Finite Group Databasebeta

← #38741843

Subgroups — #38741843

Order 25216.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#27161790C211yesmaximal, characteristic
2#27161371C211yesmaximal, characteristic, complemented
3#27161366C211yesmaximal, characteristic
4#27160828C211yesmaximal, characteristic, complemented
5#27160811C211yesmaximal, characteristic, complemented
6#27160808C211yesmaximal, characteristic
7#494×C197C211yesmaximal, characteristic, complemented
8#20263317—21no—
9#20263306—21no—
10#20263302—22no—
11C4×Dic394C2211yescharacteristic
12C4×D788C2211yescharacteristic
13#20263288—21no—
14C4⋊C4×D197—21no—
15#20263259—21no—
16#20263257—21no—
17C4⋊#13096802C2211yescharacteristic
18C4⋊#13096802C2211yescharacteristic
19#20263249—21no—
20#20263246—21no—
21C2×Q16×C197—22no—
22C4⋊Q8×C197C2211yescharacteristic
23C4⋊Q8×C197C2211yescharacteristic
24C4×C8×C197C2211yescharacteristic
25C2×Dic394D411yescharacteristic
26C2×D788D411yescharacteristic
27C2×C4×D197—22no—
28C2×C4×D197—21no—
29C2×#13096802—21no—
30C2×#13096802—21no—
31C4⋊Dic197—22no—
32C4⋊Dic197D411yescharacteristic
33C4⋊Dic197D411yescharacteristic
34#15691337—22no—
35#15691337—21no—
36#15691335—22no—
37#15691335—21no—
38C4×Dic197—21no—
39C2×Q8×C197—21no—
40C2×Q8×C197D412yes—
41Q16×C197—44no—
42C2×C8×C197—21no—
43C4⋊C4×C197—21no—
44C4⋊C4×C197—21no—
45C4⋊C4×C197—21no—
46C4⋊C4×C197—21no—
47C42×C197C2311yescharacteristic
48C22×D197—21no—
49Dic394—21no—
50Dic394—21no—
51D788—21no—
52D788Q1612yescomplemented
53C2×Dic197—21no—
54C2×Dic197—21no—
55C2×Dic197—21no—
56C4×D197—41no—
57C4×D197—41no—
58C4×D197—82no—
59#13096802—41no—
60#13096802—41no—
61Q8×C197—42no—
62Q8×C197—42no—
63C2×C4×C197—22no—
64C2×C4×C197—41no—
65C2×C4×C197C2×D411yescharacteristic
66C2×C4×C197C2×D411yescharacteristic
67C2×C4×C197C2×D411yescharacteristic
68C8×C197—22no—
69C22×C197#12011yescharacteristic
70C2×D197—22no—
71C2×D197—41no—
72C2×D197—41no—
73C4×C197—21no—
74C4×C197—42no—
75C4×C197—81no—
76C4×C197C2×Q1611yescharacteristic
77C4×C197#13511yescharacteristic
78C4×C197#13711yescharacteristic
79C4×C197#13711yescharacteristic
80Dic197—41no—
81Dic197—41no—
82Dic197—41no—
83C2×C197#45111yescharacteristic
84C2×C197#45211yescharacteristic
85C2×C197#45311yescharacteristic
86D197—42no—
87C197#163211yescharacteristic, complemented, Sylow
88#1632—1971nomaximal, Sylow
89#554—1971no—
90#494D19711yescharacteristic, complemented
91#477—1971no—
92#476—1971no—
93#333—1971no—
94Q8⋊C8—1971no—
95D4⋊C8—1971no—
96C2×Q16—22no—
97C4⋊Q8C2×D19711yescharacteristic
98C4⋊Q8C2×D19711yescharacteristic
99#125—3941no—
100C22⋊Q8—3941no—
101C22⋊Q8—3942no—
102C4×Q8—1971no—
103C4×D4—3941no—
104C4×D4—1971no—
105C2×C4⋊C4—3941no—
106#107—3941no—
107#106—3941no—
108C4⋊C8—1971no—
109C4⋊C8—1971no—
110#103—3941no—
111#102—3941no—
112C4×C8C2×D19711yescharacteristic
113C2×Q8—21no—
114C2×Q8—1971no—
115C2×Q8C22⋊D19712yes—
116C2×D4—1971no—
117C22×C4—3942no—
118C22×C4—3941no—
119Q16—44no—
120C2×C8—21no—
121C2×C8—3941no—
122C2×C8—3941no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—3942no—
128C4⋊C4—3942no—
129C4⋊C4—3941no—
130C4⋊C4—1971no—
131C4⋊C4—1971no—
132C22⋊C4—3942no—
133C22⋊C4—3941no—
134C42—3941no—
135C42C22×D19711yescharacteristic
136C23—3941no—
137Q8—42no—
138Q8—42no—
139Q8—3941no—
140Q8—3941no—
141D4—3941no—
142D4—1972no—
143C2×C4—22no—
144C2×C4—41no—
145C2×C4—7881no—
146C2×C4—7881no—
147C2×C4—15762no—
148C2×C4—3941no—
149C2×C4—3941no—
150C2×C4—3941no—
151C2×C4D4×D19711yescharacteristic
152C2×C4D4×D19711yescharacteristic
153C2×C4C2×C22⋊D19711yescharacteristic
154C8—22no—
155C8—7881no—
156C8—7881no—
157C22—7881no—
158C22—7881no—
159C22—3942no—
160C22#2026329411yescharacteristic
161C4—21no—
162C4—42no—
163C4—81no—
164C4—7881no—
165C4—7881no—
166C4—7881no—
167C4Q16×D19711yescharacteristic
168C4#2026334911yescharacteristic
169C4#2026335811yescharacteristic
170C4#2026335811yescharacteristic
171C2—7882no—
172C2#2716123311yescharacteristic
173C2#2716124011yescharacteristic
174C2#2716124611yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#494×C197C2
socleC22×C197#120
prime radicalC2×D788D4
hypercenter#494D197
hypocenterC197#1632
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
2. #27161371C2semidirect
4. #27160828C2semidirect
5. #27160811C2semidirect
7. #494×C197C2semidirect
52. D788Q16semidirect
87. C197#1632semidirect
90. #494D197semidirect

Maximal subgroups

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

subgroupindex
#271617902
#271613712
#271613662
#271608282
#271608112
#271608082
#494×C1972
#1632197