Finite Group Databasebeta

← #14530087

Subgroups — #14530087

Order 2176.

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#11762837C211yesmaximal, characteristic, complemented
2#11762805C212yesmaximal, complemented
3#11762783C212yesmaximal, complemented
4#11762566C211yesmaximal, characteristic, complemented
5C17×#575C211yesmaximal, characteristic
6C2×#41×C17C2212yes—
7C17×#137—22no—
8C17×AGL(1,ℤ/8)—22no—
9C17×#135—22no—
10C2×M16×C17C2211yescharacteristic
11#10587612C2212yes—
12#10587612C2211yescharacteristic
13C22×#3622C2211yescharacteristic
14#10587506—22no—
15#10587505—24no—
16#10587503—22no—
17#10587485—22no—
18#41×C17—22no—
19#41×C17—22no—
20#41×C17—22no—
21#41×C17—22no—
22C2×Q8×C17D412yes—
23C2×D4×C17—22no—
24C2×D4×C17D412yes—
25C22×C4×C17—22no—
26C22×C4×C17C2311yescharacteristic
27Q16×C17—22no—
28SD16×C17—24no—
29D8×C17—22no—
30M16×C17—22no—
31C2×C8×C17D412yes—
32C22⋊Dic17—22no—
33C22⋊Dic17D412yes—
34C4⋊Dic17—21no—
35C4⋊Dic17—22no—
36C4×Dic17—21no—
37C4×Dic17—22no—
38C2×#3622—21no—
39C2×#3622—22no—
40C23×C17—22no—
41C23×C17C2×D411yescharacteristic
42Q8×C17—22no—
43Q8×C17—22no—
44D4×C17—22no—
45D4×C17—22no—
46D4×C17—42no—
47D4×C17—42no—
48C2×C4×C17—22no—
49C2×C4×C17—42no—
50C2×C4×C17—42no—
51C2×C4×C17C2×D412yes—
52C2×C4×C17C2×D411yescharacteristic
53C2×C4×C17#4111yescharacteristic
54C2×C4×C17#4112yes—
55C2×Dic17—22no—
56C2×Dic17—42no—
57C8×C17—22no—
58#3622—41no—
59#2053—171nomaximal, Sylow
60C22×C17—41no—
61C22×C17—42no—
62C22×C17—82no—
63C22×C17#12111yescharacteristic
64C22×C17#12112yes—
65C4×C17—42no—
66C4×C17#12011yescharacteristic
67C4×C17#12312yes—
68C4×C17#12411yescharacteristic
69Dic17—42no—
70Dic17—82no—
71#575D1711yescharacteristic
72#424—171no—
73#416—172no—
74#410—172no—
75H(3,ℤ/4)—171no—
76C2×C17—21no—
77C2×C17—22no—
78C2×C17—82no—
79C2×C17#39411yescharacteristic
80C2×#41C2×D1712yes—
81#137—22no—
82AGL(1,ℤ/8)—22no—
83#135—22no—
84C2×M16C2×D1711yescharacteristic
85C22×C8—171no—
86#117—171no—
87#117—172no—
88C4⋊C8—342no—
89#103—342no—
90#102—342no—
91#99—344no—
92C17#205311yescharacteristic, complemented, Sylow
93#41—22no—
94#41—22no—
95#41—22no—
96#41—22no—
97C2×Q8C22⋊D1712yes—
98C2×D4—22no—
99C2×D4C22⋊D1712yes—
100C22×C4—22no—
101C22×C4C22×D1711yescharacteristic
102Q16—22no—
103SD16—24no—
104D8—22no—
105M16—22no—
106C2×C8—342no—
107C2×C8—341no—
108C2×C8C22⋊D1712yes—
109C4⋊C4—341no—
110C4⋊C4—342no—
111C22⋊C4—172no—
112C22⋊C4—342no—
113C42—341no—
114C42—342no—
115C23—22no—
116C23D4×D1711yescharacteristic
117Q8—22no—
118Q8—22no—
119D4—22no—
120D4—22no—
121D4—42no—
122D4—42no—
123C2×C4—22no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—342no—
127C2×C4—682no—
128C2×C4#6325411yescharacteristic
129C2×C4#6325412yes—
130C2×C4C2×C22⋊D1712yes—
131C2×C4C2×C22⋊D1711yescharacteristic
132C8—22no—
133C8—681no—
134C22—41no—
135C22—42no—
136C22—82no—
137C22#1058762411yescharacteristic
138C22#1058762512yes—
139C4—42no—
140C4—682no—
141C4—1362no—
142C4#1058762112yes—
143C4#1058762211yescharacteristic
144C4#1058763911yescharacteristic
145C2—21no—
146C2—22no—
147C2—82no—
148C2#1176272311yescharacteristic
subgroupG/H
centerC4#10587622
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
Fitting, prime radicalC17×#575C2
socleC2×C17#394
hypercenter#575D17
hypocenterC17#2053
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #11762837C2semidirect
2. #11762805C2semidirect
3. #11762783C2semidirect
4. #11762566C2semidirect
92. C17#2053semidirect

Maximal subgroups

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

subgroupindex
#117628372
#117628052
#117627832
#117625662
C17×#5752
#205317