Finite Group Databasebeta

← #17948906

Subgroups — #17948906

Order 4352.

374 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 72 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#14545012C211yesmaximal, characteristic
2#14544663C211yesmaximal, characteristic
3D17×#340C211yesmaximal, characteristic
4C2×#10587530—21no—
5C2×#10587535C412yescomplemented
6#11763780—21no—
7#11763704C412yescomplemented
8C22⋊#63216—21no—
9#11763685—21no—
10C22×C8×D17C2211yescharacteristic
11C2×C4⋊C4×D17—21no—
12#11762572C411yescharacteristic
13#11762571—21no—
14C17×#340C411yescharacteristic
15C22×C4×D17—41no—
16C22×C4×D17D411yescharacteristic
17C2×#63243—22no—
18C22×C8×C17C2×C411yescharacteristic
19C2×C4⋊C4×C17—21no—
20C2×C4⋊Dic17—21no—
21C2×#63225—41no—
22C22×#3622D411yescharacteristic
23C2×C8×D17—42no—
24C2×C8×D17C2×C411yescharacteristic
25C2×C8×D17C2×C411yescharacteristic
26C2×C8×D17D411yescharacteristic
27C2×C8×D17Q811yescharacteristic
28C4⋊C4×D17—24no—
29C4⋊C4×D17—42no—
30C22⋊#3624—41no—
31#10587543—41no—
32C2×#63216—22no—
33#10587535—42no—
34#10587530—42no—
35C23×D17SD1611yescharacteristic
36C22×C4×C17—41no—
37C22×C4×C17C22⋊C411yescharacteristic
38C22×Dic17—41no—
39C22×Dic17D811yescharacteristic
40C2×C4×D17—22no—
41C2×C4×D17—22no—
42C2×C4×D17—44no—
43C2×C4×D17—82no—
44C2×C4×D17—82no—
45C2×C4×D17C22⋊C411yescharacteristic
46C2×C4×D17C4⋊C411yescharacteristic
47C2×C4×D17SD1611yescharacteristic
48C2×C4×D17Q1611yescharacteristic
49C2×#3624—42no—
50#63243—42no—
51#63243—42no—
52C2×C8×C17—41no—
53C2×C8×C17C4211yescharacteristic
54C2×C8×C17C4⋊C411yescharacteristic
55C4⋊C4×C17—22no—
56C4⋊C4×C17—41no—
57C4⋊Dic17—22no—
58C4⋊Dic17—41no—
59#63225—44no—
60C2×#3622—41no—
61C2×#3622C22⋊C411yescharacteristic
62C2×#3622C4⋊C411yescharacteristic
63C8×D17—22no—
64C8×D17—22no—
65C8×D17—41no—
66C8×D17—41no—
67#63216—42no—
68#7264—171nomaximal, Sylow
69C23×C17#10211yescharacteristic
70C22×D17—21no—
71C22×D17—21no—
72C22×D17—22no—
73C22×D17—22no—
74C22×D17#10311yescharacteristic
75C22×D17#10611yescharacteristic
76C2×C4×C17—21no—
77C2×C4×C17—21no—
78C2×C4×C17—42no—
79C2×C4×C17—81no—
80C2×C4×C17—81no—
81C2×C4×C17#9511yescharacteristic
82C2×C4×C17#10311yescharacteristic
83C2×Dic17—22no—
84C2×Dic17—42no—
85C2×Dic17—81no—
86C2×Dic17—81no—
87C2×Dic17#10211yescharacteristic
88C2×Dic17#10711yescharacteristic
89C4×D17—22no—
90C4×D17—21no—
91C4×D17—21no—
92C4×D17—22no—
93C4×D17—88no—
94C4×D17#9912yes—
95C4×D17#10412yes—
96#3624—82no—
97C8×C17—21no—
98C8×C17—21no—
99#3622—21no—
100#3622—21no—
101#2112—171no—
102#1810—171no—
103C2×#340—171no—
104C22×C17—21no—
105C22×C17—21no—
106C22×C17#34011yescharacteristic
107C22×C17#36111yescharacteristic
108C22×C17#36211yescharacteristic
109C2×D17—24no—
110C2×D17—24no—
111C2×D17—21no—
112C2×D17—21no—
113C2×D17—21no—
114C2×D17—21no—
115C2×D17#32812yes—
116C2×D17#36512yes—
117C4×C17—21no—
118C4×C17—82no—
119C4×C17#33911yescharacteristic
120C4×C17#34211yescharacteristic
121Dic17—21no—
122Dic17—82no—
123Dic17#32712yes—
124C23×C8—171no—
125C22×C4⋊C4—341no—
126C2×M32—341no—
127C2×#108—172no—
128#407—341no—
129C22⋊C16—172no—
130C22⋊C16—341no—
131#341—341no—
132#340—171no—
133#340—341no—
134#340#59511yescharacteristic
135C2×C17—21no—
136C2×C17—21no—
137C2×C17#129411yescharacteristic
138C2×C17#138311yescharacteristic
139C2×C17#138711yescharacteristic
140D17—22no—
141D17—22no—
142C23×C4—171no—
143C23×C4—681no—
144C2×M16—342no—
145C22×C8—171no—
146C22×C8—171no—
147C22×C8—171no—
148C22×C8—171no—
149C22×C8—171no—
150C22×C8—682no—
151C22×C8C2×#59511yescharacteristic
152C2×C4⋊C4—21no—
153C2×C4⋊C4—344no—
154C2×C4⋊C4—341no—
155C2×C4⋊C4—682no—
156C2×C4⋊C4—681no—
157M32—682no—
158C2×C16—342no—
159#108—682no—
160C22⋊C8—681no—
161C22⋊C8—681no—
162C17#726411yescharacteristic, complemented, Sylow
163C24—171no—
164C22×C4—41no—
165C22×C4—171no—
166C22×C4—171no—
167C22×C4—171no—
168C22×C4—171no—
169C22×C4—171no—
170C22×C4—342no—
171C22×C4—342no—
172C22×C4—684no—
173C22×C4—681no—
174C22×C4—1362no—
175C22×C4—1362no—
176C22×C4C22⋊#59511yescharacteristic
177M16—682no—
178M16—682no—
179C2×C8—41no—
180C2×C8—171no—
181C2×C8—171no—
182C2×C8—342no—
183C2×C8—342no—
184C2×C8—681no—
185C2×C8—681no—
186C2×C8—681no—
187C2×C8—682no—
188C2×C8C4×#59511yescharacteristic
189C2×C8C4⋊#59511yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—41no—
192C4⋊C4—342no—
193C4⋊C4—681no—
194C4⋊C4—684no—
195C16—682no—
196C23—171no—
197C23—171no—
198C23—342no—
199C23—342no—
200C23—341no—
201C23—341no—
202C23#1058754711yescharacteristic
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—171no—
209C2×C4—172no—
210C2×C4—172no—
211C2×C4—171no—
212C2×C4—342no—
213C2×C4—342no—
214C2×C4—342no—
215C2×C4—341no—
216C2×C4—341no—
217C2×C4—682no—
218C2×C4—1368no—
219C2×C4—1361no—
220C2×C4—1361no—
221C2×C4(C2×#595)⋊C411yescharacteristic
222C2×C4#1058754911yescharacteristic
223C8—21no—
224C8—21no—
225C8—341no—
226C8—341no—
227C8—1362no—
228C22—21no—
229C22—21no—
230C22—172no—
231C22—172no—
232C22—344no—
233C22—344no—
234C22—341no—
235C22—341no—
236C22—341no—
237C22—341no—
238C22#1176368211yescharacteristic
239C22#1176372711yescharacteristic
240C22#1176372811yescharacteristic
241C4—21no—
242C4—82no—
243C4—172no—
244C4—341no—
245C4—1362no—
246C4#1176368011yescharacteristic
247C4#1176368611yescharacteristic
248C2—21no—
249C2—21no—
250C2—342no—
251C2—342no—
252C2#1454365811yescharacteristic
253C2#1454384811yescharacteristic
254C2#1454385811yescharacteristic
subgroupG/H
centerC22#11763682
commutatorC2×C8×C17C42
FrattiniC22×C8C2×#595
FittingC17×#340C4
socleC22×C17#340
prime radicalC23×D17SD16
cosocleC22×C8×D17C22
hypercenter#340#595
hypocenterC17#7264
LCSG ▷ C2×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #340

Complements

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

NH ≅ G/Ntype
5. C2×#10587535C4semidirect
7. #11763704C4semidirect
162. C17#7264semidirect

Maximal subgroups

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

subgroupindex
#145450122
#145446632
D17×#3402
#726417