Finite Group Databasebeta

← #16932164

Subgroups — #16932164

Order 4352.

542 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 135 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#14537808C211yesmaximal, characteristic
2#14528776C211yesmaximal, characteristic
3#14527213C212yesmaximal, complemented
4#14526642C212yesmaximal, complemented
5C17×#2580C211yesmaximal, characteristic
6D4×#3622—24no—
7#11762862C2212yes—
8#11762833C2212yes—
9C2×#10587504—21no—
10C2×C22⋊#3622—22no—
11#11762578—21no—
12Q8⋊#3622—24no—
13C2×C4×Q8×C17C2211yescharacteristic
14C2×C4×D4×C17C2211yescharacteristic
15C17×#399C412yes—
16C17×#393C412yes—
17C17×#390—21no—
18C17×#389—21no—
19C17×#385—22no—
20C17×#382C2211yescharacteristic
21C22×Q8×C17C2×C411yescharacteristic
22C22×D4×C17D411yescharacteristic
23C23×C4×C17D412yes—
24C4×Q8×C17—22no—
25C4×Q8×C17—22no—
26C4×D4×C17—22no—
27C4×D4×C17—22no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—22no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17—22no—
34C2×C22⋊C4×C17—22no—
35C2×C22⋊C4×C17C2×C412yes—
36C2×C42×C17—21no—
37C2×C42×C17C2311yescharacteristic
38C2×#63223—22no—
39C22×#3622—24no—
40C17×#95—22no—
41C17×#95—21no—
42C17×#95C2×C412yes—
43#10587504—42no—
44C22⋊#3622—24no—
45C22⋊#3622—22no—
46C4⋊#3622—22no—
47C4⋊#3622—22no—
48C4⋊#3622—22no—
49C4×#3622D414yes—
50C24×C17C22⋊C412yes—
51C2×Q8×C17—22no—
52C2×Q8×C17—42no—
53C2×D4×C17—22no—
54C2×D4×C17—22no—
55C2×D4×C17SD1612yes—
56C2×D4×C17Q1612yes—
57C22×C4×C17—21no—
58C22×C4×C17—22no—
59C22×C4×C17—22no—
60C22×C4×C17—22no—
61C22×C4×C17—21no—
62C22×C4×C17—82no—
63C22×C4×C17C22⋊C412yes—
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C4⋊C4×C17—22no—
68C4⋊C4×C17—22no—
69C4⋊C4×C17—41no—
70C4⋊C4×C17—41no—
71C4⋊C4×C17—42no—
72C4⋊C4×C17—82no—
73C22⋊C4×C17—24no—
74C22⋊C4×C17—44no—
75C22⋊C4×C17—82no—
76C42×C17—42no—
77C42×C17C2×D412yes—
78C42×C17#4112yes—
79#63223—42no—
80#63223—42no—
81C2×#3622—24no—
82C2×#3622—42no—
83C2×#3622—84no—
84#9483—171nomaximal, Sylow
85C23×C17—22no—
86C23×C17—22no—
87C23×C17—42no—
88C23×C17—42no—
89C23×C17#10314yes—
90C23×C17C2×C22⋊C411yescharacteristic
91Q8×C17—24no—
92Q8×C17—42no—
93D4×C17—24no—
94D4×C17—24no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—41no—
98C2×C4×C17—41no—
99C2×C4×C17—44no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17—42no—
103C2×C4×C17—42no—
104C2×C4×C17—41no—
105C2×C4×C17—42no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—42no—
109C2×C4×C17—82no—
110C2×C4×C17C4×D412yes—
111C2×C4×C17#12011yescharacteristic
112C2×C4×C17#12112yes—
113C2×C4×C17#12311yescharacteristic
114C2×C4×C17C2×SD1611yescharacteristic
115C2×C4×C17C2×Q1611yescharacteristic
116C2×C4×C17#13512yes—
117#3622—44no—
118#3622—82no—
119#2972—171no—
120#2580D1711yescharacteristic
121#1800—171no—
122#1487—172no—
123#1286—172no—
124C22×C17—21no—
125C22×C17—21no—
126C22×C17—24no—
127C22×C17—44no—
128C22×C17—42no—
129C22×C17—42no—
130C22×C17—82no—
131C22×C17#38611yescharacteristic
132C22×C17C2×#10311yescharacteristic
133C22×C17#41611yescharacteristic
134C4×C17—22no—
135C4×C17—21no—
136C4×C17—21no—
137C4×C17—42no—
138C4×C17—42no—
139C4×C17—82no—
140C4×C17—81no—
141C4×C17#46512yes—
142C4×C17#46712yes—
143C2×C4×Q8C2×D1711yescharacteristic
144C2×C4×D4C2×D1711yescharacteristic
145C8×D4—344no—
146#431—172no—
147#423—172no—
148C2×#100—341no—
149C2×C22⋊C8—342no—
150#399Dic1712yes—
151#393Dic1712yes—
152#390—21no—
153#389—21no—
154#385—22no—
155#382C2×D1711yescharacteristic
156#343—341no—
157Q8⋊C8—344no—
158C2×C17—21no—
159C2×C17—21no—
160C2×C17—44no—
161C2×C17#189111yescharacteristic
162C2×C17#189211yescharacteristic
163C2×C17#191111yescharacteristic
164C22×Q8C2×Dic1711yescharacteristic
165C22×D4C22⋊D1711yescharacteristic
166C23×C4C22⋊D1712yes—
167C2×M16—342no—
168C22×C8—344no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×D4—22no—
172C4×D4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C2×Dic1711yescharacteristic
177C2×C4⋊C4C22⋊D1711yescharacteristic
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4C2×Dic1712yes—
181C2×C42—21no—
182C2×C42C22×D1711yescharacteristic
183C4⋊C8—342no—
184C4⋊C8—342no—
185C4⋊C8—342no—
186#100—682no—
187C22⋊C8—344no—
188C22⋊C8—342no—
189C4×C8—174no—
190#95—22no—
191#95—21no—
192#95C2×Dic1712yes—
193C17#948311yescharacteristic, complemented, Sylow
194C24C22⋊Dic1712yes—
195C2×Q8—22no—
196C2×Q8—42no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4#6323012yes—
200C2×D4#6323112yes—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—82no—
207C22×C4C22⋊Dic1712yes—
208C22×C4C22×Dic1711yescharacteristic
209C22×C4C2×C22⋊D1711yescharacteristic
210C22×C4C2×C22⋊D1711yescharacteristic
211M16—682no—
212M16—682no—
213C2×C8—344no—
214C2×C8—682no—
215C2×C8—1364no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C22⋊C4—24no—
223C22⋊C4—44no—
224C22⋊C4—82no—
225C42—42no—
226C42D4×D1712yes—
227C42#6325412yes—
228C23—22no—
229C23—22no—
230C23—42no—
231C23—42no—
232C23#1058750614yes—
233C23C2×C22⋊Dic1711yescharacteristic
234Q8—24no—
235Q8—42no—
236D4—24no—
237D4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—44no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—82no—
253C2×C4D4×Dic1712yes—
254C2×C4#1058762111yescharacteristic
255C2×C4#1058762311yescharacteristic
256C2×C4#1058762512yes—
257C2×C4C2×#6323011yescharacteristic
258C2×C4C2×#6323111yescharacteristic
259C2×C4#1058763612yes—
260C8—684no—
261C8—1362no—
262C22—21no—
263C22—21no—
264C22—24no—
265C22—44no—
266C22—42no—
267C22—42no—
268C22—82no—
269C22#1176268911yescharacteristic
270C22C2×#1058750611yescharacteristic
271C22#1176280511yescharacteristic
272C4—21no—
273C4—21no—
274C4—22no—
275C4—42no—
276C4—42no—
277C4—82no—
278C4—81no—
279C4#1176304012yes—
280C4#1176304812yes—
281C2—21no—
282C2—21no—
283C2—44no—
284C2#1452920311yescharacteristic
285C2#1452920811yescharacteristic
286C2#1452931711yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2580C2
socleC22×C17#386
prime radicalC22×D4×C17D4
cosocleC2×C42×C17C23
hypercenter#2580D17
hypocenterC17#9483
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2580

Complements

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

NH ≅ G/Ntype
3. #14527213C2semidirect
4. #14526642C2semidirect
193. C17#9483semidirect

Maximal subgroups

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

subgroupindex
#145378082
#145287762
#145272132
#145266422
C17×#25802
#948317