Finite Group Databasebeta

← #36363683

Subgroups — #36363683

Order 22144.

424 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012038C211yesmaximal, characteristic
2C4×Q8×D173C211yesmaximal, characteristic, complemented
3#26011806C211yesmaximal, characteristic, complemented
4#26011803C211yesmaximal, characteristic, complemented
5#26011785C211yesmaximal, characteristic, complemented
6#26011784C211yesmaximal, characteristic, complemented
7#26011690C211yesmaximal, characteristic, complemented
8#26011689C211yesmaximal, characteristic, complemented
9#26011631C211yesmaximal, characteristic, complemented
10#26011628C211yesmaximal, characteristic, complemented
11#26011593C211yesmaximal, characteristic, complemented
12#26011591C211yesmaximal, characteristic, complemented
13#26011589C211yesmaximal, characteristic, complemented
14#26011546C211yesmaximal, characteristic, complemented
15#440×C173C211yesmaximal, characteristic, complemented
16C2×#14583826—21no—
17C2×Q8×D173C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21#19149678C2211yescharacteristic
22#19149675C2211yescharacteristic
23#19149665—21no—
24#19149654C2211yescharacteristic
25#19149652—21no—
26#19149650C2211yescharacteristic
27Q8×Dic173C2211yescharacteristic
28C4×Dic346—21no—
29C4×Dic346C2211yescharacteristic
30#19149644—21no—
31#19149644C2211yescharacteristic
32C4×D692C2211yescharacteristic
33C4×C22⋊D173—21no—
34C4⋊C4×D173—21no—
35C4⋊C4×D173C2211yescharacteristic
36C42×D173—21no—
37C42×D173C2211yescharacteristic
38#19149613C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149590C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55#107×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic
59#14583826—44no—
60C2×Dic346—21no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63Q8×D173—22no—
64Q8×D173—22no—
65C2×D692—21no—
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173D411yescharacteristic
74C2×C4×D173D411yescharacteristic
75C2×C4×D173C2311yescharacteristic
76#14583815—22no—
77#14583814—22no—
78#14583813—22no—
79C2×#13063436C2311yescharacteristic
80C2×#13063436C2311yescharacteristic
81C4⋊Dic173—21no—
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88C22⋊Dic173—21no—
89#14583803—21no—
90#14583803—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173D411yescharacteristic
95C4×Dic173D411yescharacteristic
96C4×Dic173C2311yescharacteristic
97C2×Q8×C173C2311yescharacteristic
98C2×D4×C173C2311yescharacteristic
99C22×C4×C173—21no—
100SD16×C173—22no—
101C2×C8×C173C2311yescharacteristic
102C2×C8×C173C2311yescharacteristic
103C4⋊C4×C173—21no—
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C42×C173—21no—
108C42×C173C2311yescharacteristic
109C22×D173—21no—
110C22×D173C2×D411yescharacteristic
111Dic346—24no—
112Dic346—21no—
113Dic346—42no—
114Dic346#4112yescomplemented
115D692—21no—
116D692—21no—
117D692—42no—
118C22⋊D173—42no—
119C22⋊D173—42no—
120C22⋊D173—42no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C4×D173—24no—
128C4×D173—22no—
129C4×D173—21no—
130C4×D173—21no—
131C4×D173—22no—
132C4×D173—42no—
133C4×D173—42no—
134#13063436—21no—
135#13063436—21no—
136C23×C173—21no—
137Q8×C173—21no—
138Q8×C173#4112yescomplemented
139D4×C173—21no—
140D4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—42no—
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2411yescharacteristic
148C8×C173—21no—
149C8×C173—21no—
150C22×C173—41no—
151C22×C173—41no—
152C22×C173—41no—
153C22×C173C22×D411yescharacteristic
154C2×D173—21no—
155C2×D173—21no—
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—41no—
159C2×D173AGL(1,ℤ/8)12yes—
160C4×C173—22no—
161C4×C173—21no—
162C4×C173—21no—
163C4×C173—41no—
164C4×C173—41no—
165C4×C173C2×#4111yescharacteristic
166C4×C173C2×#4111yescharacteristic
167Dic173—22no—
168Dic173—21no—
169Dic173—21no—
170Dic173—41no—
171Dic173—41no—
172Dic173#13712yes—
173C2×C173—81no—
174C2×C173#52511yescharacteristic
175C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
176C2×C173C2×#13711yescharacteristic
177D173—22no—
178D173—81no—
179C173#311211yescharacteristic, complemented, Sylow
180#3112—1731nomaximal, Sylow
181#532—1731no—
182C2×C4×Q8—1731no—
183#480—1731no—
184#480—1731no—
185#477—1731no—
186#477—1731no—
187#460—1731no—
188#460—1731no—
189#448—1731no—
190#448—1731no—
191#440—1731no—
192#440—1731no—
193#440—1731no—
194#440D17311yescharacteristic, complemented
195#432—1731no—
196C2×#41—3461no—
197C22×Q8—1731no—
198C2×SD16—1731no—
199C2×SD16—1731no—
200C2×SD16—1731no—
201C2×SD16C2×D17311yescharacteristic
202C4⋊Q8—1731no—
203#127—1731no—
204#124—3461no—
205#121—3461no—
206#121—1731no—
207#121—1731no—
208C4×Q8—3461no—
209C4×Q8—3461no—
210C4×Q8—1731no—
211C4×Q8—1731no—
212C4×Q8—1731no—
213C4×Q8C2×D17311yescharacteristic
214C4×D4—3461no—
215C4×D4—1731no—
216C4×D4C2×D17311yescharacteristic
217C2×C4⋊C4—3461no—
218C2×C4⋊C4—1731no—
219C2×C42—3461no—
220C2×C42—1731no—
221#107—1731no—
222#107—1731no—
223#107—1731no—
224#107C2×D17311yescharacteristic
225C4⋊C8—1731no—
226C4⋊C8—1731no—
227#103—1731no—
228#103—1731no—
229#103—1731no—
230#103C2×D17311yescharacteristic
231#102—1731no—
232#102—1731no—
233#102—1731no—
234#102C2×D17311yescharacteristic
235C22⋊C8—1731no—
236C22⋊C8—1731no—
237#97—1731no—
238#97C2×D17311yescharacteristic
239#41—6924no—
240C2×Q8—3461no—
241C2×Q8—3461no—
242C2×Q8—3462no—
243C2×Q8—3462no—
244C2×Q8—1731no—
245C2×Q8C22×D17311yescharacteristic
246C2×D4—3461no—
247C2×D4—3461no—
248C2×D4—3461no—
249C2×D4—3461no—
250C2×D4—1731no—
251C2×D4C22×D17311yescharacteristic
252C22×C4—21no—
253C22×C4—3461no—
254C22×C4—3461no—
255C22×C4—3461no—
256C22×C4—1731no—
257C22×C4—1731no—
258C22×C4—1731no—
259SD16—22no—
260SD16—3462no—
261SD16—3462no—
262SD16—3462no—
263C2×C8—1731no—
264C2×C8—1731no—
265C2×C8C22×D17311yescharacteristic
266C2×C8C22×D17311yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—3461no—
271C4⋊C4—3461no—
272C4⋊C4—3461no—
273C4⋊C4—1731no—
274C4⋊C4—1731no—
275C4⋊C4C22×D17311yescharacteristic
276C4⋊C4C22×D17311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C42—21no—
282C42—3461no—
283C42—3461no—
284C42—3461no—
285C42—1731no—
286C42—1731no—
287C42—1731no—
288C42C22×D17311yescharacteristic
289C23—21no—
290C23—3461no—
291C23—1731no—
292Q8—21no—
293Q8—3464no—
294Q8—3461no—
295Q8—1732no—
296Q8—6922no—
297Q8#1458382612yescomplemented
298D4—21no—
299D4—21no—
300D4—3461no—
301D4—3461no—
302D4—6922no—
303D4—6922no—
304D4—6922no—
305D4—6922no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—3464no—
311C2×C4—3462no—
312C2×C4—3462no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—3461no—
317C2×C4—3461no—
318C2×C4—1731no—
319C2×C4—1731no—
320C2×C4—1731no—
321C2×C4—6922no—
322C2×C4—6922no—
323C2×C4D4×D17311yescharacteristic
324C2×C4D4×D17311yescharacteristic
325C2×C4C23×D17311yescharacteristic
326C8—21no—
327C8—21no—
328C8—3461no—
329C8—3461no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—3461no—
334C22—3461no—
335C22—1732no—
336C22—6921no—
337C22—6921no—
338C22—6921no—
339C22C2×D4×D17311yescharacteristic
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—3462no—
346C4—3461no—
347C4—3461no—
348C4—1732no—
349C4—6921no—
350C4—6921no—
351C4#41×D17311yescharacteristic
352C4C2×#1458382611yescharacteristic
353C2—81no—
354C2—3462no—
355C2—13841no—
356C2#2601200011yescharacteristic
357C2AGL(1,ℤ/8)×D17311yescharacteristic
358C2#137×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#440×C173C2
socleC22×C173C22×D4
prime radical#26012038C2
hypercenter#440D173
hypocenterC173#3112
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D173C2semidirect
3. #26011806C2semidirect
4. #26011803C2semidirect
5. #26011785C2semidirect
6. #26011784C2semidirect
7. #26011690C2semidirect
8. #26011689C2semidirect
9. #26011631C2semidirect
10. #26011628C2semidirect
11. #26011593C2semidirect
12. #26011591C2semidirect
13. #26011589C2semidirect
14. #26011546C2semidirect
15. #440×C173C2semidirect
114. Dic346#41semidirect
138. Q8×C173#41semidirect
179. C173#3112semidirect
194. #440D173semidirect
297. Q8#14583826semidirect

Maximal subgroups

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

subgroupindex
#260120382
C4×Q8×D1732
#260118062
#260118032
#260117852
#260117842
#260116902
#260116892
#260116312
#260116282
#260115932
#260115912
#260115892
#260115462
#440×C1732
#3112173