Finite Group Databasebeta

← #14530137

Subgroups — #14530137

Order 2176.

314 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 73 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
2#11762682C212yesmaximal
3#11762576C211yesmaximal, characteristic
4#11762575C212yesmaximal
5C17×#385C211yesmaximal, characteristic
6C23×C4×C17C2211yescharacteristic
7C2×C4⋊C4×C17—21no—
8C2×C22⋊C4×C17C2211yescharacteristic
9C2×C22⋊C4×C17C2211yescharacteristic
10C2×C22⋊Dic17C2212yes—
11C2×C22⋊Dic17C2212yes—
12#10587610—21no—
13#10587610—21no—
14C2×#63225—22no—
15#10587564—22no—
16#10587564—22no—
17C17×#95—21no—
18#10587502—22no—
19C24×C17C2311yescharacteristic
20C22×C4×C17—22no—
21C22×C4×C17—21no—
22C22×C4×C17—21no—
23C22×C4×C17—42no—
24C22×C4×C17D411yescharacteristic
25C22×C4×C17D411yescharacteristic
26C22×Dic17—22no—
27C22×Dic17D412yes—
28C22×Dic17D412yes—
29C2×Dic34—21no—
30C2×Dic34—21no—
31C4⋊C4×C17—42no—
32C22⋊C4×C17—22no—
33C22⋊C4×C17—21no—
34C22⋊C4×C17Q812yescomplemented
35C22⋊Dic17—22no—
36C22⋊Dic17—22no—
37C22⋊Dic17—22no—
38C22⋊Dic17Q814yescomplemented
39C4⋊Dic17—21no—
40C4⋊Dic17—21no—
41#63225—22no—
42#63225—22no—
43#63225—44no—
44C23×C17—41no—
45C23×C17—41no—
46C23×C17C2×D412yes—
47C23×C17C2×D411yescharacteristic
48C23×C17C2×Q811yescharacteristic
49C23×C17C2×Q812yes—
50C23×C17#4111yescharacteristic
51C2×C4×C17—22no—
52C2×C4×C17—24no—
53C2×C4×C17—21no—
54C2×C4×C17—21no—
55C2×C4×C17—22no—
56C2×C4×C17—41no—
57C2×C4×C17—41no—
58C2×C4×C17—42no—
59C2×C4×C17—41no—
60C2×C4×C17—41no—
61C2×C4×C17—41no—
62C2×C4×C17—41no—
63C2×Dic17—22no—
64C2×Dic17—22no—
65C2×Dic17—24no—
66C2×Dic17—42no—
67C2×Dic17—42no—
68C2×Dic17—42no—
69C2×Dic17—42no—
70Dic34—81no—
71Dic34—81no—
72#2062—171nomaximal, Sylow
73C22×C17—22no—
74C22×C17—21no—
75C22×C17—22no—
76C22×C17—21no—
77C22×C17—41no—
78C22×C17—42no—
79C22×C17—41no—
80C22×C17#12011yescharacteristic
81C22×C17C22⋊Q812yes—
82C22×C17C22⋊Q811yescharacteristic
83C22×C17C4⋊Q812yes—
84C22×C17C4⋊Q811yescharacteristic
85C4×C17—42no—
86C4×C17—41no—
87C4×C17—42no—
88C4×C17—41no—
89Dic17—44no—
90Dic17—42no—
91Dic17—42no—
92#543—171no—
93#385—172no—
94#385D1711yescharacteristic
95#342—172no—
96#342—171no—
97C2×C17—22no—
98C2×C17—21no—
99C2×C17—22no—
100C2×C17—21no—
101C2×C17#39511yescharacteristic
102C2×C17B(4,𝔽2)11yescharacteristic
103C2×C17#45811yescharacteristic
104C23×C4C2×D1711yescharacteristic
105C22⋊Q8—342no—
106C22⋊Q8—342no—
107C22⋊Q8—341no—
108C22⋊Q8—341no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—342no—
111C2×C22⋊C4—172no—
112C2×C22⋊C4—172no—
113C2×C22⋊C4C2×D1711yescharacteristic
114C2×C22⋊C4C2×D1711yescharacteristic
115#95—21no—
116#95—342no—
117C17#206211yescharacteristic, complemented, Sylow
118C24C22×D1711yescharacteristic
119C2×Q8—341no—
120C2×Q8—341no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—42no—
125C22×C4—172no—
126C22×C4—172no—
127C22×C4—342no—
128C22×C4C22⋊D1711yescharacteristic
129C22×C4C22⋊D1711yescharacteristic
130C4⋊C4—42no—
131C4⋊C4—342no—
132C4⋊C4—342no—
133C4⋊C4—341no—
134C4⋊C4—341no—
135C4⋊C4—684no—
136C22⋊C4—21no—
137C22⋊C4—22no—
138C22⋊C4—174no—
139C22⋊C4—342no—
140C22⋊C4—342no—
141C22⋊C4—342no—
142C22⋊C4Dic3412yescomplemented
143C23—41no—
144C23—41no—
145C23C2×Dic3411yescharacteristic
146C23#6325211yescharacteristic
147C23D4×D1712yes—
148C23Q8×D1712yes—
149C23C2×C22⋊D1711yescharacteristic
150Q8—1361no—
151Q8—1361no—
152C2×C4—22no—
153C2×C4—24no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—344no—
165C2×C4—342no—
166C2×C4—342no—
167C2×C4—682no—
168C2×C4—682no—
169C2×C4—682no—
170C2×C4—682no—
171C22—21no—
172C22—22no—
173C22—21no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—42no—
178C22#1058757412yes—
179C22#1058758212yes—
180C22#1058761011yescharacteristic
181C22#1058762311yescharacteristic
182C22#1058763011yescharacteristic
183C4—42no—
184C4—42no—
185C4—41no—
186C4—41no—
187C4—684no—
188C4—682no—
189C4—682no—
190C2—21no—
191C2—22no—
192C2—22no—
193C2—21no—
194C2#1176272911yescharacteristic
195C2#1176299511yescharacteristic
196C2#1176299811yescharacteristic
subgroupG/H
centerC22#10587623
commutator, prime radical, cosocleC24×C17C23
FrattiniC24C22×D17
FittingC17×#385C2
socleC22×C17#120
hypercenter#385D17
hypocenterC17#2062
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ #385

Complements

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

NH ≅ G/Ntype
34. C22⋊C4×C17Q8semidirect
38. C22⋊Dic17Q8semidirect
117. C17#2062semidirect
142. C22⋊C4Dic34semidirect

Maximal subgroups

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

subgroupindex
#117634352
#117626822
#117625762
#117625752
C17×#3852
#206217