Finite Group Databasebeta

← #38743095

Subgroups — #38743095

Order 25216.

332 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 89 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#27161625C211yesmaximal, characteristic
2#27161095C212yesmaximal, complemented
3#27161053C211yesmaximal, characteristic, complemented
4#27160840C212yesmaximal, complemented
5#568×C197C211yesmaximal, characteristic, complemented
6C2×#15691358C2211yescharacteristic
7#20263324C411yescharacteristic, complemented
8#20263322C411yescharacteristic, complemented
9#20263315C411yescharacteristic, complemented
10#20263313C411yescharacteristic, complemented
11#20263298—22no—
12C4×C22⋊D197—22no—
13#20263285C2212yes—
14C2×C4×Dic197C2211yescharacteristic
15#20263252—24no—
16#20263241—22no—
17C2×#41×C197C2211yescharacteristic
18#131×C197—24no—
19C2×M16×C197—22no—
20C2×M16×C197C2212yes—
21#15691358—22no—
22#15691358—22no—
23C2×Dic394C2×C411yescharacteristic
24C2×D788C2×C411yescharacteristic
25C2×C22⋊D197—22no—
26C2×C22⋊D197—21no—
27C22×Dic197—21no—
28C2×C4×D197—21no—
29C4⋊Dic197—22no—
30#15691337—22no—
31C22⋊Dic197—22no—
32C22⋊Dic197—22no—
33#15691335—22no—
34C4×Dic197—21no—
35C4×Dic197—22no—
36C4×Dic197C2×C411yescharacteristic
37C4×Dic197C2×C411yescharacteristic
38#41×C197—22no—
39#41×C197—22no—
40C2×Q8×C197C2×C411yescharacteristic
41C2×D4×C197C2×C411yescharacteristic
42C2×D4×C197D412yes—
43C22×C4×C197D412yes—
44C22×C4×C197C2311yescharacteristic
45M16×C197—24no—
46M16×C197—22no—
47M16×C197D414yescomplemented
48C2×C8×C197—24no—
49C2×C8×C197—22no—
50C22×D197—21no—
51Dic394—22no—
52D788—22no—
53C22⋊D197—44no—
54C22⋊D197—42no—
55C2×Dic197—21no—
56C2×Dic197—21no—
57C2×Dic197—21no—
58C2×Dic197—42no—
59C2×Dic197—42no—
60C4×D197—41no—
61C4×D197—41no—
62C23×C197C22⋊C412yes—
63C23×C197C2×D411yescharacteristic
64Q8×C197—22no—
65D4×C197—24no—
66D4×C197—22no—
67C2×C4×C197—22no—
68C2×C4×C197—22no—
69C2×C4×C197C22⋊C412yes—
70C2×C4×C197C22×C411yescharacteristic
71C2×C4×C197C2×D412yes—
72C2×C4×C197C2×D411yescharacteristic
73C2×C4×C197#4112yes—
74C8×C197—24no—
75C22×C197—22no—
76C22×C197—41no—
77C22×C197—42no—
78C22×C197C2×C22⋊C411yescharacteristic
79C22×C197#12112yes—
80C2×D197—42no—
81C2×D197—41no—
82C4×C197—22no—
83C4×C197C4×D412yes—
84C4×C197#12011yescharacteristic
85C4×C197#12311yescharacteristic
86Dic197—22no—
87Dic197—41no—
88Dic197—41no—
89Dic197—82no—
90C2×C197—21no—
91C2×C197—22no—
92C2×C197—42no—
93C2×C197#38611yescharacteristic
94D197—81no—
95C197#188411yescharacteristic, complemented, Sylow
96#1884—1971nomaximal, Sylow
97#568D19711yescharacteristic, complemented
98#532—1971no—
99#421—1972no—
100#410—1971no—
101#339—1972no—
102C2×#41—1971no—
103C2×#41C2×D19711yescharacteristic
104#131—24no—
105C2×M16—22no—
106C2×M16C2×D19712yes—
107C4⋊Q8—1971no—
108#127—1971no—
109#124—1971no—
110#124—1971no—
111#121—3942no—
112C4×D4—3942no—
113#117—1972no—
114C2×C42—1971no—
115#104—3944no—
116#99—3942no—
117#41—22no—
118#41—22no—
119#41—3942no—
120#41—3942no—
121C2×Q8—1971no—
122C2×Q8C4×D19711yescharacteristic
123C2×D4—3942no—
124C2×D4—3941no—
125C2×D4—1971no—
126C2×D4C4×D19711yescharacteristic
127C2×D4D78812yes—
128C22×C4—3941no—
129C22×C4—3941no—
130C22×C4C22⋊D19712yes—
131C22×C4C22×D19711yescharacteristic
132M16—24no—
133M16—22no—
134M16C22⋊D19714yescomplemented
135C2×C8—24no—
136C2×C8—22no—
137C4⋊C4—3942no—
138C4⋊C4—3942no—
139C22⋊C4—3942no—
140C22⋊C4—3942no—
141C22⋊C4—3942no—
142C42—3942no—
143C42—3941no—
144C42—1971no—
145C42—1971no—
146C23—3941no—
147C23#1569133512yes—
148C23C2×D78811yescharacteristic
149Q8—22no—
150Q8—3942no—
151D4—24no—
152D4—22no—
153D4—7882no—
154D4—7884no—
155D4—3942no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—7882no—
159C2×C4—7881no—
160C2×C4—7881no—
161C2×C4—7882no—
162C2×C4—3941no—
163C2×C4—3941no—
164C2×C4—3941no—
165C2×C4#1569133512yes—
166C2×C4C2×C4×D19711yescharacteristic
167C2×C4C2×C22⋊D19712yes—
168C2×C4C2×C22⋊D19711yescharacteristic
169C2×C4#1569135812yes—
170C8—24no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—7882no—
175C22—7881no—
176C22C2×#1569133511yescharacteristic
177C22#2026330012yes—
178C4—22no—
179C4—7881no—
180C4—7881no—
181C4—15762no—
182C4—3942no—
183C4C4×C22⋊D19712yes—
184C4#2026329511yescharacteristic
185C4#2026330911yescharacteristic
186C2—21no—
187C2—22no—
188C2—42no—
189C2—15761no—
190C2#2716095811yescharacteristic
subgroupG/H
centerC4#20263295
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#568×C197C2
socleC2×C197#386
prime radical#27161625C2
cosocleC22×C4×C197C23
hypercenter#568D197
hypocenterC197#1884
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #27161095C2semidirect
3. #27161053C2semidirect
4. #27160840C2semidirect
5. #568×C197C2semidirect
7. #20263324C4semidirect
8. #20263322C4semidirect
9. #20263315C4semidirect
10. #20263313C4semidirect
47. M16×C197D4semidirect
95. C197#1884semidirect
97. #568D197semidirect
134. M16C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
#271616252
#271610952
#271610532
#271608402
#568×C1972
#1884197