Finite Group Databasebeta

← #16947705

Subgroups — #16947705

Order 4352.

274 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 59 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#14531001C212yesmaximal
2#14530359C211yesmaximal, characteristic
3#14526772C212yesmaximal
4#14526735C211yesmaximal, characteristic
5C17×#3255C211yesmaximal, characteristic
6#11762858C2211yescharacteristic
7#11762631—22no—
8#11762625—22no—
9#11762591C2212yes—
10#11762581—21no—
11#11762569—21no—
12#11762566C2211yescharacteristic
13C17×#533—21no—
14C17×#475—22no—
15C17×#470—21no—
16C17×#470C2212yes—
17C17×#462—22no—
18C17×#423C2211yescharacteristic
19C2×Q16×C17—21no—
20C2×Q16×C17D412yes—
21C2×M16×C17D411yescharacteristic
22C2×M16×C17C2311yescharacteristic
23C17×C4⋊Q8—21no—
24C17×C4⋊Q8—21no—
25C17×#125—41no—
26C17×C22⋊Q8—22no—
27C17×C22⋊Q8—41no—
28C4×Q8×C17—22no—
29C4×Q8×C17—41no—
30C17×#117—41no—
31C2×C42×C17D411yescharacteristic
32#10587612—21no—
33C2×#63223—21no—
34C17×#106—21no—
35C17×#106D412yes—
36C17×C4⋊C8—22no—
37C17×#103—22no—
38C17×#103—22no—
39#10587490—22no—
40#10587483—22no—
41C2×Q8×C17—22no—
42C2×Q8×C17—41no—
43C22×C4×C17—21no—
44C22×C4×C17C2×D411yescharacteristic
45Q16×C17—42no—
46Q16×C17—81no—
47M16×C17—41no—
48M16×C17—41no—
49C2×C8×C17—21no—
50C2×C8×C17C2×D412yes—
51C4⋊C4×C17—22no—
52C4⋊C4×C17—42no—
53C4⋊C4×C17—41no—
54C4⋊C4×C17—41no—
55C4⋊C4×C17—41no—
56C4⋊C4×C17—42no—
57C22⋊C4×C17—42no—
58C42×C17—21no—
59C42×C17—42no—
60C42×C17SD1611yescharacteristic
61C42×C17SD1611yescharacteristic
62C22⋊Dic17—41no—
63C4⋊Dic17—41no—
64C4×Dic17—41no—
65#63223—41no—
66#63223—41no—
67C2×#3622—41no—
68#63214—42no—
69#12308—171nomaximal, Sylow
70C23×C17AGL(1,ℤ/8)11yescharacteristic
71Q8×C17—42no—
72Q8×C17—42no—
73Q8×C17—82no—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—21no—
78C2×C4×C17—42no—
79C2×C4×C17—42no—
80C2×C4×C17—42no—
81C2×C4×C17#12011yescharacteristic
82C2×C4×C17C2×SD1611yescharacteristic
83C2×Dic17—81no—
84C8×C17—22no—
85C8×C17—41no—
86#3622—81no—
87#3255D1711yescharacteristic
88#2239—172no—
89#2102—171no—
90#1340—172no—
91#1327—171no—
92C22×C17—21no—
93C22×C17—41no—
94C22×C17#45011yescharacteristic
95C4×C17—22no—
96C4×C17—21no—
97C4×C17—41no—
98C4×C17—82no—
99C4×C17—82no—
100C4×C17#45411yescharacteristic
101C4×C17#45811yescharacteristic
102Dic17—161no—
103#533—21no—
104#475—22no—
105#470—21no—
106#470C2×D1712yes—
107#462—22no—
108#430—171no—
109#423C2×D1711yescharacteristic
110#365—342no—
111#362—342no—
112#349—172no—
113#344—341no—
114#339—341no—
115H(3,ℤ/4)—171no—
116C2×C17—21no—
117C2×C17—41no—
118C2×C17#161611yescharacteristic
119C2×Q16—21no—
120C2×Q16C22⋊D1712yes—
121C2×M16—341no—
122C2×M16C22⋊D1711yescharacteristic
123C2×M16C22×D1711yescharacteristic
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126#125—41no—
127C22⋊Q8—22no—
128C22⋊Q8—41no—
129C4×Q8—22no—
130C4×Q8—41no—
131#117—41no—
132#117—341no—
133C2×C42C22⋊D1711yescharacteristic
134M32—342no—
135#108—342no—
136#106—21no—
137#106C22⋊D1712yes—
138C4⋊C8—22no—
139#103—22no—
140#103—22no—
141C17#1230811yescharacteristic, complemented, Sylow
142C2×Q8—22no—
143C2×Q8—41no—
144C22×C4—21no—
145C22×C4C2×C22⋊D1711yescharacteristic
146Q16—42no—
147Q16—81no—
148M16—41no—
149M16—41no—
150M16—681no—
151M16—681no—
152C2×C8—21no—
153C2×C8—681no—
154C2×C8C2×C22⋊D1712yes—
155C4⋊C4—22no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—42no—
159C4⋊C4—41no—
160C4⋊C4—42no—
161C4⋊C4—681no—
162C22⋊C4—42no—
163C22⋊C4—681no—
164C42—21no—
165C42—42no—
166C42—681no—
167C42#6322911yescharacteristic
168C42#6322911yescharacteristic
169C16—682no—
170C23#1058761811yescharacteristic
171Q8—42no—
172Q8—42no—
173Q8—82no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—1361no—
182C2×C4C2×#6322911yescharacteristic
183C2×C4#1058763911yescharacteristic
184C8—22no—
185C8—41no—
186C8—1361no—
187C22—21no—
188C22—41no—
189C22#1176296011yescharacteristic
190C4—22no—
191C4—21no—
192C4—41no—
193C4—82no—
194C4—82no—
195C4—2721no—
196C4#1176298211yescharacteristic
197C4#1176299911yescharacteristic
198C2—21no—
199C2—41no—
200C2#1452782111yescharacteristic
subgroupG/H
centerC2#14527821
commutator, cosocleC2×M16×C17C23
FrattiniC2×M16C22×D17
FittingC17×#3255C2
socleC2×C17#1616
prime radicalC23×C17AGL(1,ℤ/8)
hypercenter#3255D17
hypocenterC17#12308
LCSG ▷ C2×M16×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3255

Complements

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

NH ≅ G/Ntype
141. C17#12308semidirect

Maximal subgroups

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

subgroupindex
#145310012
#145303592
#145267722
#145267352
C17×#32552
#1230817