Finite Group Databasebeta

← #17958304

Subgroups — #17958304

Order 4352.

460 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 100 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#14544027C212yesmaximal, complemented
2#14543724C212yesmaximal, complemented
3#14543712C212yesmaximal
4#124×Dic17C211yesmaximal, characteristic
5#11763825—22no—
6#11763654—22no—
7#11763654C2214yes—
8C2×Q8×Dic17C2211yescharacteristic
9C2×D4×Dic17C411yescharacteristic
10#11762716C411yescharacteristic
11#11762712C2211yescharacteristic
12#11762702—22no—
13#11762675C2211yescharacteristic
14C22⋊C4×Dic17—22no—
15C42×Dic17C411yescharacteristic
16C2×C17×#124C411yescharacteristic
17C22×Q8×C17C2×C411yescharacteristic
18C22×D4×C17C2×C411yescharacteristic
19C23×Dic17—21no—
20C17×#124—22no—
21C17×#124C814yescomplemented
22C2×C22⋊C4×C17—22no—
23C2×C42×C17C2×C411yescharacteristic
24C2×C22⋊Dic17—21no—
25C2×C22⋊Dic17—22no—
26Q8×Dic17—41no—
27Q8×Dic17—41no—
28D4×Dic17—41no—
29D4×Dic17—41no—
30C2×C4⋊Dic17C2×C411yescharacteristic
31C2×C4⋊Dic17D412yes—
32C2×C4⋊Dic17C2311yescharacteristic
33C2×C4×Dic17—21no—
34C2×C4×Dic17—21no—
35C2×C4×Dic17C2×C411yescharacteristic
36C2×C4×Dic17C2×C411yescharacteristic
37C2×C4×Dic17D412yes—
38C22⋊#3624—24no—
39C4⋊#3624—42no—
40C4×#3624—42no—
41#10587502—22no—
42#10587502—22no—
43C24×C17—21no—
44C2×Q8×C17—41no—
45C2×Q8×C17—41no—
46C2×Q8×C17C2×C812yes—
47C2×Q8×C17M1612yes—
48C2×D4×C17—21no—
49C2×D4×C17—41no—
50C2×D4×C17—41no—
51C2×D4×C17C2×C812yes—
52C22×C4×C17—21no—
53C22×C4×C17C22⋊C412yes—
54C22×C4×C17C22×C411yescharacteristic
55C22×Dic17—21no—
56C22×Dic17—21no—
57C22×Dic17—44no—
58C22×Dic17—41no—
59C22×Dic17—41no—
60C22×Dic17C22⋊C411yescharacteristic
61C22×Dic17C22⋊C411yescharacteristic
62C22×Dic17C2×D412yes—
63C2×#3624—44no—
64C22⋊C4×C17—28no—
65C42×C17—21no—
66C42×C17C2×C812yes—
67C22⋊Dic17—42no—
68C22⋊Dic17—44no—
69C4⋊Dic17—42no—
70C4⋊Dic17—41no—
71C4⋊Dic17—41no—
72C4×Dic17—22no—
73C4×Dic17—22no—
74C4×Dic17—21no—
75C4×Dic17—21no—
76C4×Dic17—42no—
77C4×Dic17—42no—
78C4×Dic17—41no—
79#11476—171nomaximal, Sylow
80C23×C17—22no—
81C23×C17—42no—
82C23×C17—42no—
83C23×C17—41no—
84C23×C17—41no—
85C23×C17C2×C22⋊C411yescharacteristic
86Q8×C17—42no—
87Q8×C17—42no—
88D4×C17—42no—
89D4×C17—42no—
90C2×C4×C17—22no—
91C2×C4×C17—21no—
92C2×C4×C17—22no—
93C2×C4×C17—21no—
94C2×C4×C17—22no—
95C2×C4×C17—41no—
96C2×C4×C17—41no—
97C2×C4×C17C22⋊C814yes—
98C2×C4×C17C22×C811yescharacteristic
99C2×C4×C17C2×M1611yescharacteristic
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—21no—
103C2×Dic17—22no—
104C2×Dic17—41no—
105C2×Dic17—41no—
106C2×Dic17—42no—
107C2×Dic17—41no—
108C2×Dic17—42no—
109C2×Dic17—41no—
110C2×Dic17—82no—
111C2×Dic17—82no—
112C2×Dic17#9911yescharacteristic
113C2×Dic17#9911yescharacteristic
114#3624—84no—
115C4×#124—171no—
116#1461—172no—
117#1324—172no—
118#1318—172no—
119C22×C17—21no—
120C22×C17—21no—
121C22×C17—44no—
122C22×C17—42no—
123C22×C17—82no—
124C22×C17—81no—
125C22×C17—81no—
126C22×C17C2×C22⋊C811yescharacteristic
127C22×C17C2×#9911yescharacteristic
128C22×C17#41311yescharacteristic
129C4×C17—24no—
130C4×C17—21no—
131C4×C17—21no—
132C4×C17—42no—
133Dic17—21no—
134Dic17—42no—
135Dic17—42no—
136Dic17—41no—
137Dic17—81no—
138Dic17#35512yes—
139C2×#124#59511yescharacteristic
140C2×C4×Q8—171no—
141C2×C4×D4—171no—
142#431—342no—
143#391—171no—
144#390—171no—
145#388—342no—
146#382—171no—
147C4×C22⋊C4—342no—
148C43—171no—
149#324—174no—
150#324—342no—
151C2×C17—21no—
152C2×C17—21no—
153C2×C17—82no—
154C2×C17#145911yescharacteristic
155C2×C17#212611yescharacteristic
156C2×C17C2×#35511yescharacteristic
157C22×Q8C2×#59511yescharacteristic
158C22×D4C2×#59511yescharacteristic
159C23×C4—341no—
160#124—22no—
161#124#362414yescomplemented
162C4×Q8—681no—
163C4×Q8—681no—
164C4×D4—681no—
165C4×D4—681no—
166C2×C4⋊C4—172no—
167C2×C4⋊C4—171no—
168C2×C4⋊C4—171no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—341no—
171C2×C22⋊C4—342no—
172C2×C42—172no—
173C2×C42—171no—
174C2×C42—171no—
175C2×C42—341no—
176C2×C42—341no—
177C2×C42C2×#59511yescharacteristic
178C4⋊C8—682no—
179C22⋊C8—344no—
180C4×C8—682no—
181#95—342no—
182#95—342no—
183C17#1147611yescharacteristic, complemented, Sylow
184C24—21no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8C2×#362412yes—
188C2×Q8#6324712yes—
189C2×D4—21no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4C2×#362412yes—
193C22×C4—21no—
194C22×C4—171no—
195C22×C4—171no—
196C22×C4—172no—
197C22×C4—341no—
198C22×C4—341no—
199C22×C4—681no—
200C22×C4—681no—
201C22×C4—684no—
202C22×C4C22⋊#59512yes—
203C22×C4C22×#59511yescharacteristic
204C2×C8—684no—
205C4⋊C4—681no—
206C4⋊C4—682no—
207C4⋊C4—681no—
208C22⋊C4—28no—
209C22⋊C4—684no—
210C22⋊C4—682no—
211C42—21no—
212C42—342no—
213C42—342no—
214C42—341no—
215C42—341no—
216C42—681no—
217C42—682no—
218C42—682no—
219C42C2×#362412yes—
220C23—22no—
221C23—42no—
222C23—42no—
223C23—41no—
224C23—41no—
225C23C2×C22⋊#59511yescharacteristic
226Q8—42no—
227Q8—42no—
228D4—42no—
229D4—42no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—171no—
238C2×C4—171no—
239C2×C4—342no—
240C2×C4—342no—
241C2×C4—341no—
242C2×C4—342no—
243C2×C4—681no—
244C2×C4—681no—
245C2×C4—681no—
246C2×C4—682no—
247C2×C4—682no—
248C2×C4—681no—
249C2×C4—1362no—
250C2×C4—1362no—
251C2×C4C22⋊#362414yes—
252C2×C4C22×#362411yescharacteristic
253C2×C4C2×#6324711yescharacteristic
254C8—1364no—
255C22—21no—
256C22—21no—
257C22—44no—
258C22—42no—
259C22—82no—
260C22—81no—
261C22—81no—
262C22C2×C22⋊#362411yescharacteristic
263C22D17⋊#9911yescharacteristic
264C22#1176379611yescharacteristic
265C4—21no—
266C4—21no—
267C4—24no—
268C4—42no—
269C4—172no—
270C4—341no—
271C4—682no—
272C4—682no—
273C4—681no—
274C4—1361no—
275C2—21no—
276C2—21no—
277C2—82no—
278C2#1454402011yescharacteristic
279C2#1454505611yescharacteristic
280C2D17⋊#35511yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C17×#124C4
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C4⋊Dic17C23
hypercenterC2×#124#595
hypocenterC17#11476
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #14544027C2semidirect
2. #14543724C2semidirect
21. C17×#124C8semidirect
161. #124#3624semidirect
183. C17#11476semidirect

Maximal subgroups

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

subgroupindex
#145440272
#145437242
#145437122
#124×Dic172
#1147617