Finite Group Databasebeta

← #14529948

Subgroups — #14529948

Order 2176.

466 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 69 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#11763435C211yesmaximal, characteristic, complemented
2#11762665C212yesmaximal
3#11762576C211yesmaximal, characteristic, complemented
4#11762575C212yesmaximal, complemented
5C17×#379C211yesmaximal, characteristic
6C25×C17C2211yescharacteristic
7C2×C22⋊C4×C17—22no—
8C2×C22⋊C4×C17C2211yescharacteristic
9C2×C22⋊C4×C17C2211yescharacteristic
10C2×C22⋊Dic17—24no—
11C2×C22⋊Dic17C2212yes—
12C2×C22⋊Dic17C2212yes—
13#10587610—22no—
14#10587564—24no—
15C24×C17—22no—
16C24×C17—21no—
17C24×C17—21no—
18C24×C17—22no—
19C24×C17—44no—
20C24×C17Q812yescomplemented
21C24×C17C2311yescharacteristic
22C22×C4×C17—21no—
23C22×C4×C17D412yes—
24C22×Dic17—22no—
25C22×Dic17D414yes—
26C2×Dic34—22no—
27C22⋊C4×C17—22no—
28C22⋊C4×C17—22no—
29C22⋊C4×C17—44no—
30C22⋊Dic17—22no—
31C22⋊Dic17—22no—
32C22⋊Dic17—22no—
33C22⋊Dic17—22no—
34C22⋊Dic17—48no—
35C4⋊Dic17—22no—
36#63225—24no—
37C23×C17—28no—
38C23×C17—24no—
39C23×C17—24no—
40C23×C17—22no—
41C23×C17—44no—
42C23×C17—42no—
43C23×C17—44no—
44C23×C17—42no—
45C23×C17—44no—
46C23×C17—48no—
47C23×C17—41no—
48C23×C17—42no—
49C23×C17—41no—
50C23×C17C2×D412yes—
51C23×C17C2×D411yescharacteristic
52C23×C17C2×Q811yescharacteristic
53C23×C17#4111yescharacteristic
54C23×C17#4112yes—
55C2×C4×C17—22no—
56C2×C4×C17—42no—
57C2×C4×C17—42no—
58C2×C4×C17—41no—
59C2×Dic17—24no—
60C2×Dic17—44no—
61C2×Dic17—44no—
62C2×Dic17—42no—
63Dic34—82no—
64#2029—171nomaximal, Sylow
65C22×C17—24no—
66C22×C17—22no—
67C22×C17—28no—
68C22×C17—24no—
69C22×C17—41no—
70C22×C17—41no—
71C22×C17—42no—
72C22×C17—48no—
73C22×C17—44no—
74C22×C17—44no—
75C22×C17—42no—
76C22×C17—42no—
77C22×C17—44no—
78C22×C17#12011yescharacteristic
79C22×C17C22⋊Q812yes—
80C22×C17C22⋊Q811yescharacteristic
81C22×C17#12412yes—
82C22×C17#12411yescharacteristic
83C4×C17—42no—
84C4×C17—81no—
85Dic17—44no—
86Dic17—82no—
87#543—171no—
88#379—172no—
89#379D1711yescharacteristic
90#342—172no—
91#342—171no—
92C2×C17—22no—
93C2×C17—21no—
94C2×C17—22no—
95C2×C17—21no—
96C2×C17—44no—
97C2×C17#39311yescharacteristic
98C2×C17B(4,𝔽2)12yes—
99C25C2×D1711yescharacteristic
100C22⋊Q8—344no—
101C22⋊Q8—342no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—172no—
104C2×C22⋊C4—172no—
105C2×C22⋊C4—344no—
106C2×C22⋊C4C2×D1711yescharacteristic
107C2×C22⋊C4C2×D1711yescharacteristic
108C17#202911yescharacteristic, complemented, Sylow
109C24—22no—
110C24—21no—
111C24—21no—
112C24—22no—
113C24—44no—
114C24Dic3412yescomplemented
115C24C22×D1711yescharacteristic
116C2×Q8—342no—
117C22×C4—21no—
118C22×C4—174no—
119C22×C4—342no—
120C22×C4C22⋊D1712yes—
121C4⋊C4—344no—
122C4⋊C4—342no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—44no—
126C22⋊C4—342no—
127C22⋊C4—342no—
128C22⋊C4—342no—
129C22⋊C4—342no—
130C22⋊C4—688no—
131C23—24no—
132C23—28no—
133C23—24no—
134C23—22no—
135C23—44no—
136C23—42no—
137C23—42no—
138C23—44no—
139C23—44no—
140C23—48no—
141C23—42no—
142C23—41no—
143C23—41no—
144C23C2×Dic3411yescharacteristic
145C23#6325211yescharacteristic
146C23D4×D1712yes—
147C23#6325412yes—
148C23C2×C22⋊D1711yescharacteristic
149Q8—1362no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—344no—
155C2×C4—684no—
156C2×C4—684no—
157C2×C4—682no—
158C22—22no—
159C22—24no—
160C22—28no—
161C22—24no—
162C22—41no—
163C22—41no—
164C22—42no—
165C22—44no—
166C22—48no—
167C22—44no—
168C22—42no—
169C22—42no—
170C22—44no—
171C22#1058756412yes—
172C22#1058757112yes—
173C22#1058761011yescharacteristic
174C22#1058762211yescharacteristic
175C22#1058762311yescharacteristic
176C4—42no—
177C4—81no—
178C4—684no—
179C4—1362no—
180C2—21no—
181C2—22no—
182C2—22no—
183C2—21no—
184C2—44no—
185C2#1176272111yescharacteristic
186C2#1176299512yes—
subgroupG/H
centerC22#10587623
commutator, cosocleC24×C17C23
FrattiniC24C22×D17
FittingC17×#379C2
socleC22×C17#120
prime radicalC25×C17C22
hypercenter#379D17
hypocenterC17#2029
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ #379

Complements

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

NH ≅ G/Ntype
1. #11763435C2semidirect
3. #11762576C2semidirect
4. #11762575C2semidirect
20. C24×C17Q8semidirect
108. C17#2029semidirect
114. C24Dic34semidirect

Maximal subgroups

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

subgroupindex
#117634352
#117626652
#117625762
#117625752
C17×#3792
#202917