Finite Group Databasebeta

← #36364117

Subgroups — #36364117

Order 22144.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012043C211yesmaximal, characteristic, complemented
2#26012036C211yesmaximal, characteristic
3#26011888C211yesmaximal, characteristic, complemented
4#26011862C211yesmaximal, characteristic, complemented
5#26011840C211yesmaximal, characteristic, complemented
6#26011836C211yesmaximal, characteristic, complemented
7#26011722C211yesmaximal, characteristic, complemented
8#26011720C211yesmaximal, characteristic, complemented
9#26011718C211yesmaximal, characteristic, complemented
10#26011582C211yesmaximal, characteristic, complemented
11SD16×Dic173C211yesmaximal, characteristic, complemented
12#26011578C211yesmaximal, characteristic, complemented
13#26011576C211yesmaximal, characteristic, complemented
14C2×C8×Dic173C211yesmaximal, characteristic, complemented
15#465×C173C211yesmaximal, characteristic, complemented
16C2×#14583825—21no—
17C2×#14583815C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20C22×#13063436C2211yescharacteristic
21#19149678C2211yescharacteristic
22#19149678C2211yescharacteristic
23#19149677C2211yescharacteristic
24#19149675C2211yescharacteristic
25#19149669—21no—
26#19149665—21no—
27#19149658C2211yescharacteristic
28#19149656—21no—
29#19149653C2211yescharacteristic
30#19149651—21no—
31Q8×Dic173C2211yescharacteristic
32#19149644—21no—
33#19149644C2211yescharacteristic
34D4×Dic173C2211yescharacteristic
35#19149641C2211yescharacteristic
36#19149639—21no—
37#19149635—21no—
38C2×C4×Dic173C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149610C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149602C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149599C2211yescharacteristic
48C8×Dic173C2211yescharacteristic
49C8×Dic173C2211yescharacteristic
50C8×Dic173C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#106×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583825—44no—
60C2×Dic346—21no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63C2×D692C2311yescharacteristic
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C22×Dic173—21no—
68C22×Dic173D411yescharacteristic
69C22×Dic173D411yescharacteristic
70C2×C4×D173—21no—
71#14583815—22no—
72#14583814—22no—
73#14583813—22no—
74C2×#13063436—22no—
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C4⋊Dic173—21no—
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88#14583803—21no—
89#14583803—21no—
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173D411yescharacteristic
93C4×Dic173D411yescharacteristic
94C4×Dic173C2311yescharacteristic
95C4×Dic173C2311yescharacteristic
96C2×Q8×C173C2311yescharacteristic
97C2×D4×C173—21no—
98C2×D4×C173C2311yescharacteristic
99C22×C4×C173C2311yescharacteristic
100SD16×C173—22no—
101C2×C8×C173—22no—
102C2×C8×C173C2311yescharacteristic
103C2×C8×C173C2311yescharacteristic
104C4⋊C4×C173—21no—
105C4⋊C4×C173C2311yescharacteristic
106C4⋊C4×C173C2311yescharacteristic
107C22⋊C4×C173—21no—
108C22⋊C4×C173—21no—
109C22×D173—21no—
110Dic346—21no—
111Dic346—21no—
112Dic346—42no—
113Dic346—42no—
114D692—21no—
115D692—21no—
116C22⋊D173—42no—
117C22⋊D173—42no—
118C22⋊D173—42no—
119C2×Dic173—22no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—42no—
124C2×Dic173SD1614yescomplemented
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—42no—
130#13063436—21no—
131#13063436#4112yescomplemented
132C23×C173—21no—
133C23×C173C2×D411yescharacteristic
134Q8×C173—21no—
135Q8×C173—21no—
136D4×C173—21no—
137D4×C173—21no—
138D4×C173—42no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2411yescharacteristic
146C8×C173—21no—
147C8×C173#4112yescomplemented
148C22×C173—21no—
149C22×C173—21no—
150C22×C173—41no—
151C22×C173—41no—
152C22×C173—41no—
153C22×C173C2×SD1612yes—
154C22×C173C22×D411yescharacteristic
155C2×D173—41no—
156C2×D173—41no—
157C2×D173—41no—
158C4×C173—21no—
159C4×C173—41no—
160C4×C173—41no—
161C4×C173—41no—
162C4×C173C2×#4111yescharacteristic
163C4×C173C2×#4111yescharacteristic
164Dic173—21no—
165Dic173—41no—
166Dic173—41no—
167Dic173—41no—
168Dic173C2×SD1612yes—
169Dic173C2×SD1612yes—
170Dic173#13512yes—
171C2×C173—22no—
172C2×C173—81no—
173C2×C173#53211yescharacteristic
174C2×C173C22×SD1611yescharacteristic
175C2×C173C2×#13511yescharacteristic
176D173—81no—
177C173#316711yescharacteristic, complemented, Sylow
178#3167—1731nomaximal, Sylow
179#533—1731no—
180#532—1731no—
181#498—1731no—
182#492—1731no—
183#487—1731no—
184#486—1731no—
185#465—1731no—
186#465—1731no—
187#465—1731no—
188#465D17311yescharacteristic, complemented
189C4×SD16—1731no—
190C4×SD16—1731no—
191C4×SD16—1731no—
192C4×SD16—1731no—
193C2×C4×C8—1731no—
194C2×#41—3461no—
195C2×SD16—1731no—
196C2×SD16—1731no—
197C2×SD16—1731no—
198C2×SD16C2×D17311yescharacteristic
199C22×C8—1731no—
200C22×C8C2×D17311yescharacteristic
201C4⋊Q8—1731no—
202C4⋊Q8—1731no—
203C4⋊Q8—1731no—
204#127—1731no—
205#125—3461no—
206#124—3461no—
207C22⋊Q8—3461no—
208C22⋊Q8—1731no—
209C22⋊Q8C2×D17311yescharacteristic
210#121—3461no—
211#121—1731no—
212#121C2×D17311yescharacteristic
213C4×Q8—3461no—
214C4×Q8—1731no—
215C4×Q8—1731no—
216C4×D4—3461no—
217C4×D4—1731no—
218C4×D4—1731no—
219#117—3461no—
220C2×C42—1731no—
221#106—1731no—
222#106—1731no—
223#106—1731no—
224#106C2×D17311yescharacteristic
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#103C2×D17311yescharacteristic
229#102—1731no—
230#102—1731no—
231#102—1731no—
232#102C2×D17311yescharacteristic
233C4×C8—1731no—
234C4×C8—1731no—
235C4×C8—1731no—
236C4×C8—1731no—
237#41—6924no—
238C2×Q8—3461no—
239C2×Q8—3461no—
240C2×Q8—1731no—
241C2×Q8C22×D17311yescharacteristic
242C2×D4—21no—
243C2×D4—3461no—
244C2×D4—3461no—
245C2×D4—3461no—
246C2×D4—1731no—
247C2×D4C22×D17311yescharacteristic
248C22×C4—3461no—
249C22×C4—3461no—
250C22×C4—1731no—
251C22×C4—1731no—
252C22×C4C22×D17311yescharacteristic
253SD16—22no—
254SD16—3462no—
255SD16—3462no—
256SD16—3462no—
257C2×C8—22no—
258C2×C8—3462no—
259C2×C8—1731no—
260C2×C8—1731no—
261C2×C8C22×D17311yescharacteristic
262C2×C8C22×D17311yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—3461no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—3461no—
271C4⋊C4—1731no—
272C4⋊C4—1731no—
273C4⋊C4C22×D17311yescharacteristic
274C4⋊C4C22×D17311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C42—3461no—
282C42—3461no—
283C42—1731no—
284C42—1731no—
285C42—1731no—
286C42—1731no—
287C23—21no—
288C23—3461no—
289C23D4×D17311yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—3461no—
293Q8—3461no—
294Q8—6922no—
295Q8—6922no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—3461no—
300D4—3461no—
301D4—6922no—
302D4—6922no—
303D4—6922no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—3462no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—1731no—
314C2×C4—1734no—
315C2×C4—1731no—
316C2×C4—1731no—
317C2×C4—1731no—
318C2×C4—6922no—
319C2×C4—6922no—
320C2×C4D4×D17311yescharacteristic
321C2×C4C23×D17311yescharacteristic
322C8—21no—
323C8—3461no—
324C8—1732no—
325C8#1458382512yescomplemented
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—6921no—
332C22—6921no—
333C22—6921no—
334C22SD16×D17312yes—
335C22C2×D4×D17311yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—3461no—
341C4—1732no—
342C4—1732no—
343C4—1732no—
344C4—6921no—
345C4—6921no—
346C4—6921no—
347C4#41×D17311yescharacteristic
348C4C2×#1458382511yescharacteristic
349C2—22no—
350C2—81no—
351C2—13841no—
352C2#2601203611yescharacteristic
353C2C2×SD16×D17311yescharacteristic
354C2#135×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#465×C173C2
socleC22×C173C22×D4
prime radical#26012036C2
hypercenter#465D173
hypocenterC173#3167
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #26012043C2semidirect
3. #26011888C2semidirect
4. #26011862C2semidirect
5. #26011840C2semidirect
6. #26011836C2semidirect
7. #26011722C2semidirect
8. #26011720C2semidirect
9. #26011718C2semidirect
10. #26011582C2semidirect
11. SD16×Dic173C2semidirect
12. #26011578C2semidirect
13. #26011576C2semidirect
14. C2×C8×Dic173C2semidirect
15. #465×C173C2semidirect
124. C2×Dic173SD16semidirect
131. #13063436#41semidirect
147. C8×C173#41semidirect
177. C173#3167semidirect
188. #465D173semidirect
325. C8#14583825semidirect

Maximal subgroups

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

subgroupindex
#260120432
#260120362
#260118882
#260118622
#260118402
#260118362
#260117222
#260117202
#260117182
#260115822
SD16×Dic1732
#260115782
#260115762
C2×C8×Dic1732
#465×C1732
#3167173