Finite Group Databasebeta

← #36365548

Subgroups — #36365548

Order 22144.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012150C211yesmaximal, characteristic, complemented
2#26011853C211yesmaximal, characteristic, complemented
3#26011826C211yesmaximal, characteristic, complemented
4#26011804C211yesmaximal, characteristic, complemented
5#26011783C211yesmaximal, characteristic, complemented
6#26011735C211yesmaximal, characteristic, complemented
7#26011725C211yesmaximal, characteristic, complemented
8#26011706C211yesmaximal, characteristic, complemented
9#26011634C211yesmaximal, characteristic, complemented
10#26011629C211yesmaximal, characteristic, complemented
11#26011591C211yesmaximal, characteristic, complemented
12#26011563C211yesmaximal, characteristic, complemented
13#26011490C211yesmaximal, characteristic, complemented
14C2×#19149602C211yesmaximal, characteristic, complemented
15#547×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21C2×#14583810C2211yescharacteristic
22C22×#13063436C2211yescharacteristic
23#19149667C2211yescharacteristic
24#19149665—21no—
25#19149658C2211yescharacteristic
26#19149657—21no—
27#19149653C2211yescharacteristic
28#19149651—21no—
29C4×Dic346C2211yescharacteristic
30#19149639—21no—
31#19149612C2211yescharacteristic
32#19149612C2211yescharacteristic
33#19149612C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic
35#19149602C2211yescharacteristic
36#19149602C2211yescharacteristic
37#19149602C2211yescharacteristic
38#19149602C2211yescharacteristic
39#19149602C2211yescharacteristic
40#19149602C2211yescharacteristic
41#19149600C2211yescharacteristic
42#19149599C2211yescharacteristic
43#19149599C2211yescharacteristic
44#19149599C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic
47#19149590C2211yescharacteristic
48C2×#41×C173—21no—
49C4⋊Q8×C173C2211yescharacteristic
50#123×C173—21no—
51#123×C173—21no—
52C22⋊Q8×C173—21no—
53#121×C173C2211yescharacteristic
54#121×C173C2211yescharacteristic
55C4×D4×C173C2211yescharacteristic
56C4×D4×C173C2211yescharacteristic
57C2×C4⋊C4×C173C2211yescharacteristic
58C2×C4⋊C4×C173C2211yescharacteristic
59#14583826—22no—
60#14583826—41no—
61C2×Dic346—22no—
62C2×Dic346—21no—
63C2×Dic346—41no—
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C22×Dic173—21no—
70C2×C4×D173—21no—
71#14583814—22no—
72#14583814—41no—
73#14583811—41no—
74#14583810—41no—
75C2×#13063436—41no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83C22⋊Dic173—21no—
84#14583803—21no—
85#14583803—21no—
86#14583803—21no—
87C4×Dic173—21no—
88#41×C173—44no—
89C2×Q8×C173—21no—
90C2×D4×C173—21no—
91C2×D4×C173—21no—
92C2×D4×C173D411yescharacteristic
93C2×D4×C173C2311yescharacteristic
94C22×C4×C173—21no—
95C22×C4×C173—21no—
96C22×C4×C173—21no—
97C22×C4×C173C2311yescharacteristic
98C22×C4×C173C2311yescharacteristic
99C4⋊C4×C173—21no—
100C4⋊C4×C173—21no—
101C4⋊C4×C173—21no—
102C4⋊C4×C173D411yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C22⋊C4×C173—21no—
109C22⋊C4×C173D411yescharacteristic
110C22⋊C4×C173D411yescharacteristic
111C42×C173C2311yescharacteristic
112C22×D173—21no—
113Dic346—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—42no—
117Dic346#4112yescomplemented
118D692—21no—
119D692—21no—
120C22⋊D173—42no—
121C22⋊D173—42no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—42no—
126C4×D173—42no—
127#13063436—21no—
128#13063436—21no—
129#13063436—21no—
130#13063436—21no—
131C23×C173—21no—
132C23×C173C2×D411yescharacteristic
133C23×C173C2×D411yescharacteristic
134Q8×C173—42no—
135D4×C173—22no—
136D4×C173—21no—
137D4×C173—21no—
138D4×C173—42no—
139D4×C173—42no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—42no—
148C2×C4×C173—42no—
149C2×C4×C173—42no—
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2×D411yescharacteristic
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2411yescharacteristic
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—21no—
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173AGL(1,ℤ/8)12yes—
164C22×C173C22×D411yescharacteristic
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—41no—
168C4×C173—21no—
169C4×C173—21no—
170C4×C173—21no—
171C4×C173—21no—
172C4×C173—41no—
173C4×C173—41no—
174C4×C173—41no—
175C4×C173C2×#4111yescharacteristic
176C4×C173H(5,𝔽2)11yescharacteristic
177Dic173—22no—
178Dic173—41no—
179Dic173—41no—
180C2×C173—22no—
181C2×C173—41no—
182C2×C173—81no—
183C2×C173#54611yescharacteristic
184C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C173#57811yescharacteristic
186D173—81no—
187C173#330811yescharacteristic, complemented, Sylow
188#3308—1731nomaximal, Sylow
189#548—1731no—
190#547D17311yescharacteristic, complemented
191#489—1731no—
192#484—1731no—
193#480—1731no—
194#477—1731no—
195#468—1731no—
196#466—1731no—
197#463—1731no—
198#449—1731no—
199#448—1731no—
200#440—1731no—
201#435—1731no—
202#417—1731no—
203C2×#103—1731no—
204C2×#41—21no—
205C2×#41—1731no—
206C22×Q8—1731no—
207C2×SD16—1731no—
208C2×SD16—1731no—
209C2×D8—1731no—
210C2×M16—1731no—
211C22×C8—1731no—
212C4⋊Q8C2×D17311yescharacteristic
213#124—3461no—
214#124—1731no—
215#123—21no—
216#123—21no—
217C22⋊Q8—21no—
218C22⋊Q8—3461no—
219C22⋊Q8—1731no—
220#121—3461no—
221#121—1731no—
222#121C2×D17311yescharacteristic
223#121C2×D17311yescharacteristic
224C4×Q8—1731no—
225C4×D4—3461no—
226C4×D4C2×D17311yescharacteristic
227C4×D4C2×D17311yescharacteristic
228C2×C4⋊C4C2×D17311yescharacteristic
229C2×C4⋊C4C2×D17311yescharacteristic
230#107—1731no—
231#107—1731no—
232#107—1731no—
233C4⋊C8—1731no—
234#103—1731no—
235#103—1731no—
236#103—1731no—
237#103—1731no—
238#103—1731no—
239#103—1731no—
240#102—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244C22⋊C8—1731no—
245C22⋊C8—1731no—
246#97—1731no—
247#41—44no—
248#41—3462no—
249#41—6921no—
250C2×Q8—21no—
251C2×Q8—3462no—
252C2×Q8—3461no—
253C2×Q8—1731no—
254C2×Q8—1731no—
255C2×Q8—6921no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—1731no—
261C2×D4C22⋊D17311yescharacteristic
262C2×D4C22×D17311yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4C22×D17311yescharacteristic
269C22×C4C22×D17311yescharacteristic
270SD16—3462no—
271SD16—6921no—
272D8—6921no—
273M16—6921no—
274C2×C8—1731no—
275C2×C8—1731no—
276C2×C8—1731no—
277C2×C8—1731no—
278C2×C8—6921no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—3461no—
283C4⋊C4—3461no—
284C4⋊C4—1731no—
285C4⋊C4C22⋊D17311yescharacteristic
286C4⋊C4C22×D17311yescharacteristic
287C4⋊C4C22×D17311yescharacteristic
288C4⋊C4C22×D17311yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4C22⋊D17311yescharacteristic
297C22⋊C4C22⋊D17311yescharacteristic
298C42—3461no—
299C42C22×D17311yescharacteristic
300C23—21no—
301C23—3461no—
302C23C2×C22⋊D17311yescharacteristic
303C23C2×C22⋊D17311yescharacteristic
304Q8—42no—
305Q8—3461no—
306Q8—3461no—
307Q8—3461no—
308Q8—1732no—
309Q8—6922no—
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—3461no—
316D4—3461no—
317D4—6922no—
318D4—6922no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—6922no—
333C2×C4—6922no—
334C2×C4C2×C22⋊D17311yescharacteristic
335C2×C4C2×C22⋊D17311yescharacteristic
336C2×C4C2×C22⋊D17311yescharacteristic
337C2×C4C2×C22⋊D17311yescharacteristic
338C2×C4C23×D17311yescharacteristic
339C8—3461no—
340C8—3461no—
341C8—3461no—
342C8—3461no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—6921no—
352C22—6921no—
353C22—6921no—
354C22#1914970512yes—
355C22C22×C22⋊D17311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—3462no—
364C4—6921no—
365C4—6921no—
366C4#41×D17311yescharacteristic
367C4#1914972511yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—13841no—
372C2#2601212911yescharacteristic
373C2C2×#1914970511yescharacteristic
374C2#2601230611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#547×C173C2
socleC22×C173C22×D4
hypercenter#547D173
hypocenterC173#3308
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #26012150C2semidirect
2. #26011853C2semidirect
3. #26011826C2semidirect
4. #26011804C2semidirect
5. #26011783C2semidirect
6. #26011735C2semidirect
7. #26011725C2semidirect
8. #26011706C2semidirect
9. #26011634C2semidirect
10. #26011629C2semidirect
11. #26011591C2semidirect
12. #26011563C2semidirect
13. #26011490C2semidirect
14. C2×#19149602C2semidirect
15. #547×C173C2semidirect
117. Dic346#41semidirect
187. C173#3308semidirect
190. #547D173semidirect

Maximal subgroups

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

subgroupindex
#260121502
#260118532
#260118262
#260118042
#260117832
#260117352
#260117252
#260117062
#260116342
#260116292
#260115912
#260115632
#260114902
C2×#191496022
#547×C1732
#3308173