Finite Group Databasebeta

← #16927795

Subgroups — #16927795

Order 4352.

656 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 107 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#14528712C211yesmaximal, characteristic
2#14528702C211yesmaximal, characteristic
3#14526637C214yesmaximal, complemented
4C17×#2297C211yesmaximal, characteristic
5C2×#10587507—21no—
6C2×#10587504—21no—
7C2×#10587505—21no—
8C2×#10587505—21no—
9#11762773C2212yes—
10#11762672C2212yes—
11C22×#41×C17C2211yescharacteristic
12C2×C17×#117C411yescharacteristic, complemented
13C22×C42×C17C411yescharacteristic, complemented
14C17×#391—21no—
15C17×#390—21no—
16C17×#386—21no—
17C17×#386—21no—
18C4×C22⋊C4×C17C412yescomplemented
19C4×C22⋊C4×C17C2212yes—
20C2×#41×C17—24no—
21C2×#41×C17—42no—
22C2×#41×C17—42no—
23C22×Q8×C17D411yescharacteristic
24C22×D4×C17D411yescharacteristic
25C22×D4×C17D411yescharacteristic
26C22×D4×C17D411yescharacteristic
27C23×C4×C17—21no—
28C23×C4×C17C2×C411yescharacteristic
29C23×C4×C17C2×C411yescharacteristic
30C23×C4×C17C2311yescharacteristic
31C17×#117—42no—
32C2×C4⋊C4×C17—21no—
33C2×C22⋊C4×C17—21no—
34C2×C22⋊C4×C17—22no—
35C2×C22⋊C4×C17—22no—
36C2×C42×C17—21no—
37C2×C42×C17—42no—
38C2×C42×C17C2×C412yes—
39C2×C42×C17C2×C412yes—
40C2×C22⋊Dic17—22no—
41C2×#63223—22no—
42C17×#95—22no—
43C17×#95—22no—
44#10587507—42no—
45#10587505—42no—
46#10587505—42no—
47#10587504—42no—
48#10587502—24no—
49C22⋊#3622—24no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C22⋊C411yescharacteristic
52C24×C17C2×D411yescharacteristic
53#41×C17—416no—
54C2×Q8×C17—22no—
55C2×Q8×C17—22no—
56C2×Q8×C17—41no—
57C2×D4×C17—22no—
58C2×D4×C17—22no—
59C2×D4×C17—22no—
60C2×D4×C17—22no—
61C2×D4×C17—22no—
62C2×D4×C17—22no—
63C2×D4×C17—41no—
64C2×D4×C17—41no—
65C2×D4×C17—41no—
66C22×C4×C17—22no—
67C22×C4×C17—22no—
68C22×C4×C17—22no—
69C22×C4×C17—22no—
70C22×C4×C17—22no—
71C22×C4×C17—22no—
72C22×C4×C17—41no—
73C22×C4×C17—41no—
74C22×C4×C17—41no—
75C22×C4×C17—41no—
76C22×C4×C17—44no—
77C22×C4×C17—42no—
78C22×C4×C17C22⋊C411yescharacteristic
79C22×C4×C17C22⋊C411yescharacteristic
80C22×C4×C17C22×C411yescharacteristic
81C22×C4×C17C2×D411yescharacteristic
82C22×C4×C17#4114yes—
83C22×Dic17—42no—
84C4⋊C4×C17—42no—
85C22⋊C4×C17—44no—
86C22⋊C4×C17—44no—
87C22⋊C4×C17—42no—
88C42×C17—42no—
89C42×C17—42no—
90C42×C17—42no—
91C22⋊Dic17—42no—
92C22⋊Dic17—42no—
93#63223—42no—
94#63223—42no—
95C2×#3622—42no—
96#8738—171nomaximal, Sylow
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—42no—
105C23×C17—42no—
106C23×C17—42no—
107C23×C17#9912yes—
108C23×C17C2×C22⋊C411yescharacteristic
109C23×C17#12312yes—
110Q8×C17—44no—
111D4×C17—44no—
112D4×C17—44no—
113D4×C17—44no—
114C2×C4×C17—28no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—24no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—44no—
126C2×C4×C17—44no—
127C2×C4×C17—42no—
128C2×C4×C17—48no—
129C2×C4×C17—42no—
130C2×C4×C17—42no—
131C2×C4×C17—42no—
132C2×C4×C17—42no—
133C2×C4×C17#9912yes—
134C2×C4×C17#11712yes—
135C2×C4×C17#12312yes—
136C2×Dic17—82no—
137C2×Dic17—84no—
138#3622—82no—
139#2297D1711yescharacteristic
140#1785—171no—
141#1782—171no—
142#1284—174no—
143C22×C17—22no—
144C22×C17—22no—
145C22×C17—22no—
146C22×C17—24no—
147C22×C17—42no—
148C22×C17—42no—
149C22×C17—42no—
150C22×C17—41no—
151C22×C17—42no—
152C22×C17—81no—
153C22×C17—81no—
154C22×C17—81no—
155C22×C17—81no—
156C22×C17#38111yescharacteristic
157C22×C17C2×#9911yescharacteristic
158C22×C17C2×#9911yescharacteristic
159C4×C17—22no—
160C4×C17—22no—
161C4×C17—22no—
162C4×C17—44no—
163C4×C17—41no—
164C4×C17—82no—
165Dic17—162no—
166C22×#41C2×D1711yescharacteristic
167C2×#117Dic1711yescharacteristic, complemented
168C22×C42Dic1711yescharacteristic, complemented
169C2×#101—341no—
170C2×#100—341no—
171C2×#99—341no—
172C2×#99—341no—
173#407—172no—
174#391—21no—
175#390—21no—
176#386—21no—
177#386—21no—
178#381—172no—
179C4×C22⋊C4Dic1712yescomplemented
180C4×C22⋊C4C2×D1712yes—
181C2×C17—21no—
182C2×C17—21no—
183C2×C17—42no—
184C2×C17—42no—
185C2×C17—81no—
186C2×C17#181911yescharacteristic
187C2×C17#182712yes—
188C2×#41—24no—
189C2×#41—42no—
190C2×#41—42no—
191C22×Q8C22⋊D1711yescharacteristic
192C22×D4C22⋊D1711yescharacteristic
193C22×D4C22⋊D1711yescharacteristic
194C22×D4C22⋊D1711yescharacteristic
195C23×C4—21no—
196C23×C4C2×Dic1711yescharacteristic
197C23×C4C2×Dic1711yescharacteristic
198C23×C4C22×D1711yescharacteristic
199C2×M16—342no—
200#117—42no—
201C2×C4⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—342no—
206C2×C42—21no—
207C2×C42—42no—
208C2×C42C2×Dic1712yes—
209C2×C42C2×Dic1712yes—
210#101—682no—
211#100—682no—
212#99—682no—
213#99—682no—
214C22⋊C8—344no—
215#95—22no—
216#95—22no—
217#95—344no—
218C17#873811yescharacteristic, complemented, Sylow
219C24C22⋊Dic1711yescharacteristic
220C24C22⋊Dic1711yescharacteristic
221C24C2×C22⋊D1711yescharacteristic
222#41—416no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—41no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—682no—
248C22×C4C22⋊Dic1711yescharacteristic
249C22×C4C22⋊Dic1711yescharacteristic
250C22×C4#6325214yes—
251C22×C4C22×Dic1711yescharacteristic
252C22×C4C2×C22⋊D1711yescharacteristic
253M16—682no—
254M16—682no—
255C2×C8—682no—
256C4⋊C4—42no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—42no—
260C22⋊C4—682no—
261C22⋊C4—682no—
262C42—42no—
263C42—42no—
264C42—42no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23#1058750512yes—
276C23#1058761512yes—
277C23C2×C22⋊Dic1711yescharacteristic
278Q8—44no—
279D4—44no—
280D4—44no—
281D4—44no—
282C2×C4—28no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—24no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—48no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—1362no—
302C2×C4—1364no—
303C2×C4#1058750512yes—
304C2×C4#1058761212yes—
305C2×C4#1058761512yes—
306C8—1362no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—24no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—41no—
315C22—42no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22#1176267211yescharacteristic
321C22C2×#1058750511yescharacteristic
322C22C2×#1058750511yescharacteristic
323C4—22no—
324C4—22no—
325C4—22no—
326C4—44no—
327C4—41no—
328C4—82no—
329C4—2722no—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2—81no—
335C2#1452885811yescharacteristic
336C2#1452888812yes—
subgroupG/H
centerC22#11762672
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2297C2
socleC22×C17#381
prime radicalC22×#41×C17C22
cosocleC23×C4×C17C23
hypercenter#2297D17
hypocenterC17#8738
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2297

Complements

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

NH ≅ G/Ntype
3. #14526637C2semidirect
12. C2×C17×#117C4semidirect
13. C22×C42×C17C4semidirect
18. C4×C22⋊C4×C17C4semidirect
167. C2×#117Dic17semidirect
168. C22×C42Dic17semidirect
179. C4×C22⋊C4Dic17semidirect
218. C17#8738semidirect

Maximal subgroups

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

subgroupindex
#145287122
#145287022
#145266372
C17×#22972
#873817