Finite Group Databasebeta

← #17949819

Subgroups — #17949819

Order 4352.

612 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 118 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#14544675C211yesmaximal, characteristic
2#14544447C211yesmaximal, characteristic
3#14543603C214yesmaximal, complemented
4C2×#11762705C211yesmaximal, characteristic
5C22×#63247—21no—
6C2×C22⋊#3624—21no—
7#11763675—21no—
8#11763675—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
25C2×#63247—24no—
26C2×#63247—42no—
27C2×#63247—42no—
28C22×#3624—22no—
29C22×#3624—22no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—42no—
32C2×C4⋊C4×C17C2×C412yes—
33C2×C4⋊Dic17—24no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17C2×C412yes—
36C2×#63225—22no—
37C2×#63225—22no—
38C2×#63225—42no—
39C2×#63225—42no—
40C2×C4×Dic17—42no—
41C2×C4×Dic17C2×C411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17C2311yescharacteristic
45C22⋊#3624—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—42no—
54C22×C4×C17—42no—
55C22×C4×C17—42no—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22×C411yescharacteristic
58C22×Dic17—22no—
59C22×Dic17—22no—
60C22×Dic17—24no—
61C22×Dic17—22no—
62C22×Dic17—22no—
63C22×Dic17—22no—
64C22×Dic17—41no—
65C22×Dic17—42no—
66C22×Dic17—42no—
67C22×Dic17C22⋊C411yescharacteristic
68C22×Dic17C22⋊C411yescharacteristic
69C22×Dic17D812yes—
70C22×Dic17SD1612yes—
71C22×Dic17C2×D411yescharacteristic
72C22×Dic17C2×D411yescharacteristic
73#63247—48no—
74C2×#3624—24no—
75C2×#3624—42no—
76C2×#3624—42no—
77C2×#3624—42no—
78C2×#3624—42no—
79C2×#3624—42no—
80C4⋊C4×C17—42no—
81C4⋊C4×C17—42no—
82C4⋊Dic17—24no—
83C4⋊Dic17—22no—
84C4⋊Dic17M1614yescomplemented
85#63225—44no—
86#63225—42no—
87#63225—42no—
88C4×Dic17—41no—
89C4×Dic17—41no—
90C4×Dic17—41no—
91C4×Dic17—41no—
92#7610—171nomaximal, Sylow
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17#9912yes—
98C23×C17#10212yes—
99C23×C17#10412yes—
100C23×C17C2×C22⋊C411yescharacteristic
101C2×C4×C17—28no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—84no—
110C2×C4×C17#10112yes—
111C2×C4×C17C2×M1612yes—
112C2×Dic17—24no—
113C2×Dic17—24no—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—24no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—44no—
125C2×Dic17—48no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17#10212yes—
129C2×Dic17C2×D811yescharacteristic
130C2×Dic17C2×SD1611yescharacteristic
131C2×Dic17#13712yes—
132#3624—44no—
133#3624—82no—
134C2×#389—171no—
135#1817—171no—
136#1728—171no—
137#1271—174no—
138C22×C17—22no—
139C22×C17—22no—
140C22×C17—22no—
141C22×C17—22no—
142C22×C17—22no—
143C22×C17—22no—
144C22×C17#32714yes—
145C22×C17#40711yescharacteristic
146C22×C17C2×#9911yescharacteristic
147C22×C17C2×#10111yescharacteristic
148C22×C17C2×#10211yescharacteristic
149C22×C17#41911yescharacteristic
150C22×C17C2×#10411yescharacteristic
151C22×C17#42111yescharacteristic
152C4×C17—22no—
153C4×C17—22no—
154C4×C17—82no—
155Dic17—21no—
156Dic17—22no—
157Dic17—21no—
158Dic17—22no—
159Dic17—44no—
160Dic17—41no—
161C22×M16—341no—
162C22×C4⋊C4—171no—
163C22×C4⋊C4—341no—
164C22×C4⋊C4#59511yescharacteristic
165C22×C42—171no—
166C2×C22⋊C8—341no—
167#389—172no—
168#389—172no—
169#389—341no—
170#389—341no—
171C2×#95—341no—
172#336—172no—
173#336—172no—
174#336—341no—
175#336—341no—
176C2×C17—24no—
177C2×C17#146311yescharacteristic
178C2×C17#148611yescharacteristic
179C2×C17#149111yescharacteristic
180C2×C17C2×#32711yescharacteristic
181C2×C17#151111yescharacteristic
182C2×C17#152811yescharacteristic
183C2×C17#154611yescharacteristic
184C23×C4—21no—
185C23×C4—171no—
186C23×C4—171no—
187C23×C4—341no—
188C23×C4C2×#59511yescharacteristic
189C2×M16—344no—
190C2×M16—682no—
191C2×M16—682no—
192C22×C8—342no—
193C22×C8—342no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—172no—
197C2×C4⋊C4—344no—
198C2×C4⋊C4—341no—
199C2×C4⋊C4—342no—
200C2×C4⋊C4—342no—
201C2×C4⋊C4—682no—
202C2×C4⋊C4—682no—
203C2×C4⋊C4C2×#59512yes—
204C2×C42—171no—
205C2×C42—171no—
206C2×C42—171no—
207C2×C42—171no—
208C2×C42—682no—
209C22⋊C8—682no—
210#95—342no—
211#95—342no—
212C17#761011yescharacteristic, complemented, Sylow
213C24C22⋊#59511yescharacteristic
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—171no—
222C22×C4—171no—
223C22×C4—171no—
224C22×C4—172no—
225C22×C4—172no—
226C22×C4—171no—
227C22×C4—342no—
228C22×C4—342no—
229C22×C4—342no—
230C22×C4—344no—
231C22×C4—342no—
232C22×C4—342no—
233C22×C4—681no—
234C22×C4—682no—
235C22×C4—682no—
236C22×C4C22⋊#59511yescharacteristic
237C22×C4C22×#59511yescharacteristic
238M16—688no—
239C2×C8—344no—
240C2×C8—682no—
241C2×C8—682no—
242C2×C8—682no—
243C2×C8—682no—
244C2×C8—682no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—174no—
248C4⋊C4—344no—
249C4⋊C4—342no—
250C4⋊C4—682no—
251C4⋊C4—684no—
252C4⋊C4—682no—
253C42—681no—
254C42—681no—
255C42—681no—
256C42—681no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23#1058754712yes—
262C23#1058754812yes—
263C23#1058755112yes—
264C23C2×C22⋊#59511yescharacteristic
265C2×C4—28no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—84no—
274C2×C4—171no—
275C2×C4—172no—
276C2×C4—171no—
277C2×C4—172no—
278C2×C4—341no—
279C2×C4—341no—
280C2×C4—341no—
281C2×C4—341no—
282C2×C4—341no—
283C2×C4—341no—
284C2×C4—344no—
285C2×C4—344no—
286C2×C4—344no—
287C2×C4—341no—
288C2×C4—341no—
289C2×C4—341no—
290C2×C4—684no—
291C2×C4—688no—
292C2×C4—682no—
293C2×C4—682no—
294C2×C4#1058754512yes—
295C2×C4D17⋊M1612yes—
296C8—684no—
297C8—1362no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22#1176366014yes—
305C22#1176378111yescharacteristic
306C22C2×#1058755111yescharacteristic
307C22C2×#1058754511yescharacteristic
308C22C2×#1058754711yescharacteristic
309C22#1176380211yescharacteristic
310C22C2×#1058754811yescharacteristic
311C22#1176380811yescharacteristic
312C4—22no—
313C4—22no—
314C4—82no—
315C4—341no—
316C4—342no—
317C4—341no—
318C4—342no—
319C4—681no—
320C4—684no—
321C2—24no—
322C2#1454403411yescharacteristic
323C2#1454411111yescharacteristic
324C2#1454412811yescharacteristic
325C2C2×#1176366011yescharacteristic
326C2#1454419411yescharacteristic
327C2#1454424611yescharacteristic
328C2#1454429311yescharacteristic
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#7610
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. #14543603C2semidirect
84. C4⋊Dic17M16semidirect
212. C17#7610semidirect

Maximal subgroups

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

subgroupindex
#145446752
#145444472
#145436032
C2×#117627052
#761017