Finite Group Databasebeta

← #17192008

Subgroups — #17192008

Order 4352.

472 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 165 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
1Q8×Dic68C211yesmaximal, characteristic
2#14531011C213yesmaximal
3#14530936C213yesmaximal
4#14530831C213yesmaximal
5Q8×#63214C211yesmaximal, characteristic
6C4×#10587500C213yesmaximal
7C17×#3395C211yesmaximal, characteristic
8Q8×Dic34—21no—
9C2×#10587500C2213yes—
10#11763214C2213yes—
11#11763105—23no—
12C4×Dic68C2213yes—
13#11762634C2213yes—
14C4⋊#63214C2213yes—
15#11762613C2216yes—
16#11762613C2213yes—
17#11762612C2211yescharacteristic
18C4×#63214C2213yes—
19C17×#557—21no—
20C17×#494C2213yes—
21C17×#462—23no—
22C8×Q8×C17C2211yescharacteristic
23C4×Q16×C17C2213yes—
24C2×Q16×C17—23no—
25C2×Q16×C17C2313yes—
26C17×C4⋊Q8—23no—
27C17×#125—43no—
28C4×Q8×C17—23no—
29C4×Q8×C17—41no—
30C4×Q8×C17D411yescharacteristic
31Q8×Dic17—41no—
32C2×Dic68C2311yescharacteristic
33#10587574—43no—
34#10587555—23no—
35C4×Dic34—23no—
36C17×#107C2311yescharacteristic
37C17×C4⋊C8D413yes—
38C17×#103—23no—
39C4×C8×C17C2313yes—
40#10587500—26no—
41#10587489—23no—
42#10587489C2313yes—
43#10587487—23no—
44C2×#63214C2313yes—
45C2×#63214C2311yescharacteristic
46C2×Q8×C17—23no—
47C2×Q8×C17D811yescharacteristic
48C2×Dic34—21no—
49C2×Dic34—43no—
50Q16×C17—23no—
51Q16×C17—23no—
52Q16×C17—83no—
53C2×C8×C17C2×D413yes—
54C2×C8×C17C2411yescharacteristic
55C4⋊C4×C17—21no—
56C4⋊C4×C17—43no—
57C4⋊C4×C17—43no—
58C4⋊C4×C17—43no—
59C4⋊C4×C17D813yes—
60C42×C17—43no—
61C42×C17C2×D413yes—
62C4⋊Dic17—23no—
63C4⋊Dic17—43no—
64#63225—43no—
65C4×Dic17—43no—
66Dic68—21no—
67Dic68#4112yes—
68#63214—23no—
69#63214—21no—
70#33930—171nomaximal, Sylow
71Q8×C17—43no—
72Q8×C17—43no—
73Q8×C17Q3214yescomplemented
74C2×C4×C17—41no—
75C2×C4×C17—43no—
76C2×C4×C17C2×D813yes—
77C2×C4×C17C2×D813yes—
78C2×C4×C17C22×D411yescharacteristic
79C2×Dic17—41no—
80C2×Dic17—43no—
81Dic34—22no—
82Dic34—41no—
83Dic34—163no—
84C8×C17—23no—
85C8×C17C2×#4111yescharacteristic
86C8×C172−1+411yescharacteristic
87#3395D1711yescharacteristic
88Q8×Q16—171no—
89#2241—173no—
90#2227—173no—
91#2209—173no—
92Q8×C16—171no—
93C4×Q32—173no—
94C22×C17C22×D811yescharacteristic
95C4×C17—23no—
96C4×C17—83no—
97C4×C17—81no—
98C4×C17C2×Q3216yes—
99C4×C17#54811yescharacteristic
100C4×C17#57711yescharacteristic
101Dic17—42no—
102Dic17—83no—
103Q82—341no—
104#557—21no—
105C2×Q32—173no—
106#500—173no—
107#494C2×D1713yes—
108#477—343no—
109#462—23no—
110C8×Q8C2×D1711yescharacteristic
111C4×Q16—173no—
112C4×Q16C2×D1713yes—
113#366—173no—
114C4⋊C16—173no—
115#358—173no—
116#358—176no—
117#358—171no—
118C4×C16—173no—
119C2×C17#335811yescharacteristic
120C2×C17C22×Q3211yescharacteristic
121C2×C17#341611yescharacteristic
122C2×Q16—23no—
123C2×Q16—171no—
124C2×Q16C22×D1713yes—
125C4⋊Q8—23no—
126C4⋊Q8—343no—
127C4⋊Q8—683no—
128#125—43no—
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—173no—
138#107—343no—
139#107C22×D1711yescharacteristic
140C4⋊C8C22⋊D1713yes—
141#103—23no—
142#103—343no—
143C4×C8C22×D1713yes—
144C17#3393011yescharacteristic, complemented, Sylow
145C2×Q8—23no—
146C2×Q8—341no—
147C2×Q8—683no—
148C2×Q8#6322811yescharacteristic
149Q16—23no—
150Q16—23no—
151Q16—83no—
152Q16—172no—
153Q16—341no—
154C2×C8C2×C22⋊D1713yes—
155C2×C8C23×D1711yescharacteristic
156C4⋊C4—21no—
157C4⋊C4—43no—
158C4⋊C4—43no—
159C4⋊C4—43no—
160C4⋊C4—343no—
161C4⋊C4—683no—
162C4⋊C4—683no—
163C4⋊C4#6322813yes—
164C42—43no—
165C42—683no—
166C42C2×C22⋊D1713yes—
167C16—343no—
168C16—341no—
169Q8—43no—
170Q8—43no—
171Q8—2723no—
172Q8—342no—
173Q8—681no—
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—
183C8#41×D1711yescharacteristic
184C8#1058769711yescharacteristic
185C22C22×#6322811yescharacteristic
186C4—23no—
187C4—83no—
188C4—81no—
189C4—682no—
190C4—1363no—
191C4C2×#1058750016yes—
192C4#1176346611yescharacteristic
193C4#1176362011yescharacteristic
194C2#1454141511yescharacteristic
195C2C22×#1058750011yescharacteristic
196C2#1454192811yescharacteristic
subgroupG/H
centerC22C22×#63228
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC17×#3395C2
socle, prime radicalC22×C17C22×D8
hypercenter#3395D17
hypocenterC17#33930
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3395

Complements

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

NH ≅ G/Ntype
73. 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
Q8×Dic682
#145310112
#145309362
#145308312
Q8×#632142
C4×#105875002
C17×#33952
#3393017