Finite Group Databasebeta

← #36364441

Subgroups — #36364441

Order 22144.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012070C211yesmaximal, characteristic
2#26012065C211yesmaximal, characteristic, complemented
3#26011833C211yesmaximal, characteristic, complemented
4#26011825C211yesmaximal, characteristic, complemented
5#26011820C211yesmaximal, characteristic, complemented
6#26011812C211yesmaximal, characteristic, complemented
7#26011748C211yesmaximal, characteristic, complemented
8#26011734C211yesmaximal, characteristic, complemented
9#26011731C211yesmaximal, characteristic, complemented
10#26011717C211yesmaximal, characteristic, complemented
11#26011650C211yesmaximal, characteristic, complemented
12#26011650C211yesmaximal, characteristic, complemented
13#26011647C211yesmaximal, characteristic, complemented
14#26011456C211yesmaximal, characteristic, complemented
15#448×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583816C2211yescharacteristic
20C2×#14583809C2211yescharacteristic
21C2×C8×D173C2211yescharacteristic
22#19149677—21no—
23#19149668—21no—
24#19149662—21no—
25#19149662—21no—
26#19149659C2211yescharacteristic
27#19149658C2211yescharacteristic
28#19149656—21no—
29#19149655C2211yescharacteristic
30#19149655C2211yescharacteristic
31#19149655C2211yescharacteristic
32#19149655C2211yescharacteristic
33#19149652—21no—
34#19149651—21no—
35#19149650—21no—
36#19149650C2211yescharacteristic
37#19149638C2211yescharacteristic
38C4⋊C4×D173C2211yescharacteristic
39#19149613C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149611C2211yescharacteristic
42#19149610C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C22×Q8×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—41no—
60#14583826—41no—
61#14583825—41no—
62#14583825—41no—
63C2×Dic346D411yescharacteristic
64C2×Dic346C2311yescharacteristic
65C2×D692D411yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173D411yescharacteristic
69C2×C22⋊D173D411yescharacteristic
70C22×Dic173—21no—
71C2×C4×D173—21no—
72C2×C4×D173C2311yescharacteristic
73C2×C4×D173C2311yescharacteristic
74#14583816—41no—
75#14583816—41no—
76#14583809—41no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C8×D173—41no—
80C4⋊Dic173—21no—
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98C4×Dic173—21no—
99C2×Q8×C173—21no—
100C2×Q8×C173—41no—
101C2×Q8×C173—41no—
102C2×Q8×C173C2311yescharacteristic
103C2×Q8×C173C2311yescharacteristic
104C2×D4×C173—21no—
105C2×D4×C173C2311yescharacteristic
106C22×C4×C173—21no—
107C22×C4×C173C2311yescharacteristic
108SD16×C173—41no—
109SD16×C173—41no—
110C2×C8×C173C2311yescharacteristic
111C2×C8×C173C2311yescharacteristic
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22×D173C2×D411yescharacteristic
115C22×D173C2×D411yescharacteristic
116Dic346—21no—
117Dic346—21no—
118Dic346—41no—
119D692—41no—
120C22⋊D173—41no—
121C22⋊D173—41no—
122C22⋊D173—42no—
123C22⋊D173—81no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—42no—
129C2×Dic173C2×D411yescharacteristic
130C2×Dic173C2×D411yescharacteristic
131C4×D173—21no—
132C4×D173—21no—
133C4×D173—21no—
134C4×D173—21no—
135C4×D173—81no—
136#13063436—21no—
137#13063436—21no—
138C23×C173—21no—
139C23×C173C2×D411yescharacteristic
140Q8×C173—21no—
141Q8×C173—21no—
142Q8×C173—21no—
143Q8×C173—21no—
144Q8×C173—42no—
145D4×C173—21no—
146D4×C173—21no—
147D4×C173—81no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—42no—
154C2×C4×C173C2×D411yescharacteristic
155C2×C4×C173C2411yescharacteristic
156C8×C173—21no—
157C8×C173—21no—
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—21no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173C22×D411yescharacteristic
165C2×D173—21no—
166C2×D173—21no—
167C2×D173—21no—
168C2×D173—21no—
169C2×D173—21no—
170C2×D173—21no—
171C4×C173—21no—
172C4×C173—41no—
173C4×C173—41no—
174C4×C173—41no—
175C4×C173H(5,𝔽2)11yescharacteristic
176C4×C173H(5,𝔽2)11yescharacteristic
177Dic173—21no—
178Dic173—21no—
179Dic173—41no—
180Dic173—41no—
181Dic173—41no—
182Dic173—41no—
183C2×C173—41no—
184C2×C173—81no—
185C2×C173#53411yescharacteristic
186C2×C173#57711yescharacteristic
187C2×C173#57811yescharacteristic
188D173—41no—
189D173—41no—
190C173#320111yescharacteristic, complemented, Sylow
191#3201—1731nomaximal, Sylow
192#537—1731no—
193#536—1731no—
194#485—1731no—
195#484—1731no—
196#483—1731no—
197#482—1731no—
198#470—1731no—
199#468—1731no—
200#467—1731no—
201#465—1731no—
202#452—1731no—
203#452—1731no—
204#451—1731no—
205#448D17311yescharacteristic, complemented
206#408—1731no—
207C2×#41—1731no—
208C2×#41—1731no—
209C22×Q8C2×D17311yescharacteristic
210C2×Q16—1731no—
211C2×Q16—1731no—
212C2×SD16C2×D17311yescharacteristic
213C2×SD16C2×D17311yescharacteristic
214C2×M16—1731no—
215C22×C8—1731no—
216C4⋊Q8—3461no—
217#124—3461no—
218#123—3461no—
219#123—3461no—
220C22⋊Q8—3461no—
221C22⋊Q8—1731no—
222C22⋊Q8—1731no—
223C22⋊Q8—1731no—
224C22⋊Q8—1731no—
225C22⋊Q8—1731no—
226C22⋊Q8—1731no—
227#121—3461no—
228#121—3461no—
229#121—3461no—
230#121—1731no—
231#121C2×D17311yescharacteristic
232#117—1731no—
233C2×C4⋊C4—1731no—
234#107—1731no—
235#107—1731no—
236#106—1731no—
237#106—1731no—
238#103—1731no—
239#103—1731no—
240#103—1731no—
241#103—1731no—
242#103C2×D17311yescharacteristic
243#103C2×D17311yescharacteristic
244#102—1731no—
245#102—1731no—
246C22⋊C8—1731no—
247C22⋊C8—1731no—
248C22⋊C8—1731no—
249C22⋊C8C2×D17311yescharacteristic
250#41—6921no—
251#41—6921no—
252#41—6921no—
253#41—6921no—
254C2×Q8—21no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—1731no—
258C2×Q8—1731no—
259C2×Q8C22×D17311yescharacteristic
260C2×Q8C22×D17311yescharacteristic
261C2×D4—21no—
262C2×D4—3461no—
263C2×D4—3461no—
264C2×D4—1731no—
265C2×D4—1731no—
266C2×D4—1731no—
267C2×D4C22×D17311yescharacteristic
268C22×C4—21no—
269C22×C4—3461no—
270C22×C4—3461no—
271C22×C4—1731no—
272C22×C4—1731no—
273C22×C4C22×D17311yescharacteristic
274Q16—6921no—
275Q16—6921no—
276SD16—41no—
277SD16—41no—
278M16—6921no—
279C2×C8—1731no—
280C2×C8—1731no—
281C2×C8—6921no—
282C2×C8C22×D17311yescharacteristic
283C2×C8C22×D17311yescharacteristic
284C4⋊C4—3461no—
285C4⋊C4—3461no—
286C4⋊C4—3461no—
287C4⋊C4—3461no—
288C4⋊C4—3461no—
289C4⋊C4—3461no—
290C4⋊C4—3461no—
291C4⋊C4—1731no—
292C4⋊C4—1731no—
293C4⋊C4—1731no—
294C4⋊C4C22×D17311yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—3461no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4—3461no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C42—3461no—
305C23—21no—
306C23—1731no—
307C23—1731no—
308C23D4×D17311yescharacteristic
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8—42no—
314Q8—3461no—
315Q8—3461no—
316Q8—6921no—
317D4—21no—
318D4—21no—
319D4—81no—
320D4—13841no—
321D4—6921no—
322D4—6921no—
323D4—6921no—
324D4—6922no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—3461no—
336C2×C4—3461no—
337C2×C4—3461no—
338C2×C4—3461no—
339C2×C4—13841no—
340C2×C4—1731no—
341C2×C4—1731no—
342C2×C4—6922no—
343C2×C4D4×D17311yescharacteristic
344C2×C4C23×D17311yescharacteristic
345C8—21no—
346C8—21no—
347C8—3461no—
348C8—3461no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—3461no—
356C22—3461no—
357C22—3461no—
358C22—3461no—
359C22—3461no—
360C22—3461no—
361C22C2×D4×D17311yescharacteristic
362C4—21no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—3461no—
367C4—3461no—
368C4—6921no—
369C4—6921no—
370C4—6921no—
371C4—6921no—
372C4#1914972411yescharacteristic
373C4#1914972411yescharacteristic
374C2—41no—
375C2—81no—
376C2—6921no—
377C2—6921no—
378C2#2601205111yescharacteristic
379C2#2601229711yescharacteristic
380C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#448×C173C2
socleC22×C173C22×D4
prime radical#26012070C2
hypercenter#448D173
hypocenterC173#3201
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #26012065C2semidirect
3. #26011833C2semidirect
4. #26011825C2semidirect
5. #26011820C2semidirect
6. #26011812C2semidirect
7. #26011748C2semidirect
8. #26011734C2semidirect
9. #26011731C2semidirect
10. #26011717C2semidirect
11. #26011650C2semidirect
12. #26011650C2semidirect
13. #26011647C2semidirect
14. #26011456C2semidirect
15. #448×C173C2semidirect
190. C173#3201semidirect
205. #448D173semidirect

Maximal subgroups

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

subgroupindex
#260120702
#260120652
#260118332
#260118252
#260118202
#260118122
#260117482
#260117342
#260117312
#260117172
#260116502
#260116472
#260114562
#448×C1732
#3201173