Finite Group Databasebeta

← #36366151

Subgroups — #36366151

Order 22144.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012205C211yesmaximal, characteristic, complemented
2#26012142C211yesmaximal, characteristic, complemented
3#26011888C211yesmaximal, characteristic, complemented
4#26011817C211yesmaximal, characteristic, complemented
5#26011816C211yesmaximal, characteristic, complemented
6#26011808C211yesmaximal, characteristic, complemented
7#26011780C211yesmaximal, characteristic, complemented
8#26011769C211yesmaximal, characteristic, complemented
9#26011743C211yesmaximal, characteristic, complemented
10#26011742C211yesmaximal, characteristic, complemented
11#26011571C211yesmaximal, characteristic, complemented
12#26011566C211yesmaximal, characteristic, complemented
13#26011531C211yesmaximal, characteristic, complemented
14#26011530C211yesmaximal, characteristic, complemented
15#481×C173C211yesmaximal, characteristic, complemented
16C2×#14583826—21no—
17C2×#14583811C2211yescharacteristic
18C2×#14583810C2211yescharacteristic
19C2×#14583809C2211yescharacteristic
20#19149678C2211yescharacteristic
21#19149678C2211yescharacteristic
22#19149666—21no—
23#19149665—21no—
24#19149664—21no—
25#19149662—21no—
26#19149657—21no—
27#19149654C2211yescharacteristic
28#19149653C2211yescharacteristic
29#19149651—21no—
30#19149650C2211yescharacteristic
31D4×Dic173C2211yescharacteristic
32#19149641C2211yescharacteristic
33C4×C22⋊D173C2211yescharacteristic
34#19149638C2211yescharacteristic
35#19149637C2211yescharacteristic
36#19149635—21no—
37C4⋊C4×D173C2211yescharacteristic
38#19149613C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149611C2211yescharacteristic
42#19149610C2211yescharacteristic
43#19149610C2211yescharacteristic
44#19149610C2211yescharacteristic
45#19149608C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52#121×C173C2211yescharacteristic
53C2×C4⋊C4×C173C2211yescharacteristic
54#106×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—44no—
60C2×Dic346—21no—
61C2×Dic346—21no—
62C2×D692—21no—
63C2×D692C2311yescharacteristic
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173D411yescharacteristic
67C22×Dic173—21no—
68C2×C4×D173—21no—
69C2×C4×D173—21no—
70C2×C4×D173C2311yescharacteristic
71#14583811—22no—
72#14583810—22no—
73#14583809—22no—
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805D411yescharacteristic
83C22⋊Dic173—21no—
84C22⋊Dic173—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173D411yescharacteristic
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803D411yescharacteristic
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173C2311yescharacteristic
95C2×D4×C173—21no—
96C2×D4×C173C2311yescharacteristic
97C22×C4×C173—21no—
98C22×C4×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C4⋊C4×C173—21no—
102C4⋊C4×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C22×D173—21no—
107C22×D173C2×D411yescharacteristic
108Dic346—42no—
109Dic346—81no—
110D692—21no—
111D692—21no—
112D692—42no—
113C22⋊D173—22no—
114C22⋊D173—42no—
115C22⋊D173—81no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—42no—
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C4×D173—21no—
124C4×D173—21no—
125C4×D173—42no—
126C4×D173—42no—
127#13063436—21no—
128#13063436#4112yescomplemented
129C23×C173—21no—
130C23×C173C2×D411yescharacteristic
131D4×C173—21no—
132D4×C173—21no—
133D4×C173—81no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—42no—
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2411yescharacteristic
142C8×C173—21no—
143C8×C173—21no—
144C22×C173—21no—
145C22×C173—21no—
146C22×C173—21no—
147C22×C173—41no—
148C22×C173—41no—
149C22×C173—41no—
150C22×C173C22×D411yescharacteristic
151C2×D173—21no—
152C2×D173—21no—
153C2×D173—21no—
154C2×D173—41no—
155C2×D173—41no—
156C2×D173—41no—
157C4×C173—21no—
158C4×C173—41no—
159C4×C173—41no—
160C4×C173—41no—
161C4×C173C2×#4111yescharacteristic
162C4×C1732−1+411yescharacteristic
163Dic173—21no—
164Dic173—21no—
165Dic173—41no—
166Dic173—41no—
167Dic173—41no—
168Dic173AGL(1,ℤ/8)12yes—
169C2×C173—41no—
170C2×C173—81no—
171C2×C173#54711yescharacteristic
172C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
173C2×C173#57711yescharacteristic
174D173—41no—
175D173—81no—
176C173#335411yescharacteristic, complemented, Sylow
177#3354—1731nomaximal, Sylow
178#555—1731no—
179#547—1731no—
180#498—1731no—
181#482—1731no—
182#482—1731no—
183#481—1731no—
184#481D17311yescharacteristic, complemented
185#476—1731no—
186#474—1731no—
187#469—1731no—
188#469—1731no—
189C4×D8—1731no—
190#436—1731no—
191#428—1731no—
192#428—1731no—
193C2×#41—3461no—
194C2×D8—1731no—
195C2×M16—1731no—
196C2×M16—1731no—
197C4⋊Q8—1731no—
198C4⋊Q8—1731no—
199#124—3461no—
200#124—3461no—
201#123—3461no—
202#123—3461no—
203C22⋊Q8—3461no—
204#121—3461no—
205#121—1731no—
206#121—1731no—
207#121—1731no—
208#121C2×D17311yescharacteristic
209C4×D4—1731no—
210C4×D4—1731no—
211C4×D4—1731no—
212#117—3461no—
213#117—1731no—
214#117—1731no—
215C2×C4⋊C4—1731no—
216C2×C4⋊C4C2×D17311yescharacteristic
217#107—1731no—
218#107—1731no—
219#107—1731no—
220#106—1731no—
221#106—1731no—
222#106—1731no—
223#106—1731no—
224#106C2×D17311yescharacteristic
225#106C2×D17311yescharacteristic
226C4⋊C8—1731no—
227#102—1731no—
228#102—1731no—
229#102—1731no—
230#102—1731no—
231#102C2×D17311yescharacteristic
232#102C2×D17311yescharacteristic
233C22⋊C8—1731no—
234C22⋊C8C2×D17311yescharacteristic
235C4×C8—1731no—
236#41—6924no—
237C2×Q8—3461no—
238C2×Q8—3461no—
239C2×D4—21no—
240C2×D4—3461no—
241C2×D4—3461no—
242C2×D4—3461no—
243C2×D4—1731no—
244C2×D4—1731no—
245C2×D4C22×D17311yescharacteristic
246C22×C4—21no—
247C22×C4—3461no—
248C22×C4—3461no—
249C22×C4—3461no—
250C22×C4—1731no—
251C22×C4C22×D17311yescharacteristic
252D8—3462no—
253M16—3462no—
254M16—3462no—
255C2×C8—1731no—
256C2×C8—1731no—
257C2×C8C22×D17311yescharacteristic
258C2×C8C22×D17311yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—3461no—
261C4⋊C4—3461no—
262C4⋊C4—3461no—
263C4⋊C4—1731no—
264C4⋊C4—1731no—
265C4⋊C4—1731no—
266C4⋊C4—1731no—
267C4⋊C4C22×D17311yescharacteristic
268C4⋊C4C22×D17311yescharacteristic
269C4⋊C4C22×D17311yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—3461no—
272C22⋊C4—3461no—
273C22⋊C4—3461no—
274C22⋊C4—3461no—
275C22⋊C4—3461no—
276C22⋊C4—3461no—
277C22⋊C4—3461no—
278C22⋊C4—1731no—
279C22⋊C4—1731no—
280C42—3461no—
281C42—3461no—
282C42—1731no—
283C23—21no—
284C23—3461no—
285C23—1731no—
286C23D4×D17311yescharacteristic
287Q8—13841no—
288Q8—6922no—
289D4—21no—
290D4—21no—
291D4—81no—
292D4—3462no—
293D4—3461no—
294D4—3461no—
295D4—13841no—
296D4—6922no—
297D4—6922no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—3461no—
305C2×C4—3461no—
306C2×C4—3461no—
307C2×C4—3461no—
308C2×C4—3461no—
309C2×C4—1731no—
310C2×C4—1731no—
311C2×C4—1731no—
312C2×C4—6922no—
313C2×C4—6922no—
314C2×C4—6922no—
315C2×C4D4×D17311yescharacteristic
316C2×C4C23×D17311yescharacteristic
317C8—21no—
318C8—21no—
319C8—3461no—
320C8—1732no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—3461no—
328C22—3461no—
329C22—3461no—
330C22—6921no—
331C22—6921no—
332C22—6921no—
333C22C2×D4×D17311yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—3461no—
339C4—3461no—
340C4—1732no—
341C4—6921no—
342C4—6921no—
343C4—6921no—
344C4#41×D17311yescharacteristic
345C4#1914972611yescharacteristic
346C2—41no—
347C2—81no—
348C2—13841no—
349C2—6921no—
350C2#2601213911yescharacteristic
351C2AGL(1,ℤ/8)×D17311yescharacteristic
352C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#481×C173C2
socleC22×C173C22×D4
hypercenter#481D173
hypocenterC173#3354
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #26012205C2semidirect
2. #26012142C2semidirect
3. #26011888C2semidirect
4. #26011817C2semidirect
5. #26011816C2semidirect
6. #26011808C2semidirect
7. #26011780C2semidirect
8. #26011769C2semidirect
9. #26011743C2semidirect
10. #26011742C2semidirect
11. #26011571C2semidirect
12. #26011566C2semidirect
13. #26011531C2semidirect
14. #26011530C2semidirect
15. #481×C173C2semidirect
128. #13063436#41semidirect
176. C173#3354semidirect
184. #481D173semidirect

Maximal subgroups

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

subgroupindex
#260122052
#260121422
#260118882
#260118172
#260118162
#260118082
#260117802
#260117692
#260117432
#260117422
#260115712
#260115662
#260115312
#260115302
#481×C1732
#3354173