Finite Group Databasebeta

← #36364446

Subgroups — #36364446

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#26012071C211yesmaximal, characteristic
2#26012064C211yesmaximal, characteristic, complemented
3#26011831C211yesmaximal, characteristic, complemented
4#26011825C211yesmaximal, characteristic, complemented
5#26011823C211yesmaximal, characteristic, complemented
6#26011817C211yesmaximal, characteristic, complemented
7#26011737C211yesmaximal, characteristic, complemented
8#26011730C211yesmaximal, characteristic, complemented
9#26011722C211yesmaximal, characteristic, complemented
10#26011656C211yesmaximal, characteristic, complemented
11#26011652C211yesmaximal, characteristic, complemented
12#26011645C211yesmaximal, characteristic, complemented
13#26011631C211yesmaximal, characteristic, complemented
14#26011459C211yesmaximal, characteristic, complemented
15#470×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19C2×#14583816C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21C2×#14583813C2211yescharacteristic
22C22×#13063436C2211yescharacteristic
23#19149678—21no—
24#19149665—21no—
25#19149664—21no—
26#19149661—21no—
27#19149659C2211yescharacteristic
28#19149658C2211yescharacteristic
29#19149657—21no—
30#19149655C2211yescharacteristic
31#19149655C2211yescharacteristic
32#19149654—21no—
33#19149654C2211yescharacteristic
34#19149653C2211yescharacteristic
35#19149652—21no—
36#19149652—21no—
37#19149637C2211yescharacteristic
38C4⋊C4×D173C2211yescharacteristic
39#19149613C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149610C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C2×Q16×C173C2211yescharacteristic
53C2×M16×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#106×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59#14583827—41no—
60#14583827—41no—
61#14583826—41no—
62#14583826—41no—
63C2×Dic346—21no—
64C2×Dic346D411yescharacteristic
65C2×Dic346C2311yescharacteristic
66Q8×D173—41no—
67Q8×D173—41no—
68C2×D692—21no—
69C2×D692—21no—
70C2×D692D411yescharacteristic
71C2×D692C2311yescharacteristic
72C2×C22⋊D173—21no—
73C2×C22⋊D173D411yescharacteristic
74C2×C22⋊D173D411yescharacteristic
75C2×C4×D173—21no—
76C2×C4×D173—21no—
77C2×C4×D173—21no—
78C2×C4×D173C2311yescharacteristic
79C2×C4×D173C2311yescharacteristic
80#14583816—41no—
81#14583815—41no—
82#14583813—41no—
83C2×#13063436—41no—
84C2×#13063436C2311yescharacteristic
85C2×#13063436C2311yescharacteristic
86C4⋊Dic173—21no—
87C4⋊Dic173C2311yescharacteristic
88C4⋊Dic173C2311yescharacteristic
89#14583805—21no—
90#14583805—21no—
91#14583805—21no—
92#14583805—21no—
93C22⋊Dic173—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100C4×Dic173—21no—
101C2×Q8×C173C2311yescharacteristic
102C2×Q8×C173C2311yescharacteristic
103C22×C4×C173C2311yescharacteristic
104Q16×C173—41no—
105M16×C173—41no—
106C2×C8×C173C2311yescharacteristic
107C2×C8×C173C2311yescharacteristic
108C4⋊C4×C173—21no—
109C4⋊C4×C173—21no—
110C4⋊C4×C173C2311yescharacteristic
111C4⋊C4×C173C2311yescharacteristic
112C22⋊C4×C173—21no—
113C22⋊C4×C173—21no—
114C22×D173—21no—
115C22×D173C2×D411yescharacteristic
116C22×D173C2×D411yescharacteristic
117Dic346—21no—
118Dic346—21no—
119Dic346—41no—
120Dic346—42no—
121D692—21no—
122D692—21no—
123D692—41no—
124D692—42no—
125D692—81no—
126C22⋊D173—41no—
127C22⋊D173—41no—
128C22⋊D173—81no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C4×D173—21no—
135C4×D173—21no—
136C4×D173—21no—
137C4×D173—21no—
138C4×D173—42no—
139C4×D173—42no—
140C4×D173—81no—
141#13063436—21no—
142#13063436—21no—
143C23×C173C2×D411yescharacteristic
144Q8×C173—21no—
145Q8×C173—21no—
146Q8×C173—21no—
147Q8×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173C2×D411yescharacteristic
155C2×C4×C173C2411yescharacteristic
156C8×C173—21no—
157C8×C173—21no—
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—21no—
161C22×C173C22×D411yescharacteristic
162C2×D173—21no—
163C2×D173—21no—
164C2×D173—21no—
165C2×D173—21no—
166C2×D173—21no—
167C2×D173—21no—
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C4×C173—21no—
172C4×C173—41no—
173C4×C173—41no—
174C4×C173—41no—
175C4×C173—41no—
176C4×C173H(5,𝔽2)11yescharacteristic
177C4×C173H(5,𝔽2)11yescharacteristic
178Dic173—21no—
179Dic173—21no—
180Dic173—41no—
181Dic173—41no—
182Dic173—41no—
183C2×C173—41no—
184C2×C173#53411yescharacteristic
185C2×C173#57711yescharacteristic
186C2×C173#57811yescharacteristic
187D173—41no—
188D173—41no—
189D173—81no—
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#470D17311yescharacteristic, complemented
199#468—1731no—
200#467—1731no—
201#465—1731no—
202#452—1731no—
203#452—1731no—
204#451—1731no—
205#448—1731no—
206#408—1731no—
207C2×#41—1731no—
208C2×#41—1731no—
209C22×Q8—1731no—
210C2×Q16—1731no—
211C2×Q16C2×D17311yescharacteristic
212C2×SD16—1731no—
213C2×SD16—1731no—
214C2×M16C2×D17311yescharacteristic
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⋊Q8C2×D17311yescharacteristic
226C22⋊Q8C2×D17311yescharacteristic
227#121—3461no—
228#121—3461no—
229#121—3461no—
230#121—1731no—
231#121—1731no—
232#117—1731no—
233C2×C4⋊C4—1731no—
234#107—1731no—
235#107—1731no—
236#106—1731no—
237#106C2×D17311yescharacteristic
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⋊C8—1731no—
250#41—6921no—
251#41—6921no—
252#41—6921no—
253#41—6921no—
254C2×Q8—3461no—
255C2×Q8—1731no—
256C2×Q8—1731no—
257C2×Q8—6921no—
258C2×Q8—6921no—
259C2×Q8C22×D17311yescharacteristic
260C2×Q8C22×D17311yescharacteristic
261C2×D4—3461no—
262C2×D4—3461no—
263C2×D4—3461no—
264C2×D4—1731no—
265C2×D4—1731no—
266C2×D4—1731no—
267C2×D4—1731no—
268C22×C4—3461no—
269C22×C4—3461no—
270C22×C4—3461no—
271C22×C4—1731no—
272C22×C4—1731no—
273C22×C4C22×D17311yescharacteristic
274Q16—41no—
275Q16—6921no—
276SD16—6921no—
277SD16—6921no—
278M16—41no—
279C2×C8—1731no—
280C2×C8—1731no—
281C2×C8—6921no—
282C2×C8C22×D17311yescharacteristic
283C2×C8C22×D17311yescharacteristic
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—3461no—
287C4⋊C4—3461no—
288C4⋊C4—3461no—
289C4⋊C4—3461no—
290C4⋊C4—3461no—
291C4⋊C4—1731no—
292C4⋊C4—1731no—
293C4⋊C4C22×D17311yescharacteristic
294C4⋊C4C22×D17311yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4—3461no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C42—3461no—
305C23—3461no—
306C23—1731no—
307C23—1731no—
308C23D4×D17311yescharacteristic
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8—3461no—
314Q8—3461no—
315Q8—6921no—
316Q8—6922no—
317D4—3461no—
318D4—3461no—
319D4—13841no—
320D4—13841no—
321D4—6921no—
322D4—6922no—
323D4—6921no—
324D4—6921no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—3461no—
336C2×C4—3461no—
337C2×C4—3461no—
338C2×C4—13841no—
339C2×C4—1731no—
340C2×C4—1731no—
341C2×C4—6922no—
342C2×C4—6922no—
343C2×C4D4×D17311yescharacteristic
344C2×C4C23×D17311yescharacteristic
345C8—21no—
346C8—21no—
347C8—3461no—
348C8—3461no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—3461no—
353C22—3461no—
354C22—3461no—
355C22—3461no—
356C22—3461no—
357C22—3461no—
358C22—6921no—
359C22—6921no—
360C22—6921no—
361C22C2×D4×D17311yescharacteristic
362C4—21no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—3461no—
368C4—3461no—
369C4—6921no—
370C4—6921no—
371C4—6921no—
372C4#1914972411yescharacteristic
373C4#1914972511yescharacteristic
374C2—41no—
375C2—13841no—
376C2—6921no—
377C2—6921no—
378C2#2601204911yescharacteristic
379C2#2601230011yescharacteristic
380C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#470×C173C2
socleC22×C173C22×D4
prime radical#26012071C2
hypercenter#470D173
hypocenterC173#3201
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #26012064C2semidirect
3. #26011831C2semidirect
4. #26011825C2semidirect
5. #26011823C2semidirect
6. #26011817C2semidirect
7. #26011737C2semidirect
8. #26011730C2semidirect
9. #26011722C2semidirect
10. #26011656C2semidirect
11. #26011652C2semidirect
12. #26011645C2semidirect
13. #26011631C2semidirect
14. #26011459C2semidirect
15. #470×C173C2semidirect
190. C173#3201semidirect
198. #470D173semidirect

Maximal subgroups

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

subgroupindex
#260120712
#260120642
#260118312
#260118252
#260118232
#260118172
#260117372
#260117302
#260117222
#260116562
#260116522
#260116452
#260116312
#260114592
#470×C1732
#3201173