Finite Group Databasebeta

← #36365114

Subgroups — #36365114

Order 22144.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149709C211yesmaximal, characteristic, complemented
2C2×#19149699C211yesmaximal, characteristic, complemented
3#26012150C211yesmaximal, characteristic, complemented
4#26011883C211yesmaximal, characteristic, complemented
5#26011739C211yesmaximal, characteristic, complemented
6#26011730C211yesmaximal, characteristic, complemented
7#26011706C211yesmaximal, characteristic, complemented
8#26011698C211yesmaximal, characteristic, complemented
9#26011653C211yesmaximal, characteristic, complemented
10#26011651C211yesmaximal, characteristic, complemented
11#26011646C211yesmaximal, characteristic, complemented
12#26011636C211yesmaximal, characteristic, complemented
13#26011591C211yesmaximal, characteristic, complemented
14#26011563C211yesmaximal, characteristic, complemented
15#547×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18#19149709—24no—
19#19149699—24no—
20C2×#14583816C2211yescharacteristic
21C2×#14583816C2211yescharacteristic
22C2×#14583816C2211yescharacteristic
23C2×#14583816C2211yescharacteristic
24C2×#14583814C2211yescharacteristic
25C2×#14583814C2211yescharacteristic
26C2×#14583813C2211yescharacteristic
27C2×#14583813C2211yescharacteristic
28C2×#14583811C2211yescharacteristic
29C2×#14583810C2211yescharacteristic
30C22×#13063436C2211yescharacteristic
31#19149667C2211yescharacteristic
32#19149665—21no—
33#19149658C2211yescharacteristic
34#19149657—21no—
35#19149653C2211yescharacteristic
36#19149651—21no—
37C4×Dic346C2211yescharacteristic
38#19149639—21no—
39#19149612C2211yescharacteristic
40C4⋊#13063436C2211yescharacteristic
41#19149603C2211yescharacteristic
42#19149603C2211yescharacteristic
43#19149602C2211yescharacteristic
44#19149602C2211yescharacteristic
45#19149600C2211yescharacteristic
46#19149599C2211yescharacteristic
47C22⋊#13063436C2211yescharacteristic
48C22⋊#13063436C2211yescharacteristic
49#19149590C2211yescharacteristic
50C2×#41×C173C2211yescharacteristic
51C2×#41×C173C2211yescharacteristic
52C4⋊Q8×C173C2211yescharacteristic
53#123×C173—21no—
54#123×C173—21no—
55C22⋊Q8×C173C2211yescharacteristic
56C22⋊Q8×C173C2211yescharacteristic
57#121×C173—21no—
58C4×D4×C173C2211yescharacteristic
59C4×D4×C173C2211yescharacteristic
60C2×C4⋊C4×C173—21no—
61#14583826—22no—
62#14583826—41no—
63C2×Dic346—22no—
64C2×Dic346—21no—
65C2×Dic346—41no—
66C2×Dic346C2311yescharacteristic
67C2×Dic346C2311yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C22×Dic173—21no—
72C2×C4×D173—21no—
73#14583816—22no—
74#14583816—22no—
75#14583816—22no—
76#14583816—41no—
77#14583814—22no—
78#14583814D414yescomplemented
79#14583813—22no—
80#14583813—41no—
81#14583811—22no—
82#14583810—22no—
83C2×#13063436—22no—
84C2×#13063436C2311yescharacteristic
85C2×#13063436C2311yescharacteristic
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C4⋊Dic173C2311yescharacteristic
89#14583805—21no—
90#14583805—21no—
91C22⋊Dic173—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95C4×Dic173—21no—
96#41×C173—22no—
97#41×C173—22no—
98#41×C173—41no—
99#41×C173—41no—
100C2×Q8×C173C2311yescharacteristic
101C2×Q8×C173C2311yescharacteristic
102C2×D4×C173—21no—
103C2×D4×C173—21no—
104C2×D4×C173D411yescharacteristic
105C2×D4×C173C2311yescharacteristic
106C22×C4×C173—21no—
107C22×C4×C173—21no—
108C22×C4×C173—21no—
109C22×C4×C173C2311yescharacteristic
110C22×C4×C173C2311yescharacteristic
111C4⋊C4×C173—21no—
112C4⋊C4×C173—21no—
113C4⋊C4×C173—21no—
114C4⋊C4×C173—21no—
115C4⋊C4×C173D411yescharacteristic
116C4⋊C4×C173C2311yescharacteristic
117C22⋊C4×C173—21no—
118C22⋊C4×C173—21no—
119C22⋊C4×C173—21no—
120C22⋊C4×C173D411yescharacteristic
121C22⋊C4×C173D411yescharacteristic
122C42×C173C2311yescharacteristic
123C22×D173—21no—
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—42no—
128Dic346C2×D412yes—
129D692—21no—
130D692—21no—
131C22⋊D173—42no—
132C22⋊D173—42no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—42no—
137C4×D173—42no—
138#13063436—21no—
139#13063436—21no—
140#13063436—21no—
141#13063436C2×D412yes—
142C23×C173—21no—
143C23×C173C2×D411yescharacteristic
144C23×C173C2×D411yescharacteristic
145Q8×C173—21no—
146Q8×C173—21no—
147Q8×C173—21no—
148Q8×C173—21no—
149D4×C173—21no—
150D4×C173—41no—
151D4×C173—42no—
152D4×C173—42no—
153D4×C173C2×D412yes—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—42no—
162C2×C4×C173—42no—
163C2×C4×C173—42no—
164C2×C4×C173C2×D411yescharacteristic
165C2×C4×C173C2×D411yescharacteristic
166C2×C4×C173C2×D411yescharacteristic
167C2×C4×C173C2×D411yescharacteristic
168C2×C4×C173C2411yescharacteristic
169C22×C173—21no—
170C22×C173—21no—
171C22×C173—22no—
172C22×C173—21no—
173C22×C173—21no—
174C22×C173—21no—
175C22×C173—21no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173C22×D411yescharacteristic
179C2×D173—41no—
180C2×D173—41no—
181C2×D173—41no—
182C4×C173—21no—
183C4×C173—21no—
184C4×C173—21no—
185C4×C173—21no—
186C4×C173—41no—
187C4×C173—41no—
188C4×C173—41no—
189C4×C173C22×D411yescharacteristic
190C4×C173H(5,𝔽2)11yescharacteristic
191Dic173—22no—
192Dic173—41no—
193Dic173—41no—
194C2×C173—41no—
195C2×C173—41no—
196C2×C173—42no—
197C2×C173D4211yescharacteristic
198C2×C173#57511yescharacteristic
199C2×C173#57811yescharacteristic
200D173—81no—
201C173#326911yescharacteristic, complemented, Sylow
202#3269—1731nomaximal, Sylow
203C2×#137—1731no—
204C2×#135—1731no—
205#548—1731no—
206#547D17311yescharacteristic, complemented
207#497—1731no—
208#468—1731no—
209#467—1731no—
210#463—1731no—
211#462—1731no—
212#452—1731no—
213#452—1731no—
214#451—1731no—
215#449—1731no—
216#440—1731no—
217#435—1731no—
218C2×#41—1731no—
219C2×#41C2×D17311yescharacteristic
220C2×#41C2×D17311yescharacteristic
221C22×Q8—1731no—
222#137—3464no—
223#135—3464no—
224C2×Q16—1731no—
225C2×Q16—1731no—
226C2×Q16—1731no—
227C2×Q16—1731no—
228C2×SD16—1731no—
229C2×SD16—1731no—
230C2×SD16—1731no—
231C2×SD16—1731no—
232C2×D8—1731no—
233C2×M16—1731no—
234C22×C8—1731no—
235C4⋊Q8C2×D17311yescharacteristic
236#124—3461no—
237#124—1731no—
238#123—21no—
239#123—21no—
240C22⋊Q8—3461no—
241C22⋊Q8—1731no—
242C22⋊Q8C2×D17311yescharacteristic
243C22⋊Q8C2×D17311yescharacteristic
244#121—21no—
245#121—3461no—
246#121—1731no—
247C4×Q8—1731no—
248C4×D4—3461no—
249C4×D4C2×D17311yescharacteristic
250C4×D4C2×D17311yescharacteristic
251C2×C4⋊C4—21no—
252#107—1731no—
253C4⋊C8—1731no—
254#103—1731no—
255#103—1731no—
256#103—1731no—
257#103—1731no—
258#102—1731no—
259#102—1731no—
260C22⋊C8—1731no—
261C22⋊C8—1731no—
262#97—1731no—
263#41—22no—
264#41—22no—
265#41—41no—
266#41—41no—
267#41—3462no—
268#41—6921no—
269C2×Q8—3462no—
270C2×Q8—3461no—
271C2×Q8—1731no—
272C2×Q8—1731no—
273C2×Q8—6921no—
274C2×Q8C22×D17311yescharacteristic
275C2×Q8C22×D17311yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—3461no—
279C2×D4—3461no—
280C2×D4—1731no—
281C2×D4C22⋊D17311yescharacteristic
282C2×D4C22×D17311yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—3461no—
287C22×C4—3461no—
288C22×C4C22×D17311yescharacteristic
289C22×C4C22×D17311yescharacteristic
290Q16—3462no—
291Q16—3462no—
292Q16—3462no—
293Q16—6921no—
294SD16—3462no—
295SD16—3462no—
296SD16—1734no—
297SD16—6921no—
298D8—3462no—
299M16—3462no—
300C2×C8—3462no—
301C2×C8—1731no—
302C2×C8—1731no—
303C2×C8—1731no—
304C2×C8—1731no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—3461no—
310C4⋊C4—3461no—
311C4⋊C4—1731no—
312C4⋊C4C22⋊D17311yescharacteristic
313C4⋊C4C22×D17311yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C22⋊C4—3461no—
320C22⋊C4—3461no—
321C22⋊C4C22⋊D17311yescharacteristic
322C22⋊C4C22⋊D17311yescharacteristic
323C42—3461no—
324C42C22×D17311yescharacteristic
325C23—21no—
326C23—3461no—
327C23C2×C22⋊D17311yescharacteristic
328C23C2×C22⋊D17311yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—3461no—
334Q8—3461no—
335Q8—3461no—
336Q8—1732no—
337Q8—6922no—
338D4—21no—
339D4—41no—
340D4—42no—
341D4—42no—
342D4—3461no—
343D4—3461no—
344D4—6922no—
345D4—6922no—
346D4D4×D17312yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—3461no—
358C2×C4—3461no—
359C2×C4—3461no—
360C2×C4—6922no—
361C2×C4—6922no—
362C2×C4C2×C22⋊D17311yescharacteristic
363C2×C4C2×C22⋊D17311yescharacteristic
364C2×C4C2×C22⋊D17311yescharacteristic
365C2×C4C2×C22⋊D17311yescharacteristic
366C2×C4C23×D17311yescharacteristic
367C8—3461no—
368C8—3461no—
369C8—3461no—
370C8—1732no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—41no—
379C22—41no—
380C22—6921no—
381C22—6921no—
382C22—6921no—
383C22C22×C22⋊D17311yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—3462no—
392C4—6921no—
393C4—6921no—
394C4C2×D4×D17311yescharacteristic
395C4#1914972411yescharacteristic
396C2—41no—
397C2—41no—
398C2—42no—
399C2—13841no—
400C2D4×C22⋊D17311yescharacteristic
401C2#2601228911yescharacteristic
402C2#2601230611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#547×C173C2
socleC22×C173C22×D4
hypercenter#547D173
hypocenterC173#3269
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#19149709C2semidirect
2. C2×#19149699C2semidirect
3. #26012150C2semidirect
4. #26011883C2semidirect
5. #26011739C2semidirect
6. #26011730C2semidirect
7. #26011706C2semidirect
8. #26011698C2semidirect
9. #26011653C2semidirect
10. #26011651C2semidirect
11. #26011646C2semidirect
12. #26011636C2semidirect
13. #26011591C2semidirect
14. #26011563C2semidirect
15. #547×C173C2semidirect
78. #14583814D4semidirect
201. C173#3269semidirect
206. #547D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497092
C2×#191496992
#260121502
#260118832
#260117392
#260117302
#260117062
#260116982
#260116532
#260116512
#260116462
#260116362
#260115912
#260115632
#547×C1732
#3269173