Finite Group Databasebeta

← #16937556

Subgroups — #16937556

Order 4352.

602 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 75 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#14534449C211yesmaximal, characteristic, complemented
2#14528728C212yesmaximal
3#14526692C211yesmaximal, characteristic, complemented
4#14526636C212yesmaximal, complemented
5C17×#1782C211yesmaximal, characteristic
6C2×#10587508—22no—
7C2×#10587508—22no—
8#11762773C2212yes—
9#11762745—21no—
10#11762721—21no—
11#11762713—22no—
12#11762696—22no—
13C4×C22⋊Dic17C2212yes—
14C22×#41×C17C2211yescharacteristic
15C2×C17×#99—21no—
16C2×C17×#99—21no—
17C17×#381C2211yescharacteristic
18C17×#381C2211yescharacteristic
19C2×#41×C17—22no—
20C2×#41×C17—22no—
21C2×#41×C17—42no—
22C2×#41×C17—42no—
23C22×Q8×C17Q811yescharacteristic, complemented
24C22×D4×C17—21no—
25C22×D4×C17Q811yescharacteristic, complemented
26C23×C4×C17—21no—
27C23×C4×C17C2311yescharacteristic
28C22×Dic34—21no—
29C2×C22⋊C4×C17—21no—
30C2×C22⋊C4×C17D411yescharacteristic
31C2×C22⋊C4×C17D411yescharacteristic
32C2×C22⋊Dic17—22no—
33C2×C4⋊Dic17—21no—
34C2×#63225—22no—
35C2×C4×Dic17D412yes—
36C2×C4×Dic17D412yes—
37C2×#63223—22no—
38C17×#99—42no—
39C17×#99—42no—
40C17×#95—22no—
41C17×#95—22no—
42#10587508—44no—
43#10587508—44no—
44#10587502—21no—
45#10587502—22no—
46#10587502—21no—
47#10587502—22no—
48C22⋊#3622—22no—
49C22⋊#3622—22no—
50C24×C17—21no—
51C24×C17C2×D411yescharacteristic
52#41×C17—24no—
53#41×C17—24no—
54#41×C17—42no—
55#41×C17—42no—
56#41×C17—44no—
57#41×C17—44no—
58C2×Q8×C17—22no—
59C2×Q8×C17—42no—
60C2×D4×C17—22no—
61C2×D4×C17—22no—
62C2×D4×C17—22no—
63C2×D4×C17—42no—
64C2×D4×C17—41no—
65C2×D4×C17—41no—
66C2×D4×C17—41no—
67C2×D4×C17—41no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—41no—
72C22×C4×C17—41no—
73C22×C4×C17—42no—
74C22×C4×C17—42no—
75C22×C4×C17—42no—
76C22×C4×C17—41no—
77C22×C4×C17C2×D412yes—
78C22×C4×C17C2×Q811yescharacteristic
79C22×C4×C17#4111yescharacteristic
80C22×C4×C17#4112yes—
81C22×Dic17—22no—
82C22×Dic17—22no—
83C2×Dic34—44no—
84C2×Dic34—81no—
85C22⋊C4×C17—41no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C22⋊C4×C17—41no—
89C22⋊Dic17—44no—
90C4⋊Dic17—81no—
91#63225—44no—
92C4×Dic17—42no—
93C4×Dic17—42no—
94#63223—42no—
95#63223—42no—
96C2×#3622—42no—
97#10386—171nomaximal, Sylow
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—22no—
101C23×C17—42no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17—82no—
105C23×C17#12011yescharacteristic
106C23×C17C22⋊Q811yescharacteristic
107C23×C17#12411yescharacteristic
108Q8×C17—24no—
109Q8×C17—42no—
110D4×C17—24no—
111D4×C17—42no—
112D4×C17—42no—
113D4×C17—42no—
114D4×C17—42no—
115D4×C17—42no—
116C2×C4×C17—22no—
117C2×C4×C17—22no—
118C2×C4×C17—22no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—42no—
122C2×C4×C17—44no—
123C2×C4×C17—44no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—81no—
131C2×C4×C17—82no—
132C2×C4×C17C22⋊Q812yes—
133C2×C4×C17#12412yes—
134C2×Dic17—24no—
135C2×Dic17—42no—
136C2×Dic17—42no—
137C2×Dic17—42no—
138C2×Dic17—42no—
139C2×Dic17—42no—
140Dic34—84no—
141#3622—82no—
142#2607—171no—
143#1789—172no—
144#1782D1711yescharacteristic
145#1306—171no—
146#1284—172no—
147C22×C17—22no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17—22no—
151C22×C17—22no—
152C22×C17—42no—
153C22×C17—81no—
154C22×C17—81no—
155C22×C17—82no—
156C22×C17—82no—
157C22×C17—82no—
158C22×C17#39311yescharacteristic
159C22×C17B(4,𝔽2)11yescharacteristic
160C22×C17B(4,𝔽2)11yescharacteristic
161C4×C17—22no—
162C4×C17—22no—
163C4×C17—42no—
164C4×C17—81no—
165C4×C17—81no—
166C4×C17—161no—
167Dic17—44no—
168Dic17—82no—
169C22×#41C2×D1711yescharacteristic
170C2×#104—342no—
171C2×#104—342no—
172C2×#99—21no—
173C2×#99—21no—
174#407—172no—
175#399—341no—
176#393—341no—
177#391—342no—
178#387—342no—
179#381C2×D1711yescharacteristic
180#381C2×D1711yescharacteristic
181C4×C22⋊C4—172no—
182C2×C17—21no—
183C2×C17—21no—
184C2×C17—41no—
185C2×C17—41no—
186C2×C17—82no—
187C2×C17#202911yescharacteristic
188C2×C17#203112yes—
189C2×#41—22no—
190C2×#41—22no—
191C2×#41—42no—
192C2×#41—42no—
193C22×Q8—341no—
194C22×Q8Dic3411yescharacteristic, complemented
195C22×D4—21no—
196C22×D4Dic3411yescharacteristic, complemented
197C23×C4—21no—
198C23×C4C22×D1711yescharacteristic
199C2×M16—342no—
200C2×C4⋊C4—342no—
201C2×C4⋊C4—341no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—342no—
204C2×C22⋊C4C22⋊D1711yescharacteristic
205C2×C22⋊C4C22⋊D1711yescharacteristic
206C2×C42—172no—
207C2×C42—172no—
208#104—684no—
209#104—684no—
210#99—42no—
211#99—42no—
212C22⋊C8—342no—
213C22⋊C8—342no—
214#95—22no—
215#95—22no—
216#95—341no—
217#95—342no—
218#95—341no—
219#95—342no—
220C17#1038611yescharacteristic, complemented, Sylow
221C24—21no—
222C24C2×C22⋊D1711yescharacteristic
223#41—24no—
224#41—24no—
225#41—42no—
226#41—42no—
227#41—44no—
228#41—44no—
229C2×Q8—22no—
230C2×Q8—42no—
231C2×Q8—684no—
232C2×Q8—1361no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—342no—
251C22×C4—342no—
252C22×C4C2×Dic3411yescharacteristic
253C22×C4#6325211yescharacteristic
254C22×C4D4×D1712yes—
255C22×C4#6325412yes—
256M16—682no—
257M16—682no—
258C2×C8—682no—
259C4⋊C4—684no—
260C4⋊C4—1361no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—684no—
266C42—682no—
267C42—682no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—42no—
272C23—41no—
273C23—41no—
274C23—82no—
275C23#1058761011yescharacteristic
276C23#1058762211yescharacteristic
277C23#1058762311yescharacteristic
278Q8—24no—
279Q8—42no—
280Q8—1364no—
281D4—24no—
282D4—42no—
283D4—42no—
284D4—42no—
285D4—42no—
286D4—42no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—344no—
304C2×C4—682no—
305C2×C4—682no—
306C2×C4—682no—
307C2×C4—682no—
308C2×C4—682no—
309C2×C4#1058756412yes—
310C2×C4#1058757112yes—
311C8—1362no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—81no—
319C22—81no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22#1176272111yescharacteristic
324C22#1176299511yescharacteristic
325C22#1176299511yescharacteristic
326C4—22no—
327C4—22no—
328C4—42no—
329C4—81no—
330C4—81no—
331C4—161no—
332C4—684no—
333C4—1362no—
334C2—21no—
335C2—21no—
336C2—41no—
337C2—41no—
338C2—82no—
339C2#1452994811yescharacteristic
340C2#1452995812yes—
subgroupG/H
centerC22#11762721
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#1782C2
socleC22×C17#393
prime radicalC22×#41×C17C22
hypercenter#1782D17
hypocenterC17#10386
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #14534449C2semidirect
3. #14526692C2semidirect
4. #14526636C2semidirect
23. C22×Q8×C17Q8semidirect
25. C22×D4×C17Q8semidirect
194. C22×Q8Dic34semidirect
196. C22×D4Dic34semidirect
220. C17#10386semidirect

Maximal subgroups

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

subgroupindex
#145344492
#145287282
#145266922
#145266362
C17×#17822
#1038617