Finite Group Databasebeta

← #36363366

Subgroups — #36363366

Order 22144.

376 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012047C211yesmaximal, characteristic
2C4×Q8×D173C211yesmaximal, characteristic
3#26011799C211yesmaximal, characteristic, complemented
4#26011798C211yesmaximal, characteristic, complemented
5#26011795C211yesmaximal, characteristic, complemented
6#26011785C211yesmaximal, characteristic, complemented
7#26011784C211yesmaximal, characteristic, complemented
8#26011775C211yesmaximal, characteristic, complemented
9#26011774C211yesmaximal, characteristic, complemented
10#26011532C211yesmaximal, characteristic
11#26011530C211yesmaximal, characteristic
12#26011513C211yesmaximal, characteristic
13#26011498C211yesmaximal, characteristic
14#26011498C211yesmaximal, characteristic
15#479×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C2×#14583809C2211yescharacteristic
18C2×#14583809C2211yescharacteristic
19#19149678C2211yescharacteristic
20#19149678C2211yescharacteristic
21#19149671C2211yescharacteristic
22#19149657—21no—
23#19149657—21no—
24Q8×Dic173C2211yescharacteristic
25C4×Dic346—21no—
26C4×Dic346C2211yescharacteristic
27#19149644—21no—
28#19149644—21no—
29#19149644—21no—
30#19149644C2211yescharacteristic
31#19149637C2211yescharacteristic
32#19149637C2211yescharacteristic
33C4⋊C4×D173—21no—
34C4⋊C4×D173C2211yescharacteristic
35C42×D173—21no—
36C42×D173C2211yescharacteristic
37#19149613C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149610C2211yescharacteristic
42#19149610C2211yescharacteristic
43C4⋊#13063436C2211yescharacteristic
44#19149608C2211yescharacteristic
45#19149608C2211yescharacteristic
46#19149604C2211yescharacteristic
47#19149604C2211yescharacteristic
48#19149603C2211yescharacteristic
49#19149603C2211yescharacteristic
50#19149602C2211yescharacteristic
51#19149602C2211yescharacteristic
52#125×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346—21no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63Q8×D173—22no—
64Q8×D173Q814yescomplemented
65C2×C4×D173—21no—
66C2×C4×D173—21no—
67C2×C4×D173D411yescharacteristic
68C2×C4×D173D411yescharacteristic
69C2×C4×D173C2311yescharacteristic
70#14583809—22no—
71#14583809—22no—
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C4⋊Dic173—21no—
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583803—21no—
86#14583803—21no—
87C4×Dic173—21no—
88C4×Dic173—21no—
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173D411yescharacteristic
93C4×Dic173D411yescharacteristic
94C4×Dic173C2311yescharacteristic
95C2×Q8×C173C2311yescharacteristic
96C2×C8×C173C2311yescharacteristic
97C2×C8×C173C2311yescharacteristic
98C4⋊C4×C173—21no—
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—42no—
110Dic346—42no—
111Dic346C2×Q812yes—
112C2×Dic173—21no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173C2×D411yescharacteristic
117C2×Dic173C2×D411yescharacteristic
118C2×Dic173C2×D411yescharacteristic
119C4×D173—24no—
120C4×D173—22no—
121C4×D173—41no—
122C4×D173—42no—
123C4×D173C2×Q812yes—
124C4×D173#4112yes—
125#13063436—21no—
126#13063436—21no—
127Q8×C173—21no—
128Q8×C173C2×Q812yes—
129C2×C4×C173—21no—
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173C2×D411yescharacteristic
134C2×C4×C173C2×D411yescharacteristic
135C2×C4×C173C2411yescharacteristic
136C8×C173—21no—
137C8×C173—21no—
138C22×C173C22×D411yescharacteristic
139C2×D173—21no—
140C2×D173—21no—
141C2×D173C22⋊Q812yes—
142C4×C173—22no—
143C4×C173—21no—
144C4×C173—21no—
145C4×C173—41no—
146C4×C173—41no—
147C4×C173—41no—
148C4×C173C22×Q811yescharacteristic
149C4×C173C2×#4111yescharacteristic
150Dic173—22no—
151Dic173—21no—
152Dic173—41no—
153Dic173—41no—
154Dic173—41no—
155Dic173C22⋊Q812yes—
156Dic173#13712yes—
157C2×C173C2×C22⋊Q811yescharacteristic
158C2×C173C2×#13711yescharacteristic
159C2×C173#57511yescharacteristic
160D173—22no—
161C173#307811yescharacteristic, complemented, Sylow
162#3078—1731nomaximal, Sylow
163#533—1731no—
164C2×C4×Q8—1731no—
165#479—1731no—
166#479—1731no—
167#479—1731no—
168#479D17311yescharacteristic, complemented
169#477—1731no—
170#477—1731no—
171#475—1731no—
172#475—1731no—
173#428—1731no—
174#428—1731no—
175#423—1731no—
176#419—1731no—
177#419—1731no—
178C22×Q8—1731no—
179C2×M16—1731no—
180C2×M16—1731no—
181C4⋊Q8—1731no—
182C4⋊Q8—1731no—
183#125—1731no—
184#125C2×D17311yescharacteristic
185C22⋊Q8—3461no—
186C22⋊Q8—3461no—
187C4×Q8—3461no—
188C4×Q8—3461no—
189C4×Q8—3461no—
190C4×Q8—3461no—
191C4×Q8—1731no—
192C4×Q8—1731no—
193C4×Q8—1731no—
194C4×Q8C2×D17311yescharacteristic
195#117—1731no—
196#117—1731no—
197C2×C4⋊C4—3461no—
198C2×C4⋊C4—1731no—
199C2×C42—3461no—
200C2×C42—1731no—
201#107—1731no—
202#107—1731no—
203#107—1731no—
204#107C2×D17311yescharacteristic
205#106—1731no—
206#106—1731no—
207#106—1731no—
208#106C2×D17311yescharacteristic
209C4⋊C8—1731no—
210C4⋊C8—1731no—
211C4⋊C8—1731no—
212C4⋊C8C2×D17311yescharacteristic
213#103—1731no—
214#103—1731no—
215#103—1731no—
216#103—1731no—
217#103—1731no—
218#103—1731no—
219#103C2×D17311yescharacteristic
220#103C2×D17311yescharacteristic
221C2×Q8—3461no—
222C2×Q8—3461no—
223C2×Q8—3461no—
224C2×Q8—3462no—
225C2×Q8—1734no—
226C2×Q8—1731no—
227C2×Q8C22×D17311yescharacteristic
228C22×C4—3461no—
229C22×C4—3461no—
230C22×C4—1731no—
231C22×C4—1731no—
232C22×C4—1731no—
233M16—3462no—
234M16—3462no—
235C2×C8—1731no—
236C2×C8—1731no—
237C2×C8C22×D17311yescharacteristic
238C2×C8C22×D17311yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—3461no—
243C4⋊C4—3461no—
244C4⋊C4—3461no—
245C4⋊C4—3461no—
246C4⋊C4—3461no—
247C4⋊C4—3461no—
248C4⋊C4—3461no—
249C4⋊C4—3461no—
250C4⋊C4—1731no—
251C4⋊C4—1731no—
252C4⋊C4—1731no—
253C4⋊C4C22×D17311yescharacteristic
254C4⋊C4C22×D17311yescharacteristic
255C4⋊C4C22×D17311yescharacteristic
256C22⋊C4—3461no—
257C22⋊C4—3461no—
258C42—21no—
259C42—3461no—
260C42—3461no—
261C42—3461no—
262C42—3461no—
263C42—3461no—
264C42—1731no—
265C42—1731no—
266C42—1731no—
267C42C22×D17311yescharacteristic
268C23—1731no—
269Q8—21no—
270Q8—3464no—
271Q8—3461no—
272Q8—1732no—
273Q8—6922no—
274Q8—6922no—
275Q8Q8×D17312yes—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—3464no—
281C2×C4—3462no—
282C2×C4—3461no—
283C2×C4—3461no—
284C2×C4—3461no—
285C2×C4—3461no—
286C2×C4—1731no—
287C2×C4—1731no—
288C2×C4—1731no—
289C2×C4—1732no—
290C2×C4—1732no—
291C2×C4—6921no—
292C2×C4—6922no—
293C2×C4D4×D17311yescharacteristic
294C2×C4D4×D17311yescharacteristic
295C2×C4C23×D17311yescharacteristic
296C8—21no—
297C8—21no—
298C8—3461no—
299C8—3461no—
300C22—3461no—
301C22—3461no—
302C22—1732no—
303C22C2×D4×D17311yescharacteristic
304C4—22no—
305C4—21no—
306C4—21no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—3462no—
311C4—3461no—
312C4—1732no—
313C4—1732no—
314C4—6921no—
315C4—6921no—
316C4—6921no—
317C4C2×Q8×D17311yescharacteristic
318C4#41×D17311yescharacteristic
319C2—3462no—
320C2C22⋊Q8×D17311yescharacteristic
321C2#137×D17311yescharacteristic
322C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#479×C173C2
socleC22×C173C22×D4
prime radicalC22×D173C2×D4
hypercenter#479D173
hypocenterC173#3078
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
3. #26011799C2semidirect
4. #26011798C2semidirect
5. #26011795C2semidirect
6. #26011785C2semidirect
7. #26011784C2semidirect
8. #26011775C2semidirect
9. #26011774C2semidirect
15. #479×C173C2semidirect
64. Q8×D173Q8semidirect
161. C173#3078semidirect
168. #479D173semidirect

Maximal subgroups

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

subgroupindex
#260120472
C4×Q8×D1732
#260117992
#260117982
#260117952
#260117852
#260117842
#260117752
#260117742
#260115322
#260115302
#260115132
#260114982
#479×C1732
#3078173