Finite Group Databasebeta

← #16953414

Subgroups — #16953414

Order 4352.

370 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 97 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#14539308C211yesmaximal, characteristic
2#14530505C212yesmaximal
3#14529655C211yesmaximal, characteristic
4#14526890C212yesmaximal
5C17×#1377C211yesmaximal, characteristic
6C22×Dic68C2211yescharacteristic
7#11763366—21no—
8#11763218—24no—
9#11762847—21no—
10#11762847C2212yes—
11C2×#10587488C2211yescharacteristic
12C2×#10587487—21no—
13#11762715—21no—
14#11762614—24no—
15C2×C17×M32C2212yes—
16C2×C17×#97C2211yescharacteristic
17C22×Dic34—21no—
18C22×C8×C17D411yescharacteristic
19C22×C8×C17C2311yescharacteristic
20C2×C42×C17D411yescharacteristic
21#10587612—22no—
22C2×C4⋊Dic17—21no—
23#10587610—42no—
24C2×Dic68—22no—
25C2×Dic68—21no—
26C2×Dic68—22no—
27C2×Dic68Q812yescomplemented
28#10587556—22no—
29#10587555—22no—
30C4×Dic34—42no—
31C17×M32—22no—
32C17×M32—22no—
33C2×C16×C17—22no—
34C17×#97—22no—
35#10587502—42no—
36#10587489—22no—
37#10587489D414yes—
38#10587488—22no—
39#10587488—22no—
40#10587488—22no—
41#10587487—42no—
42C22×C4×C17D812yes—
43C22×C4×C17C2×D411yescharacteristic
44C22×Dic17—41no—
45C22×Dic17—41no—
46C2×Dic34—22no—
47C2×Dic34—41no—
48C2×Dic34—41no—
49C2×Dic34—41no—
50C2×Dic34—41no—
51C2×Dic34—42no—
52C2×Dic34—41no—
53C2×C8×C17—21no—
54C2×C8×C17—21no—
55C2×C8×C17SD1612yes—
56C2×C8×C17C2×D412yes—
57C2×C8×C17C2×Q811yescharacteristic
58C2×C8×C17#4112yes—
59C2×C8×C17#4111yescharacteristic
60C42×C17—21no—
61C42×C17—21no—
62C22⋊Dic17—42no—
63C4⋊Dic17—24no—
64C4⋊Dic17—42no—
65C4⋊Dic17—41no—
66C4⋊Dic17—41no—
67#63225—44no—
68C4×Dic17—42no—
69Dic68—24no—
70Dic68—42no—
71C16×C17—42no—
72#13368—171nomaximal, Sylow
73C23×C17C2×D811yescharacteristic
74C2×C4×C17—22no—
75C2×C4×C17—42no—
76C2×C4×C17#12011yescharacteristic
77C2×C4×C17C22⋊Q811yescharacteristic
78C2×C4×C17#12411yescharacteristic
79C2×C4×C17C2×SD1611yescharacteristic
80C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
81C2×C4×C17#13711yescharacteristic
82C2×Dic17—44no—
83C2×Dic17—42no—
84C2×Dic17—81no—
85C2×Dic17—81no—
86C2×Dic17—82no—
87C2×Dic17—81no—
88Dic34—42no—
89Dic34—42no—
90Dic34—84no—
91Dic34—81no—
92Dic34—81no—
93C8×C17—22no—
94C8×C17C22⋊Q812yes—
95C8×C17#12412yes—
96#3162—171no—
97#2141—172no—
98#1971—171no—
99#1386—172no—
100#1377D1711yescharacteristic
101C22×C17—22no—
102C22×C17#44711yescharacteristic
103C22×C17#47411yescharacteristic
104C22×C17#48611yescharacteristic
105C4×C17—42no—
106C4×C17#39311yescharacteristic
107C4×C17#44811yescharacteristic
108C4×C17#47611yescharacteristic
109C4×C17#48911yescharacteristic
110Dic17—84no—
111Dic17—82no—
112Dic17—161no—
113C22×Q16—171no—
114#533—341no—
115C2×M32C2×D1712yes—
116#501—344no—
117#427—172no—
118#427—341no—
119C2×#106—171no—
120C2×#103—341no—
121C2×#97C2×D1711yescharacteristic
122#391—341no—
123#358—344no—
124C2×C17—21no—
125C2×C17—21no—
126C2×C17#202311yescharacteristic
127C2×C17#219112yes—
128C22×Q8—341no—
129C2×Q16—172no—
130C2×Q16—342no—
131C2×Q16—342no—
132C2×Q16—341no—
133C22×C8D6811yescharacteristic
134C22×C8C22×D1711yescharacteristic
135C4⋊Q8—342no—
136#125—342no—
137C22⋊Q8—682no—
138C4×Q8—682no—
139#117—342no—
140C2×C4⋊C4—341no—
141C2×C42D6811yescharacteristic
142M32—22no—
143M32—22no—
144C2×C16—22no—
145#107—174no—
146#107—342no—
147#106—342no—
148#106—342no—
149#106—342no—
150#103—682no—
151#97—22no—
152#95—682no—
153C17#1336811yescharacteristic, complemented, Sylow
154C2×Q8—342no—
155C2×Q8—682no—
156C2×Q8—681no—
157C2×Q8—681no—
158C2×Q8—681no—
159C2×Q8—681no—
160C2×Q8—681no—
161C22×C4—681no—
162C22×C4—681no—
163C22×C4D13612yes—
164C22×C4C2×D6811yescharacteristic
165Q16—344no—
166Q16—682no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#6321912yes—
170C2×C8#6325212yes—
171C2×C8D4×D1712yes—
172C2×C8#6325411yescharacteristic
173C2×C8Q8×D1711yescharacteristic
174C4⋊C4—344no—
175C4⋊C4—684no—
176C4⋊C4—682no—
177C4⋊C4—681no—
178C4⋊C4—681no—
179C22⋊C4—682no—
180C42—21no—
181C42—21no—
182C42—682no—
183C16—42no—
184C23C2×D13611yescharacteristic
185Q8—682no—
186Q8—682no—
187Q8—1364no—
188Q8—1361no—
189Q8—1361no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—684no—
193C2×C4—682no—
194C2×C4—1362no—
195C2×C4—1361no—
196C2×C4—1361no—
197C2×C4—1361no—
198C2×C4#1058756111yescharacteristic
199C2×C4#1058756811yescharacteristic
200C2×C4#1058758311yescharacteristic
201C2×C4C2×#6321911yescharacteristic
202C2×C4#1058759411yescharacteristic
203C2×C4#1058759511yescharacteristic
204C8—22no—
205C8#1058757112yes—
206C8#1058758212yes—
207C22—22no—
208C22#1176294511yescharacteristic
209C22#1176309011yescharacteristic
210C22#1176315611yescharacteristic
211C4—42no—
212C4—2721no—
213C4—1362no—
214C4—1364no—
215C4#1176272211yescharacteristic
216C4#1176295011yescharacteristic
217C4#1176310011yescharacteristic
218C4#1176316911yescharacteristic
219C2—21no—
220C2—21no—
221C2#1452992111yescharacteristic
222C2#1453073712yes—
subgroupG/H
centerC22#11762945
commutator, cosocleC22×C8×C17C23
FrattiniC22×C8C22×D17
FittingC17×#1377C2
socleC22×C17#447
prime radicalC23×C17C2×D8
hypercenter#1377D17
hypocenterC17#13368
LCSG ▷ C22×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #1377

Complements

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

NH ≅ G/Ntype
27. C2×Dic68Q8semidirect
153. C17#13368semidirect

Maximal subgroups

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

subgroupindex
#145393082
#145305052
#145296552
#145268902
C17×#13772
#1336817