Finite Group Databasebeta

← #17949909

Subgroups — #17949909

Order 4352.

520 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 102 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#14544569C212yesmaximal, complemented
2#14543604C212yesmaximal, complemented
3#14543600C212yesmaximal
4#124×Dic17C211yesmaximal, characteristic
5C2×#10587544—22no—
6C2×#10587542—22no—
7#11763675C2214yes—
8C2×Q8×Dic17C411yescharacteristic
9C2×D4×Dic17C411yescharacteristic
10#11762716C2211yescharacteristic
11#11762712C2211yescharacteristic
12#11762702—22no—
13#11762675C411yescharacteristic
14C22⋊C4×Dic17—22no—
15C42×Dic17C2211yescharacteristic
16C2×C17×#124C411yescharacteristic
17C22×Q8×C17C2×C411yescharacteristic
18C22×D4×C17C2×C411yescharacteristic
19C23×Dic17—21no—
20C22×#3624—24no—
21C17×#124—22no—
22C17×#124—22no—
23C2×C22⋊C4×C17—22no—
24C2×C42×C17C2×C411yescharacteristic
25C2×C22⋊Dic17—21no—
26C2×C22⋊Dic17—22no—
27Q8×Dic17—42no—
28D4×Dic17—42no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17C2×C411yescharacteristic
31C2×C4⋊Dic17C2×C411yescharacteristic
32C2×C4×Dic17—21no—
33C2×C4×Dic17C2×C411yescharacteristic
34C2×C4×Dic17D412yes—
35C2×C4×Dic17D412yes—
36C2×C4×Dic17C2311yescharacteristic
37#10587544—28no—
38#10587542—44no—
39#10587502—22no—
40#10587502—22no—
41C24×C17—21no—
42C2×Q8×C17—21no—
43C2×Q8×C17—41no—
44C2×Q8×C17—41no—
45C2×Q8×C17M1612yes—
46C2×D4×C17—21no—
47C2×D4×C17—41no—
48C2×D4×C17—41no—
49C2×D4×C17M1612yes—
50C22×C4×C17—21no—
51C22×C4×C17C22⋊C412yes—
52C22×C4×C17C22×C411yescharacteristic
53C22×Dic17—21no—
54C22×Dic17—21no—
55C22×Dic17—42no—
56C22×Dic17—44no—
57C22×Dic17C22⋊C411yescharacteristic
58C22×Dic17C22⋊C411yescharacteristic
59C22×Dic17C2×D412yes—
60C2×#3624—28no—
61C2×#3624—44no—
62C2×#3624—44no—
63C22⋊C4×C17—28no—
64C42×C17—21no—
65C42×C17—21no—
66C22⋊Dic17—42no—
67C22⋊Dic17—42no—
68C22⋊Dic17—42no—
69C4⋊Dic17—42no—
70C4⋊Dic17—41no—
71C4⋊Dic17—41no—
72C4×Dic17—22no—
73C4×Dic17—24no—
74C4×Dic17—22no—
75C4×Dic17—22no—
76C4×Dic17—41no—
77C4×Dic17—42no—
78#7635—171nomaximal, Sylow
79C23×C17—22no—
80C23×C17—41no—
81C23×C17—41no—
82C23×C17—42no—
83C23×C17—42no—
84C23×C17C2×C22⋊C411yescharacteristic
85Q8×C17—42no—
86Q8×C17—42no—
87D4×C17—42no—
88D4×C17—42no—
89C2×C4×C17—21no—
90C2×C4×C17—21no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—41no—
96C2×C4×C17—41no—
97C2×C4×C17C2×M1611yescharacteristic
98C2×C4×C17C2×M1611yescharacteristic
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—21no—
102C2×Dic17—22no—
103C2×Dic17—41no—
104C2×Dic17—41no—
105C2×Dic17—42no—
106C2×Dic17—42no—
107C2×Dic17—84no—
108C2×Dic17#10012yes—
109C2×Dic17#10414yes—
110C2×Dic17AGL(1,ℤ/8)12yes—
111C2×Dic17#13712yes—
112#3624—48no—
113C4×#124—171no—
114#1767—172no—
115#1272—172no—
116#1270—172no—
117C22×C17—42no—
118C22×C17—44no—
119C22×C17—81no—
120C22×C17—82no—
121C22×C17—81no—
122C22×C17#40711yescharacteristic
123C22×C17#41011yescharacteristic
124C22×C17C2×#10011yescharacteristic
125C22×C17#41712yes—
126C22×C17C2×#10412yes—
127C4×C17—21no—
128C4×C17—21no—
129C4×C17—42no—
130C4×C17—42no—
131Dic17—22no—
132Dic17—22no—
133Dic17—41no—
134Dic17—81no—
135Dic17—81no—
136Dic17#32914yes—
137C2×#124#59511yescharacteristic
138C2×C4×Q8—171no—
139C2×C4×D4—171no—
140C2×C4⋊C8—342no—
141C2×#97—342no—
142#391—171no—
143#390—171no—
144#388—342no—
145#382—171no—
146C4×C22⋊C4—342no—
147C43—171no—
148#336—174no—
149C2×C17—82no—
150C2×C17#146611yescharacteristic
151C2×C17#148912yes—
152C2×C17#150912yes—
153C2×C17C2×#32911yescharacteristic
154C2×C17#156011yescharacteristic
155C22×Q8C2×#59511yescharacteristic
156C22×D4C2×#59511yescharacteristic
157C23×C4—341no—
158C22×C8—344no—
159#124—22no—
160#124—22no—
161C4×Q8—682no—
162C4×D4—682no—
163C2×C4⋊C4—171no—
164C2×C4⋊C4—171no—
165C2×C4⋊C4—341no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—341no—
168C2×C22⋊C4—342no—
169C2×C42—172no—
170C2×C42—172no—
171C2×C42—171no—
172C2×C42—171no—
173C2×C42—341no—
174C2×C42C2×#59511yescharacteristic
175C4⋊C8—348no—
176#97—684no—
177#95—342no—
178#95—342no—
179C17#763511yescharacteristic, complemented, Sylow
180C24—21no—
181C2×Q8—21no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8#6324712yes—
185C2×D4—21no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4#6324712yes—
189C22×C4—21no—
190C22×C4—171no—
191C22×C4—172no—
192C22×C4—171no—
193C22×C4—341no—
194C22×C4—341no—
195C22×C4—682no—
196C22×C4—684no—
197C22×C4C22⋊#59512yes—
198C22×C4C22×#59511yescharacteristic
199C2×C8—348no—
200C2×C8—684no—
201C2×C8—684no—
202C4⋊C4—681no—
203C4⋊C4—681no—
204C4⋊C4—682no—
205C22⋊C4—28no—
206C22⋊C4—682no—
207C22⋊C4—682no—
208C22⋊C4—682no—
209C42—21no—
210C42—21no—
211C42—342no—
212C42—344no—
213C42—342no—
214C42—342no—
215C42—681no—
216C42—682no—
217C23—22no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23—42no—
222C23C2×C22⋊#59511yescharacteristic
223Q8—42no—
224Q8—42no—
225D4—42no—
226D4—42no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—172no—
236C2×C4—174no—
237C2×C4—172no—
238C2×C4—172no—
239C2×C4—342no—
240C2×C4—342no—
241C2×C4—341no—
242C2×C4—342no—
243C2×C4—681no—
244C2×C4—681no—
245C2×C4—682no—
246C2×C4—682no—
247C2×C4—1364no—
248C2×C4C2×#6324711yescharacteristic
249C2×C4C2×#6324711yescharacteristic
250C8—688no—
251C22—42no—
252C22—44no—
253C22—82no—
254C22—81no—
255C22—81no—
256C22#1176377911yescharacteristic
257C22#1176378911yescharacteristic
258C22D17⋊#10011yescharacteristic
259C22#1176380012yes—
260C22D17⋊#10412yes—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—174no—
266C4—342no—
267C4—342no—
268C4—681no—
269C4—1361no—
270C4—1361no—
271C2—82no—
272C2#1454404411yescharacteristic
273C2#1454411912yes—
274C2#1454418512yes—
275C2D17⋊#32911yescharacteristic
276C2#1454432211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×#124C4
socleC23×C17C2×C22⋊C4
prime radicalC22×D4×C17C2×C4
cosocleC2×C4×Dic17C23
hypercenterC2×#124#595
hypocenterC17#7635
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #14544569C2semidirect
2. #14543604C2semidirect
179. C17#7635semidirect

Maximal subgroups

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

subgroupindex
#145445692
#145436042
#145436002
#124×Dic172
#763517