Finite Group Databasebeta

← #16945813

Subgroups — #16945813

Order 4352.

306 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 85 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#14539368C211yesmaximal, characteristic
2D17×C4⋊C16C211yesmaximal, characteristic
3#14527546C211yesmaximal, characteristic
4#14526840C211yesmaximal, characteristic, complemented
5#14526838C211yesmaximal, characteristic, complemented
6#14526833C211yesmaximal, characteristic, complemented
7C17×#1362C211yesmaximal, characteristic, complemented
8#11763364—21no—
9C2×C16×D17—21no—
10#11763214C2211yescharacteristic
11#11763159—21no—
12#11763047—21no—
13#11763047C412yes—
14Q16×Dic17—21no—
15Q16×Dic17C412yes—
16#11762865—21no—
17C4×C8×D17C2211yescharacteristic
18C4⋊#63214C2211yescharacteristic
19#11762627—21no—
20#11762562C2211yescharacteristic
21#11762559—21no—
22C17×#494C2211yescharacteristic
23C17×C4⋊C16C2211yescharacteristic
24C17×#335C2211yescharacteristic
25C2×Q16×C17—21no—
26C2×Q16×C17C2×C412yes—
27C17×C4⋊Q8—21no—
28#10587632—22no—
29Q8×Dic17—22no—
30C2×C8×D17C2×C411yescharacteristic
31C2×C8×D17D411yescharacteristic
32#10587583—41no—
33#10587578—41no—
34#10587576—41no—
35#10587573—41no—
36C42×D17D411yescharacteristic
37#10587555—21no—
38C2×C16×C17—21no—
39C17×C4⋊C8—21no—
40C4×C8×C17C2311yescharacteristic
41#10587506—22no—
42#10587506—22no—
43#10587491—41no—
44#10587489—21no—
45#10587489C2×C412yes—
46#10587486—41no—
47C8×Dic17C2×C411yescharacteristic
48C8×Dic17D411yescharacteristic
49C2×#63214—21no—
50C4⋊#3622—21no—
51C4×#3622D411yescharacteristic
52C16×D17—42no—
53C2×Q8×C17—22no—
54C2×C4×D17C22⋊C411yescharacteristic
55C2×C4×D17D811yescharacteristic
56C2×C4×D17SD1611yescharacteristic
57C2×Dic34—41no—
58Q16×C17—42no—
59Q16×C17—41no—
60Q16×C17—41no—
61C2×C8×C17—41no—
62C2×C8×C17C22×C411yescharacteristic
63C2×C8×C17C2×D411yescharacteristic
64C4⋊C4×C17—41no—
65C4⋊C4×C17—41no—
66C42×C17C2×D411yescharacteristic
67#63227—42no—
68C4⋊Dic17—22no—
69C4⋊Dic17—41no—
70C4⋊Dic17—42no—
71#63225—42no—
72C4×Dic17—42no—
73C4×Dic17C22⋊C411yescharacteristic
74C4×Dic17D811yescharacteristic
75C4×Dic17SD1611yescharacteristic
76C2×#3622—41no—
77C2×#3622C22⋊C411yescharacteristic
78C2×#3622C22⋊C411yescharacteristic
79C8×D17—21no—
80C8×D17—21no—
81C8×D17—41no—
82C16×C17—41no—
83#63214—41no—
84#11984—171nomaximal, Sylow
85C22×D17#10211yescharacteristic
86Q8×C17—42no—
87Q8×C17—42no—
88C2×C4×C17—42no—
89C2×C4×C17C2×C22⋊C411yescharacteristic
90C2×C4×C17C2×D811yescharacteristic
91C2×C4×C17C2×SD1611yescharacteristic
92C2×Dic17—42no—
93C2×Dic17#10211yescharacteristic
94C2×Dic17#10211yescharacteristic
95C2×Dic17#10211yescharacteristic
96C4×D17—22no—
97C4×D17—21no—
98C4×D17—21no—
99C4×D17—21no—
100C4×D17—21no—
101Dic34—81no—
102Dic34—81no—
103C8×C17—21no—
104C8×C17—81no—
105C8×C17AGL(1,ℤ/8)11yescharacteristic
106C8×C17#13711yescharacteristic
107#3622—21no—
108#3622—21no—
109#3622—81no—
110#3170—171no—
111C2×C4⋊C16—171no—
112#1568—171no—
113#1364—171no—
114#1364—171no—
115#1362—171no—
116#1362D1711yescharacteristic, complemented
117C22×C17C2×#10211yescharacteristic
118C2×D17—21no—
119C2×D17—21no—
120C2×D17#36212yes—
121C4×C17—21no—
122C4×C17—82no—
123C4×C17#41311yescharacteristic
124C4×C17#41711yescharacteristic
125C4×C17#50811yescharacteristic
126C4×C17#50811yescharacteristic
127Dic17—21no—
128Dic17—21no—
129Dic17—21no—
130Dic17—82no—
131#533—341no—
132C22×C16—341no—
133#500—171no—
134#494C2×D1711yescharacteristic
135#487—341no—
136#467—172no—
137#467—341no—
138C4×Q16—172no—
139C4×Q16—341no—
140#432—341no—
141C2×C4×C8—171no—
142C4⋊C16—171no—
143C4⋊C16—341no—
144C4⋊C16C2×D1711yescharacteristic
145#335—171no—
146#335C2×D1711yescharacteristic
147#334—341no—
148C2×C17#154711yescharacteristic
149C2×C17#213811yescharacteristic
150C2×C17C2×#36211yescharacteristic
151D17—22no—
152C2×Q16—21no—
153C2×Q16C4×D1712yes—
154C22×C8—171no—
155C22×C8—171no—
156C4⋊Q8—21no—
157C4⋊Q8—341no—
158#125—681no—
159C22⋊Q8—342no—
160C22⋊Q8—681no—
161C4×Q8—342no—
162C4×Q8—681no—
163#117—681no—
164C2×C42—171no—
165C2×C16—21no—
166C2×C16—341no—
167C2×C16—682no—
168#107—172no—
169#107—341no—
170C4⋊C8—21no—
171C4⋊C8—341no—
172#103—342no—
173#103—342no—
174C22⋊C8—681no—
175#97—681no—
176C4×C8—171no—
177C4×C8—171no—
178C4×C8—171no—
179C4×C8C22×D1711yescharacteristic
180C17#1198411yescharacteristic, complemented, Sylow
181C2×Q8—22no—
182C2×Q8—681no—
183C22×C4—171no—
184C22×C4—171no—
185C22×C4—171no—
186Q16—42no—
187Q16—41no—
188Q16—41no—
189C2×C8—41no—
190C2×C8—171no—
191C2×C8—171no—
192C2×C8—341no—
193C2×C8—341no—
194C2×C8—681no—
195C2×C8—681no—
196C2×C8C2×C4×D1711yescharacteristic
197C2×C8D4×D1711yescharacteristic
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—342no—
201C4⋊C4—682no—
202C4⋊C4—682no—
203C4⋊C4—681no—
204C22⋊C4—682no—
205C42—171no—
206C42—171no—
207C42—171no—
208C42—682no—
209C42D4×D1711yescharacteristic
210C16—41no—
211C16—681no—
212C23—171no—
213Q8—42no—
214Q8—42no—
215Q8—1361no—
216Q8—1361no—
217C2×C4—42no—
218C2×C4—171no—
219C2×C4—171no—
220C2×C4—171no—
221C2×C4—342no—
222C2×C4—341no—
223C2×C4—341no—
224C2×C4—341no—
225C2×C4—341no—
226C2×C4—682no—
227C2×C4C22⋊C4×D1711yescharacteristic
228C2×C4D8×D1711yescharacteristic
229C2×C4SD16×D1711yescharacteristic
230C8—21no—
231C8—81no—
232C8—341no—
233C8—341no—
234C8—1361no—
235C8#1058759711yescharacteristic
236C8#1058760111yescharacteristic
237C22—172no—
238C22—341no—
239C22—341no—
240C22#102×D1711yescharacteristic
241C4—21no—
242C4—82no—
243C4—341no—
244C4—341no—
245C4—341no—
246C4—1362no—
247C4#1176279011yescharacteristic
248C4#1176280711yescharacteristic
249C4#1176324011yescharacteristic
250C4#1176324211yescharacteristic
251C2—342no—
252C2#1452745811yescharacteristic
253C2#1453049011yescharacteristic
254C2D17×#36211yescharacteristic
subgroupG/H
centerC22#102×D17
commutatorC2×C8×C17C22×C4
FrattiniC4×C8C22×D17
FittingC17×#1362C2
socleC22×C17C2×#102
prime radicalC22×D17#102
cosocleC4×C8×C17C23
hypercenter#1362D17
hypocenterC17#11984
LCSG ▷ C2×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
4. #14526840C2semidirect
5. #14526838C2semidirect
6. #14526833C2semidirect
7. C17×#1362C2semidirect
116. #1362D17semidirect
180. C17#11984semidirect

Maximal subgroups

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

subgroupindex
#145393682
D17×C4⋊C162
#145275462
#145268402
#145268382
#145268332
C17×#13622
#1198417