Finite Group Databasebeta

← #36364494

Subgroups — #36364494

Order 22144.

438 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012037C211yesmaximal, characteristic, complemented
2#26012036C211yesmaximal, characteristic, complemented
3#26011866C211yesmaximal, characteristic, complemented
4#26011862C211yesmaximal, characteristic, complemented
5#26011840C211yesmaximal, characteristic, complemented
6#26011836C211yesmaximal, characteristic, complemented
7#26011709C211yesmaximal, characteristic, complemented
8#26011708C211yesmaximal, characteristic, complemented
9#26011705C211yesmaximal, characteristic, complemented
10#26011704C211yesmaximal, characteristic, complemented
11#26011638C211yesmaximal, characteristic, complemented
12#26011636C211yesmaximal, characteristic, complemented
13#26011634C211yesmaximal, characteristic, complemented
14#26011541C211yesmaximal, characteristic, complemented
15#449×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583825—21no—
18C2×#14583815C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21C2×D1384C2211yescharacteristic
22C2×#14583811C2211yescharacteristic
23C2×#14583811C2211yescharacteristic
24#19149678C2211yescharacteristic
25#19149677C2211yescharacteristic
26#19149675C2211yescharacteristic
27#19149675C2211yescharacteristic
28#19149668C2211yescharacteristic
29#19149666C2211yescharacteristic
30#19149665—21no—
31#19149653C2211yescharacteristic
32#19149651—21no—
33#19149651—21no—
34#19149651—21no—
35D4×Dic173C2211yescharacteristic
36#19149641C2211yescharacteristic
37C4×C22⋊D173—21no—
38C4×C22⋊D173—21no—
39#19149639—21no—
40C2×C4×Dic173C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50C8×Dic173C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×D8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—41no—
60#14583826—41no—
61#14583825—44no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C22×Dic173—21no—
73C22×Dic173D411yescharacteristic
74C22×Dic173D411yescharacteristic
75C2×C4×D173—21no—
76C2×C4×D173—21no—
77#14583815—41no—
78#14583814—41no—
79#14583813—41no—
80D1384—41no—
81#14583811—41no—
82#14583811—41no—
83C2×#13063436C2311yescharacteristic
84C2×#13063436C2311yescharacteristic
85C4⋊Dic173C2311yescharacteristic
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96C4×Dic173D411yescharacteristic
97C4×Dic173D411yescharacteristic
98C4×Dic173C2311yescharacteristic
99C4×Dic173C2311yescharacteristic
100#41×C173—41no—
101#41×C173—41no—
102C2×Q8×C173C2311yescharacteristic
103C2×D4×C173—21no—
104C2×D4×C173—21no—
105C2×D4×C173C2311yescharacteristic
106C2×D4×C173C2311yescharacteristic
107C22×C4×C173—21no—
108C22×C4×C173C2311yescharacteristic
109SD16×C173—41no—
110D8×C173—41no—
111C2×C8×C173C2311yescharacteristic
112C2×C8×C173C2311yescharacteristic
113C4⋊C4×C173C2311yescharacteristic
114C22⋊C4×C173—21no—
115C22×D173—21no—
116C22×D173—21no—
117Dic346—21no—
118Dic346—21no—
119Dic346—42no—
120D692—21no—
121D692—21no—
122D692—21no—
123D692—21no—
124C22⋊D173—42no—
125C22⋊D173—42no—
126C22⋊D173—42no—
127C22⋊D173—42no—
128C22⋊D173—42no—
129C22⋊D173—42no—
130C2×Dic173—22no—
131C2×Dic173—22no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—42no—
135C2×Dic173C2×D411yescharacteristic
136C2×Dic173C2×D411yescharacteristic
137C2×Dic173C2×D411yescharacteristic
138C2×Dic173C2×D411yescharacteristic
139C4×D173—42no—
140C4×D173—42no—
141#13063436—21no—
142#13063436—21no—
143C23×C173—21no—
144C23×C173—21no—
145C23×C173C2×D411yescharacteristic
146Q8×C173—21no—
147Q8×C173—21no—
148D4×C173—21no—
149D4×C173—21no—
150D4×C173—21no—
151D4×C173—21no—
152D4×C173—42no—
153D4×C173—42no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—42no—
159C2×C4×C173C2×D411yescharacteristic
160C2×C4×C173C2411yescharacteristic
161C8×C173—21no—
162C8×C173—21no—
163C22×C173—21no—
164C22×C173—21no—
165C22×C173—21no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173C22×D411yescharacteristic
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—41no—
178C2×D173—41no—
179C4×C173—21no—
180C4×C173—41no—
181C4×C173—41no—
182C4×C173H(5,𝔽2)11yescharacteristic
183C4×C173H(5,𝔽2)11yescharacteristic
184Dic173—21no—
185Dic173—21no—
186Dic173—41no—
187Dic173—41no—
188Dic173#13512yes—
189Dic173AGL(1,ℤ/8)12yes—
190C2×C173—41no—
191C2×C173—81no—
192C2×C173—81no—
193C2×C173#53511yescharacteristic
194C2×C173C2×#13511yescharacteristic
195C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
196D173—81no—
197D173—81no—
198C173#320711yescharacteristic, complemented, Sylow
199#3207—1731nomaximal, Sylow
200#532—1731no—
201#532—1731no—
202#493—1731no—
203#492—1731no—
204#487—1731no—
205#486—1731no—
206#463—1731no—
207#463—1731no—
208#463—1731no—
209#463—1731no—
210#449—1731no—
211#449—1731no—
212#449—1731no—
213#449D17311yescharacteristic, complemented
214#431—1731no—
215C2×#41—3461no—
216C2×#41—1731no—
217C2×#41C2×D17311yescharacteristic
218C2×SD16—1731no—
219C2×SD16—1731no—
220C2×SD16—1731no—
221C2×SD16C2×D17311yescharacteristic
222C2×D8—1731no—
223C2×D8—1731no—
224C2×D8—1731no—
225C2×D8C2×D17311yescharacteristic
226C4⋊Q8—1731no—
227C4⋊Q8—1731no—
228#127—1731no—
229#127—1731no—
230#124—3461no—
231#124—1731no—
232#124—1731no—
233#121—3461no—
234#121—3461no—
235#121—3461no—
236#121—1731no—
237#121C2×D17311yescharacteristic
238C4×D4—3461no—
239C4×D4—3461no—
240C4×D4—3461no—
241C4×D4—1731no—
242C4×D4—1731no—
243C2×C42—1731no—
244C4⋊C8—1731no—
245C4⋊C8—1731no—
246#103—1731no—
247#103—1731no—
248#103—1731no—
249#103C2×D17311yescharacteristic
250#102—1731no—
251#102—1731no—
252#102—1731no—
253#102C2×D17311yescharacteristic
254C22⋊C8—1731no—
255C22⋊C8C2×D17311yescharacteristic
256C4×C8—1731no—
257C4×C8—1731no—
258#41—41no—
259#41—41no—
260#41—6921no—
261#41—6921no—
262#41—6924no—
263C2×Q8—3461no—
264C2×Q8—1731no—
265C2×Q8C22×D17311yescharacteristic
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—3461no—
272C2×D4—3461no—
273C2×D4—3461no—
274C2×D4—1731no—
275C2×D4—1731no—
276C2×D4C22×D17311yescharacteristic
277C2×D4C22×D17311yescharacteristic
278C22×C4—21no—
279C22×C4—3461no—
280C22×C4—3461no—
281C22×C4—3461no—
282C22×C4—1731no—
283C22×C4—1731no—
284C22×C4C22×D17311yescharacteristic
285SD16—41no—
286SD16—6921no—
287SD16—6921no—
288SD16—6921no—
289D8—41no—
290D8—6921no—
291D8—6921no—
292D8—6921no—
293C2×C8—1731no—
294C2×C8—1731no—
295C2×C8C22×D17311yescharacteristic
296C2×C8C22×D17311yescharacteristic
297C4⋊C4—3461no—
298C4⋊C4—3461no—
299C4⋊C4—3461no—
300C4⋊C4—1731no—
301C4⋊C4C22×D17311yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—3461no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—3461no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C42—1731no—
311C42—1731no—
312C42—1731no—
313C42—1731no—
314C23—21no—
315C23—21no—
316C23—3461no—
317C23—3461no—
318C23D4×D17311yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—3461no—
322Q8—3461no—
323Q8—6922no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—42no—
329D4—42no—
330D4—3461no—
331D4—3461no—
332D4—3461no—
333D4—3461no—
334D4—6922no—
335D4—6922no—
336D4—6922no—
337D4—6922no—
338D4—6922no—
339D4—6922no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—3462no—
346C2×C4—3462no—
347C2×C4—3461no—
348C2×C4—3461no—
349C2×C4—1731no—
350C2×C4—1731no—
351C2×C4—1731no—
352C2×C4—1731no—
353C2×C4—6922no—
354C2×C4—6922no—
355C2×C4—6922no—
356C2×C4D4×D17311yescharacteristic
357C2×C4C23×D17311yescharacteristic
358C8—21no—
359C8—21no—
360C8—3461no—
361C8—3461no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—6921no—
372C22—6921no—
373C22—6921no—
374C22—6921no—
375C22—6921no—
376C22—6921no—
377C22C2×D4×D17311yescharacteristic
378C4—21no—
379C4—41no—
380C4—41no—
381C4—3461no—
382C4—3461no—
383C4—1732no—
384C4—1732no—
385C4—6921no—
386C4—6921no—
387C4#1914972411yescharacteristic
388C4#1914972411yescharacteristic
389C2—41no—
390C2—81no—
391C2—81no—
392C2—13841no—
393C2—13841no—
394C2#2601205611yescharacteristic
395C2#135×D17311yescharacteristic
396C2AGL(1,ℤ/8)×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#449×C173C2
socleC22×C173C22×D4
hypercenter#449D173
hypocenterC173#3207
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #26012037C2semidirect
2. #26012036C2semidirect
3. #26011866C2semidirect
4. #26011862C2semidirect
5. #26011840C2semidirect
6. #26011836C2semidirect
7. #26011709C2semidirect
8. #26011708C2semidirect
9. #26011705C2semidirect
10. #26011704C2semidirect
11. #26011638C2semidirect
12. #26011636C2semidirect
13. #26011634C2semidirect
14. #26011541C2semidirect
15. #449×C173C2semidirect
198. C173#3207semidirect
213. #449D173semidirect

Maximal subgroups

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

subgroupindex
#260120372
#260120362
#260118662
#260118622
#260118402
#260118362
#260117092
#260117082
#260117052
#260117042
#260116382
#260116362
#260116342
#260115412
#449×C1732
#3207173