Finite Group Databasebeta

← #36365125

Subgroups — #36365125

Order 22144.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149712C211yesmaximal, characteristic, complemented
2C2×#19149700C211yesmaximal, characteristic, complemented
3#26012147C211yesmaximal, characteristic, complemented
4#26012143C211yesmaximal, characteristic, complemented
5#26011882C211yesmaximal, characteristic, complemented
6#26011733C211yesmaximal, characteristic, complemented
7#26011705C211yesmaximal, characteristic, complemented
8#26011700C211yesmaximal, characteristic, complemented
9#26011656C211yesmaximal, characteristic, complemented
10#26011654C211yesmaximal, characteristic, complemented
11#26011646C211yesmaximal, characteristic, complemented
12#26011634C211yesmaximal, characteristic, complemented
13#26011590C211yesmaximal, characteristic, complemented
14#26011560C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C2×#14583825C2211yescharacteristic
19C22×Dic346C2211yescharacteristic
20#19149712—24no—
21#19149700—24no—
22C2×Dic692C2211yescharacteristic
23C2×Dic692C2211yescharacteristic
24C2×#14583816C2211yescharacteristic
25C2×#14583816C2211yescharacteristic
26C2×#14583815C2211yescharacteristic
27C2×#14583815C2211yescharacteristic
28C2×#14583814C2211yescharacteristic
29C2×#14583811C2211yescharacteristic
30C2×C8×D173C2211yescharacteristic
31#19149677C2211yescharacteristic
32#19149666C2211yescharacteristic
33#19149665—21no—
34#19149664—21no—
35#19149661—21no—
36#19149659C2211yescharacteristic
37#19149659C2211yescharacteristic
38#19149656—21no—
39#19149652—21no—
40#19149651—21no—
41#19149644C2211yescharacteristic
42D4×Dic173—21no—
43#19149641C2211yescharacteristic
44C4×C22⋊D173C2211yescharacteristic
45C4⋊C4×D173—21no—
46#19149608C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149602C2211yescharacteristic
49#19149602C2211yescharacteristic
50#19149599C2211yescharacteristic
51C22⋊#13063436C2211yescharacteristic
52#19149591C2211yescharacteristic
53#19149589C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C2×M16×C173C2211yescharacteristic
56C22⋊Q8×C173C2211yescharacteristic
57#121×C173C2211yescharacteristic
58#107×C173C2211yescharacteristic
59#103×C173C2211yescharacteristic
60#102×C173C2211yescharacteristic
61#14583827—22no—
62#14583827—41no—
63#14583826—22no—
64#14583826—41no—
65#14583825—22no—
66#14583825—41no—
67C2×Dic346—22no—
68C2×Dic346—21no—
69C2×Dic346—41no—
70C2×Dic346C2311yescharacteristic
71C2×Dic346C2311yescharacteristic
72C2×Dic346C2311yescharacteristic
73C2×D692—21no—
74C2×D692C2311yescharacteristic
75C2×C22⋊D173—21no—
76C2×C22⋊D173—21no—
77C2×C22⋊D173D411yescharacteristic
78C22×Dic173—21no—
79C22×Dic173—21no—
80C2×C4×D173—21no—
81C2×C4×D173—21no—
82C2×C4×D173—21no—
83C2×C4×D173C2311yescharacteristic
84Dic692—22no—
85Dic692—22no—
86#14583816—22no—
87#14583816—41no—
88#14583815—22no—
89#14583815D414yescomplemented
90#14583814—41no—
91#14583811—22no—
92C2×#13063436C2311yescharacteristic
93C2×#13063436C2311yescharacteristic
94C8×D173—22no—
95C4⋊Dic173—21no—
96C4⋊Dic173—21no—
97#14583805—21no—
98#14583805—21no—
99#14583805—21no—
100#14583805D411yescharacteristic
101C22⋊Dic173—21no—
102C22⋊Dic173—21no—
103C22⋊Dic173D411yescharacteristic
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803D411yescharacteristic
108C4×Dic173—21no—
109C4×Dic173C2311yescharacteristic
110C2×Q8×C173C2311yescharacteristic
111C2×D4×C173—21no—
112C2×D4×C173C2311yescharacteristic
113C22×C4×C173C2311yescharacteristic
114SD16×C173—22no—
115M16×C173—22no—
116C2×C8×C173C2311yescharacteristic
117C2×C8×C173C2311yescharacteristic
118C4⋊C4×C173—21no—
119C4⋊C4×C173C2311yescharacteristic
120C4⋊C4×C173C2311yescharacteristic
121C22⋊C4×C173—21no—
122C22⋊C4×C173—21no—
123C22×D173—21no—
124C22×D173C2×D411yescharacteristic
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—21no—
129Dic346—21no—
130Dic346—42no—
131Dic346C2×D412yes—
132D692—21no—
133D692—42no—
134D692C2×D412yes—
135C22⋊D173—41no—
136C22⋊D173—42no—
137C22⋊D173—42no—
138C2×Dic173—21no—
139C2×Dic173—21no—
140C2×Dic173—21no—
141C2×Dic173—42no—
142C2×Dic173—42no—
143C2×Dic173C2×D411yescharacteristic
144C2×Dic173C2×D411yescharacteristic
145C2×Dic173C2×D411yescharacteristic
146C4×D173—21no—
147C4×D173—21no—
148C4×D173—42no—
149C4×D173—42no—
150C4×D173—42no—
151#13063436—21no—
152#13063436—21no—
153C23×C173—21no—
154C23×C173C2×D411yescharacteristic
155Q8×C173—21no—
156Q8×C173—21no—
157D4×C173—21no—
158D4×C173—21no—
159D4×C173—42no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173C2×D411yescharacteristic
166C2×C4×C173C2411yescharacteristic
167C8×C173—21no—
168C8×C173C2×D412yes—
169C22×C173—21no—
170C22×C173—21no—
171C22×C173—21no—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173—41no—
175C22×C173C22×D411yescharacteristic
176C2×D173—21no—
177C2×D173—22no—
178C2×D173—21no—
179C2×D173—21no—
180C2×D173—41no—
181C2×D173—41no—
182C4×C173—21no—
183C4×C173—41no—
184C4×C173—41no—
185C4×C173—41no—
186C4×C173C22×D411yescharacteristic
187C4×C173H(5,𝔽2)11yescharacteristic
188Dic173—21no—
189Dic173—22no—
190Dic173—21no—
191Dic173—21no—
192Dic173—41no—
193Dic173—41no—
194C2×C173—41no—
195C2×C173—81no—
196C2×C173D4211yescharacteristic
197C2×C173#57511yescharacteristic
198C2×C173#57811yescharacteristic
199D173—41no—
200D173—42no—
201C173#326911yescharacteristic, complemented, Sylow
202#3269—1731nomaximal, Sylow
203C2×#137—1731no—
204C2×#135—1731no—
205#548—1731no—
206#547—1731no—
207#497—1731no—
208#468D17311yescharacteristic, complemented
209#467—1731no—
210#463—1731no—
211#462—1731no—
212#452—1731no—
213#452—1731no—
214#451—1731no—
215#449—1731no—
216#440—1731no—
217#435—1731no—
218C2×#41—1731no—
219C2×#41—1731no—
220C2×#41—1731no—
221C22×Q8—1731no—
222#137—3464no—
223#135—3464no—
224C2×Q16—1731no—
225C2×Q16—1731no—
226C2×Q16—1731no—
227C2×Q16—1731no—
228C2×SD16—1731no—
229C2×SD16—1731no—
230C2×SD16—1731no—
231C2×SD16C2×D17311yescharacteristic
232C2×D8—1731no—
233C2×M16C2×D17311yescharacteristic
234C22×C8—1731no—
235C4⋊Q8—1731no—
236#124—3461no—
237#124—1731no—
238#123—3461no—
239#123—3461no—
240C22⋊Q8—3461no—
241C22⋊Q8—1731no—
242C22⋊Q8—1731no—
243C22⋊Q8C2×D17311yescharacteristic
244#121—3461no—
245#121—3461no—
246#121C2×D17311yescharacteristic
247C4×Q8—1731no—
248C4×D4—3461no—
249C4×D4—1731no—
250C4×D4—1731no—
251C2×C4⋊C4—3461no—
252#107C2×D17311yescharacteristic
253C4⋊C8—1731no—
254#103—1731no—
255#103—1731no—
256#103—1731no—
257#103C2×D17311yescharacteristic
258#102—1731no—
259#102C2×D17311yescharacteristic
260C22⋊C8—1731no—
261C22⋊C8—1731no—
262#97—1731no—
263#41—3462no—
264#41—3462no—
265#41—3462no—
266#41—6921no—
267#41—6921no—
268#41—6921no—
269C2×Q8—3462no—
270C2×Q8—3461no—
271C2×Q8—1731no—
272C2×Q8—1731no—
273C2×Q8—1731no—
274C2×Q8—6921no—
275C2×Q8C22×D17311yescharacteristic
276C2×D4—21no—
277C2×D4—3461no—
278C2×D4—3461no—
279C2×D4—3461no—
280C2×D4—1731no—
281C2×D4—1731no—
282C2×D4C22×D17311yescharacteristic
283C22×C4—3461no—
284C22×C4—3461no—
285C22×C4—3461no—
286C22×C4—3461no—
287C22×C4—3461no—
288C22×C4—1731no—
289C22×C4C22×D17311yescharacteristic
290Q16—3462no—
291Q16—3462no—
292Q16—3462no—
293Q16—6921no—
294SD16—22no—
295SD16—3462no—
296SD16—1734no—
297SD16—6921no—
298D8—3462no—
299M16—22no—
300C2×C8—3462no—
301C2×C8—1731no—
302C2×C8—1731no—
303C2×C8C22×D17311yescharacteristic
304C2×C8C22×D17311yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—3461no—
307C4⋊C4—3461no—
308C4⋊C4—3461no—
309C4⋊C4—3461no—
310C4⋊C4—3461no—
311C4⋊C4—1731no—
312C4⋊C4C22×D17311yescharacteristic
313C4⋊C4C22×D17311yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—3461no—
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C22⋊C4—3461no—
320C22⋊C4—3461no—
321C22⋊C4—1731no—
322C22⋊C4—1731no—
323C42—3461no—
324C42—1731no—
325C23—21no—
326C23—3461no—
327C23—1731no—
328C23D4×D17311yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—3461no—
332Q8—3461no—
333Q8—3461no—
334Q8—3461no—
335Q8—3461no—
336Q8—1732no—
337Q8—6922no—
338D4—21no—
339D4—21no—
340D4—42no—
341D4—3461no—
342D4—1732no—
343D4—6922no—
344D4—6921no—
345D4—6922no—
346D4—6922no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—3461no—
353C2×C4—3461no—
354C2×C4—3461no—
355C2×C4—3461no—
356C2×C4—3461no—
357C2×C4—1731no—
358C2×C4—1731no—
359C2×C4—1731no—
360C2×C4—6922no—
361C2×C4—6922no—
362C2×C4—6922no—
363C2×C4—6922no—
364C2×C4—6922no—
365C2×C4D4×D17311yescharacteristic
366C2×C4C23×D17311yescharacteristic
367C8—21no—
368C8—3461no—
369C8—3461no—
370C8D4×D17312yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—3461no—
378C22—3461no—
379C22—3461no—
380C22—3462no—
381C22—6921no—
382C22—6921no—
383C22C2×D4×D17311yescharacteristic
384C4—21no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—3461no—
389C4—3461no—
390C4—3461no—
391C4—3462no—
392C4—6921no—
393C4—6921no—
394C4C2×D4×D17311yescharacteristic
395C4#1914972511yescharacteristic
396C2—41no—
397C2—81no—
398C2—6921no—
399C2—6922no—
400C2#2601212511yescharacteristic
401C2#2601228711yescharacteristic
402C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
hypercenter#468D173
hypocenterC173#3269
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#19149712C2semidirect
2. C2×#19149700C2semidirect
3. #26012147C2semidirect
4. #26012143C2semidirect
5. #26011882C2semidirect
6. #26011733C2semidirect
7. #26011705C2semidirect
8. #26011700C2semidirect
9. #26011656C2semidirect
10. #26011654C2semidirect
11. #26011646C2semidirect
12. #26011634C2semidirect
13. #26011590C2semidirect
14. #26011560C2semidirect
15. #468×C173C2semidirect
89. #14583815D4semidirect
201. C173#3269semidirect
208. #468D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497122
C2×#191497002
#260121472
#260121432
#260118822
#260117332
#260117052
#260117002
#260116562
#260116542
#260116462
#260116342
#260115902
#260115602
#468×C1732
#3269173