Finite Group Databasebeta

← #38742791

Subgroups — #38742791

Order 25216.

316 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 81 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×C4⋊C4×D197C211yesmaximal, characteristic
2#27161042C212yesmaximal
3#27160848C211yesmaximal, characteristic, complemented
4#27160847C212yesmaximal, complemented
5#343×C197C211yesmaximal, characteristic, complemented
6C22×C4×D197C412yes—
7C22×C4×D197C2211yescharacteristic
8C2×#15691342C2212yes—
9C2×C4⋊Dic197C2211yescharacteristic
10C2×#15691337C412yes—
11C4⋊C4×D197—22no—
12C4⋊C4×D197—22no—
13#20263238—22no—
14#20263238—22no—
15C2×M16×C197C2212yes—
16C2×C4⋊C4×C197C2211yescharacteristic
17C23×D197C2×C411yescharacteristic
18C22×Dic197C2×C411yescharacteristic
19C22×Dic197C2×C412yes—
20C2×C4×D197—22no—
21C2×C4×D197—22no—
22C2×C4×D197—21no—
23C2×C4×D197—22no—
24C2×C4×D197—21no—
25C2×C4×D197—44no—
26C2×C4×D197D412yes—
27C2×C4×D197D412yes—
28#15691342—22no—
29#15691342—22no—
30C2×#13096802—22no—
31C4⋊Dic197—22no—
32#15691337—24no—
33C22×C4×C197C2×C412yes—
34C22×C4×C197C2311yescharacteristic
35M16×C197—22no—
36M16×C197—22no—
37C2×C8×C197—22no—
38C4⋊C4×C197—22no—
39C22×D197—21no—
40C22×D197—21no—
41C22×D197—22no—
42C22×D197—22no—
43C22×D197C22⋊C411yescharacteristic
44C22×D197C22⋊C411yescharacteristic
45C2×Dic197—21no—
46C2×Dic197—21no—
47C2×Dic197—22no—
48C2×Dic197—42no—
49C2×Dic197C22⋊C412yes—
50C4×D197—22no—
51C4×D197—22no—
52C4×D197—22no—
53C4×D197—22no—
54C4×D197—44no—
55C4×D197—44no—
56#13096802—42no—
57C23×C197C22×C411yescharacteristic
58C2×C4×C197—22no—
59C2×C4×C197—42no—
60C2×C4×C197C2×D412yes—
61C2×C4×C197#4111yescharacteristic
62C2×C4×C197#4112yes—
63C2×C4×C197#4111yescharacteristic
64C8×C197—42no—
65C22×C197—21no—
66C22×C197—21no—
67C22×C197C2×C22⋊C411yescharacteristic
68C22×C197#11711yescharacteristic
69C22×C197#11711yescharacteristic
70C2×D197—24no—
71C2×D197—24no—
72C2×D197—21no—
73C2×D197—21no—
74C2×D197—21no—
75C2×D197—21no—
76C2×D197#10012yes—
77C2×D197#10112yes—
78C4×C197—42no—
79C4×C197#12312yes—
80C4×C197#12312yes—
81Dic197—22no—
82Dic197—42no—
83C2×C197—21no—
84C2×C197—21no—
85C2×C197#38111yescharacteristic
86C2×C197C2×#10011yescharacteristic
87C2×C197C2×#10111yescharacteristic
88D197—22no—
89D197—22no—
90C197#182111yescharacteristic, complemented, Sylow
91#1821—1971nomaximal, Sylow
92C22×C4⋊C4—1971no—
93#407—1972no—
94#343—1972no—
95#343—1971no—
96#343D19711yescharacteristic, complemented
97C23×C4—1972no—
98C23×C4—1971no—
99C2×M16—1972no—
100C2×M16C2×D19712yes—
101C2×C4⋊C4—3942no—
102C2×C4⋊C4—3942no—
103C2×C4⋊C4—1972no—
104C2×C4⋊C4—1971no—
105C2×C4⋊C4C2×D19711yescharacteristic
106C22⋊C8—3942no—
107C22⋊C8—3942no—
108C24—1971no—
109C22×C4—7884no—
110C22×C4—3942no—
111C22×C4—3942no—
112C22×C4—3941no—
113C22×C4—3942no—
114C22×C4—3941no—
115C22×C4—1972no—
116C22×C4—1971no—
117C22×C4—1972no—
118C22×C4—1972no—
119C22×C4C4×D19712yes—
120C22×C4C22×D19711yescharacteristic
121M16—22no—
122M16—22no—
123M16—3942no—
124M16—3942no—
125C2×C8—22no—
126C2×C8—3942no—
127C4⋊C4—22no—
128C4⋊C4—3944no—
129C4⋊C4—3942no—
130C23—3942no—
131C23—3942no—
132C23—3941no—
133C23—3941no—
134C23—1971no—
135C23—1971no—
136C23C2×C4×D19711yescharacteristic
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—7884no—
140C2×C4—7884no—
141C2×C4—7882no—
142C2×C4—3942no—
143C2×C4—3941no—
144C2×C4—3941no—
145C2×C4—3942no—
146C2×C4—3942no—
147C2×C4—3942no—
148C2×C4—3942no—
149C2×C4—1972no—
150C2×C4D4×D19712yes—
151C2×C4#1569135711yescharacteristic
152C2×C4#1569135812yes—
153C2×C4#1569135911yescharacteristic
154C8—42no—
155C8—7882no—
156C22—21no—
157C22—21no—
158C22—3944no—
159C22—3944no—
160C22—3941no—
161C22—3941no—
162C22—3941no—
163C22—3941no—
164C22—1972no—
165C22—1972no—
166C22C22⋊C4×D19711yescharacteristic
167C22#2026328311yescharacteristic
168C22#2026328411yescharacteristic
169C4—42no—
170C4—7882no—
171C4—3942no—
172C4#2026330812yes—
173C4#2026331112yes—
174C2—21no—
175C2—21no—
176C2—3942no—
177C2—3942no—
178C2#2716094111yescharacteristic
179C2#100×D19711yescharacteristic
180C2#101×D19711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D197
commutatorC23×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#343×C197C2
socleC22×C197C2×C22⋊C4
prime radicalC23×D197C2×C4
cosocleC22×C4×C197C23
hypercenter#343D197
hypocenterC197#1821
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
3. #27160848C2semidirect
4. #27160847C2semidirect
5. #343×C197C2semidirect
90. C197#1821semidirect
96. #343D197semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1972
#271610422
#271608482
#271608472
#343×C1972
#1821197