Finite Group Databasebeta

← #36364055

Subgroups — #36364055

Order 22144.

532 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149711C211yesmaximal, characteristic, complemented
2C2×#19149708C211yesmaximal, characteristic, complemented
3C2×#19149700C211yesmaximal, characteristic
4C2×Q16×D173C211yesmaximal, characteristic, complemented
5#26012066C211yesmaximal, characteristic, complemented
6#26012061C211yesmaximal, characteristic, complemented
7#26011885C211yesmaximal, characteristic, complemented
8#26011884C211yesmaximal, characteristic, complemented
9#26011882C211yesmaximal, characteristic, complemented
10#26011873C211yesmaximal, characteristic, complemented
11#26011872C211yesmaximal, characteristic, complemented
12#26011870C211yesmaximal, characteristic, complemented
13#26011869C211yesmaximal, characteristic, complemented
14#26011448C211yesmaximal, characteristic, complemented
15#497×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19C2×Q8×D173C2211yescharacteristic
20#19149711—24no—
21#19149708—24no—
22#19149700—24no—
23C2×Dic692C2211yescharacteristic
24C2×Dic692C2211yescharacteristic
25C2×#14583816C2211yescharacteristic
26C2×#14583816C2211yescharacteristic
27C2×#14583816C2211yescharacteristic
28C2×#14583816C2211yescharacteristic
29C2×#14583816C2211yescharacteristic
30Q16×D173—24no—
31C2×#14583815C2211yescharacteristic
32C2×#14583815C2211yescharacteristic
33C2×#14583814C2211yescharacteristic
34C2×#14583813C2211yescharacteristic
35C2×#14583811C2211yescharacteristic
36C2×#14583809C2211yescharacteristic
37C2×C8×D173C2211yescharacteristic
38#19149679C2211yescharacteristic
39#19149677C2211yescharacteristic
40#19149677C2211yescharacteristic
41#19149668C2211yescharacteristic
42#19149667C2211yescharacteristic
43#19149666C2211yescharacteristic
44#19149664—22no—
45#19149661—22no—
46#19149657—22no—
47#19149656—22no—
48#19149636C2211yescharacteristic
49#19149589C2211yescharacteristic
50C4×#13063436C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×Q16×C173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C2×SD16×C173C2211yescharacteristic
56C4⋊Q8×C173C2211yescharacteristic
57#124×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic
59#14583827—22no—
60#14583827—22no—
61#14583825—22no—
62#14583825—22no—
63C2×Dic346—21no—
64C2×Dic346—21no—
65C2×Dic346C2311yescharacteristic
66C2×Dic346C2311yescharacteristic
67C2×Dic346C2311yescharacteristic
68Q8×D173—22no—
69Q8×D173—22no—
70Q8×D173—22no—
71Q8×D173—22no—
72C2×D692—21no—
73C2×D692C2311yescharacteristic
74C2×C22⋊D173—21no—
75C22×Dic173—21no—
76C2×C4×D173—21no—
77C2×C4×D173—21no—
78C2×C4×D173—21no—
79C2×C4×D173C2311yescharacteristic
80Dic692—22no—
81Dic692—22no—
82#14583816—22no—
83#14583816—22no—
84#14583816—22no—
85#14583816—22no—
86#14583816—22no—
87#14583815—22no—
88#14583815—22no—
89#14583814—22no—
90#14583813—22no—
91#14583811—22no—
92#14583809D414yes—
93C2×#13063436C2311yescharacteristic
94C2×#13063436C2311yescharacteristic
95C8×D173D414yes—
96C4⋊Dic173—22no—
97#14583805—22no—
98#14583805—22no—
99#14583805D412yes—
100C22⋊Dic173—22no—
101#14583803—22no—
102#14583803—22no—
103#14583803D412yes—
104C4×Dic173C2311yescharacteristic
105C2×Q8×C173C2311yescharacteristic
106C2×Q8×C173C2311yescharacteristic
107C2×Q8×C173C2311yescharacteristic
108C2×D4×C173C2311yescharacteristic
109Q16×C173—22no—
110Q16×C173—22no—
111SD16×C173—22no—
112SD16×C173—22no—
113C2×C8×C173C2311yescharacteristic
114C2×C8×C173C2311yescharacteristic
115C4⋊C4×C173—22no—
116C22⋊C4×C173—22no—
117C42×C173C2311yescharacteristic
118C22×D173—21no—
119C22×D173C2×D411yescharacteristic
120Dic346—21no—
121Dic346—21no—
122Dic346—21no—
123Dic346—21no—
124Dic346—21no—
125Dic346—21no—
126Dic346—42no—
127Dic346—42no—
128D692—21no—
129D692—21no—
130D692—42no—
131C22⋊D173—42no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—42no—
136C2×Dic173C2×D412yes—
137C2×Dic173C2×D411yescharacteristic
138C4×D173—42no—
139C4×D173—42no—
140C4×D173—42no—
141C4×D173C2×D412yes—
142C4×D173C2×D412yes—
143#13063436C2×D412yes—
144#13063436C2×D412yes—
145C23×C173—21no—
146Q8×C173—21no—
147Q8×C173—21no—
148Q8×C173—21no—
149Q8×C173—21no—
150Q8×C173—21no—
151Q8×C173—21no—
152D4×C173—21no—
153D4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173C2×D412yes—
158C2×C4×C173C2411yescharacteristic
159C8×C173C2×D412yes—
160C8×C173C2×D412yes—
161C22×C173—41no—
162C22×C173—42no—
163C22×C173C22×D411yescharacteristic
164C2×D173—22no—
165C2×D173—41no—
166C2×D173—42no—
167C2×D173#12712yes—
168C4×C173—22no—
169C4×C173—41no—
170C4×C173—41no—
171C4×C173—41no—
172C4×C173C22×D411yescharacteristic
173C4×C173C22×D411yescharacteristic
174Dic173—22no—
175Dic173—41no—
176Dic173—41no—
177Dic173—41no—
178Dic173#12712yes—
179C2×C173—81no—
180C2×C173C2×#12711yescharacteristic
181C2×C173#57812yes—
182D173—22no—
183D173—81no—
184C173#315611yescharacteristic, complemented, Sylow
185#3156—1731nomaximal, Sylow
186C2×#137—1731no—
187C2×#137—1731no—
188C2×#135—1731no—
189C22×Q16—1731no—
190#536—1731no—
191#536—1731no—
192#497—1731no—
193#497—1731no—
194#497—1731no—
195#497D17311yescharacteristic, complemented
196#495—1731no—
197#495—1731no—
198#494—1731no—
199#494—1731no—
200#405—1731no—
201C2×#41—1731no—
202C2×#41—1731no—
203C22×Q8—1731no—
204C22×Q8—1731no—
205#137—3464no—
206#137—3464no—
207#135—3464no—
208C2×Q16—3464no—
209C2×Q16—1731no—
210C2×Q16—1731no—
211C2×Q16—1731no—
212C2×Q16—1731no—
213C2×Q16—1731no—
214C2×Q16—1731no—
215C2×Q16—1731no—
216C2×Q16C2×D17311yescharacteristic
217C2×Q16C2×D17311yescharacteristic
218C2×SD16—1731no—
219C2×SD16—1731no—
220C2×SD16—1731no—
221C2×SD16—1731no—
222C2×SD16C2×D17311yescharacteristic
223C2×SD16C2×D17311yescharacteristic
224C2×D8—1731no—
225C2×M16—1731no—
226C22×C8—1731no—
227C4⋊Q8—1731no—
228C4⋊Q8—1731no—
229C4⋊Q8—1731no—
230C4⋊Q8C2×D17311yescharacteristic
231#124—1731no—
232#124—1731no—
233#124—1731no—
234#124C2×D17311yescharacteristic
235#123—3462no—
236#123—3462no—
237C22⋊Q8—3462no—
238C22⋊Q8—3462no—
239#117—1731no—
240#97—1731no—
241#97C2×D17311yescharacteristic
242C4×C8—1731no—
243C4×C8—1731no—
244#41—3462no—
245#41—3462no—
246#41—3462no—
247#41—3462no—
248C2×Q8—3461no—
249C2×Q8—3461no—
250C2×Q8—3462no—
251C2×Q8—3462no—
252C2×Q8—3462no—
253C2×Q8—3462no—
254C2×Q8—1731no—
255C2×Q8—1731no—
256C2×Q8—1731no—
257C2×Q8C22×D17311yescharacteristic
258C2×Q8C22×D17311yescharacteristic
259C2×Q8C22×D17311yescharacteristic
260C2×D4—3461no—
261C2×D4—3461no—
262C2×D4—1731no—
263C2×D4C22×D17311yescharacteristic
264C22×C4—3461no—
265C22×C4—3461no—
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4—1731no—
269Q16—22no—
270Q16—22no—
271Q16—3462no—
272Q16—3462no—
273Q16—3462no—
274Q16—3462no—
275Q16—3462no—
276Q16—3462no—
277Q16—3462no—
278SD16—22no—
279SD16—22no—
280SD16—3462no—
281SD16—3462no—
282SD16—3462no—
283SD16—3462no—
284D8—3462no—
285M16—1734no—
286C2×C8—1731no—
287C2×C8—1731no—
288C2×C8—1734no—
289C2×C8C22×D17311yescharacteristic
290C2×C8C22×D17311yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—3462no—
293C4⋊C4—3462no—
294C4⋊C4—3462no—
295C4⋊C4—1732no—
296C22⋊C4—22no—
297C22⋊C4—3462no—
298C22⋊C4—3462no—
299C22⋊C4—3462no—
300C22⋊C4—1732no—
301C42—1731no—
302C42C22×D17311yescharacteristic
303C23—21no—
304C23—3461no—
305C23—1731no—
306Q8—21no—
307Q8—21no—
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—3461no—
313Q8—3461no—
314Q8—3461no—
315Q8—3461no—
316Q8—3461no—
317Q8—3461no—
318Q8—6922no—
319Q8—6922no—
320D4—21no—
321D4—21no—
322D4—3461no—
323D4—3461no—
324D4—6922no—
325D4—6922no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—1732no—
333C2×C4—1731no—
334C2×C4—1732no—
335C2×C4—1732no—
336C2×C4—6922no—
337C2×C4—6922no—
338C2×C4—6922no—
339C2×C4—6922no—
340C2×C4D4×D17312yes—
341C2×C4C23×D17311yescharacteristic
342C8—1732no—
343C8—1732no—
344C8D4×D17312yes—
345C8D4×D17312yes—
346C22—41no—
347C22—42no—
348C22—3462no—
349C22—1732no—
350C22—6921no—
351C22—6922no—
352C22C2×D4×D17311yescharacteristic
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3462no—
358C4—1732no—
359C4—6921no—
360C4—6921no—
361C4—6921no—
362C4C2×D4×D17311yescharacteristic
363C4C2×D4×D17311yescharacteristic
364C2—81no—
365C2—3462no—
366C2—13841no—
367C2#127×D17311yescharacteristic
368C2#2601230712yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#497×C173C2
socleC22×C173C22×D4
prime radicalC2×#19149700C2
hypercenter#497D173
hypocenterC173#3156
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#19149711C2semidirect
2. C2×#19149708C2semidirect
4. C2×Q16×D173C2semidirect
5. #26012066C2semidirect
6. #26012061C2semidirect
7. #26011885C2semidirect
8. #26011884C2semidirect
9. #26011882C2semidirect
10. #26011873C2semidirect
11. #26011872C2semidirect
12. #26011870C2semidirect
13. #26011869C2semidirect
14. #26011448C2semidirect
15. #497×C173C2semidirect
184. C173#3156semidirect
195. #497D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497112
C2×#191497082
C2×#191497002
C2×Q16×D1732
#260120662
#260120612
#260118852
#260118842
#260118822
#260118732
#260118722
#260118702
#260118692
#260114482
#497×C1732
#3156173