Finite Group Databasebeta

← #16931737

Subgroups — #16931737

Order 4352.

570 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 115 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#14537825C211yesmaximal, characteristic
2#14533737C211yesmaximal, characteristic
3#14527470C212yesmaximal, complemented
4#14526685C212yesmaximal, complemented
5C17×#1800C211yesmaximal, characteristic
6C2×#10587555C2211yescharacteristic
7C2×#10587610—22no—
8#11762887—24no—
9#11762862C2212yes—
10#11762773—22no—
11#11762745C414yes—
12#11762712—21no—
13#11762678—21no—
14C4×C4⋊Dic17C2211yescharacteristic
15#11762554—24no—
16C2×C4×D4×C17C2211yescharacteristic
17C17×#423C2212yes—
18C2×C17×#100—21no—
19C17×#343—21no—
20C22×D4×C17D411yescharacteristic
21C23×C4×C17D412yes—
22C22×Dic34—21no—
23C2×M16×C17—22no—
24C4×D4×C17—22no—
25C4×D4×C17—22no—
26C2×C4⋊C4×C17D411yescharacteristic
27C2×C22⋊C4×C17C2×C412yes—
28C2×C42×C17C2311yescharacteristic
29C2×C22⋊Dic17—24no—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17C2×C412yes—
33#10587610—48no—
34C2×#63225—24no—
35C2×C4×Dic17—21no—
36C2×#63223—24no—
37#10587555—22no—
38#10587555—22no—
39C17×C4⋊C8—22no—
40C17×C4⋊C8—22no—
41C17×#100—42no—
42#10587502C2×C412yes—
43C22⋊#3622—24no—
44C22⋊#3622—22no—
45C4⋊#3622D414yes—
46C4×#3622—22no—
47C24×C17C22⋊C412yes—
48C2×D4×C17—22no—
49C2×D4×C17—21no—
50C2×D4×C17—21no—
51C2×D4×C17—22no—
52C22×C4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17—44no—
55C22×C4×C17C22⋊C412yes—
56C22×C4×C17C22×C411yescharacteristic
57C22×C4×C17C2×D411yescharacteristic
58C22×C4×C17C2×D411yescharacteristic
59C22×Dic17—22no—
60C22×Dic17—22no—
61C2×Dic34—44no—
62C2×Dic34—41no—
63C2×Dic34—41no—
64M16×C17—42no—
65M16×C17—42no—
66C2×C8×C17—42no—
67C4⋊C4×C17—22no—
68C22⋊C4×C17—24no—
69C42×C17C2×D412yes—
70C42×C17#4112yes—
71C22⋊Dic17—48no—
72C4⋊Dic17—24no—
73C4⋊Dic17—24no—
74C4⋊Dic17—42no—
75#63225—48no—
76C4×Dic17—41no—
77C4×Dic17—41no—
78#63223—84no—
79C2×#3622—24no—
80#9403—171nomaximal, Sylow
81C23×C17—22no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—42no—
85C23×C17—42no—
86C23×C17C2×C22⋊C411yescharacteristic
87D4×C17—42no—
88D4×C17—42no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—41no—
92C2×C4×C17—44no—
93C2×C4×C17—42no—
94C2×C4×C17—82no—
95C2×C4×C17C4×D412yes—
96C2×C4×C17#12011yescharacteristic
97C2×C4×C17#12112yes—
98C2×C4×C17#12311yescharacteristic
99C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
100C2×C4×C17#13711yescharacteristic
101C2×C4×C17#13712yes—
102C2×Dic17—24no—
103C2×Dic17—42no—
104C2×Dic17—42no—
105C2×Dic17—44no—
106C2×Dic17—42no—
107Dic34—82no—
108Dic34—82no—
109C8×C17—82no—
110#3622—44no—
111#2975—171no—
112#2542—171no—
113#1800D1711yescharacteristic
114#1549—172no—
115#1303—172no—
116C22×C17—21no—
117C22×C17—21no—
118C22×C17—44no—
119C22×C17—42no—
120C22×C17—42no—
121C22×C17—42no—
122C22×C17—82no—
123C22×C17#38611yescharacteristic
124C22×C17#41711yescharacteristic
125C22×C17#41911yescharacteristic
126C4×C17—21no—
127C4×C17—22no—
128C4×C17—42no—
129C4×C17#45312yes—
130C4×C17#46812yes—
131C4×C17#47012yes—
132Dic17—44no—
133Dic17—82no—
134C2×C4⋊Q8—171no—
135C2×C22⋊Q8—342no—
136C2×C4×D4C2×D1711yescharacteristic
137#436—344no—
138#431—172no—
139#423C2×D1712yes—
140C2×#100—21no—
141#407—342no—
142#399—174no—
143#390—341no—
144#384—341no—
145C4×C4⋊C4—171no—
146#343—21no—
147#332—344no—
148C2×C17—21no—
149C2×C17—21no—
150C2×C17—82no—
151C2×C17#188711yescharacteristic
152C2×C17#189411yescharacteristic
153C2×C17#191511yescharacteristic
154C22×Q8—341no—
155C22×D4C22⋊D1711yescharacteristic
156C23×C4D6812yes—
157C2×M16—22no—
158C2×M16—344no—
159C4⋊Q8—342no—
160C4⋊Q8—342no—
161C22⋊Q8—688no—
162C4×D4—22no—
163C4×D4—22no—
164C2×C4⋊C4—172no—
165C2×C4⋊C4—341no—
166C2×C4⋊C4—341no—
167C2×C4⋊C4—344no—
168C2×C4⋊C4C22⋊D1711yescharacteristic
169C2×C22⋊C4—344no—
170C2×C22⋊C4C4×D1712yes—
171C2×C42—341no—
172C2×C42C22×D1711yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—174no—
176#100—42no—
177C22⋊C8—344no—
178C22⋊C8—342no—
179C4×C8—342no—
180#95—172no—
181C17#940311yescharacteristic, complemented, Sylow
182C24#6322712yes—
183C2×Q8—684no—
184C2×Q8—681no—
185C2×Q8—681no—
186C2×D4—21no—
187C2×D4—22no—
188C2×D4—21no—
189C2×D4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—44no—
193C22×C4—342no—
194C22×C4—342no—
195C22×C4#6322712yes—
196C22×C4C2×C4×D1711yescharacteristic
197C22×C4C2×D6811yescharacteristic
198C22×C4C2×C22⋊D1711yescharacteristic
199M16—42no—
200M16—42no—
201M16—1364no—
202C2×C8—42no—
203C2×C8—344no—
204C4⋊C4—22no—
205C4⋊C4—344no—
206C4⋊C4—344no—
207C4⋊C4—688no—
208C4⋊C4—682no—
209C22⋊C4—24no—
210C22⋊C4—688no—
211C42—681no—
212C42—681no—
213C42D4×D1712yes—
214C42#6325412yes—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—42no—
219C23—42no—
220C23C2×#6322711yescharacteristic
221Q8—1362no—
222Q8—1362no—
223D4—42no—
224D4—42no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—82no—
231C2×C4—344no—
232C2×C4—682no—
233C2×C4—682no—
234C2×C4—684no—
235C2×C4—682no—
236C2×C4#1058756712yes—
237C2×C4#1058756811yescharacteristic
238C2×C4#1058757211yescharacteristic
239C2×C4#1058761811yescharacteristic
240C2×C4#1058762512yes—
241C2×C4#1058762811yescharacteristic
242C2×C4#1058763712yes—
243C8—82no—
244C8—684no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—82no—
252C22#1176268611yescharacteristic
253C22#1176280811yescharacteristic
254C22#1176281711yescharacteristic
255C4—22no—
256C4—21no—
257C4—42no—
258C4—684no—
259C4—1362no—
260C4#1176297912yes—
261C4#1176305312yes—
262C4#1176306512yes—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#1452918011yescharacteristic
267C2#1452921511yescharacteristic
268C2#1452933411yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1800C2
socleC22×C17#386
prime radicalC22×D4×C17D4
cosocleC2×C42×C17C23
hypercenter#1800D17
hypocenterC17#9403
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #14527470C2semidirect
4. #14526685C2semidirect
181. C17#9403semidirect

Maximal subgroups

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

subgroupindex
#145378252
#145337372
#145274702
#145266852
C17×#18002
#940317