Finite Group Databasebeta

← #16931727

Subgroups — #16931727

Order 4352.

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

normal subgroups only

This group has 534 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#14533776C211yesmaximal, characteristic
3#14527457C212yesmaximal, complemented
4#14526685C212yesmaximal, complemented
5C17×#1800C211yesmaximal, characteristic
6C2×#10587555C2211yescharacteristic
7C2×#10587622—21no—
8#11762887—24no—
9#11762862C2212yes—
10#11762773—22no—
11#11762734C414yes—
12#11762714—21no—
13#11762669—22no—
14C4×C4⋊Dic17C2211yescharacteristic
15#11762556—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—
30#10587622—44no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17C2×C412yes—
34C2×C4×Dic17—21no—
35C2×#63223—24no—
36#10587555—22no—
37#10587555—22no—
38C17×C4⋊C8—22no—
39C17×C4⋊C8—22no—
40C17×#100—42no—
41#10587502—24no—
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—22no—
51C2×D4×C17—21no—
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—42no—
62C2×Dic34—41no—
63C2×Dic34—42no—
64C2×Dic34—41no—
65M16×C17—42no—
66M16×C17—42no—
67C2×C8×C17—42no—
68C4⋊C4×C17—22no—
69C22⋊C4×C17—24no—
70C42×C17C2×D412yes—
71C42×C17#4112yes—
72C22⋊Dic17—48no—
73C4⋊Dic17—24no—
74C4⋊Dic17—42no—
75C4⋊Dic17—41no—
76C4⋊Dic17—41no—
77C4×Dic17—42no—
78#63223—84no—
79C2×#3622—24no—
80#9401—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×C17AGL(1,ℤ/8)12yes—
101C2×C4×C17#13711yescharacteristic
102C2×Dic17—42no—
103C2×Dic17—42no—
104C2×Dic17—42no—
105C2×Dic17—42no—
106C2×Dic17—42no—
107C2×Dic17—42no—
108Dic34—82no—
109Dic34—82no—
110C8×C17—82no—
111#3622—44no—
112#2975—171no—
113#2547—171no—
114#1800D1711yescharacteristic
115#1546—172no—
116#1303—172no—
117C22×C17—21no—
118C22×C17—21no—
119C22×C17—44no—
120C22×C17—42no—
121C22×C17—42no—
122C22×C17—42no—
123C22×C17—82no—
124C22×C17#38611yescharacteristic
125C22×C17#41711yescharacteristic
126C22×C17#41811yescharacteristic
127C4×C17—21no—
128C4×C17—22no—
129C4×C17—42no—
130C4×C17#45512yes—
131C4×C17#46812yes—
132C4×C17#46912yes—
133Dic17—82no—
134Dic17—82no—
135C2×C4⋊Q8—171no—
136C2×#124—341no—
137C2×C4×D4C2×D1711yescharacteristic
138#436—344no—
139#431—172no—
140#423C2×D1712yes—
141C2×#100—21no—
142#407—342no—
143#397—174no—
144#391—341no—
145#380—342no—
146C4×C4⋊C4—171no—
147#343—21no—
148#332—344no—
149C2×C17—21no—
150C2×C17—21no—
151C2×C17—82no—
152C2×C17#188511yescharacteristic
153C2×C17#189511yescharacteristic
154C2×C17#191511yescharacteristic
155C22×Q8—341no—
156C22×D4C22⋊D1711yescharacteristic
157C23×C4D6812yes—
158C2×M16—22no—
159C2×M16—344no—
160C4⋊Q8—342no—
161C4⋊Q8—342no—
162#124—684no—
163C4×D4—22no—
164C4×D4—22no—
165C2×C4⋊C4—172no—
166C2×C4⋊C4—341no—
167C2×C4⋊C4—341no—
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—
181#95—344no—
182C17#940111yescharacteristic, complemented, Sylow
183C24#6322712yes—
184C2×Q8—681no—
185C2×Q8—682no—
186C2×Q8—681no—
187C2×Q8—682no—
188C2×D4—21no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—342no—
196C22×C4—342no—
197C22×C4#6322712yes—
198C22×C4C2×C4×D1711yescharacteristic
199C22×C4C2×D6811yescharacteristic
200C22×C4C2×C22⋊D1711yescharacteristic
201M16—42no—
202M16—42no—
203M16—1364no—
204C2×C8—42no—
205C2×C8—344no—
206C4⋊C4—22no—
207C4⋊C4—344no—
208C4⋊C4—682no—
209C4⋊C4—681no—
210C4⋊C4—681no—
211C22⋊C4—24no—
212C22⋊C4—688no—
213C42—682no—
214C42D4×D1712yes—
215C42#6325412yes—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23C2×#6322711yescharacteristic
222Q8—1362no—
223Q8—1362no—
224D4—42no—
225D4—42no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—82no—
232C2×C4—682no—
233C2×C4—682no—
234C2×C4—682no—
235C2×C4—682no—
236C2×C4—682no—
237C2×C4—682no—
238C2×C4#1058756712yes—
239C2×C4#1058756811yescharacteristic
240C2×C4#1058757211yescharacteristic
241C2×C4#1058761811yescharacteristic
242C2×C4#1058762512yes—
243C2×C4#1058762811yescharacteristic
244C2×C4#1058763512yes—
245C8—82no—
246C8—684no—
247C22—21no—
248C22—21no—
249C22—44no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—82no—
254C22#1176268611yescharacteristic
255C22#1176280811yescharacteristic
256C22#1176281311yescharacteristic
257C4—22no—
258C4—21no—
259C4—42no—
260C4—1362no—
261C4—1362no—
262C4#1176298912yes—
263C4#1176305712yes—
264C4#1176306012yes—
265C2—21no—
266C2—21no—
267C2—82no—
268C2#1452917011yescharacteristic
269C2#1452922011yescharacteristic
270C2#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#9401
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. #14527457C2semidirect
4. #14526685C2semidirect
182. C17#9401semidirect

Maximal subgroups

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

subgroupindex
#145378252
#145337762
#145274572
#145266852
C17×#18002
#940117