Finite Group Databasebeta

← #36363016

Subgroups — #36363016

Order 22144.

496 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149701C211yesmaximal, characteristic, complemented
2C2×#19149699C211yesmaximal, characteristic, complemented
3#26012037C211yesmaximal, characteristic, complemented
4#26011959C211yesmaximal, characteristic, complemented
5#26011716C211yesmaximal, characteristic, complemented
6#26011712C211yesmaximal, characteristic, complemented
7#26011709C211yesmaximal, characteristic, complemented
8#26011705C211yesmaximal, characteristic, complemented
9#26011700C211yesmaximal, characteristic, complemented
10#26011695C211yesmaximal, characteristic, complemented
11#26011690C211yesmaximal, characteristic, complemented
12#26011685C211yesmaximal, characteristic, complemented
13C2×#19149608C211yesmaximal, characteristic, complemented
14#26011485C211yesmaximal, characteristic, complemented
15#416×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18#19149701—24no—
19#19149699—24no—
20C2×Dic692C2211yescharacteristic
21C2×#14583816C2211yescharacteristic
22C2×#14583815C2211yescharacteristic
23C2×#14583815C2211yescharacteristic
24C2×#14583814C2211yescharacteristic
25C2×#14583813C2211yescharacteristic
26C2×D1384C2211yescharacteristic
27C2×#14583811C2211yescharacteristic
28C22×#13063436C2211yescharacteristic
29#19149677C2211yescharacteristic
30#19149675C2211yescharacteristic
31#19149668C2211yescharacteristic
32#19149666C2211yescharacteristic
33#19149651—21no—
34#19149651—21no—
35#19149644C2211yescharacteristic
36#19149644C2211yescharacteristic
37#19149641C2211yescharacteristic
38#19149641C2211yescharacteristic
39C4×C22⋊D173—21no—
40C4×C22⋊D173—21no—
41#19149639—21no—
42#19149639—21no—
43#19149636—21no—
44C2×C4×Dic173C2211yescharacteristic
45C42×D173—21no—
46#19149608C2211yescharacteristic
47#19149608C2211yescharacteristic
48#19149608C2211yescharacteristic
49#19149608C2211yescharacteristic
50#19149602C2211yescharacteristic
51#19149602C2211yescharacteristic
52#19149599C2211yescharacteristic
53#19149599C2211yescharacteristic
54C2×#41×C173C2211yescharacteristic
55C22×C8×C173C2211yescharacteristic
56#117×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59#102×C173C2211yescharacteristic
60#102×C173C2211yescharacteristic
61#14583826—22no—
62#14583826—22no—
63#14583826—22no—
64#14583826D414yescomplemented
65C2×Dic346C2311yescharacteristic
66C2×Dic346C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C22×Dic173D411yescharacteristic
74C22×Dic173D411yescharacteristic
75C2×C4×D173—21no—
76C2×C4×D173—21no—
77C2×C4×D173—21no—
78C2×C4×D173—21no—
79Dic692—22no—
80#14583816—22no—
81#14583815—22no—
82#14583815—22no—
83#14583814—22no—
84#14583813—22no—
85D1384—22no—
86#14583811—22no—
87C2×#13063436—22no—
88C2×#13063436C2311yescharacteristic
89C2×#13063436C2311yescharacteristic
90#14583805—21no—
91#14583805—21no—
92#14583805—21no—
93#14583805—21no—
94C22⋊Dic173—21no—
95C22⋊Dic173—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100C4×Dic173—21no—
101C4×Dic173—21no—
102C4×Dic173D411yescharacteristic
103C4×Dic173D411yescharacteristic
104C4×Dic173C2311yescharacteristic
105C4×Dic173C2311yescharacteristic
106#41×C173—22no—
107#41×C173—22no—
108C2×Q8×C173C2311yescharacteristic
109C2×D4×C173—21no—
110C2×D4×C173C2311yescharacteristic
111C22×C4×C173—21no—
112C22×C4×C173C2311yescharacteristic
113C2×C8×C173—22no—
114C2×C8×C173C2311yescharacteristic
115C2×C8×C173C2311yescharacteristic
116C4⋊C4×C173C2311yescharacteristic
117C4⋊C4×C173C2311yescharacteristic
118C22⋊C4×C173—21no—
119C42×C173—21no—
120C22×D173—21no—
121C22×D173—21no—
122Dic346—21no—
123Dic346—21no—
124Dic346—21no—
125Dic346C2×D412yes—
126D692—21no—
127D692—21no—
128D692—21no—
129D692C2×D412yes—
130C22⋊D173—24no—
131C22⋊D173—42no—
132C22⋊D173—42no—
133C22⋊D173—42no—
134C2×Dic173—21no—
135C2×Dic173—22no—
136C2×Dic173—21no—
137C2×Dic173—41no—
138C2×Dic173C2×D411yescharacteristic
139C2×Dic173C2×D411yescharacteristic
140C2×Dic173C2×D411yescharacteristic
141C2×Dic173C2×D411yescharacteristic
142C4×D173—24no—
143C4×D173—42no—
144C4×D173—42no—
145C4×D173—42no—
146#13063436—21no—
147#13063436—21no—
148C23×C173—21no—
149C23×C173C2×D411yescharacteristic
150Q8×C173—21no—
151Q8×C173—21no—
152D4×C173—21no—
153D4×C173—21no—
154D4×C173—42no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—42no—
159C2×C4×C173C2×D412yes—
160C2×C4×C173C2×D411yescharacteristic
161C2×C4×C173#4112yes—
162C2×C4×C173C2411yescharacteristic
163C8×C173—21no—
164C8×C173—21no—
165C22×C173—21no—
166C22×C173—21no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173#12112yes—
171C22×C173C22×D411yescharacteristic
172C2×D173—22no—
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—41no—
178C4×C173—41no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173#12112yes—
182C4×C173C22×D411yescharacteristic
183C4×C173C2×#4111yescharacteristic
184Dic173—22no—
185Dic173—21no—
186Dic173—21no—
187Dic173—21no—
188Dic173—41no—
189Dic173#13512yescomplemented
190C2×C173—22no—
191C2×C173—81no—
192C2×C173C2×#12111yescharacteristic
193C2×C173C2×#13511yescharacteristic
194C2×C173#57511yescharacteristic
195D173—42no—
196D173—81no—
197C173#303511yescharacteristic, complemented, Sylow
198#3035—1731nomaximal, Sylow
199C2×#135—1731no—
200C2×#135—1731no—
201#532—1731no—
202C4×#41—1731no—
203#464—1731no—
204#464—1731no—
205#463—1731no—
206#463—1731no—
207#462—1731no—
208#461—1731no—
209#460—1731no—
210#459—1731no—
211C2×C4⋊C8—1731no—
212#416—1731no—
213#416D17311yescharacteristic, complemented
214C2×#41—1731no—
215C2×#41—1731no—
216C2×#41C2×D17311yescharacteristic
217#135—3464no—
218#135—3464no—
219C2×Q16—1731no—
220C2×Q16—1731no—
221C2×SD16—1731no—
222C2×SD16—1731no—
223C2×SD16—1731no—
224C2×SD16—1731no—
225C2×D8—1731no—
226C2×D8—1731no—
227C22×C8—1731no—
228C22×C8C2×D17311yescharacteristic
229C4⋊Q8—1731no—
230#127—1731no—
231#124—1731no—
232#124—1731no—
233#121—3461no—
234#121—3461no—
235C4×Q8—1731no—
236C4×Q8—1731no—
237C4×D4—3461no—
238C4×D4—3461no—
239C4×D4—3461no—
240C4×D4—3461no—
241C4×D4—1731no—
242C4×D4—1731no—
243#117—3461no—
244#117C2×D17311yescharacteristic
245C2×C42—3461no—
246C2×C42—1731no—
247C4⋊C8—1731no—
248C4⋊C8—1731no—
249C4⋊C8—1731no—
250C4⋊C8—1731no—
251#103—1731no—
252#103—1731no—
253#103C2×D17311yescharacteristic
254#103C2×D17311yescharacteristic
255#102—1731no—
256#102—1731no—
257#102C2×D17311yescharacteristic
258#102C2×D17311yescharacteristic
259#41—22no—
260#41—22no—
261#41—3462no—
262#41—3462no—
263#41—3462no—
264#41—1734no—
265C2×Q8—1731no—
266C2×Q8—1731no—
267C2×Q8C22×D17311yescharacteristic
268C2×D4—21no—
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—3461no—
272C2×D4—3461no—
273C2×D4—1731no—
274C2×D4—1731no—
275C2×D4C22×D17311yescharacteristic
276C22×C4—21no—
277C22×C4—3461no—
278C22×C4—3461no—
279C22×C4—3461no—
280C22×C4—3461no—
281C22×C4—1731no—
282C22×C4—1731no—
283C22×C4C22×D17311yescharacteristic
284Q16—3462no—
285Q16—3462no—
286SD16—3462no—
287SD16—3462no—
288SD16—3462no—
289SD16—3462no—
290D8—3462no—
291D8—3462no—
292C2×C8—22no—
293C2×C8—3462no—
294C2×C8—1731no—
295C2×C8—1731no—
296C2×C8C22×D17311yescharacteristic
297C2×C8C22×D17311yescharacteristic
298C4⋊C4—3461no—
299C4⋊C4—3461no—
300C4⋊C4—3461no—
301C4⋊C4—3461no—
302C4⋊C4C22×D17311yescharacteristic
303C4⋊C4C22×D17311yescharacteristic
304C22⋊C4—21no—
305C22⋊C4—3461no—
306C22⋊C4—3461no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C22⋊C4—3461no—
311C42—21no—
312C42—3461no—
313C42—3461no—
314C42—1731no—
315C42—1731no—
316C42—1731no—
317C42—1731no—
318C23—21no—
319C23—3461no—
320C23—3461no—
321C23D4×D17311yescharacteristic
322Q8—21no—
323Q8—21no—
324Q8—3461no—
325Q8—3461no—
326Q8—3461no—
327Q8—1732no—
328D4—21no—
329D4—21no—
330D4—42no—
331D4—3464no—
332D4—3461no—
333D4—3461no—
334D4—3461no—
335D4—1732no—
336D4—6922no—
337D4—6922no—
338D4—6922no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—3461no—
344C2×C4—3462no—
345C2×C4—3464no—
346C2×C4—3461no—
347C2×C4—1731no—
348C2×C4—1731no—
349C2×C4—1731no—
350C2×C4—1731no—
351C2×C4—6922no—
352C2×C4—6922no—
353C2×C4—6921no—
354C2×C4—6922no—
355C2×C4D4×D17312yes—
356C2×C4D4×D17311yescharacteristic
357C2×C4#1458382612yes—
358C2×C4C23×D17311yescharacteristic
359C8—21no—
360C8—21no—
361C8—3461no—
362C8—3461no—
363C22—21no—
364C22—21no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—3462no—
369C22—6921no—
370C22—6921no—
371C22—6921no—
372C22—6921no—
373C22—6921no—
374C22#1914965212yes—
375C22C2×D4×D17311yescharacteristic
376C4—41no—
377C4—41no—
378C4—41no—
379C4—3462no—
380C4—3461no—
381C4—3461no—
382C4—3461no—
383C4—1732no—
384C4—6921no—
385C4#1914965212yes—
386C4C2×D4×D17311yescharacteristic
387C4C2×#1458382611yescharacteristic
388C2—22no—
389C2—81no—
390C2—13841no—
391C2—6922no—
392C2C2×#1914965211yescharacteristic
393C2#135×D17311yescharacteristic
394C2#2601228811yescharacteristic
subgroupG/H
centerC2×C4D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#416×C173C2
socleC22×C173C22×D4
hypercenter#416D173
hypocenterC173#3035
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#19149701C2semidirect
2. C2×#19149699C2semidirect
3. #26012037C2semidirect
4. #26011959C2semidirect
5. #26011716C2semidirect
6. #26011712C2semidirect
7. #26011709C2semidirect
8. #26011705C2semidirect
9. #26011700C2semidirect
10. #26011695C2semidirect
11. #26011690C2semidirect
12. #26011685C2semidirect
13. C2×#19149608C2semidirect
14. #26011485C2semidirect
15. #416×C173C2semidirect
64. #14583826D4semidirect
189. Dic173#135semidirect
197. C173#3035semidirect
213. #416D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497012
C2×#191496992
#260120372
#260119592
#260117162
#260117122
#260117092
#260117052
#260117002
#260116952
#260116902
#260116852
C2×#191496082
#260114852
#416×C1732
#3035173