Q8×C283C2×C414yescomplemented
55C22×C4×C283C2312yes—
56C22×C4×C283C2312yes—
57C22×C4×C283C2311yescharacteristic
58C22×C4×C283C2311yescharacteristic
59C22×C4×C283C2311yescharacteristic
60C4⋊C4×C283—41no—
61C42×C283—41no—
62C22×D283—22no—
63C22×D283—22no—
64C22×D283—22no—
65C22×D283—22no—
66C22×D283—22no—
67C22×D283—22no—
68C22×D283C22×C414yes—
69D1132—44no—
70C2×Dic283—22no—
71C2×Dic283—22no—
72C2×Dic283—22no—
73C2×Dic283—22no—
74C2×Dic283—22no—
75C2×Dic283—22no—
76C2×Dic283—22no—
77C2×Dic283—22no—
78C2×Dic283C22×C414yes—
79C4×D283—48no—
80C23×C283C2411yescharacteristic
81Q8×C283—44no—
82C2×C4×C283—21no—
83C2×C4×C283—21no—
84C2×C4×C283—21no—
85C2×C4×C283—21no—
86C2×C4×C283—21no—
87C2×C4×C283—21no—
88C2×C4×C283—22no—
89C2×C4×C283—22no—
90C2×C4×C283—22no—
91C2×C4×C283—22no—
92C2×C4×C283—22no—
93C2×C4×C283—22no—
94C2×C4×C283C22×C412yes—
95C2×C4×C283C22×C412yes—
96C2×C4×C283C22×C412yes—
97C22×C283C23×C411yescharacteristic
98C22×C283H(5,𝔽2)11yescharacteristic
99C22×C283H(5,𝔽2)11yescharacteristic
100C22×C283H(5,𝔽2)12yes—
101C22×C2832−1+412yes—
102C2×D283—24no—
103C2×D283—24no—
104C2×D283—24no—
105C2×D283—42no—
106C2×D283—42no—
107C2×D283—42no—
108C2×D283—42no—
109C4×C283—22no—
110C4×C283—22no—
111C4×C283—22no—
112C4×C283—42no—
113C4×C283—42no—
114Dic283—24no—
115Dic283—42no—
116Dic283—42no—
117C2×C283#51811yescharacteristic
118C2×C283#52012yes—
119C2×C283H(3,𝔽4)11yescharacteristic
120C2×C283#56212yes—
121C2×C283#56311yescharacteristic
122D283—44no—
123C283#237911yescharacteristic, complemented, Sylow
124#2379—2831nomaximal, Sylow
125C2×#124—2831no—
126#391—2831no—
127#391D28311yescharacteristic, complemented
128#388—2832no—
129#388—2832no—
130#386—2832no—
131#385—2832no—
132#385—2832no—
133#382—2832no—
134C22×Q8C2×D28311yescharacteristic
135C22×D4—2831no—
136C23×C4—2832no—
137#124—2838no—
138C2×C4⋊C4—2832no—
139C2×C4⋊C4—2832no—
140C2×C4⋊C4—2831no—
141C2×C4⋊C4C2×D28311yescharacteristic
142C2×C22⋊C4—2832no—
143C2×C22⋊C4—2832no—
144C2×C22⋊C4—2832no—
145C2×C22⋊C4—2832no—
146C2×C22⋊C4—2832no—
147C2×C42—2831no—
148C2×C42—2831no—
149C2×C42C2×D28311yescharacteristic
150#95—2832no—
151#95—2832no—
152#95—2832no—
153#95—2832no—
154#95C2×D28312yes—
155#95C2×D28312yes—
156C24—2832no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8C4×D28314yescomplemented
160C2×D4—2834no—
161C2×D4—11321no—
162C2×D4—11321no—
163C22×C4—2832no—
164C22×C4—2832no—
165C22×C4—2832no—
166C22×C4—11322no—
167C22×C4—11322no—
168C22×C4—11322no—
169C22×C4C22×D28312yes—
170C22×C4C22×D28312yes—
171C22×C4C22×D28311yescharacteristic
172C22×C4C22×D28311yescharacteristic
173C22×C4C22×D28311yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—11322no—
176C4⋊C4—11322no—
177C4⋊C4—11321no—
178C22⋊C4—2838no—
179C22⋊C4—2838no—
180C22⋊C4—11322no—
181C22⋊C4—11322no—
182C22⋊C4—11322no—
183C42—41no—
184C42—2834no—
185C42—11321no—
186C23—2834no—
187C23—5662no—
188C23—5662no—
189C23—5662no—
190C23—5662no—
191C23—5662no—
192C23—5662no—
193C23C23×D28311yescharacteristic
194Q8—44no—
195D4—11324no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—2834no—
209C2×C4—5662no—
210C2×C4—5662no—
211C2×C4—5662no—
212C2×C4—5662no—
213C2×C4—5662no—
214C2×C4—5662no—
215C2×C4—5662no—
216C2×C4—5662no—
217C2×C4—11328no—
218C2×C4C2×C4×D28312yes—
219C2×C4C2×C4×D28312yes—
220C2×C4C2×C4×D28312yes—
221C22—5664no—
222C22—5664no—
223C22—5664no—
224C22—11322no—
225C22—11322no—
226C22—11322no—
227C22—11322no—
228C22C22×C4×D28311yescharacteristic
229C22#2258365511yescharacteristic
230C22#2258365511yescharacteristic
231C22#2258365612yes—
232C22#2258365812yes—
233C4—22no—
234C4—22no—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—5664no—
239C4—11322no—
240C4—11322no—
241C2—11324no—
242C2#3182693811yescharacteristic
243C2#3182694712yes—
244C2#3182720711yescharacteristic
245C2#3182721112yes—
246C2#3182721411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D283
commutatorC22×C283C23×C4
Fitting#391×C283C2
socle, cosocleC23×C283C24
prime radicalC22×D1132C22
hypercenter#391D283
hypocenterC283#2379
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #31826372C2semidirect
3. #31826356C2semidirect
4. #31826354C2semidirect
6. #31826339C2semidirect
7. #31826336C2semidirect
8. #31826329C2semidirect
9. #391×C283C2semidirect
12. #22583599C4semidirect
48. (C2×D283)⋊C4C2×C4semidirect
49. (C2×D283)⋊C4C2×C4semidirect
51. C4×Dic283C2×C4semidirect
54. C2×Q8×C283C2×C4semidirect
123. C283#2379semidirect
127. #391D283semidirect
159. C2×Q8C4×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225835992
#318263722
#318263562
#318263542
#318263432
#318263392
#318263362
#318263292
#391×C2832
#2379283