Finite Group Databasebeta

← #36364456

Subgroups — #36364456

Order 22144.

378 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012065C211yesmaximal, characteristic
2#26012064C211yesmaximal, characteristic
3#26011833C211yesmaximal, characteristic, complemented
4#26011831C211yesmaximal, characteristic, complemented
5#26011830C211yesmaximal, characteristic, complemented
6#26011829C211yesmaximal, characteristic, complemented
7#26011748C211yesmaximal, characteristic, complemented
8#26011746C211yesmaximal, characteristic, complemented
9#26011731C211yesmaximal, characteristic, complemented
10#26011729C211yesmaximal, characteristic, complemented
11#26011647C211yesmaximal, characteristic, complemented
12#26011645C211yesmaximal, characteristic, complemented
13#26011645C211yesmaximal, characteristic, complemented
14#26011456C211yesmaximal, characteristic
15#451×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583816C2211yescharacteristic
20C2×#14583809C2211yescharacteristic
21C2×C8×D173C2211yescharacteristic
22#19149678—21no—
23#19149677—21no—
24#19149668—21no—
25#19149665—21no—
26#19149664—21no—
27#19149662—21no—
28#19149662—21no—
29#19149661—21no—
30#19149659C2211yescharacteristic
31#19149658C2211yescharacteristic
32#19149655C2211yescharacteristic
33#19149655C2211yescharacteristic
34#19149655C2211yescharacteristic
35#19149655C2211yescharacteristic
36#19149655C2211yescharacteristic
37#19149638C2211yescharacteristic
38#19149637C2211yescharacteristic
39#19149613C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149611C2211yescharacteristic
42#19149610C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149603C2211yescharacteristic
48#19149603C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C22×Q8×C173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic
54C2×Q16×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—41no—
60#14583826—41no—
61C2×Dic346—21no—
62C2×Dic346D411yescharacteristic
63C2×Dic346C2311yescharacteristic
64Q8×D173—41no—
65Q8×D173—41no—
66C2×D692D411yescharacteristic
67C2×C22⋊D173D411yescharacteristic
68C2×C22⋊D173D411yescharacteristic
69C2×C4×D173—21no—
70C2×C4×D173C2311yescharacteristic
71C2×C4×D173C2311yescharacteristic
72#14583816—41no—
73#14583816—41no—
74#14583809—41no—
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C8×D173—41no—
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97C4×Dic173—21no—
98C4×Dic173—21no—
99C2×Q8×C173—21no—
100C2×Q8×C173—41no—
101C2×Q8×C173—41no—
102C2×Q8×C173C2311yescharacteristic
103C2×Q8×C173C2311yescharacteristic
104C2×Q8×C173C2311yescharacteristic
105C22×C4×C173—21no—
106C22×C4×C173C2311yescharacteristic
107Q16×C173—41no—
108Q16×C173—41no—
109C2×C8×C173C2311yescharacteristic
110C2×C8×C173C2311yescharacteristic
111C4⋊C4×C173—21no—
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22×D173C2×D411yescharacteristic
115C22×D173C2×D411yescharacteristic
116Dic346—21no—
117Dic346—21no—
118Dic346—41no—
119Dic346—42no—
120D692—41no—
121C22⋊D173—41no—
122C22⋊D173—41no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—21no—
130C4×D173—21no—
131C4×D173—21no—
132C4×D173—21no—
133C4×D173—42no—
134#13063436—21no—
135#13063436—21no—
136C23×C173C2×D411yescharacteristic
137Q8×C173—21no—
138Q8×C173—21no—
139Q8×C173—21no—
140Q8×C173—21no—
141Q8×C173—21no—
142Q8×C173—21no—
143Q8×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—21no—
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×C173C22×D411yescharacteristic
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—21no—
162C2×D173—21no—
163C2×D173—21no—
164C2×D173—21no—
165C4×C173—21no—
166C4×C173—41no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173—41no—
170C4×C173H(5,𝔽2)11yescharacteristic
171C4×C173H(5,𝔽2)11yescharacteristic
172Dic173—21no—
173Dic173—21no—
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177Dic173—41no—
178C2×C173—41no—
179C2×C173#53411yescharacteristic
180C2×C173#57811yescharacteristic
181C2×C173#57811yescharacteristic
182D173—41no—
183D173—41no—
184C173#320211yescharacteristic, complemented, Sylow
185#3202—1731nomaximal, Sylow
186#536—1731no—
187#536—1731no—
188#485—1731no—
189#485—1731no—
190#485—1731no—
191#485—1731no—
192#470—1731no—
193#470—1731no—
194#467—1731no—
195#467—1731no—
196#451—1731no—
197#451—1731no—
198#451—1731no—
199#451D17311yescharacteristic, complemented
200#408—1731no—
201C2×#41—1731no—
202C22×Q8—1731no—
203C22×Q8C2×D17311yescharacteristic
204C2×Q16—1731no—
205C2×Q16—1731no—
206C2×Q16C2×D17311yescharacteristic
207C2×Q16C2×D17311yescharacteristic
208C2×M16—1731no—
209C22×C8—1731no—
210C4⋊Q8—3461no—
211C4⋊Q8—3461no—
212#124—3461no—
213#124—3461no—
214#123—3461no—
215#123—3461no—
216#123—3461no—
217#123—3461no—
218C22⋊Q8—1731no—
219C22⋊Q8—1731no—
220C22⋊Q8—1731no—
221C22⋊Q8—1731no—
222C22⋊Q8—1731no—
223C22⋊Q8—1731no—
224C22⋊Q8—1731no—
225C22⋊Q8C2×D17311yescharacteristic
226#117—1731no—
227#117—1731no—
228#107—1731no—
229#107—1731no—
230#106—1731no—
231#106—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103—1731no—
236#103—1731no—
237#103—1731no—
238#103C2×D17311yescharacteristic
239#103C2×D17311yescharacteristic
240C22⋊C8—1731no—
241C22⋊C8—1731no—
242C22⋊C8—1731no—
243C22⋊C8C2×D17311yescharacteristic
244#41—6921no—
245#41—6921no—
246C2×Q8—21no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—3461no—
250C2×Q8—1731no—
251C2×Q8—1731no—
252C2×Q8—6921no—
253C2×Q8—6921no—
254C2×Q8C22×D17311yescharacteristic
255C2×Q8C22×D17311yescharacteristic
256C2×Q8C22×D17311yescharacteristic
257C2×D4—1731no—
258C2×D4—1731no—
259C2×D4—1731no—
260C22×C4—21no—
261C22×C4—3461no—
262C22×C4—1731no—
263C22×C4—1731no—
264C22×C4C22×D17311yescharacteristic
265Q16—41no—
266Q16—41no—
267Q16—6921no—
268Q16—6921no—
269M16—6921no—
270C2×C8—1731no—
271C2×C8—1731no—
272C2×C8—6921no—
273C2×C8C22×D17311yescharacteristic
274C2×C8C22×D17311yescharacteristic
275C4⋊C4—21no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—3461no—
281C4⋊C4—3461no—
282C4⋊C4—3461no—
283C4⋊C4—1731no—
284C4⋊C4—1731no—
285C4⋊C4—1731no—
286C4⋊C4C22×D17311yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C42—3461no—
298C42—3461no—
299C23—1731no—
300C23—1731no—
301C23D4×D17311yescharacteristic
302Q8—21no—
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308Q8—42no—
309Q8—3461no—
310Q8—3461no—
311Q8—6921no—
312Q8—6922no—
313D4—6921no—
314D4—6921no—
315D4—6921no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—3461no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—1731no—
332C2×C4—1731no—
333C2×C4—6922no—
334C2×C4D4×D17311yescharacteristic
335C2×C4C23×D17311yescharacteristic
336C8—21no—
337C8—21no—
338C8—3461no—
339C8—3461no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—3461no—
344C22—3461no—
345C22—3461no—
346C22—3461no—
347C22—3461no—
348C22—3461no—
349C22C2×D4×D17311yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—3461no—
356C4—3461no—
357C4—6921no—
358C4—6921no—
359C4—6921no—
360C4—6921no—
361C4#1914972411yescharacteristic
362C4#1914972411yescharacteristic
363C2—41no—
364C2—6921no—
365C2—6921no—
366C2#2601205111yescharacteristic
367C2#2601230411yescharacteristic
368C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#451×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583826C22
hypercenter#451D173
hypocenterC173#3202
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #26011833C2semidirect
4. #26011831C2semidirect
5. #26011830C2semidirect
6. #26011829C2semidirect
7. #26011748C2semidirect
8. #26011746C2semidirect
9. #26011731C2semidirect
10. #26011729C2semidirect
11. #26011647C2semidirect
12. #26011645C2semidirect
13. #26011645C2semidirect
15. #451×C173C2semidirect
184. C173#3202semidirect
199. #451D173semidirect

Maximal subgroups

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

subgroupindex
#260120652
#260120642
#260118332
#260118312
#260118302
#260118292
#260117482
#260117462
#260117312
#260117292
#260116472
#260116452
#260114562
#451×C1732
#3202173