Finite Group Databasebeta

← #14528038

Subgroups — #14528038

Order 2176.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#11763529C211yesmaximal, characteristic
2#11763212C211yesmaximal, characteristic, complemented
3#11763091C211yesmaximal, characteristic, complemented
4#11762556C211yesmaximal, characteristic, complemented
5#11762539C211yesmaximal, characteristic, complemented
6D4⋊#3622C211yesmaximal, characteristic
7C17×#461C211yesmaximal, characteristic
8C2×SD16×C17—21no—
9C17×C4⋊Q8C2211yescharacteristic
10C4×D4×C17C2211yescharacteristic
11#10587622—21no—
12#10587612—21no—
13#10587610—21no—
14#10587573—21no—
15#10587565—22no—
16#10587555C2211yescharacteristic
17C4×Dic34C2211yescharacteristic
18C17×C4⋊C8C2211yescharacteristic
19C17×#103—21no—
20#10587481—21no—
21#10587479—21no—
22#10587478—22no—
23C4⋊#3622C2211yescharacteristic
24C4×#3622C2211yescharacteristic
25C2×Q8×C17—21no—
26C2×D4×C17D411yescharacteristic
27C22×C4×C17—21no—
28C2×Dic34—21no—
29C2×Dic34D411yescharacteristic
30SD16×C17—42no—
31C2×C8×C17—21no—
32C4⋊C4×C17—21no—
33C4⋊C4×C17—21no—
34C4⋊C4×C17D411yescharacteristic
35C22⋊C4×C17—21no—
36C42×C17C2311yescharacteristic
37C22⋊Dic17—22no—
38C22⋊Dic17—22no—
39C4⋊Dic17—21no—
40C4⋊Dic17D412yes—
41C4⋊Dic17D411yescharacteristic
42#63225—22no—
43#63225—21no—
44C4×Dic17—22no—
45C4×Dic17—21no—
46C2×#3622—21no—
47C2×#3622—21no—
48C23×C17—21no—
49Q8×C17—41no—
50Q8×C17—41no—
51D4×C17—21no—
52D4×C17—21no—
53C2×C4×C17—21no—
54C2×C4×C17—41no—
55C2×C4×C17—41no—
56C2×C4×C17—41no—
57C2×C4×C17C2×D411yescharacteristic
58C2×C4×C17C2×D411yescharacteristic
59C2×C4×C17C2×D411yescharacteristic
60C2×Dic17—22no—
61C2×Dic17—21no—
62C2×Dic17—21no—
63Dic34—21no—
64Dic34—21no—
65Dic34—41no—
66Dic34—41no—
67C8×C17—41no—
68#3622—22no—
69#3622—41no—
70#1650—171nomaximal, Sylow
71C22×C17—41no—
72C22×C17—41no—
73C22×C17—41no—
74C22×C17#12011yescharacteristic
75C4×C17—21no—
76C4×C17—41no—
77C4×C17—81no—
78C4×C17#13511yescharacteristic
79C4×C17#13511yescharacteristic
80C4×C17AGL(1,ℤ/8)11yescharacteristic
81C4×C17#13711yescharacteristic
82Dic17—42no—
83Dic17—41no—
84Dic17—41no—
85#556—171no—
86#500—171no—
87#475—171no—
88#461D1711yescharacteristic
89#332—171no—
90Q8⋊C8—171no—
91D4⋊C8—171no—
92C2×C17—81no—
93C2×C17#44911yescharacteristic
94C2×C17#45211yescharacteristic
95C2×C17#45611yescharacteristic
96C2×SD16—21no—
97C4⋊Q8—171no—
98C4⋊Q8C2×D1711yescharacteristic
99#126—342no—
100#124—341no—
101C22⋊Q8—341no—
102C4×Q8—171no—
103C4×Q8—341no—
104C4×D4C2×D1711yescharacteristic
105#117—341no—
106#107—342no—
107#106—341no—
108C4⋊C8—171no—
109C4⋊C8C2×D1711yescharacteristic
110#103—21no—
111#103—341no—
112C4×C8—171no—
113C17#165011yescharacteristic, complemented, Sylow
114C2×Q8—21no—
115C2×Q8—171no—
116C2×Q8—341no—
117C2×D4C22⋊D1711yescharacteristic
118C22×C4—21no—
119SD16—42no—
120C2×C8—21no—
121C2×C8—341no—
122C2×C8—341no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—171no—
126C4⋊C4—172no—
127C4⋊C4—342no—
128C4⋊C4—341no—
129C4⋊C4—341no—
130C4⋊C4C22⋊D1711yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—342no—
133C22⋊C4—342no—
134C42—342no—
135C42—341no—
136C42C22×D1711yescharacteristic
137C23—21no—
138Q8—41no—
139Q8—41no—
140Q8—341no—
141Q8—341no—
142Q8—681no—
143Q8—681no—
144D4—21no—
145D4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—341no—
151C2×C4—342no—
152C2×C4—341no—
153C2×C4D4×D1711yescharacteristic
154C2×C4D4×D1711yescharacteristic
155C2×C4C2×C22⋊D1711yescharacteristic
156C8—41no—
157C8—342no—
158C8—681no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#1058762311yescharacteristic
163C4—21no—
164C4—41no—
165C4—81no—
166C4—682no—
167C4—681no—
168C4—681no—
169C4#1058760111yescharacteristic
170C4#1058760211yescharacteristic
171C4#1058761811yescharacteristic
172C4#1058763611yescharacteristic
173C2—81no—
174C2#1176295211yescharacteristic
175C2#1176296811yescharacteristic
176C2#1176299011yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
FittingC17×#461C2
socleC22×C17#120
prime radicalC2×D4×C17D4
hypercenter#461D17
hypocenterC17#1650
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #11763212C2semidirect
3. #11763091C2semidirect
4. #11762556C2semidirect
5. #11762539C2semidirect
113. C17#1650semidirect

Maximal subgroups

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

subgroupindex
#117635292
#117632122
#117630912
#117625562
#117625392
D4⋊#36222
C17×#4612
#165017