Finite Group Databasebeta

← #27143039

Subgroups — #27143039

Order 12416.

264 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 67 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#20260763C211yesmaximal, characteristic, complemented
2#20260731C212yesmaximal, complemented
3#20260709C212yesmaximal, complemented
4#20260492C211yesmaximal, characteristic, complemented
5#575×C97C211yesmaximal, characteristic
6C22×#11715895C2211yescharacteristic
7#15690769C2212yes—
8#15690769C2211yescharacteristic
9#15690739—22no—
10#15690734—22no—
11#15690731—22no—
12#15690725—24no—
13C2×#41×C97C2212yes—
14#137×C97—22no—
15AGL(1,ℤ/8)×C97—22no—
16#135×C97—22no—
17C2×M16×C97C2211yescharacteristic
18C2×#11715895—22no—
19C2×#11715895—21no—
20C4⋊Dic97—21no—
21C4⋊Dic97—22no—
22C22⋊Dic97—22no—
23C22⋊Dic97D412yes—
24C4×Dic97—21no—
25C4×Dic97—22no—
26#41×C97—22no—
27#41×C97—22no—
28#41×C97—22no—
29#41×C97—22no—
30C2×Q8×C97D412yes—
31C2×D4×C97—22no—
32C2×D4×C97D412yes—
33C22×C4×C97—22no—
34C22×C4×C97C2311yescharacteristic
35Q16×C97—22no—
36SD16×C97—24no—
37D8×C97—22no—
38M16×C97—22no—
39C2×C8×C97D412yes—
40C23×C97—22no—
41C23×C97C2×D411yescharacteristic
42Q8×C97—22no—
43Q8×C97—22no—
44D4×C97—22no—
45D4×C97—22no—
46D4×C97—42no—
47D4×C97—42no—
48C2×C4×C97—22no—
49C2×C4×C97—42no—
50C2×C4×C97—42no—
51C2×C4×C97C2×D412yes—
52C2×C4×C97C2×D411yescharacteristic
53C2×C4×C97#4111yescharacteristic
54C2×C4×C97#4112yes—
55C2×Dic97—22no—
56C2×Dic97—42no—
57C8×C97—22no—
58#11715895—41no—
59C22×C97—41no—
60C22×C97—42no—
61C22×C97—82no—
62C22×C97#12111yescharacteristic
63C22×C97#12112yes—
64C4×C97—42no—
65C4×C97#12011yescharacteristic
66C4×C97#12312yes—
67C4×C97#12411yescharacteristic
68Dic97—42no—
69Dic97—82no—
70C2×C97—21no—
71C2×C97—22no—
72C2×C97—82no—
73C2×C97#39411yescharacteristic
74#2053—971nomaximal, Sylow
75C97#205311yescharacteristic, complemented, Sylow
76#575D9711yescharacteristic
77#424—971no—
78#416—972no—
79#410—972no—
80H(3,ℤ/4)—971no—
81C2×#41C2×D9712yes—
82#137—22no—
83AGL(1,ℤ/8)—22no—
84#135—22no—
85C2×M16C2×D9711yescharacteristic
86C22×C8—971no—
87#117—971no—
88#117—972no—
89C4⋊C8—1942no—
90#103—1942no—
91#102—1942no—
92#99—1944no—
93#41—22no—
94#41—22no—
95#41—22no—
96#41—22no—
97C2×Q8C22⋊D9712yes—
98C2×D4—22no—
99C2×D4C22⋊D9712yes—
100C22×C4—22no—
101C22×C4C22×D9711yescharacteristic
102Q16—22no—
103SD16—24no—
104D8—22no—
105M16—22no—
106C2×C8—1942no—
107C2×C8—1941no—
108C2×C8C22⋊D9712yes—
109C4⋊C4—1942no—
110C4⋊C4—1941no—
111C22⋊C4—972no—
112C22⋊C4—1942no—
113C42—1942no—
114C42—1941no—
115C23—22no—
116C23D4×D9711yescharacteristic
117Q8—22no—
118Q8—22no—
119D4—22no—
120D4—22no—
121D4—42no—
122D4—42no—
123C2×C4—22no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—3882no—
127C2×C4—1942no—
128C2×C4C2×C22⋊D9712yes—
129C2×C4C2×C22⋊D9711yescharacteristic
130C2×C4#1309551011yescharacteristic
131C2×C4#1309551012yes—
132C8—22no—
133C8—3881no—
134C22—41no—
135C22—42no—
136C22—82no—
137C22#1569078111yescharacteristic
138C22#1569078212yes—
139C4—42no—
140C4—7762no—
141C4—3882no—
142C4#1569077911yescharacteristic
143C4#1569079112yes—
144C4#1569079711yescharacteristic
145C2—22no—
146C2—21no—
147C2—82no—
148C2#2026064911yescharacteristic
subgroupG/H
centerC4#15690797
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting, prime radical#575×C97C2
socleC2×C97#394
hypercenter#575D97
hypocenterC97#2053
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #20260763C2semidirect
2. #20260731C2semidirect
3. #20260709C2semidirect
4. #20260492C2semidirect
75. C97#2053semidirect

Maximal subgroups

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

subgroupindex
#202607632
#202607312
#202607092
#202604922
#575×C972
#205397