Finite Group Databasebeta

← #17949257

Subgroups — #17949257

Order 4352.

282 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 36 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#14543878C211yesmaximal, characteristic
2#14543713C211yesmaximal, characteristic
3#14529867C211yesmaximal, characteristic
4C2×#10587539—21no—
5#11763654—21no—
6C22×#63230—21no—
7C2×#10587506—21no—
8#11762711C2211yescharacteristic
9#11762564C411yescharacteristic, complemented
10C2×C17×C4⋊Q8C411yescharacteristic, complemented
11C22×Q8×C17—21no—
12C22×D68D411yescharacteristic
13C2×C22⋊#595—41no—
14C17×C4⋊Q8—42no—
15C2×C4⋊C4×C17—41no—
16C2×C42×C17C2×C411yescharacteristic
17C2×#63230—42no—
18C2×#63230—42no—
19C2×#63230—41no—
20C2×#63230—41no—
21C2×#63227—41no—
22C2×C4⋊Dic17D411yescharacteristic
23C22×#3622—21no—
24C22⋊#3624—41no—
25#10587539—22no—
26#10587539—41no—
27#10587506—42no—
28C23×D17—21no—
29C22×#595—81no—
30C2×Q8×C17—22no—
31C2×Q8×C17—41no—
32C2×Q8×C17—41no—
33C2×Q8×C17—41no—
34C2×Q8×C17—41no—
35C2×Q8×C17—41no—
36C22×C4×C17—21no—
37C22×C4×C17—41no—
38C22×C4×C17C22⋊C411yescharacteristic
39C22×Dic17—21no—
40C2×D68—21no—
41C2×D68—41no—
42C2×D68—41no—
43C2×D68D811yescharacteristic
44C2×D68SD1611yescharacteristic
45C22⋊#595—42no—
46C22⋊#595—81no—
47C2×#3624—81no—
48C4⋊C4×C17—42no—
49C4⋊C4×C17—42no—
50C42×C17—41no—
51#63230—44no—
52#63230—44no—
53#63227—82no—
54C4⋊Dic17—41no—
55C2×#3622—22no—
56C2×#3622—41no—
57C2×#3622—41no—
58#7403—171nomaximal, Sylow
59C23×C17#9911yescharacteristic
60C22×D17—21no—
61C22×D17—21no—
62C22×D17—41no—
63C22×D17—41no—
64C22×D17—81no—
65C22×D17—81no—
66C2×#595—82no—
67C2×#595—161no—
68C2×#595—161no—
69Q8×C17—42no—
70Q8×C17—42no—
71Q8×C17—81no—
72Q8×C17—81no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—22no—
76C2×C4×C17—41no—
77C2×C4×C17—41no—
78C2×C4×C17—42no—
79C2×C4×C17—81no—
80C2×C4×C17—81no—
81C2×C4×C17#10211yescharacteristic
82C2×C4×C17#10411yescharacteristic
83C2×Dic17—41no—
84C2×Dic17—81no—
85D68—41no—
86D68—41no—
87D68—81no—
88#3624—161no—
89#3622—42no—
90#2013—171no—
91#1393—171no—
92#1318—171no—
93C22×C17—21no—
94C22×C17—21no—
95C22×C17#32711yescharacteristic
96C22×C17AΣL(1,𝔽16)11yescharacteristic
97C22×C17#35311yescharacteristic
98C2×D17—41no—
99C2×D17—41no—
100C2×D17—81no—
101C2×D17—81no—
102C2×D17—81no—
103C2×D17—81no—
104C2×D17—161no—
105#595—162no—
106C4×C17—21no—
107C4×C17—21no—
108C4×C17—42no—
109C4×C17—82no—
110Dic17—81no—
111C22×SD16—341no—
112C2×C4⋊Q8#59511yescharacteristic, complemented
113C2×#103—341no—
114C2×#99—341no—
115#390—171no—
116#336—171no—
117#324—341no—
118C2×C17—21no—
119C2×C17—21no—
120C2×C17#134311yescharacteristic
121C2×C17#140711yescharacteristic
122C2×C17#140911yescharacteristic
123D17—81no—
124D17—81no—
125C22×Q8—21no—
126C22×D4—171no—
127C2×SD16—682no—
128C2×SD16—681no—
129C2×SD16—681no—
130C2×SD16—682no—
131C22×C8—341no—
132C4⋊Q8—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—171no—
135C2×C22⋊C4—681no—
136C2×C22⋊C4—681no—
137C2×C42C2×#59511yescharacteristic
138#103—682no—
139#99—342no—
140#99—681no—
141C22⋊C8—681no—
142C17#740311yescharacteristic, complemented, Sylow
143C24—341no—
144C2×Q8—22no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×D4—171no—
151C2×D4—171no—
152C2×D4—341no—
153C2×D4—681no—
154C2×D4—681no—
155C22×C4—21no—
156C22×C4—41no—
157C22×C4—341no—
158C22×C4—1361no—
159C22×C4C22⋊#59511yescharacteristic
160SD16—684no—
161SD16—684no—
162C2×C8—342no—
163C2×C8—681no—
164C2×C8—681no—
165C2×C8—1361no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—681no—
169C22⋊C4—682no—
170C22⋊C4—1362no—
171C22⋊C4—1361no—
172C42—41no—
173C23—341no—
174C23—341no—
175C23—681no—
176C23—681no—
177C23—1361no—
178C23—1361no—
179C23#1058755111yescharacteristic
180Q8—42no—
181Q8—42no—
182Q8—81no—
183Q8—81no—
184D4—681no—
185D4—681no—
186D4—1361no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—2721no—
196C2×C4—2721no—
197C2×C4—681no—
198C2×C4—1362no—
199C2×C4—1361no—
200C2×C4#1058754711yescharacteristic
201C2×C4#1058754811yescharacteristic
202C8—2721no—
203C8—682no—
204C22—21no—
205C22—21no—
206C22—2721no—
207C22—681no—
208C22—681no—
209C22—1361no—
210C22—1361no—
211C22—1361no—
212C22—1361no—
213C22#1176366011yescharacteristic
214C22#1176370911yescharacteristic
215C22#1176371311yescharacteristic
216C4—21no—
217C4—21no—
218C4—42no—
219C4—82no—
220C4—2722no—
221C4—1361no—
222C2—21no—
223C2—21no—
224C2—1361no—
225C2—1361no—
226C2#1454376211yescharacteristic
227C2#1454391311yescharacteristic
228C2#1454391711yescharacteristic
subgroupG/H
centerC22#11763713
commutatorC2×C42×C17C2×C4
FrattiniC2×C42C2×#595
FittingC2×C17×C4⋊Q8C4
socleC22×C17#353
prime radicalC22×D68D4
cosocle#11762711C22
hypercenterC2×C4⋊Q8#595
hypocenterC17#7403
LCSG ▷ C2×C42×C17 ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
9. #11762564C4semidirect
10. C2×C17×C4⋊Q8C4semidirect
112. C2×C4⋊Q8#595semidirect
142. C17#7403semidirect

Maximal subgroups

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

subgroupindex
#145438782
#145437132
#145298672
#740317