Finite Group Databasebeta

← #36364564

Subgroups — #36364564

Order 22144.

400 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012045C211yesmaximal, characteristic, complemented
2#124×D173C211yesmaximal, characteristic
3#26011884C211yesmaximal, characteristic, complemented
4#26011882C211yesmaximal, characteristic, complemented
5#26011852C211yesmaximal, characteristic, complemented
6#26011849C211yesmaximal, characteristic, complemented
7#26011714C211yesmaximal, characteristic, complemented
8#26011713C211yesmaximal, characteristic, complemented
9#26011710C211yesmaximal, characteristic, complemented
10#26011648C211yesmaximal, characteristic, complemented
11#26011645C211yesmaximal, characteristic, complemented
12#26011643C211yesmaximal, characteristic
13#26011640C211yesmaximal, characteristic
14#26011547C211yesmaximal, characteristic, complemented
15#464×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C2×Dic692C2211yescharacteristic, complemented
19C2×#14583816C2211yescharacteristic, complemented
20C2×#14583816C2211yescharacteristic, complemented
21C2×#14583815C2211yescharacteristic
22C2×#14583814C2211yescharacteristic
23C2×#14583813C2211yescharacteristic
24#19149678C2211yescharacteristic, complemented
25#19149677C2211yescharacteristic, complemented
26#19149669—21no—
27#19149668C2211yescharacteristic
28#19149667C2211yescharacteristic
29#19149666C2211yescharacteristic
30#19149665—21no—
31#19149665—21no—
32#19149659C2211yescharacteristic
33#19149657—21no—
34#19149655C2211yescharacteristic
35C4×Dic346—21no—
36C4×Dic346C2211yescharacteristic, complemented
37#19149636—21no—
38C22⋊C4×D173—21no—
39C22⋊C4×D173—21no—
40C42×D173C2211yescharacteristic
41C4⋊#13063436C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic, complemented
51#19149589C2211yescharacteristic, complemented
52C2×Q16×C173C2211yescharacteristic, complemented
53C2×SD16×C173C2211yescharacteristic
54#124×C173C2211yescharacteristic
55C4×Q8×C173C2211yescharacteristic, complemented
56C4⋊C8×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic
59C23×D173—21no—
60C2×Dic346—21no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63C2×Dic346C2311yescharacteristic
64Q8×D173—41no—
65Q8×D173—41no—
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68D4×D173—41no—
69D4×D173—41no—
70C2×C4×D173—21no—
71C2×C4×D173D411yescharacteristic
72C2×C4×D173D411yescharacteristic
73C2×C4×D173C2311yescharacteristic
74Dic692—41no—
75#14583816—41no—
76#14583816—41no—
77#14583815—41no—
78#14583814—41no—
79#14583813—41no—
80C2×#13063436C2311yescharacteristic
81C2×#13063436C2311yescharacteristic
82C4⋊Dic173—21no—
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97C4×Dic173—21no—
98C4×Dic173D411yescharacteristic
99C4×Dic173D411yescharacteristic
100C4×Dic173C2311yescharacteristic
101C2×Q8×C173C2311yescharacteristic
102C2×Q8×C173C2311yescharacteristic
103C2×D4×C173C2311yescharacteristic
104Q16×C173—41no—
105SD16×C173—41no—
106C2×C8×C173C2311yescharacteristic
107C2×C8×C173C2311yescharacteristic
108C4⋊C4×C173—21no—
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22⋊C4×C173—21no—
112C42×C173—21no—
113C42×C173C2311yescharacteristic
114C22×D173—21no—
115C22×D173—42no—
116C22×D173—42no—
117C22×D173—42no—
118C22×D173C2×D411yescharacteristic
119Dic346—21no—
120Dic346—21no—
121Dic346—21no—
122Dic346—21no—
123Dic346—42no—
124Dic346—42no—
125D692—21no—
126D692—21no—
127C22⋊D173—42no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C4×D173—22no—
135C4×D173—21no—
136C4×D173—21no—
137C4×D173—41no—
138C4×D173—42no—
139#13063436—21no—
140#13063436—21no—
141C23×C173—21no—
142Q8×C173—21no—
143Q8×C173—21no—
144Q8×C173—21no—
145Q8×C173—21no—
146D4×C173—21no—
147D4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2×D411yescharacteristic
153C2×C4×C173C2411yescharacteristic
154C8×C173—21no—
155C8×C173—21no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173C22×D411yescharacteristic
160C2×D173—21no—
161C2×D173—21no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C2×D173—84no—
166C2×D173#13712yes—
167C4×C173—21no—
168C4×C173—21no—
169C4×C173—41no—
170C4×C173—41no—
171C4×C173—41no—
172C4×C173H(5,𝔽2)11yescharacteristic
173C4×C173H(5,𝔽2)11yescharacteristic
174Dic173—21no—
175Dic173—21no—
176Dic173—21no—
177Dic173—41no—
178Dic173—41no—
179Dic173—41no—
180C2×C173—81no—
181C2×C173#53511yescharacteristic
182C2×C173C2×#13711yescharacteristic
183C2×C173#57511yescharacteristic
184D173—22no—
185D173—81no—
186C173#321511yescharacteristic, complemented, Sylow
187#3215—1731nomaximal, Sylow
188#533—1731no—
189C2×#124—1731no—
190#497—1731no—
191#497—1731no—
192#489—1731no—
193#489—1731no—
194#464—1731no—
195#464—1731no—
196#464—1731no—
197#464D17311yescharacteristic, complemented
198#451—1731no—
199#451—1731no—
200#450—1731no—
201#450—1731no—
202#432—1731no—
203C22×Q8—1731no—
204C22×D4—1731no—
205C2×Q16—1731no—
206C2×Q16—1731no—
207C2×Q16—1731no—
208C2×Q16C2×D17311yescharacteristic, complemented
209C2×SD16—1731no—
210C2×SD16—1731no—
211C2×SD16—1731no—
212C2×SD16C2×D17311yescharacteristic
213C4⋊Q8—1731no—
214C4⋊Q8—1731no—
215#125—3461no—
216#124—3461no—
217#124—3461no—
218#124—1731no—
219#124—1731no—
220#124—1731no—
221#124C2×D17311yescharacteristic
222C22⋊Q8—3461no—
223C22⋊Q8—1731no—
224C22⋊Q8—1731no—
225C4×Q8—3461no—
226C4×Q8—1731no—
227C4×Q8C2×D17311yescharacteristic, complemented
228#117—3461no—
229C2×C22⋊C4—3461no—
230C2×C22⋊C4—3461no—
231C2×C42—1731no—
232C4⋊C8—1731no—
233C4⋊C8C2×D17311yescharacteristic, complemented
234#103—1731no—
235#103—1731no—
236#103—1731no—
237#103C2×D17311yescharacteristic, complemented
238#102—1731no—
239#102—1731no—
240#102—1731no—
241#102C2×D17311yescharacteristic
242C22⋊C8—1731no—
243C22⋊C8—1731no—
244#97—1731no—
245#97—1731no—
246C24—3461no—
247C2×Q8—3461no—
248C2×Q8—3461no—
249C2×Q8—1731no—
250C2×Q8—1731no—
251C2×Q8—6921no—
252C2×Q8—6921no—
253C2×Q8C22×D17311yescharacteristic
254C2×Q8C22×D17311yescharacteristic
255C2×D4—3461no—
256C2×D4—1731no—
257C2×D4—6921no—
258C2×D4—6921no—
259C2×D4C22×D17311yescharacteristic
260C22×C4—3461no—
261C22×C4—1731no—
262C22×C4—1731no—
263C22×C4—1731no—
264Q16—41no—
265Q16—6921no—
266Q16—6921no—
267Q16—6921no—
268SD16—41no—
269SD16—6921no—
270SD16—6921no—
271SD16—6921no—
272C2×C8—1731no—
273C2×C8—1731no—
274C2×C8C22×D17311yescharacteristic
275C2×C8C22×D17311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—3461no—
281C4⋊C4—3461no—
282C4⋊C4—3461no—
283C4⋊C4—1731no—
284C4⋊C4C22×D17311yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C42—21no—
296C42—3461no—
297C42—1731no—
298C42—1731no—
299C42—1731no—
300C42C22×D17311yescharacteristic
301C23—21no—
302C23—3461no—
303C23—1731no—
304C23—6922no—
305C23—6922no—
306C23—6922no—
307Q8—21no—
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—3461no—
312Q8—3461no—
313Q8—3461no—
314Q8—3461no—
315Q8—6922no—
316Q8—6922no—
317D4—21no—
318D4—21no—
319D4—3461no—
320D4—3461no—
321D4—6922no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—3462no—
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—6921no—
335C2×C4—6922no—
336C2×C4D4×D17311yescharacteristic
337C2×C4D4×D17311yescharacteristic
338C2×C4C23×D17311yescharacteristic
339C8—21no—
340C8—21no—
341C8—3461no—
342C8—3461no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—3461no—
347C22—3461no—
348C22—13844no—
349C22—1732no—
350C22—6921no—
351C22—6921no—
352C22—6921no—
353C22C2×D4×D17311yescharacteristic
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—3461no—
360C4—3461no—
361C4—3461no—
362C4—6921no—
363C4—6921no—
364C4—6921no—
365C4#1914972411yescharacteristic
366C4#1914972511yescharacteristic
367C2—81no—
368C2—3462no—
369C2—13841no—
370C2#2601205711yescharacteristic
371C2#137×D17311yescharacteristic
372C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#464×C173C2
socleC22×C173C22×D4
prime radicalC2×D4×D173C22
hypercenter#464D173
hypocenterC173#3215
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #26012045C2semidirect
3. #26011884C2semidirect
4. #26011882C2semidirect
5. #26011852C2semidirect
6. #26011849C2semidirect
7. #26011714C2semidirect
8. #26011713C2semidirect
9. #26011710C2semidirect
10. #26011648C2semidirect
11. #26011645C2semidirect
14. #26011547C2semidirect
15. #464×C173C2semidirect
18. C2×Dic692C22semidirect
19. C2×#14583816C22semidirect
20. C2×#14583816C22semidirect
24. #19149678C22semidirect
25. #19149677C22semidirect
36. C4×Dic346C22semidirect
41. C4⋊#13063436C22semidirect
42. #19149604C22semidirect
43. #19149603C22semidirect
44. #19149602C22semidirect
50. #19149589C22semidirect
51. #19149589C22semidirect
52. C2×Q16×C173C22semidirect
55. C4×Q8×C173C22semidirect
56. C4⋊C8×C173C22semidirect
57. #103×C173C22semidirect
186. C173#3215semidirect
197. #464D173semidirect
208. C2×Q16C2×D173semidirect
227. C4×Q8C2×D173semidirect
233. C4⋊C8C2×D173semidirect
237. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120452
#124×D1732
#260118842
#260118822
#260118522
#260118492
#260117142
#260117132
#260117102
#260116482
#260116452
#260116432
#260116402
#260115472
#464×C1732
#3215173