Finite Group Databasebeta

← #17949689

Subgroups — #17949689

Order 4352.

604 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 146 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#14544638C211yesmaximal, characteristic
2C4×C22⋊#3624C211yesmaximal, characteristic
3#14543603C214yesmaximal, complemented
4C2×#11762705C211yesmaximal, characteristic
5C2×C4⋊#3624—21no—
6C2×C22⋊#3624—21no—
7C2×C22⋊#3624—21no—
8C2×C4×#3624—21no—
9#11763675C2212yes—
10#11763675C2212yes—
11C22×C4⋊Dic17C411yescharacteristic
12C22×#63225—21no—
13C22×C4×Dic17C2211yescharacteristic
14#11762705—21no—
15#11762705—21no—
16#11762705C412yes—
17#11762705C2212yes—
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—42no—
29C2×C4⋊C4×C17C2×C412yes—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—22no—
32C2×C4⋊Dic17C814yescomplemented
33C2×C4⋊Dic17C2×C412yes—
34C2×#63225—22no—
35C2×#63225—22no—
36C2×#63225—42no—
37C2×#63225—42no—
38C2×C4×Dic17—42no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17C2311yescharacteristic
43C22⋊#3624—24no—
44C22⋊#3624—42no—
45C4⋊#3624—42no—
46C4×#3624—42no—
47#10587502—22no—
48#10587502—22no—
49C24×C17C22⋊C411yescharacteristic
50C22×C4×C17—22no—
51C22×C4×C17—22no—
52C22×C4×C17—22no—
53C22×C4×C17—42no—
54C22×C4×C17—42no—
55C22×C4×C17—42no—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C2×C812yes—
58C22×C4×C17M1612yes—
59C22×C4×C17C22×C411yescharacteristic
60C22×Dic17—24no—
61C22×Dic17—22no—
62C22×Dic17—22no—
63C22×Dic17—22no—
64C22×Dic17—41no—
65C22×Dic17—41no—
66C22×Dic17—41no—
67C22×Dic17—42no—
68C22×Dic17—42no—
69C22×Dic17C22⋊C411yescharacteristic
70C22×Dic17C22⋊C411yescharacteristic
71C22×Dic17D812yes—
72C22×Dic17SD1612yes—
73C22×Dic17C2×D411yescharacteristic
74C22×Dic17C2×D411yescharacteristic
75C2×#3624—24no—
76C2×#3624—42no—
77C2×#3624—42no—
78C2×#3624—42no—
79C2×#3624—42no—
80C2×#3624—42no—
81C4⋊C4×C17—42no—
82C4⋊C4×C17—42no—
83C4⋊Dic17—24no—
84C4⋊Dic17—22no—
85C4⋊Dic17C2×C814yes—
86#63225—44no—
87#63225—42no—
88#63225—42no—
89C4×Dic17—22no—
90C4×Dic17—22no—
91C4×Dic17—41no—
92C4×Dic17—41no—
93#7573—171nomaximal, Sylow
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17C22⋊C812yes—
99C23×C17#10212yes—
100C23×C17#10412yes—
101C23×C17C2×C22⋊C411yescharacteristic
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—24no—
105C2×C4×C17—24no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—84no—
110C2×C4×C17C22⋊C812yes—
111C2×C4×C17#9912yes—
112C2×C4×C17#10112yes—
113C2×C4×C17C22×C811yescharacteristic
114C2×C4×C17C2×M1611yescharacteristic
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—24no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17—44no—
128C2×Dic17—48no—
129C2×Dic17—42no—
130C2×Dic17—42no—
131C2×Dic17#10212yes—
132C2×Dic17C2×D811yescharacteristic
133C2×Dic17C2×SD1611yescharacteristic
134C2×Dic17#13712yes—
135#3624—44no—
136#3624—82no—
137C2×#389—171no—
138#1797—171no—
139C4×C22⋊C8—171no—
140#1271—174no—
141C22×C17—22no—
142C22×C17—22no—
143C22×C17—22no—
144C22×C17—22no—
145C22×C17—22no—
146C22×C17—22no—
147C22×C17D4⋊C814yes—
148C22×C17C2×C22⋊C811yescharacteristic
149C22×C17C2×#9911yescharacteristic
150C22×C17C2×#10111yescharacteristic
151C22×C17C2×#10211yescharacteristic
152C22×C17#41911yescharacteristic
153C22×C17C2×#10411yescharacteristic
154C22×C17#42111yescharacteristic
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—82no—
158C4×C17#32414yes—
159Dic17—22no—
160Dic17—21no—
161Dic17—22no—
162Dic17—21no—
163Dic17—44no—
164Dic17—41no—
165C22×C4⋊C4—171no—
166C22×C4⋊C4—341no—
167C22×C4⋊C4#59511yescharacteristic
168C22×C42—171no—
169C2×C4⋊C8—341no—
170C2×C22⋊C8—341no—
171C2×C22⋊C8—341no—
172C2×C4×C8—341no—
173#389—172no—
174#389—172no—
175#389—341no—
176#389—341no—
177C2×#95—341no—
178#336—172no—
179#336—172no—
180C2×C17—24no—
181C2×C17C2×#32411yescharacteristic
182C2×C17C2×D4⋊C811yescharacteristic
183C2×C17#147911yescharacteristic
184C2×C17#150811yescharacteristic
185C2×C17#151911yescharacteristic
186C2×C17#152811yescharacteristic
187C2×C17#154611yescharacteristic
188C23×C4—21no—
189C23×C4—171no—
190C23×C4—171no—
191C23×C4—341no—
192C23×C4C2×#59511yescharacteristic
193C22×C8—342no—
194C22×C8—342no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—174no—
198C2×C4⋊C4—172no—
199C2×C4⋊C4—342no—
200C2×C4⋊C4—341no—
201C2×C4⋊C4—342no—
202C2×C4⋊C4—342no—
203C2×C4⋊C4—682no—
204C2×C4⋊C4—682no—
205C2×C4⋊C4C2×#59512yes—
206C2×C42—171no—
207C2×C42—171no—
208C2×C42—171no—
209C2×C42—171no—
210C2×C42—682no—
211C4⋊C8—682no—
212C22⋊C8—344no—
213C22⋊C8—682no—
214C4×C8—682no—
215#95—342no—
216#95—342no—
217C17#757311yescharacteristic, complemented, Sylow
218C24C22⋊#59511yescharacteristic
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—171no—
226C22×C4—171no—
227C22×C4—171no—
228C22×C4—172no—
229C22×C4—172no—
230C22×C4—171no—
231C22×C4—342no—
232C22×C4—342no—
233C22×C4—344no—
234C22×C4—342no—
235C22×C4—681no—
236C22×C4—681no—
237C22×C4—681no—
238C22×C4—682no—
239C22×C4—682no—
240C22×C4#6324212yes—
241C22×C4#6324312yes—
242C22×C4C22⋊#59511yescharacteristic
243C22×C4C22×#59511yescharacteristic
244C2×C8—344no—
245C2×C8—682no—
246C2×C8—682no—
247C2×C8—682no—
248C2×C8—682no—
249C2×C8—682no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—174no—
253C4⋊C4—342no—
254C4⋊C4—344no—
255C4⋊C4—682no—
256C4⋊C4—684no—
257C4⋊C4—682no—
258C42—342no—
259C42—342no—
260C42—681no—
261C42—681no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23#1058754312yes—
267C23#1058754712yes—
268C23#1058755012yes—
269C23C2×C22⋊#59511yescharacteristic
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—84no—
278C2×C4—171no—
279C2×C4—172no—
280C2×C4—171no—
281C2×C4—172no—
282C2×C4—341no—
283C2×C4—341no—
284C2×C4—341no—
285C2×C4—341no—
286C2×C4—341no—
287C2×C4—341no—
288C2×C4—344no—
289C2×C4—341no—
290C2×C4—341no—
291C2×C4—341no—
292C2×C4—682no—
293C2×C4—682no—
294C2×C4—684no—
295C2×C4—682no—
296C2×C4—682no—
297C2×C4—688no—
298C2×C4#1058754312yes—
299C2×C4#1058754512yes—
300C2×C4#1058755112yes—
301C2×C4C2×#6324211yescharacteristic
302C2×C4C2×#6324311yescharacteristic
303C8—684no—
304C8—1362no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22#1176365714yes—
312C22C2×#1058754311yescharacteristic
313C22C2×#1058755111yescharacteristic
314C22C2×#1058754511yescharacteristic
315C22C2×#1058754711yescharacteristic
316C22#1176380211yescharacteristic
317C22C2×#1058755011yescharacteristic
318C22#1176380711yescharacteristic
319C4—21no—
320C4—21no—
321C4—82no—
322C4—342no—
323C4—341no—
324C4—342no—
325C4—341no—
326C4—681no—
327C4—684no—
328C4#1176365514yes—
329C2—24no—
330C2C2×#1176365511yescharacteristic
331C2C2×#1176365711yescharacteristic
332C2#1454408811yescharacteristic
333C2#1454418411yescharacteristic
334C2#1454422011yescharacteristic
335C2#1454424811yescharacteristic
336C2#1454429311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#7573
LCSG ▷ C2×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. #14543603C2semidirect
32. C2×C4⋊Dic17C8semidirect
217. C17#7573semidirect

Maximal subgroups

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

subgroupindex
#145446382
C4×C22⋊#36242
#145436032
C2×#117627052
#757317