Finite Group Databasebeta

← #16932178

Subgroups — #16932178

Order 4352.

542 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 131 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#14533689C211yesmaximal, characteristic
2#14528780C211yesmaximal, characteristic, complemented
3#14527210C212yesmaximal
4#14526643C212yesmaximal, complemented
5C17×#2982C211yesmaximal, characteristic, complemented
6C2×#10587632—21no—
7C2×C4×D68C2211yescharacteristic
8#11762833C2212yes—
9C2×#10587507—21no—
10#11762739C412yes—
11#11762726C412yes—
12#11762700—22no—
13#11762675C2211yescharacteristic
14#11762666—21no—
15#11762578—21no—
16#11762538—24no—
17C2×C4×Q8×C17C2211yescharacteristic
18C8×Q8×C17—24no—
19C17×#431—22no—
20C17×#431C2212yes—
21C22×Q8×C17D411yescharacteristic
22C22×D68C2×C411yescharacteristic
23C22×C4×D17—21no—
24C4×Q8×C17—22no—
25C4×Q8×C17—22no—
26C2×C4⋊C4×C17C2×C412yes—
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C42×C17D412yes—
29C2×C42×C17C2311yescharacteristic
30#10587632—42no—
31#10587632—42no—
32C2×#63227—22no—
33C2×#63227—22no—
34C2×#63227—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×#63225—22no—
38C2×#63223—22no—
39C4×D68—22no—
40C4×D68—22no—
41C17×C4⋊C8—24no—
42C17×C4⋊C8—22no—
43C17×C22⋊C8—22no—
44C4×C8×C17—24no—
45C4×C8×C17D414yes—
46#10587507—42no—
47#10587502—22no—
48#10587502—21no—
49#10587502C2×C412yes—
50C4⋊#3622—22no—
51C4⋊#3622—22no—
52C23×D17—21no—
53C2×Q8×C17—22no—
54C2×Q8×C17—22no—
55C2×Q8×C17D812yes—
56C2×Q8×C17SD1612yes—
57C22×C4×C17C22⋊C412yes—
58C22×C4×C17C22⋊C412yes—
59C22×C4×C17C22×C411yescharacteristic
60C22×C4×C17C2×D411yescharacteristic
61C22×C4×C17C2×D411yescharacteristic
62C22×Dic17—21no—
63C22×Dic17—22no—
64C2×D68—22no—
65C2×D68—42no—
66C2×C4×D17—42no—
67C2×C4×D17—41no—
68C2×C4×D17—41no—
69C2×C4×D17—42no—
70C2×C8×C17—24no—
71C4⋊C4×C17—24no—
72C4⋊C4×C17—22no—
73C42×C17—22no—
74C42×C17—22no—
75C42×C17C2×D412yes—
76C42×C17#4112yes—
77#63227—44no—
78#63227—44no—
79#63227—41no—
80#63227—41no—
81C4⋊Dic17—22no—
82C4⋊Dic17—41no—
83C4⋊Dic17—41no—
84#63225—44no—
85#63223—42no—
86#63223—42no—
87C2×#3622—42no—
88#9486—171nomaximal, Sylow
89C23×C17C2×C22⋊C411yescharacteristic
90C22×D17—22no—
91C22×D17—41no—
92C22×D17—41no—
93C22×D17—41no—
94C22×D17—41no—
95C22×D17—41no—
96C22×D17—41no—
97Q8×C17—24no—
98Q8×C17—24no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—41no—
103C2×C4×C17—41no—
104C2×C4×C17—42no—
105C2×C4×C17#10214yes—
106C2×C4×C17C4×D412yes—
107C2×C4×C17#12011yescharacteristic
108C2×C4×C17#12112yes—
109C2×C4×C17#12311yescharacteristic
110C2×C4×C17C2×D811yescharacteristic
111C2×C4×C17C2×SD1611yescharacteristic
112C2×C4×C17#13512yes—
113C2×Dic17—42no—
114C2×Dic17—42no—
115C2×Dic17—41no—
116C2×Dic17—41no—
117C2×Dic17—42no—
118C2×Dic17—41no—
119D68—24no—
120D68—42no—
121C4×D17—82no—
122C4×D17—82no—
123C8×C17—44no—
124#3622—82no—
125#2982D1711yescharacteristic, complemented
126#2536—171no—
127#1801—171no—
128#1486—172no—
129#1286—172no—
130C22×C17—21no—
131C22×C17—21no—
132C22×C17#38611yescharacteristic
133C22×C17C2×#10211yescharacteristic
134C22×C17#41611yescharacteristic
135C2×D17—42no—
136C2×D17—42no—
137C2×D17—42no—
138C2×D17—81no—
139C2×D17—81no—
140C2×D17—81no—
141C2×D17—81no—
142C4×C17—22no—
143C4×C17—24no—
144C4×C17—21no—
145C4×C17—21no—
146C4×C17—42no—
147C4×C17#46512yes—
148C4×C17#46612yes—
149Dic17—82no—
150Dic17—81no—
151C2×C22⋊Q8—341no—
152C2×C4×Q8C2×D1711yescharacteristic
153C2×C4×D4—171no—
154C8×Q8—24no—
155#431—22no—
156#431C2×D1712yes—
157#423—172no—
158C2×#101—341no—
159#398—172no—
160#394—172no—
161#388—342no—
162#382—171no—
163#380—341no—
164#343—341no—
165D4⋊C8—344no—
166C2×C17—21no—
167C2×C17—21no—
168C2×C17#189111yescharacteristic
169C2×C17#189311yescharacteristic
170C2×C17#191211yescharacteristic
171D17—82no—
172C22×Q8D6811yescharacteristic
173C22×D4—171no—
174C23×C4—341no—
175C2×M16—342no—
176C22⋊Q8—682no—
177C22⋊Q8—682no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×D4—342no—
181C4×D4—342no—
182C2×C4⋊C4—171no—
183C2×C4⋊C4—341no—
184C2×C4⋊C4—342no—
185C2×C4⋊C4C4×D1712yes—
186C2×C4⋊C4D6811yescharacteristic
187C2×C22⋊C4—341no—
188C2×C22⋊C4—342no—
189C2×C22⋊C4—342no—
190C2×C42C22⋊D1712yes—
191C2×C42C22×D1711yescharacteristic
192C4⋊C8—24no—
193C4⋊C8—22no—
194C4⋊C8—342no—
195C4⋊C8—342no—
196#101—682no—
197C22⋊C8—22no—
198C4×C8—24no—
199C4×C8C22⋊D1714yes—
200#95—172no—
201#95—341no—
202#95—342no—
203C17#948611yescharacteristic, complemented, Sylow
204C24—341no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8#6321912yes—
208C2×Q8D13612yes—
209C2×D4—342no—
210C2×D4—682no—
211C22×C4—342no—
212C22×C4—341no—
213C22×C4—681no—
214C22×C4—681no—
215C22×C4—682no—
216C22×C4—682no—
217C22×C4#6322712yes—
218C22×C4#6322712yes—
219C22×C4C2×C4×D1711yescharacteristic
220C22×C4C2×D6811yescharacteristic
221C22×C4C2×C22⋊D1711yescharacteristic
222M16—682no—
223M16—682no—
224C2×C8—24no—
225C2×C8—682no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—342no—
229C4⋊C4—684no—
230C4⋊C4—681no—
231C4⋊C4—681no—
232C22⋊C4—684no—
233C22⋊C4—684no—
234C22⋊C4—681no—
235C22⋊C4—681no—
236C42—22no—
237C42—22no—
238C42#6325212yes—
239C42C2×C22⋊D1712yes—
240C23—342no—
241C23—681no—
242C23—681no—
243C23—681no—
244C23—681no—
245C23—681no—
246C23—681no—
247C23C2×#6322711yescharacteristic
248Q8—24no—
249Q8—24no—
250D4—344no—
251D4—682no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—682no—
259C2×C4—682no—
260C2×C4—681no—
261C2×C4—682no—
262C2×C4—681no—
263C2×C4—681no—
264C2×C4—1362no—
265C2×C4—1362no—
266C2×C4#1058749214yes—
267C2×C4C2×#6321911yescharacteristic
268C2×C4C2×D13611yescharacteristic
269C2×C4#1058759012yes—
270C2×C4C4×C22⋊D1712yes—
271C2×C4#1058761511yescharacteristic
272C2×C4#1058761612yes—
273C2×C4#1058763911yescharacteristic
274C8—44no—
275C8—1362no—
276C22—21no—
277C22—21no—
278C22—682no—
279C22—682no—
280C22—682no—
281C22—1361no—
282C22—1361no—
283C22—1361no—
284C22—1361no—
285C22#1176268811yescharacteristic
286C22C2×#1058749211yescharacteristic
287C22#1176280611yescharacteristic
288C4—21no—
289C4—21no—
290C4—22no—
291C4—24no—
292C4—42no—
293C4—1362no—
294C4—1361no—
295C4#1176303812yes—
296C4#1176304512yes—
297C2—21no—
298C2—21no—
299C2—1362no—
300C2#1452920211yescharacteristic
301C2#1452921211yescharacteristic
302C2#1452932111yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2982C2
socleC22×C17#386
prime radicalC22×D68C2×C4
cosocleC2×C42×C17C23
hypercenter#2982D17
hypocenterC17#9486
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #14528780C2semidirect
4. #14526643C2semidirect
5. C17×#2982C2semidirect
125. #2982D17semidirect
203. C17#9486semidirect

Maximal subgroups

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

subgroupindex
#145336892
#145287802
#145272102
#145266432
C17×#29822
#948617