Finite Group Databasebeta

← #36365213

Subgroups — #36365213

Order 22144.

476 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×#19149708C211yesmaximal, characteristic, complemented
3#26012148C211yesmaximal, characteristic, complemented
4#26012147C211yesmaximal, characteristic, complemented
5#26011874C211yesmaximal, characteristic, complemented
6#26011736C211yesmaximal, characteristic, complemented
7#26011735C211yesmaximal, characteristic, complemented
8#26011712C211yesmaximal, characteristic, complemented
9#26011711C211yesmaximal, characteristic, complemented
10#26011649C211yesmaximal, characteristic, complemented
11#26011648C211yesmaximal, characteristic, complemented
12#26011633C211yesmaximal, characteristic, complemented
13#26011583C211yesmaximal, characteristic, complemented
14#26011566C211yesmaximal, characteristic, complemented
15#449×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19#19149709—24no—
20#19149708—24no—
21C2×Dic692C2211yescharacteristic
22C2×Dic692C2211yescharacteristic
23C2×#14583816C2211yescharacteristic
24C2×#14583816C2211yescharacteristic
25C2×#14583814C2211yescharacteristic
26C2×#14583814C2211yescharacteristic
27C2×#14583814C2211yescharacteristic
28C2×#14583810C2211yescharacteristic
29C2×#14583809C2211yescharacteristic
30#19149666—21no—
31#19149666C2211yescharacteristic
32#19149666C2211yescharacteristic
33#19149665—21no—
34#19149659C2211yescharacteristic
35#19149658—21no—
36#19149658C2211yescharacteristic
37#19149656—21no—
38#19149651—21no—
39#19149650—21no—
40#19149650C2211yescharacteristic
41Q8×Dic173C2211yescharacteristic
42#19149644C2211yescharacteristic
43D4×Dic173—21no—
44C4×C22⋊D173—21no—
45C4×C22⋊D173C2211yescharacteristic
46#19149612C2211yescharacteristic
47#19149608C2211yescharacteristic
48#19149604C2211yescharacteristic
49#19149604C2211yescharacteristic
50#19149602C2211yescharacteristic
51#19149600C2211yescharacteristic
52#19149591C2211yescharacteristic
53C8×Dic173C2211yescharacteristic
54C2×#41×C173C2211yescharacteristic
55C2×SD16×C173C2211yescharacteristic
56C2×D8×C173C2211yescharacteristic
57#121×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59#102×C173C2211yescharacteristic
60C22⋊C8×C173C2211yescharacteristic
61#14583825—22no—
62#14583825—41no—
63C2×Dic346—21no—
64C2×Dic346—22no—
65C2×Dic346—21no—
66C2×Dic346—41no—
67C2×Dic346C2311yescharacteristic
68C2×Dic346C2311yescharacteristic
69C2×Dic346C2311yescharacteristic
70Q8×D173—22no—
71Q8×D173—41no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173D411yescharacteristic
75C22×Dic173—21no—
76C22×Dic173—21no—
77C2×C4×D173—21no—
78C2×C4×D173C2311yescharacteristic
79Dic692—22no—
80Dic692—41no—
81#14583816—22no—
82#14583816D414yescomplemented
83#14583814—22no—
84#14583814—22no—
85#14583814—22no—
86#14583810—22no—
87#14583809—22no—
88C2×#13063436C2311yescharacteristic
89C2×#13063436C2311yescharacteristic
90C4⋊Dic173—21no—
91C4⋊Dic173—21no—
92C4⋊Dic173C2311yescharacteristic
93#14583805—21no—
94#14583805—21no—
95#14583805D411yescharacteristic
96C22⋊Dic173—21no—
97C22⋊Dic173—21no—
98C22⋊Dic173—21no—
99C22⋊Dic173—21no—
100C22⋊Dic173—21no—
101C22⋊Dic173—21no—
102C22⋊Dic173D411yescharacteristic
103#14583803—21no—
104#14583803D411yescharacteristic
105C4×Dic173—21no—
106C4×Dic173—21no—
107C4×Dic173C2311yescharacteristic
108#41×C173—22no—
109#41×C173—41no—
110C2×Q8×C173C2311yescharacteristic
111C2×D4×C173—21no—
112C2×D4×C173—21no—
113C2×D4×C173C2311yescharacteristic
114C2×D4×C173C2311yescharacteristic
115C22×C4×C173—21no—
116C22×C4×C173C2311yescharacteristic
117SD16×C173—22no—
118D8×C173—41no—
119C2×C8×C173C2311yescharacteristic
120C2×C8×C173C2311yescharacteristic
121C4⋊C4×C173C2311yescharacteristic
122C22⋊C4×C173—21no—
123C22×D173C2×D411yescharacteristic
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—21no—
129Dic346—42no—
130Dic346—42no—
131Dic346C2×D412yes—
132C22⋊D173—41no—
133C22⋊D173—42no—
134C22⋊D173—42no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—42no—
140C2×Dic173—42no—
141C2×Dic173C2×D411yescharacteristic
142C2×Dic173C2×D411yescharacteristic
143C2×Dic173C2×D411yescharacteristic
144C4×D173—21no—
145C4×D173—21no—
146C4×D173—42no—
147#13063436—21no—
148#13063436C2×D412yes—
149C23×C173—21no—
150C23×C173—21no—
151C23×C173C2×D411yescharacteristic
152Q8×C173—21no—
153Q8×C173C2×D412yes—
154D4×C173—21no—
155D4×C173—21no—
156D4×C173—21no—
157D4×C173—21no—
158D4×C173—42no—
159D4×C173—42no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—42no—
165C2×C4×C173C2×D411yescharacteristic
166C2×C4×C173C2411yescharacteristic
167C8×C173—21no—
168C8×C173—21no—
169C22×C173—21no—
170C22×C173—21no—
171C22×C173—21no—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173—41no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173C22×D411yescharacteristic
179C2×D173—21no—
180C2×D173—21no—
181C2×D173—21no—
182C4×C173—21no—
183C4×C173—22no—
184C4×C173—41no—
185C4×C173C22×D411yescharacteristic
186C4×C173H(5,𝔽2)11yescharacteristic
187Dic173—21no—
188Dic173—22no—
189Dic173—21no—
190Dic173—21no—
191Dic173—41no—
192Dic173—41no—
193Dic173—41no—
194C2×C173—41no—
195C2×C173—81no—
196C2×C173—81no—
197C2×C173D4211yescharacteristic
198C2×C173#57511yescharacteristic
199C2×C173#57811yescharacteristic
200D173—41no—
201C173#327711yescharacteristic, complemented, Sylow
202#3277—1731nomaximal, Sylow
203C2×#137—1731no—
204C2×#137—1731no—
205#548—1731no—
206#548—1731no—
207#495—1731no—
208#468—1731no—
209#468—1731no—
210#464—1731no—
211#464—1731no—
212#451—1731no—
213#451—1731no—
214#449—1731no—
215#449D17311yescharacteristic, complemented
216C4×Q16—1731no—
217#436—1731no—
218C2×#41—1731no—
219C2×#41C2×D17311yescharacteristic
220C22×Q8—1731no—
221C22×Q8—1731no—
222#137—3464no—
223#137—3464no—
224C2×Q16—1731no—
225C2×Q16—1731no—
226C2×Q16—1731no—
227C2×Q16—1731no—
228C2×SD16—1731no—
229C2×SD16—1731no—
230C2×SD16—1731no—
231C2×SD16C2×D17311yescharacteristic
232C2×D8C2×D17311yescharacteristic
233C2×M16—1731no—
234C2×M16—1731no—
235#124—3461no—
236#124—3461no—
237#124—1731no—
238#124—1731no—
239C22⋊Q8—3461no—
240C22⋊Q8—3461no—
241C22⋊Q8—1731no—
242C22⋊Q8—1731no—
243#121—3461no—
244#121—3461no—
245#121—1731no—
246#121C2×D17311yescharacteristic
247C4×Q8—1731no—
248C4×Q8—1731no—
249C4×D4—3461no—
250C4×D4—3461no—
251C4×D4—1731no—
252#107—1731no—
253C4⋊C8—1731no—
254#103—1731no—
255#103—1731no—
256#103—1731no—
257#103C2×D17311yescharacteristic
258#102—1731no—
259#102C2×D17311yescharacteristic
260C22⋊C8—1731no—
261C22⋊C8C2×D17311yescharacteristic
262C4×C8—1731no—
263#41—22no—
264#41—41no—
265#41—3462no—
266#41—6921no—
267C2×Q8—3461no—
268C2×Q8—3462no—
269C2×Q8—3461no—
270C2×Q8—3462no—
271C2×Q8—1731no—
272C2×Q8—1731no—
273C2×Q8—1731no—
274C2×Q8—6921no—
275C2×Q8—6921no—
276C2×Q8C22×D17311yescharacteristic
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—3461no—
280C2×D4—3461no—
281C2×D4—1731no—
282C2×D4C22×D17311yescharacteristic
283C2×D4C22×D17311yescharacteristic
284C22×C4—21no—
285C22×C4—3461no—
286C22×C4—3461no—
287C22×C4—3461no—
288C22×C4—1731no—
289C22×C4C22×D17311yescharacteristic
290Q16—3462no—
291Q16—3462no—
292Q16—1734no—
293Q16—6921no—
294SD16—22no—
295SD16—3462no—
296SD16—3462no—
297SD16—3462no—
298D8—41no—
299M16—3462no—
300M16—3462no—
301C2×C8—1731no—
302C2×C8—1731no—
303C2×C8C22×D17311yescharacteristic
304C2×C8C22×D17311yescharacteristic
305C4⋊C4—3461no—
306C4⋊C4—3461no—
307C4⋊C4—3461no—
308C4⋊C4—3461no—
309C4⋊C4—1731no—
310C4⋊C4—1731no—
311C4⋊C4C22×D17311yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—3461no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4—3461no—
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C22⋊C4—3461no—
320C22⋊C4—1731no—
321C22⋊C4—1731no—
322C42—3461no—
323C42—3461no—
324C42—1731no—
325C23—21no—
326C23—21no—
327C23—1731no—
328C23D4×D17311yescharacteristic
329Q8—21no—
330Q8—3461no—
331Q8—3461no—
332Q8—3461no—
333Q8—3461no—
334Q8—3461no—
335Q8—1732no—
336Q8—6922no—
337Q8—6922no—
338Q8D4×D17312yes—
339D4—21no—
340D4—21no—
341D4—21no—
342D4—21no—
343D4—42no—
344D4—42no—
345D4—6921no—
346D4—6922no—
347D4—6922no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—3461no—
354C2×C4—3461no—
355C2×C4—3461no—
356C2×C4—3461no—
357C2×C4—3461no—
358C2×C4—3461no—
359C2×C4—1731no—
360C2×C4—1731no—
361C2×C4—1731no—
362C2×C4—6922no—
363C2×C4—6922no—
364C2×C4—6922no—
365C2×C4D4×D17311yescharacteristic
366C2×C4C23×D17311yescharacteristic
367C8—21no—
368C8—21no—
369C8—3461no—
370C8—1732no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—3461no—
381C22—3461no—
382C22—3461no—
383C22C2×D4×D17311yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—3461no—
388C4—3461no—
389C4—3461no—
390C4—3462no—
391C4—6921no—
392C4—6921no—
393C4—6921no—
394C4C2×D4×D17311yescharacteristic
395C4#1914972411yescharacteristic
396C2—41no—
397C2—81no—
398C2—81no—
399C2—6921no—
400C2#2601212311yescharacteristic
401C2#2601228711yescharacteristic
402C2#2601230511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#449×C173C2
socleC22×C173C22×D4
hypercenter#449D173
hypocenterC173#3277
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#19149709C2semidirect
2. C2×#19149708C2semidirect
3. #26012148C2semidirect
4. #26012147C2semidirect
5. #26011874C2semidirect
6. #26011736C2semidirect
7. #26011735C2semidirect
8. #26011712C2semidirect
9. #26011711C2semidirect
10. #26011649C2semidirect
11. #26011648C2semidirect
12. #26011633C2semidirect
13. #26011583C2semidirect
14. #26011566C2semidirect
15. #449×C173C2semidirect
82. #14583816D4semidirect
201. C173#3277semidirect
215. #449D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497092
C2×#191497082
#260121482
#260121472
#260118742
#260117362
#260117352
#260117122
#260117112
#260116492
#260116482
#260116332
#260115832
#260115662
#449×C1732
#3277173