Finite Group Databasebeta

← #27158029

Subgroups — #27158029

Order 12416.

250 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 54 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#20261721C212yesmaximal, complemented
2#20261646C212yesmaximal, complemented
3#20261644C212yesmaximal, complemented
4#20261285C211yesmaximal, characteristic
5D97⋊M16—22no—
6#15690871C2214yes—
7#15690870—22no—
8C2×#13095511C411yescharacteristic
9C2×#13095511C2211yescharacteristic
10#15690810C2211yescharacteristic
11#15690807C411yescharacteristic
12#15690798C411yescharacteristic
13#15690798C2211yescharacteristic
14#15690784—22no—
15C4×D388—22no—
16C2×C42×C97C411yescharacteristic
17#13095521—42no—
18#13095520—24no—
19#13095519—42no—
20#13095511—41no—
21#13095511—41no—
22#13095511—41no—
23#13095511—41no—
24C2×Dic194C2×C411yescharacteristic
25C2×Dic194C2311yescharacteristic
26C2×D388—22no—
27C2×D388C2×C411yescharacteristic
28C2×D388C2×C411yescharacteristic
29C2×C22⋊D97—21no—
30C2×C22⋊D97D412yes—
31C2×C4×D97—21no—
32C2×C4×D97D412yes—
33C4⋊Dic97—22no—
34#13095488—22no—
35#13095488—22no—
36C22×C4×C97—21no—
37C22×C4×C97C2×C411yescharacteristic
38C42×C97—21no—
39C42×C97C2×C411yescharacteristic
40C42×C97C2×C411yescharacteristic
41C23×C97C22⋊C411yescharacteristic
42C22×D97—21no—
43C22×D97C22⋊C412yes—
44C2×C4×C97—21no—
45C2×C4×C97—21no—
46C2×C4×C97—21no—
47C2×C4×C97—21no—
48C2×C4×C97—42no—
49C2×C4×C97C22⋊C411yescharacteristic
50C2×C4×C97C22×C411yescharacteristic
51C22⋊D97—42no—
52C22⋊D97—42no—
53C2×Dic97—21no—
54C2×Dic97C2×D412yes—
55D388—41no—
56D388—41no—
57D388—44no—
58C4×D97—22no—
59C4×D97—22no—
60C4×D97—41no—
61C4×D97—41no—
62Dic194—41no—
63Dic194—41no—
64#11715897—44no—
65C22×C97—21no—
66C22×C97—41no—
67C22×C97C2×C22⋊C411yescharacteristic
68C2×D97—22no—
69C2×D97—42no—
70C2×D97—42no—
71C2×D97—41no—
72C4×C97—22no—
73C4×C97—21no—
74C4×C97—41no—
75C4×C97#9911yescharacteristic
76C4×C97#9911yescharacteristic
77Dic97—22no—
78Dic97—41no—
79C2×C97—21no—
80C2×C97—41no—
81C2×C97C2×#9911yescharacteristic
82D97—42no—
83D97—81no—
84#2132—971nomaximal, Sylow
85C97#213211yescharacteristic, complemented, Sylow
86#532—971no—
87#413—972no—
88#356—972no—
89#355—972no—
90C2×#41—971no—
91C2×#41—971no—
92C2×M16—1942no—
93C4⋊Q8—971no—
94#127—971no—
95#124—971no—
96#124—971no—
97#121—1942no—
98C4×D4—1942no—
99C2×C42#8760411yescharacteristic
100#101—974no—
101#100—1942no—
102#41—3881no—
103#41—3881no—
104#41—3881no—
105#41—3881no—
106C2×Q8—971no—
107C2×Q8—971no—
108C2×D4—972no—
109C2×D4—971no—
110C2×D4—971no—
111C2×D4—1942no—
112C2×D4—1941no—
113C22×C4—21no—
114C22×C4—972no—
115C22×C4—1941no—
116C22×C4C2×#8760411yescharacteristic
117M16—3882no—
118M16—1944no—
119C2×C8—3882no—
120C4⋊C4—1942no—
121C22⋊C4—1942no—
122C22⋊C4—1942no—
123C42—21no—
124C42C2×#8760411yescharacteristic
125C42C2×#8760411yescharacteristic
126C23—972no—
127C23—1941no—
128C23C22⋊#8760411yescharacteristic
129Q8—3881no—
130Q8—3881no—
131D4—3881no—
132D4—3884no—
133D4—3881no—
134D4—3882no—
135D4—3882no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—42no—
141C2×C4—972no—
142C2×C4—3881no—
143C2×C4—3881no—
144C2×C4—1941no—
145C2×C4—1942no—
146C2×C4—1942no—
147C2×C4C22⋊#8760411yescharacteristic
148C2×C4C22×#8760411yescharacteristic
149C8—3884no—
150C22—21no—
151C22—41no—
152C22—3882no—
153C22—3882no—
154C22—3881no—
155C22—1942no—
156C22C2×C22⋊#8760411yescharacteristic
157C4—22no—
158C4—21no—
159C4—41no—
160C4—3881no—
161C4—1942no—
162C4#1569086911yescharacteristic
163C4#1569086911yescharacteristic
164C2—21no—
165C2—41no—
166C2—7761no—
167C2—3882no—
168C2C2×#1569086911yescharacteristic
subgroupG/H
centerC4#15690869
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×C42×C97C4
socleC2×C97C2×#99
prime radical#20261285C2
cosocleC2×Dic194C23
hypercenterC2×C42#87604
hypocenterC97#2132
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. #20261721C2semidirect
2. #20261646C2semidirect
3. #20261644C2semidirect
85. C97#2132semidirect

Maximal subgroups

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

subgroupindex
#202617212
#202616462
#202616442
#202612852
#213297