Finite Group Databasebeta

← #36364604

Subgroups — #36364604

Order 22144.

396 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012037C211yesmaximal, characteristic, complemented
2#26012027C211yesmaximal, characteristic, complemented
3#26011884C211yesmaximal, characteristic, complemented
4#26011877C211yesmaximal, characteristic, complemented
5#26011858C211yesmaximal, characteristic, complemented
6#26011854C211yesmaximal, characteristic, complemented
7#26011715C211yesmaximal, characteristic, complemented
8#26011711C211yesmaximal, characteristic, complemented
9#26011709C211yesmaximal, characteristic, complemented
10#26011705C211yesmaximal, characteristic, complemented
11#26011651C211yesmaximal, characteristic, complemented
12#26011638C211yesmaximal, characteristic, complemented
13#26011636C211yesmaximal, characteristic, complemented
14#26011545C211yesmaximal, characteristic, complemented
15#452×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21C2×D1384C2211yescharacteristic
22C2×#14583811C2211yescharacteristic
23#19149677C2211yescharacteristic
24#19149675C2211yescharacteristic
25#19149674—21no—
26#19149670C2211yescharacteristic
27#19149668C2211yescharacteristic
28#19149668C2211yescharacteristic
29#19149666C2211yescharacteristic
30#19149663—21no—
31#19149653C2211yescharacteristic
32#19149651—21no—
33#19149651—21no—
34Q8×Dic173C2211yescharacteristic
35#19149641C2211yescharacteristic
36C4×C22⋊D173—21no—
37C4×C22⋊D173—21no—
38#19149639—21no—
39#19149637—21no—
40C2×C4×Dic173C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149589C2211yescharacteristic
51#19149589C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—41no—
60#14583826—41no—
61C2×Dic346C2311yescharacteristic
62C2×D692C2311yescharacteristic
63C2×D692C2311yescharacteristic
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C22×Dic173D411yescharacteristic
69C22×Dic173D411yescharacteristic
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72#14583816—41no—
73#14583815—41no—
74#14583813—41no—
75#14583813—41no—
76D1384—41no—
77#14583811—41no—
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C4⋊Dic173—21no—
81C4⋊Dic173—21no—
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94C4×Dic173—21no—
95C4×Dic173D411yescharacteristic
96C4×Dic173D411yescharacteristic
97C4×Dic173C2311yescharacteristic
98C4×Dic173C2311yescharacteristic
99#41×C173—41no—
100#41×C173—41no—
101C2×Q8×C173C2311yescharacteristic
102C2×Q8×C173C2311yescharacteristic
103C2×D4×C173—21no—
104C2×D4×C173C2311yescharacteristic
105C22×C4×C173—21no—
106C22×C4×C173C2311yescharacteristic
107Q16×C173—41no—
108SD16×C173—41no—
109C2×C8×C173C2311yescharacteristic
110C2×C8×C173C2311yescharacteristic
111C4⋊C4×C173—21no—
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22×D173—21no—
115C22×D173—21no—
116Dic346—21no—
117Dic346—21no—
118D692—21no—
119D692—21no—
120D692—21no—
121D692—21no—
122C22⋊D173—42no—
123C22⋊D173—42no—
124C22⋊D173—42no—
125C22⋊D173—42no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—41no—
129C2×Dic173—41no—
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C4×D173—42no—
135C4×D173—42no—
136#13063436—21no—
137#13063436—21no—
138C23×C173—21no—
139C23×C173C2×D411yescharacteristic
140Q8×C173—21no—
141Q8×C173—21no—
142Q8×C173—21no—
143Q8×C173—21no—
144D4×C173—21no—
145D4×C173—21no—
146D4×C173—42no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—42no—
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2411yescharacteristic
155C8×C173—21no—
156C8×C173—21no—
157C22×C173—21no—
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173C22×D411yescharacteristic
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C4×C173—21no—
171C4×C173—41no—
172C4×C173—41no—
173C4×C173—41no—
174C4×C173H(5,𝔽2)11yescharacteristic
175C4×C173H(5,𝔽2)11yescharacteristic
176Dic173—21no—
177Dic173—21no—
178Dic173—21no—
179Dic173—21no—
180Dic173—41no—
181Dic173—41no—
182C2×C173—41no—
183C2×C173—81no—
184C2×C173#53511yescharacteristic
185C2×C173#57511yescharacteristic
186C2×C173#57511yescharacteristic
187D173—81no—
188D173—81no—
189C173#321911yescharacteristic, complemented, Sylow
190#3219—1731nomaximal, Sylow
191#532—1731no—
192#529—1731no—
193#497—1731no—
194#496—1731no—
195#491—1731no—
196#490—1731no—
197#464—1731no—
198#464—1731no—
199#463—1731no—
200#463—1731no—
201#452—1731no—
202#452D17311yescharacteristic, complemented
203#449—1731no—
204#449—1731no—
205#432—1731no—
206C2×#41—1731no—
207C2×#41C2×D17311yescharacteristic
208C2×Q16—1731no—
209C2×Q16C2×D17311yescharacteristic
210C2×SD16—1731no—
211C2×SD16—1731no—
212C2×SD16—1731no—
213C2×SD16C2×D17311yescharacteristic
214C2×D8—1731no—
215C2×D8—1731no—
216C4⋊Q8—1731no—
217#127—1731no—
218#126—3461no—
219#125—1731no—
220#124—1731no—
221#124—1731no—
222#124—1731no—
223#123—3461no—
224C22⋊Q8C2×D17311yescharacteristic
225#121—3461no—
226#121—3461no—
227#121—1731no—
228C4×Q8—1731no—
229C4×D4—3461no—
230C4×D4—3461no—
231C4×D4—3461no—
232C4×D4—1731no—
233#117—3461no—
234C2×C42—1731no—
235C4⋊C8—1731no—
236C4⋊C8—1731no—
237#103—1731no—
238#103—1731no—
239#103—1731no—
240#103C2×D17311yescharacteristic
241#102—1731no—
242#102—1731no—
243#102—1731no—
244#102C2×D17311yescharacteristic
245C22⋊C8—1731no—
246C22⋊C8C2×D17311yescharacteristic
247#97—1731no—
248#97—1731no—
249#41—41no—
250#41—41no—
251#41—6921no—
252#41—6921no—
253C2×Q8—1731no—
254C2×Q8C22×D17311yescharacteristic
255C2×Q8C22×D17311yescharacteristic
256C2×D4—21no—
257C2×D4—3461no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—3461no—
261C2×D4—1731no—
262C2×D4—1731no—
263C2×D4C22×D17311yescharacteristic
264C22×C4—21no—
265C22×C4—3461no—
266C22×C4—3461no—
267C22×C4—1731no—
268C22×C4—1731no—
269C22×C4C22×D17311yescharacteristic
270Q16—41no—
271Q16—6921no—
272SD16—41no—
273SD16—6921no—
274SD16—6921no—
275SD16—6921no—
276D8—6921no—
277D8—6921no—
278C2×C8—1731no—
279C2×C8—1731no—
280C2×C8C22×D17311yescharacteristic
281C2×C8C22×D17311yescharacteristic
282C4⋊C4—21no—
283C4⋊C4—3461no—
284C4⋊C4—3461no—
285C4⋊C4—3461no—
286C4⋊C4—3461no—
287C4⋊C4—3461no—
288C4⋊C4—1731no—
289C4⋊C4C22×D17311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C42—3461no—
300C42—1731no—
301C42—1731no—
302C42—1731no—
303C42—1731no—
304C23—21no—
305C23—3461no—
306C23—3461no—
307C23D4×D17311yescharacteristic
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—3461no—
313Q8—3461no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—3461no—
318D4—3461no—
319D4—3461no—
320D4—3461no—
321D4—6922no—
322D4—6922no—
323D4—6922no—
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—1731no—
334C2×C4—1731no—
335C2×C4—1731no—
336C2×C4—1731no—
337C2×C4—6922no—
338C2×C4—6922no—
339C2×C4—6921no—
340C2×C4—6921no—
341C2×C4D4×D17311yescharacteristic
342C2×C4C23×D17311yescharacteristic
343C8—21no—
344C8—21no—
345C8—3461no—
346C8—3461no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—6921no—
354C22—6921no—
355C22—6921no—
356C22—6921no—
357C22—6921no—
358C22—6921no—
359C22C2×D4×D17311yescharacteristic
360C4—21no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—3461no—
365C4—3461no—
366C4—3461no—
367C4—3461no—
368C4—6921no—
369C4—6921no—
370C4#1914972411yescharacteristic
371C4#1914972411yescharacteristic
372C2—41no—
373C2—81no—
374C2—13841no—
375C2—13841no—
376C2#2601205611yescharacteristic
377C2#2601228811yescharacteristic
378C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#452×C173C2
socleC22×C173C22×D4
hypercenter#452D173
hypocenterC173#3219
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #26012037C2semidirect
2. #26012027C2semidirect
3. #26011884C2semidirect
4. #26011877C2semidirect
5. #26011858C2semidirect
6. #26011854C2semidirect
7. #26011715C2semidirect
8. #26011711C2semidirect
9. #26011709C2semidirect
10. #26011705C2semidirect
11. #26011651C2semidirect
12. #26011638C2semidirect
13. #26011636C2semidirect
14. #26011545C2semidirect
15. #452×C173C2semidirect
189. C173#3219semidirect
202. #452D173semidirect

Maximal subgroups

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

subgroupindex
#260120372
#260120272
#260118842
#260118772
#260118582
#260118542
#260117152
#260117112
#260117092
#260117052
#260116512
#260116382
#260116362
#260115452
#452×C1732
#3219173