Finite Group Databasebeta

← #36365147

Subgroups — #36365147

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×#19149708C211yesmaximal, characteristic, complemented
2C2×#19149698C211yesmaximal, characteristic, complemented
3#26012161C211yesmaximal, characteristic, complemented
4#26012149C211yesmaximal, characteristic, complemented
5#26011741C211yesmaximal, characteristic, complemented
6#26011717C211yesmaximal, characteristic, complemented
7#26011710C211yesmaximal, characteristic, complemented
8#26011688C211yesmaximal, characteristic, complemented
9#26011650C211yesmaximal, characteristic, complemented
10#26011633C211yesmaximal, characteristic, complemented
11#26011633C211yesmaximal, characteristic, complemented
12#26011628C211yesmaximal, characteristic, complemented
13#26011599C211yesmaximal, characteristic, complemented
14#26011564C211yesmaximal, characteristic, complemented
15#496×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×#14583825C2211yescharacteristic
19C2×Q8×D173C2211yescharacteristic
20#19149708—24no—
21#19149698—24no—
22C2×Dic692C2211yescharacteristic
23C2×#14583816C2211yescharacteristic
24C2×#14583814C2211yescharacteristic
25C2×#14583814C2211yescharacteristic
26C2×#14583814C2211yescharacteristic
27C2×#14583809C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic
29#19149675—21no—
30#19149665—21no—
31#19149656—21no—
32#19149655C2211yescharacteristic
33#19149652—21no—
34#19149651—21no—
35#19149651—21no—
36#19149650C2211yescharacteristic
37#19149650C2211yescharacteristic
38#19149650C2211yescharacteristic
39C4×Dic346C2211yescharacteristic
40C4×Dic346C2211yescharacteristic
41C4×D692C2211yescharacteristic
42D4×Dic173—21no—
43#19149639—21no—
44#19149611C2211yescharacteristic
45C4⋊#13063436C2211yescharacteristic
46#19149604C2211yescharacteristic
47#19149604C2211yescharacteristic
48#19149603C2211yescharacteristic
49#19149600C2211yescharacteristic
50#19149600C2211yescharacteristic
51#19149600C2211yescharacteristic
52#19149591C2211yescharacteristic
53#19149591C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C2×SD16×C173C2211yescharacteristic
56C2×D8×C173C2211yescharacteristic
57C2×D8×C173C2211yescharacteristic
58#127×C173C2211yescharacteristic
59#124×C173C2211yescharacteristic
60#97×C173C2211yescharacteristic
61#14583825—22no—
62#14583825—22no—
63#14583825—22no—
64#14583825—41no—
65#14583825—41no—
66#14583825—41no—
67C2×Dic346—21no—
68C2×Dic346C2311yescharacteristic
69C2×Dic346C2311yescharacteristic
70Q8×D173—22no—
71Q8×D173—41no—
72C2×D692D411yescharacteristic
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76C2×C22⋊D173—21no—
77C2×C22⋊D173—21no—
78C2×C22⋊D173—21no—
79C22×Dic173—21no—
80C22×Dic173—21no—
81C22×Dic173—21no—
82C2×C4×D173—21no—
83C2×C4×D173C2311yescharacteristic
84C2×C4×D173C2311yescharacteristic
85Dic692D414yescomplemented
86#14583816—22no—
87#14583814—22no—
88#14583814—22no—
89#14583814—22no—
90#14583809—22no—
91C2×#13063436C2311yescharacteristic
92C2×#13063436C2311yescharacteristic
93C8×D173—22no—
94C4⋊Dic173D411yescharacteristic
95C4⋊Dic173C2311yescharacteristic
96C4⋊Dic173C2311yescharacteristic
97#14583805—21no—
98#14583805—21no—
99#14583805—21no—
100C22⋊Dic173—21no—
101C22⋊Dic173—21no—
102C22⋊Dic173—21no—
103#14583803—21no—
104#14583803D411yescharacteristic
105#14583803D411yescharacteristic
106C4×Dic173—21no—
107C4×Dic173—21no—
108C2×Q8×C173C2311yescharacteristic
109C2×D4×C173—21no—
110C2×D4×C173—21no—
111C2×D4×C173C2311yescharacteristic
112C2×D4×C173C2311yescharacteristic
113C2×D4×C173C2311yescharacteristic
114SD16×C173—22no—
115SD16×C173—41no—
116D8×C173—22no—
117D8×C173—41no—
118C2×C8×C173C2311yescharacteristic
119C2×C8×C173C2311yescharacteristic
120C22⋊C4×C173—21no—
121C22⋊C4×C173—21no—
122C42×C173C2311yescharacteristic
123C22×D173C2×D411yescharacteristic
124C22×D173C2×D411yescharacteristic
125Dic346—21no—
126Dic346—21no—
127Dic346—42no—
128Dic346C2×D412yes—
129Dic346C2×D412yes—
130D692—41no—
131C22⋊D173—42no—
132C22⋊D173—42no—
133C22⋊D173—42no—
134C22⋊D173—42no—
135C22⋊D173—81no—
136C22⋊D173—81no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—21no—
140C2×Dic173—21no—
141C2×Dic173—42no—
142C2×Dic173—42no—
143C2×Dic173—42no—
144C2×Dic173C2×D411yescharacteristic
145C2×Dic173C2×D411yescharacteristic
146C4×D173—21no—
147C4×D173—21no—
148C4×D173—21no—
149C4×D173—21no—
150C4×D173—42no—
151#13063436—21no—
152#13063436—21no—
153C23×C173—21no—
154C23×C173—21no—
155C23×C173—21no—
156Q8×C173—21no—
157Q8×C173—21no—
158D4×C173—21no—
159D4×C173—21no—
160D4×C173—21no—
161D4×C173—21no—
162D4×C173—21no—
163D4×C173—21no—
164D4×C173—42no—
165D4×C173—81no—
166C2×C4×C173—21no—
167C2×C4×C173C2×D411yescharacteristic
168C2×C4×C173C2×D411yescharacteristic
169C2×C4×C173C2411yescharacteristic
170C8×C173—21no—
171C8×C173C2×D412yes—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173—41no—
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—21no—
186C2×D173—21no—
187C2×D173—21no—
188C4×C173—21no—
189C4×C173—21no—
190C4×C173—41no—
191C4×C173C22×D411yescharacteristic
192C4×C173H(5,𝔽2)11yescharacteristic
193Dic173—21no—
194Dic173—21no—
195Dic173—22no—
196Dic173—22no—
197Dic173—41no—
198Dic173—41no—
199C2×C173—81no—
200C2×C173—81no—
201C2×C173—81no—
202C2×C173D4211yescharacteristic
203C2×C173#57511yescharacteristic
204C2×C173#57711yescharacteristic
205D173—41no—
206D173—41no—
207C173#327111yescharacteristic, complemented, Sylow
208#3271—1731nomaximal, Sylow
209C2×#137—1731no—
210C2×#135—1731no—
211#550—1731no—
212#548—1731no—
213#496D17311yescharacteristic, complemented
214#469—1731no—
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×SD16C2×D17311yescharacteristic
236C2×SD16C2×D17311yescharacteristic
237C2×D8C2×D17311yescharacteristic
238C2×D8C2×D17311yescharacteristic
239C2×M16—1731no—
240C22×C8—1731no—
241#127—3461no—
242#127C2×D17311yescharacteristic
243#124—3461no—
244#124C2×D17311yescharacteristic
245C22⋊Q8—3461no—
246C22⋊Q8—1731no—
247#121—3461no—
248#121—3461no—
249#121—3461no—
250#121—1731no—
251#121—1731no—
252#121—1731no—
253C4×Q8—1731no—
254C4×Q8—1731no—
255C4×D4—3461no—
256C4×D4—3461no—
257C4×D4—1731no—
258#106—1731no—
259C4⋊C8—1731no—
260#103—1731no—
261#103—1731no—
262#103—1731no—
263#102—1731no—
264#102—1731no—
265#102—1731no—
266C22⋊C8—1731no—
267C22⋊C8—1731no—
268#97C2×D17311yescharacteristic
269#41—3462no—
270#41—3462no—
271#41—3462no—
272#41—6921no—
273#41—6921no—
274#41—6921no—
275C2×Q8—3461no—
276C2×Q8—3462no—
277C2×Q8—1731no—
278C2×Q8—1731no—
279C2×Q8—6921no—
280C2×Q8C22×D17311yescharacteristic
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×D4C22×D17311yescharacteristic
291C2×D4C22×D17311yescharacteristic
292C2×D4C22×D17311yescharacteristic
293C22×C4—3461no—
294C22×C4—3461no—
295C22×C4—3461no—
296C22×C4—3461no—
297C22×C4—1731no—
298C22×C4—1731no—
299Q16—3462no—
300Q16—1734no—
301SD16—22no—
302SD16—41no—
303SD16—3462no—
304SD16—3462no—
305SD16—3462no—
306D8—22no—
307D8—41no—
308M16—3462no—
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⋊C4—1731no—
319C4⋊C4—1731no—
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—
330C42C22×D17311yescharacteristic
331C23—21no—
332C23—21no—
333C23—21no—
334C23—1731no—
335C23—1731no—
336Q8—21no—
337Q8—21no—
338Q8—3461no—
339Q8—3461no—
340Q8—1732no—
341Q8—1732no—
342Q8—6922no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—81no—
351D4—13841no—
352D4—13841no—
353D4—6921no—
354D4—6922no—
355D4—6922no—
356D4—6922no—
357D4—6922no—
358C2×C4—21no—
359C2×C4—3461no—
360C2×C4—3461no—
361C2×C4—3461no—
362C2×C4—3461no—
363C2×C4—3461no—
364C2×C4—3461no—
365C2×C4—3461no—
366C2×C4—3461no—
367C2×C4—1731no—
368C2×C4—1731no—
369C2×C4—6922no—
370C2×C4—6922no—
371C2×C4—6922no—
372C2×C4—6922no—
373C2×C4D4×D17311yescharacteristic
374C2×C4D4×D17311yescharacteristic
375C2×C4C23×D17311yescharacteristic
376C8—21no—
377C8—3461no—
378C8—3461no—
379C8D4×D17312yes—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—3461no—
390C22—3461no—
391C22—3461no—
392C22—3461no—
393C22—3461no—
394C22—3461no—
395C22C2×D4×D17311yescharacteristic
396C4—21no—
397C4—21no—
398C4—41no—
399C4—3461no—
400C4—3461no—
401C4—3462no—
402C4—3462no—
403C4—6921no—
404C4—6921no—
405C4C2×D4×D17311yescharacteristic
406C4#1914972411yescharacteristic
407C2—81no—
408C2—81no—
409C2—81no—
410C2—6921no—
411C2—6921no—
412C2#2601212611yescharacteristic
413C2#2601228711yescharacteristic
414C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#496×C173C2
socleC22×C173C22×D4
hypercenter#496D173
hypocenterC173#3271
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#19149708C2semidirect
2. C2×#19149698C2semidirect
3. #26012161C2semidirect
4. #26012149C2semidirect
5. #26011741C2semidirect
6. #26011717C2semidirect
7. #26011710C2semidirect
8. #26011688C2semidirect
9. #26011650C2semidirect
10. #26011633C2semidirect
11. #26011633C2semidirect
12. #26011628C2semidirect
13. #26011599C2semidirect
14. #26011564C2semidirect
15. #496×C173C2semidirect
85. Dic692D4semidirect
207. C173#3271semidirect
213. #496D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497082
C2×#191496982
#260121612
#260121492
#260117412
#260117172
#260117102
#260116882
#260116502
#260116332
#260116282
#260115992
#260115642
#496×C1732
#3271173