Finite Group Databasebeta

← #36365149

Subgroups — #36365149

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×#19149710C211yesmaximal, characteristic, complemented
2C2×#19149699C211yesmaximal, characteristic, complemented
3#26012162C211yesmaximal, characteristic, complemented
4#26011881C211yesmaximal, characteristic, complemented
5#26011742C211yesmaximal, characteristic, complemented
6#26011722C211yesmaximal, characteristic, complemented
7#26011710C211yesmaximal, characteristic, complemented
8#26011691C211yesmaximal, characteristic, complemented
9#26011651C211yesmaximal, characteristic, complemented
10#26011636C211yesmaximal, characteristic, complemented
11#26011634C211yesmaximal, characteristic, complemented
12#26011630C211yesmaximal, characteristic, complemented
13#26011596C211yesmaximal, characteristic, complemented
14#26011563C211yesmaximal, characteristic, complemented
15#548×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18#19149710—24no—
19#19149699—24no—
20C2×#14583816C2211yescharacteristic
21C2×#14583816C2211yescharacteristic
22C2×#14583814C2211yescharacteristic
23C2×#14583813C2211yescharacteristic
24C2×#14583813C2211yescharacteristic
25C2×#14583813C2211yescharacteristic
26C2×#14583813C2211yescharacteristic
27C2×#14583811C2211yescharacteristic
28C2×#14583811C2211yescharacteristic
29C2×#14583810C2211yescharacteristic
30C22×#13063436C2211yescharacteristic
31#19149676C2211yescharacteristic
32#19149675—21no—
33#19149653C2211yescharacteristic
34#19149653C2211yescharacteristic
35#19149652—21no—
36#19149652—21no—
37C4×Dic346C2211yescharacteristic
38#19149641—21no—
39#19149610C2211yescharacteristic
40C4⋊#13063436C2211yescharacteristic
41#19149603C2211yescharacteristic
42#19149603C2211yescharacteristic
43#19149602C2211yescharacteristic
44#19149600C2211yescharacteristic
45#19149599C2211yescharacteristic
46#19149599C2211yescharacteristic
47C22⋊#13063436C2211yescharacteristic
48C22⋊#13063436C2211yescharacteristic
49#19149590C2211yescharacteristic
50C2×#41×C173C2211yescharacteristic
51C22×Q8×C173C2211yescharacteristic
52#124×C173—21no—
53#124×C173C2211yescharacteristic
54C22⋊Q8×C173—21no—
55C22⋊Q8×C173C2211yescharacteristic
56#121×C173—21no—
57#121×C173C2211yescharacteristic
58C4×Q8×C173C2211yescharacteristic
59C4×D4×C173—21no—
60C4×D4×C173C2211yescharacteristic
61#14583826—22no—
62#14583826—22no—
63#14583826—41no—
64#14583826—41no—
65C2×Dic346C2311yescharacteristic
66C2×D692—21no—
67C2×D692—21no—
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173—21no—
76#14583816—22no—
77#14583816D414yescomplemented
78#14583814—22no—
79#14583813—22no—
80#14583813—22no—
81#14583813—22no—
82#14583813—41no—
83#14583811—22no—
84#14583811—41no—
85#14583810—22no—
86C2×#13063436—22no—
87C2×#13063436C2311yescharacteristic
88C2×#13063436C2311yescharacteristic
89C2×#13063436C2311yescharacteristic
90C2×#13063436C2311yescharacteristic
91C4⋊Dic173C2311yescharacteristic
92#14583805—21no—
93#14583803—21no—
94#14583803—21no—
95C4×Dic173—21no—
96#41×C173—22no—
97#41×C173—41no—
98C2×Q8×C173—21no—
99C2×Q8×C173—22no—
100C2×Q8×C173—41no—
101C2×Q8×C173C2311yescharacteristic
102C2×Q8×C173C2311yescharacteristic
103C2×D4×C173—21no—
104C2×D4×C173—21no—
105C2×D4×C173D411yescharacteristic
106C2×D4×C173C2311yescharacteristic
107C22×C4×C173—21no—
108C22×C4×C173—21no—
109C22×C4×C173C2311yescharacteristic
110C22×C4×C173C2311yescharacteristic
111C4⋊C4×C173—21no—
112C4⋊C4×C173—21no—
113C4⋊C4×C173D411yescharacteristic
114C4⋊C4×C173C2311yescharacteristic
115C22⋊C4×C173—21no—
116C22⋊C4×C173—21no—
117C22⋊C4×C173—21no—
118C22⋊C4×C173—21no—
119C22⋊C4×C173D411yescharacteristic
120C22⋊C4×C173D411yescharacteristic
121C42×C173—21no—
122C42×C173C2311yescharacteristic
123C22×D173—21no—
124C22×D173—21no—
125Dic346—21no—
126Dic346C2×D412yes—
127D692—21no—
128D692—21no—
129D692—21no—
130D692—21no—
131D692—42no—
132D692—81no—
133C22⋊D173—42no—
134C22⋊D173—42no—
135C22⋊D173—81no—
136C22⋊D173—81no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C4×D173—42no—
140C4×D173—42no—
141#13063436—21no—
142#13063436—21no—
143#13063436—21no—
144#13063436C2×D412yes—
145C23×C173—21no—
146C23×C173C2×D411yescharacteristic
147C23×C173C2×D411yescharacteristic
148Q8×C173—21no—
149Q8×C173—21no—
150Q8×C173—21no—
151Q8×C173—42no—
152Q8×C173C2×D412yes—
153D4×C173—21no—
154D4×C173—21no—
155D4×C173—41no—
156D4×C173—42no—
157D4×C173—42no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—42no—
166C2×C4×C173—42no—
167C2×C4×C173C2×D411yescharacteristic
168C2×C4×C173C2×D411yescharacteristic
169C2×C4×C173C2×D411yescharacteristic
170C2×C4×C173C2×D411yescharacteristic
171C2×C4×C173C2411yescharacteristic
172C22×C173—21no—
173C22×C173—21no—
174C22×C173—21no—
175C22×C173—21no—
176C22×C173—21no—
177C22×C173—21no—
178C22×C173—41no—
179C22×C173—41no—
180C22×C173—41no—
181C22×C173C22×D411yescharacteristic
182C2×D173—41no—
183C2×D173—41no—
184C2×D173—41no—
185C2×D173—41no—
186C2×D173—41no—
187C2×D173—41no—
188C4×C173—21no—
189C4×C173—21no—
190C4×C173—22no—
191C4×C173—21no—
192C4×C173—21no—
193C4×C173—41no—
194C4×C173—41no—
195C4×C173C22×D411yescharacteristic
196C4×C173H(5,𝔽2)11yescharacteristic
197Dic173—22no—
198Dic173—41no—
199C2×C173—41no—
200C2×C173—41no—
201C2×C173—81no—
202C2×C173D4211yescharacteristic
203C2×C173#57511yescharacteristic
204C2×C173#57711yescharacteristic
205D173—81no—
206D173—81no—
207C173#327111yescharacteristic, complemented, Sylow
208#3271—1731nomaximal, Sylow
209C2×#137—1731no—
210C2×#135—1731no—
211#550—1731no—
212#548D17311yescharacteristic, complemented
213#496—1731no—
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×#41C2×D17311yescharacteristic
227C22×Q8C2×D17311yescharacteristic
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×D8—1731no—
239C2×M16—1731no—
240C22×C8—1731no—
241#127—3461no—
242#127—1731no—
243#124—21no—
244#124C2×D17311yescharacteristic
245C22⋊Q8—21no—
246C22⋊Q8C2×D17311yescharacteristic
247#121—21no—
248#121—3461no—
249#121—3461no—
250#121—1731no—
251#121—1731no—
252#121C2×D17311yescharacteristic
253C4×Q8—1731no—
254C4×Q8C2×D17311yescharacteristic
255C4×D4—21no—
256C4×D4—3461no—
257C4×D4C2×D17311yescharacteristic
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#97—1731no—
269#41—22no—
270#41—41no—
271#41—3462no—
272#41—3462no—
273#41—6921no—
274#41—6921no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—1731no—
279C2×Q8C22×D17311yescharacteristic
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×D4—1731no—
291C2×D4C22⋊D17311yescharacteristic
292C2×D4C22×D17311yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—3461no—
296C22×C4—3461no—
297C22×C4C22×D17311yescharacteristic
298C22×C4C22×D17311yescharacteristic
299Q16—3462no—
300Q16—1734no—
301SD16—3462no—
302SD16—3462no—
303SD16—3462no—
304SD16—3462no—
305SD16—6921no—
306D8—3462no—
307D8—6921no—
308M16—3462no—
309C2×C8—3462no—
310C2×C8—1731no—
311C2×C8—1731no—
312C2×C8—1731no—
313C2×C8—1731no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—3461no—
317C4⋊C4—1731no—
318C4⋊C4C22⋊D17311yescharacteristic
319C4⋊C4C22×D17311yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—3461no—
325C22⋊C4—3461no—
326C22⋊C4C22⋊D17311yescharacteristic
327C22⋊C4C22⋊D17311yescharacteristic
328C42—21no—
329C42—3461no—
330C42C22×D17311yescharacteristic
331C23—21no—
332C23—3461no—
333C23—3461no—
334C23C2×C22⋊D17311yescharacteristic
335C23C2×C22⋊D17311yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—42no—
340Q8—3461no—
341Q8—1732no—
342Q8D4×D17312yes—
343D4—21no—
344D4—21no—
345D4—41no—
346D4—42no—
347D4—42no—
348D4—3461no—
349D4—3461no—
350D4—3461no—
351D4—3461no—
352D4—13841no—
353D4—13841no—
354D4—13841no—
355D4—6922no—
356D4—6922no—
357D4—6922no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—3461no—
368C2×C4—3461no—
369C2×C4—6922no—
370C2×C4—6922no—
371C2×C4C2×C22⋊D17311yescharacteristic
372C2×C4C2×C22⋊D17311yescharacteristic
373C2×C4C2×C22⋊D17311yescharacteristic
374C2×C4C2×C22⋊D17311yescharacteristic
375C2×C4C23×D17311yescharacteristic
376C8—3461no—
377C8—3461no—
378C8—3461no—
379C8—1732no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—6921no—
390C22—6921no—
391C22—6921no—
392C22—6921no—
393C22—6921no—
394C22—6921no—
395C22C22×C22⋊D17311yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—3462no—
404C4—6921no—
405C4C2×D4×D17311yescharacteristic
406C4#1914972411yescharacteristic
407C2—41no—
408C2—41no—
409C2—81no—
410C2—13841no—
411C2—13841no—
412C2D4×C22⋊D17311yescharacteristic
413C2#2601228911yescharacteristic
414C2#2601229911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#548×C173C2
socleC22×C173C22×D4
hypercenter#548D173
hypocenterC173#3271
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#19149710C2semidirect
2. C2×#19149699C2semidirect
3. #26012162C2semidirect
4. #26011881C2semidirect
5. #26011742C2semidirect
6. #26011722C2semidirect
7. #26011710C2semidirect
8. #26011691C2semidirect
9. #26011651C2semidirect
10. #26011636C2semidirect
11. #26011634C2semidirect
12. #26011630C2semidirect
13. #26011596C2semidirect
14. #26011563C2semidirect
15. #548×C173C2semidirect
77. #14583816D4semidirect
207. C173#3271semidirect
212. #548D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497102
C2×#191496992
#260121622
#260118812
#260117422
#260117222
#260117102
#260116912
#260116512
#260116362
#260116342
#260116302
#260115962
#260115632
#548×C1732
#3271173