Finite Group Databasebeta

← #17949949

Subgroups — #17949949

Order 4352.

584 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 104 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, complemented
2#14544640C211yesmaximal, characteristic, complemented
3#14543602C212yesmaximal, complemented
4#14543600C212yesmaximal, complemented
5#121×Dic17C211yesmaximal, characteristic
6C2×#10587544—21no—
7C2×C22⋊#3624—21no—
8C2×C22⋊#3624—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×D4×Dic17—21no—
13C2×D4×Dic17C411yescharacteristic
14C22×C4×Dic17C2211yescharacteristic
15#11762712C2211yescharacteristic
16#11762708C2211yescharacteristic
17#11762702—21no—
18#11762689—21no—
19#11762669C411yescharacteristic
20C4⋊C4×Dic17C411yescharacteristic
21C22⋊C4×Dic17—21no—
22C2×C17×#121C411yescharacteristic
23C22×D4×C17—21no—
24C22×D4×C17C2×C411yescharacteristic
25C23×C4×C17C2×C411yescharacteristic
26C23×Dic17—21no—
27C23×Dic17D411yescharacteristic
28C23×Dic17D411yescharacteristic
29C22×#3624—22no—
30C22×#3624—22no—
31C17×#121—22no—
32C17×#121—22no—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C22⋊C4×C17—21no—
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—21no—
39D4×Dic17—42no—
40D4×Dic17—42no—
41D4×Dic17—42no—
42C2×C4⋊Dic17—21no—
43C2×C4⋊Dic17C2×C411yescharacteristic
44C2×C4⋊Dic17C2×C411yescharacteristic
45C2×C4×Dic17—21no—
46C2×C4×Dic17—42no—
47C2×C4×Dic17C2×C411yescharacteristic
48C2×C4×Dic17D411yescharacteristic
49C2×C4×Dic17D411yescharacteristic
50C2×C4×Dic17C2311yescharacteristic
51C22⋊#3624—42no—
52C22⋊#3624—42no—
53#10587544—42no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57C24×C17—21no—
58C24×C17C22⋊C411yescharacteristic
59C2×D4×C17—24no—
60C2×D4×C17—21no—
61C2×D4×C17—41no—
62C2×D4×C17—41no—
63C2×D4×C17—42no—
64C2×D4×C17—42no—
65C2×D4×C17M1612yes—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—41no—
70C22×C4×C17—41no—
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22×C411yescharacteristic
73C22×Dic17—22no—
74C22×Dic17—22no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—41no—
78C22×Dic17—41no—
79C22×Dic17—41no—
80C22×Dic17—42no—
81C22×Dic17—42no—
82C22×Dic17—42no—
83C22×Dic17C22⋊C411yescharacteristic
84C22×Dic17C22⋊C411yescharacteristic
85C22×Dic17D812yes—
86C22×Dic17SD1612yes—
87C22×Dic17C2×D411yescharacteristic
88C22×Dic17C2×D411yescharacteristic
89C2×#3624—42no—
90C2×#3624—42no—
91C2×#3624—42no—
92C2×#3624—42no—
93C2×#3624—42no—
94C2×#3624—42no—
95C4⋊C4×C17—21no—
96C4⋊C4×C17M1612yes—
97C22⋊C4×C17—24no—
98C22⋊Dic17—42no—
99C22⋊Dic17—42no—
100C22⋊Dic17—41no—
101C22⋊Dic17—41no—
102C22⋊Dic17—42no—
103C4⋊Dic17—41no—
104C4⋊Dic17—41no—
105C4⋊Dic17—41no—
106C4⋊Dic17—41no—
107C4×Dic17—22no—
108C4×Dic17—22no—
109C4×Dic17—41no—
110C4×Dic17—41no—
111C4×Dic17—42no—
112#7647—171nomaximal, Sylow
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17—21no—
118C23×C17—21no—
119C23×C17—22no—
120C23×C17—41no—
121C23×C17—41no—
122C23×C17—41no—
123C23×C17—41no—
124C23×C17—41no—
125C23×C17—41no—
126C23×C17#10212yes—
127C23×C17C2×C22⋊C411yescharacteristic
128D4×C17—42no—
129D4×C17—42no—
130D4×C17—48no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—22no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—42no—
138C2×C4×C17—42no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17C2×M1611yescharacteristic
142C2×C4×C17C2×M1611yescharacteristic
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—42no—
152C2×Dic17—42no—
153C2×Dic17—44no—
154C2×Dic17—41no—
155C2×Dic17—41no—
156C2×Dic17—41no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159C2×Dic17—41no—
160C2×Dic17—84no—
161C2×Dic17#9912yes—
162C2×Dic17#10212yes—
163C2×Dic17C2×D811yescharacteristic
164C2×Dic17C2×SD1611yescharacteristic
165C2×Dic17#13512yes—
166#3624—82no—
167#3624—82no—
168C4×#121—171no—
169#1817—171no—
170#1797—171no—
171#1271—172no—
172#1270—172no—
173C22×C17—22no—
174C22×C17—22no—
175C22×C17—22no—
176C22×C17—41no—
177C22×C17—41no—
178C22×C17—41no—
179C22×C17—41no—
180C22×C17—42no—
181C22×C17—42no—
182C22×C17—42no—
183C22×C17—81no—
184C22×C17—81no—
185C22×C17—81no—
186C22×C17—81no—
187C22×C17#40711yescharacteristic
188C22×C17C2×#9911yescharacteristic
189C22×C17#41311yescharacteristic
190C22×C17C2×#10211yescharacteristic
191C22×C17#41611yescharacteristic
192C22×C17#42111yescharacteristic
193C22×C17#42111yescharacteristic
194C4×C17—21no—
195C4×C17—21no—
196C4×C17—41no—
197C4×C17—42no—
198Dic17—22no—
199Dic17—22no—
200Dic17—21no—
201Dic17—22no—
202Dic17—21no—
203Dic17—81no—
204Dic17—81no—
205C2×#121#59511yescharacteristic
206C2×C4×D4—171no—
207C2×C4×D4—341no—
208C22×C42—171no—
209C2×C4⋊C8—341no—
210C2×C22⋊C8—341no—
211C2×C22⋊C8—341no—
212#390—171no—
213#389—171no—
214#388—341no—
215#386—341no—
216#380—171no—
217C4×C4⋊C4—171no—
218C4×C22⋊C4—341no—
219#336—172no—
220#336—172no—
221#336—341no—
222C2×C17—42no—
223C2×C17—82no—
224C2×C17#146511yescharacteristic
225C2×C17#148611yescharacteristic
226C2×C17#148811yescharacteristic
227C2×C17#151611yescharacteristic
228C2×C17#151911yescharacteristic
229C2×C17#153511yescharacteristic
230C2×C17#154011yescharacteristic
231C22×D4—21no—
232C22×D4C2×#59511yescharacteristic
233C23×C4—171no—
234C23×C4—171no—
235C23×C4—341no—
236C23×C4C2×#59511yescharacteristic
237C22×C8—342no—
238C22×C8—342no—
239#121—22no—
240#121—22no—
241C4×D4—682no—
242C4×D4—682no—
243C4×D4—682no—
244C2×C4⋊C4—171no—
245C2×C4⋊C4—171no—
246C2×C4⋊C4—341no—
247C2×C4⋊C4C2×#59511yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—341no—
250C2×C22⋊C4—341no—
251C2×C22⋊C4—341no—
252C2×C22⋊C4—341no—
253C2×C42—171no—
254C2×C42—171no—
255C2×C42—171no—
256C2×C42—171no—
257C2×C42—341no—
258C2×C42—682no—
259C4⋊C8—682no—
260C22⋊C8—682no—
261C22⋊C8—682no—
262#95—341no—
263#95—341no—
264#95—341no—
265C17#764711yescharacteristic, complemented, Sylow
266C24—21no—
267C24C22⋊#59511yescharacteristic
268C2×D4—24no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4#6324712yes—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—171no—
281C22×C4—171no—
282C22×C4—171no—
283C22×C4—171no—
284C22×C4—172no—
285C22×C4—172no—
286C22×C4—342no—
287C22×C4—342no—
288C22×C4—341no—
289C22×C4—341no—
290C22×C4—682no—
291C22×C4—681no—
292C22×C4—681no—
293C22×C4—681no—
294C22×C4—682no—
295C22×C4—682no—
296C22×C4C22⋊#59511yescharacteristic
297C22×C4C22×#59511yescharacteristic
298C2×C8—682no—
299C2×C8—682no—
300C2×C8—682no—
301C2×C8—682no—
302C2×C8—682no—
303C2×C8—682no—
304C4⋊C4—21no—
305C4⋊C4—681no—
306C4⋊C4—681no—
307C4⋊C4—681no—
308C4⋊C4—681no—
309C4⋊C4#6324712yes—
310C22⋊C4—24no—
311C22⋊C4—682no—
312C22⋊C4—682no—
313C22⋊C4—681no—
314C22⋊C4—681no—
315C22⋊C4—682no—
316C42—342no—
317C42—342no—
318C42—681no—
319C42—681no—
320C42—682no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23#1058754712yes—
335C23C2×C22⋊#59511yescharacteristic
336D4—42no—
337D4—42no—
338D4—48no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—172no—
350C2×C4—172no—
351C2×C4—171no—
352C2×C4—171no—
353C2×C4—172no—
354C2×C4—341no—
355C2×C4—341no—
356C2×C4—341no—
357C2×C4—341no—
358C2×C4—341no—
359C2×C4—341no—
360C2×C4—341no—
361C2×C4—341no—
362C2×C4—682no—
363C2×C4—682no—
364C2×C4—684no—
365C2×C4—681no—
366C2×C4—681no—
367C2×C4—681no—
368C2×C4—681no—
369C2×C4—681no—
370C2×C4—681no—
371C2×C4—1364no—
372C2×C4C2×#6324711yescharacteristic
373C2×C4C2×#6324711yescharacteristic
374C8—1362no—
375C8—1362no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22#1176377911yescharacteristic
391C22D17⋊#9911yescharacteristic
392C22#1176379611yescharacteristic
393C22C2×#1058754711yescharacteristic
394C22#1176379911yescharacteristic
395C22#1176380611yescharacteristic
396C22#1176380711yescharacteristic
397C4—21no—
398C4—21no—
399C4—41no—
400C4—42no—
401C4—342no—
402C4—342no—
403C4—341no—
404C4—342no—
405C4—341no—
406C4—1361no—
407C4—1361no—
408C2—42no—
409C2—82no—
410C2#1454404011yescharacteristic
411C2#1454410911yescharacteristic
412C2#1454411511yescharacteristic
413C2#1454420911yescharacteristic
414C2#1454421911yescharacteristic
415C2#1454426911yescharacteristic
416C2#1454428211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
Fitting, prime radicalC2×C17×#121C4
socleC23×C17C2×C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC2×#121#595
hypocenterC17#7647
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #14544675C2semidirect
2. #14544640C2semidirect
3. #14543602C2semidirect
4. #14543600C2semidirect
265. C17#7647semidirect

Maximal subgroups

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

subgroupindex
#145446752
#145446402
#145436022
#145436002
#121×Dic172
#764717