Finite Group Databasebeta

← #36363398

Subgroups — #36363398

Order 22144.

362 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012118C211yesmaximal, characteristic
2#26011967C211yesmaximal, characteristic
3#26011801C211yesmaximal, characteristic, complemented
4#26011799C211yesmaximal, characteristic, complemented
5#26011798C211yesmaximal, characteristic, complemented
6#26011796C211yesmaximal, characteristic, complemented
7#26011784C211yesmaximal, characteristic, complemented
8#26011783C211yesmaximal, characteristic, complemented
9#26011775C211yesmaximal, characteristic, complemented
10#26011530C211yesmaximal, characteristic
11#26011526C211yesmaximal, characteristic
12#26011516C211yesmaximal, characteristic
13#26011498C211yesmaximal, characteristic
14#103×D173C211yesmaximal, characteristic
15#475×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C2×#14583809C2211yescharacteristic
18C2×C8×D173C2211yescharacteristic
19#19149678C2211yescharacteristic
20#19149677—21no—
21#19149671C2211yescharacteristic
22#19149670C2211yescharacteristic
23#19149669—21no—
24#19149657—21no—
25#19149655—21no—
26Q8×Dic173C2211yescharacteristic
27C4×Dic346—21no—
28C4×Dic346C2211yescharacteristic
29#19149644—21no—
30#19149644C2211yescharacteristic
31#19149637—21no—
32#19149637C2211yescharacteristic
33#19149637C2211yescharacteristic
34#19149636—21no—
35#19149636C2211yescharacteristic
36C4⋊C4×D173C2211yescharacteristic
37#19149613C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149611C2211yescharacteristic
42#19149610C2211yescharacteristic
43C4⋊#13063436C2211yescharacteristic
44#19149608C2211yescharacteristic
45#19149608C2211yescharacteristic
46#19149604C2211yescharacteristic
47#19149604C2211yescharacteristic
48#19149603C2211yescharacteristic
49#19149603C2211yescharacteristic
50#19149602C2211yescharacteristic
51#19149602C2211yescharacteristic
52C4⋊Q8×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54#106×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346—21no—
61C2×Dic346C2311yescharacteristic
62Q8×D173—22no—
63Q8×D173Q814yescomplemented
64C2×C4×D173—21no—
65C2×C4×D173—21no—
66C2×C4×D173C2311yescharacteristic
67#14583809—22no—
68C2×#13063436C2311yescharacteristic
69C2×#13063436C2311yescharacteristic
70C8×D173—22no—
71C4⋊Dic173—21no—
72C4⋊Dic173—21no—
73C4⋊Dic173C2311yescharacteristic
74C4⋊Dic173C2311yescharacteristic
75C4⋊Dic173C2311yescharacteristic
76#14583805—21no—
77#14583805—21no—
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805D411yescharacteristic
84#14583805D411yescharacteristic
85#14583803—21no—
86#14583803—21no—
87#14583803D411yescharacteristic
88#14583803D411yescharacteristic
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173C2311yescharacteristic
95C2×Q8×C173—21no—
96C2×Q8×C173C2311yescharacteristic
97C2×C8×C173C2311yescharacteristic
98C2×C8×C173C2311yescharacteristic
99C4⋊C4×C173—21no—
100C4⋊C4×C173—21no—
101C4⋊C4×C173C2311yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C42×C173—21no—
105C42×C173C2311yescharacteristic
106C22×D173C2×D411yescharacteristic
107Dic346—21no—
108Dic346—24no—
109Dic346—81no—
110Dic346C2×Q812yes—
111C2×Dic173—21no—
112C2×Dic173—21no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173C2×D411yescharacteristic
116C2×Dic173C2×D411yescharacteristic
117C2×Dic173C2×D411yescharacteristic
118C4×D173—24no—
119C4×D173—81no—
120C4×D173C2×Q812yes—
121C4×D173#4112yes—
122#13063436—21no—
123#13063436—21no—
124Q8×C173—21no—
125Q8×C173—81no—
126Q8×C173C2×Q812yes—
127C2×C4×C173—21no—
128C2×C4×C173—21no—
129C2×C4×C173—21no—
130C2×C4×C173—21no—
131C2×C4×C173C2×D411yescharacteristic
132C2×C4×C173C2×D411yescharacteristic
133C2×C4×C173C2411yescharacteristic
134C8×C173—21no—
135C8×C173—21no—
136C22×C173C22×D411yescharacteristic
137C2×D173—21no—
138C2×D173—21no—
139C2×D173C22⋊Q812yes—
140C4×C173—22no—
141C4×C173—21no—
142C4×C173—21no—
143C4×C173—41no—
144C4×C173—41no—
145C4×C173—41no—
146C4×C173C22×Q811yescharacteristic
147C4×C173C2×#4111yescharacteristic
148Dic173—22no—
149Dic173—21no—
150Dic173—21no—
151Dic173—41no—
152Dic173—41no—
153Dic173—41no—
154Dic173C22⋊Q812yes—
155C2×C173C2×C22⋊Q811yescharacteristic
156C2×C173#57711yescharacteristic
157C2×C173#57811yescharacteristic
158D173—22no—
159C173#308211yescharacteristic, complemented, Sylow
160#3082—1731nomaximal, Sylow
161#544—1731no—
162#519—1731no—
163#479—1731no—
164#479—1731no—
165#479—1731no—
166#479—1731no—
167#477—1731no—
168#477—1731no—
169#475—1731no—
170#475D17311yescharacteristic, complemented
171#428—1731no—
172#427—1731no—
173#424—1731no—
174#419—1731no—
175C2×#103—1731no—
176C22×Q8—1731no—
177C2×M16—1731no—
178C22×C8—1731no—
179C4⋊Q8—3461no—
180C4⋊Q8—1731no—
181C4⋊Q8C2×D17311yescharacteristic
182#125—3461no—
183#125—1731no—
184#125—1731no—
185C22⋊Q8—3461no—
186C22⋊Q8—3461no—
187C4×Q8—3461no—
188C4×Q8—3461no—
189C4×Q8—1731no—
190C4×Q8—1731no—
191C4×Q8—1731no—
192C4×Q8C2×D17311yescharacteristic
193#117—3461no—
194#117—3461no—
195#117—1731no—
196#117—1731no—
197#117—1731no—
198C2×C4⋊C4—1731no—
199#107—1731no—
200#107—1731no—
201#107—1731no—
202#107—1731no—
203#106—1731no—
204#106—1731no—
205#106C2×D17311yescharacteristic
206#106C2×D17311yescharacteristic
207C4⋊C8—1731no—
208C4⋊C8—1731no—
209C4⋊C8—1731no—
210C4⋊C8C2×D17311yescharacteristic
211#103—1731no—
212#103—1731no—
213#103—1731no—
214#103—1731no—
215#103—1731no—
216#103—1731no—
217#103C2×D17311yescharacteristic
218#103C2×D17311yescharacteristic
219C2×Q8—21no—
220C2×Q8—3461no—
221C2×Q8—3461no—
222C2×Q8—3462no—
223C2×Q8—1734no—
224C2×Q8—1731no—
225C2×Q8C22×D17311yescharacteristic
226C22×C4—3461no—
227C22×C4—3461no—
228C22×C4—1731no—
229M16—3462no—
230C2×C8—3462no—
231C2×C8—1731no—
232C2×C8—1731no—
233C2×C8C22×D17311yescharacteristic
234C2×C8C22×D17311yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—3461no—
238C4⋊C4—3461no—
239C4⋊C4—3461no—
240C4⋊C4—3461no—
241C4⋊C4—3461no—
242C4⋊C4—3461no—
243C4⋊C4—3461no—
244C4⋊C4—3461no—
245C4⋊C4—3461no—
246C4⋊C4—1731no—
247C4⋊C4—1731no—
248C4⋊C4—1731no—
249C4⋊C4—1731no—
250C4⋊C4—1731no—
251C4⋊C4C22×D17311yescharacteristic
252C4⋊C4C22×D17311yescharacteristic
253C4⋊C4C22×D17311yescharacteristic
254C22⋊C4—3461no—
255C22⋊C4—3461no—
256C22⋊C4—1731no—
257C22⋊C4—1731no—
258C42—21no—
259C42—3461no—
260C42—3461no—
261C42—3461no—
262C42—3461no—
263C42—3461no—
264C42—1731no—
265C42C22×D17311yescharacteristic
266C23—1731no—
267Q8—21no—
268Q8—81no—
269Q8—3464no—
270Q8—3461no—
271Q8—13841no—
272Q8—1732no—
273Q8Q8×D17312yes—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—3464no—
279C2×C4—3461no—
280C2×C4—3461no—
281C2×C4—3461no—
282C2×C4—3461no—
283C2×C4—13841no—
284C2×C4—1731no—
285C2×C4—1731no—
286C2×C4—1731no—
287C2×C4—1732no—
288C2×C4—1732no—
289C2×C4D4×D17311yescharacteristic
290C2×C4D4×D17311yescharacteristic
291C2×C4C23×D17311yescharacteristic
292C8—21no—
293C8—21no—
294C8—3461no—
295C8—3461no—
296C22—3461no—
297C22—3461no—
298C22—1732no—
299C22C2×D4×D17311yescharacteristic
300C4—22no—
301C4—21no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—3462no—
307C4—3461no—
308C4—3461no—
309C4—1732no—
310C4—6921no—
311C4—6921no—
312C4—6921no—
313C4C2×Q8×D17311yescharacteristic
314C4#41×D17311yescharacteristic
315C2—3462no—
316C2C22⋊Q8×D17311yescharacteristic
317C2#2601229711yescharacteristic
318C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#475×C173C2
socleC22×C173C22×D4
prime radicalC22×D173C2×D4
hypercenter#475D173
hypocenterC173#3082
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
3. #26011801C2semidirect
4. #26011799C2semidirect
5. #26011798C2semidirect
6. #26011796C2semidirect
7. #26011784C2semidirect
8. #26011783C2semidirect
9. #26011775C2semidirect
15. #475×C173C2semidirect
63. Q8×D173Q8semidirect
159. C173#3082semidirect
170. #475D173semidirect

Maximal subgroups

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

subgroupindex
#260121182
#260119672
#260118012
#260117992
#260117982
#260117962
#260117842
#260117832
#260117752
#260115302
#260115262
#260115162
#260114982
#103×D1732
#475×C1732
#3082173