Finite Group Databasebeta

← #16947765

Subgroups — #16947765

Order 4352.

384 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 71 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
1Q16×D68C211yesmaximal, characteristic, complemented
2#14531040C211yesmaximal, characteristic, complemented
3#14530824C211yesmaximal, characteristic, complemented
4#14528140C211yesmaximal, characteristic
5#14526832C211yesmaximal, characteristic, complemented
6#14526740C211yesmaximal, characteristic, complemented
7C17×#1332C211yesmaximal, characteristic, complemented
8C2×Q16×D17—21no—
9Q8×D68—21no—
10C2×#10587471—22no—
11C2×#10587499—21no—
12#11763189C2211yescharacteristic
13#11763186C2211yescharacteristic
14#11763051—21no—
15#11763019—21no—
16#11763005—21no—
17#11762966—22no—
18C8×D68C2211yescharacteristic
19C4⋊#63214C2211yescharacteristic
20#11762609—21no—
21#11762552—21no—
22C4×Q16×C17C2211yescharacteristic
23C4×C16×C17C2211yescharacteristic
24C17×#335C2211yescharacteristic
25C2×Q8×D17—22no—
26C2×Q16×C17D411yescharacteristic
27C4×Q8×C17—21no—
28C2×#63231—22no—
29Q16×D17—42no—
30Q16×D17—42no—
31C2×Dic68—21no—
32C2×Dic68D412yes—
33C2×D136—21no—
34C2×D136—21no—
35C2×D136—41no—
36C2×C8×D17—21no—
37#10587583—22no—
38#10587583—42no—
39#10587560—21no—
40C4×D68—41no—
41C4×D68D411yescharacteristic
42#10587555—21no—
43#10587555—41no—
44C2×C16×C17—21no—
45C17×#107D411yescharacteristic
46C17×C4⋊C8—21no—
47C17×#103—21no—
48C4×C8×C17C2311yescharacteristic
49#10587499—42no—
50#10587492—41no—
51#10587491—21no—
52C2×#63214—21no—
53#10587481—22no—
54C4⋊#3622D411yescharacteristic
55#10587471—44no—
56C2×Q8×C17—21no—
57Q8×D17—44no—
58Q8×D17—82no—
59C2×D68—41no—
60C2×D68—41no—
61C2×D68—41no—
62C2×D68D811yescharacteristic
63C2×C4×D17—21no—
64C2×C4×D17—42no—
65C2×Dic34—22no—
66C2×Dic34—42no—
67Q16×C17—21no—
68Q16×C17D812yescomplemented
69C2×C8×C17—41no—
70C2×C8×C17C2×D411yescharacteristic
71C2×C8×C17C2×D411yescharacteristic
72C4⋊C4×C17—21no—
73C4⋊C4×C17—41no—
74C42×C17—41no—
75C42×C17C2×D411yescharacteristic
76#63231—44no—
77#63227—21no—
78#63227—42no—
79C4⋊Dic17—42no—
80C4⋊Dic17—41no—
81C4⋊Dic17—41no—
82C4⋊Dic17D811yescharacteristic
83C2×#3622—21no—
84Dic68—42no—
85Dic68—41no—
86Dic68—41no—
87D136—42no—
88D136—41no—
89D136—41no—
90D136—82no—
91C8×D17—41no—
92C8×D17—41no—
93C16×C17—22no—
94#63214—41no—
95#12318—171nomaximal, Sylow
96C22×D17—21no—
97C22×D17—81no—
98Q8×C17—22no—
99Q8×C17—41no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17#12011yescharacteristic
103C2×C4×C17C2×D811yescharacteristic
104C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
105C2×Dic17—21no—
106C2×Dic17—42no—
107D68—21no—
108D68—81no—
109D68—81no—
110D68—81no—
111D68—81no—
112D68—161no—
113D68SD3212yescomplemented
114C4×D17—41no—
115C4×D17—41no—
116C4×D17—84no—
117Dic34—42no—
118Dic34—42no—
119Dic34—84no—
120C8×C17—21no—
121C8×C17—81no—
122C8×C17C2×D811yescharacteristic
123C8×C17AGL(1,ℤ/8)11yescharacteristic
124#3622—41no—
125D4×Q16—171no—
126#2248—171no—
127#2207—171no—
128#1667—171no—
129#1362—171no—
130#1332D1711yescharacteristic, complemented
131D4⋊C16—171no—
132C22×C17#44711yescharacteristic
133C2×D17—22no—
134C2×D17—41no—
135C2×D17—41no—
136C2×D17—161no—
137C2×D17—161no—
138C2×D17—161no—
139C4×C17—21no—
140C4×C17—42no—
141C4×C17—81no—
142C4×C17#44711yescharacteristic
143C4×C17#45311yescharacteristic
144C4×C17C2×SD3211yescharacteristic
145C4×C17#51011yescharacteristic
146Dic17—41no—
147Dic17—82no—
148C22×Q16—341no—
149D4×Q8—341no—
150C2×SD32—342no—
151C2×SD32—341no—
152#494—171no—
153#493—171no—
154#467—341no—
155#462—341no—
156#459—341no—
157#451—342no—
158C4×Q16C2×D1711yescharacteristic
159C8×D4—171no—
160C4⋊C16—171no—
161#357—341no—
162C4×C16C2×D1711yescharacteristic
163#335C2×D1711yescharacteristic
164#331—341no—
165C2×C17#161911yescharacteristic
166C2×C17#218511yescharacteristic
167C2×C17#219311yescharacteristic
168D17—42no—
169D17—321no—
170C22×Q8—342no—
171C2×Q16—172no—
172C2×Q16—342no—
173C2×Q16—341no—
174C2×Q16—682no—
175C2×Q16—682no—
176C2×Q16D6811yescharacteristic
177C2×D8—341no—
178C2×D8—341no—
179C2×D8—681no—
180C22×C8—341no—
181C4⋊Q8—341no—
182C4⋊Q8—681no—
183#127—341no—
184C22⋊Q8—342no—
185C22⋊Q8—682no—
186C4×Q8—21no—
187C4×D4—171no—
188C4×D4—681no—
189SD32—684no—
190SD32—682no—
191C2×C16—21no—
192C2×C16—341no—
193#107D6811yescharacteristic
194C4⋊C8—21no—
195C4⋊C8—171no—
196#103—21no—
197#103—342no—
198#102—681no—
199C22⋊C8—341no—
200C4×C8C22×D1711yescharacteristic
201C17#1231811yescharacteristic, complemented, Sylow
202C2×Q8—21no—
203C2×Q8—342no—
204C2×Q8—682no—
205C2×Q8—684no—
206C2×Q8—1362no—
207C2×D4—171no—
208C2×D4—681no—
209C2×D4—681no—
210C2×D4—681no—
211C22×C4—341no—
212C22×C4—682no—
213Q16—21no—
214Q16—682no—
215Q16—684no—
216Q16—681no—
217Q16—681no—
218Q16D13612yescomplemented
219D8—682no—
220D8—681no—
221D8—681no—
222D8—1362no—
223C2×C8—41no—
224C2×C8—341no—
225C2×C8—681no—
226C2×C8—681no—
227C2×C8C2×D6811yescharacteristic
228C2×C8D4×D1711yescharacteristic
229C4⋊C4—21no—
230C4⋊C4—41no—
231C4⋊C4—171no—
232C4⋊C4—682no—
233C4⋊C4—681no—
234C4⋊C4—681no—
235C22⋊C4—341no—
236C22⋊C4—682no—
237C42—41no—
238C42D4×D1711yescharacteristic
239C16—22no—
240C16—681no—
241C23—341no—
242C23—1361no—
243Q8—22no—
244Q8—41no—
245Q8—682no—
246Q8—682no—
247Q8—1364no—
248D4—2721no—
249D4—172no—
250D4—341no—
251D4—1361no—
252D4—1361no—
253D4—1361no—
254D4—1361no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—341no—
258C2×C4—681no—
259C2×C4—681no—
260C2×C4—682no—
261C2×C4—1364no—
262C2×C4#1058756811yescharacteristic
263C2×C4D8×D1711yescharacteristic
264C2×C4#1058760011yescharacteristic
265C8—21no—
266C8—81no—
267C8—681no—
268C8C2×D13611yescharacteristic
269C8#1058759411yescharacteristic
270C22—2721no—
271C22—2721no—
272C22—2721no—
273C22—342no—
274C22—681no—
275C22—681no—
276C22#1176294411yescharacteristic
277C4—21no—
278C4—42no—
279C4—81no—
280C4—681no—
281C4—1362no—
282C4#1176294511yescharacteristic
283C4#1176297811yescharacteristic
284C4SD32×D1711yescharacteristic
285C4#1176325111yescharacteristic
286C2—5441no—
287C2—682no—
288C2#1452784011yescharacteristic
289C2#1453070211yescharacteristic
290C2#1453074911yescharacteristic
subgroupG/H
centerC22#11762944
commutator, cosocleC4×C8×C17C23
FrattiniC4×C8C22×D17
FittingC17×#1332C2
socleC22×C17#447
prime radical#14528140C2
hypercenter#1332D17
hypocenterC17#12318
LCSG ▷ C4×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1332

Complements

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

NH ≅ G/Ntype
1. Q16×D68C2semidirect
2. #14531040C2semidirect
3. #14530824C2semidirect
5. #14526832C2semidirect
6. #14526740C2semidirect
7. C17×#1332C2semidirect
68. Q16×C17D8semidirect
113. D68SD32semidirect
130. #1332D17semidirect
201. C17#12318semidirect
218. Q16D136semidirect

Maximal subgroups

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

subgroupindex
Q16×D682
#145310402
#145308242
#145281402
#145268322
#145267402
C17×#13322
#1231817