Finite Group Databasebeta

← #36370429

Subgroups — #36370429

Order 22144.

412 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012526C211yesmaximal, characteristic, complemented
2#26012524C211yesmaximal, characteristic, complemented
3#26012523C211yesmaximal, characteristic, complemented
4#26012522C211yesmaximal, characteristic, complemented
5#26012522C211yesmaximal, characteristic, complemented
6#26012520C211yesmaximal, characteristic, complemented
7#26012518C211yesmaximal, characteristic, complemented
8D4×#13063443C211yesmaximal, characteristic, complemented
9#26012515C211yesmaximal, characteristic, complemented
10#26012515C211yesmaximal, characteristic, complemented
11#26012465C211yesmaximal, characteristic, complemented
12#26012464C211yesmaximal, characteristic, complemented
13#26012461C211yesmaximal, characteristic, complemented
14C2×C22⋊#13063443C211yesmaximal, characteristic, complemented
15#26011997C211yesmaximal, characteristic
16C2×#14583836C2211yescharacteristic, complemented
17C2×#14583835C2211yescharacteristic
18C2×#14583835C2211yescharacteristic
19C2×#14583835C2211yescharacteristic, complemented
20C2×D173⋊C8C2211yescharacteristic, complemented
21C22×#13063443C2211yescharacteristic
22C22×#13063443C2211yescharacteristic
23C22×#13063443C2211yescharacteristic, complemented
24#19149746C2211yescharacteristic, complemented
25#19149746C2211yescharacteristic, complemented
26C4⋊#13063443C2211yescharacteristic, complemented
27C4⋊#13063443C2211yescharacteristic, complemented
28#19149736C2211yescharacteristic, complemented
29#19149736C2211yescharacteristic, complemented
30C22⋊#13063443C2211yescharacteristic
31C22⋊#13063443C2211yescharacteristic
32C22⋊#13063443C2211yescharacteristic
33C22⋊#13063443C2211yescharacteristic
34C22⋊#13063443C2211yescharacteristic
35C22⋊#13063443C2211yescharacteristic
36C22⋊#13063443C2211yescharacteristic
37C22⋊#13063443C2211yescharacteristic
38C22⋊#13063443C2211yescharacteristic, complemented
39C22⋊#13063443C2211yescharacteristic, complemented
40#19149734C2211yescharacteristic, complemented
41#19149734C2211yescharacteristic, complemented
42C4×#13063443C2211yescharacteristic, complemented
43C4×#13063443C2211yescharacteristic, complemented
44C2×#14583825C2211yescharacteristic
45C23×Dic173C2211yescharacteristic
46#19149663C411yescharacteristic
47#19149660C411yescharacteristic
48#19149656C411yescharacteristic
49#19149656C411yescharacteristic
50#19149651C411yescharacteristic
51#19149651C411yescharacteristic
52#19149648C411yescharacteristic
53D4×Dic173C2211yescharacteristic
54D4×Dic173C2211yescharacteristic
55#19149639C2211yescharacteristic
56#19149639C2211yescharacteristic
57#19149635C2211yescharacteristic
58#120×C173C411yescharacteristic
59#14583836—41no—
60#14583835—41no—
61#14583835—41no—
62#14583835—41no—
63D173⋊C8—41no—
64C2×#13063443—41no—
65C2×#13063443—41no—
66C2×#13063443—41no—
67C2×#13063443C2311yescharacteristic
68C2×#13063443C2311yescharacteristic
69C2×#13063443C2311yescharacteristic
70C2×#13063443C2311yescharacteristic
71C2×#13063443C2311yescharacteristic
72C2×#13063443C2311yescharacteristic
73C2×#13063443C2311yescharacteristic
74C2×#13063443C2311yescharacteristic
75#14583825—41no—
76#14583825—41no—
77C2×Dic346C2×C411yescharacteristic
78C2×C22⋊D173C2×C411yescharacteristic
79C2×C22⋊D173C2×C411yescharacteristic
80C2×C22⋊D173C2×C411yescharacteristic
81C2×C22⋊D173C2×C411yescharacteristic
82C22×Dic173—41no—
83C22×Dic173—41no—
84C22×Dic173C2×C411yescharacteristic
85C22×Dic173C2×C411yescharacteristic
86C22×Dic173C2×C411yescharacteristic
87C22×Dic173C2311yescharacteristic
88C22×Dic173C2311yescharacteristic
89C22×Dic173C2311yescharacteristic
90C22×Dic173C2311yescharacteristic
91C2×C4×D173C2311yescharacteristic
92C4⋊Dic173C2×C411yescharacteristic
93C4⋊Dic173C2×C411yescharacteristic
94#14583805C2×C411yescharacteristic
95#14583805C2×C411yescharacteristic
96#14583805C2×C411yescharacteristic
97#14583805C2×C411yescharacteristic
98C22⋊Dic173C2×C411yescharacteristic
99C22⋊Dic173C2×C411yescharacteristic
100C22⋊Dic173C2×C411yescharacteristic
101C22⋊Dic173C2×C411yescharacteristic
102C22⋊Dic173C2×C411yescharacteristic
103#14583803C2×C411yescharacteristic
104#14583803C2×C411yescharacteristic
105C4×Dic173C2311yescharacteristic
106C4×Dic173C2311yescharacteristic
107C24×C173C2×C411yescharacteristic
108C2×D4×C173C2×C411yescharacteristic
109C2×D4×C173C2×C411yescharacteristic
110C2×D4×C173C2×C411yescharacteristic
111C22⋊C4×C173C2×C411yescharacteristic
112C22⋊C4×C173C2×C411yescharacteristic
113C22⋊C4×C173C2×C411yescharacteristic
114#13063443—21no—
115#13063443—21no—
116#13063443—21no—
117#13063443—21no—
118#13063443—21no—
119#13063443—21no—
120#13063443—21no—
121#13063443—21no—
122C22×D173C22×C411yescharacteristic
123Dic346—41no—
124C22⋊D173—22no—
125C22⋊D173—41no—
126C22⋊D173—41no—
127C22⋊D173—41no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—41no—
137C2×Dic173—41no—
138C2×Dic173—41no—
139C2×Dic173C22×C411yescharacteristic
140C2×Dic173C22×C411yescharacteristic
141C2×Dic173C22×C411yescharacteristic
142C2×Dic173C22×C411yescharacteristic
143C2×Dic173C22×C411yescharacteristic
144C2×Dic173C22×C411yescharacteristic
145C2×Dic173C2411yescharacteristic
146C4×D173—21no—
147C4×D173—21no—
148C23×C173—22no—
149C23×C173—41no—
150C23×C173—41no—
151C23×C173C22×C411yescharacteristic
152C23×C173C22×C411yescharacteristic
153C23×C173C22×C411yescharacteristic
154C23×C173C22×C411yescharacteristic
155D4×C173—22no—
156D4×C173—41no—
157D4×C173—41no—
158C2×C4×C173C22×C411yescharacteristic
159C2×C4×C173C22×C411yescharacteristic
160C2×C4×C173C22×C411yescharacteristic
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—21no—
164C22×C173—21no—
165C22×C173—21no—
166C22×C173—21no—
167C22×C173—21no—
168C22×C173—21no—
169C22×C173—21no—
170C22×C173—21no—
171C22×C173—21no—
172C22×C173—44no—
173C22×C173#13112yes—
174C22×C173C23×C411yescharacteristic
175C2×D173—21no—
176C2×D173—21no—
177C2×D173—21no—
178C4×C173—21no—
179C4×C173—21no—
180C4×C173—21no—
181Dic173—21no—
182Dic173—21no—
183Dic173—21no—
184Dic173—21no—
185Dic173—21no—
186Dic173—21no—
187Dic173H(5,𝔽2)11yescharacteristic
188Dic173H(5,𝔽2)11yescharacteristic
189C2×C173—22no—
190C2×C173—41no—
191C2×C173—41no—
192C2×C173—41no—
193C2×C173#51811yescharacteristic
194C2×C173C2×#13111yescharacteristic
195C2×C173#56811yescharacteristic
196D173—41no—
197C173#297611yescharacteristic, complemented, Sylow
198#2976—1731nomaximal, Sylow
199#525—1731no—
200#436—1731no—
201#436—1731no—
202#435—1731no—
203#435—1731no—
204#435—1731no—
205#435—1731no—
206C8×D4—1731no—
207C8×D4—1731no—
208#433—1731no—
209#433—1731no—
210#408—1731no—
211#408—1731no—
212#407—1731no—
213C2×C22⋊C8—1731no—
214C2×#41—1731no—
215C23×C4—1731no—
216C2×M16—1731no—
217C2×M16—1731no—
218C2×M16—1731no—
219C2×M16—1731no—
220C22×C8—1731no—
221C22×C8—1731no—
222C22×C8—1731no—
223C22×C8—1731no—
224#123—1731no—
225#123—1731no—
226C22⋊Q8—1731no—
227C22⋊Q8—1731no—
228#121—1731no—
229#121—1731no—
230#120—1731no—
231#120#1062192411yescharacteristic
232C4×D4—1731no—
233C4×D4—1731no—
234C4×D4—1731no—
235C4×D4—1731no—
236#117—1731no—
237C4⋊C8—1731no—
238C4⋊C8—1731no—
239C4⋊C8—1731no—
240C4⋊C8—1731no—
241C22⋊C8—1731no—
242C22⋊C8—1731no—
243C22⋊C8—1731no—
244C22⋊C8—1731no—
245C22⋊C8—1731no—
246C22⋊C8—1731no—
247C22⋊C8—1731no—
248C22⋊C8—1731no—
249C22⋊C8—1731no—
250C22⋊C8—1731no—
251C22⋊C8—1731no—
252C22⋊C8—1731no—
253#97—1731no—
254#97—1731no—
255C4×C8—1731no—
256C4×C8—1731no—
257C24C2×#1062192411yescharacteristic
258#41—6921no—
259#41—6921no—
260C2×Q8—1731no—
261C2×D4—1731no—
262C2×D4—1731no—
263C2×D4—1731no—
264C2×D4—1731no—
265C2×D4C2×#1062192411yescharacteristic
266C2×D4C2×#1062192411yescharacteristic
267C2×D4C2×#1062192411yescharacteristic
268C22×C4—1731no—
269C22×C4—1731no—
270C22×C4—1731no—
271C22×C4—1731no—
272C22×C4—1731no—
273C22×C4—1731no—
274C22×C4—1731no—
275C22×C4—1731no—
276C22×C4—6921no—
277C22×C4—6921no—
278M16—6921no—
279M16—6921no—
280M16—6921no—
281M16—6921no—
282C2×C8—1731no—
283C2×C8—1731no—
284C2×C8—1731no—
285C2×C8—1731no—
286C2×C8—1731no—
287C2×C8—1731no—
288C2×C8—1731no—
289C2×C8—1731no—
290C2×C8—6921no—
291C2×C8—6921no—
292C2×C8—6921no—
293C2×C8—6921no—
294C4⋊C4—1731no—
295C4⋊C4—1731no—
296C4⋊C4—1731no—
297C4⋊C4—1731no—
298C4⋊C4—1731no—
299C4⋊C4—1731no—
300C22⋊C4—1731no—
301C22⋊C4—1731no—
302C22⋊C4—1731no—
303C22⋊C4—1731no—
304C22⋊C4—1731no—
305C22⋊C4—1731no—
306C22⋊C4—1731no—
307C22⋊C4C2×#1062192411yescharacteristic
308C22⋊C4C2×#1062192411yescharacteristic
309C22⋊C4C2×#1062192411yescharacteristic
310C42—1731no—
311C42—1731no—
312C23—22no—
313C23—41no—
314C23—41no—
315C23—1731no—
316C23C22×#1062192411yescharacteristic
317C23C22×#1062192411yescharacteristic
318C23C22×#1062192411yescharacteristic
319C23C22×#1062192411yescharacteristic
320Q8—6921no—
321D4—22no—
322D4—41no—
323D4—41no—
324D4—3462no—
325D4—6921no—
326D4—6921no—
327D4—6921no—
328C2×C4—3461no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—3461no—
336C2×C4—3461no—
337C2×C4—3461no—
338C2×C4—1731no—
339C2×C4—1731no—
340C2×C4—1731no—
341C2×C4—1731no—
342C2×C4—1731no—
343C2×C4—1731no—
344C2×C4—1731no—
345C2×C4—6921no—
346C2×C4—6921no—
347C2×C4—6921no—
348C2×C4C22×#1062192411yescharacteristic
349C2×C4C22×#1062192411yescharacteristic
350C2×C4C22×#1062192411yescharacteristic
351C8—3461no—
352C8—3461no—
353C8—3461no—
354C8—3461no—
355C8—3461no—
356C8—3461no—
357C8—3461no—
358C8—3461no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—44no—
371C22—3461no—
372C22—3461no—
373C22—3461no—
374C22#1914976612yes—
375C22C23×#1062192411yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—3461no—
380C4—3461no—
381C4—3461no—
382C4—3461no—
383C4—3461no—
384C4—3461no—
385C4—1731no—
386C4—1731no—
387C2—22no—
388C2—41no—
389C2—41no—
390C2—41no—
391C2—6921no—
392C2#2601255811yescharacteristic
393C2C2×#1914976611yescharacteristic
394C2#2601256911yescharacteristic
subgroupG/H
center, FrattiniC22C23×#10621924
commutator, socleC22×C173C23×C4
Fitting#120×C173C4
prime radical#26011997C2
cosocleC2×Dic173C24
hypercenter#120#10621924
hypocenterC173#2976
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #26012526C2semidirect
2. #26012524C2semidirect
3. #26012523C2semidirect
4. #26012522C2semidirect
5. #26012522C2semidirect
6. #26012520C2semidirect
7. #26012518C2semidirect
8. D4×#13063443C2semidirect
9. #26012515C2semidirect
10. #26012515C2semidirect
11. #26012465C2semidirect
12. #26012464C2semidirect
13. #26012461C2semidirect
14. C2×C22⋊#13063443C2semidirect
16. C2×#14583836C22semidirect
19. C2×#14583835C22semidirect
20. C2×D173⋊C8C22semidirect
23. C22×#13063443C22semidirect
24. #19149746C22semidirect
25. #19149746C22semidirect
26. C4⋊#13063443C22semidirect
27. C4⋊#13063443C22semidirect
28. #19149736C22semidirect
29. #19149736C22semidirect
38. C22⋊#13063443C22semidirect
39. C22⋊#13063443C22semidirect
40. #19149734C22semidirect
41. #19149734C22semidirect
42. C4×#13063443C22semidirect
43. C4×#13063443C22semidirect
197. C173#2976semidirect

Maximal subgroups

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

subgroupindex
#260125262
#260125242
#260125232
#260125222
#260125202
#260125182
D4×#130634432
#260125152
#260124652
#260124642
#260124612
C2×C22⋊#130634432
#260119972
#2976173