Finite Group Databasebeta

← #36363723

Subgroups — #36363723

Order 22144.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012045C211yesmaximal, characteristic, complemented
2C4×#14583827C211yesmaximal, characteristic
3#26011835C211yesmaximal, characteristic, complemented
4#26011831C211yesmaximal, characteristic, complemented
5#26011784C211yesmaximal, characteristic, complemented
6#26011780C211yesmaximal, characteristic, complemented
7#26011699C211yesmaximal, characteristic, complemented
8#26011695C211yesmaximal, characteristic, complemented
9#26011656C211yesmaximal, characteristic
10#26011652C211yesmaximal, characteristic
11#26011596C211yesmaximal, characteristic, complemented
12#26011593C211yesmaximal, characteristic, complemented
13#26011589C211yesmaximal, characteristic, complemented
14#26011546C211yesmaximal, characteristic, complemented
15#441×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20#19149678C2211yescharacteristic
21#19149677C2211yescharacteristic
22#19149669—21no—
23#19149659C2211yescharacteristic
24#19149657—21no—
25#19149655C2211yescharacteristic
26Q8×Dic173C2211yescharacteristic
27C4×Dic346—21no—
28C4×Dic346C2211yescharacteristic
29C4×D692—21no—
30C4×D692C2211yescharacteristic
31#19149641—21no—
32#19149641C2211yescharacteristic
33#19149637—21no—
34#19149637C2211yescharacteristic
35#19149636—21no—
36C42×D173—21no—
37C42×D173C2211yescharacteristic
38#19149613C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149590C2211yescharacteristic
52C2×Q16×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54C4×Q8×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic
59#14583827—22no—
60#14583827—22no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63C2×D692—21no—
64C2×D692C2311yescharacteristic
65C2×C4×D173—21no—
66C2×C4×D173—21no—
67C2×C4×D173—21no—
68C2×C4×D173D411yescharacteristic
69C2×C4×D173D411yescharacteristic
70C2×C4×D173C2311yescharacteristic
71#14583816—22no—
72#14583815—22no—
73#14583813—22no—
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C4⋊Dic173—21no—
77C4⋊Dic173—21no—
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173D411yescharacteristic
94C4×Dic173D411yescharacteristic
95C4×Dic173C2311yescharacteristic
96C2×Q8×C173C2311yescharacteristic
97C2×Q8×C173C2311yescharacteristic
98Q16×C173—22no—
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C4⋊C4×C173—21no—
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C42×C173—21no—
106C42×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173C2×D411yescharacteristic
110Dic346—21no—
111Dic346—21no—
112Dic346—42no—
113D692—24no—
114D692—21no—
115D692#4112yes—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173C2×D411yescharacteristic
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C4×D173—24no—
123C4×D173—21no—
124C4×D173—21no—
125C4×D173—22no—
126C4×D173—41no—
127C4×D173—42no—
128C4×D173—42no—
129#13063436—21no—
130#13063436—21no—
131Q8×C173—21no—
132Q8×C173—21no—
133Q8×C173—21no—
134Q8×C173#4112yes—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173C2×D411yescharacteristic
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2411yescharacteristic
142C8×C173—21no—
143C8×C173—21no—
144C22×C173C22×D411yescharacteristic
145C2×D173—21no—
146C2×D173—22no—
147C2×D173—21no—
148C2×D173—21no—
149C2×D173—41no—
150C2×D173—41no—
151C4×C173—22no—
152C4×C173—21no—
153C4×C173—21no—
154C4×C173—41no—
155C4×C173—41no—
156C4×C173—41no—
157C4×C173C2×#4111yescharacteristic
158C4×C173C2×#4111yescharacteristic
159Dic173—21no—
160Dic173—21no—
161Dic173—41no—
162Dic173—41no—
163Dic173—41no—
164Dic173#13712yes—
165C2×C173#52511yescharacteristic
166C2×C173C2×#13711yescharacteristic
167C2×C173#57511yescharacteristic
168D173—41no—
169D173—42no—
170C173#311611yescharacteristic, complemented, Sylow
171#3116—1731nomaximal, Sylow
172#533—1731no—
173C4×#41—1731no—
174#485—1731no—
175#485—1731no—
176#477—1731no—
177#476—1731no—
178#462—1731no—
179#461—1731no—
180#452—1731no—
181#452—1731no—
182#441—1731no—
183#441D17311yescharacteristic, complemented
184#440—1731no—
185#440—1731no—
186#432—1731no—
187C2×#41—1731no—
188C2×Q16—1731no—
189C2×Q16C2×D17311yescharacteristic
190C2×SD16—1731no—
191C2×SD16—1731no—
192C4⋊Q8—1731no—
193C4⋊Q8—1731no—
194#125—3461no—
195C22⋊Q8—3461no—
196C22⋊Q8—1731no—
197C22⋊Q8—1731no—
198C4×Q8—3461no—
199C4×Q8—1731no—
200C4×Q8—1731no—
201C4×Q8C2×D17311yescharacteristic
202C4×Q8C2×D17311yescharacteristic
203C4×D4—3461no—
204C4×D4—3461no—
205C4×D4—1731no—
206C4×D4—1731no—
207#117—3461no—
208#117—3461no—
209#117—1731no—
210C2×C42—3461no—
211C2×C42—1731no—
212#107—1731no—
213#107—1731no—
214#106—1731no—
215#106C2×D17311yescharacteristic
216C4⋊C8—1731no—
217C4⋊C8—1731no—
218#103—1731no—
219#103—1731no—
220#103—1731no—
221#103—1731no—
222#103C2×D17311yescharacteristic
223#103C2×D17311yescharacteristic
224#102—1731no—
225#102—1731no—
226C22⋊C8—1731no—
227C22⋊C8—1731no—
228#97—1731no—
229#97C2×D17311yescharacteristic
230#41—3462no—
231#41—3462no—
232C2×Q8—3461no—
233C2×Q8—1731no—
234C2×Q8C22×D17311yescharacteristic
235C2×Q8C22×D17311yescharacteristic
236C2×D4—3461no—
237C2×D4—1731no—
238C22×C4—3461no—
239C22×C4—3461no—
240C22×C4—3461no—
241C22×C4—1731no—
242C22×C4—1731no—
243C22×C4—1731no—
244Q16—22no—
245Q16—3462no—
246SD16—3462no—
247SD16—3462no—
248C2×C8—1731no—
249C2×C8—1731no—
250C2×C8C22×D17311yescharacteristic
251C2×C8C22×D17311yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—3461no—
255C4⋊C4—3461no—
256C4⋊C4—3461no—
257C4⋊C4—3461no—
258C4⋊C4—3461no—
259C4⋊C4—3461no—
260C4⋊C4—3461no—
261C4⋊C4—1731no—
262C4⋊C4—1731no—
263C4⋊C4C22×D17311yescharacteristic
264C4⋊C4C22×D17311yescharacteristic
265C22⋊C4—3461no—
266C22⋊C4—3461no—
267C22⋊C4—3461no—
268C22⋊C4—3461no—
269C22⋊C4—3461no—
270C42—21no—
271C42—21no—
272C42—3461no—
273C42—3461no—
274C42—3461no—
275C42—1731no—
276C42—1731no—
277C42—1731no—
278C42C22×D17311yescharacteristic
279C23—3461no—
280C23—1731no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—3461no—
285Q8—3461no—
286Q8—6922no—
287Q8#1458382612yes—
288D4—3464no—
289D4—3461no—
290D4—1732no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—3464no—
296C2×C4—3462no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—3461no—
300C2×C4—3461no—
301C2×C4—3461no—
302C2×C4—1731no—
303C2×C4—1731no—
304C2×C4—1731no—
305C2×C4—6921no—
306C2×C4—6922no—
307C2×C4—6922no—
308C2×C4D4×D17311yescharacteristic
309C2×C4D4×D17311yescharacteristic
310C2×C4C23×D17311yescharacteristic
311C8—21no—
312C8—21no—
313C8—3461no—
314C8—3461no—
315C22—3461no—
316C22—3462no—
317C22—3461no—
318C22—3461no—
319C22—6921no—
320C22—6921no—
321C22C2×D4×D17311yescharacteristic
322C4—22no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—3461no—
329C4—3461no—
330C4—1732no—
331C4—6921no—
332C4—6921no—
333C4—6921no—
334C4#41×D17311yescharacteristic
335C4C2×#1458382611yescharacteristic
336C2—6921no—
337C2—6922no—
338C2#2601200011yescharacteristic
339C2#137×D17311yescharacteristic
340C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#441×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583827C22
hypercenter#441D173
hypocenterC173#3116
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. #26012045C2semidirect
3. #26011835C2semidirect
4. #26011831C2semidirect
5. #26011784C2semidirect
6. #26011780C2semidirect
7. #26011699C2semidirect
8. #26011695C2semidirect
11. #26011596C2semidirect
12. #26011593C2semidirect
13. #26011589C2semidirect
14. #26011546C2semidirect
15. #441×C173C2semidirect
170. C173#3116semidirect
183. #441D173semidirect

Maximal subgroups

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

subgroupindex
#260120452
C4×#145838272
#260118352
#260118312
#260117842
#260117802
#260116992
#260116952
#260116562
#260116522
#260115962
#260115932
#260115892
#260115462
#441×C1732
#3116173