Finite Group Databasebeta

← #16923578

Subgroups — #16923578

Order 4352.

422 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 87 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#14530312C211yesmaximal, characteristic
2#14530192C212yesmaximal, complemented
3#14526610C212yesmaximal, complemented
4#14526605C211yesmaximal, characteristic
5C17×#2804C211yesmaximal, characteristic
6C2×#10587478—21no—
7C2×#10587506—22no—
8C2×#10587503—21no—
9#11762679C2211yescharacteristic
10#11762571—22no—
11#11762564C2211yescharacteristic
12#11762564C2212yes—
13C17×#400—21no—
14C17×#393—22no—
15C17×#391C2212yes—
16C17×#390C2211yescharacteristic
17C17×#386—21no—
18C17×#385—22no—
19C22×Q8×C17D412yes—
20C22×D4×C17D411yescharacteristic
21C23×C4×C17—21no—
22C2×C4⋊C4×C17—21no—
23C2×C4⋊C4×C17D412yes—
24C2×C4⋊C4×C17D411yescharacteristic
25C2×C22⋊C4×C17—22no—
26C2×C22⋊C4×C17—22no—
27C2×C22⋊C4×C17—21no—
28C2×C42×C17C2311yescharacteristic
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17—22no—
31C22×#3622—21no—
32C22×#3622—22no—
33C17×#95—22no—
34C17×#95—21no—
35C17×#95—22no—
36C17×#95—22no—
37#10587506—44no—
38#10587503—42no—
39#10587478—42no—
40C24×C17—21no—
41C2×Q8×C17—22no—
42C2×Q8×C17—22no—
43C2×Q8×C17—42no—
44C2×Q8×C17—42no—
45C2×D4×C17—21no—
46C2×D4×C17—41no—
47C2×D4×C17—41no—
48C2×D4×C17Q1612yes—
49C22×C4×C17—22no—
50C22×C4×C17—22no—
51C22×C4×C17—21no—
52C22×C4×C17—81no—
53C22×C4×C17—82no—
54C22×C4×C17C2×D411yescharacteristic
55C22×C4×C17C2×D412yes—
56C22×Dic17—41no—
57C4⋊C4×C17—41no—
58C4⋊C4×C17—42no—
59C4⋊C4×C17—81no—
60C22⋊C4×C17—42no—
61C22⋊C4×C17—42no—
62C22⋊C4×C17—82no—
63C22⋊C4×C17—81no—
64C42×C17—41no—
65C4⋊Dic17—41no—
66C4⋊Dic17—41no—
67C4⋊Dic17—42no—
68C4⋊Dic17—42no—
69C2×#3622—22no—
70C2×#3622—41no—
71C2×#3622—41no—
72C2×#3622—42no—
73C2×#3622—42no—
74C2×#3622—42no—
75#7821—171nomaximal, Sylow
76C23×C17—22no—
77C23×C17—42no—
78C23×C17—41no—
79C23×C17—42no—
80C23×C17—41no—
81C23×C17#12011yescharacteristic
82Q8×C17—44no—
83Q8×C17—82no—
84D4×C17—42no—
85D4×C17—81no—
86C2×C4×C17—21no—
87C2×C4×C17—22no—
88C2×C4×C17—21no—
89C2×C4×C17—22no—
90C2×C4×C17—42no—
91C2×C4×C17—42no—
92C2×C4×C17—41no—
93C2×C4×C17—42no—
94C2×C4×C17—42no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17—84no—
99C2×C4×C17C2×Q1611yescharacteristic
100C2×C4×C17#13512yes—
101C2×C4×C17AGL(1,ℤ/8)12yes—
102C2×C4×C17#13711yescharacteristic
103C2×Dic17—82no—
104C2×Dic17—81no—
105#3622—42no—
106#3622—82no—
107#2804D1711yescharacteristic
108#2093—171no—
109#2072—172no—
110#1272—172no—
111#1270—171no—
112C22×C17—42no—
113C22×C17—44no—
114C22×C17—82no—
115C22×C17—81no—
116C22×C17—81no—
117C22×C17#44912yes—
118C22×C17#45111yescharacteristic
119C22×C17#45311yescharacteristic
120C22×C17#45512yes—
121C22×C17#45811yescharacteristic
122C4×C17—22no—
123C4×C17—22no—
124C4×C17—21no—
125C4×C17—21no—
126C4×C17—82no—
127C4×C17—81no—
128C4×C17—82no—
129Dic17—161no—
130C2×#107—341no—
131C2×#103—342no—
132C2×#102—341no—
133#400—21no—
134#393—22no—
135#391C2×D1712yes—
136#390C2×D1711yescharacteristic
137#386—21no—
138#385—22no—
139#384—171no—
140#340—342no—
141#336—172no—
142#336—171no—
143C2×C17—82no—
144C2×C17#161212yes—
145C2×C17#161411yescharacteristic
146C2×C17#162011yescharacteristic
147C2×C17#164412yes—
148C2×C17#165911yescharacteristic
149C22×Q8C22⋊D1712yes—
150C22×D4C22⋊D1711yescharacteristic
151C23×C4—21no—
152C22×C8—341no—
153C22×C8—342no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—342no—
156C2×C4⋊C4—341no—
157C2×C4⋊C4C22⋊D1712yes—
158C2×C4⋊C4C22⋊D1711yescharacteristic
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—21no—
162C2×C42C22×D1711yescharacteristic
163#107—682no—
164#103—684no—
165#102—682no—
166#95—22no—
167#95—21no—
168#95—22no—
169#95—22no—
170C17#782111yescharacteristic, complemented, Sylow
171C24—21no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4#6323112yes—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—81no—
184C22×C4—82no—
185C22×C4—681no—
186C22×C4C2×C22⋊D1711yescharacteristic
187C22×C4C2×C22⋊D1712yes—
188C2×C8—342no—
189C2×C8—681no—
190C2×C8—681no—
191C2×C8—682no—
192C2×C8—682no—
193C2×C8—682no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4—681no—
198C4⋊C4—681no—
199C4⋊C4—682no—
200C4⋊C4—682no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—82no—
204C22⋊C4—81no—
205C42—41no—
206C23—22no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23#1058763911yescharacteristic
212Q8—44no—
213Q8—82no—
214D4—42no—
215D4—81no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—84no—
229C2×C4—1362no—
230C2×C4—1361no—
231C2×C4#1058762811yescharacteristic
232C2×C4C2×#6323111yescharacteristic
233C2×C4#1058763512yes—
234C2×C4#1058763612yes—
235C8—682no—
236C8—1362no—
237C22—42no—
238C22—44no—
239C22—82no—
240C22—81no—
241C22—81no—
242C22#1176295412yes—
243C22#1176296511yescharacteristic
244C22#1176297711yescharacteristic
245C22#1176298712yes—
246C22#1176299911yescharacteristic
247C4—22no—
248C4—22no—
249C4—21no—
250C4—21no—
251C4—82no—
252C4—82no—
253C4—81no—
254C4—2721no—
255C2—82no—
256C2#1452779812yes—
257C2#1452780911yescharacteristic
258C2#1452784311yescharacteristic
259C2#1452799812yes—
260C2#1452809011yescharacteristic
subgroupG/H
centerC23#10587639
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#2804C2
socleC23×C17#120
prime radicalC22×D4×C17D4
hypercenter#2804D17
hypocenterC17#7821
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2804

Complements

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

NH ≅ G/Ntype
2. #14530192C2semidirect
3. #14526610C2semidirect
170. C17#7821semidirect

Maximal subgroups

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

subgroupindex
#145303122
#145301922
#145266102
#145266052
C17×#28042
#782117