Finite Group Databasebeta

← #36365622

Subgroups — #36365622

Order 22144.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012212C211yesmaximal, characteristic, complemented
2#26012152C211yesmaximal, characteristic
3#26011852C211yesmaximal, characteristic, complemented
4#26011830C211yesmaximal, characteristic, complemented
5#26011816C211yesmaximal, characteristic, complemented
6#26011784C211yesmaximal, characteristic, complemented
7#26011739C211yesmaximal, characteristic, complemented
8#26011729C211yesmaximal, characteristic, complemented
9#26011715C211yesmaximal, characteristic, complemented
10#26011652C211yesmaximal, characteristic, complemented
11#26011598C211yesmaximal, characteristic, complemented
12#26011565C211yesmaximal, characteristic, complemented
13#26011500C211yesmaximal, characteristic, complemented
14#26011486C211yesmaximal, characteristic, complemented
15#451×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583813C2211yescharacteristic
18C2×#14583810C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic
20#19149678C2211yescharacteristic
21#19149674—21no—
22#19149672—21no—
23#19149668—21no—
24#19149668C2211yescharacteristic
25#19149665—21no—
26#19149655—21no—
27#19149655C2211yescharacteristic
28#19149655C2211yescharacteristic
29#19149653—21no—
30#19149653C2211yescharacteristic
31Q8×Dic173C2211yescharacteristic
32Q8×Dic173C2211yescharacteristic
33#19149644—21no—
34C4×C22⋊D173—21no—
35C4×C22⋊D173C2211yescharacteristic
36#19149638C2211yescharacteristic
37#19149637C2211yescharacteristic
38#19149613C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149603C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149589C2211yescharacteristic
52C22×Q8×C173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic
54C2×Q16×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583827—22no—
60#14583827—41no—
61C2×Dic346—21no—
62C2×D692—21no—
63C2×D692C2311yescharacteristic
64C2×C22⋊D173—21no—
65C2×C22⋊D173D411yescharacteristic
66C2×C4×D173—21no—
67C2×C4×D173C2311yescharacteristic
68#14583813—41no—
69#14583810—41no—
70C2×#13063436C2311yescharacteristic
71C2×#13063436C2311yescharacteristic
72C8×D173—41no—
73C4⋊Dic173—21no—
74C4⋊Dic173—21no—
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805D411yescharacteristic
82C22⋊Dic173—21no—
83C22⋊Dic173D411yescharacteristic
84#14583803—21no—
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803D411yescharacteristic
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96C2×Q8×C173—21no—
97C2×Q8×C173—22no—
98C2×Q8×C173—41no—
99C2×Q8×C173C2311yescharacteristic
100C2×Q8×C173C2311yescharacteristic
101C2×Q8×C173C2311yescharacteristic
102C22×C4×C173—21no—
103C22×C4×C173C2311yescharacteristic
104Q16×C173—22no—
105Q16×C173—41no—
106C2×C8×C173C2311yescharacteristic
107C2×C8×C173C2311yescharacteristic
108C4⋊C4×C173—21no—
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22×D173—21no—
112C22×D173C2×D411yescharacteristic
113Dic346—42no—
114D692—21no—
115D692—21no—
116D692—42no—
117C22⋊D173—41no—
118C22⋊D173—42no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173C2×D411yescharacteristic
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C4×D173—21no—
126C4×D173—21no—
127C4×D173—42no—
128#13063436—21no—
129#13063436—21no—
130C23×C173C2×D411yescharacteristic
131Q8×C173—21no—
132Q8×C173—21no—
133Q8×C173—21no—
134Q8×C173—21no—
135Q8×C173—21no—
136Q8×C173—42no—
137Q8×C173#4112yes—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—42no—
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2411yescharacteristic
147C8×C173—21no—
148C8×C173—21no—
149C22×C173—21no—
150C22×C173—21no—
151C22×C173—21no—
152C22×C173C22×D411yescharacteristic
153C2×D173—21no—
154C2×D173—21no—
155C2×D173—21no—
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—41no—
159C4×C173—21no—
160C4×C173—22no—
161C4×C173—41no—
162C4×C173—41no—
163C4×C173—41no—
164C4×C173C2×#4111yescharacteristic
165C4×C173H(5,𝔽2)11yescharacteristic
166Dic173—21no—
167Dic173—21no—
168Dic173—21no—
169Dic173—41no—
170Dic173—41no—
171Dic173—41no—
172C2×C173—41no—
173C2×C173#54611yescharacteristic
174C2×C173#57511yescharacteristic
175C2×C173#57811yescharacteristic
176D173—41no—
177D173—81no—
178C173#331311yescharacteristic, complemented, Sylow
179#3313—1731nomaximal, Sylow
180#556—1731no—
181#548—1731no—
182#489—1731no—
183#485—1731no—
184#482—1731no—
185#477—1731no—
186#468—1731no—
187#467—1731no—
188#464—1731no—
189#452—1731no—
190#451D17311yescharacteristic, complemented
191#441—1731no—
192#435—1731no—
193#419—1731no—
194#416—1731no—
195C2×#41—1731no—
196C22×Q8C2×D17311yescharacteristic
197C2×Q16C2×D17311yescharacteristic
198C2×Q16C2×D17311yescharacteristic
199C2×SD16—1731no—
200C2×M16—1731no—
201C22×C8—1731no—
202C4⋊Q8—1731no—
203#126—3461no—
204#126—3461no—
205#124—3461no—
206#124—3461no—
207#124—1731no—
208C22⋊Q8—3461no—
209C22⋊Q8—1731no—
210C22⋊Q8—1731no—
211C22⋊Q8C2×D17311yescharacteristic
212#121—3461no—
213#121—1731no—
214C4×Q8—3461no—
215C4×Q8—1731no—
216C4×Q8—1731no—
217C4×D4—3461no—
218C4×D4—1731no—
219#117—1731no—
220#117—1731no—
221#107—1731no—
222#107—1731no—
223#106—1731no—
224C4⋊C8—1731no—
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#103—1731no—
229#103—1731no—
230#103C2×D17311yescharacteristic
231#103C2×D17311yescharacteristic
232#102—1731no—
233#102—1731no—
234#102—1731no—
235C22⋊C8—1731no—
236C22⋊C8C2×D17311yescharacteristic
237#97—1731no—
238#41—3462no—
239#41—6921no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—3461no—
244C2×Q8C22×D17311yescharacteristic
245C2×Q8C22×D17311yescharacteristic
246C2×Q8C22×D17311yescharacteristic
247C2×D4—3461no—
248C2×D4—3461no—
249C2×D4—1731no—
250C2×D4—1731no—
251C22×C4—21no—
252C22×C4—3461no—
253C22×C4—1731no—
254C22×C4C22×D17311yescharacteristic
255Q16—22no—
256Q16—41no—
257SD16—6921no—
258M16—6921no—
259C2×C8—1731no—
260C2×C8—1731no—
261C2×C8—6921no—
262C2×C8C22×D17311yescharacteristic
263C2×C8C22×D17311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—1731no—
271C4⋊C4—1731no—
272C4⋊C4—1731no—
273C4⋊C4—1731no—
274C4⋊C4C22×D17311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—3461no—
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—1731no—
284C22⋊C4—1731no—
285C42—3461no—
286C42—3461no—
287C42—3461no—
288C42—3461no—
289C42—1731no—
290C23—3461no—
291C23—1731no—
292C23D4×D17311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8—6922no—
300Q8#1458382512yes—
301D4—3461no—
302D4—3461no—
303D4—6922no—
304D4—6921no—
305D4—6922no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
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×C4D4×D17311yescharacteristic
323C2×C4C23×D17311yescharacteristic
324C8—21no—
325C8—21no—
326C8—3461no—
327C8—3461no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—3461no—
332C22—3461no—
333C22—3461no—
334C22—6921no—
335C22—6921no—
336C22—6921no—
337C22C2×D4×D17311yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3461no—
344C4—3461no—
345C4—3461no—
346C4—6921no—
347C4—6921no—
348C4—6921no—
349C4C2×#1458382511yescharacteristic
350C4#1914972411yescharacteristic
351C2—41no—
352C2—13841no—
353C2—6921no—
354C2#2601212811yescharacteristic
355C2#2601229011yescharacteristic
356C2#2601230411yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#451×C173C2
socleC22×C173C22×D4
prime radical#26012152C2
hypercenter#451D173
hypocenterC173#3313
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #26012212C2semidirect
3. #26011852C2semidirect
4. #26011830C2semidirect
5. #26011816C2semidirect
6. #26011784C2semidirect
7. #26011739C2semidirect
8. #26011729C2semidirect
9. #26011715C2semidirect
10. #26011652C2semidirect
11. #26011598C2semidirect
12. #26011565C2semidirect
13. #26011500C2semidirect
14. #26011486C2semidirect
15. #451×C173C2semidirect
178. C173#3313semidirect
190. #451D173semidirect

Maximal subgroups

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

subgroupindex
#260122122
#260121522
#260118522
#260118302
#260118162
#260117842
#260117392
#260117292
#260117152
#260116522
#260115982
#260115652
#260115002
#260114862
#451×C1732
#3313173