Finite Group Databasebeta

← #36365555

Subgroups — #36365555

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#26012148C211yesmaximal, characteristic, complemented
2#26012142C211yesmaximal, characteristic, complemented
3#26011849C211yesmaximal, characteristic, complemented
4#26011826C211yesmaximal, characteristic, complemented
5#26011803C211yesmaximal, characteristic, complemented
6#26011784C211yesmaximal, characteristic, complemented
7#26011737C211yesmaximal, characteristic, complemented
8#26011726C211yesmaximal, characteristic, complemented
9#26011705C211yesmaximal, characteristic, complemented
10#26011636C211yesmaximal, characteristic, complemented
11#26011631C211yesmaximal, characteristic, complemented
12#26011594C211yesmaximal, characteristic, complemented
13#26011560C211yesmaximal, characteristic, complemented
14#103×D173C211yesmaximal, characteristic, complemented
15#417×C173C211yesmaximal, characteristic, complemented
16C2×#14583826—21no—
17C2×Q8×D173C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21C2×C8×D173C2211yescharacteristic
22#19149678C2211yescharacteristic
23#19149666—21no—
24#19149666C2211yescharacteristic
25#19149664—21no—
26#19149662—21no—
27#19149658—21no—
28#19149658C2211yescharacteristic
29#19149657—21no—
30#19149654C2211yescharacteristic
31#19149653C2211yescharacteristic
32#19149650—21no—
33#19149650C2211yescharacteristic
34Q8×Dic173C2211yescharacteristic
35#19149641C2211yescharacteristic
36C4×C22⋊D173—21no—
37C4×C22⋊D173C2211yescharacteristic
38C4⋊C4×D173C2211yescharacteristic
39#19149613C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149612C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149589C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic
53C2×M16×C173C2211yescharacteristic
54C2×C4⋊C4×C173C2211yescharacteristic
55#103×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583826—44no—
60C2×Dic346—21no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63Q8×D173—22no—
64Q8×D173—41no—
65C2×D692—21no—
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173D411yescharacteristic
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173C2311yescharacteristic
74#14583815—41no—
75#14583813—22no—
76#14583811—41no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C8×D173—41no—
80C4⋊Dic173—21no—
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805D411yescharacteristic
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173D411yescharacteristic
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803D411yescharacteristic
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98#41×C173—22no—
99#41×C173—41no—
100C2×Q8×C173C2311yescharacteristic
101C2×D4×C173—21no—
102C2×D4×C173C2311yescharacteristic
103C22×C4×C173—21no—
104C22×C4×C173—21no—
105C22×C4×C173C2311yescharacteristic
106M16×C173—41no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C4⋊C4×C173—21no—
110C4⋊C4×C173C2311yescharacteristic
111C4⋊C4×C173C2311yescharacteristic
112C22×D173—21no—
113C22×D173C2×D411yescharacteristic
114Dic346—21no—
115Dic346—21no—
116Dic346—42no—
117Dic346—42no—
118D692—21no—
119D692—21no—
120D692—42no—
121C22⋊D173—22no—
122C22⋊D173—42no—
123C22⋊D173—42no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C2×Dic173C2×D411yescharacteristic
130C4×D173—21no—
131C4×D173—21no—
132C4×D173—42no—
133C4×D173—42no—
134C4×D173—42no—
135#13063436—21no—
136#13063436—21no—
137C23×C173—21no—
138C23×C173C2×D411yescharacteristic
139Q8×C173—21no—
140Q8×C173#4112yescomplemented
141D4×C173—21no—
142D4×C173—21no—
143D4×C173—42no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—42no—
150C2×C4×C173—42no—
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2411yescharacteristic
153C8×C173—21no—
154C8×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—21no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173C22×D411yescharacteristic
162C2×D173—21no—
163C2×D173—21no—
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173AGL(1,ℤ/8)12yes—
168C4×C173—21no—
169C4×C173—22no—
170C4×C173—41no—
171C4×C173—41no—
172C4×C173C2×#4111yescharacteristic
173C4×C173H(5,𝔽2)11yescharacteristic
174Dic173—21no—
175Dic173—21no—
176Dic173—21no—
177Dic173—41no—
178Dic173—41no—
179Dic173—41no—
180C2×C173—41no—
181C2×C173—81no—
182C2×C173#54611yescharacteristic
183C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C173#57811yescharacteristic
185D173—22no—
186D173—81no—
187C173#330811yescharacteristic, complemented, Sylow
188#3308—1731nomaximal, Sylow
189#548—1731no—
190#547—1731no—
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#417D17311yescharacteristic, complemented
203C2×#103—1731no—
204C2×#41—3461no—
205C2×#41C2×D17311yescharacteristic
206C22×Q8—1731no—
207C2×SD16—1731no—
208C2×SD16—1731no—
209C2×D8—1731no—
210C2×M16C2×D17311yescharacteristic
211C22×C8—1731no—
212C4⋊Q8—1731no—
213#124—3461no—
214#124—1731no—
215#123—3461no—
216#123—3461no—
217C22⋊Q8—3461no—
218C22⋊Q8—3461no—
219C22⋊Q8—1731no—
220#121—3461no—
221#121—1731no—
222#121—1731no—
223#121—1731no—
224C4×Q8—1731no—
225C4×D4—3461no—
226C4×D4—1731no—
227C4×D4—1731no—
228C2×C4⋊C4—1731no—
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#103C2×D17311yescharacteristic
239#103C2×D17311yescharacteristic
240#102—1731no—
241#102—1731no—
242#102C2×D17311yescharacteristic
243#102C2×D17311yescharacteristic
244C22⋊C8—1731no—
245C22⋊C8—1731no—
246#97—1731no—
247#41—22no—
248#41—41no—
249#41—6924no—
250C2×Q8—3461no—
251C2×Q8—3461no—
252C2×Q8—3462no—
253C2×Q8—1731no—
254C2×Q8—6921no—
255C2×Q8C22×D17311yescharacteristic
256C2×D4—21no—
257C2×D4—3461no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—1731no—
261C2×D4—1731no—
262C2×D4C22×D17311yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—3461no—
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4—1731no—
269C22×C4C22×D17311yescharacteristic
270SD16—3462no—
271SD16—6921no—
272D8—6921no—
273M16—41no—
274C2×C8—1731no—
275C2×C8—1731no—
276C2×C8—6921no—
277C2×C8C22×D17311yescharacteristic
278C2×C8C22×D17311yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—3461no—
281C4⋊C4—3461no—
282C4⋊C4—3461no—
283C4⋊C4—3461no—
284C4⋊C4—1731no—
285C4⋊C4—1731no—
286C4⋊C4—1731no—
287C4⋊C4C22×D17311yescharacteristic
288C4⋊C4C22×D17311yescharacteristic
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—1731no—
297C22⋊C4—1731no—
298C42—3461no—
299C42—1731no—
300C23—21no—
301C23—3461no—
302C23—1731no—
303C23D4×D17311yescharacteristic
304Q8—21no—
305Q8—3461no—
306Q8—3461no—
307Q8—6922no—
308Q8—6922no—
309Q8#1458382612yescomplemented
310D4—21no—
311D4—21no—
312D4—42no—
313D4—3462no—
314D4—3461no—
315D4—3461no—
316D4—6922no—
317D4—6922no—
318D4—6922no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—3461no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—1731no—
332C2×C4—1731no—
333C2×C4—1731no—
334C2×C4—6922no—
335C2×C4—6922no—
336C2×C4—6922no—
337C2×C4D4×D17311yescharacteristic
338C2×C4C23×D17311yescharacteristic
339C8—21no—
340C8—21no—
341C8—3461no—
342C8—3461no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—3461no—
350C22—3461no—
351C22—1732no—
352C22—6921no—
353C22—6921no—
354C22—6921no—
355C22C2×D4×D17311yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—3461no—
361C4—3461no—
362C4—3461no—
363C4—6921no—
364C4—6921no—
365C4—6921no—
366C4C2×#1458382611yescharacteristic
367C4#1914972411yescharacteristic
368C2—41no—
369C2—81no—
370C2—3462no—
371C2—13841no—
372C2#2601213111yescharacteristic
373C2AGL(1,ℤ/8)×D17311yescharacteristic
374C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#417×C173C2
socleC22×C173C22×D4
hypercenter#417D173
hypocenterC173#3308
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #26012148C2semidirect
2. #26012142C2semidirect
3. #26011849C2semidirect
4. #26011826C2semidirect
5. #26011803C2semidirect
6. #26011784C2semidirect
7. #26011737C2semidirect
8. #26011726C2semidirect
9. #26011705C2semidirect
10. #26011636C2semidirect
11. #26011631C2semidirect
12. #26011594C2semidirect
13. #26011560C2semidirect
14. #103×D173C2semidirect
15. #417×C173C2semidirect
140. Q8×C173#41semidirect
187. C173#3308semidirect
202. #417D173semidirect
309. Q8#14583826semidirect

Maximal subgroups

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

subgroupindex
#260121482
#260121422
#260118492
#260118262
#260118032
#260117842
#260117372
#260117262
#260117052
#260116362
#260116312
#260115942
#260115602
#103×D1732
#417×C1732
#3308173