Finite Group Databasebeta

← #17192012

Subgroups — #17192012

Order 4352.

472 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 167 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#14541790C211yesmaximal, characteristic
2#14531013C213yesmaximal
3#14530933C213yesmaximal
4#14530834C213yesmaximal
5Q8×#63214C211yesmaximal, characteristic
6C4×#10587500C213yesmaximal
7Q8×Q16×C17C211yesmaximal, characteristic
8#11763538—21no—
9C2×#10587500C2213yes—
10#11763189C2213yes—
11#11763020—23no—
12C4×Dic68C2213yes—
13#11762634C2213yes—
14C4⋊#63214C2213yes—
15#11762613C2211yescharacteristic
16#11762612C2216yes—
17#11762612C2213yes—
18C4×#63214C2213yes—
19Q82×C17—21no—
20C17×#500C2213yes—
21C17×#477—23no—
22C8×Q8×C17C2211yescharacteristic
23C4×Q16×C17C2213yes—
24C2×Q16×C17C2311yescharacteristic
25C17×C4⋊Q8—23no—
26C17×C4⋊Q8—43no—
27C4×Q8×C17—23no—
28C4×Q8×C17—41no—
29C4×Q8×C17D411yescharacteristic
30Q8×Dic17—41no—
31C2×Dic68—23no—
32C2×Dic68C2313yes—
33#10587576—43no—
34#10587555—23no—
35C4×Dic34—23no—
36C17×#107—23no—
37C17×#107C2313yes—
38C17×C4⋊C8D413yes—
39C17×#103—23no—
40C4×C8×C17C2313yes—
41#10587500—26no—
42#10587489C2311yescharacteristic
43#10587487—23no—
44C2×#63214C2313yes—
45C2×#63214C2311yescharacteristic
46C2×Q8×C17—21no—
47C2×Q8×C17—43no—
48C2×Q8×C17D811yescharacteristic
49C2×Dic34—23no—
50Q16×C17—21no—
51Q16×C17#4112yes—
52C2×C8×C17C2×D413yes—
53C2×C8×C17C2411yescharacteristic
54C4⋊C4×C17—23no—
55C4⋊C4×C17—43no—
56C4⋊C4×C17—43no—
57C4⋊C4×C17D813yes—
58C42×C17—43no—
59C42×C17C2×D413yes—
60C4⋊Dic17—21no—
61C4⋊Dic17—43no—
62C4⋊Dic17—43no—
63#63225—43no—
64C4×Dic17—43no—
65Dic68—23no—
66Dic68—23no—
67Dic68—83no—
68#63214—23no—
69#63214—21no—
70#33930—171nomaximal, Sylow
71Q8×C17—22no—
72Q8×C17—41no—
73Q8×C17—163no—
74Q8×C17Q3214yescomplemented
75C2×C4×C17—41no—
76C2×C4×C17—43no—
77C2×C4×C17C2×D813yes—
78C2×C4×C17C2×D813yes—
79C2×C4×C17C22×D411yescharacteristic
80C2×Dic17—43no—
81C2×Dic17—41no—
82Dic34—43no—
83Dic34—43no—
84C8×C17—23no—
85C8×C17C2×#4111yescharacteristic
86C8×C172−1+411yescharacteristic
87#3395—171no—
88Q8×Q16D1711yescharacteristic
89#2241—173no—
90#2227—173no—
91#2209—173no—
92Q8×C16—171no—
93C4×Q32—173no—
94C22×C17C22×D811yescharacteristic
95C4×C17—23no—
96C4×C17—42no—
97C4×C17—83no—
98C4×C17C2×Q3216yes—
99C4×C17#54811yescharacteristic
100C4×C17#57711yescharacteristic
101Dic17—83no—
102Dic17—81no—
103Q82—21no—
104#557—341no—
105C2×Q32—173no—
106#500C2×D1713yes—
107#494—173no—
108#477—23no—
109#462—343no—
110C8×Q8C2×D1711yescharacteristic
111C4×Q16—173no—
112C4×Q16C2×D1713yes—
113#366—173no—
114C4⋊C16—173no—
115#358—171no—
116#358—173no—
117#358—176no—
118C4×C16—173no—
119C2×C17#335811yescharacteristic
120C2×C17C22×Q3211yescharacteristic
121C2×C17#341611yescharacteristic
122C2×Q16—173no—
123C2×Q16—343no—
124C2×Q16C22×D1711yescharacteristic
125C4⋊Q8—23no—
126C4⋊Q8—43no—
127C4⋊Q8—343no—
128#125—683no—
129C4×Q8—23no—
130C4×Q8—41no—
131C4×Q8—343no—
132C4×Q8—681no—
133C4×Q8C22⋊D1711yescharacteristic
134Q32—346no—
135C2×C16—171no—
136C2×C16—173no—
137#107—23no—
138#107—171no—
139#107C22×D1713yes—
140C4⋊C8C22⋊D1713yes—
141#103—23no—
142#103—343no—
143C4×C8C22×D1713yes—
144C17#3393011yescharacteristic, complemented, Sylow
145C2×Q8—21no—
146C2×Q8—43no—
147C2×Q8—343no—
148C2×Q8#6322811yescharacteristic
149Q16—21no—
150Q16—343no—
151Q16—343no—
152Q16—1363no—
153Q16#6325612yes—
154C2×C8C2×C22⋊D1713yes—
155C2×C8C23×D1711yescharacteristic
156C4⋊C4—23no—
157C4⋊C4—43no—
158C4⋊C4—43no—
159C4⋊C4—341no—
160C4⋊C4—683no—
161C4⋊C4—683no—
162C4⋊C4—683no—
163C4⋊C4#6322813yes—
164C42—43no—
165C42—683no—
166C42C2×C22⋊D1713yes—
167C16—343no—
168C16—341no—
169Q8—22no—
170Q8—41no—
171Q8—163no—
172Q8—683no—
173Q8—683no—
174Q8#1058750014yescomplemented
175C2×C4—41no—
176C2×C4—43no—
177C2×C4—681no—
178C2×C4—683no—
179C2×C4C2×#6322813yes—
180C2×C4C2×#6322813yes—
181C2×C4C22×C22⋊D1711yescharacteristic
182C8—23no—
183C8C2×#6325611yescharacteristic
184C8#1058769411yescharacteristic
185C22C22×#6322811yescharacteristic
186C4—23no—
187C4—42no—
188C4—83no—
189C4—1363no—
190C4—1361no—
191C4C2×#1058750016yes—
192C4#1176347011yescharacteristic
193C4#1176361711yescharacteristic
194C2#1454141911yescharacteristic
195C2C22×#1058750011yescharacteristic
196C2#1454193111yescharacteristic
subgroupG/H
centerC22C22×#63228
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingQ8×Q16×C17C2
socle, prime radicalC22×C17C22×D8
hypercenterQ8×Q16D17
hypocenterC17#33930
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ Q8×Q16

Complements

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

NH ≅ G/Ntype
74. Q8×C17Q32semidirect
144. C17#33930semidirect
174. Q8#10587500semidirect

Maximal subgroups

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

subgroupindex
#145417902
#145310132
#145309332
#145308342
Q8×#632142
C4×#105875002
Q8×Q16×C172
#3393017