Finite Group Databasebeta

← #36363707

Subgroups — #36363707

Order 22144.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012036C211yesmaximal, characteristic, complemented
2#41×Dic173C211yesmaximal, characteristic, complemented
3#26011816C211yesmaximal, characteristic, complemented
4#26011812C211yesmaximal, characteristic, complemented
5#26011774C211yesmaximal, characteristic, complemented
6#26011769C211yesmaximal, characteristic, complemented
7#26011689C211yesmaximal, characteristic, complemented
8#26011684C211yesmaximal, characteristic, complemented
9#26011636C211yesmaximal, characteristic, complemented
10#26011603C211yesmaximal, characteristic, complemented
11#26011600C211yesmaximal, characteristic, complemented
12#26011594C211yesmaximal, characteristic, complemented
13#26011592C211yesmaximal, characteristic, complemented
14#26011545C211yesmaximal, characteristic, complemented
15#449×C173C211yesmaximal, characteristic, complemented
16C2×#14583825—21no—
17C2×#14583813C2211yescharacteristic
18C2×#14583811C2211yescharacteristic
19#19149678C2211yescharacteristic
20#19149675C2211yescharacteristic
21#19149665—21no—
22#19149653C2211yescharacteristic
23#19149651—21no—
24Q8×Dic173C2211yescharacteristic
25Q8×Dic173C2211yescharacteristic
26D4×Dic173—21no—
27D4×Dic173—21no—
28D4×Dic173C2211yescharacteristic
29D4×Dic173C2211yescharacteristic
30D4×Dic173C2211yescharacteristic
31#19149641C2211yescharacteristic
32#19149639—21no—
33#19149638—21no—
34#19149638C2211yescharacteristic
35C2×C4×Dic173—21no—
36C2×C4×Dic173C2211yescharacteristic
37#19149613C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149589C2211yescharacteristic
51#19149589C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×D8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583825—44no—
60C2×Dic346—21no—
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C22×Dic173—21no—
66C22×Dic173—21no—
67C22×Dic173—21no—
68C22×Dic173D411yescharacteristic
69C22×Dic173D411yescharacteristic
70C2×C4×D173—21no—
71#14583813—22no—
72#14583811—22no—
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C4⋊Dic173—21no—
76C4⋊Dic173—21no—
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80#14583805—21no—
81C22⋊Dic173—21no—
82C22⋊Dic173—21no—
83C22⋊Dic173—21no—
84C22⋊Dic173—21no—
85#14583803—21no—
86#14583803—21no—
87C4×Dic173—21no—
88C4×Dic173—21no—
89C4×Dic173—21no—
90C4×Dic173D411yescharacteristic
91C4×Dic173D411yescharacteristic
92C4×Dic173C2311yescharacteristic
93C4×Dic173C2311yescharacteristic
94#41×C173—22no—
95#41×C173—22no—
96C2×Q8×C173C2311yescharacteristic
97C2×D4×C173—21no—
98C2×D4×C173—21no—
99C2×D4×C173C2311yescharacteristic
100C2×D4×C173C2311yescharacteristic
101C22×C4×C173—21no—
102C22×C4×C173C2311yescharacteristic
103SD16×C173—22no—
104D8×C173—22no—
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173C2311yescharacteristic
108C22⋊C4×C173—21no—
109C22×D173—21no—
110Dic346—42no—
111D692—21no—
112D692—21no—
113C22⋊D173—42no—
114C22⋊D173—42no—
115C22⋊D173—42no—
116C2×Dic173—22no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—41no—
121C2×Dic173—42no—
122C2×Dic173—42no—
123C2×Dic173—42no—
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C4×D173—42no—
129#13063436—21no—
130#13063436—21no—
131C23×C173—21no—
132C23×C173—21no—
133C23×C173C2×D411yescharacteristic
134Q8×C173—21no—
135Q8×C173#4112yescomplemented
136D4×C173—24no—
137D4×C173—21no—
138D4×C173—21no—
139D4×C173—21no—
140D4×C173—42no—
141D4×C173#4112yescomplemented
142C2×C4×C173—24no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2411yescharacteristic
149C8×C173—21no—
150C8×C173—21no—
151C22×C173—21no—
152C22×C173—22no—
153C22×C173—21no—
154C22×C173—21no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173C22×D411yescharacteristic
161C2×D173—41no—
162C2×D173—41no—
163C2×D173—41no—
164C4×C173—21no—
165C4×C173—22no—
166C4×C173—41no—
167C4×C173C2×#4111yescharacteristic
168C4×C173C2×#4111yescharacteristic
169Dic173—21no—
170Dic173—21no—
171Dic173—21no—
172Dic173—41no—
173Dic173—41no—
174Dic173—41no—
175Dic173AGL(1,ℤ/8)12yes—
176C2×C173—41no—
177C2×C173—42no—
178C2×C173—81no—
179C2×C173#52511yescharacteristic
180C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C173#57511yescharacteristic
182D173—81no—
183C173#311511yescharacteristic, complemented, Sylow
184#3115—1731nomaximal, Sylow
185#532—1731no—
186C4×#41—1731no—
187#482—1731no—
188#482—1731no—
189#475—1731no—
190#474—1731no—
191#460—1731no—
192#459—1731no—
193#449—1731no—
194#449D17311yescharacteristic, complemented
195#442—1731no—
196#442—1731no—
197#440—1731no—
198#440—1731no—
199#432—1731no—
200C2×#41—3461no—
201C2×#41C2×D17311yescharacteristic
202C2×SD16—1731no—
203C2×SD16C2×D17311yescharacteristic
204C2×D8—1731no—
205C2×D8C2×D17311yescharacteristic
206C4⋊Q8—1731no—
207#127—1731no—
208#124—3461no—
209#121—3461no—
210#121—1731no—
211#121C2×D17311yescharacteristic
212C4×Q8—1731no—
213C4×Q8—1731no—
214C4×D4—3461no—
215C4×D4—3461no—
216C4×D4—3461no—
217C4×D4—1731no—
218C4×D4—1731no—
219C4×D4—1731no—
220C4×D4—1731no—
221#117—3461no—
222#117—1731no—
223C2×C42—3461no—
224C2×C42—1731no—
225#107—1731no—
226#107—1731no—
227#106—1731no—
228#106—1731no—
229C4⋊C8—1731no—
230C4⋊C8—1731no—
231#103—1731no—
232#103C2×D17311yescharacteristic
233#102—1731no—
234#102—1731no—
235#102—1731no—
236#102—1731no—
237#102—1731no—
238#102C2×D17311yescharacteristic
239C22⋊C8—1731no—
240C22⋊C8C2×D17311yescharacteristic
241#97—1731no—
242#97—1731no—
243#41—22no—
244#41—22no—
245#41—6924no—
246C2×Q8—3461no—
247C2×Q8C22×D17311yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—3461no—
251C2×D4—3461no—
252C2×D4—3461no—
253C2×D4—1731no—
254C2×D4C22×D17311yescharacteristic
255C2×D4C22×D17311yescharacteristic
256C22×C4—21no—
257C22×C4—3461no—
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—1731no—
262C22×C4—1731no—
263C22×C4C22×D17311yescharacteristic
264SD16—22no—
265SD16—3462no—
266D8—22no—
267D8—3462no—
268C2×C8—1731no—
269C2×C8—1731no—
270C2×C8C22×D17311yescharacteristic
271C2×C8C22×D17311yescharacteristic
272C4⋊C4—3461no—
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—1731no—
276C4⋊C4—1731no—
277C4⋊C4—1731no—
278C4⋊C4C22×D17311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C42—3461no—
287C42—3461no—
288C42—3461no—
289C42—1731no—
290C42—1731no—
291C42—1731no—
292C42—1731no—
293C23—21no—
294C23—21no—
295C23—3461no—
296C23D4×D17311yescharacteristic
297Q8—21no—
298Q8—6922no—
299Q8#1458382512yescomplemented
300D4—24no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—42no—
305D4—3461no—
306D4—3461no—
307D4—6922no—
308D4—6922no—
309D4—6922no—
310D4#1458382512yescomplemented
311C2×C4—24no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—3462no—
317C2×C4—3461no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—1731no—
321C2×C4—1731no—
322C2×C4—1731no—
323C2×C4—1731no—
324C2×C4—6922no—
325C2×C4—6921no—
326C2×C4—6922no—
327C2×C4—6922no—
328C2×C4—6922no—
329C2×C4D4×D17311yescharacteristic
330C2×C4C23×D17311yescharacteristic
331C8—21no—
332C8—21no—
333C8—3461no—
334C8—3461no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—6921no—
345C22—6921no—
346C22—6921no—
347C22C2×D4×D17311yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—3461no—
352C4—3461no—
353C4—3461no—
354C4—1732no—
355C4—6921no—
356C4—6921no—
357C4—6921no—
358C4C2×#1458382511yescharacteristic
359C4C2×#1458382511yescharacteristic
360C2—41no—
361C2—42no—
362C2—81no—
363C2—13841no—
364C2#2601199711yescharacteristic
365C2AGL(1,ℤ/8)×D17311yescharacteristic
366C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#449×C173C2
socleC22×C173C22×D4
hypercenter#449D173
hypocenterC173#3115
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #26012036C2semidirect
2. #41×Dic173C2semidirect
3. #26011816C2semidirect
4. #26011812C2semidirect
5. #26011774C2semidirect
6. #26011769C2semidirect
7. #26011689C2semidirect
8. #26011684C2semidirect
9. #26011636C2semidirect
10. #26011603C2semidirect
11. #26011600C2semidirect
12. #26011594C2semidirect
13. #26011592C2semidirect
14. #26011545C2semidirect
15. #449×C173C2semidirect
135. Q8×C173#41semidirect
141. D4×C173#41semidirect
183. C173#3115semidirect
194. #449D173semidirect
299. Q8#14583825semidirect
310. D4#14583825semidirect

Maximal subgroups

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

subgroupindex
#260120362
#41×Dic1732
#260118162
#260118122
#260117742
#260117692
#260116892
#260116842
#260116362
#260116032
#260116002
#260115942
#260115922
#260115452
#449×C1732
#3115173