Finite Group Databasebeta

← #36365316

Subgroups — #36365316

Order 22144.

504 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149700C211yesmaximal, characteristic, complemented
2C2×#19149699C211yesmaximal, characteristic, complemented
3#26012160C211yesmaximal, characteristic, complemented
4#26012159C211yesmaximal, characteristic, complemented
5#26011866C211yesmaximal, characteristic, complemented
6#26011726C211yesmaximal, characteristic, complemented
7#26011725C211yesmaximal, characteristic, complemented
8#26011690C211yesmaximal, characteristic, complemented
9#26011689C211yesmaximal, characteristic, complemented
10#26011639C211yesmaximal, characteristic, complemented
11#26011638C211yesmaximal, characteristic, complemented
12#26011633C211yesmaximal, characteristic, complemented
13#26011583C211yesmaximal, characteristic, complemented
14#26011554C211yesmaximal, characteristic, complemented
15#449×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C2×#14583825C2211yescharacteristic
19#19149700—24no—
20#19149699—24no—
21C2×#14583816C2211yescharacteristic
22C2×#14583815C2211yescharacteristic
23C2×#14583814C2211yescharacteristic
24C2×#14583813C2211yescharacteristic
25C2×D1384C2211yescharacteristic
26C2×#14583811C2211yescharacteristic
27C2×#14583811C2211yescharacteristic
28C22×#13063436C2211yescharacteristic
29C2×C8×D173C2211yescharacteristic
30#19149675—21no—
31#19149675—21no—
32#19149675C2211yescharacteristic
33#19149675C2211yescharacteristic
34#19149654C2211yescharacteristic
35#19149653—21no—
36#19149653C2211yescharacteristic
37#19149652—21no—
38#19149651—21no—
39#19149650—21no—
40#19149650C2211yescharacteristic
41Q8×Dic173C2211yescharacteristic
42#19149644C2211yescharacteristic
43C4×C22⋊D173—21no—
44C4×C22⋊D173C2211yescharacteristic
45#19149639—21no—
46#19149612C2211yescharacteristic
47#19149608C2211yescharacteristic
48#19149604C2211yescharacteristic
49#19149601C2211yescharacteristic
50#19149600C2211yescharacteristic
51#19149599C2211yescharacteristic
52#19149591C2211yescharacteristic
53C8×Dic173C2211yescharacteristic
54C2×#41×C173C2211yescharacteristic
55C2×SD16×C173C2211yescharacteristic
56C2×D8×C173C2211yescharacteristic
57#121×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59#102×C173C2211yescharacteristic
60C22⋊C8×C173C2211yescharacteristic
61#14583827—22no—
62#14583827—41no—
63#14583826—22no—
64#14583826—41no—
65#14583825—22no—
66#14583825—41no—
67C2×Dic346C2311yescharacteristic
68C2×D692—21no—
69C2×D692—21no—
70C2×D692C2311yescharacteristic
71C2×D692C2311yescharacteristic
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76C2×C22⋊D173—21no—
77C2×C22⋊D173—21no—
78C2×C22⋊D173—21no—
79C2×C22⋊D173—21no—
80C2×C22⋊D173D411yescharacteristic
81C22×Dic173—21no—
82C2×C4×D173—21no—
83C2×C4×D173—21no—
84C2×C4×D173C2311yescharacteristic
85#14583816D414yescomplemented
86#14583815—22no—
87#14583814—22no—
88#14583813—22no—
89D1384—41no—
90#14583811—22no—
91#14583811—22no—
92C2×#13063436—22no—
93C2×#13063436C2311yescharacteristic
94C2×#13063436C2311yescharacteristic
95C8×D173—22no—
96C4⋊Dic173—21no—
97C4⋊Dic173C2311yescharacteristic
98#14583805—21no—
99#14583805D411yescharacteristic
100C22⋊Dic173—21no—
101C22⋊Dic173D411yescharacteristic
102#14583803—21no—
103#14583803—21no—
104#14583803D411yescharacteristic
105C4×Dic173—21no—
106C4×Dic173—21no—
107C4×Dic173C2311yescharacteristic
108#41×C173—22no—
109#41×C173—41no—
110C2×Q8×C173C2311yescharacteristic
111C2×D4×C173—21no—
112C2×D4×C173—21no—
113C2×D4×C173C2311yescharacteristic
114C2×D4×C173C2311yescharacteristic
115C22×C4×C173—21no—
116C22×C4×C173C2311yescharacteristic
117SD16×C173—22no—
118D8×C173—41no—
119C2×C8×C173C2311yescharacteristic
120C2×C8×C173C2311yescharacteristic
121C4⋊C4×C173C2311yescharacteristic
122C22⋊C4×C173—21no—
123C22×D173—21no—
124C22×D173—21no—
125C22×D173C2×D411yescharacteristic
126Dic346—21no—
127Dic346C2×D412yes—
128D692—21no—
129D692—21no—
130D692—21no—
131D692—21no—
132D692—42no—
133D692—81no—
134C22⋊D173—22no—
135C22⋊D173—42no—
136C22⋊D173—42no—
137C22⋊D173—42no—
138C22⋊D173—42no—
139C22⋊D173—81no—
140C22⋊D173—81no—
141C22⋊D173—81no—
142C22⋊D173—81no—
143C2×Dic173—21no—
144C2×Dic173—21no—
145C2×Dic173—42no—
146C2×Dic173C2×D411yescharacteristic
147C2×Dic173C2×D411yescharacteristic
148C2×Dic173C2×D411yescharacteristic
149C4×D173—21no—
150C4×D173—21no—
151C4×D173—42no—
152C4×D173—42no—
153#13063436—21no—
154#13063436C2×D412yes—
155C23×C173—21no—
156C23×C173—21no—
157C23×C173C2×D411yescharacteristic
158Q8×C173—21no—
159Q8×C173C2×D412yes—
160D4×C173—21no—
161D4×C173—21no—
162D4×C173—21no—
163D4×C173—21no—
164D4×C173—42no—
165D4×C173—81no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173—21no—
169C2×C4×C173—21no—
170C2×C4×C173—42no—
171C2×C4×C173C2×D411yescharacteristic
172C2×C4×C173C2411yescharacteristic
173C8×C173—21no—
174C8×C173—21no—
175C22×C173—21no—
176C22×C173—21no—
177C22×C173—21no—
178C22×C173—41no—
179C22×C173—41no—
180C22×C173—41no—
181C22×C173—41no—
182C22×C173—41no—
183C22×C173—41no—
184C22×C173C22×D411yescharacteristic
185C2×D173—21no—
186C2×D173—21no—
187C2×D173—21no—
188C2×D173—41no—
189C2×D173—41no—
190C2×D173—41no—
191C2×D173—41no—
192C2×D173—41no—
193C2×D173—41no—
194C4×C173—21no—
195C4×C173—22no—
196C4×C173—41no—
197C4×C173C22×D411yescharacteristic
198C4×C173H(5,𝔽2)11yescharacteristic
199Dic173—21no—
200Dic173—22no—
201Dic173—21no—
202Dic173—41no—
203Dic173#13512yescomplemented
204C2×C173—41no—
205C2×C173—81no—
206C2×C173—81no—
207C2×C173D4211yescharacteristic
208C2×C173C2×#13511yescharacteristic
209C2×C173#57611yescharacteristic
210D173—41no—
211D173—81no—
212D173—81no—
213C173#328711yescharacteristic, complemented, Sylow
214#3287—1731nomaximal, Sylow
215C2×#135—1731no—
216C2×#135—1731no—
217#550—1731no—
218#550—1731no—
219#493—1731no—
220#466—1731no—
221#466—1731no—
222#460—1731no—
223#460—1731no—
224#449—1731no—
225#449—1731no—
226#449—1731no—
227#449D17311yescharacteristic, complemented
228C4×Q16—1731no—
229C8×D4—1731no—
230C2×#41—1731no—
231C2×#41—1731no—
232C2×#41—1731no—
233C2×#41C2×D17311yescharacteristic
234#135—3464no—
235#135—3464no—
236C2×Q16—1731no—
237C2×SD16—1731no—
238C2×SD16—1731no—
239C2×SD16—1731no—
240C2×SD16C2×D17311yescharacteristic
241C2×D8—1731no—
242C2×D8—1731no—
243C2×D8—1731no—
244C2×D8C2×D17311yescharacteristic
245C22×C8—1731no—
246C22×C8—1731no—
247#127—3461no—
248#127—3461no—
249#127—1731no—
250#127—1731no—
251#121—3461no—
252#121—3461no—
253#121—3461no—
254#121—3461no—
255#121—1731no—
256#121—1731no—
257#121—1731no—
258#121C2×D17311yescharacteristic
259C4×Q8—1731no—
260C4×Q8—1731no—
261C4×D4—3461no—
262C4×D4—3461no—
263C4×D4—1731no—
264#107—1731no—
265C4⋊C8—1731no—
266#103—1731no—
267#103C2×D17311yescharacteristic
268#102—1731no—
269#102—1731no—
270#102—1731no—
271#102C2×D17311yescharacteristic
272C22⋊C8—1731no—
273C22⋊C8C2×D17311yescharacteristic
274C4×C8—1731no—
275#41—22no—
276#41—41no—
277#41—3462no—
278#41—3462no—
279#41—3462no—
280#41—6921no—
281#41—6921no—
282#41—6921no—
283C2×Q8—1731no—
284C2×Q8C22×D17311yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—3461no—
288C2×D4—3461no—
289C2×D4—3461no—
290C2×D4—3461no—
291C2×D4—3461no—
292C2×D4—3461no—
293C2×D4—3461no—
294C2×D4—3461no—
295C2×D4—3461no—
296C2×D4—3461no—
297C2×D4—1731no—
298C2×D4—1731no—
299C2×D4—1731no—
300C2×D4C22×D17311yescharacteristic
301C2×D4C22×D17311yescharacteristic
302C22×C4—21no—
303C22×C4—3461no—
304C22×C4—3461no—
305C22×C4—3461no—
306C22×C4—1731no—
307C22×C4C22×D17311yescharacteristic
308Q16—1734no—
309SD16—22no—
310SD16—3462no—
311SD16—3462no—
312SD16—3462no—
313D8—41no—
314D8—3462no—
315D8—3462no—
316D8—6921no—
317C2×C8—3462no—
318C2×C8—3462no—
319C2×C8—1731no—
320C2×C8—1731no—
321C2×C8C22×D17311yescharacteristic
322C2×C8C22×D17311yescharacteristic
323C4⋊C4—3461no—
324C4⋊C4—3461no—
325C4⋊C4—1731no—
326C4⋊C4—1731no—
327C4⋊C4C22×D17311yescharacteristic
328C22⋊C4—21no—
329C22⋊C4—3461no—
330C22⋊C4—3461no—
331C22⋊C4—3461no—
332C22⋊C4—1731no—
333C22⋊C4—1731no—
334C42—3461no—
335C42—3461no—
336C42—1731no—
337C23—21no—
338C23—21no—
339C23—3461no—
340C23—3461no—
341C23—1731no—
342C23D4×D17311yescharacteristic
343Q8—21no—
344Q8—3461no—
345Q8—1732no—
346Q8D4×D17312yes—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—81no—
353D4—3462no—
354D4—3461no—
355D4—3461no—
356D4—3461no—
357D4—3461no—
358D4—13841no—
359D4—13841no—
360D4—13841no—
361D4—13841no—
362D4—13841no—
363D4—6922no—
364D4—6922no—
365D4—6922no—
366D4—6922no—
367D4—6922no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—3461no—
374C2×C4—3461no—
375C2×C4—3461no—
376C2×C4—3461no—
377C2×C4—1731no—
378C2×C4—1731no—
379C2×C4—1731no—
380C2×C4—6922no—
381C2×C4—6922no—
382C2×C4—6922no—
383C2×C4D4×D17311yescharacteristic
384C2×C4C23×D17311yescharacteristic
385C8—21no—
386C8—21no—
387C8—3461no—
388C8—1732no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—3461no—
399C22—3461no—
400C22—3461no—
401C22—6921no—
402C22—6921no—
403C22—6921no—
404C22—6921no—
405C22—6921no—
406C22—6921no—
407C22C2×D4×D17311yescharacteristic
408C4—21no—
409C4—22no—
410C4—41no—
411C4—3461no—
412C4—3461no—
413C4—3462no—
414C4—1732no—
415C4—6921no—
416C4C2×D4×D17311yescharacteristic
417C4#1914972411yescharacteristic
418C2—41no—
419C2—81no—
420C2—81no—
421C2—13841no—
422C2—13841no—
423C2—6921no—
424C2#2601212311yescharacteristic
425C2#135×D17311yescharacteristic
426C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#449×C173C2
socleC22×C173C22×D4
hypercenter#449D173
hypocenterC173#3287
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. C2×#19149700C2semidirect
2. C2×#19149699C2semidirect
3. #26012160C2semidirect
4. #26012159C2semidirect
5. #26011866C2semidirect
6. #26011726C2semidirect
7. #26011725C2semidirect
8. #26011690C2semidirect
9. #26011689C2semidirect
10. #26011639C2semidirect
11. #26011638C2semidirect
12. #26011633C2semidirect
13. #26011583C2semidirect
14. #26011554C2semidirect
15. #449×C173C2semidirect
85. #14583816D4semidirect
203. Dic173#135semidirect
213. C173#3287semidirect
227. #449D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497002
C2×#191496992
#260121602
#260121592
#260118662
#260117262
#260117252
#260116902
#260116892
#260116392
#260116382
#260116332
#260115832
#260115542
#449×C1732
#3287173