Finite Group Databasebeta

← #36365155

Subgroups — #36365155

Order 22144.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149712C211yesmaximal, characteristic, complemented
2C2×#19149698C211yesmaximal, characteristic, complemented
3#26012159C211yesmaximal, characteristic, complemented
4#26012147C211yesmaximal, characteristic, complemented
5#26011879C211yesmaximal, characteristic, complemented
6#26011721C211yesmaximal, characteristic, complemented
7#26011712C211yesmaximal, characteristic, complemented
8#26011690C211yesmaximal, characteristic, complemented
9#26011654C211yesmaximal, characteristic, complemented
10#26011637C211yesmaximal, characteristic, complemented
11#26011634C211yesmaximal, characteristic, complemented
12#26011629C211yesmaximal, characteristic, complemented
13#26011597C211yesmaximal, characteristic, complemented
14#26011560C211yesmaximal, characteristic, complemented
15#469×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×#14583825C2211yescharacteristic
19C22×Dic346C2211yescharacteristic
20#19149712—24no—
21#19149698—24no—
22C2×Dic692C2211yescharacteristic
23C2×Dic692C2211yescharacteristic
24C2×#14583815C2211yescharacteristic
25C2×#14583815C2211yescharacteristic
26C2×#14583814C2211yescharacteristic
27C2×#14583814C2211yescharacteristic
28C2×#14583814C2211yescharacteristic
29C2×#14583811C2211yescharacteristic
30C2×C8×D173C2211yescharacteristic
31#19149675—21no—
32#19149675C2211yescharacteristic
33#19149666C2211yescharacteristic
34#19149665—21no—
35#19149659C2211yescharacteristic
36#19149656—21no—
37#19149654C2211yescharacteristic
38#19149652—21no—
39#19149651—21no—
40#19149651—21no—
41#19149644C2211yescharacteristic
42#19149644C2211yescharacteristic
43D4×Dic173—21no—
44C4×C22⋊D173C2211yescharacteristic
45#19149639—21no—
46#19149608C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149602C2211yescharacteristic
49#19149602C2211yescharacteristic
50#19149599C2211yescharacteristic
51C22⋊#13063436C2211yescharacteristic
52#19149591C2211yescharacteristic
53#19149589C2211yescharacteristic
54C2×D8×C173C2211yescharacteristic
55C2×M16×C173C2211yescharacteristic
56#121×C173C2211yescharacteristic
57#121×C173C2211yescharacteristic
58#106×C173C2211yescharacteristic
59#102×C173C2211yescharacteristic
60#102×C173C2211yescharacteristic
61#14583826—22no—
62#14583826—41no—
63#14583825—22no—
64#14583825—22no—
65#14583825—41no—
66#14583825—41no—
67C2×Dic346—22no—
68C2×Dic346—21no—
69C2×Dic346—41no—
70C2×Dic346C2311yescharacteristic
71C2×Dic346C2311yescharacteristic
72C2×Dic346C2311yescharacteristic
73C2×D692—21no—
74C2×D692C2311yescharacteristic
75C2×C22⋊D173—21no—
76C2×C22⋊D173—21no—
77C2×C22⋊D173—21no—
78C2×C22⋊D173—21no—
79C2×C22⋊D173—21no—
80C2×C22⋊D173D411yescharacteristic
81C22×Dic173—21no—
82C22×Dic173—21no—
83C22×Dic173—21no—
84C2×C4×D173—21no—
85C2×C4×D173C2311yescharacteristic
86Dic692—22no—
87Dic692D414yescomplemented
88#14583815—22no—
89#14583815—22no—
90#14583814—22no—
91#14583814—22no—
92#14583814—41no—
93#14583811—41no—
94C2×#13063436C2311yescharacteristic
95C2×#13063436C2311yescharacteristic
96C8×D173—22no—
97C4⋊Dic173—21no—
98#14583805—21no—
99#14583805—21no—
100#14583805D411yescharacteristic
101C22⋊Dic173—21no—
102C22⋊Dic173—21no—
103C22⋊Dic173D411yescharacteristic
104#14583803—21no—
105#14583803—21no—
106#14583803D411yescharacteristic
107C4×Dic173—21no—
108C4×Dic173—21no—
109C4×Dic173C2311yescharacteristic
110C2×D4×C173—21no—
111C2×D4×C173—21no—
112C2×D4×C173C2311yescharacteristic
113C2×D4×C173C2311yescharacteristic
114C22×C4×C173C2311yescharacteristic
115D8×C173—22no—
116M16×C173—22no—
117C2×C8×C173C2311yescharacteristic
118C2×C8×C173C2311yescharacteristic
119C4⋊C4×C173C2311yescharacteristic
120C4⋊C4×C173C2311yescharacteristic
121C22⋊C4×C173—21no—
122C22⋊C4×C173—21no—
123C22×D173—21no—
124C22×D173C2×D411yescharacteristic
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—21no—
129Dic346—42no—
130Dic346C2×D412yes—
131Dic346C2×D412yes—
132D692—21no—
133D692—21no—
134D692—81no—
135C22⋊D173—41no—
136C22⋊D173—42no—
137C22⋊D173—42no—
138C22⋊D173—42no—
139C22⋊D173—42no—
140C22⋊D173—81no—
141C2×Dic173—21no—
142C2×Dic173—21no—
143C2×Dic173—21no—
144C2×Dic173—42no—
145C2×Dic173—42no—
146C2×Dic173—42no—
147C2×Dic173C2×D411yescharacteristic
148C2×Dic173C2×D411yescharacteristic
149C2×Dic173C2×D411yescharacteristic
150C4×D173—21no—
151C4×D173—21no—
152C4×D173—42no—
153#13063436—21no—
154#13063436—21no—
155C23×C173—21no—
156C23×C173—21no—
157C23×C173C2×D411yescharacteristic
158D4×C173—21no—
159D4×C173—21no—
160D4×C173—21no—
161D4×C173—21no—
162D4×C173—42no—
163D4×C173—81no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173C2×D411yescharacteristic
169C2×C4×C173C2411yescharacteristic
170C8×C173—21no—
171C8×C173C2×D412yes—
172C22×C173—21no—
173C22×C173—21no—
174C22×C173—21no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173—41no—
179C22×C173—41no—
180C22×C173—41no—
181C22×C173C22×D411yescharacteristic
182C2×D173—21no—
183C2×D173—21no—
184C2×D173—21no—
185C2×D173—41no—
186C2×D173—41no—
187C2×D173—41no—
188C4×C173—21no—
189C4×C173—41no—
190C4×C173—41no—
191C4×C173C22×D411yescharacteristic
192C4×C173H(5,𝔽2)11yescharacteristic
193Dic173—21no—
194Dic173—22no—
195Dic173—22no—
196Dic173—21no—
197Dic173—21no—
198Dic173—41no—
199C2×C173—41no—
200C2×C173—81no—
201C2×C173—81no—
202C2×C173D4211yescharacteristic
203C2×C173#57511yescharacteristic
204C2×C173#57711yescharacteristic
205D173—41no—
206D173—81no—
207C173#327111yescharacteristic, complemented, Sylow
208#3271—1731nomaximal, Sylow
209C2×#137—1731no—
210C2×#135—1731no—
211#550—1731no—
212#548—1731no—
213#496—1731no—
214#469D17311yescharacteristic, complemented
215#465—1731no—
216#464—1731no—
217#460—1731no—
218#452—1731no—
219#449—1731no—
220#449—1731no—
221#448—1731no—
222#441—1731no—
223#435—1731no—
224C2×#41—1731no—
225C2×#41—1731no—
226C2×#41—1731no—
227C22×Q8—1731no—
228#137—3464no—
229#135—3464no—
230C2×Q16—1731no—
231C2×Q16—1731no—
232C2×SD16—1731no—
233C2×SD16—1731no—
234C2×SD16—1731no—
235C2×SD16—1731no—
236C2×SD16—1731no—
237C2×D8—1731no—
238C2×D8C2×D17311yescharacteristic
239C2×M16C2×D17311yescharacteristic
240C22×C8—1731no—
241#127—3461no—
242#127—1731no—
243#124—3461no—
244#124—1731no—
245C22⋊Q8—3461no—
246C22⋊Q8—1731no—
247#121—3461no—
248#121—3461no—
249#121—3461no—
250#121—1731no—
251#121C2×D17311yescharacteristic
252#121C2×D17311yescharacteristic
253C4×Q8—1731no—
254C4×Q8—1731no—
255C4×D4—3461no—
256C4×D4—3461no—
257C4×D4—1731no—
258#106C2×D17311yescharacteristic
259C4⋊C8—1731no—
260#103—1731no—
261#103—1731no—
262#103—1731no—
263#102—1731no—
264#102C2×D17311yescharacteristic
265#102C2×D17311yescharacteristic
266C22⋊C8—1731no—
267C22⋊C8—1731no—
268#97—1731no—
269#41—3462no—
270#41—3462no—
271#41—3462no—
272#41—6921no—
273#41—6921no—
274#41—6921no—
275C2×Q8—3462no—
276C2×Q8—3461no—
277C2×Q8—1731no—
278C2×Q8—1731no—
279C2×Q8—1731no—
280C2×Q8—6921no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—3461no—
284C2×D4—3461no—
285C2×D4—3461no—
286C2×D4—3461no—
287C2×D4—3461no—
288C2×D4—3461no—
289C2×D4—1731no—
290C2×D4—1731no—
291C2×D4C22×D17311yescharacteristic
292C2×D4C22×D17311yescharacteristic
293C22×C4—3461no—
294C22×C4—3461no—
295C22×C4—3461no—
296C22×C4—3461no—
297C22×C4—1731no—
298C22×C4C22×D17311yescharacteristic
299Q16—3462no—
300Q16—1734no—
301SD16—3462no—
302SD16—3462no—
303SD16—3462no—
304SD16—3462no—
305SD16—6921no—
306D8—22no—
307D8—6921no—
308M16—22no—
309C2×C8—3462no—
310C2×C8—1731no—
311C2×C8—1731no—
312C2×C8C22×D17311yescharacteristic
313C2×C8C22×D17311yescharacteristic
314C4⋊C4—3461no—
315C4⋊C4—3461no—
316C4⋊C4—3461no—
317C4⋊C4—1731no—
318C4⋊C4C22×D17311yescharacteristic
319C4⋊C4C22×D17311yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—3461no—
323C22⋊C4—3461no—
324C22⋊C4—3461no—
325C22⋊C4—3461no—
326C22⋊C4—1731no—
327C22⋊C4—1731no—
328C42—3461no—
329C42—3461no—
330C42—1731no—
331C23—21no—
332C23—21no—
333C23—3461no—
334C23—1731no—
335C23D4×D17311yescharacteristic
336Q8—3461no—
337Q8—3461no—
338Q8—3461no—
339Q8—3461no—
340Q8—1732no—
341Q8—1732no—
342Q8—6922no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—42no—
348D4—81no—
349D4—3461no—
350D4—3461no—
351D4—13841no—
352D4—13841no—
353D4—6921no—
354D4—6922no—
355D4—6922no—
356D4—6922no—
357D4—6922no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—3461no—
363C2×C4—3461no—
364C2×C4—3461no—
365C2×C4—3461no—
366C2×C4—3461no—
367C2×C4—1731no—
368C2×C4—1731no—
369C2×C4—1731no—
370C2×C4—6922no—
371C2×C4—6922no—
372C2×C4—6922no—
373C2×C4—6922no—
374C2×C4D4×D17311yescharacteristic
375C2×C4C23×D17311yescharacteristic
376C8—21no—
377C8—3461no—
378C8—3461no—
379C8D4×D17312yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—3461no—
390C22—3461no—
391C22—3461no—
392C22—6921no—
393C22—6921no—
394C22—6921no—
395C22C2×D4×D17311yescharacteristic
396C4—21no—
397C4—41no—
398C4—41no—
399C4—3461no—
400C4—3461no—
401C4—3461no—
402C4—3462no—
403C4—3462no—
404C4—6921no—
405C4C2×D4×D17311yescharacteristic
406C4#1914972511yescharacteristic
407C2—41no—
408C2—81no—
409C2—81no—
410C2—13841no—
411C2—6921no—
412C2#2601212511yescharacteristic
413C2#2601228711yescharacteristic
414C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#469×C173C2
socleC22×C173C22×D4
hypercenter#469D173
hypocenterC173#3271
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#19149712C2semidirect
2. C2×#19149698C2semidirect
3. #26012159C2semidirect
4. #26012147C2semidirect
5. #26011879C2semidirect
6. #26011721C2semidirect
7. #26011712C2semidirect
8. #26011690C2semidirect
9. #26011654C2semidirect
10. #26011637C2semidirect
11. #26011634C2semidirect
12. #26011629C2semidirect
13. #26011597C2semidirect
14. #26011560C2semidirect
15. #469×C173C2semidirect
87. Dic692D4semidirect
207. C173#3271semidirect
214. #469D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497122
C2×#191496982
#260121592
#260121472
#260118792
#260117212
#260117122
#260116902
#260116542
#260116372
#260116342
#260116292
#260115972
#260115602
#469×C1732
#3271173