Finite Group Databasebeta

← #36363253

Subgroups — #36363253

Order 22144.

472 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149711C211yesmaximal, characteristic, complemented
2C2×#19149685C211yesmaximal, characteristic, complemented
3#26012071C211yesmaximal, characteristic
4#26012064C211yesmaximal, characteristic, complemented
5#26011747C211yesmaximal, characteristic, complemented
6#26011746C211yesmaximal, characteristic, complemented
7#26011739C211yesmaximal, characteristic, complemented
8#26011738C211yesmaximal, characteristic, complemented
9#26011731C211yesmaximal, characteristic, complemented
10#26011723C211yesmaximal, characteristic, complemented
11#26011720C211yesmaximal, characteristic, complemented
12#26011532C211yesmaximal, characteristic, complemented
13#26011486C211yesmaximal, characteristic, complemented
14#103×D173C211yesmaximal, characteristic, complemented
15#467×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19#19149711—24no—
20C2×#14583816C2211yescharacteristic
21C2×#14583815C2211yescharacteristic
22C2×#14583813C2211yescharacteristic
23#19149685—24no—
24C2×#14583810C2211yescharacteristic
25C2×#14583809C2211yescharacteristic
26C2×#14583809C2211yescharacteristic
27C2×C8×D173C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic
29#19149678—21no—
30#19149665—21no—
31#19149664—21no—
32#19149661—21no—
33#19149659C2211yescharacteristic
34#19149658C2211yescharacteristic
35#19149657—21no—
36#19149655C2211yescharacteristic
37#19149655C2211yescharacteristic
38#19149654—21no—
39#19149654C2211yescharacteristic
40#19149653C2211yescharacteristic
41#19149652—21no—
42#19149652—21no—
43#19149637C2211yescharacteristic
44C4⋊C4×D173C2211yescharacteristic
45#19149612C2211yescharacteristic
46#19149611C2211yescharacteristic
47#19149610C2211yescharacteristic
48#19149604C2211yescharacteristic
49#19149603C2211yescharacteristic
50#19149603C2211yescharacteristic
51#19149602C2211yescharacteristic
52#19149601C2211yescharacteristic
53#19149599C2211yescharacteristic
54C2×Q16×C173C2211yescharacteristic
55C22×C8×C173C2211yescharacteristic
56C22⋊Q8×C173C2211yescharacteristic
57C22⋊Q8×C173C2211yescharacteristic
58#107×C173C2211yescharacteristic
59#103×C173C2211yescharacteristic
60#103×C173C2211yescharacteristic
61#14583827—22no—
62#14583827—22no—
63#14583826—22no—
64#14583826—22no—
65C2×Dic346—21no—
66C2×Dic346D411yescharacteristic
67C2×Dic346C2311yescharacteristic
68Q8×D173—22no—
69Q8×D173—22no—
70C2×D692—21no—
71C2×D692—21no—
72C2×D692D411yescharacteristic
73C2×D692C2311yescharacteristic
74C2×C22⋊D173—21no—
75C2×C22⋊D173D411yescharacteristic
76C2×C22⋊D173D411yescharacteristic
77C2×C4×D173—21no—
78C2×C4×D173—21no—
79C2×C4×D173—21no—
80C2×C4×D173C2311yescharacteristic
81C2×C4×D173C2311yescharacteristic
82#14583816—22no—
83#14583815—22no—
84#14583813—22no—
85#14583810—22no—
86#14583809—22no—
87#14583809D414yescomplemented
88C2×#13063436C2311yescharacteristic
89C2×#13063436C2311yescharacteristic
90C8×D173—22no—
91C8×D173—22no—
92C4⋊Dic173—21no—
93C4⋊Dic173C2311yescharacteristic
94C4⋊Dic173C2311yescharacteristic
95#14583805—21no—
96#14583805—21no—
97#14583805—21no—
98#14583805—21no—
99C22⋊Dic173—21no—
100#14583803—21no—
101#14583803—21no—
102#14583803—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803—21no—
106C4×Dic173—21no—
107C2×Q8×C173C2311yescharacteristic
108C2×Q8×C173C2311yescharacteristic
109C22×C4×C173C2311yescharacteristic
110Q16×C173—22no—
111C2×C8×C173—22no—
112C2×C8×C173C2311yescharacteristic
113C2×C8×C173C2311yescharacteristic
114C4⋊C4×C173—21no—
115C4⋊C4×C173—21no—
116C4⋊C4×C173C2311yescharacteristic
117C4⋊C4×C173C2311yescharacteristic
118C22⋊C4×C173—21no—
119C22⋊C4×C173—21no—
120C22×D173—21no—
121C22×D173C2×D411yescharacteristic
122C22×D173C2×D411yescharacteristic
123Dic346—22no—
124Dic346—21no—
125Dic346—21no—
126Dic346—42no—
127D692—22no—
128D692—21no—
129D692—21no—
130D692—42no—
131D692—81no—
132C22⋊D173—22no—
133C22⋊D173—22no—
134C22⋊D173—81no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173C2×D411yescharacteristic
139C2×Dic173C2×D411yescharacteristic
140C4×D173—21no—
141C4×D173—21no—
142C4×D173—42no—
143C4×D173—42no—
144C4×D173—81no—
145C4×D173C2×D412yes—
146C4×D173#4112yes—
147#13063436—21no—
148#13063436C2×D412yes—
149C23×C173C2×D411yescharacteristic
150Q8×C173—21no—
151Q8×C173—21no—
152Q8×C173—21no—
153Q8×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173C2×D411yescharacteristic
161C2×C4×C173C2411yescharacteristic
162C8×C173—21no—
163C8×C173C2×D412yes—
164C22×C173—21no—
165C22×C173—21no—
166C22×C173—21no—
167C22×C173C22×D411yescharacteristic
168C2×D173—21no—
169C2×D173—21no—
170C2×D173—21no—
171C2×D173—21no—
172C2×D173—21no—
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173#12112yes—
177C4×C173—21no—
178C4×C173—41no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173—41no—
182C4×C173C22×D411yescharacteristic
183C4×C173C2×#4111yescharacteristic
184Dic173—21no—
185Dic173—41no—
186Dic173—41no—
187Dic173—41no—
188Dic173#12112yes—
189C2×C173—41no—
190C2×C173C2×#12111yescharacteristic
191C2×C173#57711yescharacteristic
192C2×C173#57811yescharacteristic
193D173—22no—
194D173—41no—
195D173—81no—
196C173#306311yescharacteristic, complemented, Sylow
197#3063—1731nomaximal, Sylow
198C2×#137—1731no—
199C2×#131—1731no—
200#537—1731no—
201#536—1731no—
202#470—1731no—
203#470—1731no—
204#468—1731no—
205#468—1731no—
206#467—1731no—
207#467D17311yescharacteristic, complemented
208#465—1731no—
209#465—1731no—
210#428—1731no—
211#416—1731no—
212C2×#103—1731no—
213C2×#41—1731no—
214C2×#41—1731no—
215C22×Q8—1731no—
216#137—3464no—
217C2×Q16—1731no—
218C2×Q16C2×D17311yescharacteristic
219C2×SD16—1731no—
220C2×SD16—1731no—
221#131—3464no—
222C2×M16—1731no—
223C2×M16—1731no—
224C2×M16—1731no—
225C22×C8—1731no—
226C22×C8—1731no—
227C22×C8C2×D17311yescharacteristic
228C4⋊Q8—3461no—
229#124—3461no—
230#123—3461no—
231#123—3461no—
232C22⋊Q8—3461no—
233C22⋊Q8—1731no—
234C22⋊Q8—1731no—
235C22⋊Q8—1731no—
236C22⋊Q8—1731no—
237C22⋊Q8C2×D17311yescharacteristic
238C22⋊Q8C2×D17311yescharacteristic
239#121—3461no—
240#121—3461no—
241#121—3461no—
242#121—1731no—
243#121—1731no—
244#117—1731no—
245C2×C4⋊C4—1731no—
246#107—1731no—
247#107C2×D17311yescharacteristic
248#106—1731no—
249#106—1731no—
250#103—1731no—
251#103—1731no—
252#103—1731no—
253#103—1731no—
254#103C2×D17311yescharacteristic
255#103C2×D17311yescharacteristic
256#102—1731no—
257#102—1731no—
258#41—3462no—
259#41—3462no—
260#41—3462no—
261#41—3462no—
262C2×Q8—3461no—
263C2×Q8—3462no—
264C2×Q8—3462no—
265C2×Q8—1731no—
266C2×Q8—1731no—
267C2×Q8C22×D17311yescharacteristic
268C2×Q8C22×D17311yescharacteristic
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—3461no—
272C2×D4—1731no—
273C2×D4—1731no—
274C2×D4—1731no—
275C2×D4—1731no—
276C22×C4—3461no—
277C22×C4—3461no—
278C22×C4—3461no—
279C22×C4—1731no—
280C22×C4—1731no—
281C22×C4C22×D17311yescharacteristic
282Q16—22no—
283Q16—3462no—
284SD16—3462no—
285SD16—3462no—
286M16—3462no—
287M16—3462no—
288M16—1734no—
289C2×C8—22no—
290C2×C8—3462no—
291C2×C8—3462no—
292C2×C8—1731no—
293C2×C8—1731no—
294C2×C8C22×D17311yescharacteristic
295C2×C8C22×D17311yescharacteristic
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—3461no—
299C4⋊C4—3461no—
300C4⋊C4—3461no—
301C4⋊C4—3461no—
302C4⋊C4—3461no—
303C4⋊C4—1731no—
304C4⋊C4—1731no—
305C4⋊C4C22×D17311yescharacteristic
306C4⋊C4C22×D17311yescharacteristic
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—3461no—
310C22⋊C4—3461no—
311C22⋊C4—3461no—
312C22⋊C4—3461no—
313C22⋊C4—3461no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C42—3461no—
317C23—3461no—
318C23—1731no—
319C23—1731no—
320C23D4×D17311yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—21no—
325Q8—3462no—
326Q8—3461no—
327Q8—3461no—
328Q8—6922no—
329D4—3462no—
330D4—3462no—
331D4—3462no—
332D4—3461no—
333D4—3461no—
334D4—13841no—
335D4—13841no—
336D4—6922no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—3461no—
347C2×C4—3461no—
348C2×C4—13841no—
349C2×C4—1731no—
350C2×C4—1731no—
351C2×C4—1732no—
352C2×C4—1732no—
353C2×C4—6922no—
354C2×C4—6922no—
355C2×C4D4×D17311yescharacteristic
356C2×C4C23×D17311yescharacteristic
357C8—21no—
358C8—3461no—
359C8—1732no—
360C8D4×D17312yes—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—3461no—
365C22—3461no—
366C22—3461no—
367C22—3461no—
368C22—3461no—
369C22—1732no—
370C22—6921no—
371C22—6921no—
372C22—6921no—
373C22C2×D4×D17311yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—3461no—
380C4—1732no—
381C4—6921no—
382C4—6921no—
383C4—6921no—
384C4C2×D4×D17311yescharacteristic
385C4#41×D17311yescharacteristic
386C2—41no—
387C2—3462no—
388C2—13841no—
389C2—6921no—
390C2#121×D17311yescharacteristic
391C2#2601229811yescharacteristic
392C2#2601230411yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#467×C173C2
socleC22×C173C22×D4
prime radical#26012071C2
hypercenter#467D173
hypocenterC173#3063
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#19149711C2semidirect
2. C2×#19149685C2semidirect
4. #26012064C2semidirect
5. #26011747C2semidirect
6. #26011746C2semidirect
7. #26011739C2semidirect
8. #26011738C2semidirect
9. #26011731C2semidirect
10. #26011723C2semidirect
11. #26011720C2semidirect
12. #26011532C2semidirect
13. #26011486C2semidirect
14. #103×D173C2semidirect
15. #467×C173C2semidirect
87. #14583809D4semidirect
196. C173#3063semidirect
207. #467D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497112
C2×#191496852
#260120712
#260120642
#260117472
#260117462
#260117392
#260117382
#260117312
#260117232
#260117202
#260115322
#260114862
#103×D1732
#467×C1732
#3063173