Finite Group Databasebeta

← #36366154

Subgroups — #36366154

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#26012204C211yesmaximal, characteristic, complemented
2#26012142C211yesmaximal, characteristic, complemented
3#26011888C211yesmaximal, characteristic, complemented
4#26011818C211yesmaximal, characteristic, complemented
5#26011815C211yesmaximal, characteristic, complemented
6#26011811C211yesmaximal, characteristic, complemented
7#26011809C211yesmaximal, characteristic, complemented
8#26011780C211yesmaximal, characteristic, complemented
9#26011770C211yesmaximal, characteristic, complemented
10#26011745C211yesmaximal, characteristic, complemented
11#26011744C211yesmaximal, characteristic, complemented
12#26011573C211yesmaximal, characteristic, complemented
13#26011567C211yesmaximal, characteristic, complemented
14#26011532C211yesmaximal, characteristic, complemented
15#428×C173C211yesmaximal, characteristic, complemented
16C2×#14583826—21no—
17C2×D1384C2211yescharacteristic
18C2×#14583809C2211yescharacteristic
19#19149678C2211yescharacteristic
20#19149678C2211yescharacteristic
21#19149667—21no—
22#19149665—21no—
23#19149664—21no—
24#19149662—21no—
25#19149657—21no—
26#19149654C2211yescharacteristic
27#19149654C2211yescharacteristic
28#19149653C2211yescharacteristic
29#19149653C2211yescharacteristic
30#19149652—21no—
31#19149641C2211yescharacteristic
32#19149641C2211yescharacteristic
33C4×C22⋊D173C2211yescharacteristic
34#19149637C2211yescharacteristic
35#19149636—21no—
36C4⋊C4×D173C2211yescharacteristic
37#19149612C2211yescharacteristic
38#19149611C2211yescharacteristic
39#19149610C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149610C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149601C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149599C2211yescharacteristic
46#19149599C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×M16×C173C2211yescharacteristic
53#117×C173C2211yescharacteristic
54C2×C4⋊C4×C173C2211yescharacteristic
55#107×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic
57#106×C173C2211yescharacteristic
58#106×C173C2211yescharacteristic
59#14583826—44no—
60C2×Dic346—21no—
61C2×Dic346—21no—
62C2×D692—21no—
63C2×D692—21no—
64C2×D692C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173D411yescharacteristic
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173C2311yescharacteristic
73D1384—22no—
74#14583809—22no—
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805D411yescharacteristic
83C22⋊Dic173D411yescharacteristic
84#14583803—21no—
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803D411yescharacteristic
92C4×Dic173—21no—
93C4×Dic173C2311yescharacteristic
94C22×C4×C173—21no—
95C22×C4×C173C2311yescharacteristic
96M16×C173—22no—
97C2×C8×C173C2311yescharacteristic
98C2×C8×C173C2311yescharacteristic
99C4⋊C4×C173—21no—
100C4⋊C4×C173C2311yescharacteristic
101C4⋊C4×C173C2311yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C22⋊C4×C173—21no—
105C42×C173—21no—
106C22×D173—21no—
107C22×D173—21no—
108C22×D173C2×D411yescharacteristic
109Dic346—42no—
110Dic346—81no—
111D692—21no—
112D692—21no—
113D692—21no—
114D692—21no—
115D692—42no—
116D692—81no—
117C22⋊D173—22no—
118C22⋊D173—42no—
119C22⋊D173—81no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173C2×D411yescharacteristic
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C4×D173—21no—
126C4×D173—21no—
127C4×D173—42no—
128C4×D173—42no—
129C4×D173—42no—
130#13063436—21no—
131#13063436—21no—
132C23×C173C2×D411yescharacteristic
133C2×C4×C173—21no—
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#4112yescomplemented
144C22×C173—21no—
145C22×C173—21no—
146C22×C173—21no—
147C22×C173C22×D411yescharacteristic
148C2×D173—21no—
149C2×D173—21no—
150C2×D173—21no—
151C2×D173—41no—
152C2×D173—41no—
153C2×D173—41no—
154C2×D173—41no—
155C2×D173—41no—
156C2×D173—41no—
157C4×C173—21no—
158C4×C173—41no—
159C4×C173—41no—
160C4×C173—41no—
161C4×C173—41no—
162C4×C173C2×#4111yescharacteristic
163C4×C1732−1+411yescharacteristic
164Dic173—21no—
165Dic173—21no—
166Dic173—41no—
167Dic173—41no—
168Dic173AGL(1,ℤ/8)12yes—
169C2×C173—41no—
170C2×C173#54711yescharacteristic
171C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C173#57711yescharacteristic
173D173—41no—
174D173—81no—
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#481—1731no—
185#476—1731no—
186#474—1731no—
187#469—1731no—
188#469—1731no—
189C4×D8—1731no—
190#436—1731no—
191#428—1731no—
192#428D17311yescharacteristic, complemented
193C2×#41—3461no—
194C2×D8—1731no—
195C2×M16—1731no—
196C2×M16C2×D17311yescharacteristic
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#121—1731no—
209C4×D4—1731no—
210C4×D4—1731no—
211C4×D4—1731no—
212#117—3461no—
213#117—1731no—
214#117C2×D17311yescharacteristic
215C2×C4⋊C4—1731no—
216C2×C4⋊C4C2×D17311yescharacteristic
217#107—1731no—
218#107C2×D17311yescharacteristic
219#107C2×D17311yescharacteristic
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#102—1731no—
232#102—1731no—
233C22⋊C8—1731no—
234C22⋊C8—1731no—
235C4×C8—1731no—
236#41—6924no—
237C2×Q8—3461no—
238C2×Q8—3461no—
239C2×D4—3461no—
240C2×D4—3461no—
241C2×D4—3461no—
242C2×D4—3461no—
243C2×D4—1731no—
244C2×D4—1731no—
245C2×D4—1731no—
246C22×C4—21no—
247C22×C4—3461no—
248C22×C4—3461no—
249C22×C4—3461no—
250C22×C4—1731no—
251C22×C4C22×D17311yescharacteristic
252D8—3462no—
253M16—22no—
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⋊C4C22×D17311yescharacteristic
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—21no—
281C42—3461no—
282C42—1731no—
283C23—3461no—
284C23—3461no—
285C23—1731no—
286C23D4×D17311yescharacteristic
287Q8—13841no—
288Q8—6922no—
289D4—3462no—
290D4—3461no—
291D4—3461no—
292D4—3461no—
293D4—3461no—
294D4—13841no—
295D4—13841no—
296D4—6922no—
297D4—6922no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
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—3461no—
319C8—3461no—
320C8#1458382612yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—3461no—
325C22—3461no—
326C22—3461no—
327C22—6921no—
328C22—6921no—
329C22—6921no—
330C22—6921no—
331C22—6921no—
332C22—6921no—
333C22C2×D4×D17311yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—3461no—
340C4—3461no—
341C4—1732no—
342C4—6921no—
343C4—6921no—
344C4C2×#1458382611yescharacteristic
345C4#1914972711yescharacteristic
346C2—41no—
347C2—13841no—
348C2—13841no—
349C2—6921no—
350C2#2601214211yescharacteristic
351C2AGL(1,ℤ/8)×D17311yescharacteristic
352C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#428×C173C2
socleC22×C173C22×D4
hypercenter#428D173
hypocenterC173#3354
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #26012204C2semidirect
2. #26012142C2semidirect
3. #26011888C2semidirect
4. #26011818C2semidirect
5. #26011815C2semidirect
6. #26011811C2semidirect
7. #26011809C2semidirect
8. #26011780C2semidirect
9. #26011770C2semidirect
10. #26011745C2semidirect
11. #26011744C2semidirect
12. #26011573C2semidirect
13. #26011567C2semidirect
14. #26011532C2semidirect
15. #428×C173C2semidirect
143. C8×C173#41semidirect
176. C173#3354semidirect
192. #428D173semidirect
320. C8#14583826semidirect

Maximal subgroups

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

subgroupindex
#260122042
#260121422
#260118882
#260118182
#260118152
#260118112
#260118092
#260117802
#260117702
#260117452
#260117442
#260115732
#260115672
#260115322
#428×C1732
#3354173