Finite Group Databasebeta

← #17958935

Subgroups — #17958935

Order 4352.

366 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 94 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#14545144C211yesmaximal, characteristic
2#14544342C211yesmaximal, characteristic
3#14543741C212yesmaximal, complemented
4#14543740C212yesmaximal, complemented
5C4×Q16×D17C211yesmaximal, characteristic
6#11763813—21no—
7C4⋊#63216—21no—
8#11763704—21no—
9#11763695C2212yes—
10#11763672—21no—
11#11763672C2212yes—
12C2×Q16×D17C411yescharacteristic
13C4×Q8×D17—21no—
14C4×Dic68C411yescharacteristic
15Q16×Dic17C2211yescharacteristic
16#11762903C2211yescharacteristic
17C4×#63231—21no—
18#11762901—21no—
19#107×D17C411yescharacteristic
20#103×D17—21no—
21C4×C8×D17C2211yescharacteristic
22C4×Q16×C17C411yescharacteristic
23C2×Q8×D17—21no—
24C2×#63243—41no—
25C2×Q16×C17C2×C411yescharacteristic
26C4×Q8×C17—21no—
27Q8×Dic17—21no—
28C2×#63231—21no—
29Q16×D17—22no—
30Q16×D17—22no—
31C2×Dic68C2×C411yescharacteristic
32C2×C8×D17C2×C411yescharacteristic
33C2×C8×D17D411yescharacteristic
34C4⋊C4×D17—21no—
35C4⋊C4×D17—41no—
36#10587573—21no—
37#10587573—41no—
38C42×D17—41no—
39C42×D17D411yescharacteristic
40C4×Dic34—21no—
41C4×Dic34—41no—
42C4⋊#3624—22no—
43C4⋊#3624—41no—
44C2×#63216—22no—
45C17×#107C2×C411yescharacteristic
46C17×#103—21no—
47C4×C8×C17C2×C411yescharacteristic
48#10587506—21no—
49#10587489C2×C411yescharacteristic
50#10587487—21no—
51C8×Dic17D411yescharacteristic
52C8×Dic17C2311yescharacteristic
53#10587481—21no—
54#10587478—21no—
55C4×#3622D411yescharacteristic
56C2×Q8×C17—21no—
57Q8×D17—24no—
58Q8×D17—42no—
59C2×C4×D17—41no—
60C2×C4×D17—41no—
61C2×C4×D17C22⋊C411yescharacteristic
62C2×C4×D17D811yescharacteristic
63C2×C4×D17SD1611yescharacteristic
64C2×Dic34—21no—
65C2×Dic34—41no—
66C2×#3624—42no—
67#63243—82no—
68Q16×C17—21no—
69Q16×C17M1612yescomplemented
70C2×C8×C17C22⋊C411yescharacteristic
71C2×C8×C17C22×C411yescharacteristic
72C4⋊C4×C17—21no—
73C4⋊C4×C17—41no—
74C42×C17—41no—
75C42×C17C22⋊C411yescharacteristic
76#63231—24no—
77C4⋊Dic17—21no—
78C4⋊Dic17—41no—
79#63225—41no—
80#63225—41no—
81#63225—41no—
82C4×Dic17—41no—
83C4×Dic17—41no—
84C4×Dic17—41no—
85C4×Dic17D811yescharacteristic
86C4×Dic17SD1611yescharacteristic
87C4×Dic17C2×D411yescharacteristic
88C2×#3622C22⋊C411yescharacteristic
89C2×#3622C2×D411yescharacteristic
90Dic68—21no—
91Dic68M1612yescomplemented
92C8×D17—21no—
93C8×D17—21no—
94C8×D17—41no—
95#63216—42no—
96#11658—171nomaximal, Sylow
97C22×D17#10211yescharacteristic
98Q8×C17—22no—
99Q8×C17—41no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17#10211yescharacteristic
103C2×C4×C17#10211yescharacteristic
104C2×C4×C17C2×C22⋊C411yescharacteristic
105C2×Dic17—41no—
106C2×Dic17—41no—
107C2×Dic17#10211yescharacteristic
108C2×Dic17C2×D811yescharacteristic
109C2×Dic17C2×SD1611yescharacteristic
110C4×D17—21no—
111C4×D17—21no—
112C4×D17—41no—
113C4×D17—44no—
114C4×D17—82no—
115C4×D17SD3212yes—
116C4×D17Q3212yes—
117Dic34—22no—
118Dic34—44no—
119Dic34—41no—
120#3624—82no—
121C8×C17—21no—
122C8×C17C2×M1611yescharacteristic
123C8×C17C2×M1611yescharacteristic
124#3622—21no—
125#3622—21no—
126C2×C4×Q16—171no—
127#2162—171no—
128#1567—171no—
129#1332—172no—
130#1332—172no—
131C22×C17C2×#10211yescharacteristic
132C2×D17—21no—
133C2×D17—21no—
134C2×D17#35812yes—
135C4×C17—21no—
136C4×C17—42no—
137C4×C17—81no—
138C4×C17#35812yes—
139C4×C17#40711yescharacteristic
140C4×C17#42111yescharacteristic
141Dic17—21no—
142Dic17—21no—
143Dic17—42no—
144Dic17—81no—
145Dic17C2×SD3211yescharacteristic
146Dic17C2×Q3211yescharacteristic
147C22×Q16—171no—
148C2×C4×Q8—341no—
149C4×Q16—171no—
150C4×Q16—171no—
151C4×Q16—171no—
152C4×Q16—341no—
153C4×Q16—341no—
154C4×Q16#59511yescharacteristic
155C2×#107—171no—
156#423—341no—
157C2×#103—341no—
158C2×C4×C8—171no—
159C4⋊C16—341no—
160C22⋊C16—341no—
161C4×C16—172no—
162#335—172no—
163#335—341no—
164C2×C17#148611yescharacteristic
165C2×C17C2×#35811yescharacteristic
166C2×C17#214311yescharacteristic
167D17—22no—
168C22×Q8—341no—
169C2×Q16—171no—
170C2×Q16—341no—
171C2×Q16—342no—
172C2×Q16—342no—
173C2×Q16C2×#59511yescharacteristic
174C2×M16—681no—
175C22×C8—171no—
176C22×C8—171no—
177C4×Q8—21no—
178C4×Q8—341no—
179C4×Q8—341no—
180C4×Q8—341no—
181C4×Q8—681no—
182C4×Q8—681no—
183C2×C4⋊C4—341no—
184C2×C4⋊C4—681no—
185C2×C42—171no—
186C2×C42—681no—
187C2×C16—342no—
188#107—171no—
189#107—341no—
190#107C2×#59511yescharacteristic
191C4⋊C8—342no—
192C4⋊C8—681no—
193#103—21no—
194#103—341no—
195#103—341no—
196#103—341no—
197C4×C8—171no—
198C4×C8—171no—
199C4×C8—171no—
200C4×C8C2×#59511yescharacteristic
201C17#1165811yescharacteristic, complemented, Sylow
202C2×Q8—21no—
203C2×Q8—344no—
204C2×Q8—341no—
205C2×Q8—681no—
206C2×Q8—682no—
207C22×C4—171no—
208C22×C4—171no—
209C22×C4—171no—
210C22×C4—681no—
211C22×C4—681no—
212Q16—21no—
213Q16—172no—
214Q16—344no—
215Q16—341no—
216Q16#6324312yescomplemented
217M16—1362no—
218C2×C8—171no—
219C2×C8—171no—
220C2×C8—341no—
221C2×C8—341no—
222C2×C8—681no—
223C2×C8—682no—
224C2×C8C22⋊#59511yescharacteristic
225C2×C8C22×#59511yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—41no—
228C4⋊C4—341no—
229C4⋊C4—681no—
230C4⋊C4—681no—
231C4⋊C4—681no—
232C4⋊C4—681no—
233C42—41no—
234C42—171no—
235C42—171no—
236C42—171no—
237C42—681no—
238C42—681no—
239C42—681no—
240C42C22⋊#59511yescharacteristic
241C16—682no—
242C23—171no—
243Q8—22no—
244Q8—41no—
245Q8—342no—
246Q8—684no—
247Q8—681no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—171no—
251C2×C4—172no—
252C2×C4—172no—
253C2×C4—171no—
254C2×C4—171no—
255C2×C4—341no—
256C2×C4—341no—
257C2×C4—681no—
258C2×C4—684no—
259C2×C4—681no—
260C2×C4—681no—
261C2×C4—1362no—
262C2×C4#1058754711yescharacteristic
263C2×C4#1058754711yescharacteristic
264C2×C4C2×C22⋊#59511yescharacteristic
265C8—21no—
266C8—341no—
267C8—341no—
268C8—1362no—
269C8C2×#6324311yescharacteristic
270C8D17⋊M1611yescharacteristic
271C22—172no—
272C22—341no—
273C22—341no—
274C22C2×#1058754711yescharacteristic
275C4—21no—
276C4—42no—
277C4—81no—
278C4—171no—
279C4—171no—
280C4—341no—
281C4—341no—
282C4—682no—
283C4—1361no—
284C4#1176372212yes—
285C4#1176378011yescharacteristic
286C4#1176380611yescharacteristic
287C2—342no—
288C2#1454411011yescharacteristic
289C2C2×#1176372211yescharacteristic
290C2#1454509811yescharacteristic
subgroupG/H
centerC22C2×#10587547
commutatorC2×C8×C17C22×C4
FrattiniC2×C8C22×#595
FittingC4×Q16×C17C4
socleC22×C17C2×#102
prime radicalC22×D17#102
cosocleC8×Dic17C23
hypercenterC4×Q16#595
hypocenterC17#11658
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
3. #14543741C2semidirect
4. #14543740C2semidirect
69. Q16×C17M16semidirect
91. Dic68M16semidirect
201. C17#11658semidirect
216. Q16#63243semidirect

Maximal subgroups

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

subgroupindex
#145451442
#145443422
#145437412
#145437402
C4×Q16×D172
#1165817