Finite Group Databasebeta

← #17956495

Subgroups — #17956495

Order 4352.

600 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 126 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#14544640C211yesmaximal, characteristic
2#14544510C211yesmaximal, characteristic
3#14543622C214yesmaximal
4C2×#11762708C211yesmaximal, characteristic
5C2×#10587544—21no—
6C2×C22⋊#3624C2212yes—
7C2×C22⋊#3624C2212yes—
8C2×#10587542—21no—
9#11763675—21no—
10#11763675—21no—
11C22×C4⋊Dic17C412yes—
12C22×C4⋊Dic17C411yescharacteristic
13C22×C4×Dic17C2211yescharacteristic
14#11762708—21no—
15#11762708—21no—
16#11762708—21no—
17#11762708—21no—
18C2×#10587502C2212yes—
19C22×C4⋊C4×C17C411yescharacteristic
20C23×C4×C17C2×C411yescharacteristic
21C23×C4×C17C2×C412yes—
22C23×Dic17C2×C411yescharacteristic
23C23×Dic17C2×C412yes—
24C23×Dic17C2311yescharacteristic
25C22×#3624—22no—
26C22×#3624—22no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—24no—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17—22no—
33C2×C4⋊Dic17—22no—
34C2×C4⋊Dic17—44no—
35C2×C4⋊Dic17—42no—
36C2×C4×Dic17—42no—
37C2×C4×Dic17D411yescharacteristic
38C2×C4×Dic17D411yescharacteristic
39C2×C4×Dic17D411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C22⋊#3624—22no—
42C22⋊#3624—24no—
43C22⋊#3624Q814yes—
44#10587544—42no—
45#10587542—42no—
46#10587502—22no—
47#10587502—22no—
48C24×C17C22×C411yescharacteristic
49C22×C4×C17—22no—
50C22×C4×C17—22no—
51C22×C4×C17—22no—
52C22×C4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17—42no—
55C22×C4×C17—42no—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17M1614yes—
59C22×Dic17—22no—
60C22×Dic17—22no—
61C22×Dic17—22no—
62C22×Dic17—24no—
63C22×Dic17—41no—
64C22×Dic17—41no—
65C22×Dic17—41no—
66C22×Dic17—42no—
67C22×Dic17—42no—
68C22×Dic17C22⋊C411yescharacteristic
69C22×Dic17C22⋊C411yescharacteristic
70C22×Dic17C2×D411yescharacteristic
71C22×Dic17C2×D411yescharacteristic
72C22×Dic17C2×Q812yes—
73C22×Dic17#4112yes—
74C2×#3624—24no—
75C2×#3624—42no—
76C2×#3624—42no—
77C2×#3624—42no—
78C2×#3624—42no—
79C2×#3624—42no—
80C4⋊C4×C17—24no—
81C4⋊C4×C17—24no—
82C4⋊Dic17—44no—
83C4⋊Dic17—42no—
84C4⋊Dic17—42no—
85C4⋊Dic17—44no—
86C4×Dic17—22no—
87C4×Dic17—22no—
88C4×Dic17—41no—
89C4×Dic17—41no—
90#9883—171nomaximal, Sylow
91C23×C17—21no—
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17C2×C22⋊C411yescharacteristic
96C23×C17C4×Q812yes—
97C23×C17C2×M1612yes—
98C23×C17#13112yes—
99C2×C4×C17—28no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—24no—
105C2×C4×C17—44no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—44no—
109C2×C4×C17#9912yes—
110C2×C4×C17#10112yes—
111C2×Dic17—21no—
112C2×Dic17—21no—
113C2×Dic17—21no—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—42no—
121C2×Dic17—42no—
122C2×Dic17—44no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—84no—
127C2×Dic17#10012yes—
128C2×Dic17C22⋊Q812yes—
129C2×Dic17#12411yescharacteristic
130C2×Dic17C4⋊Q811yescharacteristic
131#3624—44no—
132#3624—82no—
133C2×#389—171no—
134#1797—171no—
135#1749—171no—
136#1280—174no—
137C22×C17—22no—
138C22×C17—22no—
139C22×C17—22no—
140C22×C17—22no—
141C22×C17—22no—
142C22×C17—22no—
143C22×C17#39111yescharacteristic
144C22×C17#40711yescharacteristic
145C22×C17#40811yescharacteristic
146C22×C17C2×#9911yescharacteristic
147C22×C17#41011yescharacteristic
148C22×C17C2×#10011yescharacteristic
149C22×C17C2×#10111yescharacteristic
150C22×C17#44614yes—
151C4×C17—22no—
152C4×C17—22no—
153C4×C17—44no—
154Dic17—22no—
155Dic17—22no—
156Dic17—21no—
157Dic17—21no—
158Dic17—41no—
159Dic17—82no—
160C22×C4⋊C4—172no—
161C22×C4⋊C4—171no—
162C22×C4⋊C4#59511yescharacteristic
163C22×C42—171no—
164C2×C4⋊C8—341no—
165C2×C22⋊C8—172no—
166C2×C22⋊C8—172no—
167C2×#97—341no—
168#389—341no—
169#389—341no—
170#389—341no—
171#389—341no—
172C2×#95—172no—
173#336—341no—
174#336—341no—
175C2×C17—24no—
176C2×C17#146311yescharacteristic
177C2×C17#146611yescharacteristic
178C2×C17#147011yescharacteristic
179C2×C17#147111yescharacteristic
180C2×C17#198511yescharacteristic
181C2×C17#199611yescharacteristic
182C2×C17#199711yescharacteristic
183C23×C4—171no—
184C23×C4—171no—
185C23×C4—172no—
186C23×C4C2×#59512yes—
187C23×C4C2×#59511yescharacteristic
188C22×C8—342no—
189C22×C8—342no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—342no—
194C2×C4⋊C4—342no—
195C2×C4⋊C4—341no—
196C2×C4⋊C4—341no—
197C2×C4⋊C4—682no—
198C2×C4⋊C4—684no—
199C2×C42—171no—
200C2×C42—171no—
201C2×C42—171no—
202C2×C42—171no—
203C2×C42—682no—
204C4⋊C8—682no—
205C22⋊C8—174no—
206C22⋊C8—342no—
207C22⋊C8—344no—
208#97—682no—
209#95—342no—
210#95—342no—
211C17#988311yescharacteristic, complemented, Sylow
212C24C22×#59511yescharacteristic
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—171no—
221C22×C4—171no—
222C22×C4—171no—
223C22×C4—172no—
224C22×C4—172no—
225C22×C4—171no—
226C22×C4—342no—
227C22×C4—342no—
228C22×C4—342no—
229C22×C4—344no—
230C22×C4—681no—
231C22×C4—681no—
232C22×C4—681no—
233C22×C4—682no—
234C22×C4—682no—
235C22×C4#6324714yes—
236C22×C4C22⋊#59511yescharacteristic
237C22×C4C22⋊#59511yescharacteristic
238C2×C8—344no—
239C2×C8—682no—
240C2×C8—682no—
241C2×C8—682no—
242C2×C8—682no—
243C2×C8—682no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—682no—
247C4⋊C4—684no—
248C4⋊C4—684no—
249C4⋊C4—682no—
250C42—342no—
251C42—342no—
252C42—681no—
253C42—681no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23#1058768012yes—
259C23Q8×#59512yes—
260C23C2×#6324712yes—
261C23C2×C22⋊#59511yescharacteristic
262C2×C4—28no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—24no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—172no—
273C2×C4—171no—
274C2×C4—171no—
275C2×C4—172no—
276C2×C4—341no—
277C2×C4—341no—
278C2×C4—341no—
279C2×C4—341no—
280C2×C4—341no—
281C2×C4—341no—
282C2×C4—341no—
283C2×C4—341no—
284C2×C4—341no—
285C2×C4—682no—
286C2×C4—682no—
287C2×C4—684no—
288C2×C4—682no—
289C2×C4—682no—
290C2×C4—682no—
291C2×C4—1364no—
292C2×C4#1058753912yes—
293C2×C4#1058754512yes—
294C8—684no—
295C8—1362no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22(C2×#595)⋊Q811yescharacteristic
303C22#1176377911yescharacteristic
304C22#1176378411yescharacteristic
305C22C2×#1058753911yescharacteristic
306C22#1176378911yescharacteristic
307C22D17⋊#10011yescharacteristic
308C22C2×#1058754511yescharacteristic
309C22#1176386014yes—
310C4—22no—
311C4—22no—
312C4—44no—
313C4—342no—
314C4—341no—
315C4—341no—
316C4—342no—
317C4—681no—
318C4—1362no—
319C2—24no—
320C2#1454403211yescharacteristic
321C2#1454404411yescharacteristic
322C2#1454405811yescharacteristic
323C2#1454406111yescharacteristic
324C2#1454495811yescharacteristic
325C2#1454497411yescharacteristic
326C2#1454497711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutator, prime radicalC24×C17C22×C4
FrattiniC24C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
cosocleC23×Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#9883
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
211. C17#9883semidirect

Maximal subgroups

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

subgroupindex
#145446402
#145445102
#145436222
C2×#117627082
#988317