Finite Group Databasebeta

← #36364167

Subgroups — #36364167

Order 22144.

364 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012045C211yesmaximal, characteristic, complemented
2#26012028C211yesmaximal, characteristic
3#26011899C211yesmaximal, characteristic, complemented
4#26011884C211yesmaximal, characteristic, complemented
5#26011858C211yesmaximal, characteristic, complemented
6#26011852C211yesmaximal, characteristic, complemented
7#26011748C211yesmaximal, characteristic, complemented
8#26011746C211yesmaximal, characteristic, complemented
9#26011740C211yesmaximal, characteristic
10#26011738C211yesmaximal, characteristic
11C4×#14583816C211yesmaximal, characteristic, complemented
12C4×#14583815C211yesmaximal, characteristic, complemented
13C4×#14583813C211yesmaximal, characteristic, complemented
14C4×#14583809C211yesmaximal, characteristic, complemented
15C4×Q16×C173C211yesmaximal, characteristic, complemented
16C2×#14583816C2211yescharacteristic
17C2×#14583815C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19C2×#14583809C2211yescharacteristic
20C2×#14583809C2211yescharacteristic
21#19149678C2211yescharacteristic
22#19149677C2211yescharacteristic
23#19149674—21no—
24#19149670C2211yescharacteristic
25#19149669—21no—
26#19149668C2211yescharacteristic
27#19149664—21no—
28#19149659C2211yescharacteristic
29#19149657—21no—
30#19149655C2211yescharacteristic
31#19149655C2211yescharacteristic
32#19149654C2211yescharacteristic
33C4×Dic346—21no—
34C4×Dic346C2211yescharacteristic
35C4×D692—21no—
36C4×D692C2211yescharacteristic
37#19149636—21no—
38#19149636—21no—
39C42×D173C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149611C2211yescharacteristic
42#19149610C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149589C2211yescharacteristic
50#19149589C2211yescharacteristic
51C4×#13063436C2211yescharacteristic
52C2×Q16×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54C4×Q8×C173C2211yescharacteristic
55#107×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61C2×D692—21no—
62C2×D692C2311yescharacteristic
63C2×C4×D173—21no—
64C2×C4×D173D411yescharacteristic
65C2×C4×D173D411yescharacteristic
66C2×C4×D173C2311yescharacteristic
67#14583816—22no—
68#14583815—22no—
69#14583813—22no—
70#14583809—22no—
71#14583809—22no—
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C4⋊Dic173—21no—
75C4⋊Dic173—21no—
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—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—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—21no—
114D692—21no—
115D692—42no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173C2×D411yescharacteristic
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C4×D173—22no—
123C4×D173—21no—
124C4×D173—21no—
125C4×D173—41no—
126C4×D173—42no—
127#13063436—21no—
128#13063436#4112yes—
129Q8×C173—21no—
130Q8×C173—21no—
131Q8×C173—21no—
132Q8×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173C2×D411yescharacteristic
138C2×C4×C173C2×D411yescharacteristic
139C2×C4×C173C2411yescharacteristic
140C8×C173—21no—
141C8×C173#4112yes—
142C22×C173C22×D411yescharacteristic
143C2×D173—21no—
144C2×D173—21no—
145C2×D173—21no—
146C2×D173—41no—
147C2×D173—41no—
148C2×D173—41no—
149C4×C173—21no—
150C4×C173—41no—
151C4×C173—41no—
152C4×C173—41no—
153C4×C173—41no—
154C4×C173#13712yes—
155C4×C173C2×#4111yescharacteristic
156C4×C173C2×#4111yescharacteristic
157Dic173—21no—
158Dic173—21no—
159Dic173—21no—
160Dic173—41no—
161Dic173—41no—
162Dic173—41no—
163C2×C173#53211yescharacteristic
164C2×C173C2×#13711yescharacteristic
165C2×C173#57511yescharacteristic
166D173—41no—
167D173—81no—
168C173#317211yescharacteristic, complemented, Sylow
169#3172—1731nomaximal, Sylow
170#533—1731no—
171#529—1731no—
172#501—1731no—
173#497—1731no—
174#491—1731no—
175#489—1731no—
176#470—1731no—
177#470—1731no—
178#468—1731no—
179#468—1731no—
180C4×Q16—1731no—
181C4×Q16D17311yescharacteristic, complemented
182C4×SD16—1731no—
183C4×SD16—1731no—
184C4×M16—1731no—
185C2×Q16—1731no—
186C2×Q16C2×D17311yescharacteristic
187C2×SD16—1731no—
188C2×SD16—1731no—
189C2×M16—1731no—
190C2×M16—1731no—
191C4⋊Q8—1731no—
192C4⋊Q8—1731no—
193#126—3461no—
194#125—3461no—
195#125—1731no—
196#124—1731no—
197#123—3461no—
198C22⋊Q8—3461no—
199C22⋊Q8—1731no—
200C22⋊Q8—1731no—
201C22⋊Q8—1731no—
202#121—1731no—
203C4×Q8—3461no—
204C4×Q8—1731no—
205C4×Q8C2×D17311yescharacteristic
206C4×Q8C2×D17311yescharacteristic
207C4×D4—3461no—
208C4×D4—1731no—
209#117—3461no—
210#117—3461no—
211C2×C42—1731no—
212#107—1731no—
213#107C2×D17311yescharacteristic
214#106—1731no—
215#106—1731no—
216#103—1731no—
217#103—1731no—
218#103—1731no—
219#103—1731no—
220#103C2×D17311yescharacteristic
221#103C2×D17311yescharacteristic
222#102—1731no—
223#102—1731no—
224#97—1731no—
225#97—1731no—
226C4×C8—1731no—
227C4×C8C2×D17311yescharacteristic
228C2×Q8—3461no—
229C2×Q8—1731no—
230C2×Q8C22×D17311yescharacteristic
231C2×Q8C22×D17311yescharacteristic
232C2×D4—3461no—
233C2×D4—1731no—
234C22×C4—3461no—
235C22×C4—1731no—
236C22×C4—1731no—
237C22×C4—1731no—
238Q16—22no—
239Q16—3462no—
240SD16—3462no—
241SD16—3462no—
242M16—3462no—
243M16—3462no—
244C2×C8—1731no—
245C2×C8—1731no—
246C2×C8C22×D17311yescharacteristic
247C2×C8C22×D17311yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—3461no—
251C4⋊C4—3461no—
252C4⋊C4—3461no—
253C4⋊C4—3461no—
254C4⋊C4—3461no—
255C4⋊C4—3461no—
256C4⋊C4—3461no—
257C4⋊C4—3461no—
258C4⋊C4—3461no—
259C4⋊C4—1731no—
260C4⋊C4—1731no—
261C4⋊C4C22×D17311yescharacteristic
262C4⋊C4C22×D17311yescharacteristic
263C22⋊C4—3461no—
264C22⋊C4—3461no—
265C22⋊C4—3461no—
266C22⋊C4—3461no—
267C22⋊C4—3461no—
268C22⋊C4—3461no—
269C22⋊C4—3461no—
270C42—21no—
271C42—21no—
272C42—3461no—
273C42—1731no—
274C42—1731no—
275C42—1731no—
276C42C22×D17311yescharacteristic
277C23—3461no—
278C23—1731no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—3461no—
284Q8—3461no—
285Q8—6922no—
286D4—3461no—
287D4—3461no—
288D4—6922no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—3462no—
294C2×C4—3461no—
295C2×C4—3461no—
296C2×C4—3461no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—1731no—
300C2×C4—1731no—
301C2×C4—1731no—
302C2×C4—6921no—
303C2×C4—6922no—
304C2×C4D4×D17311yescharacteristic
305C2×C4D4×D17311yescharacteristic
306C2×C4C23×D17311yescharacteristic
307C8—21no—
308C8—3461no—
309C8—1732no—
310C8#1458382612yes—
311C22—3461no—
312C22—3461no—
313C22—3461no—
314C22—6921no—
315C22—6921no—
316C22—6921no—
317C22C2×D4×D17311yescharacteristic
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—3461no—
324C4—3461no—
325C4—3461no—
326C4—6921no—
327C4—6921no—
328C4—6921no—
329C4#1914971112yes—
330C4#41×D17311yescharacteristic
331C4C2×#1458382611yescharacteristic
332C2—13841no—
333C2—6921no—
334C2#2601203811yescharacteristic
335C2C2×#1914971111yescharacteristic
336C2#2601228811yescharacteristic
subgroupG/H
centerC2×C4D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC4×Q16×C173C2
socleC22×C173C22×D4
prime radical#19149654C22
hypercenterC4×Q16D173
hypocenterC173#3172
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. #26012045C2semidirect
3. #26011899C2semidirect
4. #26011884C2semidirect
5. #26011858C2semidirect
6. #26011852C2semidirect
7. #26011748C2semidirect
8. #26011746C2semidirect
11. C4×#14583816C2semidirect
12. C4×#14583815C2semidirect
13. C4×#14583813C2semidirect
14. C4×#14583809C2semidirect
15. C4×Q16×C173C2semidirect
168. C173#3172semidirect
181. C4×Q16D173semidirect

Maximal subgroups

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

subgroupindex
#260120452
#260120282
#260118992
#260118842
#260118582
#260118522
#260117482
#260117462
#260117402
#260117382
C4×#145838162
C4×#145838152
C4×#145838132
C4×#145838092
C4×Q16×C1732
#3172173