Finite Group Databasebeta

← #36363014

Subgroups — #36363014

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×#19149699C211yesmaximal, characteristic, complemented
2#26012037C211yesmaximal, characteristic, complemented
3#41×Dic173C211yesmaximal, characteristic, complemented
4#26011715C211yesmaximal, characteristic, complemented
5#26011711C211yesmaximal, characteristic, complemented
6#26011708C211yesmaximal, characteristic, complemented
7#26011704C211yesmaximal, characteristic, complemented
8#26011699C211yesmaximal, characteristic, complemented
9#26011694C211yesmaximal, characteristic, complemented
10#26011689C211yesmaximal, characteristic, complemented
11#26011684C211yesmaximal, characteristic, complemented
12C2×#19149608C211yesmaximal, characteristic, complemented
13#26011488C211yesmaximal, characteristic, complemented
14#26011487C211yesmaximal, characteristic, complemented
15C2×#135×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17#19149699—24no—
18C2×#14583816C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21C2×#14583811C2211yescharacteristic
22C22×#13063436C2211yescharacteristic
23#19149677C2211yescharacteristic
24#19149675C2211yescharacteristic
25#19149668C2211yescharacteristic
26#19149666C2211yescharacteristic
27#19149651—21no—
28#19149651—21no—
29Q8×Dic173C2211yescharacteristic
30Q8×Dic173C2211yescharacteristic
31D4×Dic173—21no—
32D4×Dic173—21no—
33D4×Dic173C2211yescharacteristic
34D4×Dic173C2211yescharacteristic
35C4×C22⋊D173—21no—
36C4×C22⋊D173—21no—
37#19149638—21no—
38#19149638C2211yescharacteristic
39C2×C4×Dic173—21no—
40C2×C4×Dic173C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149608C2211yescharacteristic
44#19149608C2211yescharacteristic
45#19149604C2211yescharacteristic
46#19149604C2211yescharacteristic
47#19149603C2211yescharacteristic
48#19149603C2211yescharacteristic
49#19149601C2211yescharacteristic
50#19149601C2211yescharacteristic
51#19149600C2211yescharacteristic
52#19149600C2211yescharacteristic
53C2×#41×C173C2211yescharacteristic
54C2×#41×C173C2211yescharacteristic
55#135×C173—24no—
56C2×Q16×C173C2211yescharacteristic
57C2×SD16×C173C2211yescharacteristic
58C2×SD16×C173C2211yescharacteristic
59C2×D8×C173C2211yescharacteristic
60C22×C8×C173C2211yescharacteristic
61#14583826—22no—
62#14583826—22no—
63C2×Dic346C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C22×Dic173—21no—
69C22×Dic173—21no—
70C22×Dic173D411yescharacteristic
71C22×Dic173D411yescharacteristic
72C2×C4×D173—21no—
73#14583816—22no—
74#14583814—22no—
75#14583813—22no—
76#14583811—22no—
77C2×#13063436—22no—
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C4⋊Dic173—21no—
81C4⋊Dic173—21no—
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91#14583803—21no—
92#14583803—21no—
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173D411yescharacteristic
97C4×Dic173D411yescharacteristic
98C4×Dic173C2311yescharacteristic
99C4×Dic173C2311yescharacteristic
100#41×C173—22no—
101#41×C173—22no—
102#41×C173—22no—
103#41×C173D414yescomplemented
104C2×Q8×C173C2311yescharacteristic
105C2×Q8×C173C2311yescharacteristic
106C2×D4×C173—21no—
107C2×D4×C173—21no—
108C2×D4×C173C2311yescharacteristic
109C2×D4×C173C2311yescharacteristic
110C22×C4×C173—21no—
111C22×C4×C173—21no—
112C22×C4×C173C2311yescharacteristic
113Q16×C173—22no—
114SD16×C173—22no—
115SD16×C173—22no—
116D8×C173—22no—
117C2×C8×C173—22no—
118C2×C8×C173C2311yescharacteristic
119C2×C8×C173C2311yescharacteristic
120C22×D173—21no—
121Dic346—21no—
122Dic346—21no—
123D692—21no—
124D692—21no—
125C22⋊D173—42no—
126C22⋊D173—42no—
127C22⋊D173—42no—
128C2×Dic173—22no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—41no—
133C2×Dic173—42no—
134C2×Dic173—42no—
135C2×Dic173C2×D411yescharacteristic
136C2×Dic173C2×D411yescharacteristic
137C2×Dic173C2×D411yescharacteristic
138C2×Dic173C2×D411yescharacteristic
139C4×D173—42no—
140#13063436—21no—
141#13063436—21no—
142C23×C173—21no—
143C23×C173—21no—
144C23×C173C2×D411yescharacteristic
145Q8×C173—21no—
146Q8×C173—21no—
147Q8×C173—21no—
148Q8×C173C2×D412yes—
149D4×C173—24no—
150D4×C173—21no—
151D4×C173—21no—
152D4×C173—21no—
153D4×C173—42no—
154D4×C173C2×D412yes—
155C2×C4×C173—21no—
156C2×C4×C173—24no—
157C2×C4×C173—21no—
158C2×C4×C173—42no—
159C2×C4×C173C2×D411yescharacteristic
160C2×C4×C173C2×D412yes—
161C2×C4×C173#4112yes—
162C2×C4×C173C2411yescharacteristic
163C8×C173—21no—
164C8×C173—21no—
165C22×C173—21no—
166C22×C173—21no—
167C22×C173—22no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173—41no—
173C22×C173#12112yes—
174C22×C173C22×D411yescharacteristic
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—41no—
178C4×C173—22no—
179C4×C173—41no—
180C4×C173#12112yes—
181C4×C173C22×D411yescharacteristic
182C4×C173C2×#4111yescharacteristic
183Dic173—21no—
184Dic173—21no—
185Dic173—21no—
186Dic173—41no—
187Dic173—41no—
188Dic173—41no—
189Dic173#13512yescomplemented
190C2×C173—22no—
191C2×C173—42no—
192C2×C173—81no—
193C2×C173C2×#12111yescharacteristic
194C2×C173C2×#13511yescharacteristic
195C2×C173#57511yescharacteristic
196D173—81no—
197C173#303511yescharacteristic, complemented, Sylow
198#3035—1731nomaximal, Sylow
199C2×#135—1731no—
200C2×#135D17311yescharacteristic, complemented
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#416—1731no—
214C2×#41—1731no—
215C2×#41C2×D17311yescharacteristic
216C2×#41C2×D17311yescharacteristic
217#135—24no—
218#135—3464no—
219C2×Q16—1731no—
220C2×Q16C2×D17311yescharacteristic
221C2×SD16—1731no—
222C2×SD16—1731no—
223C2×SD16C2×D17311yescharacteristic
224C2×SD16C2×D17311yescharacteristic
225C2×D8—1731no—
226C2×D8C2×D17311yescharacteristic
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#117—1731no—
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#103—1731no—
254#103—1731no—
255#102—1731no—
256#102—1731no—
257#102—1731no—
258#102—1731no—
259#41—22no—
260#41—22no—
261#41—22no—
262#41—3462no—
263#41—3462no—
264#41C22⋊D17314yescomplemented
265C2×Q8—1731no—
266C2×Q8C22×D17311yescharacteristic
267C2×Q8C22×D17311yescharacteristic
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—3461no—
271C2×D4—3461no—
272C2×D4—3461no—
273C2×D4—1731no—
274C2×D4C22×D17311yescharacteristic
275C2×D4C22×D17311yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—3461no—
279C22×C4—3461no—
280C22×C4—3461no—
281C22×C4—1731no—
282C22×C4—1731no—
283C22×C4C22×D17311yescharacteristic
284Q16—22no—
285Q16—3462no—
286SD16—22no—
287SD16—22no—
288SD16—3462no—
289SD16—3462no—
290D8—22no—
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⋊C4—1731no—
303C4⋊C4—1731no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—3461no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C22⋊C4—3461no—
311C42—3461no—
312C42—3461no—
313C42—3461no—
314C42—1731no—
315C42—1731no—
316C42—1731no—
317C42—1731no—
318C23—21no—
319C23—21no—
320C23—3461no—
321C23D4×D17311yescharacteristic
322Q8—21no—
323Q8—21no—
324Q8—21no—
325Q8—3461no—
326Q8—3461no—
327Q8C2×C22⋊D17312yes—
328D4—24no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—42no—
333D4—3461no—
334D4—3461no—
335D4—6922no—
336D4—6922no—
337D4—6922no—
338D4C2×C22⋊D17312yes—
339C2×C4—21no—
340C2×C4—24no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—3462no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—3461no—
347C2×C4—1731no—
348C2×C4—1731no—
349C2×C4—1731no—
350C2×C4—1731no—
351C2×C4—6921no—
352C2×C4—6922no—
353C2×C4—6922no—
354C2×C4—6922no—
355C2×C4D4×D17311yescharacteristic
356C2×C4C2×C22⋊D17312yes—
357C2×C4#1458382512yes—
358C2×C4C23×D17311yescharacteristic
359C8—21no—
360C8—21no—
361C8—3461no—
362C8—3461no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—6921no—
372C22—6921no—
373C22—6921no—
374C22#1914965112yes—
375C22C2×D4×D17311yescharacteristic
376C4—22no—
377C4—41no—
378C4—3461no—
379C4—3461no—
380C4—3461no—
381C4—1732no—
382C4—6921no—
383C4—6921no—
384C4—6921no—
385C4#1914965112yes—
386C4C22×C22⋊D17311yescharacteristic
387C4C2×#1458382511yescharacteristic
388C2—22no—
389C2—42no—
390C2—81no—
391C2—13841no—
392C2C2×#1914965111yescharacteristic
393C2#135×D17311yescharacteristic
394C2#2601228811yescharacteristic
subgroupG/H
centerC2×C4D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#135×C173C2
socleC22×C173C22×D4
hypercenterC2×#135D173
hypocenterC173#3035
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#19149699C2semidirect
2. #26012037C2semidirect
3. #41×Dic173C2semidirect
4. #26011715C2semidirect
5. #26011711C2semidirect
6. #26011708C2semidirect
7. #26011704C2semidirect
8. #26011699C2semidirect
9. #26011694C2semidirect
10. #26011689C2semidirect
11. #26011684C2semidirect
12. C2×#19149608C2semidirect
13. #26011488C2semidirect
14. #26011487C2semidirect
15. C2×#135×C173C2semidirect
103. #41×C173D4semidirect
189. Dic173#135semidirect
197. C173#3035semidirect
200. C2×#135D173semidirect
264. #41C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496992
#260120372
#41×Dic1732
#260117152
#260117112
#260117082
#260117042
#260116992
#260116942
#260116892
#260116842
C2×#191496082
#260114882
#260114872
C2×#135×C1732
#3035173