Finite Group Databasebeta

← #38741345

Subgroups — #38741345

Order 25216.

238 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 59 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#27161628C211yesmaximal, characteristic
2#27161134C212yesmaximal, complemented
3#27160819C211yesmaximal, characteristic, complemented
4#27160818C212yesmaximal, complemented
5#329×C197C211yesmaximal, characteristic, complemented
6C2×#15691359C411yescharacteristic
7C2×#15691357C411yescharacteristic
8#20263324C411yescharacteristic
9#20263322C411yescharacteristic
10#20263314C2211yescharacteristic
11#20263299—22no—
12#20263286—22no—
13C42×D197C2211yescharacteristic
14#20263255—22no—
15#20263238—22no—
16#20263237C2212yes—
17#124×C197C2211yescharacteristic
18#97×C197C2212yes—
19#15691359—42no—
20#15691357—42no—
21C2×Dic394C2×C411yescharacteristic
22C2×D788—21no—
23C2×D788C2×C411yescharacteristic
24C2×C22⋊D197—21no—
25C2×C22⋊D197—22no—
26C22×Dic197—21no—
27C2×C4×D197—21no—
28C2×C4×D197C2×C411yescharacteristic
29C2×C4×D197D412yes—
30C2×#13096802—22no—
31#15691337—22no—
32#15691335—22no—
33C4×Dic197C2×C411yescharacteristic
34C4×Dic197D412yes—
35C2×Q8×C197C2×C411yescharacteristic
36C2×D4×C197C2×C411yescharacteristic
37C2×C8×C197—22no—
38C22⋊C4×C197—22no—
39C42×C197C2311yescharacteristic
40C22×D197—21no—
41C22×D197C22⋊C411yescharacteristic
42Dic394—41no—
43D788—41no—
44D788—42no—
45C22⋊D197—44no—
46C22⋊D197—42no—
47C2×Dic197—21no—
48C2×Dic197—42no—
49C2×Dic197C22⋊C412yes—
50C2×Dic197C22⋊C411yescharacteristic
51C4×D197—22no—
52C4×D197—42no—
53C4×D197—42no—
54#13096802—42no—
55C23×C197—21no—
56Q8×C197—41no—
57D4×C197—41no—
58C2×C4×C197—21no—
59C2×C4×C197C22×C411yescharacteristic
60C2×C4×C197C2×D412yes—
61C8×C197—42no—
62C22×C197—41no—
63C22×C197—42no—
64C22×C197C2×C22⋊C411yescharacteristic
65C2×D197—21no—
66C2×D197—22no—
67C2×D197—41no—
68C2×D197—42no—
69C4×C197—21no—
70C4×C197—22no—
71C4×C197—41no—
72Dic197—22no—
73Dic197—41no—
74Dic197#10012yes—
75C2×C197—81no—
76C2×C197C2×#10011yescharacteristic
77C2×C197#42112yes—
78D197—41no—
79D197—81no—
80C197#153611yescharacteristic, complemented, Sylow
81#1536—1971nomaximal, Sylow
82#532—1971no—
83#432—1972no—
84#329—1971no—
85#329—1972no—
86#329D19711yescharacteristic, complemented
87C2×#41—1971no—
88C2×#41—1971no—
89C4⋊Q8—1971no—
90#127—1971no—
91#124—1971no—
92#124C2×D19711yescharacteristic
93#121—3942no—
94C4×D4—3942no—
95C2×C42—1971no—
96C4⋊C8—3942no—
97C22⋊C8—3942no—
98#97—1972no—
99#97C2×D19712yes—
100#41—7882no—
101#41—7882no—
102C2×Q8—1971no—
103C2×Q8C4×D19711yescharacteristic
104C2×D4—3941no—
105C2×D4—3941no—
106C2×D4—3942no—
107C2×D4—1971no—
108C2×D4C4×D19711yescharacteristic
109C22×C4—3941no—
110C22×C4—3941no—
111C22×C4—1972no—
112C22×C4—1971no—
113C2×C8—22no—
114C2×C8—3942no—
115C4⋊C4—3942no—
116C22⋊C4—22no—
117C22⋊C4—3942no—
118C42—1971no—
119C42—1972no—
120C42C22×D19711yescharacteristic
121C23—21no—
122C23—3941no—
123C23—1971no—
124Q8—41no—
125Q8—7881no—
126D4—41no—
127D4—7881no—
128D4—7882no—
129D4—7882no—
130D4—7884no—
131C2×C4—21no—
132C2×C4—7882no—
133C2×C4—7882no—
134C2×C4—7882no—
135C2×C4—3942no—
136C2×C4—3941no—
137C2×C4—1972no—
138C2×C4—1971no—
139C2×C4C2×C4×D19711yescharacteristic
140C2×C4D4×D19712yes—
141C8—42no—
142C8—7882no—
143C22—42no—
144C22—41no—
145C22—7882no—
146C22—7881no—
147C22—3942no—
148C22—3941no—
149C22C22⋊C4×D19711yescharacteristic
150C4—22no—
151C4—21no—
152C4—41no—
153C4—7881no—
154C4—3942no—
155C4—1972no—
156C2—81no—
157C2—7881no—
158C2—15761no—
159C2#100×D19711yescharacteristic
160C2#2716109412yes—
subgroupG/H
centerC22C22⋊C4×D197
commutatorC2×C4×C197C22×C4
FrattiniC42C22×D197
Fitting#329×C197C2
socleC22×C197C2×C22⋊C4
prime radical#27161628C2
cosocleC42×C197C23
hypercenter#329D197
hypocenterC197#1536
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #27161134C2semidirect
3. #27160819C2semidirect
4. #27160818C2semidirect
5. #329×C197C2semidirect
80. C197#1536semidirect
86. #329D197semidirect

Maximal subgroups

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

subgroupindex
#271616282
#271611342
#271608192
#271608182
#329×C1972
#1536197