Finite Group Databasebeta

← #17949879

Subgroups — #17949879

Order 4352.

594 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 116 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#14544510C211yesmaximal, characteristic
2#14544510C211yesmaximal, characteristic
3#14543602C214yesmaximal, complemented
4C2×#11762708C211yesmaximal, characteristic
5C2×C22⋊#3624—21no—
6C2×C22⋊#3624—21no—
7C2×#10587542—21no—
8C2×#10587542—21no—
9#11763675C2212yes—
10#11763675C2212yes—
11C22×C4⋊Dic17—21no—
12C22×C4⋊Dic17C411yescharacteristic
13C22×C4×Dic17C2211yescharacteristic
14#11762708—21no—
15#11762708—21no—
16#11762708C412yes—
17#11762708C2212yes—
18C2×#10587502—21no—
19C22×C4⋊C4×C17C411yescharacteristic
20C23×C4×C17—21no—
21C23×C4×C17C2×C411yescharacteristic
22C23×Dic17—21no—
23C23×Dic17D411yescharacteristic
24C23×Dic17D411yescharacteristic
25C22×#3624—22no—
26C22×#3624—22no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17—24no—
29C2×C4⋊C4×C17C2×C412yes—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—22no—
32C2×C4⋊Dic17—22no—
33C2×C4⋊Dic17—44no—
34C2×C4⋊Dic17—42no—
35C2×C4⋊Dic17C2×C412yes—
36C2×C4×Dic17—42no—
37C2×C4×Dic17C2×C411yescharacteristic
38C2×C4×Dic17D411yescharacteristic
39C2×C4×Dic17D411yescharacteristic
40C2×C4×Dic17C2311yescharacteristic
41C22⋊#3624—24no—
42C22⋊#3624—24no—
43#10587542—42no—
44#10587542—42no—
45#10587502—22no—
46#10587502—22no—
47C24×C17C22⋊C411yescharacteristic
48C22×C4×C17—22no—
49C22×C4×C17—22no—
50C22×C4×C17—22no—
51C22×C4×C17—24no—
52C22×C4×C17—22no—
53C22×C4×C17—42no—
54C22×C4×C17—42no—
55C22×C4×C17C22⋊C411yescharacteristic
56C22×C4×C17C22×C411yescharacteristic
57C22×Dic17—22no—
58C22×Dic17—22no—
59C22×Dic17—22no—
60C22×Dic17—22no—
61C22×Dic17—22no—
62C22×Dic17—41no—
63C22×Dic17—41no—
64C22×Dic17—42no—
65C22×Dic17—42no—
66C22×Dic17—42no—
67C22×Dic17C22⋊C411yescharacteristic
68C22×Dic17C22⋊C411yescharacteristic
69C22×Dic17C2×D411yescharacteristic
70C22×Dic17C2×D411yescharacteristic
71C2×#3624—24no—
72C2×#3624—24no—
73C2×#3624—42no—
74C2×#3624—42no—
75C2×#3624—42no—
76C2×#3624—42no—
77C4⋊C4×C17—24no—
78C4⋊C4×C17—22no—
79C4⋊C4×C17M1614yes—
80C4⋊Dic17—44no—
81C4⋊Dic17—42no—
82C4⋊Dic17—44no—
83C4⋊Dic17—42no—
84C4×Dic17—22no—
85C4×Dic17—22no—
86C4×Dic17—41no—
87C4×Dic17—41no—
88#7627—171nomaximal, Sylow
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—21no—
93C23×C17#10112yes—
94C23×C17#10412yes—
95C23×C17#10412yes—
96C23×C17C2×C22⋊C411yescharacteristic
97C2×C4×C17—28no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—24no—
103C2×C4×C17—48no—
104C2×C4×C17—42no—
105C2×C4×C17—42no—
106C2×C4×C17#9912yes—
107C2×C4×C17C2×M1612yes—
108C2×Dic17—21no—
109C2×Dic17—21no—
110C2×Dic17—21no—
111C2×Dic17—21no—
112C2×Dic17—21no—
113C2×Dic17—21no—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—44no—
119C2×Dic17—44no—
120C2×Dic17—42no—
121C2×Dic17—42no—
122C2×Dic17—42no—
123C2×Dic17—84no—
124C2×Dic17AGL(1,ℤ/8)12yes—
125C2×Dic17#13712yes—
126#3624—44no—
127#3624—44no—
128C2×#389—171no—
129#1749—171no—
130#1749—171no—
131#1271—174no—
132C22×C17—22no—
133C22×C17—22no—
134C22×C17—22no—
135C22×C17—22no—
136C22×C17—22no—
137C22×C17—22no—
138C22×C17#33014yes—
139C22×C17#40711yescharacteristic
140C22×C17C2×#9911yescharacteristic
141C22×C17C2×#10111yescharacteristic
142C22×C17#41811yescharacteristic
143C22×C17#41911yescharacteristic
144C22×C17C2×#10411yescharacteristic
145C22×C17C2×#10411yescharacteristic
146C4×C17—22no—
147C4×C17—22no—
148C4×C17—44no—
149Dic17—22no—
150Dic17—22no—
151Dic17—41no—
152Dic17—41no—
153Dic17—82no—
154C22×C4⋊C4—171no—
155C22×C4⋊C4—341no—
156C22×C4⋊C4#59511yescharacteristic
157C22×C42—171no—
158C2×C22⋊C8—341no—
159C2×C22⋊C8—341no—
160C2×#97—341no—
161C2×#97—341no—
162#389—172no—
163#389—172no—
164#389—341no—
165#389—341no—
166C2×#95—341no—
167#336—172no—
168#336—172no—
169C2×C17—24no—
170C2×C17#146311yescharacteristic
171C2×C17#149011yescharacteristic
172C2×C17#149111yescharacteristic
173C2×C17#150711yescharacteristic
174C2×C17#150811yescharacteristic
175C2×C17C2×#33011yescharacteristic
176C2×C17#155611yescharacteristic
177C23×C4—21no—
178C23×C4—171no—
179C23×C4—171no—
180C23×C4—341no—
181C23×C4C2×#59511yescharacteristic
182C22×C8—342no—
183C22×C8—342no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—172no—
187C2×C4⋊C4—342no—
188C2×C4⋊C4—342no—
189C2×C4⋊C4—341no—
190C2×C4⋊C4—684no—
191C2×C4⋊C4—682no—
192C2×C4⋊C4C2×#59512yes—
193C2×C42—171no—
194C2×C42—171no—
195C2×C42—171no—
196C2×C42—171no—
197C2×C42—682no—
198C22⋊C8—344no—
199C22⋊C8—344no—
200#97—682no—
201#97—682no—
202#95—342no—
203#95—342no—
204C17#762711yescharacteristic, complemented, Sylow
205C24C22⋊#59511yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—171no—
214C22×C4—171no—
215C22×C4—171no—
216C22×C4—171no—
217C22×C4—342no—
218C22×C4—342no—
219C22×C4—342no—
220C22×C4—342no—
221C22×C4—342no—
222C22×C4—681no—
223C22×C4—681no—
224C22×C4—682no—
225C22×C4—682no—
226C22×C4—682no—
227C22×C4C22⋊#59511yescharacteristic
228C22×C4C22×#59511yescharacteristic
229C2×C8—344no—
230C2×C8—344no—
231C2×C8—682no—
232C2×C8—682no—
233C2×C8—682no—
234C2×C8—682no—
235C4⋊C4—24no—
236C4⋊C4—22no—
237C4⋊C4—682no—
238C4⋊C4—684no—
239C4⋊C4—684no—
240C4⋊C4—682no—
241C4⋊C4#6324714yes—
242C42—342no—
243C42—342no—
244C42—681no—
245C42—681no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23#1058754512yes—
251C23#1058754812yes—
252C23#1058755012yes—
253C23C2×C22⋊#59511yescharacteristic
254C2×C4—28no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—48no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—172no—
264C2×C4—172no—
265C2×C4—341no—
266C2×C4—341no—
267C2×C4—341no—
268C2×C4—341no—
269C2×C4—341no—
270C2×C4—341no—
271C2×C4—341no—
272C2×C4—341no—
273C2×C4—341no—
274C2×C4—341no—
275C2×C4—684no—
276C2×C4—684no—
277C2×C4—682no—
278C2×C4—682no—
279C2×C4—682no—
280C2×C4—1364no—
281C2×C4#1058753912yes—
282C2×C4C2×#6324712yes—
283C8—684no—
284C8—684no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22#1176366614yes—
292C22#1176377911yescharacteristic
293C22C2×#1058753911yescharacteristic
294C22C2×#1058754511yescharacteristic
295C22#1176380111yescharacteristic
296C22#1176380211yescharacteristic
297C22C2×#1058754811yescharacteristic
298C22C2×#1058755011yescharacteristic
299C4—22no—
300C4—22no—
301C4—44no—
302C4—342no—
303C4—342no—
304C4—681no—
305C4—681no—
306C4—1362no—
307C2—24no—
308C2#1454403211yescharacteristic
309C2#1454412311yescharacteristic
310C2#1454412611yescharacteristic
311C2#1454418111yescharacteristic
312C2#1454418311yescharacteristic
313C2C2×#1176366611yescharacteristic
314C2#1454431311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#7627
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #14543602C2semidirect
204. C17#7627semidirect

Maximal subgroups

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

subgroupindex
#145445102
#145436022
C2×#117627082
#762717