Finite Group Databasebeta

← #14529941

Subgroups — #14529941

Order 2176.

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

normal subgroups only

This group has 282 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#11763440C211yesmaximal, characteristic, complemented
2#11762815C212yesmaximal
3#11762710C211yesmaximal, characteristic
4#11762570C212yesmaximal, complemented
5C17×#343C211yesmaximal, characteristic, complemented
6C22×D68C2211yescharacteristic
7C2×M16×C17C2212yes—
8C2×C4⋊C4×C17C2211yescharacteristic
9C2×#63227—22no—
10#10587612C2212yes—
11#10587610—22no—
12C2×C4×Dic17C2211yescharacteristic
13#10587576—22no—
14#10587574—22no—
15#10587492—22no—
16#10587492—22no—
17C23×D17—21no—
18C22×C4×C17D412yes—
19C22×C4×C17C2311yescharacteristic
20C22×Dic17—21no—
21C2×D68—22no—
22C2×D68—22no—
23C2×D68—21no—
24C2×D68Q812yescomplemented
25C2×Dic34—22no—
26M16×C17—22no—
27M16×C17—22no—
28C2×C8×C17—22no—
29C4⋊C4×C17—22no—
30C22⋊Dic17—22no—
31#63227—44no—
32C4⋊Dic17—22no—
33C4⋊Dic17D412yes—
34C4⋊Dic17D412yes—
35#63225—24no—
36C4×Dic17—22no—
37C4×Dic17—22no—
38C23×C17C2×D411yescharacteristic
39C22×D17—22no—
40C22×D17—42no—
41C22×D17—42no—
42C22×D17—41no—
43C22×D17—41no—
44C2×C4×C17—22no—
45C2×C4×C17—42no—
46C2×C4×C17C2×D412yes—
47C2×C4×C17C2×Q811yescharacteristic
48C2×C4×C17#4112yes—
49C2×C4×C17#4111yescharacteristic
50C2×Dic17—22no—
51C2×Dic17—22no—
52C2×Dic17—22no—
53C2×Dic17—41no—
54C2×Dic17—41no—
55D68—24no—
56D68—42no—
57Dic34—82no—
58C8×C17—42no—
59#2027—171nomaximal, Sylow
60C22×C17—21no—
61C22×C17—21no—
62C22×C17#12011yescharacteristic
63C22×C17C22⋊Q811yescharacteristic
64C22×C17#12411yescharacteristic
65C2×D17—44no—
66C2×D17—42no—
67C2×D17—82no—
68C2×D17—81no—
69C2×D17—81no—
70C4×C17—42no—
71C4×C17C22⋊Q812yes—
72C4×C17#12412yes—
73Dic17—42no—
74Dic17—42no—
75Dic17—42no—
76#544—171no—
77#418—172no—
78#390—171no—
79#343D1711yescharacteristic, complemented
80#339—172no—
81C2×C17—21no—
82C2×C17—21no—
83C2×C17#39311yescharacteristic
84C2×C17#45311yescharacteristic
85C2×C17#45411yescharacteristic
86D17—82no—
87C22×D4—171no—
88C2×M16C2×D1712yes—
89C4⋊Q8—342no—
90#125—342no—
91C22⋊Q8—342no—
92#117—172no—
93C2×C4⋊C4C2×D1711yescharacteristic
94C2×C22⋊C4—342no—
95C2×C42—171no—
96#102—342no—
97#102—342no—
98C17#202711yescharacteristic, complemented, Sylow
99C24—341no—
100C2×Q8—342no—
101C2×D4—172no—
102C2×D4—342no—
103C2×D4—342no—
104C2×D4—341no—
105C22×C4—341no—
106C22×C4D6812yes—
107C22×C4C22×D1711yescharacteristic
108M16—22no—
109M16—22no—
110C2×C8—22no—
111C4⋊C4—22no—
112C4⋊C4—172no—
113C4⋊C4—172no—
114C4⋊C4—342no—
115C4⋊C4—344no—
116C22⋊C4—342no—
117C22⋊C4—684no—
118C42—342no—
119C42—342no—
120C23—342no—
121C23—682no—
122C23—682no—
123C23—681no—
124C23—681no—
125C23C2×D6811yescharacteristic
126Q8—1362no—
127D4—344no—
128D4—682no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—342no—
132C2×C4—342no—
133C2×C4—342no—
134C2×C4—681no—
135C2×C4—681no—
136C2×C4#6325212yes—
137C2×C4D4×D1712yes—
138C2×C4#6325411yescharacteristic
139C2×C4Q8×D1711yescharacteristic
140C8—42no—
141C22—21no—
142C22—21no—
143C22—684no—
144C22—682no—
145C22—1362no—
146C22—1361no—
147C22—1361no—
148C22#1058756111yescharacteristic
149C22#1058756811yescharacteristic
150C22#1058758311yescharacteristic
151C4—42no—
152C4—682no—
153C4—682no—
154C4—682no—
155C4#1058757112yes—
156C4#1058758212yes—
157C2—21no—
158C2—21no—
159C2—1362no—
160C2#1176272211yescharacteristic
161C2#1176297911yescharacteristic
162C2#1176298411yescharacteristic
subgroupG/H
centerC22#10587568
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#343C2
socleC22×C17#120
prime radicalC22×D68C22
hypercenter#343D17
hypocenterC17#2027
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. #11763440C2semidirect
4. #11762570C2semidirect
5. C17×#343C2semidirect
24. C2×D68Q8semidirect
79. #343D17semidirect
98. C17#2027semidirect

Maximal subgroups

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

subgroupindex
#117634402
#117628152
#117627102
#117625702
C17×#3432
#202717