Finite Group Databasebeta

← #36363389

Subgroups — #36363389

Order 22144.

398 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012118C211yesmaximal, characteristic, complemented
2#26011963C211yesmaximal, characteristic
3#26011793C211yesmaximal, characteristic, complemented
4#26011792C211yesmaximal, characteristic, complemented
5#26011791C211yesmaximal, characteristic, complemented
6#26011790C211yesmaximal, characteristic, complemented
7#26011780C211yesmaximal, characteristic, complemented
8#26011769C211yesmaximal, characteristic, complemented
9#26011768C211yesmaximal, characteristic, complemented
10#26011530C211yesmaximal, characteristic, complemented
11#26011526C211yesmaximal, characteristic, complemented
12#26011516C211yesmaximal, characteristic, complemented
13#26011494C211yesmaximal, characteristic
14#102×D173C211yesmaximal, characteristic
15#476×C173C211yesmaximal, characteristic, complemented
16C2×D4×D173C2211yescharacteristic
17C2×#14583809C2211yescharacteristic
18C2×C8×D173C2211yescharacteristic
19#19149678C2211yescharacteristic, complemented
20#19149677—21no—
21#19149671C2211yescharacteristic, complemented
22#19149670C2211yescharacteristic, complemented
23#19149669—21no—
24#19149657—21no—
25#19149655—21no—
26C4×D692C2211yescharacteristic
27D4×Dic173C2211yescharacteristic
28#19149641C2211yescharacteristic
29C4×C22⋊D173—21no—
30#19149639—21no—
31#19149637C2211yescharacteristic
32#19149637C2211yescharacteristic
33#19149636C2211yescharacteristic
34C4⋊C4×D173C2211yescharacteristic
35C2×#14583803—21no—
36C22⋊C4×D173—21no—
37#19149613C2211yescharacteristic, complemented
38#19149612C2211yescharacteristic, complemented
39#19149612C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41#19149611C2211yescharacteristic, complemented
42#19149610C2211yescharacteristic, complemented
43C4⋊#13063436C2211yescharacteristic, complemented
44#19149608C2211yescharacteristic, complemented
45#19149608C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149600C2211yescharacteristic
50#19149599C2211yescharacteristic
51#19149599C2211yescharacteristic
52C4⋊Q8×C173C2211yescharacteristic, complemented
53C4×D4×C173C2211yescharacteristic
54#106×C173C2211yescharacteristic, complemented
55#106×C173C2211yescharacteristic, complemented
56C4⋊C8×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59C23×D173—21no—
60C2×Dic346—21no—
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63D4×D173—22no—
64D4×D173Q814yescomplemented
65C22×Dic173—21no—
66C2×C4×D173—21no—
67C2×C4×D173—21no—
68C2×C4×D173—21no—
69C2×C4×D173C2311yescharacteristic
70#14583809—22no—
71C2×#13063436C2311yescharacteristic
72C2×#13063436C2311yescharacteristic
73C8×D173—22no—
74C4⋊Dic173—21no—
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805D411yescharacteristic
84#14583805D411yescharacteristic
85C22⋊Dic173—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803D411yescharacteristic
91#14583803D411yescharacteristic
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173C2311yescharacteristic
95C2×Q8×C173—21no—
96C2×D4×C173C2311yescharacteristic
97C22×C4×C173—21no—
98C2×C8×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C4⋊C4×C173—21no—
101C4⋊C4×C173C2311yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C22⋊C4×C173—21no—
105C42×C173C2311yescharacteristic
106C22×D173—21no—
107C22×D173—24no—
108C22×D173—81no—
109C22×D173—81no—
110C22×D173C2×D411yescharacteristic
111Dic346—81no—
112D692—21no—
113D692C2×Q812yes—
114C22⋊D173—24no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—81no—
119C2×Dic173C2×D411yescharacteristic
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C4×D173—81no—
123C4×D173—81no—
124C4×D173—81no—
125C4×D173C2×Q812yes—
126C4×D173#4112yes—
127#13063436—21no—
128#13063436—21no—
129C23×C173—21no—
130Q8×C173—81no—
131D4×C173—21no—
132D4×C173C2×Q812yes—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—81no—
137C2×C4×C173C2×D411yescharacteristic
138C2×C4×C173C2×D411yescharacteristic
139C2×C4×C173C2411yescharacteristic
140C8×C173—21no—
141C8×C173—21no—
142C22×C173—22no—
143C22×C173—41no—
144C22×C173—41no—
145C22×C173C22×D411yescharacteristic
146C2×D173—21no—
147C2×D173—21no—
148C2×D173—22no—
149C2×D173—41no—
150C2×D173—41no—
151C2×D173—84no—
152C2×D173C22⋊Q812yes—
153C4×C173—21no—
154C4×C173—21no—
155C4×C173—41no—
156C4×C173—41no—
157C4×C173—41no—
158C4×C173C22×Q811yescharacteristic
159C4×C173C2×#4111yescharacteristic
160Dic173—21no—
161Dic173—21no—
162Dic173—41no—
163Dic173—41no—
164Dic173—41no—
165Dic173C22⋊Q812yes—
166C2×C173—42no—
167C2×C173C2×C22⋊Q811yescharacteristic
168C2×C173#57611yescharacteristic
169C2×C173#57711yescharacteristic
170D173—22no—
171D173—42no—
172C173#308111yescharacteristic, complemented, Sylow
173#3081—1731nomaximal, Sylow
174#544—1731no—
175#518—1731no—
176#478—1731no—
177#478—1731no—
178#478—1731no—
179#478—1731no—
180#476—1731no—
181#476D17311yescharacteristic, complemented
182#474—1731no—
183#474—1731no—
184#428—1731no—
185#427—1731no—
186#424—1731no—
187#418—1731no—
188C2×#102—1731no—
189C22×D4—1731no—
190C2×M16—1731no—
191C22×C8—1731no—
192C4⋊Q8—3461no—
193C4⋊Q8—1731no—
194C4⋊Q8C2×D17311yescharacteristic, complemented
195#125—3461no—
196#125—1731no—
197#125—1731no—
198C22⋊Q8—3461no—
199C22⋊Q8—3461no—
200C4×D4—3461no—
201C4×D4—3461no—
202C4×D4—1731no—
203C4×D4—1731no—
204C4×D4—1731no—
205C4×D4C2×D17311yescharacteristic
206#117—1731no—
207#117—1731no—
208#117—1731no—
209C2×C4⋊C4—1731no—
210C2×C22⋊C4—3461no—
211C2×C22⋊C4—3461no—
212#107—1731no—
213#107—1731no—
214#107—1731no—
215#107—1731no—
216#106—1731no—
217#106—1731no—
218#106C2×D17311yescharacteristic, complemented
219#106C2×D17311yescharacteristic, complemented
220C4⋊C8—1731no—
221C4⋊C8—1731no—
222C4⋊C8—1731no—
223C4⋊C8C2×D17311yescharacteristic, complemented
224#102—1731no—
225#102—1731no—
226#102—1731no—
227#102—1731no—
228#102—1731no—
229#102—1731no—
230#102C2×D17311yescharacteristic
231#102C2×D17311yescharacteristic
232C24—3461no—
233C2×Q8—21no—
234C2×Q8—3461no—
235C2×D4—3462no—
236C2×D4—3461no—
237C2×D4—1734no—
238C2×D4—1731no—
239C2×D4C22×D17311yescharacteristic
240C22×C4—21no—
241C22×C4—3461no—
242C22×C4—3461no—
243C22×C4—3461no—
244C22×C4—3461no—
245C22×C4—1731no—
246M16—3462no—
247C2×C8—3462no—
248C2×C8—1731no—
249C2×C8—1731no—
250C2×C8C22×D17311yescharacteristic
251C2×C8C22×D17311yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—3461no—
254C4⋊C4—3461no—
255C4⋊C4—3461no—
256C4⋊C4—3461no—
257C4⋊C4—3461no—
258C4⋊C4—3461no—
259C4⋊C4—1731no—
260C4⋊C4—1731no—
261C4⋊C4—1731no—
262C4⋊C4—1731no—
263C4⋊C4—1731no—
264C4⋊C4C22×D17311yescharacteristic
265C4⋊C4C22×D17311yescharacteristic
266C4⋊C4C22×D17311yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—3461no—
269C22⋊C4—3461no—
270C22⋊C4—3461no—
271C22⋊C4—3461no—
272C22⋊C4—3461no—
273C22⋊C4—1731no—
274C22⋊C4—1731no—
275C42—3461no—
276C42—3461no—
277C42—1731no—
278C42C22×D17311yescharacteristic
279C23—21no—
280C23—3464no—
281C23—3461no—
282C23—13841no—
283C23—13841no—
284C23—1731no—
285Q8—81no—
286Q8—13841no—
287D4—21no—
288D4—3464no—
289D4—3461no—
290D4—1732no—
291D4Q8×D17312yes—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—81no—
296C2×C4—3461no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—13841no—
300C2×C4—13841no—
301C2×C4—13841no—
302C2×C4—13841no—
303C2×C4—1731no—
304C2×C4—1731no—
305C2×C4—1731no—
306C2×C4—1732no—
307C2×C4—1732no—
308C2×C4D4×D17311yescharacteristic
309C2×C4D4×D17311yescharacteristic
310C2×C4C23×D17311yescharacteristic
311C8—21no—
312C8—21no—
313C8—3461no—
314C8—3461no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—3461no—
319C22—3461no—
320C22—3462no—
321C22—13844no—
322C22—1732no—
323C22—6921no—
324C22—6921no—
325C22C2×D4×D17311yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—3461no—
332C4—3461no—
333C4—1732no—
334C4—6921no—
335C4—6921no—
336C4—6921no—
337C4C2×Q8×D17311yescharacteristic
338C4#41×D17311yescharacteristic
339C2—42no—
340C2—3462no—
341C2—6922no—
342C2C22⋊Q8×D17311yescharacteristic
343C2#2601229511yescharacteristic
344C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#476×C173C2
socleC22×C173C22×D4
prime radicalC2×D4×D173C22
hypercenter#476D173
hypocenterC173#3081
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #26012118C2semidirect
3. #26011793C2semidirect
4. #26011792C2semidirect
5. #26011791C2semidirect
6. #26011790C2semidirect
7. #26011780C2semidirect
8. #26011769C2semidirect
9. #26011768C2semidirect
10. #26011530C2semidirect
11. #26011526C2semidirect
12. #26011516C2semidirect
15. #476×C173C2semidirect
19. #19149678C22semidirect
21. #19149671C22semidirect
22. #19149670C22semidirect
37. #19149613C22semidirect
38. #19149612C22semidirect
39. #19149612C22semidirect
40. #19149612C22semidirect
41. #19149611C22semidirect
42. #19149610C22semidirect
43. C4⋊#13063436C22semidirect
44. #19149608C22semidirect
45. #19149608C22semidirect
52. C4⋊Q8×C173C22semidirect
54. #106×C173C22semidirect
55. #106×C173C22semidirect
56. C4⋊C8×C173C22semidirect
64. D4×D173Q8semidirect
172. C173#3081semidirect
181. #476D173semidirect
194. C4⋊Q8C2×D173semidirect
218. #106C2×D173semidirect
219. #106C2×D173semidirect
223. C4⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121182
#260119632
#260117932
#260117922
#260117912
#260117902
#260117802
#260117692
#260117682
#260115302
#260115262
#260115162
#260114942
#102×D1732
#476×C1732
#3081173