Finite Group Databasebeta

← #36364517

Subgroups — #36364517

Order 22144.

394 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012043C211yesmaximal, characteristic, complemented
2#26012037C211yesmaximal, characteristic
3#26011869C211yesmaximal, characteristic, complemented
4#26011862C211yesmaximal, characteristic, complemented
5#26011840C211yesmaximal, characteristic, complemented
6#26011836C211yesmaximal, characteristic, complemented
7#26011716C211yesmaximal, characteristic, complemented
8#26011715C211yesmaximal, characteristic, complemented
9#26011712C211yesmaximal, characteristic, complemented
10#26011711C211yesmaximal, characteristic, complemented
11#26011655C211yesmaximal, characteristic, complemented
12#26011653C211yesmaximal, characteristic, complemented
13#26011651C211yesmaximal, characteristic, complemented
14#26011541C211yesmaximal, characteristic, complemented
15#452×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×Dic692C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583816C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21C2×#14583814C2211yescharacteristic
22C2×#14583813C2211yescharacteristic
23#19149678C2211yescharacteristic
24#19149677C2211yescharacteristic
25#19149677C2211yescharacteristic
26#19149675C2211yescharacteristic
27#19149669—21no—
28#19149668C2211yescharacteristic
29#19149666C2211yescharacteristic
30#19149658C2211yescharacteristic
31#19149656—21no—
32#19149651—21no—
33#19149651—21no—
34Q8×Dic173C2211yescharacteristic
35#19149644—21no—
36#19149644C2211yescharacteristic
37C4×C22⋊D173—21no—
38C4×C22⋊D173—21no—
39#19149635—21no—
40C2×C4×Dic173C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50C8×Dic173C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—41no—
60#14583826—41no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63C2×Dic346C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C22×Dic173D411yescharacteristic
69C22×Dic173D411yescharacteristic
70C2×C4×D173—21no—
71Dic692—41no—
72#14583816—41no—
73#14583816—41no—
74#14583815—41no—
75#14583814—41no—
76#14583813—41no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C4⋊Dic173—21no—
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91#14583803—21no—
92#14583803—21no—
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173D411yescharacteristic
96C4×Dic173D411yescharacteristic
97C4×Dic173C2311yescharacteristic
98C4×Dic173C2311yescharacteristic
99#41×C173—41no—
100#41×C173—41no—
101C2×Q8×C173C2311yescharacteristic
102C2×Q8×C173C2311yescharacteristic
103C2×D4×C173—21no—
104C2×D4×C173C2311yescharacteristic
105C22×C4×C173—21no—
106C22×C4×C173C2311yescharacteristic
107Q16×C173—41no—
108SD16×C173—41no—
109C2×C8×C173C2311yescharacteristic
110C2×C8×C173C2311yescharacteristic
111C4⋊C4×C173—21no—
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22×D173—21no—
115Dic346—21no—
116Dic346—21no—
117Dic346—21no—
118Dic346—21no—
119Dic346—42no—
120D692—21no—
121D692—21no—
122C22⋊D173—42no—
123C22⋊D173—42no—
124C22⋊D173—42no—
125C2×Dic173—22no—
126C2×Dic173—22no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C4×D173—42no—
135#13063436—21no—
136#13063436—21no—
137C23×C173—21no—
138C23×C173C2×D411yescharacteristic
139Q8×C173—21no—
140Q8×C173—21no—
141Q8×C173—21no—
142Q8×C173—21no—
143D4×C173—21no—
144D4×C173—21no—
145D4×C173—42no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—42no—
152C2×C4×C173C2×D411yescharacteristic
153C2×C4×C173C2411yescharacteristic
154C8×C173—21no—
155C8×C173—21no—
156C22×C173—21no—
157C22×C173—21no—
158C22×C173—21no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173C22×D411yescharacteristic
163C2×D173—41no—
164C2×D173—41no—
165C2×D173—41no—
166C4×C173—21no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173—41no—
170C4×C173H(5,𝔽2)11yescharacteristic
171C4×C173H(5,𝔽2)11yescharacteristic
172Dic173—21no—
173Dic173—21no—
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177Dic173#13512yes—
178Dic173#13712yes—
179C2×C173—41no—
180C2×C173—81no—
181C2×C173#53511yescharacteristic
182C2×C173C2×#13511yescharacteristic
183C2×C173C2×#13711yescharacteristic
184D173—81no—
185C173#321011yescharacteristic, complemented, Sylow
186#3210—1731nomaximal, Sylow
187#533—1731no—
188#532—1731no—
189#494—1731no—
190#492—1731no—
191#487—1731no—
192#486—1731no—
193#464—1731no—
194#464—1731no—
195#464—1731no—
196#464—1731no—
197#452—1731no—
198#452—1731no—
199#452—1731no—
200#452D17311yescharacteristic, complemented
201#431—1731no—
202C2×#41—1731no—
203C2×#41C2×D17311yescharacteristic
204C2×Q16—1731no—
205C2×Q16—1731no—
206C2×Q16—1731no—
207C2×Q16C2×D17311yescharacteristic
208C2×SD16—1731no—
209C2×SD16—1731no—
210C2×SD16—1731no—
211C2×SD16C2×D17311yescharacteristic
212C4⋊Q8—1731no—
213C4⋊Q8—1731no—
214C4⋊Q8—1731no—
215#127—1731no—
216#125—3461no—
217#124—1731no—
218#124—1731no—
219C22⋊Q8—3461no—
220C22⋊Q8—1731no—
221C22⋊Q8C2×D17311yescharacteristic
222#121—3461no—
223#121—3461no—
224C4×Q8—3461no—
225C4×Q8—1731no—
226C4×Q8—1731no—
227C4×D4—3461no—
228C4×D4—3461no—
229#117—3461no—
230C2×C42—1731no—
231C4⋊C8—1731no—
232C4⋊C8—1731no—
233#103—1731no—
234#103—1731no—
235#103—1731no—
236#103C2×D17311yescharacteristic
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102C2×D17311yescharacteristic
241C22⋊C8—1731no—
242C22⋊C8C2×D17311yescharacteristic
243C4×C8—1731no—
244C4×C8—1731no—
245#41—41no—
246#41—41no—
247#41—6921no—
248#41—6921no—
249C2×Q8—3461no—
250C2×Q8—1731no—
251C2×Q8—1731no—
252C2×Q8C22×D17311yescharacteristic
253C2×Q8C22×D17311yescharacteristic
254C2×D4—21no—
255C2×D4—3461no—
256C2×D4—3461no—
257C2×D4—3461no—
258C2×D4—1731no—
259C2×D4C22×D17311yescharacteristic
260C22×C4—21no—
261C22×C4—3461no—
262C22×C4—1731no—
263C22×C4—1731no—
264C22×C4C22×D17311yescharacteristic
265Q16—41no—
266Q16—6921no—
267Q16—6921no—
268Q16—6921no—
269SD16—41no—
270SD16—6921no—
271SD16—6921no—
272SD16—6921no—
273C2×C8—1731no—
274C2×C8—1731no—
275C2×C8C22×D17311yescharacteristic
276C2×C8C22×D17311yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—3461no—
281C4⋊C4—3461no—
282C4⋊C4—3461no—
283C4⋊C4—3461no—
284C4⋊C4—3461no—
285C4⋊C4—3461no—
286C4⋊C4—1731no—
287C4⋊C4C22×D17311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C42—3461no—
295C42—3461no—
296C42—1731no—
297C42—1731no—
298C42—1731no—
299C42—1731no—
300C23—21no—
301C23—3461no—
302C23D4×D17311yescharacteristic
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—3461no—
308Q8—3461no—
309Q8—3461no—
310Q8—3461no—
311Q8—6922no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—3461no—
316D4—3461no—
317D4—6922no—
318D4—6922no—
319D4—6922no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—3462no—
327C2×C4—3462no—
328C2×C4—3461no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—1731no—
332C2×C4—1731no—
333C2×C4—1731no—
334C2×C4—1731no—
335C2×C4—6922no—
336C2×C4D4×D17311yescharacteristic
337C2×C4C23×D17311yescharacteristic
338C8—21no—
339C8—21no—
340C8—3461no—
341C8—3461no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—6921no—
349C22—6921no—
350C22—6921no—
351C22C2×D4×D17311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—3461no—
357C4—3461no—
358C4—1732no—
359C4—1732no—
360C4—6921no—
361C4—6921no—
362C4—6921no—
363C4#1914972411yescharacteristic
364C4#1914972411yescharacteristic
365C2—41no—
366C2—81no—
367C2—13841no—
368C2#2601205611yescharacteristic
369C2#135×D17311yescharacteristic
370C2#137×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#452×C173C2
socleC22×C173C22×D4
prime radical#26012037C2
hypercenter#452D173
hypocenterC173#3210
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #26012043C2semidirect
3. #26011869C2semidirect
4. #26011862C2semidirect
5. #26011840C2semidirect
6. #26011836C2semidirect
7. #26011716C2semidirect
8. #26011715C2semidirect
9. #26011712C2semidirect
10. #26011711C2semidirect
11. #26011655C2semidirect
12. #26011653C2semidirect
13. #26011651C2semidirect
14. #26011541C2semidirect
15. #452×C173C2semidirect
185. C173#3210semidirect
200. #452D173semidirect

Maximal subgroups

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

subgroupindex
#260120432
#260120372
#260118692
#260118622
#260118402
#260118362
#260117162
#260117152
#260117122
#260117112
#260116552
#260116532
#260116512
#260115412
#452×C1732
#3210173