Finite Group Databasebeta

← #36363813

Subgroups — #36363813

Order 22144.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012102C211yesmaximal, characteristic, complemented
2#26011963C211yesmaximal, characteristic
3#26011831C211yesmaximal, characteristic, complemented
4#26011830C211yesmaximal, characteristic, complemented
5#26011792C211yesmaximal, characteristic, complemented
6#26011791C211yesmaximal, characteristic, complemented
7#26011693C211yesmaximal, characteristic, complemented
8#26011640C211yesmaximal, characteristic
9#26011640C211yesmaximal, characteristic
10#26011594C211yesmaximal, characteristic, complemented
11#26011592C211yesmaximal, characteristic, complemented
12SD16×Dic173C211yesmaximal, characteristic, complemented
13#26011576C211yesmaximal, characteristic, complemented
14#26011551C211yesmaximal, characteristic, complemented
15#461×C173C211yesmaximal, characteristic, complemented
16C2×D4×D173C2211yescharacteristic
17C2×#14583813C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19#19149674—21no—
20#19149674—21no—
21#19149671C2211yescharacteristic, complemented
22#19149669—21no—
23#19149669—21no—
24#19149655C2211yescharacteristic
25#19149655C2211yescharacteristic
26Q8×Dic173C2211yescharacteristic, complemented
27Q8×Dic173C2211yescharacteristic, complemented
28C4×D692C2211yescharacteristic
29D4×Dic173C2211yescharacteristic
30#19149641C2211yescharacteristic
31C4×C22⋊D173—21no—
32#19149639—21no—
33#19149637C2211yescharacteristic
34#19149636C2211yescharacteristic
35C2×#14583803—21no—
36C22⋊C4×D173—21no—
37#19149613C2211yescharacteristic, complemented
38#19149612C2211yescharacteristic, complemented
39#19149611C2211yescharacteristic, complemented
40#19149610C2211yescharacteristic, complemented
41C4⋊#13063436C2211yescharacteristic, complemented
42#19149603C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C2×SD16×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C4⋊Q8×C173C2211yescharacteristic, complemented
55C4×D4×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#103×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C2×D692C2311yescharacteristic
61C2×C22⋊D173—21no—
62D4×D173—22no—
63D4×D173—22no—
64C22×Dic173—21no—
65C2×C4×D173—21no—
66C2×C4×D173—21no—
67C2×C4×D173C2311yescharacteristic
68#14583813—22no—
69#14583813—22no—
70C2×#13063436C2311yescharacteristic
71C2×#13063436C2311yescharacteristic
72C4⋊Dic173—21no—
73C4⋊Dic173—21no—
74C4⋊Dic173C2311yescharacteristic
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77#14583805—21no—
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805D411yescharacteristic
82#14583805D411yescharacteristic
83C22⋊Dic173—21no—
84#14583803—21no—
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803D411yescharacteristic
90#14583803D411yescharacteristic
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173C2311yescharacteristic
95C2×Q8×C173C2311yescharacteristic
96C2×Q8×C173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C22×C4×C173—21no—
99SD16×C173—22no—
100SD16×C173—22no—
101C2×C8×C173C2311yescharacteristic
102C2×C8×C173C2311yescharacteristic
103C4⋊C4×C173—21no—
104C4⋊C4×C173—21no—
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173—24no—
110C22×D173—81no—
111C22×D173—81no—
112C22×D173C2×D411yescharacteristic
113D692—21no—
114D692#4112yes—
115C22⋊D173—24no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—81no—
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C4×D173—21no—
124C4×D173—21no—
125C4×D173—81no—
126C4×D173—81no—
127#13063436—21no—
128#13063436—21no—
129C23×C173—21no—
130Q8×C173—21no—
131Q8×C173—21no—
132Q8×C173—21no—
133Q8×C173—21no—
134D4×C173—21no—
135D4×C173#4112yes—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—81no—
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2411yescharacteristic
143C8×C173—21no—
144C8×C173—21no—
145C22×C173—22no—
146C22×C173—41no—
147C22×C173—41no—
148C22×C173C22×D411yescharacteristic
149C2×D173—21no—
150C2×D173—22no—
151C2×D173—21no—
152C2×D173—21no—
153C2×D173—41no—
154C2×D173—41no—
155C2×D173—84no—
156C4×C173—21no—
157C4×C173—21no—
158C4×C173—41no—
159C4×C173—41no—
160C4×C173—41no—
161C4×C173C2×#4111yescharacteristic
162C4×C173C2×#4111yescharacteristic
163Dic173—21no—
164Dic173—21no—
165Dic173—21no—
166Dic173—41no—
167Dic173—41no—
168Dic173—41no—
169C2×C173—42no—
170C2×C173#52511yescharacteristic
171C2×C173#57711yescharacteristic
172C2×C173#57711yescharacteristic
173D173—41no—
174D173—42no—
175C173#312311yescharacteristic, complemented, Sylow
176#3123—1731nomaximal, Sylow
177#541—1731no—
178#518—1731no—
179#485—1731no—
180#485—1731no—
181#478—1731no—
182#478—1731no—
183#461—1731no—
184#461D17311yescharacteristic, complemented
185#450—1731no—
186#450—1731no—
187#440—1731no—
188#440—1731no—
189C4×SD16—1731no—
190C4×SD16—1731no—
191#433—1731no—
192C22×D4—1731no—
193C2×SD16—1731no—
194C2×SD16—1731no—
195C2×SD16C2×D17311yescharacteristic
196C2×SD16C2×D17311yescharacteristic
197C4⋊Q8C2×D17311yescharacteristic, complemented
198#126—3461no—
199#126—3461no—
200#125—3461no—
201#125—3461no—
202#125—1731no—
203C22⋊Q8—1731no—
204C22⋊Q8—1731no—
205C4×Q8—1731no—
206C4×Q8—1731no—
207C4×D4—3461no—
208C4×D4—3461no—
209C4×D4—1731no—
210C4×D4—1731no—
211C4×D4—1731no—
212C4×D4C2×D17311yescharacteristic
213#117—1731no—
214#117—1731no—
215C2×C22⋊C4—3461no—
216C2×C22⋊C4—3461no—
217#107—1731no—
218#107—1731no—
219#106—1731no—
220#106—1731no—
221C4⋊C8—1731no—
222C4⋊C8C2×D17311yescharacteristic, complemented
223#103—1731no—
224#103—1731no—
225#103C2×D17311yescharacteristic, complemented
226#103C2×D17311yescharacteristic, complemented
227#102—1731no—
228#102—1731no—
229#102—1731no—
230#102—1731no—
231C22⋊C8—1731no—
232C22⋊C8—1731no—
233#97—1731no—
234C4×C8—1731no—
235C24—3461no—
236C2×Q8C22×D17311yescharacteristic
237C2×Q8C22×D17311yescharacteristic
238C2×D4—3462no—
239C2×D4—3462no—
240C2×D4—3461no—
241C2×D4—1731no—
242C2×D4C22×D17311yescharacteristic
243C22×C4—21no—
244C22×C4—3461no—
245C22×C4—3461no—
246C22×C4—3461no—
247C22×C4—1731no—
248SD16—22no—
249SD16—22no—
250SD16—3462no—
251SD16—3462no—
252C2×C8—1731no—
253C2×C8—1731no—
254C2×C8C22×D17311yescharacteristic
255C2×C8C22×D17311yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—3461no—
259C4⋊C4—3461no—
260C4⋊C4—3461no—
261C4⋊C4—3461no—
262C4⋊C4—3461no—
263C4⋊C4—3461no—
264C4⋊C4—1731no—
265C4⋊C4—1731no—
266C4⋊C4—1731no—
267C4⋊C4—1731no—
268C4⋊C4—1731no—
269C4⋊C4C22×D17311yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—3461no—
272C22⋊C4—3461no—
273C22⋊C4—3461no—
274C22⋊C4—3461no—
275C22⋊C4—3461no—
276C22⋊C4—3461no—
277C22⋊C4—1731no—
278C22⋊C4—1731no—
279C42—3461no—
280C42—3461no—
281C42—3461no—
282C42—1731no—
283C42C22×D17311yescharacteristic
284C23—21no—
285C23—3464no—
286C23—3461no—
287C23—13841no—
288C23—13841no—
289C23—1731no—
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294D4—21no—
295D4—3464no—
296D4—3461no—
297D4—1732no—
298D4#1458382512yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—3461no—
304C2×C4—3461no—
305C2×C4—3461no—
306C2×C4—3461no—
307C2×C4—3461no—
308C2×C4—13841no—
309C2×C4—13841no—
310C2×C4—13841no—
311C2×C4—1731no—
312C2×C4—1731no—
313C2×C4—1731no—
314C2×C4D4×D17311yescharacteristic
315C2×C4D4×D17311yescharacteristic
316C2×C4C23×D17311yescharacteristic
317C8—21no—
318C8—21no—
319C8—3461no—
320C8—3461no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—3461no—
325C22—3462no—
326C22—3461no—
327C22—3461no—
328C22—13844no—
329C22—6921no—
330C22—6921no—
331C22C2×D4×D17311yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—3461no—
338C4—3461no—
339C4—3461no—
340C4—6921no—
341C4—6921no—
342C4—6921no—
343C4#41×D17311yescharacteristic
344C4C2×#1458382511yescharacteristic
345C2—42no—
346C2—6921no—
347C2—6922no—
348C2#2601200111yescharacteristic
349C2#2601229711yescharacteristic
350C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#461×C173C2
socleC22×C173C22×D4
prime radicalC2×D4×D173C22
hypercenter#461D173
hypocenterC173#3123
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #26012102C2semidirect
3. #26011831C2semidirect
4. #26011830C2semidirect
5. #26011792C2semidirect
6. #26011791C2semidirect
7. #26011693C2semidirect
10. #26011594C2semidirect
11. #26011592C2semidirect
12. SD16×Dic173C2semidirect
13. #26011576C2semidirect
14. #26011551C2semidirect
15. #461×C173C2semidirect
21. #19149671C22semidirect
26. Q8×Dic173C22semidirect
27. Q8×Dic173C22semidirect
37. #19149613C22semidirect
38. #19149612C22semidirect
39. #19149611C22semidirect
40. #19149610C22semidirect
41. C4⋊#13063436C22semidirect
42. #19149603C22semidirect
43. #19149603C22semidirect
50. #19149589C22semidirect
51. C8×Dic173C22semidirect
54. C4⋊Q8×C173C22semidirect
56. C4⋊C8×C173C22semidirect
57. #103×C173C22semidirect
58. #103×C173C22semidirect
175. C173#3123semidirect
184. #461D173semidirect
197. C4⋊Q8C2×D173semidirect
222. C4⋊C8C2×D173semidirect
225. #103C2×D173semidirect
226. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121022
#260119632
#260118312
#260118302
#260117922
#260117912
#260116932
#260116402
#260115942
#260115922
SD16×Dic1732
#260115762
#260115512
#461×C1732
#3123173