Finite Group Databasebeta

← C2×#22611998 #33240354

Subgroups — C2×#22611998 #33240354

Order 18688.

706 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989027C211yesmaximal, characteristic, complemented
2C2×#17988970C211yesmaximal, characteristic, complemented
3C2×#17988704C211yesmaximal, characteristic, complemented
4#22611998C218yesmaximal, direct product factor
5C2×#17988407C211yesmaximal, characteristic, complemented
6C2×#17988404C211yesmaximal, characteristic, complemented
7C2×#17988400C211yesmaximal, characteristic, complemented
8C2×#509×C73C211yesmaximal, characteristic
9C22×Dic292C2211yescharacteristic
10C22×#11924016—21no—
11C2×#14556598—21no—
12C2×#14556578—21no—
13#17989027—22no—
14#17989027C2214yescomplemented
15C2×#14556537—21no—
16C2×#14556535—21no—
17C2×#14556531C2211yescharacteristic
18#17988970—22no—
19#17988970C2214yescomplemented
20#17988704—22no—
21#17988704C2214yescomplemented
22C2×#14556527C2211yescharacteristic
23C2×#14556518—21no—
24C2×C8×Dic73C2211yescharacteristic
25#17988407C2214yescomplemented
26#17988404C2214yescomplemented
27#17988400C2214yescomplemented
28C2×#135×C73C2211yescharacteristic
29C22×D8×C73C2211yescharacteristic
30#509×C73—22no—
31#509×C73C2214yes—
32C2×SD32×C73—21no—
33C2×D16×C73—21no—
34C2×M32×C73C2211yescharacteristic
35C22×Dic146—21no—
36C2×Dic292—21no—
37C2×Dic292—21no—
38C2×Dic292—21no—
39C2×Dic292—21no—
40C2×Dic292—21no—
41C2×Dic292C2312yes—
42C2×#11924016—24no—
43C2×#11924016—41no—
44C2×#11924016—41no—
45C2×#11924016—42no—
46#14556598—24no—
47#14556598—42no—
48C2×#11924012—21no—
49C2×#11924012—41no—
50C22×#10596831—41no—
51C22×#10596831D411yescharacteristic
52#14556578—24no—
53#14556578—42no—
54C2×C22⋊Dic73—42no—
55C2×C4×Dic73D411yescharacteristic
56#14556537—24no—
57#14556535—24no—
58#14556531—21no—
59#14556531C2312yes—
60C2×#11924003—21no—
61#14556527—21no—
62#14556527C2312yes—
63#14556518—24no—
64C8×Dic73D412yes—
65C8×Dic73C2312yes—
66C2×#41×C73—21no—
67C22×D4×C73—21no—
68#135×C73—21no—
69#135×C73—21no—
70#135×C73D412yes—
71#135×C73C2312yes—
72C2×Q16×C73D411yescharacteristic
73C2×SD16×C73—21no—
74C2×D8×C73—21no—
75C2×D8×C73—21no—
76C2×D8×C73—21no—
77C2×D8×C73—21no—
78C2×D8×C73—21no—
79C2×D8×C73D411yescharacteristic
80C2×D8×C73C2312yes—
81C22×C8×C73C2311yescharacteristic
82SD32×C73—24no—
83D16×C73—24no—
84M32×C73—21no—
85M32×C73C2312yes—
86C2×C16×C73—21no—
87C2×Dic146—22no—
88C2×Dic146—41no—
89C2×Dic146—41no—
90C2×Dic146—41no—
91C2×Dic146—41no—
92C2×Dic146—41no—
93C22×Dic73—21no—
94C22×Dic73—41no—
95Dic292—22no—
96Dic292—22no—
97Dic292—41no—
98Dic292—41no—
99#11924016—44no—
100#11924016—44no—
101#11924012—22no—
102#11924012—41no—
103#11924012—44no—
104C2×#10596831—41no—
105C2×#10596831—44no—
106C2×#10596831D811yescharacteristic
107C2×#10596831D811yescharacteristic
108C2×#10596831D812yes—
109C2×#10596831C2×D412yes—
110C22⋊Dic73—48no—
111C4×Dic73—21no—
112C4×Dic73C2×D412yes—
113#11924003—22no—
114C24×C73—41no—
115#41×C73—22no—
116#41×C73—22no—
117#41×C73—41no—
118#41×C73—41no—
119C2×Q8×C73—21no—
120C2×D4×C73—21no—
121C2×D4×C73—22no—
122C2×D4×C73—41no—
123C2×D4×C73—41no—
124C2×D4×C73—41no—
125C2×D4×C73—41no—
126C2×D4×C73—41no—
127C2×D4×C73—41no—
128C22×C4×C73—41no—
129C22×C4×C73C2×D411yescharacteristic
130Q16×C73—21no—
131Q16×C73C2×D412yes—
132SD16×C73—22no—
133SD16×C73—22no—
134D8×C73—21no—
135D8×C73—22no—
136D8×C73—22no—
137D8×C73—41no—
138D8×C73—41no—
139D8×C73C2×D412yes—
140C2×C8×C73C2×D411yescharacteristic
141C2×C8×C73C2×D411yescharacteristic
142C2×C8×C73C2×D411yescharacteristic
143C2×C8×C73#4111yescharacteristic
144C2×C8×C73#4111yescharacteristic
145C2×C8×C73C2411yescharacteristic
146C16×C73—22no—
147C23×C73—41no—
148C23×C73—42no—
149C23×C73—84no—
150C23×C73—81no—
151C23×C73—81no—
152C23×C73AGL(1,ℤ/8)11yescharacteristic
153Q8×C73—22no—
154Q8×C73—41no—
155D4×C73—22no—
156D4×C73—42no—
157D4×C73—42no—
158D4×C73—42no—
159D4×C73—42no—
160D4×C73—41no—
161D4×C73—81no—
162D4×C73—81no—
163C2×C4×C73—41no—
164C2×C4×C73—42no—
165C2×C4×C73—42no—
166C2×C4×C73#12111yescharacteristic
167C2×C4×C73#12111yescharacteristic
168C2×C4×C73C2×D811yescharacteristic
169C2×C4×C73C2×D811yescharacteristic
170C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C73C22×D411yescharacteristic
172C2×Dic73—22no—
173C2×Dic73—42no—
174C2×Dic73—42no—
175C2×Dic73—81no—
176C2×Dic73—81no—
177Dic146—42no—
178Dic146—42no—
179Dic146—81no—
180Dic146—81no—
181C8×C73#12111yescharacteristic
182C8×C73#12111yescharacteristic
183C8×C73C22×D411yescharacteristic
184C8×C73C2×#4111yescharacteristic
185#10596831—42no—
186#10596831C2×D812yes—
187#10596831C2×D812yes—
188C22×C73—22no—
189C22×C73—42no—
190C22×C73—84no—
191C22×C73—82no—
192C22×C73—82no—
193C22×C73—162no—
194C22×C73—162no—
195C22×C73#45911yescharacteristic
196C22×C73#45911yescharacteristic
197C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
198C4×C73—42no—
199C4×C73#45911yescharacteristic
200C4×C73#45911yescharacteristic
201C4×C73C2×#12111yescharacteristic
202C4×C73C22×D811yescharacteristic
203Dic73—42no—
204Dic73—82no—
205C2×#2212—731nomaximal, Sylow
206C2×C73—21no—
207C2×C73—21no—
208C2×C73—82no—
209C2×C73—162no—
210C2×C73#221212yescomplemented
211C2×C73C2×#45911yescharacteristic
212C2×#510—731no—
213C2×#509D7311yescharacteristic
214C2×#495—731no—
215C2×#443—731no—
216#2212—738no—
217C2×#364—731no—
218C2×#362—731no—
219C2×#361—731no—
220C73C2×#221211yescharacteristic, complemented, Sylow
221C2×#135C2×D7311yescharacteristic
222C22×Q16—731no—
223C22×SD16—1461no—
224C22×D8C2×D7311yescharacteristic
225C2×#131—1461no—
226C2×#124—1461no—
227#510—734no—
228#510—1462no—
229#509—22no—
230#509C2×D7314yes—
231C2×Q32—1461no—
232C2×SD32—21no—
233C2×SD32—1461no—
234C2×D16—21no—
235C2×M32—731no—
236C2×M32C2×D7311yescharacteristic
237#495—734no—
238#495—1462no—
239#443—734no—
240#443—1462no—
241C2×#108—731no—
242C2×#104—1461no—
243C2×C4×C8—731no—
244#364—734no—
245#362—734no—
246#361—734no—
247C2×#41—21no—
248C22×Q8—1461no—
249C22×D4—21no—
250#135—21no—
251#135—21no—
252#135C22⋊D7312yes—
253#135C22×D7312yes—
254C2×Q16—732no—
255C2×Q16—1461no—
256C2×Q16—1461no—
257C2×Q16—1461no—
258C2×Q16—1461no—
259C2×Q16—1461no—
260C2×Q16C22⋊D7311yescharacteristic
261C2×SD16—21no—
262C2×SD16—2921no—
263C2×SD16—2921no—
264C2×SD16—2922no—
265C2×SD16—1464no—
266C2×D8—21no—
267C2×D8—21no—
268C2×D8—21no—
269C2×D8—21no—
270C2×D8—21no—
271C2×D8C22⋊D7311yescharacteristic
272C2×D8C22×D7312yes—
273#131—2922no—
274#131—1464no—
275C2×M16—2921no—
276C2×M16—1461no—
277C22×C8—2921no—
278C22×C8—731no—
279C22×C8C22×D7311yescharacteristic
280#124—2922no—
281#124—1464no—
282C2×C22⋊C4—2922no—
283C2×C42—731no—
284Q32—1464no—
285SD32—24no—
286SD32—1464no—
287D16—24no—
288M32—21no—
289M32—732no—
290M32—1461no—
291M32C22×D7312yes—
292C2×C16—21no—
293C2×C16—1461no—
294#108—732no—
295#108—1461no—
296#104—1464no—
297C4×C8—732no—
298C4×C8—732no—
299C24—41no—
300#41—22no—
301#41—22no—
302#41—41no—
303#41—41no—
304C2×Q8—21no—
305C2×Q8—2921no—
306C2×Q8—2921no—
307C2×Q8—2921no—
308C2×Q8—2921no—
309C2×Q8—2921no—
310C2×Q8—1462no—
311C2×D4—21no—
312C2×D4—22no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C22×C4—41no—
320C22×C4—2921no—
321C22×C4—1461no—
322C22×C4D4×D7311yescharacteristic
323Q16—21no—
324Q16—2921no—
325Q16—2921no—
326Q16—1462no—
327Q16—1462no—
328Q16C2×C22⋊D7312yes—
329SD16—22no—
330SD16—22no—
331SD16—2924no—
332SD16—2924no—
333D8—21no—
334D8—22no—
335D8—22no—
336D8—41no—
337D8—41no—
338D8C2×C22⋊D7312yes—
339M16—2921no—
340M16—2924no—
341M16—1462no—
342C2×C8—2921no—
343C2×C8—2924no—
344C2×C8—731no—
345C2×C8—731no—
346C2×C8—732no—
347C2×C8—732no—
348C2×C8D4×D7311yescharacteristic
349C2×C8C2×C22⋊D7311yescharacteristic
350C2×C8C2×C22⋊D7311yescharacteristic
351C2×C8#1192402711yescharacteristic
352C2×C8#1192402711yescharacteristic
353C2×C8C23×D7311yescharacteristic
354C22⋊C4—2928no—
355C42—732no—
356C42—1461no—
357C16—22no—
358C16—1462no—
359C23—41no—
360C23—42no—
361C23—84no—
362C23—81no—
363C23—81no—
364C23#1455661511yescharacteristic
365Q8—22no—
366Q8—41no—
367Q8—2922no—
368Q8—2922no—
369Q8—5841no—
370Q8—5841no—
371D4—22no—
372D4—42no—
373D4—42no—
374D4—41no—
375D4—42no—
376D4—42no—
377D4—81no—
378D4—81no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—2922no—
383C2×C4—2922no—
384C2×C4—5841no—
385C2×C4—5841no—
386C2×C4—1462no—
387C2×C4#1455656311yescharacteristic
388C2×C4#1455656311yescharacteristic
389C2×C4D8×D7311yescharacteristic
390C2×C4D8×D7311yescharacteristic
391C2×C4#1455661511yescharacteristic
392C2×C4C2×D4×D7311yescharacteristic
393C8—2922no—
394C8—732no—
395C8—732no—
396C8#1455656311yescharacteristic
397C8#1455656311yescharacteristic
398C8C22×C22⋊D7311yescharacteristic
399C8C2×#1192402711yescharacteristic
400C22—22no—
401C22—42no—
402C22—84no—
403C22—82no—
404C22—82no—
405C22—162no—
406C22—162no—
407C22#1798878011yescharacteristic
408C22#1798878011yescharacteristic
409C22C2×#1455661511yescharacteristic
410C4—42no—
411C4—2922no—
412C4—5842no—
413C4#1798878011yescharacteristic
414C4#1798878011yescharacteristic
415C4C2×#1455656311yescharacteristic
416C4C2×D8×D7311yescharacteristic
417C2—21no—
418C2—21no—
419C2—82no—
420C2—162no—
421C2#2261199812yesdirect product factor
422C2C2×#1798878011yescharacteristic
subgroupG/H
centerC22#17988780
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
Fitting, prime radicalC2×#509×C73C2
socleC22×C73#459
hypercenterC2×#509D73
hypocenterC73C2×#2212
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. C2×#17989027C2semidirect
2. C2×#17988970C2semidirect
3. C2×#17988704C2semidirect
4. #22611998C2direct (N × H)
5. C2×#17988407C2semidirect
6. C2×#17988404C2semidirect
7. C2×#17988400C2semidirect
14. #17989027C22semidirect
19. #17988970C22semidirect
21. #17988704C22semidirect
25. #17988407C22semidirect
26. #17988404C22semidirect
27. #17988400C22semidirect
210. C2×C73#2212semidirect
220. C73C2×#2212semidirect
421. C2#22611998direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179890272
C2×#179889702
C2×#179887042
#226119982
C2×#179884072
C2×#179884042
C2×#179884002
C2×#509×C732
C2×#221273