Finite Group Databasebeta

← #38755429

Subgroups — #38755429

Order 25216.

372 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 115 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#27161799C211yesmaximal, characteristic, complemented
2#27161744C211yesmaximal, characteristic, complemented
3#27161437C212yesmaximal, complemented
4#27161431C211yesmaximal, characteristic, complemented
5#27161373C212yesmaximal, complemented
6#27161293C212yesmaximal
7#27161285C211yesmaximal, characteristic, complemented
8#27161206C211yesmaximal, characteristic, complemented
9#27161178C212yesmaximal, complemented
10C4×#15691343C211yesmaximal, characteristic
11#474×C197C211yesmaximal, characteristic, complemented
12C2×Q8×D197—21no—
13C2×#15691347C2212yes—
14C2×#15691343C2211yescharacteristic
15#20263326C2211yescharacteristic
16#20263325C2212yes—
17#20263319—22no—
18#20263313C2212yes—
19#20263305—21no—
20#20263304—22no—
21#20263302—21no—
22C4×Dic394C2211yescharacteristic
23#20263291C2212yes—
24C4×D788C2211yescharacteristic
25#20263288C2212yes—
26#20263285—21no—
27#20263259C2211yescharacteristic
28#20263255C2212yes—
29#20263251C2212yes—
30#20263249C2212yes—
31#20263249C2212yes—
32#20263247C2211yescharacteristic
33#20263246C2211yescharacteristic
34#20263236C2212yes—
35C4×#13096802C2211yescharacteristic
36C4⋊Q8×C197C2211yescharacteristic
37C4×D4×C197C2211yescharacteristic
38#107×C197C2212yes—
39C4⋊C8×C197C2211yescharacteristic
40#102×C197C2212yes—
41C2×Dic394—22no—
42C2×Dic394D411yescharacteristic
43C2×Dic394C2312yes—
44Q8×D197—44no—
45C2×D788C2311yescharacteristic
46C2×C4×D197—22no—
47C2×C4×D197—21no—
48#15691347—24no—
49#15691343—22no—
50C2×#13096802C2311yescharacteristic
51C2×#13096802C2311yescharacteristic
52C4⋊Dic197—21no—
53C4⋊Dic197—21no—
54C4⋊Dic197D411yescharacteristic
55C4⋊Dic197C2311yescharacteristic
56#15691337—22no—
57#15691337—22no—
58#15691337D412yes—
59C22⋊Dic197—22no—
60C22⋊Dic197—22no—
61#15691335—22no—
62#15691335—21no—
63C4×Dic197—22no—
64C4×Dic197C2312yes—
65C2×Q8×C197—21no—
66C2×D4×C197C2311yescharacteristic
67C22×C4×C197—21no—
68C2×C8×C197C2312yes—
69C4⋊C4×C197—21no—
70C4⋊C4×C197C2312yes—
71C4⋊C4×C197C2311yescharacteristic
72C22⋊C4×C197—21no—
73C42×C197C2311yescharacteristic
74C22×D197—21no—
75Dic394—22no—
76Dic394—22no—
77Dic394—22no—
78Dic394—44no—
79D788—21no—
80D788#4112yes—
81C2×Dic197—22no—
82C2×Dic197—21no—
83C2×Dic197C2×D412yes—
84C2×Dic197C2×D412yes—
85C4×D197—44no—
86C4×D197—42no—
87#13096802—21no—
88#13096802—21no—
89C23×C197—21no—
90Q8×C197—42no—
91D4×C197—21no—
92D4×C197—21no—
93C2×C4×C197—21no—
94C2×C4×C197—22no—
95C2×C4×C197—42no—
96C2×C4×C197C2×D411yescharacteristic
97C2×C4×C197C2×D411yescharacteristic
98C2×C4×C197C2411yescharacteristic
99C8×C197—22no—
100C22×C197—41no—
101C22×C197—41no—
102C22×C197—41no—
103C22×C197C22×D411yescharacteristic
104C2×D197—22no—
105C2×D197—41no—
106C2×D197—41no—
107C4×C197—21no—
108C4×C197—41no—
109C4×C197—42no—
110C4×C197AGL(1,ℤ/8)12yes—
111C4×C197C2×#4111yescharacteristic
112C4×C1972−1+411yescharacteristic
113Dic197—22no—
114Dic197—22no—
115Dic197—42no—
116Dic197—41no—
117C2×C197—81no—
118C2×C197#54811yescharacteristic
119C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C197#57811yescharacteristic
121D197—42no—
122C197#336711yescharacteristic, complemented, Sylow
123#3367—1971nomaximal, Sylow
124#556—1971no—
125D4×Q8—1971no—
126#489—1972no—
127#487—1971no—
128#477—1972no—
129#474D19711yescharacteristic, complemented
130#463—1972no—
131#461—1971no—
132#446—1971no—
133#440—1972no—
134C4×D8—1971no—
135C22×Q8—3941no—
136C2×SD16—1972no—
137C2×D8—1971no—
138C4⋊Q8—1972no—
139C4⋊Q8—1971no—
140C4⋊Q8C2×D19711yescharacteristic
141#126—3942no—
142#124—1972no—
143C22⋊Q8—3942no—
144C22⋊Q8—3941no—
145C22⋊Q8—3941no—
146C4×Q8—1972no—
147C4×Q8—1971no—
148C4×D4—1972no—
149C4×D4—1971no—
150C4×D4C2×D19711yescharacteristic
151#117—3941no—
152#107—1971no—
153#107C2×D19712yes—
154C4⋊C8—1972no—
155C4⋊C8C2×D19711yescharacteristic
156#103—1972no—
157#103—1972no—
158#103—1972no—
159#102—1971no—
160#102—1971no—
161#102C2×D19712yes—
162#97—1972no—
163C4×C8—1971no—
164C2×Q8—21no—
165C2×Q8—7884no—
166C2×Q8—3942no—
167C2×Q8—1972no—
168C2×Q8—1971no—
169C2×D4—1971no—
170C2×D4C22×D19711yescharacteristic
171C22×C4—21no—
172C22×C4—3942no—
173C22×C4—3941no—
174SD16—3944no—
175D8—3942no—
176C2×C8—1971no—
177C2×C8—1971no—
178C2×C8C22×D19712yes—
179C4⋊C4—21no—
180C4⋊C4—3942no—
181C4⋊C4—3942no—
182C4⋊C4—3941no—
183C4⋊C4—3941no—
184C4⋊C4—1971no—
185C4⋊C4—1972no—
186C4⋊C4—1971no—
187C4⋊C4C22×D19711yescharacteristic
188C4⋊C4C22×D19712yes—
189C22⋊C4—21no—
190C22⋊C4—3942no—
191C22⋊C4—3942no—
192C22⋊C4—3942no—
193C22⋊C4—3941no—
194C42—3942no—
195C42—1972no—
196C42C22×D19711yescharacteristic
197C23—21no—
198C23—3941no—
199Q8—42no—
200Q8—7884no—
201Q8—3942no—
202Q8—3942no—
203Q8—3942no—
204D4—21no—
205D4—21no—
206D4—3941no—
207D4—1972no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—7884no—
212C2×C4—7882no—
213C2×C4—3942no—
214C2×C4—3941no—
215C2×C4—1972no—
216C2×C4—1972no—
217C2×C4D4×D19711yescharacteristic
218C2×C4D4×D19711yescharacteristic
219C2×C4C23×D19711yescharacteristic
220C8—22no—
221C8—3941no—
222C8—3941no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—7881no—
227C22—7881no—
228C22—3942no—
229C22C2×D4×D19711yescharacteristic
230C4—21no—
231C4—41no—
232C4—42no—
233C4—7881no—
234C4—7882no—
235C4—3942no—
236C4—3942no—
237C4#2026335012yes—
238C4#41×D19711yescharacteristic
239C4#2026337411yescharacteristic
240C2—81no—
241C2—7882no—
242C2#2716173911yescharacteristic
243C2C2×#2026335011yescharacteristic
244C2#2716189511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#474×C197C2
socleC22×C197C22×D4
prime radicalC2×#15691343C22
hypercenter#474D197
hypocenterC197#3367
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #27161799C2semidirect
2. #27161744C2semidirect
3. #27161437C2semidirect
4. #27161431C2semidirect
5. #27161373C2semidirect
7. #27161285C2semidirect
8. #27161206C2semidirect
9. #27161178C2semidirect
11. #474×C197C2semidirect
122. C197#3367semidirect
129. #474D197semidirect

Maximal subgroups

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

subgroupindex
#271617992
#271617442
#271614372
#271614312
#271613732
#271612932
#271612852
#271612062
#271611782
C4×#156913432
#474×C1972
#3367197