Finite Group Databasebeta

← C2×#21426167 #28743063

Subgroups — C2×#21426167 #28743063

Order 15104.

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×#16837837C211yesmaximal, characteristic
2C2×#16837787C211yesmaximal, characteristic, complemented
3C2×#16837519C211yesmaximal, characteristic, complemented
4#21426167C218yesmaximal, direct product factor
5C2×#16837222C211yesmaximal, characteristic, complemented
6C2×#16837218C211yesmaximal, characteristic, complemented
7C2×#16837216C211yesmaximal, characteristic, complemented
8C2×C59×#510C211yesmaximal, characteristic, complemented
9C22×#11745671—21no—
10C22×D472C2211yescharacteristic
11C2×#14273734—21no—
12C2×#14273716—21no—
13#16837837—22no—
14#16837837C2214yes—
15C2×#14273670—21no—
16C2×#14273668—21no—
17C2×#14273667C2211yescharacteristic
18#16837787—22no—
19#16837787C2214yescomplemented
20#16837519—22no—
21#16837519C2214yescomplemented
22C2×#14273663C2211yescharacteristic
23C2×#14273654—21no—
24C2×C8×Dic59C2211yescharacteristic
25#16837222C2214yescomplemented
26#16837218C2214yescomplemented
27#16837216C2214yescomplemented
28C2×#135×C59C2211yescharacteristic
29C22×Q16×C59C2211yescharacteristic
30C59×#510—22no—
31C59×#510C2214yescomplemented
32C2×Q32×C59—21no—
33C2×SD32×C59—21no—
34C2×M32×C59C2211yescharacteristic
35C22×D236—21no—
36C2×#11745671—24no—
37C2×#11745671—41no—
38C2×#11745671—41no—
39C2×#11745671—42no—
40C2×D472—21no—
41C2×D472—21no—
42C2×D472—21no—
43C2×D472—21no—
44C2×D472—21no—
45C2×D472C2312yes—
46#14273734—24no—
47#14273734—42no—
48C2×#11745664—21no—
49C2×#11745664—41no—
50C22×C59⋊C8—41no—
51C22×C59⋊C8D411yescharacteristic
52#14273716—24no—
53#14273716—42no—
54C2×(C2×D59)⋊C4—42no—
55C2×C4×Dic59D411yescharacteristic
56#14273670—24no—
57#14273668—24no—
58#14273667—21no—
59#14273667C2312yes—
60C2×C59⋊C16—21no—
61#14273663—21no—
62#14273663C2312yes—
63#14273654—24no—
64C8×Dic59D412yes—
65C8×Dic59C2312yes—
66C2×#41×C59—21no—
67C22×Q8×C59—21no—
68#135×C59—21no—
69#135×C59—21no—
70#135×C59D412yes—
71#135×C59C2312yes—
72C2×Q16×C59—21no—
73C2×Q16×C59—21no—
74C2×Q16×C59—21no—
75C2×Q16×C59—21no—
76C2×Q16×C59—21no—
77C2×Q16×C59D411yescharacteristic
78C2×Q16×C59C2312yes—
79C2×SD16×C59—21no—
80C2×D8×C59D411yescharacteristic
81C22×C8×C59C2311yescharacteristic
82Q32×C59—24no—
83SD32×C59—24no—
84M32×C59—21no—
85M32×C59C2312yes—
86C2×C16×C59—21no—
87C23×D59—41no—
88#41×C59—22no—
89#41×C59—22no—
90#41×C59—41no—
91#41×C59—41no—
92C2×Q8×C59—21no—
93C2×Q8×C59—22no—
94C2×Q8×C59—41no—
95C2×Q8×C59—41no—
96C2×Q8×C59—41no—
97C2×Q8×C59—41no—
98C2×Q8×C59—41no—
99C2×D4×C59—21no—
100C2×D4×C59—41no—
101C22×C4×C59—41no—
102C22×C4×C59—41no—
103C22×C4×C59C2×D411yescharacteristic
104C22×Dic59—21no—
105C2×D236—22no—
106C2×D236—41no—
107C2×D236—41no—
108C2×D236—41no—
109C2×D236—41no—
110C2×D236—41no—
111Q16×C59—21no—
112Q16×C59—22no—
113Q16×C59—22no—
114Q16×C59—41no—
115Q16×C59—41no—
116Q16×C59C2×D412yes—
117SD16×C59—22no—
118SD16×C59—22no—
119D8×C59—21no—
120D8×C59C2×D412yes—
121C2×C8×C59C2×D411yescharacteristic
122C2×C8×C59C2×D411yescharacteristic
123C2×C8×C59C2×D411yescharacteristic
124C2×C8×C59#4111yescharacteristic
125C2×C8×C59#4111yescharacteristic
126C2×C8×C59C2411yescharacteristic
127#11745671—44no—
128#11745671—44no—
129(C2×D59)⋊C4—48no—
130C4×Dic59—21no—
131C4×Dic59C2×D412yes—
132#11745664—22no—
133#11745664—41no—
134#11745664—44no—
135C2×C59⋊C8—41no—
136C2×C59⋊C8—44no—
137C2×C59⋊C8D811yescharacteristic
138C2×C59⋊C8D811yescharacteristic
139C2×C59⋊C8D812yes—
140C2×C59⋊C8C2×D412yes—
141D472—22no—
142D472—22no—
143D472—41no—
144D472—41no—
145C16×C59—22no—
146C59⋊C16—22no—
147C23×C59—41no—
148C23×C59AGL(1,ℤ/8)11yescharacteristic
149C22×D59—42no—
150C22×D59—84no—
151C22×D59—81no—
152C22×D59—81no—
153Q8×C59—22no—
154Q8×C59—41no—
155Q8×C59—42no—
156Q8×C59—42no—
157Q8×C59—81no—
158Q8×C59—81no—
159D4×C59—22no—
160D4×C59—42no—
161D4×C59—42no—
162D4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—42no—
165C2×C4×C59—42no—
166C2×C4×C59—42no—
167C2×C4×C59—81no—
168C2×C4×C59—81no—
169C2×C4×C59#12111yescharacteristic
170C2×C4×C59#12111yescharacteristic
171C2×C4×C59C2×D811yescharacteristic
172C2×C4×C59C2×D811yescharacteristic
173C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C59C22×D411yescharacteristic
175C2×Dic59—22no—
176C2×Dic59—42no—
177D236—42no—
178D236—42no—
179D236—81no—
180D236—81no—
181C8×C59#12111yescharacteristic
182C8×C59#12111yescharacteristic
183C8×C59C22×D411yescharacteristic
184C8×C59C2×#4111yescharacteristic
185C59⋊C8—42no—
186C59⋊C8C2×D812yes—
187C59⋊C8C2×D812yes—
188C2×#2212—591nomaximal, Sylow
189C22×C59—22no—
190C22×C59—42no—
191C22×C59—82no—
192C22×C59#45911yescharacteristic
193C22×C59#45911yescharacteristic
194C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
195C2×D59—84no—
196C2×D59—82no—
197C2×D59—162no—
198C2×D59—162no—
199C4×C59—42no—
200C4×C59—82no—
201C4×C59#45911yescharacteristic
202C4×C59#45911yescharacteristic
203C4×C59C2×#12111yescharacteristic
204C4×C59C22×D811yescharacteristic
205Dic59—42no—
206C2×#510D5911yescharacteristic, complemented
207C2×#509—591no—
208C2×#495—591no—
209C2×#443—591no—
210#2212—598no—
211C2×#364—591no—
212C2×#362—591no—
213C2×#361—591no—
214C2×C59—21no—
215C2×C59—21no—
216C2×C59—82no—
217C2×C59#221212yescomplemented
218C2×C59C2×#45911yescharacteristic
219D59—162no—
220C2×#135C2×D5911yescharacteristic
221C22×Q16C2×D5911yescharacteristic
222C22×SD16—1181no—
223C22×D8—591no—
224C2×#131—1181no—
225C2×#124—1181no—
226#510—22no—
227#510C2×D5914yescomplemented
228#509—594no—
229#509—1182no—
230C2×Q32—21no—
231C2×SD32—21no—
232C2×SD32—1181no—
233C2×D16—1181no—
234C2×M32—591no—
235C2×M32C2×D5911yescharacteristic
236#495—594no—
237#495—1182no—
238#443—594no—
239#443—1182no—
240C2×#108—591no—
241C2×#104—1181no—
242C2×C4×C8—591no—
243#364—594no—
244#362—594no—
245#361—594no—
246C59C2×#221211yescharacteristic, complemented, Sylow
247C2×#41—21no—
248C22×Q8—21no—
249C22×D4—1181no—
250#135—21no—
251#135—21no—
252#135C22⋊D5912yes—
253#135C22×D5912yes—
254C2×Q16—21no—
255C2×Q16—21no—
256C2×Q16—21no—
257C2×Q16—21no—
258C2×Q16—21no—
259C2×Q16C22⋊D5911yescharacteristic
260C2×Q16C22×D5912yes—
261C2×SD16—21no—
262C2×SD16—1184no—
263C2×SD16—2361no—
264C2×SD16—2361no—
265C2×SD16—2362no—
266C2×D8—592no—
267C2×D8—1181no—
268C2×D8—1181no—
269C2×D8—1181no—
270C2×D8—1181no—
271C2×D8—1181no—
272C2×D8C22⋊D5911yescharacteristic
273#131—1184no—
274#131—2362no—
275C2×M16—1181no—
276C2×M16—2361no—
277C22×C8—591no—
278C22×C8—2361no—
279C22×C8C22×D5911yescharacteristic
280#124—1184no—
281#124—2362no—
282C2×C22⋊C4—2362no—
283C2×C42—591no—
284Q32—24no—
285SD32—24no—
286SD32—1184no—
287D16—1184no—
288M32—21no—
289M32—592no—
290M32—1181no—
291M32C22×D5912yes—
292C2×C16—21no—
293C2×C16—1181no—
294#108—592no—
295#108—1181no—
296#104—1184no—
297C4×C8—592no—
298C4×C8—592no—
299C24—2361no—
300#41—22no—
301#41—22no—
302#41—41no—
303#41—41no—
304C2×Q8—21no—
305C2×Q8—22no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×D4—21no—
312C2×D4—41no—
313C2×D4—1182no—
314C2×D4—2361no—
315C2×D4—2361no—
316C2×D4—2361no—
317C2×D4—2361no—
318C2×D4—2361no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—1181no—
322C22×C4D4×D5911yescharacteristic
323Q16—21no—
324Q16—22no—
325Q16—22no—
326Q16—41no—
327Q16—41no—
328Q16C2×C22⋊D5912yes—
329SD16—22no—
330SD16—22no—
331SD16—2364no—
332SD16—2364no—
333D8—21no—
334D8—1182no—
335D8—1182no—
336D8—2361no—
337D8—2361no—
338D8C2×C22⋊D5912yes—
339M16—1182no—
340M16—2364no—
341M16—2361no—
342C2×C8—591no—
343C2×C8—591no—
344C2×C8—592no—
345C2×C8—592no—
346C2×C8—2364no—
347C2×C8—2361no—
348C2×C8D4×D5911yescharacteristic
349C2×C8#1174568711yescharacteristic
350C2×C8#1174568711yescharacteristic
351C2×C8C2×C22⋊D5911yescharacteristic
352C2×C8C2×C22⋊D5911yescharacteristic
353C2×C8C23×D5911yescharacteristic
354C22⋊C4—2368no—
355C42—592no—
356C42—1181no—
357C16—22no—
358C16—1182no—
359C23—41no—
360C23—4724no—
361C23—4721no—
362C23—4721no—
363C23—2362no—
364C23#1427375111yescharacteristic
365Q8—22no—
366Q8—41no—
367Q8—42no—
368Q8—42no—
369Q8—81no—
370Q8—81no—
371D4—22no—
372D4—42no—
373D4—42no—
374D4—41no—
375D4—4721no—
376D4—4721no—
377D4—2362no—
378D4—2362no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—1182no—
386C2×C4—2362no—
387C2×C4#1427369911yescharacteristic
388C2×C4#1427369911yescharacteristic
389C2×C4D8×D5911yescharacteristic
390C2×C4D8×D5911yescharacteristic
391C2×C4#1427375111yescharacteristic
392C2×C4C2×D4×D5911yescharacteristic
393C8—592no—
394C8—592no—
395C8—2362no—
396C8#1427369911yescharacteristic
397C8#1427369911yescharacteristic
398C8C22×C22⋊D5911yescharacteristic
399C8C2×#1174568711yescharacteristic
400C22—22no—
401C22—42no—
402C22—82no—
403C22—9442no—
404C22—9442no—
405C22—4724no—
406C22—4722no—
407C22#1683759511yescharacteristic
408C22#1683759511yescharacteristic
409C22C2×#1427375111yescharacteristic
410C4—42no—
411C4—82no—
412C4—2362no—
413C4#1683759511yescharacteristic
414C4#1683759511yescharacteristic
415C4C2×#1427369911yescharacteristic
416C4C2×D8×D5911yescharacteristic
417C2—21no—
418C2—21no—
419C2—82no—
420C2—9442no—
421C2#2142616712yesdirect product factor
422C2C2×#1683759511yescharacteristic
subgroupG/H
centerC22#16837595
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC2×C59×#510C2
socleC22×C59#459
prime radicalC2×#16837837C2
hypercenterC2×#510D59
hypocenterC59C2×#2212
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#510

Complements

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

NH ≅ G/Ntype
2. C2×#16837787C2semidirect
3. C2×#16837519C2semidirect
4. #21426167C2direct (N × H)
5. C2×#16837222C2semidirect
6. C2×#16837218C2semidirect
7. C2×#16837216C2semidirect
8. C2×C59×#510C2semidirect
19. #16837787C22semidirect
21. #16837519C22semidirect
25. #16837222C22semidirect
26. #16837218C22semidirect
27. #16837216C22semidirect
31. C59×#510C22semidirect
206. C2×#510D59semidirect
217. C2×C59#2212semidirect
227. #510C2×D59semidirect
246. C59C2×#2212semidirect
421. C2#21426167direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168378372
C2×#168377872
C2×#168375192
#214261672
C2×#168372222
C2×#168372182
C2×#168372162
C2×C59×#5102
C2×#221259